重构body api，性能分析，项目整理
This commit is contained in:
Frank14f
@@ -0,0 +1,247 @@
|
||||
# Rotating cylinder validation against [Kan99b]
|
||||
|
||||
## Goal
|
||||
|
||||
This validation should stay small, direct, and defensible.
|
||||
|
||||
The main design rules are:
|
||||
|
||||
- use the paper's direct numeric anchor at `Re = 100, alpha = 1.0` as the main hard benchmark
|
||||
- use a low-rotation case to test the lift trend
|
||||
- use suppression cases to test flow classification, not exact threshold fitting
|
||||
- do not treat values read from figures near `alpha_L` as tight amplitude targets
|
||||
|
||||
This keeps the matrix representative without overfitting to sensitive threshold points.
|
||||
|
||||
## Strong numeric anchors from [Kan99b]
|
||||
|
||||
The strongest exact benchmark in the paper is the convergence case at `Re = 100, alpha = 1.0`.
|
||||
|
||||
| Quantity | Reference value |
|
||||
|---|---:|
|
||||
| `St` | 0.1655 |
|
||||
| `mean C_L` | -2.4881 |
|
||||
| `mean C_D` | 1.1040 |
|
||||
| `C'_L` | 0.3631 |
|
||||
| `C'_D` | 0.0993 |
|
||||
|
||||
For low rotation at `Re = 100`, the paper also gives the mean lift trend
|
||||
|
||||
\[
|
||||
\overline{C_L} \approx -2.48\alpha
|
||||
\]
|
||||
|
||||
which is a good secondary benchmark for small `alpha`.
|
||||
|
||||
The suppression thresholds are given as trends:
|
||||
|
||||
| Reynolds number | Expected `alpha_L` |
|
||||
|---|---:|
|
||||
| 60 | about 1.4 |
|
||||
| 100 | about 1.8 |
|
||||
| 160 | about 1.9 |
|
||||
|
||||
These threshold values should be used as regime guides, not as tight one-point numeric targets. In the suppression curve from [Kan99b] shown above, the boundary is exactly the kind of place where a small solver difference can change the observed state.
|
||||
|
||||
## Fixed solver setup
|
||||
|
||||
| Item | Setting |
|
||||
|---|---|
|
||||
| Dimension | 2D |
|
||||
| Lattice | D2Q9 |
|
||||
| Streaming | double buffer |
|
||||
| Curved boundary | current Bouzidi moving wall implementation |
|
||||
| Inlet profile | uniform |
|
||||
| Top and bottom boundaries | free slip |
|
||||
| Outlet | neq extrapolation |
|
||||
| LES | off |
|
||||
| Precision | FP32 |
|
||||
| Cylinder diameter | `D = 30` lattice units |
|
||||
| Cylinder radius | `R = 15` lattice units |
|
||||
| Rotation input | update body omega only |
|
||||
|
||||
The baseline domain remains the current medium far field unless a later boundary sensitivity check shows otherwise.
|
||||
|
||||
## Inlet recommendation by collision model
|
||||
|
||||
Kan99b is an open-flow validation, not a confined-channel benchmark.
|
||||
|
||||
| Collision | Recommended inlet | Secondary choice | Avoid as primary |
|
||||
|---|---|---|---|
|
||||
| SRT | `equilibrium` | `regularized` | `zou_he_local` |
|
||||
| TRT | `regularized` | `equilibrium` | `zou_he_local` until the anchor is stable |
|
||||
| MRT | `regularized` or `zou_he_local` | `equilibrium` | `channel_stabilized` |
|
||||
|
||||
Keep one inlet family per collision model across the primary matrix.
|
||||
|
||||
## Lattice-unit mapping
|
||||
|
||||
Use
|
||||
|
||||
\[
|
||||
U_\infty = 0.03
|
||||
\]
|
||||
|
||||
With `D = 30`,
|
||||
|
||||
\[
|
||||
\nu = \frac{U_\infty D}{Re} = \frac{0.9}{Re}
|
||||
\]
|
||||
|
||||
| `Re` | `nu` | SRT equivalent `omega` |
|
||||
|---|---:|---:|
|
||||
| 60 | 0.015000 | 1.83486 |
|
||||
| 100 | 0.009000 | 1.89753 |
|
||||
| 160 | 0.005625 | 1.93470 |
|
||||
|
||||
The body rotation rate is
|
||||
|
||||
\[
|
||||
\omega_{body} = \frac{2 \alpha U_\infty}{D} = 0.002\alpha
|
||||
\]
|
||||
|
||||
| `alpha` | body omega |
|
||||
|---|---:|
|
||||
| 0.5 | 0.0010 |
|
||||
| 1.0 | 0.0020 |
|
||||
| 1.6 | 0.0032 |
|
||||
| 2.0 | 0.0040 |
|
||||
|
||||
## Primary matrix
|
||||
|
||||
This is the recommended main validation set.
|
||||
|
||||
| Case | `Re` | `alpha` | Role |
|
||||
|---|---:|---:|---|
|
||||
| K1 | 100 | 0.5 | low-rotation lift trend check |
|
||||
| K2 | 100 | 1.0 | strongest hard anchor |
|
||||
| K3 | 60 | 1.6 | low-Re suppression classification |
|
||||
| K4 | 100 | 2.0 | mid-Re suppression classification |
|
||||
| K5 | 160 | 2.0 | high-Re suppression classification |
|
||||
|
||||
Optional baseline if needed for debugging or plots:
|
||||
|
||||
| Case | `Re` | `alpha` | Status |
|
||||
|---|---:|---:|---|
|
||||
| K0 | 100 | 0.0 | optional |
|
||||
|
||||
This matrix covers:
|
||||
|
||||
- one periodic low-rotation trend point
|
||||
- one exact hard anchor with full force data
|
||||
- suppression behavior at low, medium, and high Reynolds number
|
||||
|
||||
## How to judge each case
|
||||
|
||||
### K1
|
||||
|
||||
Use K1 to check the low-rotation lift law.
|
||||
|
||||
Target:
|
||||
|
||||
\[
|
||||
\overline{C_L} \approx -2.48 \times 0.5 \approx -1.24
|
||||
\]
|
||||
|
||||
This is a trend check, not a strict fluctuation-amplitude benchmark.
|
||||
|
||||
### K2
|
||||
|
||||
Use K2 as the hard benchmark case.
|
||||
|
||||
Preferred agreement band:
|
||||
|
||||
| Quantity | Preferred band |
|
||||
|---|---:|
|
||||
| `St` | within 3 percent |
|
||||
| `mean C_L` | within 4 percent |
|
||||
| `mean C_D` | within 5 percent |
|
||||
| `C'_L` | within 8 percent |
|
||||
| `C'_D` | within 10 percent |
|
||||
|
||||
### K3 to K5
|
||||
|
||||
Use K3 to K5 as suppression classification cases.
|
||||
|
||||
Primary success signature:
|
||||
|
||||
- `C'_L` collapses toward zero in the final window
|
||||
- no sustained alternating wake remains
|
||||
- flow classification agrees with the expected suppressed regime
|
||||
|
||||
These are not exact threshold-fitting cases. Do not over-interpret a small residual fluctuation if the wake is otherwise clearly in the suppressed class.
|
||||
|
||||
## Optional threshold bracket check
|
||||
|
||||
If later you want a more explicit threshold study, use pairs around `alpha_L` rather than a single point on the boundary.
|
||||
|
||||
Recommended pairs:
|
||||
|
||||
| `Re` | Lower point | Upper point |
|
||||
|---|---:|---:|
|
||||
| 60 | 1.3 | 1.5 |
|
||||
| 100 | 1.7 | 1.9 |
|
||||
| 160 | 1.8 | 2.0 |
|
||||
|
||||
These should still be treated as regime-location checks, not hard force targets.
|
||||
|
||||
## Run policy
|
||||
|
||||
| Case type | Total steps | Warmup | Statistics |
|
||||
|---|---:|---:|---:|
|
||||
| K1 and K2 | 180000 to 220000 | first 40 percent | last 60 percent |
|
||||
| K3 to K5 | 220000 to 280000 | first 50 percent | last 50 percent |
|
||||
|
||||
The final statistics window should contain at least 20 shedding periods whenever the case remains periodic.
|
||||
|
||||
## TRT re-entry rule
|
||||
|
||||
Bring TRT back in this order:
|
||||
|
||||
1. K2 only
|
||||
2. if K2 is stable and credible, run K1
|
||||
3. only then run K3 to K5
|
||||
|
||||
This prevents TRT from expanding the matrix before the hard anchor is trustworthy.
|
||||
|
||||
## Deliverables
|
||||
|
||||
For each collision model, deliver:
|
||||
|
||||
- one table of run settings including collision, inlet scheme, wall type, `Re`, `alpha`, `nu`, and body omega
|
||||
- one CSV per run with force history
|
||||
- selected field images for wake classification
|
||||
- one summary table with `mean C_D`, `mean C_L`, `C'_D`, `C'_L`, and `St`
|
||||
- one short note stating whether suppression behavior matches [Kan99b]
|
||||
|
||||
## Recommended primary settings summary
|
||||
|
||||
| Collision | Wall | Inlet | Status |
|
||||
|---|---|---|---|
|
||||
| SRT | free slip | `equilibrium` | primary |
|
||||
| TRT | free slip | `regularized` | primary if K2 is stable |
|
||||
| MRT | free slip | `regularized` or `zou_he_local` | primary |
|
||||
|
||||
## MRT-only runner mapping
|
||||
|
||||
The current executable entrypoint is `tests/run_kan99b_rotating_cylinder.py`, and this round uses MRT-only scheduling:
|
||||
|
||||
- primary matrix is `K1-K5` with `MRT + regularized` inlet
|
||||
- one extra control run is added at K2 with `MRT + zou_he_local`
|
||||
- all runs keep `uniform` inlet profile, `free_slip` y-wall, `neq_extrap` outlet
|
||||
- output rows include `case_id`, `variant`, `collision`, `inlet_scheme`, `grid`, `steps`, `burn_in`, `St`, `St_error_pct` (for K2), and force metrics
|
||||
- K2 gate uses this document's per-metric tolerances for `St`, `mean C_L`, `mean C_D`, `C'_L`, `C'_D`
|
||||
|
||||
Example commands:
|
||||
|
||||
```bash
|
||||
conda run -n pycuda_3_10 python tests/run_kan99b_rotating_cylinder.py \
|
||||
--json-out tests/output/kan99b_validation/summary_runs.json
|
||||
|
||||
conda run -n pycuda_3_10 python tests/run_kan99b_rotating_cylinder.py \
|
||||
--case K2 --save-vorticity
|
||||
```
|
||||
|
||||
## Reference
|
||||
|
||||
[Kan99b] S. Kang, H. Choi, and S. Lee, “Laminar flow past a rotating circular cylinder,” 1999.
|
||||
@@ -0,0 +1,243 @@
|
||||
# Sah04 confined-cylinder validation against [Sah04]
|
||||
|
||||
## Goal
|
||||
|
||||
This validation should use a small set of direct periodic anchors from [Sah04], plus a small secondary block for near-onset flow-state checks.
|
||||
|
||||
The main design rules are:
|
||||
|
||||
- do not use interpolated values from figures as hard targets
|
||||
- do not use points on or extremely near critical curves as primary pass fail anchors
|
||||
- report realized blockage and realized Reynolds number, not just nominal inputs
|
||||
- use finer grids for high blockage so the narrow wall gaps are not under-resolved
|
||||
|
||||
This keeps the primary matrix small but still representative across moderate and high confinement.
|
||||
|
||||
## What counts as a hard benchmark from [Sah04]
|
||||
|
||||
The strongest periodic-flow anchors are the direct DNS values stated in the paper for developed unsteady states.
|
||||
|
||||
| Case | Blockage `beta` | Reynolds number | `St` target | Why it is a hard anchor |
|
||||
|---|---:|---:|---:|---|
|
||||
| S1 | 0.3 | 100 | 0.2115 | direct periodic DNS value |
|
||||
| S2 | 0.5 | 200 | 0.3513 | direct periodic DNS value |
|
||||
| S3 | 0.8 | 160 | `T ≈ 1.806`, so `St ≈ 0.5537` | direct case given in the paper |
|
||||
| S4 | 0.9 | 200 | 0.5314 | direct periodic DNS value |
|
||||
|
||||
These should be the primary Sah04 validation anchors.
|
||||
|
||||
## What should not be a hard target
|
||||
|
||||
The following are useful for qualitative or secondary checks, but not for the main validation gate:
|
||||
|
||||
- critical onset values from Table IV
|
||||
- any `St` or force value obtained by reading or interpolating a plot
|
||||
- cases chosen only to complete a rectangular parameter grid
|
||||
- points too close to the codimension two region or nearby neutral stability boundaries
|
||||
|
||||
This matters because the paper's stability map has several sensitive regions, especially at high blockage and near symmetry breaking. In the stability figure from [Sah04] shown above, those boundaries are exactly where a small setup difference can change the observed state.
|
||||
|
||||
## Geometry and blockage mapping
|
||||
|
||||
Keep the confined-channel layout and no-slip walls.
|
||||
|
||||
The validation table must always report both nominal and realized blockage.
|
||||
|
||||
With `D = 30` lattice units, the recommended realizations are:
|
||||
|
||||
| Case | `beta_nominal` | Suggested `H` | `beta_real` | Notes |
|
||||
|---|---:|---:|---:|---|
|
||||
| S1 | 0.3 | 100 | 0.3000 | exact |
|
||||
| S2 | 0.5 | 60 | 0.5000 | exact |
|
||||
| S3 | 0.8 | 38 or 37 | 0.7895 or 0.8108 | pick one and report it explicitly |
|
||||
| S4 | 0.9 | 33 | 0.9091 | use this, not `H = 35` |
|
||||
|
||||
Do not silently rename `beta_real` as the paper blockage.
|
||||
|
||||
## Grid density policy
|
||||
|
||||
High blockage cases need more wall-normal resolution than the base grid.
|
||||
|
||||
Minimum rule:
|
||||
|
||||
- for `beta < 0.8`, the baseline grid with `D = 30` is acceptable for the first pass
|
||||
- for `beta >= 0.8`, increase grid density by at least 2 times in each spatial direction before treating the result as validation quality
|
||||
|
||||
A practical way to do this is to keep geometry similarity while doubling the characteristic resolution:
|
||||
|
||||
- base cases: `D = 30`
|
||||
- high blockage validation cases: at least `D = 60`
|
||||
|
||||
Recommended high blockage realizations on the refined grid:
|
||||
|
||||
| Case | `beta_nominal` | Suggested refined `D` | Suggested refined `H` | `beta_real` |
|
||||
|---|---:|---:|---:|---:|
|
||||
| S3 | 0.8 | 60 | 75 | 0.8000 |
|
||||
| S4 | 0.9 | 60 | 67 | 0.8955 |
|
||||
|
||||
The point of this refinement is not only bulk accuracy. It is also to resolve the narrow cylinder wall gaps and reduce the risk that blockage effects are dominated by lattice geometry error.
|
||||
|
||||
## Primary matrix
|
||||
|
||||
This is the main Sah04 validation set.
|
||||
|
||||
| Case | `beta_nominal` | Primary target | Role |
|
||||
|---|---:|---|---|
|
||||
| S1 | 0.3 | `Re = 100`, `St = 0.2115` | moderate blockage periodic anchor |
|
||||
| S2 | 0.5 | `Re = 200`, `St = 0.3513` | medium blockage periodic anchor |
|
||||
| S3 | 0.8 | `Re = 160`, `St ≈ 0.5537` | high blockage periodic anchor |
|
||||
| S4 | 0.9 | `Re = 200`, `St = 0.5314` | very high blockage periodic anchor |
|
||||
|
||||
This matrix is smaller than the older grid, but it covers:
|
||||
|
||||
- moderate confinement
|
||||
- stronger confinement
|
||||
- high blockage periodic shedding
|
||||
- very high blockage periodic shedding
|
||||
|
||||
## Secondary onset block
|
||||
|
||||
These cases are recommended as flow-state checks, not hard `St` benchmarks.
|
||||
|
||||
| Case | `beta_nominal` | Suggested `Re` | Why it is useful | How to judge it |
|
||||
|---|---:|---:|---|---|
|
||||
| SO1 | 0.5 | about 130 | safely above first onset without sitting on the boundary | confirm sustained periodic state |
|
||||
| SO2 | 0.7 | about 120 | tests near-onset behavior in a more sensitive blockage range | confirm sustained periodic state |
|
||||
|
||||
These points exist to answer a different question from the primary matrix:
|
||||
|
||||
- does the solver enter and maintain the right flow regime once slightly above onset
|
||||
|
||||
For SO1 and SO2, judge by:
|
||||
|
||||
- persistent nonzero `C'_L`
|
||||
- a clean dominant spectral peak
|
||||
- repeatable periodic wake structure
|
||||
|
||||
Do not fail these runs because the measured `St` differs slightly from a value read from a nearby figure.
|
||||
|
||||
## Inlet and wall policy
|
||||
|
||||
Sah04 is a confined-channel benchmark, so inlet consistency matters more than inlet variety.
|
||||
|
||||
| Collision | Wall | Inlet | Status |
|
||||
|---|---|---|---|
|
||||
| SRT | no slip | `channel_stabilized` | primary |
|
||||
| TRT | no slip | `channel_stabilized` | primary |
|
||||
| MRT | no slip | `channel_stabilized` | primary |
|
||||
|
||||
Keep the inlet family fixed across collision models in the primary matrix.
|
||||
|
||||
Secondary inlet comparison, only after the primary set is working:
|
||||
|
||||
| Collision | Optional inlet | Status |
|
||||
|---|---|---|
|
||||
| MRT | `regularized` or `zou_he_local` | exploratory |
|
||||
| SRT or TRT | `equilibrium` or `regularized` | exploratory |
|
||||
|
||||
## Realized Reynolds number check
|
||||
|
||||
This is mandatory for Sah04.
|
||||
|
||||
For each run, record:
|
||||
|
||||
- nominal inlet definition
|
||||
- developed downstream velocity profile
|
||||
- measured `U_max,real`
|
||||
- measured bulk velocity if available
|
||||
- `beta_nominal`
|
||||
- `beta_real`
|
||||
- `Re_nominal`
|
||||
- `Re_real`
|
||||
|
||||
Use the paper-consistent label
|
||||
|
||||
\[
|
||||
Re_{real} = \frac{U_{max,real} D}{\nu}
|
||||
\]
|
||||
|
||||
for the final comparison table.
|
||||
|
||||
If `Re_real` drifts materially from the intended target, treat that as a setup problem before treating it as a Strouhal miss.
|
||||
|
||||
## Run policy
|
||||
|
||||
| Case block | Total steps | Burn | Statistics |
|
||||
|---|---:|---:|---:|
|
||||
| S1 and S2 | 100000 to 160000 | first 35 to 40 percent | last 60 to 65 percent |
|
||||
| S3 and S4 | 180000 to 260000 | first 45 percent | last 55 percent |
|
||||
| SO1 and SO2 | 140000 to 220000 | first 45 percent | last 55 percent |
|
||||
|
||||
For high blockage refined runs, prefer the longer end of the window.
|
||||
|
||||
## Evaluation rule
|
||||
|
||||
Use a two-layer rule.
|
||||
|
||||
### Primary periodic anchors
|
||||
|
||||
| Case | Hard target use |
|
||||
|---|---|
|
||||
| S1 to S4 | hard periodic benchmark anchors |
|
||||
|
||||
Preferred agreement band:
|
||||
|
||||
- within 5 percent when `Re_real` is close to target and the spectrum is clean
|
||||
- within 10 percent still acceptable if the run is clearly periodic and the residual mismatch is explainable by `Re_real` drift or geometry realization
|
||||
|
||||
### Secondary onset block
|
||||
|
||||
| Case | Hard target use |
|
||||
|---|---|
|
||||
| SO1 and SO2 | no hard `St` gate |
|
||||
|
||||
Success means:
|
||||
|
||||
- the flow is clearly unsteady and periodic
|
||||
- the dominant frequency is stable over long windows
|
||||
- the wake classification is consistent with being above onset
|
||||
|
||||
## Deliverables
|
||||
|
||||
For each run, deliver:
|
||||
|
||||
- one row with `beta_nominal`, `beta_real`, `Re_nominal`, `Re_real`, `nu`, collision, wall type, inlet scheme, and grid resolution
|
||||
- one downstream velocity-profile plot
|
||||
- one force-history CSV
|
||||
- one `St` estimate with the exact analysis window stated
|
||||
- selected wake images for flow classification
|
||||
|
||||
## Recommended minimum set
|
||||
|
||||
If compute budget is tight, run this order first:
|
||||
|
||||
| Priority | Runs |
|
||||
|---|---|
|
||||
| 1 | MRT on S1 to S4 |
|
||||
| 2 | SRT on S2 and S4 |
|
||||
| 3 | TRT on S2 and S4 |
|
||||
| 4 | SO1 and SO2 only after the primary anchors are behaving |
|
||||
|
||||
## MRT-only runner mapping
|
||||
|
||||
The current executable entrypoint is `tests/run_sah04_st_matrix.py`, and it is now aligned to this document's primary S1-S4 matrix:
|
||||
|
||||
- collision is fixed to `MRT`
|
||||
- inlet is fixed to `parabolic + channel_stabilized`
|
||||
- case set is `S1-S4` only
|
||||
- output rows include `case_id`, `collision`, `inlet_scheme`, `grid`, `steps`, `burn_in`, `St`, `St_error_pct`, `Re_real`, `beta_real`
|
||||
- default hard gate is 5% (`--gate-pct` can relax it to 10%)
|
||||
|
||||
Example commands:
|
||||
|
||||
```bash
|
||||
conda run -n pycuda_3_10 python tests/run_sah04_st_matrix.py \
|
||||
--json-out tests/output/sah04_mrt/summary.json
|
||||
|
||||
conda run -n pycuda_3_10 python tests/run_sah04_st_matrix.py \
|
||||
--case S3 --gate-pct 10 --final-vorticity-dir tests/output/sah04_mrt/vorticity
|
||||
```
|
||||
|
||||
## Reference
|
||||
|
||||
[Sah04] M. Sahin and R. G. Owens, “A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder,” 2004.
|
||||
@@ -0,0 +1,181 @@
|
||||
229：无控制:
|
||||
```
|
||||
SIGNAL_FEATURES0 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.1354, 0.0, 1.600),
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.0,
|
||||
'components': [
|
||||
(0.1354, 0.0, 2.099),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.1354, 0.0, 1.639),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
234：隐身:
|
||||
```
|
||||
SIGNAL_FEATURES1 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.1354, 0.0, 1.600),
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.01806,
|
||||
'components': [
|
||||
(0.1354, 0.0, 2.099),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.01806,
|
||||
'components': [
|
||||
(0.1354, 0.0, 1.639),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
欺骗：
|
||||
237 238 253：
|
||||
```
|
||||
SIGNAL_FEATURES2 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.1354, 0.0026, 1.600), # 主频
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.008730,
|
||||
'components': [
|
||||
(0.1354, 0.0045, 2.099), # 主频
|
||||
(0.2708, 0.0010, 0.612),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.008730,
|
||||
'components': [
|
||||
(0.1354, 0.0045, 1.639), # 主频
|
||||
(0.2708, 0.0010, -2.962),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
257:
|
||||
```
|
||||
SIGNAL_FEATURES3 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.1354, 0.0029, -2.619), # 主频
|
||||
(0.2708, 0.0008, 2.856),
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.0140,
|
||||
'components': [
|
||||
(0.1354, 0.0050, -0.933), # 主频
|
||||
(0.2708, 0.0010, 0.801),
|
||||
(0.1806, 0.0003, 1.854),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.014,
|
||||
'components': [
|
||||
(0.1354, 0.0050, -1.398), # 主频
|
||||
(0.2708, 0.0010, 2.208),
|
||||
(0.1806, 0.0003, 1.810),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
260 (f*1.5):
|
||||
```
|
||||
SIGNAL_FEATURES4 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.2031, 0.0026, 1.600), # 主频
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.008730,
|
||||
'components': [
|
||||
(0.2031, 0.0045, 2.099), # 主频
|
||||
(0.4062, 0.0010, 0.612),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.008730,
|
||||
'components': [
|
||||
(0.2031, 0.0045, 1.639), # 主频
|
||||
(0.4062, 0.0010, -2.962),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
262: (f*1.5):
|
||||
```
|
||||
SIGNAL_FEATURES5 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.2031, 0.0029, -2.619), # 主频
|
||||
(0.4062, 0.0008, 2.856),
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.0140,
|
||||
'components': [
|
||||
(0.2031, 0.0050, -0.933), # 主频
|
||||
(0.4062, 0.0010, 0.801),
|
||||
(0.2709, 0.0003, 1.854),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.014,
|
||||
'components': [
|
||||
(0.2031, 0.0050, -1.398), # 主频
|
||||
(0.4062, 0.0010, 2.208),
|
||||
(0.2709, 0.0003, 1.810),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
270 (f*2):
|
||||
```
|
||||
SIGNAL_FEATURES6 = {
|
||||
'action1': {
|
||||
'mean': 0.0,
|
||||
'components': [
|
||||
(0.2708, 0.0044, -2.619), # 主频
|
||||
(0.8124, 0.0012, 2.856),
|
||||
]
|
||||
},
|
||||
'action2': {
|
||||
'mean': -0.014,
|
||||
'components': [
|
||||
(0.2708, 0.0075, -0.933), # 主频
|
||||
(0.8124, 0.0015, 0.801),
|
||||
(0.5418, 0.0005, 1.854),
|
||||
]
|
||||
},
|
||||
'action3': {
|
||||
'mean': 0.014,
|
||||
'components': [
|
||||
(0.2708, 0.0075, -1.398), # 主频
|
||||
(0.8124, 0.0015, 2.208),
|
||||
(0.5418, 0.0005, 1.810),
|
||||
]
|
||||
}
|
||||
}
|
||||
```
|
||||
@@ -0,0 +1,555 @@
|
||||
RESEARCH ARTICLE | NOVEMBER 01 1999
|
||||
|
||||
# Laminar flow past a rotating circular cylinder
|
||||
|
||||
|
||||
|
||||
|
||||
Sangmo Kang; Haecheon Choi; Sangsan Lee
|
||||
|
||||
|
||||
|
||||
|
||||
Check for updates
|
||||
|
||||
Physics of Fluids 11, 3312–3321 (1999)
|
||||
|
||||
https://doi.org/10.1063/1.870190
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
View Online
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Export Citation
|
||||
|
||||
|
||||
# Articles You May Be Interested In
|
||||
|
||||
Lagrangian analysis of the laminar flat plate boundary layer
|
||||
|
||||
Physics of Fluids (October 2016)
|
||||
|
||||
Laminar flow instability in a rectangular channel with a cylindrical core
|
||||
|
||||
Physics of Fluids (April 2006)
|
||||
|
||||
Characteristics of laminar flow past a sphere in uniform shear
|
||||
|
||||
Physics of Fluids (October 2005)
|
||||
|
||||
|
||||
|
||||
|
||||
# AIP Advances
|
||||
|
||||
# Why Publish With Us?
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
21DAYS average time to1st decision
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
OVER 4 MILLION views in the last year
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
INCLUSIVE scope
|
||||
|
||||
|
||||
Learn More
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
AIP Publishing
|
||||
|
||||
|
||||
# Laminar flow past a rotating circular cylinder
|
||||
|
||||
Sangmo Kanga) and Haecheon Choib)
|
||||
|
||||
National CRI Center for Turbulence and Flow Control Research, Institute of Advanced Machinery and Design, Seoul National University, Seoul 151-742, Korea
|
||||
|
||||
Sangsan Lee
|
||||
|
||||
ETRI Supercomputer Center, P.O. Box 1, Yusong, Taejeon 305-600, Korea
|
||||
|
||||
~Received 7 October 1998; accepted 20 July 1999!
|
||||
|
||||
The present study numerically investigates two-dimensional laminar flow past a circular cylinder rotating with a constant angular velocity, for the purpose of controlling vortex shedding and understanding the underlying flow mechanism. Numerical simulations are performed for flows with Re560, 100, and 160 in the range of $0 { \leqslant } \alpha { \leqslant } 2 . 5 ,$ , where is the circumferential speed at the cylinder surface normalized by the free-stream velocity. Results show that the rotation of a cylinder can suppress vortex shedding effectively. Vortex shedding exists at low rotational speeds and completely disappears at $\alpha > \alpha _ { L }$ , where $\alpha _ { L }$ is the critical rotational speed which shows a logarithmic dependence on Re. The Strouhal number remains nearly constant regardless of while vortex shedding exists. With increasing , the mean lift increases linearly and the mean drag decreases, which differ significantly from those predicted by the potential flow theory. On the other hand, the amplitude of lift fluctuation stays nearly constant with increasing $( < \alpha _ { L } )$ , while that of drag fluctuation increases. Further studies from the instantaneous flow fields demonstrate again that the rotation of a cylinder makes a substantial effect on the flow pattern. © 1999 American Institute of Physics. @S1070-6631~99!01211-8#
|
||||
|
||||
# I. INTRODUCTION
|
||||
|
||||
Flow past a circular cylinder has been accepted as a building-block problem for understanding the vortex dynamics in bluff body wakes. At a low Reynolds number ~Re ,47!, the wake behind a ~nonrotating! cylinder comprises a steady recirculation region with two vortices symmetrically attached to the cylinder, whose size grows with increasing Reynolds number. Here, the Reynolds number Re is defined as $\mathrm { R e } { = } U _ { \infty } d / \nu ,$ where $U _ { \infty }$ is the free-stream velocity, d the diameter of the cylinder, and the kinematic viscosity. At a Reynolds number of 47<Re,200, vortex shedding occurs in the near wake behind a cylinder due to the flow instability accompanying a large fluctuating pressure and, thus, a periodically oscillating lift force. At a higher Reynolds number ~Re>200!, the flow becomes three-dimensional and turbulent, and vortex shedding also occurs but in more complicated patterns. Such fluctuating forces induced from vortex shedding may cause structural vibrations, acoustic noise or resonance, which in some cases can trigger structure failure or enhance mixing in the wake.1 Therefore, controlling vortex shedding appropriately is critical in practical engineering environments. Many attempts have been made for controlling the wake behind a circular cylinder in recent years, especially for the purpose of suppressing vortex shedding, using passive or ~nonfeedback or feedback! active controls.
|
||||
|
||||
The rotation of a cylinder in a viscous uniform flow is expected to modify wake flow pattern and vortex shedding configuration, which may reduce a flow-induced oscillation or augment a lift force. The basic rationale behind the rotation effect is that as a cylinder rotates, the flow is accelerated on one side of the cylinder and decelerated on the other side. Hence, the pressure on the accelerated side becomes smaller than that on the decelerated side, resulting in a mean lift force. Such a phenomenon is referred to as the Magnus effect. As a result, the rotation significantly alters the flow pattern, and probably has an effect on a flow-induced oscillation. Most of such researches performed to date are classified largely into two categories, a rotary oscillation and a unidirectional rotation. Flow past a circular cylinder performing a rotary oscillation has been investigated by many researchers; for example, experimentally by Taneda,2 Tokumaru and Dimotakis,3 and Filler et al.4 and numerically by Wu et al.5 and Baek and Sung.6
|
||||
|
||||
There have also been numerous investigations of flow past a circular cylinder rotating with a constant angular velocity, which is the main concern of the present study. The flow past a circular cylinder started impulsively into rotation and translation from rest becomes transient in an early time, and then reaches either a steady or unsteady ~time-periodic! state depending on two parameters, Re and . Here, the rotational speed is defined as $\alpha { = } \dot { \theta } d / ( 2 U _ { \infty } )$ , where ˙ is the angular velocity of the cylinder. Ingham,7 Badr et $a l . , ^ { 8 }$ and Ingham and Tang9 numerically investigated the rotatingcylinder flow in the laminar steady regime ~Re,47! for relatively small rotational speeds ~ <3!. They found that, although vortex shedding does not occur in the wake, the rotation delays and even inhibits the boundary-layer separation. The flow in the laminar vortex shedding regime $( 4 7 { \leqslant } \mathrm { R e } < 2 0 0 )$ has also been investigated in the literature. Badr $e t a l . ^ { 8 }$ numerically investigated unsteady flow at Re 560, 100, and 200, focusing on the flow pattern during an early time period after the impulsive rotation and translation of a cylinder. Subsequently, Tang and Ingham10 examined steady flow at $\mathrm { R e } { = } 6 0$ and 100 in the range of $0 { \leqslant } \alpha { \leqslant } 1$ by solving time-independent governing equations, and presented the changes in the flow variables. To the best of our knowledge, there is no literature that investigates the rotation effect on the laminar flow in the fully developed stage involving vortex shedding. This has mainly motivated the present numerical simulation of flow past a constantly rotating cylinder in the laminar vortex shedding regime.
|
||||
|
||||
Flow at a higher Reynolds number $( \mathrm { R e } { \geqslant } 2 0 0 )$ is more difficult to numerically simulate because of its intrinsic three-dimensionality and complexity. Tokumaru and Dimotakis11 experimentally investigated flow past a circular cylinder with both net steady rotation and forced oscillations at $\mathrm { R e } { \approx } 1 0 ^ { 3 }$ by suggesting the so-called virtual vortex method. They found that a higher cylinder aspect ratio yields a higher maximum lift coefficient in the case of steady rotation and the addition of forced rotary oscillations to the steady rotation increases or decreases the lift coefficient. Most investigations of unsteady flow at a high Reynolds number are limited to an early transient period because the flow at a later time becomes too complex to analyze. Badr and $\mathrm { D e n n i s } ^ { 1 2 }$ numerically investigated unsteady flow for $\mathbf { R e } { \geqslant } 2 0 0$ , while Coutanceau and $\mathrm { M e n a r d } ^ { 1 3 }$ experimentally investigated the same flow ~ 50.5 and 1! for comparison. Subsequently, Badr et al.14 carried out a comparative study of both the time evolutions of two-dimensional unsteady flow obtained numerically and experimentally for $1 0 ^ { 3 } { \leqslant } \mathrm { R e } { \leqslant } 1 0 ^ { 4 }$ and $0 . 5 { \leqslant } \alpha { \leqslant } 3$ . They found that vortex shedding completely disappears for $\alpha > \alpha _ { L }$ , where the critical value $\alpha _ { L }$ is almost independent of Re and is about 2. Chen et al.15 investigated the same unsteady flow at $ { \mathrm { R e } } { = } 2 0 0$ for $\alpha { \leqslant } 3 . 2 5$ , and reported that shedding of more than one vortex occurs even for $\alpha { = } 3 . 2 5$ , which is in contrast to the previous result of Badr et al.14 In company with these conflicting observations, there have been a few systematic researches about $\alpha _ { L }$ and its dependence on Re when a cylinder rotates at a constant angular velocity. Chang and Chern16 and Chen et al.15 suggested different basic modes of vortex shedding depending on Re and , respectively, by identifying the behavior of stagnation point and by characterizing the asymptotic stability of the flow. Recently, Hu et al.17 systematically studied the Hopf bifurcation describing the transition from a steady to timeperiodic solution through a Galerkin method and a system theory. They obtained the transition curve on a plane of Re and and reported that the rotation may delay the onset of vortex shedding and decrease the vortex-shedding frequency.
|
||||
|
||||
The objectives of the present study are to numerically investigate the effect of the rotation on flow past a constantly rotating circular cylinder in the laminar vortex shedding regime $( 4 7 { \leqslant } \mathrm { R e } < 2 0 0 )$ and to understand the underlying mechanism of vortex shedding suppression. This is accomplished by presenting the Strouhal number, mean flow quantities at the cylinder surface, lift and drag coefficients, and instantaneous flow fields in the fully developed stage, at various Re and . The two-dimensional unsteady Navier–Stokes equations are solved using a fully-implicit, fractional-step method in time18 and a second-order central difference scheme in space. The numerical simulations are performed for flows at Re540, 60, 100, and 160 in the range of $0 { \leqslant } \alpha { \leqslant } 2 . 5$ . Note that, in case of a stationary cylinder, the flow is steady at $\mathrm { R e } { = } 4 0$ and unsteady ~time-periodic! at Re 560, 100, and 160. All the numerical simulations are continued until the flow reaches a fully developed state, where all the flow characteristics are analyzed. The paper is organized as follows: Sec. II explains details of the numerical method. Subsequently, in Sec. III, we discuss the results from the simulations. Finally, a summary is presented in Sec. IV.
|
||||
|
||||
# II. NUMERICAL METHOD
|
||||
|
||||
The two-dimensional unsteady governing equations for an incompressible flow can be written as
|
||||
|
||||
$$
|
||||
\frac {\partial u _ {i}}{\partial t} + \frac {\partial (u _ {i} u _ {j})}{\partial x _ {j}} = - \frac {\partial p}{\partial x _ {i}} + \frac {1}{\mathrm{Re}} \frac {\partial^ {2} u _ {i}}{\partial x _ {j} \partial x _ {j}}, \tag {1}
|
||||
$$
|
||||
|
||||
$$
|
||||
\frac {\partial u _ {i}}{\partial x _ {i}} = 0, \tag {2}
|
||||
$$
|
||||
|
||||
where $x _ { i }$ are the Cartesian coordinates and $u _ { i }$ are the corresponding velocity components. All variables are nondimensionalized by the cylinder diameter d and the free-stream velocity $U _ { \infty }$ . Notice that the notation sets $( u , v )$ and $( x , y )$ are used interchangeably with $( u _ { 1 } , u _ { 2 } )$ and $( x _ { 1 } , x _ { 2 } )$ , respectively, in this paper.
|
||||
|
||||
Equations ~1! and ~2! are transformed to, by introducing generalized coordinates $\eta ^ { i }$ and volume fluxes $q ^ { i }$ across the faces of the cell,
|
||||
|
||||
$$
|
||||
\frac {\partial q ^ {i}}{\partial t} + N ^ {i} (\mathbf {q}) = - G ^ {i} (p) + L _ {1} ^ {i} (\mathbf {q}) + L _ {2} ^ {i} (\mathbf {q}), \tag {3}
|
||||
$$
|
||||
|
||||
$$
|
||||
D ^ {i} q ^ {i} = \frac {1}{\mathbf {J}} \left(\frac {\partial q ^ {1}}{\partial \eta^ {1}} + \frac {\partial q ^ {2}}{\partial \eta^ {2}}\right) = 0, \tag {4}
|
||||
$$
|
||||
|
||||
where $ { \mathbf { q } } = ( q ^ { 1 } , q ^ { 2 } )$ , Ni is the convection term, $G ^ { i } ( p )$ is the pressure gradient term, $L _ { 1 } ^ { i }$ and $L _ { 2 } ^ { i }$ are the diffusion terms without and with cross-derivatives, respectively, and $D ^ { i }$ is the divergence operator. More details are described in Choi et $a l . ^ { 1 8 }$ The transformed equations ~3! and ~4! are integrated in time using a fully-implicit, fractional-step method,18 composed of four-step time splitting. All terms in ~3! including cross-derivative diffusion terms are advanced with the Crank–Nicolson method in time and are resolved with a second-order central difference scheme in space. A Newton method is used to solve the discretized nonlinear equation.
|
||||
|
||||
The flow geometry and coordinate system along with boundary conditions are shown in Fig. 1. Here, notation sets $( r , \theta )$ and $( u _ { r } , u _ { \theta } )$ are also introduced to address, respectively, the radial and azimuthal directions and their corresponding velocity components. In this study, we use an O-type mesh, rather than a C-type mesh which may not be appropriate in the case of ~oscillatory! rotation of a cylinder.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 1. Flow geometry and coordinate system along with boundary conditions.
|
||||
|
||||
|
||||
In the C-type mesh in case of rotation, the nonzero circumferential velocity of a cylinder, unless particularly considered, makes the solution singular at the base point, causing a fatal spurious effect on the flow field. The O-type mesh used in this study is created following Beaudan and $\mathrm { M o i n } . ^ { 1 9 }$ For the computational domain with $0 . 5 { \leqslant } r { < } R _ { D }$ and $0 ^ { \circ } { \leqslant } \theta$ $< 3 6 0 ^ { \circ }$ , the grid points $r _ { m } ~ ( m { = } 1 , 2 , . . . , M )$ in the radial direction are
|
||||
|
||||
$$
|
||||
r _ {m} = 0. 5 + (R _ {D} - 0. 5) \frac {1 - s ^ {(m - 1)}}{1 - s ^ {M - 1}}, \tag {5}
|
||||
$$
|
||||
|
||||
where $R _ { D }$ is the radius of the computational domain, s the stretching factor and M the number of the radial grid points. The grid points $\theta _ { n } \ ( n { = } 1 , 2 , . . . , N )$ in the azimuthal direction are more clustered in the wake region than in the potential region, where N is the number of the azimuthal grid points. The grid boundary, or envelope, of the wake region $( x _ { b } ( h ) , y _ { b } ( h ) )$ is specified analytically as
|
||||
|
||||
$$
|
||||
\left\{ \begin{array}{c} x _ {b} (h) = \left(R _ {D} \cos \theta_ {0} - \cos \theta_ {i} + G\right) h ^ {2} - G h + \cos \theta_ {i}, \\ y _ {b} (h) = \left(R _ {D} \sin \theta_ {0} - \sin \theta_ {i} + G \tan \theta_ {i}\right) \frac {e ^ {b h} - 1 - b h}{e ^ {b} - 1 - b} \\ - G h \tan \theta_ {i} + \sin \theta_ {i}, \end{array} \right. \tag {6}
|
||||
$$
|
||||
|
||||
where h is a continuous parameter and $\theta _ { i }$ and $\theta _ { 0 }$ are the azimuthal positions of the grid boundary at $r = 0 . 5$ and $R _ { D }$ , respectively. Within the wake region, the azimuthal grid points are equispaced. In this study, the grid points $( r _ { m } , \theta _ { n } )$ are generated with $s = 1 . 0 1 6 , G = - 5 , b = 1 0 , \theta _ { i } = 8 5 ^ { \circ }$ , and $\theta _ { 0 } = 3 0 ^ { \circ }$ .
|
||||
|
||||
The condition of a constant circumferential velocity, $u _ { r }$ ${ } = 0$ and $u _ { \theta } { = } \alpha$ , is imposed on the cylinder surface. The farfield boundary is divided into inflow and outflow boundaries, that is $x { \leqslant } 0$ and $x { > } 0$ , respectively. The former boundary has a Dirichlet boundary condition, $u = 1$ and $\scriptstyle { V = 0 }$ , while the latter boundary has a convective outflow condition, $\partial u _ { i } / \partial t$ $+ c \partial u _ { i } / \partial x = 0 , ^ { 2 0 }$ where c is the space-averaged streamwise ~x-direction! exit velocity.
|
||||
|
||||
In the present study, a parameter set of $R _ { D } { = } 5 0 , M { \times } N$ $= 2 4 1 \times 2 4 1$ and $\Delta t = 0 . 0 2$ has been chosen to solve the governing equations on the O-type mesh. The computational time step $\Delta t = 0 . 0 2$ corresponds to a maximum $\mathrm { C F L } { = } 1 . 5 { - }$ 4.5. To confirm the chosen parameter values, parametric studies at Re5100 and 51 have been performed and the typical results are presented in Table I. The studies are accomplished by successively varying one parameter while keeping the others unchanged. The relative errors in the table show that the results obtained with the chosen parameter values are well converged with respect to the domain size, and spatial and temporal resolutions. The appropriateness of the boundary condition $u = 1$ and $\scriptstyle { V = 0 }$ imposed on the farfield boundary $( x { \leqslant } 0 )$ has also been examined because there may be an induced flow due to the cylinder rotation even at large distances away from the cylinder. This is accomplished by replacing the uniform flow on the far-field boundary (x $\leqslant 0 )$ with the potential flow involving the rotation effect as follows:
|
||||
|
||||
$$
|
||||
\left\{ \begin{array}{l} u = 1 - \frac {\cos 2 \theta}{4 R _ {D} ^ {2}} - \frac {\alpha \sin \theta}{2 R _ {D}}, \\ v = - \frac {\sin 2 \theta}{4 R _ {D} ^ {2}} + \frac {\alpha \cos \theta}{2 R _ {D}}. \end{array} \right. \tag {7}
|
||||
$$
|
||||
|
||||
In the case of $\alpha { = } 2 . 5$ ~the largest investigated in this paper! at Re5100, the inclusion of the rotation effect in the far-field boundary condition changed the lift coefficient, surface vorticity and surface pressure by less than 0.5%. This proves the applicability of the uniform inflow boundary condition at sufficiently large distances from the rotating cylinder.
|
||||
|
||||
To further validate the choice of numerical method and mesh, numerical simulations at the identical conditions used in previous publications have also been carried out and the results are presented in Figs. 2 and 3. Figure 2~a! shows the variation of St with Re at $\scriptstyle { \alpha = 0 } .$ , compared with the results of Park et $a l . ^ { 2 1 }$ and Williamson’s correlations,22 while Fig. 2~b! shows the variation of $C _ { L }$ with at $ { \mathrm { R e } } { = } 2 0 .$ , compared with the results of Badr et $a l . ^ { 8 }$ and Ingham and $\mathrm { T a n g . } ^ { 9 }$ Here, the Strouhal number St is defined as ${ \mathrm { S t } } { = } f d / U _ { \infty }$ , where f is the vortex-shedding frequency. The lift and drag coefficients, $C _ { L }$ and $C _ { D }$ , are defined respectively as $C _ { L } { = } 2 L / ( \rho U _ { \infty } ^ { 2 } d )$ and $C _ { D } { = } 2 D / ( \rho U _ { \infty } ^ { 2 } d )$ , where L and D are the lift and drag forces per unit cylinder length, respectively. It is clearly shown that the present results are in excellent agreement with the previous ones. Figure 3 shows time developments of velocity profiles in an early time after the impulsive rotation and translation of a cylinder at Re5200 and $\alpha { = } 0 . 5 ,$ compared with the experimental measurements of Coutanceau and Me´nard.13 The results, v-velocity along $\theta { = } 0 ^ { \circ }$ and u-velocity along $\theta { = } 9 0 ^ { \circ }$ , are not only in excellent agreement with the experimental measurements, but also with the computational results of Badr and $\mathrm { D e n n i s } ^ { 1 2 }$ ~not shown here!. The comparisons made in Figs. 2 and 3 indicate the appropriateness of the numerical method and mesh used in this study.
|
||||
|
||||
|
||||
TABLE I. Parametric studies at $\mathbf { R e } { = } 1 0 0$ and $\alpha { = } 1 . 0$ . Here, the relative errors ~%! with respect to the result from $R _ { D } = 5 0 , M \times N = 2 4 1 \times 2 4 1$ and $\Delta t = 0 . 0 2$ are parenthesized. $\Delta r _ { b }$ and $\Delta \theta _ { b }$ denote the relative magnitudes of grid-spacings at the base point. St is the Strouhal number, and $C _ { L }$ and $C _ { D }$ are the lift and drag coefficients, respectively. The overbar denotes the time averaging and the prime indicates the amplitude of fluctuation @see Eq. ~12!#.
|
||||
|
||||
|
||||
<table><tr><td><eq>R_D</eq>M×N</td><td><eq>\Delta r_b</eq>Δ <eq>\theta_b</eq></td><td><eq>\Delta t</eq></td><td>St</td><td><eq>\bar{C}_L</eq></td><td><eq>\bar{C}_D</eq></td><td><eq>C'_L</eq></td><td><eq>C'_D</eq></td></tr><tr><td>50</td><td>1</td><td>0.02</td><td>0.1655</td><td>-2.4881</td><td>1.1040</td><td>0.3631</td><td>0.0993</td></tr><tr><td>241×241</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>100</td><td>0.99</td><td>0.02</td><td>0.1650</td><td>-2.4833</td><td>1.0979</td><td>0.3603</td><td>0.0988</td></tr><tr><td>285×241</td><td>1</td><td></td><td>(0.30)</td><td>(0.19)</td><td>(0.55)</td><td>(0.77)</td><td>(0.50)</td></tr><tr><td>50</td><td>0.38</td><td>0.02</td><td>0.1656</td><td>-2.5027</td><td>1.0993</td><td>0.3576</td><td>0.0980</td></tr><tr><td>301×301</td><td>0.80</td><td></td><td>(0.06)</td><td>(0.59)</td><td>(0.43)</td><td>(1.51)</td><td>(1.31)</td></tr><tr><td>50</td><td>1</td><td>0.01</td><td>0.1656</td><td>-2.4881</td><td>1.1039</td><td>0.3630</td><td>0.0991</td></tr><tr><td>241×241</td><td>1</td><td></td><td>(0.06)</td><td>(0.00)</td><td>(0.01)</td><td>(0.03)</td><td>(0.20)</td></tr></table>
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 2. ~a! St vs Re at $\alpha { = } 0 ; \bigcirc ,$ the present study; L, Park et al. ~Ref. 21! ~6413241, C grid!; ¯, Williamson’s correlations ~Ref. 22!. ~b! $C _ { L }$ vs at $\ R e { = } 2 0 \colon { \mathcal { O } } ,$ the present study; – • h• – , Badr et al. ~Ref. $8 ) \ - \textcircled { < } -$ , Ingham and Tang ~Ref. 9!.
|
||||
|
||||
|
||||
After verifying the numerical method, we have conducted numerical simulations of flow past a constantly rotating cylinder at Re540, 60, 100, and 160 by successively increasing from 0 to 2.5. Since the fully developed flow is independent of initial conditions ~Fig. 4!, all the simulations may be started with arbitrary initial conditions only if the fully developed flow fields are to be analyzed.
|
||||
|
||||
# III. RESULTS
|
||||
|
||||
# A. Strouhal number
|
||||
|
||||
In the case of no rotation ~ 50!, the flow at $\mathrm { R e } { = } 4 0$ is steady, while the flows at Re560, 100, and 160 are timeperiodic on account of vortex shedding ~it was shown by Park et $a l . ^ { 2 1 }$ and Fey et $a l . ^ { 2 3 }$ that vortex shedding occurs at Re>47!. Simulations show that vortex shedding exists at low rotational speeds and then completely disappears at $> \alpha _ { L }$ , where the critical rotational speed $\alpha _ { L }$ depends on the Reynolds number, for example, $\alpha _ { L } { \approx } 1 . 4$ , 1.8, and 1.9 for
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 3. Time developments of velocity profiles at Re5200 and 50.5. ~a! v-velocity along 50°; ~b! u-velocity along 590°. —, the present study; s, h, n, and L, Coutanceau and Me´nard ~Ref. 13!.
|
||||
|
||||
|
||||
Re560, 100, and 160, respectively. No vortex shedding develops regardless of in the case of Re540. Variation of the Strouhal number with respect to while vortex shedding exists is shown in Fig. 5. Rotation of a cylinder does not significantly alter St in the range of $\alpha { < } \alpha _ { L }$ . More specifically, the Strouhal number stays nearly constant at low $\alpha ^ { \prime } \mathrm { s } ,$ , decreases slightly as approaches $\alpha _ { L }$ , and sharply reduces to zero at $\alpha > \alpha _ { L }$ . The present result supports the assumption of Badr et al.8 that St is more or less independent of ~St '0.14 and 0.16, respectively, for $\mathrm { R e } { = } 6 0$ and 100, consistent with the present result!, but contradicts the result of Hu et $a l . ^ { 1 7 }$ that St decreases steadily with increasing as shown in Fig. 5. However, the result of Hu et al. seems to be inaccurate, which shows a poor agreement with those of Park $e t a l . ^ { 2 \dot { 1 } }$ and Williamson22 at Re560 and 50.
|
||||
|
||||
# B. Mean flow quantities
|
||||
|
||||
The mean pressure coefficient and vorticity around the cylinder surface for cases with various combinations of the rotational speed and Reynolds number are shown in Figs. 6 and 7. In these figures, the effect of on the mean flow quantities, denoted by the solid line, is examined by fixing Re at $\mathbf { R e } { = } 1 0 0$ , while the effect of Re, denoted by the dashed line, is done by fixing at 51.0. The pressure distributions obtained from the potential flow theory are also plotted for¯ comparison. The mean flow quantity, denoted by $\overline { { ( ) } }$ , is computed by taking the temporal average of the flow quantities over one complete time cycle after the flow becomes fully developed.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 4. Variation of $C _ { L }$ according to the initial condition at $t = 0 \ \mathrm { ( R e = } 1 0 0$ and $\alpha { = } 1 . 0 ) ; ~ ( { \mathrm { a } } ) ~ - 2 0 { \leqslant } t { \leqslant } 7 0$ and ~b! $7 0 { \leqslant } t { \leqslant } 1 0 0 . -$ , started impulsively into rotation and translation from rest; –•–, started from steady flow at $\alpha { = } 2 . 0 ; \mathrm { ~ -- ~ }$ , started from time-periodic flow at $\scriptstyle \alpha = 0 .$ . This figure shows that the flow becomes fully time-periodic after a long enough time, irrespective of the initial conditions.
|
||||
|
||||
|
||||
Figure 6 shows the mean pressure coefficients, defined as $\overline { { C } } _ { p } = 2 ( \bar { p } - p _ { \infty } ) / \rho U _ { \infty } ^ { 2 }$ , where $p _ { \infty }$ is the free-stream pressure, around the cylinder surface with increasing in the range of $0 { \leqslant } \alpha { \leqslant } 2 . 5$ at $\mathrm { R e } { = } 1 0 0 .$ . As expected, the mean pressure for $\scriptstyle { \alpha = 0 }$ is symmetric about $\theta { = } 1 8 0 ^ { \circ }$ ~stagnation point!, leading to a zero mean lift. As increases, the flow becomes asymmetric and at the same time the pressure on the lower ~or the accelerated flow! side of the cylinder ~ '270°! decreases, resulting in a negative ~downward! mean lift. In addition, the stagnation point adhered to the cylinder surface moves in the reverse direction of rotation with increasing and then departs from the cylinder surface when $\alpha { \gtrsim } 0 . 5$ . Such observations are also found for all the other Reynolds numbers $\mathrm { R e } { = } 4 0 , 6 0$ , and 160. In Fig. 6 the results are also compared with those from the potential flow theory, given as
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 5. Variation of St according to at Re560, 100, and 160: $\mathrm { O , R e = 1 6 0 ; }$ $\scriptstyle \prod , \mathrm { R e } = 1 0 0 ; \Delta , \mathrm { R e } = 6 0 ; \emptyset$ , Hu et al. ~Ref. 17! ~Re560!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 6. Mean pressure coefficients around the cylinder surface at $\alpha { = } 0 , 0 . 5 ,$ , 1.0, 1.5, 2.0, and 2.5 for Re5100, compared with those from the potential flow theory for $\scriptstyle { \alpha = 0 }$ and $1 . 0 ; -$ , the present study; $- \cdot - .$ the potential theory. Also shown are the mean pressure coefficients at $\mathrm { R e } { = } 6 0$ and 160 for $\alpha { = } 1 . 0 ,$ , denoted by – – –.
|
||||
|
||||
|
||||
$$
|
||||
\bar {C} _ {p} = 1 - 4 \sin^ {2} \theta + 4 \alpha \sin \theta - \alpha^ {2}, \tag {8}
|
||||
$$
|
||||
|
||||
for cases with $\scriptstyle { \alpha = 0 }$ and 1.0. The comparison reveals that there is a big discrepancy between the potential and real viscous flows. The mean pressure coefficients at $\mathrm { R e } { = } 6 0$ and 160 for $\alpha { = } 1 . 0$ are also shown in Fig. 6. The results indicate that a change of the Reynolds number has a negligible effect on the pressure coefficient at the cylinder surface.
|
||||
|
||||
Figure 7 shows the mean vorticities around the cylinder surface for the same conditions as in Fig. 6. In this study, the mean surface vorticity is computed from $\overline { { \omega } } = 2 \alpha + \partial \overline { { u } } _ { \theta } / \partial r$ in the $( r , \theta )$ coordinate. In the case of $\alpha { = } 0 ,$ , the vorticity has the positive and negative peak values, respectively, at $\theta { \approx } 2 3 0 ^ { \circ }$ and $\theta \approx 1 3 0 ^ { \circ }$ , while it is nearly zero in the wake. With increasing , the vorticity increases in its magnitude, significantly deviating from zero in the wake region. The local peak in the wake region $( 3 0 0 ^ { \circ } < \theta < 3 6 0 ^ { \circ } )$ also increases with the peak point moving in the direction of rotation as increases, and then becomes of a maximum ~negative! value when *1.5. On the other hand, increasing Re involves an overall growth of the vorticity magnitude around the cylinder surface. The shear stress ~or friction coefficient! around the cylinder surface, defined as $\overline { { \tau } } _ { r \theta } = ( - 2 \alpha + \partial \overline { { u } } _ { \theta } / \partial r ) / \mathrm { R e }$ , can be explained from the surface vorticity shown in Fig. 7. Now that ${ \partial \overline { { u } } _ { \theta } } / { \partial r }$ is much larger than 2 for all the rotational speeds investigated, the distribution of the surface shear stress is very similar to that of the surface vorticity.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 7. Mean vorticities around the cylinder surface at $\alpha { = } 0 , 0 . 5 , 1 . 0 , 1 . 5 ,$ 2.0, and 2.5 for Re5100. Also shown are the mean vorticities at Re560 and 160 for $\alpha { = } 1 . 0 ,$ , denoted by – – –.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 8. Mean lift and drag coefficients according to the rotational speed; ~a! $\overline { { C } } _ { L } ; ( { \bf b } ) \ : \overline { { C } } _ { D } . \mathrm { ~ O , R e = 1 6 0 ; } \bigstar \bigstar \mathrm { , ~ } \forall \mathrm { e = 1 0 0 ; } \bigtriangleup , \mathrm { R e = 6 0 ; } \bigtriangledown , \mathrm { R e = 4 0 ; } \bigtriangleup$ , potential flow theory. Solid symbols ~j! are obtained from Badr et al. ~Ref. 8! at $\mathbf { R e } { = } 1 0 0$ . Also shown are the contributions of the pressure and friction forces; —, total; –•–, pressure; ¯, friction.
|
||||
|
||||
|
||||
# C. Lift and drag coefficients
|
||||
|
||||
The characteristics of lift and drag forces exerted on the cylinder rotating steadily in a viscous uniform flow is investigated in terms of the lift and drag coefficients, $C _ { L }$ and $C _ { D }$ . The ~total! drag and lift forces are composed of both the pressure and friction forces as follows:
|
||||
|
||||
$$
|
||||
C _ {L} = C _ {L p} + C _ {L f}, \quad C _ {D} = C _ {D p} + C _ {D f}. \tag {9}
|
||||
$$
|
||||
|
||||
Variations of the lift and drag coefficients for cases with various combinations of the rotational speed and Reynolds number are shown in Figs. 8–10.
|
||||
|
||||
Figure 8 shows the mean lift and drag coefficients for all the cases investigated, compared with those computed from the potential flow theory,
|
||||
|
||||
$$
|
||||
C _ {L} = - 2 \pi \alpha , \quad C _ {D} = 0. \tag {10}
|
||||
$$
|
||||
|
||||
Also shown in the same figure are the contributions of the pressure and friction forces. First, the mean lift coefficients, $\bar { C } _ { L } , \bar { C } _ { L p }$ , and $\bar { C } _ { L f }$ , are shown in Fig. 8~a!. In the figure, all the ~negative! lift coefficients increase linearly in proportion to for low rotational speeds and the least-square fit provides ~for <1.0!
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 9. Amplitudes of the lift and drag fluctuations according to the rotational speed; ~a! $C _ { L } ^ { \prime } ; ( { \bf b } ) \ C _ { D } ^ { \prime } . \bigcirc , \mathrm { R e } = 1 6 0 ; \bigcirc , \mathrm { R e } = 1 0 0 ; \bigtriangleup , \mathrm { R e } = 6 0$ . Solid symbols ~j! are obtained from Badr et al. ~Ref. 8! at Re5100.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 10. Phase diagrams of $C _ { L }$ with $C _ { D }$ for various values of ; ~a! Re 5100; ~b! Re5160. Time advances in the clockwise direction ~on the upper branch in the case of 50!.
|
||||
|
||||
|
||||
$$
|
||||
\bar {C} _ {L p} \sim - 2. 3 3 \alpha , \quad \bar {C} _ {L f} \sim - 0. 1 3 \alpha ,
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L} \sim - 2. 4 6 \alpha , \quad \text { for } \mathrm{Re} = 1 6 0,
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L p} \sim - 2. 3 1 \alpha , \quad \bar {C} _ {L f} \sim - 0. 1 7 \alpha ,
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L} \sim - 2. 4 8 \alpha , \quad \text { for } \mathrm{Re} = 1 0 0,
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L p} \sim - 2. 2 9 \alpha , \quad \bar {C} _ {L f} \sim - 0. 2 1 \alpha , \tag {11}
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L} \sim - 2. 5 0 \alpha , \quad \text { for } \mathrm{Re} = 6 0,
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L p} \sim - 2. 3 1 \alpha , \quad \bar {C} _ {L f} \sim - 0. 2 6 \alpha ,
|
||||
$$
|
||||
|
||||
$$
|
||||
\bar {C} _ {L} \sim - 2. 5 7 \alpha , \quad \text { for } \mathrm{Re} = 4 0.
|
||||
$$
|
||||
|
||||
The present result is in good agreement with the prediction of Badr et al. $\mathrm { , } \ \bar { C } _ { L } { \sim } - 2 . 5 5 \alpha$ for $\mathrm { R e } { = } 6 0$ , but in relatively poor agreement with the predictions of Tang and Ingham,10 $\bar { C } _ { L } { \sim } - 2 . 1 7 \alpha$ and 21.86 for $ { \mathrm { R e } } { = } 6 0$ and 100, respectively. Such discrepancies are due to the fact that Badr et al. investigated unsteady flow by solving time-dependent governing equations as in the present study, while Tang and Ingham investigated steady flow by solving time-independent governing equations. As shown in Eq. ~11!, the lift force mostly comes from the pressure force and its contribution increases with increasing Re, for example 92%, 93%, and 95%, respectively, for $\mathrm { R e } { = } 6 0 , 1 0 0$ , and 160. Since a change in the Reynolds number mostly affects the friction, as implied in Figs. 6 and 7, which has a negligible contribution to the total lift force, the total lift coefficient is nearly independent of Re. It is also observed that the lift force differs significantly from the result obtained from the potential flow theory @see Fig. 8~a!#.
|
||||
|
||||
The mean drag coefficients, $\bar { C } _ { D } , \ \bar { C } _ { D p }$ , and $\bar { C } _ { D f }$ , are shown in Fig. 8~b!. The drag force seems to be more complicated than the lift force. It is seen that the friction drag is of the same order of magnitude as the pressure drag and thus the total drag has a relatively large dependence on Re. As Re increases, the friction and total drag coefficients decrease. As increases, the pressure drag decreases with the increase in the friction drag, resulting in the net decrease in the total drag force. Even though the total drag forces at $\mathbf { R e } { = } 1 0 0$ and 160 are similar, their drag decompositions are different. It is also worth noting that the pressure drag becomes negative when *2.
|
||||
|
||||
In view of controlling vortex shedding, the lift and drag fluctuations are of more practical importance in engineering environments. Here, the amplitudes in the fluctuation are defined, respectively, as
|
||||
|
||||
$$
|
||||
C _ {L} ^ {\prime} = \frac {C _ {L , \max} - C _ {L , \min}}{2}, \quad C _ {D} ^ {\prime} = \frac {C _ {D , \max} - C _ {D , \min}}{2}, \tag {12}
|
||||
$$
|
||||
|
||||
where the subscripts min and max denote the minimum and maximum values, respectively, in a period. The variations of $C _ { L } ^ { \prime }$ and $C _ { D } ^ { \prime }$ with at $R e = 6 0$ , 100 and 160 are shown in Fig. 9, together with those of Badr et $a l . ^ { 8 }$ Excellent agreement is seen between the present and Badr et al.’s studies. The figure implies again that vortex shedding appears at low rotational speeds and then completrange of vortex shedding, disappears at stays nearly c $\alpha > \alpha _ { L }$ . In and $C _ { L } ^ { \prime }$ $C _ { D } ^ { \prime }$ increases linearly with increasing . On the other hand, the two fluctuation amplitudes, $C _ { L } ^ { \prime }$ and $C _ { D } ^ { \prime }$ , increase with increasing Re.
|
||||
|
||||
Behaviors of the lift and drag forces presented in Figs. 8 and 9 can be represented more clearly in the form of the phase diagram, by plotting $C _ { L }$ as a function of $C _ { D }$ at Re $= 1 0 0$ and 160 as shown in Fig. 10. The closed phase diagram indicates that the flow becomes fully time-periodic. As evident in the figure, the position of a phase diagram denotes the mean lift and drag and the size denotes the corresponding amplitudes of fluctuation. The phase diagrams confirm again that the mean lift and drag forces increase and decrease, respectively, with increasing . At the same time, the amplitude of the lift fluctuation stays almost constant and that of the drag fluctuation increases in the range of vortex shedding. It is also clear that the phase diagram collapses to a point ~denoted by the asterisk! at $\alpha > \alpha _ { L }$ , for example $\alpha { = } 1 . 9$ for $\mathbf { R e } { = } 1 0 0$ and $\alpha { = } 2 . 0$ for $\mathrm { R e } { = } 1 6 0$ . At 50, vortex shedding occurs at the frequency of $C _ { L }$ being half that of $C _ { D }$ , resulting in a symmetry and a crossline in the phase diagram. When the cylinder rotates, however, the flow becomes asymmetric due to the unidirectional rotation. Consequently, $C _ { D }$ has the same frequency as $C _ { L }$ when $\alpha { > } \alpha _ { s }$ at occurrence of vortex shedding, for example $\alpha _ { s } \approx 0 . 1$ at Re $= 1 0 0$ . The value $\alpha _ { s }$ tends to increase with increasing Re.
|
||||
|
||||
# D. Instantaneous flow fields
|
||||
|
||||
Instantaneous flow fields are investigated to identify the underlying mechanism of the suppression of vortex shedding due to the rotation. Figure 11 shows the vorticity contours at two different times, $t / T { = } 1 / 2$ and 1 in one complete time cycle, with increasing at $\mathbf { R e } { = } 1 0 0$ , where T is the nondimensional period or a reciprocal of the Strouhal number. Here, $t / T { = } 0$ is the time corresponding to the maximum lift, which does not necessarily mean that $t / T { = } 1 / 2$ corresponds to the minimum lift. As expected, it is clearly shown that vortex shedding which develops at $\scriptstyle { \alpha = 0 }$ also occurs at low values of $\alpha ,$ for example $\alpha { < } 2 . 0$ for $\mathrm { R e } { = } 1 0 0$ . While vortex shedding exists, the vorticity contours away from the cylinder surface are similar in overall shape, which indicates that the rotation effect is confined to the flow in the vicinity of the cylinder surface. In the near-surface flow, with increasing , the negative vorticity on the upper side of the cylinder becomes more dominant than the positive vorticity on the lower side. Subsequently, for $\alpha { \geq } 2 . 0$ , vortex shedding completely disappears and the flow has two stationary vorticity bubbles attached to the cylinder. As increases further, the bubbles become thinner and more inclined in the direction of rotation. In the range of $\alpha { \gtrsim } 1$ , the negative isovorticity lines in front of the cylinder form an acute curved tail, which grows around the cylinder in the direction of rotation with increasing .
|
||||
|
||||
The corresponding streamlines are shown in Fig. 12 for the same conditions. Overall, the figure shows that the rotation of a cylinder makes a substantial effect on the flow pattern. When the cylinder rotates at low rotational speeds ~ ,2.0!, two vortices are alternately shed on each side of the cylinder but the vortex shedding configuration varies according to . As increases, the vortex on the upper side in the wake becomes stronger, while that on the lower side becomes weaker. Finally, the lower vortex completely disappears when >2.0, resulting in no vortex shedding. In addition, it is valuable to compare the flow pattern to the corresponding potential flow, especially at a high rotational speed. The potential flow theory24 gives that closed streamlines surrounding the cylinder exist only when .2.0, where the stagnation point is detached from the cylinder surface. The same phenomenon can be observed in the viscous flow but for different conditions. In the present study, the flow has the closed streamlines surrounding the cylinder when $\alpha { \gtrsim } 0 . 5$ , where the separation point departs from the cylinder surface. The flow pattern for 52.5 is very similar to that of the potential flow.24
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 11. Vorticity contours at t/T51/2 ~left-hand side! and 1 ~right-hand side! with increasing at Re5100. 5~a! 0.0, ~b! 0.5, ~c! 1.0, ~d! 1.5, ~e! 2.0, and ~f! 2.5. Contour levels are from 23 to 3 in increments of 0.2. Negative values are shown as dashed.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 12. Streamlines at t/T51/2 ~left-hand side! and 1 ~right-hand side! with increasing at Re5100. 5~a! 0.0, ~b! 0.5, ~c! 1.0, ~d! 1.5, ~e! 2.0, and ~f! 2.5. Contour levels are from 20.3 to 0.3 in increments of 0.02 and from 60.3 to 61 in increments of 60.05.
|
||||
|
||||
|
||||
Figure 13 shows the time evolution of the streamlines at Re5100 and $\alpha { = } 1 . 0$ . This figure indicates the way how the shed vortices are formed and convected downstream. It is seen that there are two alternate vortices over one complete time cycle with one vortex $[ ^ { 6 } \mathrm { \AA } ^ { , 3 }$ vortex in ~e!# stronger than the other $( ^ { 6 } \mathrm { \Delta ^ { 6 } B ^ { \prime } } )$ vortex! because of the unidirectional rotation, and the stronger vortex lasts longer than the other. Such an observation becomes more pronounced as the rotational speed increases.
|
||||
|
||||
# E. Stability and bifurcation
|
||||
|
||||
Results have shown repeatedly that in the laminar vortex shedding regime ~47<Re,200! vortex shedding exists at low rotational speeds and completely disappears at $\alpha > \alpha _ { L }$ . This indicates that there should be a bifurcation curve between the existence of vortex shedding and its disappearance on a two-dimensional plane of Re and . Here, such a bifurcation curve based on the onset of vortex shedding is a typical result of flow stability problem. Figure 14 shows the bifurcation curve, or borderline, between steady and unsteady ~time-periodic! regimes obtained with varying by an increment of 0.1 at several chosen Reynolds numbers. Notice that the real bifurcation curve should reside in between open circle and cross symbols in the figure. The figure shows that the $\alpha _ { L }$ ~denoted by open circles! increases logarithmically as Re increases. The observation was also found qualitatively in Badr and Dennis,12 Coutanceau and Me´nard,13 and Badr et al., 14 who all investigated unsteady flow at high Reynolds numbers $( \mathrm { R e } { = } 2 0 0 { - } 1 0 ^ { 4 } )$ and found the disappearance of vortex shedding at $\alpha > \alpha _ { L }$ . Then, they claimed that $\alpha _ { L }$ is nearly independent of Re and is about 2, which is apparently inconsistent with the present result but may be valid at high Reynolds numbers. Recently, Hu et al.17 found analytically the logarithmic dependence of $\alpha _ { L }$ on Re in the range of $4 5 { \leqslant } \mathrm { R e } { \leqslant } 5 0$ and <1.0 using a Galerkin method and a system theory, as also depicted in Fig. 14. Agreement between the results of the present study and Hu et al. is clearly seen. It can be concluded from Fig. 14 that the Reynolds number tends to destabilize flow past a circular cylinder, while the rotation of the cylinder tends to stabilize it.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 13. Time evolution of the streamlines at Re5100 and $\alpha { = } 1 . 0 ; t / T { = } ( \mathrm { a } )$ 1/8, ~b! 2/8, ~c! 3/8, ~d! 4/8, ~e! 5/8, ~f! 6/8, ~g! 7/8, and ~h! 1. Contour levels are from 20.3 to 0.3 in increments of 0.02 and from 60.3 to 61 in increments of 60.05.
|
||||
|
||||
|
||||
# IV. SUMMARY
|
||||
|
||||
In this paper, we have performed a numerical study of fully developed two-dimensional laminar flow past a circular cylinder rotating with a constant angular velocity for the purpose of controlling vortex shedding and understanding the underlying flow mechanism. The rotation of a circular cylinder in a viscous uniform flow may significantly modify flow patterns and, thus, reduce a flow-induced oscillation from vortex shedding. Numerical simulations were performed for flows with Re540, 60, 100, and 160 in the range of $0 { \leqslant } \alpha { \leqslant } 2 . 5$ . In the case of no rotation, the flow at $\mathrm { R e } { = } 4 0$ is steady, while the flows at Re560, 100, and 160 are timeperiodic with vortex shedding.
|
||||
|
||||
Results showed that vortex shedding exists, resulting in a time-periodic flow, at low rotational speeds and completely disappears at $\alpha > \alpha _ { L }$ , where the critical rotational speed $\alpha _ { L }$ depends on the Reynolds number, for example $\alpha _ { L } { \approx } 1 . 4 ,$ , 1.8, and 1.9 for $\mathrm { R e } { = } 6 0 .$ , 100, and 160, respectively. In other words, the value of $\alpha _ { L }$ increases logarithmically as Re increases. When vortex shedding exists in the range of $\leqslant \alpha _ { L }$ , notable changes in the flow pattern are observed. First, the Strouhal number is nearly independent of the rotational speed, but strongly dependent on the Reynolds number. Second, as the rotational speed increases, the mean lift force increases almost linearly with and the mean drag force decreases. At the same time, the amplitude of the lift fluctuation stays nearly constant and that of the drag fluctuation increases linearly with . The drag and lift fluctuations also vanish at $\alpha > \alpha _ { L }$ . Further studies from the instantaneous flow fields demonstrate again that the rotation of a cylinder makes a substantial effect on the flow pattern.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 14. Bifurcation curve dividing into the steady and time-periodic regimes on a plane of Re and , compared to Hu et al. ~Ref. 17!. ~L!: s, vortex shedding; 3, no vortex shedding.
|
||||
|
||||
|
||||
# ACKNOWLEDGMENTS
|
||||
|
||||
This work was supported by National Creative Research Initiatives of the Korean Ministry of Science and Technology. The computations were performed on the CRAY YMP C90 at the Electronics and Telecommunications Research Institute. The supports are gratefully acknowledged.
|
||||
|
||||
|
||||
|
||||
1C. H. K. Williamson, ‘‘Vortex dynamics in the cylinder wake,’’ Annu. Rev. Fluid Mech. 28, 477 ~1996!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
2S. Taneda, ‘‘Visual observations of the flow past a circular cylinder performing a rotary oscillation,’’ J. Phys. Soc. Jpn. 45, 1038 ~1978!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
3P. T. Tokumaru and P. E. Dimotakis, ‘‘Rotary oscillation control of a cylinder wake,’’ J. Fluid Mech. 224, 77 ~1991!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
4J. R. Filler, P. L. Marston, and W. C. Mih, ‘‘Response of the shear layers separating from a circular cylinder to small-amplitude rotational oscillations,’’ J. Fluid Mech. 231, 481 ~1991!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
5J. M. Wu, J. D. Mo, and A. D. Vakili, ‘‘On the wake of cylinder with rotational oscillations,’’ AIAA Paper No. AIAA-89-1024, 1989.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
6S. Baek and H. J. Sung, ‘‘Numerical simulation of the flow behind a rotary oscillating circular cylinder,’’ Phys. Fluids 10, 869 ~1998!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
7D. B. Ingham, ‘‘Steady flow past a rotating cylinder,’’ Comput. Fluids 11, 351 ~1983!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
8H. M. Badr, S. C. R. Dennis, and P. J. S. Young, ‘‘Steady and unsteady flow past a rotating circular cylinder at low Reynolds numbers,’’ Comput. Fluids 17, 579 ~1989!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
9D. B. Ingham and T. Tang, ‘‘A numerical investigation into the steady flow past a rotating circular cylinder at low and intermediate Reynolds numbers,’’ J. Comput. Phys. 87, 91 ~1990!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
10T. Tang and D. B. Ingham, ‘‘On steady flow past a rotating circular cylinder at Reynolds numbers 60 and 100,’’ Comput. Fluids 19, 217 ~1991!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
11P. T. Tokumaru and P. E. Dimotakis, ‘‘The lift of a cylinder executing rotary motions in a uniform flow,’’ J. Fluid Mech. 255, 1 ~1993!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
12H. M. Badr and S. C. R. Dennis, ‘‘Time-dependent viscous flow past an impulsively started rotating and translating circular cylinder,’’ J. Fluid Mech. 158, 447 ~1985!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
13M. Coutanceau and C. Me´nard, ‘‘Influence of rotation on the near-wake development behind an impulsively started circular cylinder,’’ J. Fluid Mech. 158, 399 ~1985!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
14H. M. Badr, M. Coutanceau, S. C. R. Dennis, and C. Me´nard, ‘‘Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104,’’ J. Fluid Mech. 220, 459 ~1990!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
15Y. Chen, Y. Ou, and A. J. Pearlstein, ‘‘Development of the wake behind a circular cylinder impulsively started into rotary and rectilinear motion,’’ J. Fluid Mech. 253, 449 ~1993!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
16C. Chang and R. Chern, ‘‘Vortex shedding from an impulsively started rotating and translating circular cylinder,’’ J. Fluid Mech. 233, 265 ~1991!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
17G. Hu, D. Sun, X. Yin, and B. Tong, ‘‘Hopf bifurcation in wakes behind a rotating and translating circular cylinder,’’ Phys. Fluids 8, 1972 ~1996!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
18H. Choi, P. Moin, and J. Kim, ‘‘Direct numerical simulation of turbulent flow over riblets,’’ J. Fluid Mech. 255, 503 ~1993!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
19P. Beaudan and P. Moin, ‘‘Numerical experiments on the flow past a
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
circular cylinder at subcritical Reynolds number,’’ Report No. TF-62, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1994.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
20L. L. Pauley, P. Moin, and W. C. Reynolds, ‘‘The structure of twodimensional separation,’’ J. Fluid Mech. 220, 397 ~1990!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
21J. Park, K. Kwon, and H. Choi, ‘‘Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160,’’ KSME Int. J. 12, 1200 ~1998!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
22C. H. K. Williamson, ‘‘Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers,’’ J. Fluid Mech. 206, 579 ~1989!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
23U. Fey, M. Ko¨nig, and H. Eckelmann, ‘‘A new Strouhal–Reynoldsnumber relationship for the circular cylinder in the range 47,Re,2 3105,’’ Phys. Fluids 10, 1547 ~1998!.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
24F. M. White, Fluid Mechanics ~McGraw–Hill, New York, 1994!.
|
||||
|
||||
@@ -0,0 +1,628 @@
|
||||
Physics of Fluids
|
||||
[Non-Text]
|
||||
AIP Publishing
|
||||
09 May 2026 02:44:09
|
||||
RESEARCH ARTICLE | MAY 01 2004
|
||||
# A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder
|
||||
Mehmet Sahin; Robert G. Owens
|
||||
|
||||
|
||||
Check for updates
|
||||
Physics of Fluids 16, 1305–1320 (2004)
|
||||
https://doi.org/10.1063/1.1668285
|
||||
|
||||
|
||||
|
||||
View Online
|
||||
|
||||
|
||||
|
||||
|
||||
Export Citation
|
||||
|
||||
# Articles You May Be Interested In
|
||||
Shear-induced autorotation of freely rotatable cylinder in a channel flow at moderate Reynolds number
|
||||
Physics of Fluids (April 2018)
|
||||
Frequency lock-in mechanism in the presence of blockage effects
|
||||
Physics of Fluids (July 2024)
|
||||
The blockage and erosion characteristics of woody debris flow on an erodible gully bed: Insight from a small-scale model experiment
|
||||
Physics of Fluids (December 2024)
|
||||
|
||||
|
||||
# AIP Advances
|
||||
# Why Publish With Us?
|
||||
|
||||
|
||||
|
||||
21DAYS average time to1st decision
|
||||
|
||||
|
||||
|
||||
|
||||
OVER 4 MILLION views in the last year
|
||||
|
||||
|
||||
|
||||
|
||||
INCLUSIVE scope
|
||||
|
||||
Learn More
|
||||
|
||||
|
||||
|
||||
AIP Publishing
|
||||
|
||||
09 May 2026 02:44:09
|
||||
HTmL AB:STRACT * LINKS
|
||||
PHYSICS OF FLUIDS
|
||||
VOLUME 16, NUMBER 5
|
||||
MAY 2004
|
||||
# A numerical investigation of wall effects up to high blockage ratios on two-dimensional flow past a confined circular cylinder
|
||||
Mehmet Sahin and Robert G. Owensa)
|
||||
LMF-ISE-FSTI, Ecole Polytechnique Fe´de´rale de Lausanne, CH 1015 Lausanne, Switzerland
|
||||
~Received 25 September 2003; accepted 20 January 2004; published online 2 April 2004!
|
||||
A finite volume method based on a velocity-only formulation is used to solve the flow field around a confined circular cylinder in a channel in order to investigate lateral wall proximity effects on stability, Strouhal number, hydrodynamic forces and wake structure behind the cylinder for a wide range of blockage ratios (0.1,b<0.9) and Reynolds numbers $( 0 < R e \leqslant 2 8 0 )$ . For blockage ratios less than approximately 0.85 a first critical Reynolds number is identified at which a supercritical Hopf bifurcation of the symmetric solution occurs. For blockage ratios greater than about 0.687 and at Reynolds numbers exceeding the first critical Reynolds number a second curve of neutral stability is seen, representing a pitchfork bifurcation of the steady symmetric solution to one of two possible steady asymmetric solutions. Either side of the neutral stability curve for the pitchfork bifurcation our linear stability analysis and direct numerical simulations demonstrate that although the flow is linearly stable it is unstable to finite two-dimensional perturbations. At blockage ratios larger than about 0.82 the steady asymmetric solutions also become unstable through a Hopf bifurcation. In contrast with the first Hopf bifurcation of the symmetric solution at lower Reynolds numbers numerical calculations of the lift coefficient reveal that the oscillations are no longer symmetric in the rising and falling parts of each cycle. Very strong vortices shed from the cylinder and the wall cause drastic increases in the amplitudes of the lift and drag coefficients. A co-dimension 2 point where pitchfork and Hopf bifurcations occur simultaneously has been located in parameter space. Altogether, four distinct regions in the parameter space $( \beta , R e ) \in ( 0 , 0 . 9 ] \times ( 0 , 2 8 0 ]$ have been identified, each corresponding to a different class of flow: ~i! Steady symmetric flow, ~ii! symmetric vortex shedding, ~iii! steady asymmetric flow, and ~iv! asymmetric vortex shedding, where a periodic-in-time flow is classed as symmetric or asymmetric depending on whether the time-average over one cycle of the lift coefficient is zero or not. Numerical solutions are computed on meshes having up to 1.8 million degrees of freedom. Extensive comparisons are made with the results available in the literature. © 2004 American Institute of Physics. @DOI: 10.1063/1.1668285#
|
||||
# I. INTRODUCTION
|
||||
It is no exaggeration to say that an enormous ~and still rapidly growing! corpus of literature on the subject of bluff body wakes has developed since the pioneering work of von Ka´rma´n early last century. This fact is an attestation to both the difficulty in understanding and adequately describing the flow bifurcations that occur at various values of the Reynolds number in viscous flows and the interest in doing so. Flows having particularly simple setups such as those past a sphere or cylinder have succeeded in drawing experimentalists, theoreticians and computational fluid dynamicists into the fray that has gone on through the decades and only very recently are consensuses emerging.
|
||||
Details of recent theoretical, experimental and computational developments for unbounded flow past a cylinder may be found in the review paper of Williamson,1 where particular attention is paid to the vortex dynamics in the cylinder wake. Our interest in this paper is a careful analysis of lateral wall effects on viscous flow past a confined cylinder. What we have in mind is depicted in Fig. 1. In this figure an infinitely long cylinder of diameter D is placed symmetrically between parallel lateral walls a distance H apart. The parameter $\beta { \equiv } D / H$ is usually termed the blockage ratio. In stark contrast to the wealth of insight and commentary available on vortex dynamics in the wake of an unbounded cylinder we find ourselves with only a handful of papers offering a serious treatment of the blockage ratio effects present in the confined cylinder problem. This paucity of scientific literature should not be interpreted as implying that the problem is an unimportant one, however. On the contrary, even for unbounded flow past a cylinder the ~infinite! flow domain has to be replaced with ~or mapped onto! a finite one, thus introducing numerical or experimental blockage effects that may have considerable influence over the determined values of the flow parameters.2,3 Many of the blockage ratio effects described in the literature are more or less evident:
|
||||
|
||||
~1! In the steady flow regime, bringing the walls closer to the cylinder results in the appearance of the twin vortices in the cylinder wake at higher Reynolds numbers.
|
||||
~2! At any given modest (&50) Reynolds number and for
|
||||
a! Author to whom correspondence should be addressed. Electronic mail: robert.owens@epfl.ch
|
||||
1070-6631/2004/16(5)/1305/16/$22.00
|
||||
1305
|
||||
© 2004 American Institute of Physics
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
|
||||
|
||||
FIG. 1. Schematic of a cylinder placed symmetrically in a plane channel. The cylinder diameter is D and the channel height H.
|
||||
|
||||
$\beta { \leqslant } 0 . 2$ the length of the closed vortex bubble decreases with wall proximity, while remaining a linear function of R e . 2– 4 $R e . ^ { 2 - 4 }$
|
||||
~3! For increasing blockage ratios $\beta$ up to 0.5 the steady two-dimensional base flow is stabilized with respect to infinitesimal perturbations due to constraint by the confining walls of the separating shear layer that exists between the cylinder wake and the wall boundary layer vorticity.3,4
|
||||
~4! Once the critical Reynolds number for the primary instability has been exceeded the frequency with which periodic two-dimensional vortex shedding takes place at a given Reynolds number is an increasing function of $\bar { \boldsymbol { \beta } } . ^ { 2 , 4 - 6 }$ ~Note that the spurious result obtained by Stansby and Slaouti6 for $\beta = 0 . 5$ is thought to be due to neglect of the boundary layers in their numerical simulation using random vortex methods.!
|
||||
~5! At $R e = O ( 1 0 0 )$ both the mean drag coefficient $C _ { d }$ and the separation angle of the vortex bubble increase as the walls approach the cylinder.2,3,5,6
|
||||
In addition to the obvious interest of wall blockage effects and as observed by Chen $e t a l . , { ^ 4 }$ the choice of a bounded domain allows a more definitive specification of the flow ~both numerically and experimentally! than is possible in the unbounded case, whilst conserving the essential features of the latter.
|
||||
A problem bearing some similarities to that of flow past a confined cylinder is that of flow around a cylinder placed at various heights above a plane boundary. A recent literature survey of experimental investigations into this problem may be found in the paper of Lei et al.7 These studies have sought to address the issue of how forces on the cylinder and vortex shedding frequency depend on the ratio $g / D$ of the gap between the cylinder and the wall, g, and the cylinder diameter, D. They have also been concerned with understanding the effect on these quantities of the boundary layer thickness and the velocity gradient. Most of the experiments have been conducted at Reynolds numbers in the sub-critical regime $\left[ R e = O ( 1 \times 1 0 ^ { 4 } ) \right]$ in which the boundary layer is still laminar. Lei et $a l . ^ { 7 }$ found that the drag coefficient $C _ { d }$ increased with increasing gap ratio because of the reduction in the base pressure. The same trend in base pressure dependence had been observed by Bearman and Zdravkovich.8 The latter authors further found that the Strouhal number for $g / D { \gtrsim } 0 . 3$ was more or less constant in their experiments at a Reynolds number of $4 . 8 \times 1 0 ^ { 4 }$ . Lei et $a l . ^ { 7 }$ also noted only slight fluctuations in a Strouhal number computed from the free-stream velocity for a similar range of gap to diameter ratios. However, for gaps less than 0.3 cylinder diameters8 or $0 . 2 \mathrm { - } 0 . 3$ diameters ~depending on the boundary layer thickness7 !, vortex shedding was suppressed. Differences in the quantification of the vortex shedding suppression gap ratio were due possibly to differences in the boundary layer thicknesses generated by the experimentalists and also to the manner in which the critical gap ratio was identified: Bearman and $\mathrm { Z d r a v k o v i c h } ^ { 8 }$ using a spectral analysis of hot-wire signals in the cylinder wake whereas the method of Lei $e t a l . ^ { 7 }$ was based on observation of the spectrum of the lift coefficient. Suppression of vortices for a sufficiently small gap ratio was also confirmed by Zovatto and Pedrizzetti,9 who used a finite element method based on a vorticity-streamfunction formulation to analyze flow around a cylinder positioned eccentrically between two lateral walls. For very small gap ratios Zovatto and Pedrizzetti9 found a recirculating bubble on the wall downstream of the cylinder. A separation bubble on the wall had also been seen earlier by Bearman and Zdravkovich8 for gap ratios smaller than the critical value for vortex suppression. We will return to some of these flow phenomena in our discussion of our numerical results in Sec. IV for large blockage ratios.
|
||||
|
||||
The motivation for the present study is twofold. First, the rich fluid dynamics in the wake and near the lateral walls deserves to be investigated with greater numerical accuracy than has been possible with the computational resources available to other researchers at the time at which they prepared their manuscripts. Computations on meshes allowing for only tens of thousands of degrees of freedom have been typical ~for example, Refs. 4 and 10!. In the present study a novel finite volume method $\begin{array} { r l } { { 1 1 - 1 3 } } & { { } } \\ { . } \end{array}$ is used in a parallel implementation, permitting up to 1.8 million degrees of freedom and thus a higher resolution of the wake and boundary layer structures. Second, the only previous numerical linear stability analysis of flow past a confined cylinder available to $\mathrm { u s } ^ { 4 }$ went no further than a blockage ratio of $\beta { = } 0 . 7$ . From the results of this publication the trend seemed to be one of decreasing linear stability of the two-dimensional flow for $\beta { > } 0 . 5$ . Stability was always lost over the range of blockage ratios considered through a symmetry-breaking supercritical Hopf bifurcation. We wish in this paper to investigate the effect on the critical Reynolds number of choosing $\beta { > } 0 . 7$ and to identify the nature of the flow instabilities by means of an Arnoldi method.
|
||||
The outline of the present paper is as follows: In Sec. II we describe the problem to be solved and furnish the reader with a brief description of the numerical method used to analyze flow past a confined cylinder at Reynolds numbers up to 280. Section III is dedicated to validation of our numerical scheme for the classical problem of unbounded twodimensional flow past a circular cylinder. Extensive comparison with other results in the literature is made. In particular we find excellent agreement with previously obtained values for the drag coefficients, first critical Reynolds numbers $R e _ { \mathrm { c r i t 1 } }$ and the corresponding critical Strouhal numbers. In
|
||||
1306
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
Sec. IV we are concerned with a detailed description of wake dynamics and interactions of the wake and wall boundary layers for blockage ratios up to 0.9. For blockage ratios below approximately 0.85 the locus of a supercritical Hopf bifurcation may be traced out in parameter space. At higher Reynolds numbers and for blockage ratios sufficiently large there is a pitchfork bifurcation of the steady symmetric state to one of two asymmetric steady states. Either side of the curve of neutral stability for the pitchfork bifurcation the steady solutions are linearly stable but appear on the basis of direct numerical simulations to be unstable to finite twodimensional perturbations. For yet larger Reynolds numbers and $\beta { \gtrsim } 0 . 8 2$ a Hopf bifurcation of the asymmetric state occurs. The oscillations are now quite different from those associated with the first symmetry-breaking instability, the amplitude of the drag and lift coefficients being much stronger and the oscillations are now asymmetric in time. Finally, we draw some conclusions.
|
||||
# II. MATHEMATICAL PROBLEM AND NUMERICAL SCHEME
|
||||
An infinitely long cylinder of diameter D is placed midway between two parallel planes which are a distance H apart, as shown in Fig. 1. Let us denote by $U _ { \mathrm { m a x } }$ the maximum inlet fluid speed. The incompressible unsteady Navier– Stokes equations may be written in dimensionless form as
|
||||
$$
|
||||
\frac {\partial \mathbf {u}}{\partial t} + (\mathbf {u} \cdot \nabla) \mathbf {u} = - \nabla p + \frac {1}{R e} \nabla^ {2} \mathbf {u}, \tag {1}
|
||||
$$
|
||||
$$
|
||||
\nabla \cdot \mathbf {u} = 0, \tag {2}
|
||||
$$
|
||||
where, in the usual notation, $\mathbf { u } { = } ( u _ { 1 } , u _ { 2 } )$ denotes the velocity field, p the pressure and Re is a Reynolds number. In the present work the Reynolds number is defined as Re $= U _ { \mathrm { m a x } } D / v$ where v is the kinematic viscosity. In the presentation of results in Secs. III and IV for those flows exhibiting periodic vortex shedding, a Strouhal number St is defined by $S t { = } D / ( T U _ { \operatorname* { m a x } } ) .$ , where T is the period of vortex shedding. We denote by $( \mathbf { x } , t ) { = } ( ( x _ { 1 } , x _ { 2 } ) , t )$ a generic point in space and time.
|
||||
In Sec. III we approximate the unbounded cylinder geometry by choosing $\beta = 0 . 0 1$ and the following boundary conditions:
|
||||
$\mathrm { C y l i n d e r ~ s u r f a c e : } ~ { \bf u } { = } ( 0 , 0 ) ,$ ~3!
|
||||
$\mathrm { L a t e r a l ~ w a l l s } \colon \mathrm { ~ } \mathbf { u } = ( 1 , 0 ) ,$ ~4!
|
||||
$\mathrm { I n f l o w : } \qquad \mathbf { u } = ( 1 , 0 ) ,$ ~5!
|
||||
$\mathrm { O u t f l o w : } \qquad { \frac { \partial ^ { 2 } u _ { 1 } } { \partial x _ { 1 } ^ { 2 } } } = 0 , \ { \frac { \partial u _ { 2 } } { \partial x _ { 1 } } } = 0 .$ 9u1 du2 ~6! x1
|
||||
For the confined cylinder problem ~see Sec. IV! Eqs. ~1! and ~2! are solved subject to the following boundary conditions on the components of velocity:
|
||||
$\mathrm { C y l i n d e r ~ s u r f a c e : } ~ { \bf u } { = } ( 0 , 0 ) ,$ ~7!
|
||||
$\mathrm { L a t e r a l ~ w a l l s } \colon \mathrm { ~ } \mathbf { u } = ( 0 , 0 ) ,$ ~8!
|
||||
$\mathrm { I n f l o w : } \qquad \mathbf { u } = ( 1 - x _ { 2 } ^ { 2 } , 0 ) ,$ ~9!
|
||||
$$
|
||||
\text { Outflow: } \quad \frac {\partial^ {2} u _ {1}}{\partial x _ {1} ^ {2}} = 0, \frac {\partial u _ {2}}{\partial x _ {1}} = 0. \tag {10}
|
||||
$$
|
||||
Some care needs to be taken with how the second normal derivative outflow condition is imposed, due to possible linear dependence in the discrete equation set of this condition with the discrete form of the continuity equation. More precisely stated, the second normal derivative outflow condition will be automatically satisfied at $x _ { 2 } = 0$ upon imposition of the continuity equation ~2! within each finite volume. For all the results presented in Sec. III the dimensionless upstream and downstream channel lengths were set equal to 400 D. In Sec. IV these lengths were chosen to both be equal to 40 D. The choice of outflow boundary conditions ~6! and ~10! was motivated by the fact that our numerical method uses a velocity-only formulation so that the usual traction-free conditions could not easily be implemented. The free boundary layer type of conditions ~6! and ~10! were used successfully by Kourta et al.14 in finite volume simulations of a twodimensional plane mixing layer. Although Jin and Braza15 later developed a nonreflecting outlet condition that greatly reduced feedback noise when compared with the outlet condition of Kourta et al.,14 the outlet length used for the calculations in the present paper are considered sufficiently great that the difference between the influence of the one set of exit conditions and the other on drag, linear stability and Strouhal number would be negligible. The more complicated exit conditions of Jin and Braza15 are therefore not implemented.
|
||||
Let n denote a unit outward pointing normal vector to the boundary ]V of a finite volume V. Then integration of ~2! over V and taking the vector product of ~1! with n, followed by integration around ]V leads, respectively, to
|
||||
$$
|
||||
\oint_ {\partial \Omega} \mathbf {n} \cdot \mathbf {u} d s = 0 \tag {11}
|
||||
$$
|
||||
and
|
||||
$$
|
||||
\oint_ {\partial \Omega} \mathbf {n} \times \left[ \frac {\partial \mathbf {u}}{\partial t} + (\nabla \times \mathbf {u}) \times \mathbf {u} + \frac {1}{R e} \nabla \times (\nabla \times \mathbf {u}) \right] d s = \mathbf {0}. \tag {12}
|
||||
$$
|
||||
In our numerical scheme the continuity equation ~11! is satisfied within each finite volume while ~12! is applied to each finite volume except the finite volumes next to the wall. Therefore, vorticity creation is allowed within these finite volumes in order to satisfy the no-slip boundary conditions. Equations ~11! and ~12! with no-slip boundary conditions are enough to solve the problem in a simply connected domain ~such as that found in the lid-driven cavity problem, for $\mathrm { e x a m p l e } ^ { 1 2 } )$ . However, if the domain is not simply connected there is a need for additional equations. This is because there is a potential problem in our velocity-only formulation with multi-valuedness of the pressure field, even though the pressure does not appear explicitly as a dependent variable in our formulation. To rectify this a Kutta-type condition
|
||||
$$
|
||||
\oint_ {\Gamma} \mathbf {n} \times \left[ \frac {\partial \mathbf {u}}{\partial t} + (\nabla \times \mathbf {u}) \times \mathbf {u} + \frac {1}{R e} \nabla \times (\nabla \times \mathbf {u}) \right] d s = \mathbf {0}, \tag {13}
|
||||
$$
|
||||
1307
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
TABLE I. Values of grid parameters $i _ { \mathrm { m a x } } , k _ { \mathrm { m a x } } , k _ { \mathrm { w a l l } }$ , and N .
|
||||
|
||||
<table><tr><td rowspan="2"><eq>\beta</eq></td><td colspan="4">M1</td><td colspan="4">M2</td><td colspan="4">M3</td></tr><tr><td><eq>i_{\text{max}}</eq></td><td><eq>k_{\text{max}}</eq></td><td><eq>k_{\text{wall}}</eq></td><td>N</td><td><eq>i_{\text{max}}</eq></td><td><eq>k_{\text{max}}</eq></td><td><eq>k_{\text{wall}}</eq></td><td>N</td><td><eq>i_{\text{max}}</eq></td><td><eq>k_{\text{max}}</eq></td><td><eq>k_{\text{wall}}</eq></td><td>N</td></tr><tr><td>0.01</td><td>181</td><td>301</td><td>137</td><td>89 336</td><td>361</td><td>601</td><td>273</td><td>355 312</td><td>721</td><td>1201</td><td>545</td><td>1 417 184</td></tr><tr><td>0.1</td><td>181</td><td>441</td><td>77</td><td>116 536</td><td>361</td><td>881</td><td>153</td><td>462 512</td><td>721</td><td>1761</td><td>305</td><td>1 842 784</td></tr><tr><td>0.2</td><td>181</td><td>421</td><td>57</td><td>109 336</td><td>361</td><td>841</td><td>113</td><td>433 712</td><td>721</td><td>1681</td><td>225</td><td>1 727 584</td></tr><tr><td>0.3</td><td>181</td><td>411</td><td>47</td><td>105 736</td><td>361</td><td>821</td><td>93</td><td>419 312</td><td>721</td><td>1641</td><td>185</td><td>1 669 984</td></tr><tr><td>0.5</td><td>181</td><td>401</td><td>37</td><td>102 136</td><td>361</td><td>801</td><td>73</td><td>404 912</td><td>721</td><td>1601</td><td>145</td><td>1 612 384</td></tr><tr><td>0.7</td><td>181</td><td>391</td><td>27</td><td>98 536</td><td>361</td><td>781</td><td>53</td><td>390 512</td><td>721</td><td>1561</td><td>105</td><td>1 554 784</td></tr><tr><td>0.9</td><td>181</td><td>381</td><td>17</td><td>94 936</td><td>361</td><td>761</td><td>33</td><td>376 112</td><td>721</td><td>1521</td><td>65</td><td>1 497 184</td></tr></table>
|
||||
is imposed around the closed path G formed from the union of the outer edges of the finite volumes on the cylinder surface. The condition ~13! guarantees that
|
||||
$$
|
||||
\oint_ {\Gamma} \mathbf {n} \times \nabla p d s = \mathbf {k} [ p ] = \mathbf {0}, \tag {14}
|
||||
$$
|
||||
where k is a unit vector normal to the plane of the flow, and @ p# denotes the jump in the pressure on passing once around G. Since ~12! is satisfied in every interior finite volume, satisfaction of ~13! ensures that $p$ is single-valued at every interior finite volume vertex. The pressure can be obtained by integrating the two components of the pressure gradient appearing in the equations of linear momentum in a manner analogous to that used in finding a streamfunction from a given velocity. The values of p on the domain boundaries, when required, are determined by first computing ]p/]n from ~1!.
|
||||
A fully implicit second-order cell-vertex finite volume method based on a velocity-only formulation is used for the discretization of ~1! and ~2!. Discretization of the integrals appearing in ~11! and ~12! is effected by using the mid-point rule on cell faces. Full details of the method are supplied in two recent papers by the present authors11,12 and, in the interests of brevity, will not be reproduced here. For the timedependent computations presented in Secs. III and IV we discretize in time using an Euler implicit method and for computing steady-state base flows a Newton method is employed.
|
||||
A major part of the present paper is concerned with the linear stability of two-dimensional flow at various different blockage ratios. Consider the perturbed flow
|
||||
$$
|
||||
\mathbf {u} (\mathbf {x}, t) = \mathbf {U} (\mathbf {x}) + \mathbf {v} (\mathbf {x}) \exp (\sigma t), \tag {15}
|
||||
$$
|
||||
where U(x) is the ~numerically determined! steady base flow at a given Reynolds number. Then discretizing the dimensionless Navier–Stokes equations as described above leads to an algebraic system of equations
|
||||
$$
|
||||
\mathbf {A} \mathbf {x} = \sigma \mathbf {M} \mathbf {x}, \tag {16}
|
||||
$$
|
||||
for the nodal values of the perturbation velocity v. The matrices A and M in ~16! are block quad-diagonal and block bi-diagonal, respectively. The GEVP ~16! may be solved by applying Arnoldi’s method16,17 to the equivalent system
|
||||
$$
|
||||
\mathbf {C} \mathbf {x} = \mu \mathbf {x}, \tag {17}
|
||||
$$
|
||||
where $\mathbf { C } = ( \mathbf { A } - \lambda \mathbf { M } ) ^ { - 1 } \mathbf { M }$ and $\mu { = } \left( \sigma { - } \lambda \right) ^ { - 1 }$ . Application of the Arnoldi method results in the construction of an upper Hessenberg matrix whose eigenvalues are approximations to a subset of the eigenvalues $\mu$ of C. From the properties of Arnoldi’s method and in the absence of a shift $\lambda , ^ { 1 8 }$ best resolution of the s-spectrum is expected to be near the origin.
|
||||
The coefficient matrix A in ~16! is almost identical ~by construction! to that which arises in the computations of the steady base flow using Newton’s method. Solutions to all the discrete algebraic equations that arise in the steady, unsteady or eigenvalue problems of this paper have been obtained by implementing the MUltifrontal Massively Parallel Solver ~MUMPS! of Amestoy et al.19,20 The multifrontal method used is a direct method based on LU decomposition for the solution of sparse systems of linear equations with optimum fill in. The algorithms employed by MUMPS use a dynamic distributed task scheduling technique that permits numerical pivoting and the transfer of computational tasks to lightly loaded processors. The calculations have been performed on an SGI Origin 3800 parallel machine with 124 processors and on a Linux cluster with 22 processors.
|
||||
Three different finite volume grids (M 1 – M 3) have been used for each value of the blockage ratio considered in this paper. Each of the meshes has been generated algebraically and then smoothed by solving elliptic partial differential equations for the spatial variables $x _ { 1 }$ and $x _ { 2 }$ where derivatives are with respect to mapped variables in a space in which the mesh appears rectangular.21 For the present problem the physical grid is ‘‘cut’’ along the line $x _ { 2 } = 0$ from the rear stagnation point to the outlet before being mapped. The method of Steger and Sorenson21 allows both grid cell sizes and grid cell skewness to be controlled at the inner and outer boundaries. Meshes M 1 to M 3 are characterized by $i _ { \mathrm { m a x } }$ : The number of nodes on the surface of the cylinder, $k _ { \mathrm { m a x } } \colon$ The number of nodes along the line $x _ { 2 } = 0$ from the rear stagnation point on the cylinder to the outflow boundary and $k _ { \mathrm { w a l l } }$ : The number of nodes in the gap between the cylinder and a lateral wall. The values of $i _ { \mathrm { m a x } } , k _ { \mathrm { m a x } } , k _ { \mathrm { w a l l } }$ , and N ~the number of degrees of freedom! for the three meshes are supplied in Table I for different blockage ratios.
|
||||
# III. FLOW PAST AN UNBOUNDED CIRCULAR CYLINDER
|
||||
Flow around an unbounded circular cylinder is a classical benchmark problem for which a large number of numerical and experimental results exists. In this problem, and in approximation to the case of an unbounded flow domain, a circular cylinder of diameter D51.00 is placed symmetrically in a channel with blockage ratio $\beta = 0 . 0 1$ . For the numerical linear stability analysis the three meshes M 1 to M 3 were used, with $i _ { \mathrm { m a x } } , \ : k _ { \mathrm { m a x } }$ , and $k _ { \mathrm { w a l l } }$ as given in Table I. However, for unsteady time-dependent simulations we were only able to afford to use M1 and M2, the unsteady calculations on M3 proving to be prohibitively expensive. On the boundaries of the computational domain the conditions ~3!– ~6! were imposed.
|
||||
1308
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
|
||||
TABLE II. Unbounded flow past a cylinder. Comparison of critical Reynolds numbers computed on M1–M3 with others in the literature.
|
||||
|
||||
<table><tr><td></td><td>M1</td><td>M2</td><td>M3</td><td>Extrapolated</td><td>Jackson</td><td>Ding and Kawahara</td><td>Noack and Eckelmann</td><td>Chen et al.</td></tr><tr><td><eq>Re_{\text{crit1}}</eq></td><td>47.08</td><td>46.82</td><td>46.76</td><td>46.74</td><td>46.184</td><td>46.389</td><td>50</td><td>47.9</td></tr><tr><td><eq>St_{\text{crit1}}</eq></td><td>0.1163</td><td>0.1166</td><td>0.1167</td><td>0.1167</td><td>0.13804</td><td>0.12619</td><td>0.132</td><td>0.138</td></tr></table>
|
||||
|
||||
The linear stability analysis predictions of the critical Reynolds and Strouhal numbers corresponding to the onset of the first flow instability are supplied in Table II, as computed on meshes M 1 – M 3. Also shown are the values of these quantities when extrapolated to zero mesh size. The extrapolated critical Reynolds number is found to be $R e _ { \mathrm { c r i t 1 } }$ 546.74 with a corresponding Strouhal number of $S t _ { \mathrm { c r i t 1 } }$ 50.1167. These values are compared with others in the literature in the same table. Although we find good agreement for the critical Reynolds number with the result of Ding and Kawahara22 $( R e _ { \mathrm { c r i t 1 } } = 4 6 . 3 8 9 )$ , and Jackson10 $( R e _ { \mathrm { c r i t 1 } }$ 546.184), the critical Strouhal number manifests wider scatter in the cited references. Issues such as the blockage ratios chosen, distances from the cylinder of the upstream and downstream boundaries, boundary conditions, mesh resolution and number of eigenvalues determined may be amongst the reasons for discrepancies in the numerical results. In addition to our mesh convergence study, we present a convergence study of the leading eigenvalues on mesh M 1 with the Krylov subspace dimension m and shift parameter l in Table III in order to show that our leading eigenvalue is essentially independent of both m and l for sufficiently large values of these two parameters. Although the leading eigenvalue converges very rapidly with a suitably chosen complex shift around the leading eigenvalue, it requires complex arithmetic. A real shift also dramatically improves the convergence of the leading eigenvalue while avoiding complex arithmetic which significantly increases the memory requirements during LU factorization. Our calculations show that a Krylov subspace dimension as low as 250 can be enough to compute the leading eigenvalue with l50.5060.00i while with no shift a Krylov subspace dimension larger than 1000 may be required.
|
||||
|
||||
The computed eigenspectrum on mesh M 3 at the critical Reynolds number is given in Fig. 2. Although we present the first 250 computed eigenvalues, calculations with higher Krylov subspace dimensions showed that only the leading eigenvalues and the eigenvalues around the origin were properly converged. As may be seen, the most dangerous eigenvalue pair is well separated from the rest of the spectrum, unlike the eigenspectrum for the two-dimensional liddriven cavity problem, for example.13 This is likely to be the reason for well-developed periodic flow observed far beyond the critical Reynolds number. Our critical Strouhal number ~0.1167! compares very well with the Strouhal number St 50.1179 computed at the same Reynolds number ~46.74! from a curve fit of the two-dimensional experimental data of Williamson.23 In addition, our critical Strouhal number agrees quite well with the Strouhal number $( S t = 0 . 1 1 8 3 4 )$ of the direct numerical simulation of Posdziech and Grundmann,24 even though their critical Strouhal number and ours were computed at two slightly different Reynolds numbers $( R e = 4 7 . 5 0$ and $R e = 4 6 . 7 4 ,$ , respectively!.
|
||||
In Fig. 3 we present comparisons of the Strouhal number versus Reynolds number and in Fig. 4 comparisons of the drag coefficient $C _ { d } { = } F _ { x } / 0 . 5 U _ { \mathrm { m a x } } ^ { 2 } D$ versus Reynolds number, in further verification of our numerical scheme. Our Strouhal numbers are seen to be in very good agreement with those from the experimental work of Williamson23 for Reynolds numbers up to 200. Beyond this point the flow becomes three-dimensional and we do not expect to have agreement with the experimental results. Good agreement for St and Cd with results from the two-dimensional numerical simulations of Henderson25 and Posdziech and Grundmann24 may also be seen from Figs. 3 and 4. Our computed lift coefficients $C _ { l }$ at Reynolds numbers of 100 and 200 are 60.3333 and 60.6861 and these are in satisfactory agreement with Posdziech and Grundmann’s values of 60.321 04 and 60.673 15, respectively. Although both Henderson25 and Posdziech and Grundmann24 used high-order spectral elements the differences between their two sets of results are due to the use of different blockage ratios in their calculations. However, as the Reynolds number increases the difference in their computed results becomes smaller. An interesting convergence study on the extension of the computational domain boundary is given by Posdziech and Grundmann24 at $R e = 2 0 0 . 0 0$ . The authors concluded that the lateral boundaries should be set at a distance of at least 70 diameters away in order to obtain a Strouhal number independent of yet smaller blockage ratios. At lower Reynolds number the effect of the lateral boundaries becomes more severe ~see Fornberg,26 for example!. In addition, Zisis and Mitsoulis27 showed that the convergence of the total drag at $R e { = } 0 . 0 0$ may be very poor as $\beta$ goes to zero.
|
||||
|
||||
TABLE III. Unbounded flow past a cylinder. Convergence of the leading eigenvalue at Re547.08 on M1 with the Krylov space dimension m and shift parameter l.
|
||||
|
||||
<table><tr><td>m</td><td><eq>\lambda = 0.00 \pm 0.00i</eq></td><td><eq>\lambda = 0.25 \pm 0.00i</eq></td><td><eq>\lambda = 0.50 \pm 0.00i</eq></td></tr><tr><td>250</td><td><eq>-2.499\ 530 \times 10^{-3} \pm 0.716\ 816</eq></td><td><eq>-2.362\ 255 \times 10^{-6} \pm 0.730\ 913</eq></td><td><eq>-2.666\ 849 \times 10^{-6} \pm 0.730\ 912</eq></td></tr><tr><td>500</td><td><eq>+3.029\ 965 \times 10^{-3} \pm 0.726\ 480</eq></td><td><eq>-2.668\ 530 \times 10^{-6} \pm 0.730\ 912</eq></td><td><eq>-2.668\ 473 \times 10^{-6} \pm 0.730\ 912</eq></td></tr><tr><td>1000</td><td><eq>+2.230\ 634 \times 10^{-4} \pm 0.729\ 683</eq></td><td><eq>-2.668\ 474 \times 10^{-6} \pm 0.730\ 912</eq></td><td><eq>-2.668\ 473 \times 10^{-6} \pm 0.730\ 912</eq></td></tr></table>
|
||||
1309
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
|
||||
|
||||
FIG. 2. Reciprocal Ritz values for unbounded flow around a circular cylinder at Re546.76 computed on mesh M3 with Krylov space dimension m 5250 and shift parameter l50.5060.00i (b50.01).
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 3. Comparison of Strouhal number versus Reynolds number for unbounded flow around a circular cylinder with other results in the literature: $( - ) .$ experimental work of Williamson ~Ref. 23!; ~¯!, numerical results of Henderson ~Ref. 25!; ~s!, numerical results of Posdziech and Grundmann ~Ref. 24!; ~h!, present ( b50.01, mesh M2!.
|
||||
|
||||
|
||||
|
||||
|
||||
FIG. 4. Comparison of drag coefficient versus Reynolds number for unbounded flow around a circular cylinder with other results in the literature: $( - ) ,$ , numerical results of Henderson ~Ref. 25!; ~s!, numerical results of Posdziech and Grundmann ~Ref. 24!; ~h!, present (b50.01, mesh M2!.
|
||||
|
||||
|
||||
# IV. FLOW PAST A CONFINED CIRCULAR CYLINDER „0.1ËbË0.9…
|
||||
Flow around a confined circular cylinder ~as opposed to the unbounded case! is an attractive benchmark problem in numerical simulation since it does not suffer from any of the difficulties associated with far-field boundary conditions ~particularly at very low Reynolds numbers! and permits the use of grid points more efficiently in smaller computational domains. Somewhat surprising, therefore, is that the only numerical linear stability analysis of Newtonian flow past a confined cylinder available in the literature would seem to be that of Chen $e t a l . ^ { 4 }$ These authors went no further than identifying the curve of neutral stability for the supercritical Hopf bifurcation at blockage ratios up to $\beta { = } 0 . 7$ . This is regrettable, because as we shall see in the paragraphs to follow, the linear stability properties of the flow become rich and therefore interesting at higher blockage ratios and Reynolds numbers than those considered by Chen et al. In the present study we consider two-dimensional flow at Reynolds numbers up to 280 and for blockage ratios in the range 0.1–0.9.
|
||||
# A. Linear stability analysis
|
||||
The curves of neutral stability computed from the GEVP with a Krylov subspace dimension $m = 2 5 0$ on mesh M2 for $\beta \in \left[ 0 . 1 , 0 . 9 \right]$ and $R e < 2 8 0$ are presented in Fig.5. Our discussion of these curves will focus on the five distinct curve sections labeled AB, BC, CD, CE, and $F G$ in the same
|
||||
1310
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
https://cdn-mineru.openxlab.org.cn/result/2026-05-11/e29ea199-8ede-444f-8a3a-062a2f820b92/7143dec957189df04b69e5cf9e774f69213002910deed08e6e9428081f7c6fee.jpg
|
||||
FIG. 5. Change of critical Reynolds number corresponding to both Hopf and pitchfork bifurcations with blockage ratio $\beta ,$ computed on $M 2 . A C \mathrm { : }$ Curve of neutral stability for Hopf bifurcations about symmetric solution; CD: Transition curve from asymmetric vortex shedding ~smaller $\beta )$ to a steady asymmetric solution ~larger $\beta ) ; C E \colon$ : Neutral stability curve for pitchfork bifurcation of steady symmetric solution ~smaller $\beta )$ to a steady asymmetric state ~larger $\beta ) ; F G ;$ Hopf bifurcation of an asymmetric solution ~smaller $\beta )$ to asymmetric vortex shedding ~larger $\beta ) . ~ C$ is a co-dimension 2 point where Hopf and pitchfork bifurcations occur simultaneously.
|
||||
|
||||
TABLE IV. Convergence of critical Reynolds number for different blockage ratios with $\lambda = 0 . 0 0 \pm 0 . 0 0 i$ .
|
||||
|
||||
<table><tr><td rowspan="2">Curve section(see Fig. 5)</td><td rowspan="2"><eq>\beta</eq></td><td rowspan="2">m</td><td colspan="2">M1</td><td colspan="2">M2</td><td colspan="2">M3</td><td colspan="2">Chen et al.</td></tr><tr><td><eq>Re_{\text{crit}}</eq></td><td><eq>St_{\text{crit}}</eq></td><td><eq>Re_{\text{crit}}</eq></td><td><eq>St_{\text{crit}}</eq></td><td><eq>Re_{\text{crit}}</eq></td><td><eq>St_{\text{crit}}</eq></td><td><eq>Re_{\text{crit}}</eq></td><td><eq>St_{\text{crit}}</eq></td></tr><tr><td rowspan="12">AC</td><td>0.10</td><td>500</td><td>51.00</td><td>0.1206</td><td>50.81</td><td>0.1210</td><td>50.75</td><td>0.1211</td><td>51.77</td><td>0.1116</td></tr><tr><td>0.20</td><td>250</td><td>69.86</td><td>0.1559</td><td>69.43</td><td>0.1566</td><td>69.34</td><td>0.1567</td><td>69.93</td><td>0.1559</td></tr><tr><td>0.30</td><td>250</td><td>95.24</td><td>0.2079</td><td>94.56</td><td>0.2090</td><td>94.40</td><td>0.2093</td><td>94.85</td><td>0.2085</td></tr><tr><td>0.50</td><td>250</td><td>125.23</td><td>0.3369</td><td>124.09</td><td>0.3393</td><td>123.75</td><td>0.3399</td><td>124.58</td><td>0.3382</td></tr><tr><td>0.70</td><td>250</td><td>111.32</td><td>0.4714</td><td>110.29</td><td>0.4752</td><td>110.04</td><td>0.4762</td><td>111.04</td><td>0.4744</td></tr><tr><td>0.80</td><td>250</td><td>111.45</td><td>0.5324</td><td>110.24</td><td>0.5363</td><td>109.98</td><td>0.5374</td><td></td><td></td></tr><tr><td>0.84</td><td>250</td><td>114.44</td><td>0.5530</td><td>113.69</td><td>0.5568</td><td></td><td></td><td></td><td></td></tr><tr><td>0.84</td><td>250</td><td>130.92</td><td>0.5510</td><td>126.64</td><td>0.5557</td><td></td><td></td><td></td><td></td></tr><tr><td>0.80</td><td>250</td><td>148.24</td><td>0.5324</td><td>144.19</td><td>0.5383</td><td>143.29</td><td>0.5398</td><td></td><td></td></tr><tr><td>0.76</td><td>250</td><td>169.75</td><td>0.5115</td><td>165.49</td><td>0.5186</td><td></td><td></td><td></td><td></td></tr><tr><td>0.72</td><td>250</td><td>198.94</td><td>0.4872</td><td>193.25</td><td>0.4955</td><td></td><td></td><td></td><td></td></tr><tr><td>0.70</td><td>250</td><td>218.03</td><td>0.4737</td><td>211.01</td><td>0.4827</td><td>209.40</td><td>0.4851</td><td></td><td></td></tr><tr><td rowspan="7">CE</td><td>0.70</td><td>250</td><td>221.87</td><td></td><td>216.75</td><td></td><td>215.53</td><td></td><td></td><td></td></tr><tr><td>0.72</td><td>250</td><td>210.17</td><td></td><td>205.95</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0.76</td><td>250</td><td>190.65</td><td></td><td>187.01</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0.80</td><td>250</td><td>173.97</td><td></td><td>169.49</td><td></td><td>168.29</td><td></td><td></td><td></td></tr><tr><td>0.84</td><td>250</td><td>161.57</td><td></td><td>158.15</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0.88</td><td>250</td><td>152.93</td><td></td><td>149.84</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0.90</td><td>250</td><td>147.78</td><td></td><td>145.27</td><td></td><td>144.70</td><td></td><td></td><td></td></tr><tr><td rowspan="4">CD</td><td>0.68</td><td>250</td><td>237.33</td><td>0.4596</td><td>231.06</td><td>0.4695</td><td></td><td></td><td></td><td></td></tr><tr><td>0.66</td><td>250</td><td>259.55</td><td>0.4477</td><td>253.08</td><td>0.4566</td><td></td><td></td><td></td><td></td></tr><tr><td>0.64</td><td>250</td><td>284.56</td><td>0.4351</td><td>278.01</td><td>0.4441</td><td></td><td></td><td></td><td></td></tr><tr><td>0.62</td><td>250</td><td>312.66</td><td>0.4235</td><td>306.27</td><td>0.4326</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="6">FG</td><td>0.82</td><td>250</td><td></td><td></td><td>319.80</td><td>0.4664</td><td></td><td></td><td></td><td></td></tr><tr><td>0.82</td><td>250</td><td></td><td></td><td>227.44</td><td>0.4719</td><td></td><td></td><td></td><td></td></tr><tr><td>0.84</td><td>250</td><td>331.02</td><td>0.4954</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>0.84</td><td>250</td><td>214.00</td><td>0.4794</td><td>194.30</td><td>0.4979</td><td></td><td></td><td></td><td></td></tr><tr><td>0.88</td><td>250</td><td>180.43</td><td>0.5097</td><td>171.28</td><td>0.5234</td><td></td><td></td><td></td><td></td></tr><tr><td>0.90</td><td>250</td><td>169.44</td><td>0.5146</td><td>162.82</td><td>0.5202</td><td>160.50</td><td>0.5212</td><td></td><td></td></tr></table>
|
||||
1311
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
|
||||
|
||||
FIG. 6. Change of base flow with critical Reynolds number and blockage ratio b, computed on M 2.
|
||||
|
||||
figure. The critical Reynolds numbers and corresponding Strouhal numbers ~where appropriate! for points on each curve section and computed on meshes M 1 – M 3 are supplied in Table IV.
|
||||
# 1. Curve section AB
|
||||
Validation of our numerical stability analysis and direct numerical simulations for flow past an unbounded cylinder $\beta { \approx } 0$ has been described in Sec. II. For the confined cylinder problem we have been able to compare our critical Reynolds and Strouhal numbers for the bifurcation for the symmetric state with the values for these quantities computed by Chen et $a l . ^ { 4 }$ The available results $( 0 . 1 { \leqslant } \beta { \leqslant } 0 . 7 )$ of the critical Reynolds number calculations of Chen et al. are plotted in Fig. 5 and agreement between our results and theirs over this limited section of the curve AB is excellent. Similarly excellent agreement in the computed Strouhal numbers was seen over the same range of blockage ratios, both our results and those of Chen et al. revealing a monotonic increase in the critical Strouhal number with the blockage ratio ~see Table IV!.
|
||||
Up to a blockage ratio b50.5 Table IV and the neutral stability curve AB of Fig. 5 indicate that the flow becomes more stable to two-dimensional infinitesimal disturbances as the blockage ratio increases. All along AB the flow loses stability to a Hopf bifurcation and the Strouhal number over this range of blockage ratios is increasing. Between $\beta$ 50.75 and 0.85 it may be seen from Table IV and section AB of Fig. 5 that the flow restabilizes slightly leading up to point B $( \beta = \beta _ { B } \approx 0 . 8 5 5 )$ .
|
||||
In Fig. 6 we show the streamlines of the steady base flow at seven points on the neutral stability curves. Those corresponding to point 1 are typical of those at points on and below curve AB in Fig. 5 where the steady solution is symmetric and the only recirculatory region observed is the vortex pair immediately in the wake of the cylinder itself. That is, for solutions corresponding to parameter space on and below AB in Fig. 5 no flow separation on the walls is observed.
|
||||
# 2. Curve sections BC and CE
|
||||
In Fig. 5 section BC represents the part of the critical $\beta - R e$ curve on which the time-dependent state (symmetric periodic oscillations!, passed into by crossing AB in the direction of increasing Reynolds number, restabilizes to a symmetric steady state once more. Further increases in the Reynolds number for blockage ratios in the range $\beta _ { C }$ to $\beta _ { B }$ or for choices of $\beta$ greater than $\beta _ { B }$ may result in the steady symmetric solution becoming unstable to two-dimensional perturbations via a pitchfork bifurcation into one of two asymmetric states. The curve of neutral stability for this transition is labeled C E in Fig. 5.
|
||||
|
||||
|
||||
|
||||
FIG. 7. Streamlines of unstable symmetric and stable asymmetric solutions at $R e = 1 5 0 . 0 0$ for $\beta = 0 . 9$ computed on M3.
|
||||
|
||||
1312
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
|
||||
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
|
||||
(b)
|
||||
|
||||
|
||||
FIG. 8. Streamlines for the disturbance velocity corresponding to ~a! the first and ~b! the second leading eigenvectors at $R e = 1 4 4 . 7 0$ for b50.9 computed on M3.
|
||||
|
||||
|
||||
The point C is a co-dimension 2 point where Hopf and pitchfork bifurcations occur simultaneously. We are able to estimate the coordinates $( \beta _ { C } , R e _ { C } )$ of this point by considering it to be the point of intersection of two straight lines drawn through the pairs of points on AC, CD, and EC that correspond to $\beta = 0 . 6 8$ and 0.7. Since b50.68 is outside the range of blockage ratios corresponding to EC the ordinate for this value of $\beta$ is computed to be that at which the leading real eigenvalue in the spectrum of linear perturbations about the ~linearly unstable! steady symmetric solution is at the origin. The critical Reynolds numbers at b50.68 and 0.7 computed on curves AC, CD, and EC are detailed in Table IV and lead to the estimate $( \beta _ { C } , R e _ { C } ) = ( 0 . 6 8 7 , 2 2 4 . 1 4 2 )$ .
|
||||
The occurrence of the transition from a symmetric steady state to an asymmetric one on CE is preceded ~in Reynolds number! by the appearance in the streamlines of a pair of downstream separation bubbles on the walls. For example, in the eigenspectrum we observed that for b50.9 and at Reynolds numbers increasing up to approximately 110 the complex conjugate pair of leading eigenvalues moved in the direction of the positive real part of the spectrum. At a Reynolds number of around 110 separation bubbles appeared on the walls and with the appearance of the separation bubbles the leading eigenpair now started to move in the opposite direction while the eigenvalue on the real axis having largest real part moved right towards the origin. In Fig. 6 we plot the streamlines on BC at points 2 and 3 to demonstrate how the size of these recirculatory regions as well as their attachment distance downstream of the cylinder increase as C is approached along the curve BC. In the context of a circular cylinder near a plane boundary such downstream separation bubbles have been observed both experimentally8 and numerically9 for cylinders sufficiently close to the boundary.
|
||||
|
||||
To gain further insight into the flow transition from steady symmetric flow ~between BC and CE and for $\beta$ $\geqslant _ { \beta _ { B } ) }$ via a pitchfork bifurcation to steady asymmetric flow ~between DE and FG) we plot in Fig. 7 the streamlines of two solutions at a Reynolds number of 150 and blockage ratio of 0.9. It may be seen from Fig. 5 that this point lies between E $( R e = 1 4 4 . 7 , \beta = \beta _ { E } = 0 . 9 )$ and $\textit { F } ( R e = 1 6 0 . 5 , \beta$ $= \beta _ { F } = 0 . 9 )$ . Thus, the symmetric solution in the upper plot in Fig. 7 is linearly unstable and the lower plot represents the streamlines of one of the stable asymmetric solutions. The disturbance velocity v in equation ~II! is, of course, solenoidal. In Fig. 8~a! we plot the streamlines associated with the disturbance field and corresponding to the dominant eigenvalue at E. The addition of a multiple of the eigenvector shown in Fig. 8~a! to the symmetric steady base flow leads to one or other of the two asymmetric steady flows, the choice dependent on the direction of circulation around the symmetric streamlines in Fig. 8~a!. An anti-clockwise direction leads to reinforcement of the lower recirculation region and reduction in the size of the upper bubble. A clockwise direction has the opposite effect. The drag coefficient associated with the steady asymmetric solution shown in Fig. 7 is slightly larger than that of the corresponding unstable symmetric one.
|
||||
1313
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
TABLE V. Comparison with the results of Zisis and Mitsoulis ~Ref. 27! and Liu et al. ~Ref. 35! of computed total drag at Re50.0 for different blockage ratios b.
|
||||
|
||||
<table><tr><td><eq>\beta</eq></td><td>M1</td><td>M2</td><td>M3</td><td>Extrapolated</td><td>Zisis and Mitsoulis</td><td>Liu et al.</td></tr><tr><td>0.1</td><td>8.9125</td><td>8.9100</td><td>8.9089</td><td>8.9080</td><td>8.912</td><td>8.9067</td></tr><tr><td>0.2</td><td><eq>1.6221 \times 10^{1}</eq></td><td><eq>1.6215 \times 10^{1}</eq></td><td><eq>1.6211 \times 10^{1}</eq></td><td><eq>1.6200 \times 10^{1}</eq></td><td>...</td><td>...</td></tr><tr><td>0.3</td><td><eq>2.7923 \times 10^{1}</eq></td><td><eq>2.7910 \times 10^{1}</eq></td><td><eq>2.7902 \times 10^{1}</eq></td><td><eq>2.7886 \times 10^{1}</eq></td><td>...</td><td>...</td></tr><tr><td>0.5</td><td><eq>8.8354 \times 10^{1}</eq></td><td><eq>8.8294 \times 10^{1}</eq></td><td><eq>8.8263 \times 10^{1}</eq></td><td><eq>8.8227 \times 10^{1}</eq></td><td><eq>8.8207 \times 10^{1}</eq></td><td><eq>8.8227 \times 10^{1}</eq></td></tr><tr><td>0.7</td><td><eq>4.0347 \times 10^{2}</eq></td><td><eq>4.0318 \times 10^{2}</eq></td><td><eq>4.0299 \times 10^{2}</eq></td><td><eq>4.0257 \times 10^{2}</eq></td><td>...</td><td>...</td></tr><tr><td>0.9</td><td><eq>7.7057 \times 10^{3}</eq></td><td><eq>7.6988 \times 10^{3}</eq></td><td><eq>7.6959 \times 10^{3}</eq></td><td><eq>7.6941 \times 10^{3}</eq></td><td>...</td><td>...</td></tr></table>
|
||||
# 3. Curve section CD
|
||||
The section of the neutral stability curves labeled CD in Fig. 5 represents a transition curve with increasing b from periodic vortex shedding to the left of this curve ~smaller b! to a steady asymmetric state ~larger b!. At point C the steady solution is symmetric but moving along the curve CD towards D causes the growth of one of the recirculatory wall regions relative to the other.
|
||||
Since all curves of neutral stability in Fig. 5 have been determined using a linear stability analysis about a steady flow as described in Sec. II we have been unable to plot the precise boundaries of the transition region that must exist from symmetric oscillations to asymmetric oscillations as the curve CD is approached in parameter space from the left ~smaller b!.
|
||||
# 4. Curve section FG
|
||||
Finally, the steady asymmetric solution of the region between curves DE and FG can become unstable via a Hopf bifurcation to asymmetric vortex shedding ~see discussion in Sec. IV B!. The transition curve is plotted as FG and the streamlines of two steady base flows at points 6 and 7 of this curve are shown in Fig. 6. In Fig. 8~b! we plot the streamlines of the disturbance velocity corresponding to the leading complex eigenvalue pair in the spectrum at E. It is the addition of a mode similar in form to this ~but at a higher Reynolds number! that leads to vortex shedding about the asymmetric state. If the Reynolds number is further increased on the curve FG additional separation bubbles appear on the wall further downstream. We also remark that on the curve FG the separation bubble just behind the cylinder is generally shorter and more rounded than that computed on the curves BC and CD.
|
||||
A strong parallel is thus seen in the present results with those of numerous other authors ~see, for example, those of Battaglia et al., 28 Drikakis,29 Fearn et al., 30 Hawa and Rusak,31 Mishra and Jayaraman,32 and Oliveira33! for flows through both two-dimensional and three-dimensional symmetric expansions. All the cited authors report that steady flow with symmetric recirculatory regions through an expansion geometry encounters a supercritical pitchfork bifurcation at a certain Reynolds number ~dependent, of course, on the channel geometry! and becomes asymmetric. The difference in the streamwise attachment length of the two recirculatory regions ~still in the steady regime! becomes larger as the Reynolds number is further increased from the critical value. In the two-dimensional case, increasing the expansion ratio decreases the critical Reynolds number.28 In the threedimensional case, Schrek and Scha¨fer34 found that fixing the expansion ratio at 1:3 and decreasing the width of the channel relative to the downstream channel height from ` ~twodimensional flow! through 5 to 2 resulted in a stabilization of the flow.
|
||||
|
||||
It will be noted from Table IV that for the choices of β=0.82 and 0.84 critical Reynolds numbers of 319.8 and 331.02, respectively, are added to those that are shown in Fig. 5. This is to indicate how the curve FG would continue if the range of Reynolds numbers were to be extended in Fig. 5, although at these higher Reynolds numbers it is highly unlikely that the flow would in reality remain twodimensional.
|
||||
# B. Direct numerical simulations
|
||||
A few verifications were performed on the results of our direct numerical simulations in order to establish their reliability. First, Strouhal numbers near the critical Reynolds numbers corresponding to the onset of periodic vortex shedding and computed with direct numerical simulation were found to be in good agreement in a couple of cases with those predicted on the basis of the eigenvalue analysis of
|
||||
|
||||
|
||||
|
||||
FIG. 9. Computed drag coefficient versus Reynolds number at blockage ratios b5 0.1, 0.3, 0.5, 0.7, and 0.9.
|
||||
|
||||
1314
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
|
||||
|
||||
|
||||
FIG. 10. Change of time-dependent flow with Reynolds number and blockage ratio b, computed on M 2.
|
||||
|
||||
Sec. IV A. For example, at a blockage ratio of $\beta = 0 . 3$ and at $R e = 1 0 0$ the corresponding Strouhal number was computed from the lift coefficient data over extended time intervals and found to be equal to 0.2115. This compares well with the value of 0.2090 supplied in Table IV and computed at $R e _ { \mathrm { c r i t } } { = } 9 4 . 5 6$ . Second, we present in Table V results of computations using the three meshes M 1 to M 3 of the drag on the cylinder for various blockage ratios. These are compared with the recent numerical data of Zisis and Mitsoulis27 and Liu $e t a l . ^ { 3 5 }$ and the agreement is convincing. It should be further added that the drag result of Liu et al. for $\beta { = } 0 . 1$ is within 0.007% of the theoretically predicted value of Faxe´n.36 The drag coefficient versus Reynolds number is given in Fig. 9 for several blockage ratios. At all the blockage ratios considered here the drag coefficient behaves like $1 / R e$ at low Reynolds numbers. As the blockage ratio increases the range of values of Re over which this remains true gets smaller.
|
||||
|
||||
|
||||
|
||||
|
||||
(a)t=0
|
||||
|
||||
|
||||
|
||||
|
||||
(b)t=T/3
|
||||
|
||||
|
||||
|
||||
|
||||
(c)t=2T/3
|
||||
|
||||
|
||||
FIG. 11. Vorticity contours of the periodic flow at $R e = 2 0 0 . 0 0$ and b50.5 ~point 1 in Fig. 10! computed on M2. t50 corresponds to the solution having minimum lift coefficient and the period T'2.85.
|
||||
|
||||
1315
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
|
||||
(b)
|
||||
|
||||
|
||||
|
||||
|
||||
（c）
|
||||
|
||||
|
||||
|
||||
|
||||
(d）
|
||||
|
||||
|
||||
|
||||
|
||||
(e）
|
||||
|
||||
|
||||
FIG. 12. Phase space plots of lift and drag coefficients parametrized with nondimensional time $( t U _ { \mathrm { m a x } } / D )$ , computed on M2. ~a! $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 5$ ~point 1 in Fig. 10!, ~b! $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 7$ ~point 2 in Fig. 10!, ~c! $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 9$ ~point 3 in Fig. 10!, ~d! $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 8$ ~point 4 in Fig. 10!, ~e! $R e = 1 6 0 . 0 0$ and $\beta { = } 0 . 8$ ~point 5 in Fig. 10!.
|
||||
|
||||
1316
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
|
||||
|
||||
|
||||
(a)t=0
|
||||
|
||||
|
||||
|
||||
|
||||
(b) $t = T / 3$
|
||||
|
||||
|
||||
|
||||
|
||||
$( \mathrm { c } ) \ t = 2 T / 3$
|
||||
|
||||
|
||||
FIG. 13. Vorticity contours of the periodic flow at $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 7$ ~point 2 in Fig. 10! computed on $M 2 . \ t = 0$ corresponds to the solution having minimum lift coefficient and the period T'2.05.
|
||||
|
||||
|
||||
|
||||
|
||||
(a)t=0
|
||||
|
||||
|
||||
|
||||
|
||||
$( \mathbf { b } ) \ t = T / 3$
|
||||
|
||||
|
||||
|
||||
|
||||
$( \mathrm { c } ) \ t = 2 T / 3$
|
||||
|
||||
|
||||
FIG. 14. Vorticity contours of the periodic solution at $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 9$ ~point 3 in Fig. 10! computed on $M 2 , t = 0$ corresponds to the solution having minimum lift coefficient and the period T'1.88.
|
||||
|
||||
|
||||
|
||||
|
||||
(a） $t = 0$
|
||||
|
||||
|
||||
|
||||
|
||||
(b) $t = T / 3$
|
||||
|
||||
|
||||
|
||||
|
||||
$( \mathrm { c } ) \ t = 2 T / 3$
|
||||
|
||||
|
||||
FIG. 15. Vorticity contours of the periodic solution at $R e = 2 0 0 . 0 0$ and $\beta { = } 0 . 8$ ~point 4 in Fig. 10! computed on $M 2 , t = 0$ corresponds to the solution having minimum lift coefficient and the period $T { \approx } 1 . 8 1 5 .$ .
|
||||
|
||||
1317
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
|
||||
|
||||
(a）t=0
|
||||
|
||||
|
||||
|
||||
|
||||
(b)t=T/3
|
||||
|
||||
|
||||
|
||||
|
||||
(c)t=2T/3
|
||||
|
||||
|
||||
FIG. 16. Vorticity contours of the periodic solution at $R e = 1 6 0 . 0 0$ and b50.8 ~point 5 in Fig. 10! computed on M2. t50 corresponds to the solution having minimum lift coefficient and the period T'1.806.
|
||||
|
||||
In order to elucidate the variation in the critical Reynolds number with blockage ratio observed from the eigenvalue analysis of Sec. IV A we used direct numerical simulation to investigate the wake structure at five different locations in the $\beta - R e$ parameter space and labeled 1–5 in Fig. 10. As an aside, and before discussing our results in detail, we simply note that at blockage ratios $\beta { < } 0 . 5$ direct numerical simulations revealed that the vortex shedding over the cylinder was quite similar to that of the unbounded case, although the vortex street is shorter due to shear in the free stream. We also note that although blockage effects are expected to delay transition of the cylinder wake to threedimensional flow, it is possible that at some of the points labeled 1–4 in Fig. 10 the local velocity is so high that a three-dimensional transition occurs for the highly decelerated, separated boundary layers on the channel walls. Verification of this will have to await fully three-dimensional simulations, however.
|
||||
Time-dependent solutions are presented in Fig. 11 for $\beta = 0 . 5$ at a Reynolds number of 200 (point 1 of Fig.10).At this Reynolds number the flow has lost its stability to twodimensional disturbances and has become time-periodic with ${ S t = 0 . 3 5 1 3 }$ , which is higher than for unbounded flow around a circular cylinder $( S t = 0 . 1 9 7 7 )$ . The sequences of three ‘‘snapshots’’ in Fig. 11 are taken at times t50, T/3 and 2T/3, the nondimensional period T being approximately equal to 2.85 and determined from the lift coefficient data over long time periods. t50 corresponds to a minimum in the lift coefficient once fully periodic vortex shedding is established. It may be seen from Figs. 11~a!–11~c! that vortex shedding occurs both from the cylinder and the channel walls. As these vortices move downstream the trajectories of clockwise vortices shed from the upper part of the cylinder cross those having opposite sign ~and shed from the lower part of the cylinder! so that wall proximity effects are seen to give rise to a reverse von Ka´rma´n street. The same phenomenon has been documented by other authors.9 In Fig. 12~a! we show the $C _ { d } { - } C _ { l }$ phase space plot at point 1 of Fig. 10, once fully periodic conditions have been established. The average lift is zero and $C _ { d }$ and $C _ { l }$ are both symmetric in the rising and falling parts of each cycle.
|
||||
|
||||
At a blockage ratio of 0.7 the flow is periodic at a Reynolds number of 200.00 ~point 2 of Fig. 10! with St 50.4881. The vorticity contours at times t50, T/3 and 2T/3 are shown in Figs. 13~a!–13~c!, with the period T'2.05. Unlike in the case of $\beta = 0 . 5$ described in the paragraph above, vortex shedding from the cylinder seems to be almost suppressed at this Reynolds number, due to the proximity of point 2 to the curve of neutral stability BC. However, there are very weak vortices shed from both upper and lower lateral walls. These are well separated from each other and their interaction is weak. The phase space plot of the lift and drag coefficients at this blockage ratio are shown in Fig. 12~b!. Although the time-averaged value of the drag coefficient $C _ { d }$ has increased it is notable that the amplitude of the $C _ { d }$ oscillations is an order of magnitude less than that seen at $\beta$ 50.5.
|
||||
At a blockage ratio of 0.9 and $R e > 1 6 0 . 5$ the flow is unsteady and very strong vortices are shed from both the cylinder and the walls. The computed streamlines and vorticity contours at $R e = 2 0 0 . 0 0$ ~point 3 of Fig. 10! are shown in Fig. 14, each separated from the previous in the series by a third of a period T/3. At this Reynolds number the flow is periodic with St50.5314 (T'1.88) and this compares reasonably well ~see Table IV! with the Strouhal number of 0.5202 computed from the GEVP on mesh M 2 at the critical Reynolds number $R e _ { \mathrm { c r i t } } { = } 1 6 2 . 8 2$ . Vortices having the same sign merge just behind the cylinder and are then transported downstream. However, the vortex street formed behind the cylinder is quite different from the well-known von Ka´rma´n street in that very strong opposite-sign vortices with smaller structure move downstream and interact with the wall, creating strong vortices there. Although the streamlines of the eigenvector shown in Fig. 8~b! will be slightly modified by the time the Reynolds number reaches 200, the size of the cellular structures in the wake of the eigenvector are close to what is seen in the direct numerical solution for the vorticity in Fig. 14. Oscillations are now about an asymmetric state and lead to drastic increases in both the lift and drag coefficient values. Their variation over a cycle with dimensionless time is shown in Fig. 12~c!. In this figure it may be seen that not only is the time-averaged value of $C _ { d }$ greater than for the two previous blockage ratios considered, but the amplitudes of oscillation of both coefficients has dramatically increased after a tendency observed up to $\beta { \approx } 0 . 7 5$ of successively diminishing amplitudes. We also note for the first time that the lift coefficient is no longer symmetric in the rising and falling parts of each cycle ~the ‘‘figure of $8 ^ { \circ }$ is distorted!, although the average value of $C _ { l }$ over one cycle is zero.
|
||||
1318
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
A numerical investigation of wall effects
|
||||
|
||||
The flow behavior in the parametric region between curves BD and FG is particularly interesting since although the steady base flow solutions are linearly stable they appear to be finitely unstable. Although our primary concern in this paper is the study of wall effects on the linear stability of flow past a confined cylinder, in Fig. 10 we plot the instantaneous streamlines of two pairs of different solutions, both pairs being for a blockage ratio $\beta { = } 0 . 8$ but corresponding to two different values of $R e \colon$ at point 4 $R e = 2 0 0$ whereas at point 5 $R e = 1 6 0$ . Shown in the uppermost plots at points 4 and 5 are the steady and linearly stable solutions, but these may be sent into permanently unsteady states by running, for example, the time-dependent code with the geometrically rescaled steady base flow of point $2 \ ( \beta = 0 . 7 , R e = 2 0 0 )$ as the initial ‘‘guess.’’ Snapshots of unsteady flow at times t $= 0 , T / 3$ , and 2T/3 at point 4 are presented in Figs. 15~a!– 15~c!, with the period $T { \approx } 1 . 8 1 5$ . Unlike the other unsteady cases considered so far in this paper, the recirculation regions on the upper wall are much larger than those on the lower wall and they can move far downstream. Additionally, the recirculatory region at the lower part of the cylinder is larger than that at the upper part. This asymmetry may be seen from the phase space plot in Fig. 12~d!. The time average of the lift coefficient is no longer zero and the lift curve is asymmetric in the rising and falling part of each cycle. However, if the Reynolds number is chosen equal to 160 ~point 5 in Fig. 10! the flow becomes symmetric again since point 5 is below the curve CE. The computed vorticity contours are given in Fig. 16 with T'1.806. At this point the flow structure is quite similar to that of point 2 with vortex shedding from the upper and lower walls. The lift and drag coefficients which are supplied in the phase space plot in Fig. 12~e! are also similar to those of point 2 with zero time average of the lift coefficient.
|
||||
Numerical experiments at a blockage ratio of 0.9 and a Reynolds number of 500 indicated that the flow had become chaotic. However, it seems unlikely that the flow is still twodimensional at this Reynolds number and presentation of our results will have to await a fully three-dimensional analysis.
|
||||
# V. CONCLUSIONS
|
||||
In this paper we have computed with greater accuracy and over a larger range of blockage ratios than has proved possible in the past the effects on the drag and linear stability of lateral wall proximity for flow past a cylinder at Reynolds numbers up to 280.
|
||||
Some of the rich and complex dynamics of the system for sufficiently high Reynolds numbers and blockage ratios have been uncovered and discussed. In particular, we have found that for $R e \leqslant 2 8 0$ and $\beta { \leqslant } 0 . 9$ there are ~at least! three separate curves of neutral stability: ~a! Hopf bifurcation of a symmetric state, ~b! pitchfork bifurcation of a symmetric state to one of two asymmetric states, and ~c! Hopf bifurcation of an asymmetric state leading to asymmetric oscillations thereafter. In addition, we have drawn attention to a transition region from symmetric vortex shedding to asymmetric vortex shedding with increasing blockage ratio. Further increases in the blockage ratio ~crossing CD in Fig. 10! leads to restabilization to a steady asymmetric solution.
|
||||
A co-dimension 2 point where pitchfork and Hopf bifurcations occur simultaneously has been identified and a region in parameter space ~either side of the locus of the pitchfork bifurcation! seems to exist where the steady solution is linearly stable but unstable to finite two-dimensional perturbations.
|
||||
# ACKNOWLEDGMENTS
|
||||
The authors wish to thank Peter Monkewitz for sharing with them interesting and illuminating insights into the primary and secondary instability mechanisms. The work of the first author was supported by the Swiss National Science Foundation, Grant No. 21-61865.00.
|
||||
|
||||
|
||||
1C. H. K. Williamson, ‘‘Vortex dynamics in the cylinder wake,’’ Annu. Rev. Fluid Mech. 28, 477 ~1996!.
|
||||
|
||||
|
||||
|
||||
|
||||
2P. Anagnostopoulos, G. Iliadis, and S. Richardson, ‘‘Numerical study of the blockage effect on viscous flow past a circular cylinder,’’ Int. J. Numer. Methods Fluids 22, 1061 ~1996!.
|
||||
|
||||
|
||||
|
||||
|
||||
3M. Coutanceau and R. Bouard, ‘‘Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 1. Steady flow,’’ J. Fluid Mech. 79, 231 ~1977!.
|
||||
|
||||
|
||||
|
||||
|
||||
4J.-H. Chen, W. G. Pritchard, and S. J. Tavener, ‘‘Bifurcation for flow past a cylinder between parallel planes,’’ J. Fluid Mech. 284, 23 ~1995!.
|
||||
|
||||
|
||||
|
||||
|
||||
5M. Behr, S. Hastreiter, S. Mittal, and T. E. Tezduyar, ‘‘Incompressible flow past a circular cylinder: Dependence of the computed flow field on the location of the lateral boundaries,’’ Comput. Methods Appl. Mech. Eng. 123, 309 ~1995!.
|
||||
|
||||
|
||||
|
||||
|
||||
6P. K. Stansby and A. Slaouti, ‘‘Simulation of vortex shedding including blockage by the random-vortex and other methods,’’ Int. J. Numer. Methods Fluids 17, 1003 ~1993!.
|
||||
|
||||
|
||||
|
||||
|
||||
7C. Lei, L. Cheng, and K. Kavanagh, ‘‘Re-examination of the effect of a plane boundary on force and vortex shedding of a circular cylinder,’’ J. Wind. Eng. Ind. Aerodyn. 80, 263 ~1999!.
|
||||
|
||||
|
||||
|
||||
|
||||
8P. W. Bearman and M. M. Zdravkovich, ‘‘Flow around a circular cylinder near a plane boundary,’’ J. Fluid Mech. 89, 33 ~1978!.
|
||||
|
||||
|
||||
|
||||
|
||||
9L. Zovatto and G. Pedrizzetti, ‘‘Flow around a circular cylinder between parallel walls,’’ J. Fluid Mech. 440, 1 ~2001!.
|
||||
|
||||
|
||||
|
||||
|
||||
10C. P. Jackson, ‘‘A finite-element study of the onset of vortex shedding in flow past variously shaped bodies,’’ J. Fluid Mech. 182, 23 ~1987!.
|
||||
|
||||
|
||||
|
||||
|
||||
11M. Sahin, ‘‘Solution of the incompressible unsteady Navier–Stokes equations only in terms of the velocity components,’’ Int. J. Comput. Fluid Dyn. 17, 199 ~2003!.
|
||||
|
||||
|
||||
|
||||
|
||||
12M. Sahin and R. G. Owens, ‘‘A novel fully-implicit finite volume method applied to the lid-driven cavity problem. Part I. High Reynolds number
|
||||
|
||||
|
||||
1319
|
||||
09 May 2026 02:44:09
|
||||
Phys. Fluids, Vol. 16, No. 5, May 2004
|
||||
M. Sahin and R. G. Owens
|
||||
|
||||
|
||||
flow calculations,’’ Int. J. Numer. Methods Fluids 42, 57 ~2003!.
|
||||
|
||||
|
||||
|
||||
|
||||
13M. Sahin and R. G. Owens, ‘‘A novel fully-implicit finite volume method applied to the lid-driven cavity problem. Part II. Linear stability analysis,’’ Int. J. Numer. Methods Fluids 42, 79 ~2003!.
|
||||
|
||||
|
||||
|
||||
|
||||
14A. Kourta, M. Braza, P. Chassaing, and H. Haminh, ‘‘Numerical analysis of a natural and excited two-dimensional mixing layer,’’ AIAA J. 25, 279 ~1987!.
|
||||
|
||||
|
||||
|
||||
|
||||
15G. Jin and M. Braza, ‘‘A nonreflecting outlet boundary condition for incompressible unsteady Navier–Stokes calculations,’’ J. Comput. Phys. 107, 239 ~1993!.
|
||||
|
||||
|
||||
|
||||
|
||||
16W. E. Arnoldi, ‘‘The principle of minimized iterations in the solution of the matrix eigenvalue problem,’’ Q. Appl. Math. 9, 17 ~1951!.
|
||||
|
||||
|
||||
|
||||
|
||||
17Y. Saad, ‘‘Variations on Arnoldi’s method for computing eigen elements of large unsymmetric matrices,’’ Linear Algebr. Appl. 34, 269 ~1980!.
|
||||
|
||||
|
||||
|
||||
|
||||
18R. Natarajan, ‘‘An Arnoldi-based iterative scheme for nonsymmetric matrix pencils arising in finite element stability problems,’’ J. Comput. Phys. 100, 128 ~1992!.
|
||||
|
||||
|
||||
|
||||
|
||||
19P. R. Amestoy, I. S. Duff, and J.-Y. L’Excellent, ‘‘Multifrontal parallel distributed symmetric and unsymmetric solvers,’’ Comput. Methods Appl. Mech. Eng. 184, 501 ~2000!.
|
||||
|
||||
|
||||
|
||||
|
||||
20P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent, ‘‘A fully asynchronous multifrontal solver using distributed dynamic scheduling,’’ SIAM J. Matrix Anal. Appl. 23, 15 ~2001!.
|
||||
|
||||
|
||||
|
||||
|
||||
21J. L. Steger and R. L. Sorenson, ‘‘Automatic mesh-point clustering near a boundary in grid generation with elliptic partial differential equations,’’ J. Comput. Phys. 33, 405 ~1979!.
|
||||
|
||||
|
||||
|
||||
|
||||
22Y. Ding and M. Kawahara, ‘‘Three-dimensional linear stability analysis of incompressible viscous flows using the finite element method,’’ Int. J. Numer. Methods Fluids 31, 451 ~1999!.
|
||||
|
||||
|
||||
|
||||
|
||||
23C. H. K. Williamson, ‘‘Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers,’’ J. Fluid Mech. 206, 579 ~1989!.
|
||||
|
||||
|
||||
|
||||
|
||||
24O. Posdziech and R. Grundmann, ‘‘Numerical simulation of the flow around an infinitely long circular cylinder in the transition regime,’’ Theor. Comput. Fluid Dyn. 15, 121 ~2001!.
|
||||
|
||||
|
||||
|
||||
|
||||
25R. D. Henderson, ‘‘Detail of the drag curve near the onset of vortex shedding,’’ Phys. Fluids 7, 2102 ~1995!.
|
||||
|
||||
|
||||
|
||||
|
||||
26B. Fornberg, ‘‘A numerical study of steady viscous flow past a circular cylinder,’’ J. Fluid Mech. 98, 819 ~1980!.
|
||||
|
||||
|
||||
|
||||
|
||||
27Th. Zisis and E. Mitsoulis, ‘‘Viscoplastic flow around a cylinder kept between parallel plates,’’ J. Non-Newtonian Fluid Mech. 105, 1 ~2002!.
|
||||
|
||||
|
||||
|
||||
|
||||
28F. Battaglia, S. J. Tavener, A. K. Kulkarni, and C. L. Merkle, ‘‘Bifurcation of low Reynolds number flows in symmetric channels,’’ AIAA J. 35, 99 ~1997!.
|
||||
|
||||
|
||||
|
||||
|
||||
29D. Drikakis, ‘‘Bifurcation phenomena in incompressible sudden expansion flows,’’ Phys. Fluids 9, 76 ~1997!.
|
||||
|
||||
|
||||
|
||||
|
||||
30R. M. Fearn, T. Mullin, and K. A. Cliffe, ‘‘Nonlinear flow phenomena in a symmetric sudden expansion,’’ J. Fluid Mech. 211, 595 ~1990!.
|
||||
|
||||
|
||||
|
||||
|
||||
31T. Hawa and Z. Rusak, ‘‘The dynamics of a laminar flow in a symmetric channel with a sudden expansion,’’ J. Fluid Mech. 436, 283 ~2001!.
|
||||
|
||||
|
||||
|
||||
|
||||
32S. Mishra and K. Jayaraman, ‘‘Asymmetric flows in planar symmetric channels with large expansion ratio,’’ Int. J. Numer. Methods Fluids 38, 945 ~2002!.
|
||||
|
||||
|
||||
|
||||
|
||||
33P. J. Oliveira, ‘‘Asymmetric flows of viscoelastic fluids in symmetric planar expansion geometries,’’ J. Non-Newtonian Fluid Mech. 114, 33 ~2003!.
|
||||
|
||||
|
||||
|
||||
|
||||
34E. Schreck and M. Scha¨fer, ‘‘Numerical study of bifurcation in threedimensional sudden channel expansions,’’ Comput. Fluids 29, 583 ~2000!.
|
||||
|
||||
|
||||
|
||||
|
||||
35A. W. Liu, D. E. Bornside, R. C. Armstrong, and R. A. Brown, ‘‘Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields,’’ J. Non-Newtonian Fluid Mech. 77, 153 ~1998!.
|
||||
|
||||
|
||||
|
||||
|
||||
36O. H. Faxe´n, ‘‘Forces exerted on a rigid cylinder in a viscous fluid between two parallel fixed planes,’’ R. Swed. Acad. Eng. Sci. 187, 1 ~1946!.
|
||||
|
||||
|
||||
1320
|
||||
@@ -0,0 +1,493 @@
|
||||
## Streakline postprocessing design
|
||||
|
||||
这份说明面向一个最小可实现的离线后处理程序。输入是按时间保存的二维速度场序列，输出是接近实验染色图的 streakline 图像。核心路线是连续释粒，因为 streakline 正是“固定位置持续释放染料后，在某个时刻所有已释放粒子的位置集合” [Lan96]。对非定常尾迹，这比单帧 streamline 更贴近实验图像 [Lan96]。
|
||||
|
||||
搜索和全文阅读给出的信息已经足够支撑第一版实现。最直接的方案是按保存时刻连续注入粒子，用时空插值后的速度场推进粒子，再将粒子云渲染为图像 [Lan96, Ken96]。如果纯粒子结果过于尖锐，再在推进后加入一个很小的随机扩散项，作为对真实染色液扩散的近似。更完整的被动标量法当然更物理，但实现成本更高，不适合作为第一版 [Kim04]。
|
||||
|
||||
## Core recommendation
|
||||
|
||||
| 方法 | 与实验染色图的对应 | 实现复杂度 | 适合作为第一版 |
|
||||
|---|---|---:|---|
|
||||
| 单帧 streamline | 弱 | 很低 | 不推荐 |
|
||||
| 连续释粒 streakline | 强 | 低 | 最推荐 |
|
||||
| streakline 加少量扩散 | 很强 | 低到中 | 推荐作为第二步 |
|
||||
| 被动标量对流扩散 | 最强 | 中到高 | 暂不作为第一版 |
|
||||
|
||||
最小可实现路线如下：
|
||||
|
||||
- 从 CelerisLab 导出一串二维速度场快照 `u_x(x,y,t_k), u_y(x,y,t_k)`
|
||||
- 选定一个固定释放点或一小段释放线
|
||||
- 在每个保存时刻都注入新粒子
|
||||
- 在两个相邻快照之间，用时间插值和空间插值计算粒子速度
|
||||
- 用二阶或四阶时间积分推进粒子
|
||||
- 删去出域粒子与进入固体的粒子
|
||||
- 将当前时刻存活粒子按位置和年龄渲染成图像
|
||||
|
||||
这个程序逻辑直接对应 [Lan96] 的 streakline 算法，插值与推进细节由 [Ken96] 和 [Dar96] 支撑。
|
||||
|
||||
## Data contract
|
||||
|
||||
默认输入是规则 Cartesian 网格上的二维时间序列速度场。对当前项目，最自然的输入约定是：
|
||||
|
||||
| 名称 | 含义 |
|
||||
|---|---|
|
||||
| `t_k` | 第 `k` 个保存时刻 |
|
||||
| `ux[k, j, i]` | 时刻 `t_k` 的 x 方向速度 |
|
||||
| `uy[k, j, i]` | 时刻 `t_k` 的 y 方向速度 |
|
||||
| `mask[j, i]` | 可选，固体与流体标记 |
|
||||
| `x_i, y_j` | 网格坐标 |
|
||||
|
||||
第一版最好满足下面三条：
|
||||
|
||||
- 时间快照间隔固定或至少已知
|
||||
- 网格坐标固定不动
|
||||
- 固体几何位置已知，至少能判断粒子是否进入圆柱内部
|
||||
|
||||
如果输出来自 CelerisLab，本质上只需要把每个保存时刻的 `ux, uy` 和对应时间写出来即可。算法本身是通用的，并不依赖 LBM 本身。
|
||||
|
||||
## Mathematical model for the base streakline
|
||||
|
||||
### Particle motion
|
||||
|
||||
粒子位置满足拉格朗日运动方程 [Lan96]：
|
||||
|
||||
\[
|
||||
\frac{d \boldsymbol{x}}{dt} = \boldsymbol{u}(\boldsymbol{x}, t)
|
||||
\]
|
||||
|
||||
其中
|
||||
|
||||
\[
|
||||
\boldsymbol{x} = (x, y), \qquad \boldsymbol{u} = (u_x, u_y)
|
||||
\]
|
||||
|
||||
积分形式为 [Lan96]：
|
||||
|
||||
\[
|
||||
\boldsymbol{x}(t + \Delta t) = \boldsymbol{x}(t) + \int_t^{t+\Delta t} \boldsymbol{u}(\boldsymbol{x}(\tau), \tau) \, d\tau
|
||||
\]
|
||||
|
||||
streakline 在观察时刻 \(t_n\) 的定义是：从固定释放位置 \(\boldsymbol{x}_s\) 在过去各时刻持续注入的粒子，在 \(t_n\) 时刻的全部位置集合 [Lan96]。
|
||||
|
||||
### Time interpolation
|
||||
|
||||
速度场只在离散时刻存储，所以必须做时间插值。对位于 \(t_k \le t \le t_{k+1}\) 的任意子步，最简单的做法是线性时间插值 [Ken96]：
|
||||
|
||||
\[
|
||||
\delta = \frac{t - t_k}{t_{k+1} - t_k}
|
||||
\]
|
||||
|
||||
\[
|
||||
\boldsymbol{u}(\boldsymbol{x}, t) = (1 - \delta) \, \boldsymbol{u}_k(\boldsymbol{x}) + \delta \, \boldsymbol{u}_{k+1}(\boldsymbol{x})
|
||||
\]
|
||||
|
||||
这里 \(\boldsymbol{u}_k(\boldsymbol{x})\) 和 \(\boldsymbol{u}_{k+1}(\boldsymbol{x})\) 仍需通过空间插值得到。
|
||||
|
||||
### Spatial interpolation
|
||||
|
||||
在规则网格上，第一版直接用双线性插值即可。虽然 [Ken96] 讨论的是非结构网格上的点定位与线性插值，但其一般原则完全适用于规则网格：
|
||||
|
||||
- 先定位粒子所在单元
|
||||
- 再用单元顶点速度插值得到粒子点速度 [Ken96]
|
||||
|
||||
若粒子位于单元局部坐标 \((\xi, \eta) \in [0,1]^2\)，四个角点速度为 \(\boldsymbol{u}_{00}, \boldsymbol{u}_{10}, \boldsymbol{u}_{01}, \boldsymbol{u}_{11}\)，则
|
||||
|
||||
\[
|
||||
\boldsymbol{u}(\xi, \eta) = (1-\xi)(1-\eta) \boldsymbol{u}_{00}
|
||||
+ \xi (1-\eta) \boldsymbol{u}_{10}
|
||||
+ (1-\xi)\eta \boldsymbol{u}_{01}
|
||||
+ \xi\eta \boldsymbol{u}_{11}
|
||||
\]
|
||||
|
||||
## Time integration choice
|
||||
|
||||
[Lan96] 和 [Ken96] 都直接使用四阶 Runge Kutta。对你的第一版程序，推荐两个选项：
|
||||
|
||||
| 积分器 | 优点 | 缺点 | 建议 |
|
||||
|---|---|---|---|
|
||||
| RK2 | 简洁，容易调试 | 精度一般 | 最小原型可用 |
|
||||
| RK4 | 文献最一致，精度更稳 | 每步插值次数更多 | 默认推荐 |
|
||||
|
||||
RK4 的更新式为 [Lan96, Ken96]：
|
||||
|
||||
\[
|
||||
\boldsymbol{x}_{n+1} = \boldsymbol{x}_n + \frac{1}{6}(\boldsymbol{a} + 2\boldsymbol{b} + 2\boldsymbol{c} + \boldsymbol{d})
|
||||
\]
|
||||
|
||||
其中
|
||||
|
||||
\[
|
||||
\boldsymbol{a} = \Delta t \, \boldsymbol{u}(\boldsymbol{x}_n, t_n)
|
||||
\]
|
||||
|
||||
\[
|
||||
\boldsymbol{b} = \Delta t \, \boldsymbol{u}(\boldsymbol{x}_n + \tfrac{1}{2}\boldsymbol{a}, t_n + \tfrac{1}{2}\Delta t)
|
||||
\]
|
||||
|
||||
\[
|
||||
\boldsymbol{c} = \Delta t \, \boldsymbol{u}(\boldsymbol{x}_n + \tfrac{1}{2}\boldsymbol{b}, t_n + \tfrac{1}{2}\Delta t)
|
||||
\]
|
||||
|
||||
\[
|
||||
\boldsymbol{d} = \Delta t \, \boldsymbol{u}(\boldsymbol{x}_n + \boldsymbol{c}, t_n + \Delta t)
|
||||
\]
|
||||
|
||||
[Ken96] 特别强调，RK4 的每一个子步都要重新做点定位与时空插值。这在规则网格上不难实现，只是意味着一次完整步进需要四次速度查询 [Ken96]。
|
||||
|
||||
## Practical timestep guidance
|
||||
|
||||
第一版不必上自适应步长，但步长不能随意设大。搜索里最重要的精度提醒来自 [Dar96]：
|
||||
|
||||
- 非定常粒子积分的误差常常由时间离散控制，而不是概念本身
|
||||
- 合理步长必须不大于流动主要非定常时间尺度的同量级 [Dar96]
|
||||
- 若时间步过大，粒子轨迹会在拓扑上都出错，而不仅是位置略偏 [Dar96]
|
||||
|
||||
对当前项目，最简单的工程准则是：
|
||||
|
||||
\[
|
||||
\Delta t_{trace} \le \min \bigl(\alpha_t \, \Delta t_{save},\; \alpha_x \, \frac{\Delta x}{\max |u|} \bigr)
|
||||
\]
|
||||
|
||||
其中 \(\Delta t_{save}\) 是两个保存快照之间的时间间隔，\(\Delta x\) 是网格尺度。第一版可取：
|
||||
|
||||
- \(\alpha_t = 0.1 \sim 0.25\)
|
||||
- \(\alpha_x = 0.25 \sim 0.5\)
|
||||
|
||||
这不是文献中的严格上界，而是结合 [Dar96] 的结论给出的实现准则：粒子推进子步既要显著小于保存间隔，也不要一子步跨过太多网格。
|
||||
|
||||
[Ken96] 还给了一个很实用的自适应思路：根据相邻速度方向夹角调节步长。如果速度方向变化太快就减半，变化很小就加倍 [Ken96]。这很适合后续增强版，但不属于第一版必需项。
|
||||
|
||||
## Base streakline algorithm
|
||||
|
||||
### Release strategy
|
||||
|
||||
最符合染色实验的是连续释粒 [Lan96]。第一版可用两种释放方式：
|
||||
|
||||
| 方式 | 适用场景 | 建议 |
|
||||
|---|---|---|
|
||||
| 单点释放 | 针头式染料入口 | 最简单 |
|
||||
| 短线段释放 | 更像细缝或细带注入 | 更稳健 |
|
||||
|
||||
如果实验是在圆柱上游某一点持续释放染色液，第一版就直接使用单点释放。
|
||||
|
||||
### Particle state
|
||||
|
||||
每个粒子只需保存：
|
||||
|
||||
| 字段 | 含义 |
|
||||
|---|---|
|
||||
| `x, y` | 当前坐标 |
|
||||
| `t_birth` | 释放时刻 |
|
||||
| `age` | 当前年龄 |
|
||||
| `alive` | 是否仍在域内 |
|
||||
|
||||
第一版不需要保存整条历史轨迹。因为你要的是当前时刻的 streakline 图，而不是每个粒子的 pathline 曲线。
|
||||
|
||||
**常见误区（务必区分）**
|
||||
|
||||
| 概念 | 画什么 | 是否等于水洞染色迹线 |
|
||||
|---|---|---|
|
||||
| **Streakline（迹线）** | 固定源点**持续释粒**，在观察时刻 \(t_n\) 画出**所有仍存活粒子的当前位置**（可带年龄衰减权重） | 是 |
|
||||
| **Pathline（迹线/轨道）** | 单个粒子从释放到当前的**整条历史轨迹** | 否 |
|
||||
| **错误实现** | 每隔一段时间放一个粒子，再把**所有粒子走过的路径段**全部叠加到图上 | 否（这是 pathline 叠加，不是 streakline） |
|
||||
|
||||
水洞实验：针头在固定点连续注 dye → 某一时刻拍照 → 看到的是“此刻染料粒子在流场里的分布”，一条下游色带由**不同释放时刻、当前仍在本流场中的粒子**共同构成，而不是把每个粒子从出生到现在的轨迹都画出来。
|
||||
|
||||
### Main loop pseudocode
|
||||
|
||||
```text
|
||||
given velocity snapshots U[k] = {ux[k], uy[k]} at times t[k]
|
||||
given seeding point or seeding segment S
|
||||
initialize empty particle list P
|
||||
|
||||
for k = 0 to N-2:
|
||||
|
||||
inject new particles at source S at time t[k]
|
||||
|
||||
set t_local = t[k]
|
||||
while t_local < t[k+1]:
|
||||
dt = min(dt_trace, t[k+1] - t_local)
|
||||
|
||||
for each particle p in P with p.alive:
|
||||
v = interpolate_velocity(p.position, t_local)
|
||||
advance p by one integration step using RK2 or RK4
|
||||
update p.age
|
||||
if p leaves domain:
|
||||
p.alive = false
|
||||
if p enters solid body:
|
||||
p.alive = false
|
||||
|
||||
t_local = t_local + dt
|
||||
|
||||
optionally remove very old particles
|
||||
optionally render current particle cloud
|
||||
```
|
||||
|
||||
这个结构与 [Lan96] 的离散 streakline 算法一致，只是把“从 `t_k` 到 `t_{k+1}` 的一次推进”细化成多个更小的粒子子步，以满足 [Dar96] 对精度的要求。
|
||||
|
||||
### Velocity query pseudocode
|
||||
|
||||
```text
|
||||
function interpolate_velocity(position x, time t):
|
||||
find k such that t[k] <= t <= t[k+1]
|
||||
compute delta = (t - t[k]) / (t[k+1] - t[k])
|
||||
|
||||
u_k = bilinear_interpolation(U[k], x)
|
||||
u_k1 = bilinear_interpolation(U[k+1], x)
|
||||
|
||||
return (1 - delta) * u_k + delta * u_k1
|
||||
```
|
||||
|
||||
这是第一版最核心的数值部件。只要这部分实现正确，streakline 程序主体就很直接。
|
||||
|
||||
## Small diffusion particle model
|
||||
|
||||
当纯 streakline 太细、太锐利、不像染色液图像时，可以给每个粒子加一个很小的随机扩散项。这不是严格的被动标量求解，但能以很低代价增加染色带宽度。
|
||||
|
||||
### Guiding idea
|
||||
|
||||
[Kim04] 表明，染料图像更接近一个被动标量浓度场，其基本控制方程是对流扩散方程：
|
||||
|
||||
\[
|
||||
\frac{\partial c}{\partial t} + \boldsymbol{u} \cdot \nabla c = D \nabla^2 c
|
||||
\]
|
||||
|
||||
其中 \(c\) 是染料浓度，\(D\) 是扩散系数。对第一版粒子法，一个常见近似是将每个粒子的位置更新写成“对流加随机扩散”：
|
||||
|
||||
\[
|
||||
\boldsymbol{x}_{n+1} = \boldsymbol{x}_n + \Delta \boldsymbol{x}_{adv} + \Delta \boldsymbol{x}_{diff}
|
||||
\]
|
||||
|
||||
其中 \(\Delta \boldsymbol{x}_{adv}\) 由 RK2 或 RK4 给出，扩散项取二维各向同性随机增量：
|
||||
|
||||
\[
|
||||
\Delta \boldsymbol{x}_{diff} = \sqrt{2 D \Delta t}
|
||||
\begin{bmatrix}
|
||||
\eta_x \\
|
||||
\eta_y
|
||||
\end{bmatrix}
|
||||
\]
|
||||
|
||||
这里 \(\eta_x, \eta_y \sim \mathcal{N}(0,1)\)。
|
||||
|
||||
这个形式本身是对扩散过程的标准随机游走近似。搜索结果提醒，随机游走模型如果处理不当会产生假漂移与错误浓度偏置 [Mac92]。因此第一版的使用原则应当很克制：
|
||||
|
||||
- 只用于加入少量模糊和厚度
|
||||
- 不把粒子密度当成严格浓度
|
||||
- 不在强近壁统计上过度解读结果
|
||||
|
||||
### Diffusive particle update pseudocode
|
||||
|
||||
```text
|
||||
for each particle p in P with p.alive:
|
||||
x_adv = RK4_step(p.position, t_local, dt)
|
||||
|
||||
sigma = sqrt(2 * D * dt)
|
||||
dx_rand = sigma * normal(0, 1)
|
||||
dy_rand = sigma * normal(0, 1)
|
||||
|
||||
x_new = x_adv + [dx_rand, dy_rand]
|
||||
|
||||
if x_new leaves domain:
|
||||
p.alive = false
|
||||
else if x_new enters solid:
|
||||
p.alive = false
|
||||
else:
|
||||
p.position = x_new
|
||||
p.age += dt
|
||||
```
|
||||
|
||||
### Choosing the diffusion level
|
||||
|
||||
第一版不必试图从真实染料物性严格标定 \(D\)。更实用的做法是把 \(D\) 当作视觉匹配参数，并保持它足够小，使图像结构仍主要由对流控制。
|
||||
|
||||
[Kim04] 用 Schmidt 数控制扩散强弱。若已有参考速度 \(U\) 和长度尺度 \(L\)，则
|
||||
|
||||
\[
|
||||
Sc = \frac{\nu}{D}
|
||||
\]
|
||||
|
||||
也可写成
|
||||
|
||||
\[
|
||||
D = \frac{\nu}{Sc}
|
||||
\]
|
||||
|
||||
对第一版，可以用下面的思路选扩散强度：
|
||||
|
||||
| 目标效果 | 建议 |
|
||||
|---|---|
|
||||
| 只想让线条略微变厚 | 取较大 `Sc`，即很小的 `D` |
|
||||
| 想模拟明显洗开和模糊 | 取较小 `Sc`，即较大的 `D` |
|
||||
| 不确定 | 先从几乎看不出的弱扩散开始 |
|
||||
|
||||
因为当前目标是“少量扩散”，所以推荐先把扩散当成弱修饰，而不是主导机制。
|
||||
|
||||
## Rendering logic
|
||||
|
||||
最终图像不需要把每个粒子的完整轨迹都画出来。更像实验染色图的做法是把当前存活粒子投影到图像网格上，生成粒子密度图或带年龄权重的强度图。
|
||||
|
||||
### Minimal rendering choices
|
||||
|
||||
| 方法 | 图像风格 | 实现难度 |
|
||||
|---|---|---:|
|
||||
| 直接散点 | 最简陋 | 很低 |
|
||||
| 网格计数直方图 | 像浓度图 | 低 |
|
||||
| 高斯核累积 | 更平滑 | 低到中 |
|
||||
|
||||
第一版推荐：
|
||||
|
||||
- 把每个粒子投到像素网格
|
||||
- 对像素做计数或加权累积
|
||||
- 最后做一次轻微 Gaussian blur
|
||||
|
||||
这与先做完整被动标量相比便宜很多，但视觉上已经会很接近实验染色图。
|
||||
|
||||
### Optional particle weighting
|
||||
|
||||
可以给粒子一个简单权重：
|
||||
|
||||
\[
|
||||
I = \sum_p w_p K(\boldsymbol{x} - \boldsymbol{x}_p)
|
||||
\]
|
||||
|
||||
其中 \(K\) 是像素核或 Gaussian 核。第一版里，权重 \(w_p\) 可直接取 1，也可对年龄做衰减，例如：
|
||||
|
||||
\[
|
||||
w_p = \exp\left(- \frac{\mathrm{age}_p}{\tau_f} \right)
|
||||
\]
|
||||
|
||||
这样旧粒子会逐渐淡出，图像不会无限堆积。
|
||||
|
||||
## Boundary handling
|
||||
|
||||
这部分先按最小原则处理即可。
|
||||
|
||||
| 情况 | 第一版处理 |
|
||||
|---|---|
|
||||
| 粒子出计算域 | 直接删除 |
|
||||
| 粒子进入圆柱内部 | 直接删除 |
|
||||
| 粒子贴近边界滑动 | 暂不专门处理 |
|
||||
| 扩散后跨入固体 | 直接删除 |
|
||||
|
||||
这样做的优点是简单稳妥。后续若发现近壁 streakline 误差明显，再考虑反射、投影或更物理的壁面处理。
|
||||
|
||||
## Minimal implementation plan
|
||||
|
||||
### Version 1
|
||||
|
||||
目标是尽快得到可信的 streakline 图：
|
||||
|
||||
- 读取 `ux, uy, t`
|
||||
- 单点连续释粒
|
||||
- 双线性空间插值
|
||||
- 线性时间插值
|
||||
- RK4 粒子推进
|
||||
- 出域删除与入固体删除
|
||||
- 粒子计数成图
|
||||
|
||||
### Version 2
|
||||
|
||||
在不改变主体架构的前提下增强视觉效果：
|
||||
|
||||
- 增加短线段释放
|
||||
- 增加年龄衰减
|
||||
- 增加弱随机扩散
|
||||
- 用 Gaussian 核代替简单计数
|
||||
|
||||
### Version 3
|
||||
|
||||
若后续发现实验图像明显受扩散与混合主导，再转向更重的模型：
|
||||
|
||||
- 被动标量对流扩散
|
||||
- 用 \(Sc\) 或 \(D\) 做参数标定 [Kim04]
|
||||
|
||||
## Design decisions that are already justified by the literature
|
||||
|
||||
下面这些设计已经有足够文献支撑，可以直接采用：
|
||||
|
||||
- 用 streakline 而不是 streamline 对应持续染色实验 [Lan96]
|
||||
- 用连续释放粒子离线重构 streakline [Lan96]
|
||||
- 用时空插值从离散速度快照查询粒子速度 [Ken96]
|
||||
- 用 RK4 作为默认推进器 [Lan96, Ken96]
|
||||
- 把步长取得明显小于保存时间间隔，并避免一次跨过太多网格 [Dar96]
|
||||
- 若需要更像染料图，可以加入弱扩散，或后续上升到被动标量 [Kim04]
|
||||
|
||||
## Recommended default choices
|
||||
|
||||
| 项目 | 默认选择 |
|
||||
|---|---|
|
||||
| 释放方式 | 单点连续释粒 |
|
||||
| 空间插值 | 双线性 |
|
||||
| 时间插值 | 线性 |
|
||||
| 积分器 | RK4 |
|
||||
| 粒子子步 | 保存步长的 0.1 到 0.25 |
|
||||
| 固体处理 | 入固体即删除 |
|
||||
| 输出图像 | 粒子密度图加轻微模糊 |
|
||||
| 扩散 | 默认关闭，作为第二步 |
|
||||
|
||||
## Bottom line
|
||||
|
||||
对当前项目，最简洁且足够准确的程序逻辑不是去解新的染料场，而是先做一个离线连续释粒 streakline 后处理器。它只依赖时间序列速度场，程序结构清楚，数值风险也集中在可控的几个环节：时空插值、积分步长和边界删除 [Lan96, Dar96, Ken96]。在此基础上，再增加一个弱随机扩散项，就能以很小代价把图像从“几何上正确的粒子线”推进到“更像实验染色照片”的粒子云 [Kim04]。
|
||||
|
||||
## CelerisLab implementation mapping
|
||||
|
||||
Current implementation lives in:
|
||||
|
||||
- library: `src/CelerisLab/common/streakline.py`
|
||||
- Kan99b demo CLI: `tests/run_kan99b_streakline.py`
|
||||
- experiment notebook: `tests/experiment.ipynb` (CLEAN 5/6)
|
||||
|
||||
Available runtime modes:
|
||||
|
||||
- `online`: sample `Simulation.get_macroscopic()` directly in memory (no velocity snapshot files).
|
||||
- `offline`: replay from an existing snapshot directory and run the same streakline integrator.
|
||||
|
||||
Dense release support:
|
||||
|
||||
- base upstream points are expanded by `release_mode` (`point|line|strip`).
|
||||
- `strip` mode densifies both cross-stream (`line_count`, `line_span`) and downstream (`downstream_count`, `downstream_spacing`) directions.
|
||||
|
||||
### Performance notes (why it can feel slow)
|
||||
|
||||
Streakline post-processing is **CPU-side** in the current stack:
|
||||
|
||||
1. **GPU→host velocity copies** dominate for large grids (`nx=6000, ny=1200`). Each `get_macroscopic()` pulls full `ux/uy` arrays.
|
||||
2. Particle RK4 + bilinear interpolation runs in NumPy on CPU (vectorized over particles, not GPU).
|
||||
3. Rendering accumulates polylines into an image (optionally multi-threaded via `ThreadPoolExecutor`).
|
||||
|
||||
Practical tuning:
|
||||
|
||||
| knob | effect |
|
||||
|---|---|
|
||||
| `sample_every` | larger → fewer host copies, faster, coarser streaks |
|
||||
| `max_particle_age` | `None` → no hard cutoff; trails survive until boundary/solid |
|
||||
| `num_threads` | `0` = auto; sets OpenBLAS/OMP threads for blur/host ops |
|
||||
| `blur_sigma` | `0` disables Gaussian blur pass |
|
||||
|
||||
For `experiment.ipynb` triangle case, use **`render_streakline_density(..., minimal_axes=True)`** on the **final alive particle cloud** (`positions`, `ages`). Do **not** accumulate and draw full path history (`render_snapshot_trails` is pathline-style debug only).
|
||||
|
||||
### Clean render style (experiment)
|
||||
|
||||
`render_snapshot_trails` defaults:
|
||||
|
||||
- white background
|
||||
- red streaks, brighter toward the downstream end (`fade_along_trail=True`)
|
||||
- black filled cylinders
|
||||
- no axes, labels, colorbar, or release-point markers
|
||||
|
||||
Example commands:
|
||||
|
||||
```bash
|
||||
# Online in-memory streakline (no snapshot write)
|
||||
conda run -n pycuda_3_10 python tests/run_kan99b_streakline.py \
|
||||
--mode online --domain M --re 100 --alpha 1.0 \
|
||||
--sample-every 300 --n-snapshots 20 \
|
||||
--release-mode strip --line-count 7 --downstream-count 6
|
||||
|
||||
# Offline replay from existing velocity snapshots
|
||||
conda run -n pycuda_3_10 python tests/run_kan99b_streakline.py \
|
||||
--mode offline \
|
||||
--snapshot-dir tests/output/final_validation_round/streakline_kan99b_k2/velocity_snapshots \
|
||||
--release-mode strip --line-count 7 --downstream-count 6
|
||||
```
|
||||
Reference in New Issue
Block a user