Flow around a cylinder
Flow around a cylinder: Theoretical Foundations
Overview
Professor, is flow around a cylinder just about wind hitting a pole?
It looks simple, but the flow structure changes dramatically depending on the Reynolds number. It's a classic and fundamental problem in fluid mechanics textbooks, ranging from steady symmetric separation to Kármán vortex streets and even turbulent transition.
The Kármán vortex street is that thing where vortices shed alternately, right? Why do they alternate?
When Re exceeds about 47, the symmetry of the wake becomes absolutely unstable. A feedback loop is created as the separating shear layer on one side entrains vorticity from the opposite side, causing vortices to shed alternately from the top and bottom. This is the Benard-von Kármán vortex street.
Flow Classification by Reynolds Number
How much does it change with the Reynolds number?
It can be summarized as follows.
| Re Range | Flow State | Characteristics |
|---|---|---|
| Re < 5 | Creeping Flow | Fore-aft symmetric, no separation |
| 5 < Re < 47 | Steady Twin Vortices | Symmetric recirculation zone forms |
| 47 < Re < 190 | 2D Kármán Vortex Street | Periodic vortex shedding, Laminar Flow |
| 190 < Re < 260 | 3D Transition (Mode A/B) | Spanwise instabilities appear |
| 260 < Re < 1000 | Transition to Turbulence | Wake becomes turbulent, separation point remains laminar |
| $10^3$ < Re < $3 \times 10^5$ | Subcritical Regime | Laminar separation, $C_D \approx 1.2$ |
| $3 \times 10^5$ < Re < $10^6$ | Critical Regime (Drag Crisis) | Transition occurs before separation, $C_D$ drops sharply |
| Re > $10^6$ | Supercritical Regime | Turbulent boundary layer separation |
The vortex street appears at Re = 47? That's surprisingly low.
Yes. This threshold corresponds to a Hopf bifurcation. Linear stability analysis yields a precise critical Re value of $Re_{cr} \approx 46.7$.
Governing Equations
Then, please tell me the governing equations.
The Incompressible Navier-Stokes equations.
Here, $\nu = \mu / \rho$ is the kinematic viscosity, $\mathbf{u}$ is the velocity vector, and $p$ is pressure. Non-dimensionalization yields the Reynolds number $Re = U_\infty D / \nu$ as the sole governing parameter.
Strouhal Number and Vortex Shedding Frequency
Can the vortex shedding frequency be predicted?
It is organized using the Strouhal number.
In the subcritical regime ($300 < Re < 3 \times 10^5$), $\mathrm{St} \approx 0.2$ remains nearly constant. This is a famous result confirmed by Roshko's experiments. On the low Re side, the following empirical formula is often used as a function of Re.
Drag and Lift Coefficient
Let me also confirm the definition of the drag coefficient.
The definition per unit span length is as follows.
The steady flow $C_D$ varies greatly with Re. In the Stokes regime, $C_D \propto Re^{-1}$; in the subcritical regime, $C_D \approx 1.0 \text{--} 1.2$; in the critical regime, it drops to about $C_D \approx 0.3$; and in the supercritical regime, it recovers to about $C_D \approx 0.6 \text{--} 0.7$.
Does the lift fluctuate at the same frequency as the vortex shedding?
The lift fluctuation frequency is exactly the vortex shedding frequency $f$. On the other hand, the drag fluctuation frequency is $2f$. This is because drag increases and decreases with each vortex shed alternately, while lift changes sign with the shedding from one side.
Drag being $2f$ is interesting. Now that you mention it, it makes sense.
Kármán Vortices and Chimney Collapse—How the Strouhal Number Protects Buildings
The Kármán vortex street generated in the wake of a cylinder can sometimes cause serious engineering problems. When the vortex shedding frequency ($f = St \cdot U/D$) matches the natural frequency of a structure, resonance occurs, amplifying vibrations. The 1965 collapse of three cooling towers in a row at the Ferrybridge Power Station in the UK was also determined to be caused by vortex-induced vibration from Kármán vortices. Whether CFD can accurately calculate the Strouhal number directly impacts cost (or safety) in the design of chimneys, bridge girders, and marine riser pipes. "Kármán vortices appear even in 2D laminar CFD at Re≈100" serves as a useful benchmark for verifying the dynamic behavior of code.
Computational Methods for Flow around a cylinder
Selection of Numerical Method
When solving flow around a cylinder with CFD, which method should I use?
The optimal method changes depending on the Reynolds number. It can be summarized as follows.
| Re Range | Recommended Method | Reason |
|---|---|---|
| Re < 200 | DNS (Direct Numerical Simulation) | 2D calculation is sufficient; all scales can be resolved |
| 200 < Re < 1000 | DNS (3D) | 3D instabilities need to be resolved |
| $10^3$ < Re < $10^4$ | LES | Directly resolves turbulent structures in the wake |
| $10^4$ < Re < $10^6$ | URANS / DES / DDES | Wall resolution cost for LES becomes prohibitive |
| Re > $10^6$ | RANS (SST $k$-$\omega$) | Sufficient for engineering accuracy in many cases |
Wouldn't it be best to solve everything with DNS?
High Re DNS scales with grid points as $Re^{9/4}$. For Re=$10^6$, over $10^{13}$ points are needed, which is unrealistic even with current supercomputers.
Pressure-Velocity Coupling
For incompressible cases, how is pressure solved?
In incompressible Navier-Stokes, the pressure Poisson equation must be solved. Representative methods are as follows.
- SIMPLE Method: Patankar's semi-implicit method. Suitable for steady calculations. Pressure correction is found iteratively.
- PISO Method: Suitable for unsteady calculations. Performs two pressure corrections within one time step.
- Coupled Solver: Solves velocity and pressure simultaneously. Fast convergence but high memory consumption.
- Fractional Step Method: Finds an intermediate velocity, then corrects via pressure Poisson. Widely used in DNS.