Flow around a cylinder
Theory and Physics
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.
$$ C_D = \frac{F_D}{\frac{1}{2}\rho U_\infty^2 D}, \quad C_L = \frac{F_L}{\frac{1}{2}\rho U_\infty^2 D} $$
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.
Coffee Break Trivia
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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow with each heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost is significantly reduced, solving first with a steady-state assumption is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term becomes dramatically stronger, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever left milk in coffee without stirring? Even without mixing, after a while, they naturally blend. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re flows, convection overwhelms, and diffusion plays a minor role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to confusing absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (less dense) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget the source term? In a natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
Variable SI Unit Notes / Conversion Memo Velocity $u$ m/s When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units Pressure $p$ Pa Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis Density $\rho$ kg/m³ Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C Viscosity Coefficient $\mu$ Pa·s Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] Reynolds Number $Re$ Dimensionless $Re = \rho u L / \mu$. Criterion for Laminar/turbulent transition CFL Number Dimensionless $CFL = u \Delta t / \Delta x$. Directly related to time step stability
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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow with each heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost is significantly reduced, solving first with a steady-state assumption is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term becomes dramatically stronger, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever left milk in coffee without stirring? Even without mixing, after a while, they naturally blend. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re flows, convection overwhelms, and diffusion plays a minor role.
- Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to confusing absolute/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (less dense) than its surroundings, so it's pushed up by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by source terms. What happens if you forget the source term? In a natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for Laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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.
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