Wake
Theory and Physics
Overview
Teacher, a wake is the flow that forms behind an object, right?
Exactly. When an object moves through a fluid (or when an object is placed in a flow), the region of velocity deficit formed on the downstream side of the object is called a wake. Its applications are extremely wide-ranging, including aircraft wake turbulence, wind turbine wake interference, automobile aerodynamic drag, and slipstreaming in sports.
Wake Structure
Let's consider the far wake, sufficiently far from the object. The velocity profile of the wake takes the form of the freestream velocity $U_\infty$ minus the velocity deficit $u_{def}(x, y)$.
In the far wake, we can assume $u_{def} \ll U_\infty$, allowing the application of linearized boundary layer equations.
Self-Similar Solutions
Do wakes also have self-similar solutions?
The self-similarity of far wakes was established by Townsend's theory.
2D Wake (e.g., 2D objects like cylinders):
Axisymmetric Wake (e.g., 3D objects like spheres):
For jets, it was $u_c \propto x^{-1}$, right? So the wake decays more slowly.
Correct. A wake is a "hole" in a uniform flow, lacking the strong self-induction of a jet, so its diffusion is slower.
Momentum Integral and Drag
I heard you can determine an object's drag from its wake. Is that true?
This is a very important relationship. Integrating the momentum deficit in the wake yields the drag force on the object.
For a 2D object (per unit span length):
For an axisymmetric object:
This relationship is also called Jones' formula and forms the basis for non-contact drag measurement in wind tunnel experiments. The velocity distribution is obtained from a Pitot tube traverse in the wake, and drag is calculated via the momentum integral.
So you can find the drag without directly measuring the force?
Correct. It's a consequence of the momentum conservation law and is also used in CFD as a method to calculate drag from the momentum balance of a control volume. It should agree with the integral of pressure and friction forces on the wall surfaces.
Wake Stability
Is stability analysis of wakes also important?
Stability analysis of the wake profile can explain the characteristics of the Kármán vortex street. Temporal stability analysis of the wake velocity profile yields:
- Antisymmetric mode (sinuous mode): Corresponds to the meandering motion of the vortex street, associated with Kármán vortices.
- Symmetric mode (varicose mode): Corresponds to pulsation of the wake width. Typically less unstable than the antisymmetric mode.
Monkewitz (1988) showed the condition for a wake to become "absolutely unstable" (when the velocity deficit is sufficiently large). An absolutely unstable wake undergoes self-excited oscillations, and a Kármán vortex street forms spontaneously even without upstream disturbances.
Truck Platooning and Wake Energy Saving – Slipstream Calculations
When large trucks drive in a platoon on a highway, the following vehicles enter the wake of the leading vehicle, reducing aerodynamic drag by 20-30%. This is the "slipstream" effect, and its practical application is advancing as automated platooning for fuel efficiency. How far the wake's velocity deficit recovers (wake recovery length) depends on the Reynolds number and object shape, directly influencing the design of the optimal inter-vehicle distance. Coupled analysis, where "the velocity distribution of the leading vehicle's wake is used as the inlet boundary condition for the following vehicle," is employed in CFD. Calculations confirm that the energy-saving effect for following vehicles increases as more vehicles (2, 3, etc.) are linked.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out unstable and splashing, but after a while, the flow becomes steady, right? This term describes the "state of change." The pulsation of blood flow from a 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 this term is set to zero. This significantly reduces computational cost, so trying a steady-state solution first 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 something. Warm air from a heater reaching the far side 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 rapidly strengthens, 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 put milk in coffee and left it? Even without stirring, after a while it naturally mixes, right? 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. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re flows, convection overwhelmingly dominates, 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 pushing 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 become strange immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
- Source term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings and is 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" and are expressed by source terms. What happens if you forget a source term? In natural convection analysis, forgetting to include buoyancy means the fluid doesn't move at all—a physically impossible result, like turning on a heater in a winter room but the warm air doesn't rise.
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 flows (shock capturing required), free surface flows (VOF/Level Set, etc., required).
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 | $Re = \rho u L / \mu$. Indicator 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 Methods
What methods are used for CFD of wakes?
Wake analysis has two aspects: analysis of the object itself and analysis of the wake region.
| Purpose | Method | Remarks |
|---|---|---|
| Separation near object and near wake | RANS / DES / LES | Wall resolution required. |
| Diffusion and recovery of far wake | RANS / LES | Large computational domain required. |
| Wake stability analysis | DNS + Floquet / BiGlobal | Precise calculation of the base state is prerequisite. |
| Wind turbine wake interference | Actuator Line/Disk + LES | Model the wind turbine to focus on the wake. |
Wake Region Mesh Design
What should I be careful about when meshing the wake region?
The wake expands downstream, so the mesh must follow that expansion.
- Immediately behind object: Finest mesh. Maintain resolution comparable to the object surface mesh up to about the length of the recirculation region.
- Intermediate wake ($5D\text{--}20D$): Process of vortex structure breakdown. Gradually coarsen mesh in flow direction (growth rate $< 1.1$).
- Far wake ($> 20D$): Self-similar region. Place at least 10 cells across the wake width.
- Lateral direction: Ensure domain width at least 3 times the wake width.
How do we deal with the problem of the wake disappearing too quickly due to numerical diffusion?
The velocity deficit in a wake becomes very small downstream, making it susceptible to numerical diffusion effects. Countermeasures are:
1. Higher-order schemes: At least 2nd order accuracy. For LES, use central-difference type schemes.
2. Mesh isotropy: Avoid stretching cells too much in the flow direction. Aspect ratio $< 5$.
3. Sufficient resolution: Mesh capable of resolving the deficit profile even in regions where the deficit is below $1\%$.
4. AMR (Adaptive Mesh Refinement): Dynamically refine mesh based on vorticity or velocity gradients.
Drag Calculation via Momentum Integral Method
Please teach me how to calculate drag from the wake in CFD.
Obtain the velocity distribution at a cross-section sufficiently far downstream of the object (e.g., $10D$ downstream) and perform the momentum integral.
The second pressure term is small far away but cannot be ignored at cross-sections near the object. Verifying that the drag obtained by this method matches the drag obtained by direct integration of pressure and friction forces on the wall surfaces is a good CFD validation practice.
OpenFOAM for Wake Analysis
How can I obtain wake statistics in OpenFOAM?
Use the fieldAverage function object to compute time-averaged fields.
How can I obtain wake statistics in OpenFOAM?
Use the fieldAverage function object to compute time-averaged fields.
```
functions
{
fieldAverage1
{
type fieldAverage;
libs ("libfieldFunctionObjects.so");
writeControl writeTime;
fields
(
U { mean on; prime2Mean on; base time; }
p { mean on; prime2Mean on; base time; }
);
}
}
```
This outputs UMean (time-averaged velocity) and UPrime2Mean (Reynolds stress tensor). The wake velocity deficit profile is obtained by subtracting $U_\infty$ from UMean.
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