Wake
Wake: Theoretical Foundations
Overview
Teacher, a wake is the flow that forms behind an object, correct?
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.
Computational Methods for Wake
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.
```
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.