Mixing Layer
Mixing Layer: Theoretical Foundations
Overview
Professor, what is a mixing layer?
When two parallel flows with different velocities merge, the shear layer formed at their interface is called a mixing layer. It is one of the simplest free shear flows and a classic example of Kelvin-Helmholtz instability.
In what situations does it appear?
At the interface between an aircraft engine's exhaust and bypass flow, at river confluences, in atmospheric fronts, at the outer edges of jets issuing from nozzles, etc. It is an important flow directly linked to mixing and transport efficiency.
Basic Parameters
Let the velocities of the two flows be $U_1$ (high-speed side) and $U_2$ (low-speed side). The important parameters are as follows.
- Velocity Ratio: $r = U_2 / U_1$
- Velocity Difference: $\Delta U = U_1 - U_2$
- Convection Velocity: $U_c = (U_1 + U_2) / 2$
- Reynolds Number: $Re = \Delta U \cdot \delta_\omega / \nu$ (where $\delta_\omega$ is the vorticity thickness)
The definition of vorticity thickness is,
This is the representative thickness of the mixing layer defined from the maximum gradient of the velocity profile.
Kelvin-Helmholtz Instability
Please explain the theory of K-H instability.
We start from the linear stability analysis of a velocity discontinuity surface (vortex sheet). Adding a small wavy disturbance $\eta \propto e^{i(kx - \omega t)}$ to the discontinuity surface, the dispersion relation is,
Since the imaginary part is positive, disturbances grow for all wavenumbers $k$. That is, the velocity discontinuity surface is unstable for all wavenumbers. The growth rate is $\sigma = k \Delta U / 2$, with shorter wavelengths growing faster.
Unstable for all wavenumbers means even the smallest disturbance will grow?
Theoretically yes, but in reality, for shear layers with finite thickness, viscosity and thickness effects suppress short-wavelength instabilities. The most amplified wavelength is around $\lambda \approx 7 \delta_\omega$, and waves shorter than that are stabilized.
Self-Similar Solution and Spreading Rate
The mixing layer develops downstream, right?
Correct. Far enough downstream, the mixing layer spreads self-similarly. The vorticity thickness increases proportionally to $x$.
The spreading rate $d\delta_\omega / dx$ depends on the velocity ratio. For $r = 0$ (one side stationary), the experimental value is around $d\delta_\omega / dx \approx 0.16\text{--}0.18$. The visualization experiment by Brown & Roshko (1974) is a monumental study in this field.
What about the velocity profile in the self-similar region?
Non-dimensionalizing with $\eta = y / \delta_\omega$,
Here, $F$ has an error function-like shape. In Goertler's analytical solution, it is expressed as $F(\eta) = \frac{1}{2}[1 + \text{erf}(\sigma \eta)]$.
Pioneers of Mixing Layer TheoryโThe Achievements of Lord Kelvin and Helmholtz (1868)
The instability of a "mixing layer" where two fluids with a velocity difference meet was theoretically analyzed independently in 1868 by Hermann von Helmholtz and Lord Kelvin (William Thomson). The Kelvin-Helmholtz (KH) instability, named after them, is a pioneering achievement in linear stability theory, showing that instability occurs for any velocity difference when interfacial tension is zero. Interestingly, this theory was initially discussed in the context of "atmospheric science" (the formation mechanism of cirrus clouds). Its expansion to engineering applications (jet/mixing layer design) had to wait for the development of experimental fluid mechanics half a century later. The fact that modern CFD can quantitatively verify century-old theories is thanks to the dramatic improvement in computational power.
Computational Methods for Mixing Layers
Numerical Methods
What methods are used for CFD of mixing layers?
| Method | Application Scenario | Remarks |
|---|---|---|
| DNS | Fundamental research. $Re_{\delta_\omega} < 10^4$ | Fully resolves K-H vortex roll-up and pairing |
| LES | Medium to high Re number mixing processes | SGS model influence is relatively small (free shear flow) |
| RANS | Time-averaged spreading prediction | Detailed vortex structure is lost, but spreading rate is predictable |
| Temporal Mixing Layer | Vortex dynamics research | Simulates spatial mixing layer in a convective coordinate system. Computed with periodic boundaries |
Temporal Mixing Layer vs Spatial Mixing Layer
What is a temporal mixing layer?
A spatial mixing layer develops in the flow direction, while a temporal mixing layer evolves in time with uniform shear as the initial condition. The computational domain has periodic boundaries in the flow and span directions, corresponding to a Galilean transformation of a spatial mixing layer.
It's a technique often used in DNS. Computational cost is low, and spatial averaging of statistics is easy. However, it cannot simulate the "propagation of disturbances from upstream" of a spatial mixing layer.
Initial Conditions and Disturbance Setting
How do you set the initial conditions?
For DNS/LES of temporal mixing layers,
1. Base Flow: $\bar{u}(y) = \frac{\Delta U}{2} \tanh(2y / \delta_{\omega,0})$
2. 2D Disturbance: Superimpose the most amplified wavelength mode. Fundamental mode and subharmonic (to induce vortex pairing)
3. 3D Disturbance: Add spanwise modes (oblique modes). Necessary for 3D transition
4. Random Noise: Broadband disturbance. Reproduces natural turbulent transition
From Michalke's (1964) linear stability analysis, the wavenumber of the most amplified mode is found to be $k_{max} \delta_{\omega,0} / 2 \approx 0.4457$.
Mesh Design
How should the mesh be generated?
Key points for temporal mixing layer mesh design.
- Flow Direction ($x$): Periodic boundary. Length should be at least 4 times the fundamental mode wavelength (to track pairing twice)
- Normal Direction ($y$): Take it wide enough to track the growth of the mixing layer. Fine near the center, coarse far away (stretching)
- Span Direction ($z$): Periodic boundary. At least 2 times the wavelength of the 3D structure. $L_z \geq 2\lambda_z$
- Resolution: For DNS, $\Delta x \approx \Delta z \approx \delta_{\omega,0} / 10$. $\Delta y_{min} \approx \delta_{\omega,0} / 20$
Convection Term Scheme
What scheme is good for the convection term?
Low-dissipation schemes are essential for mixing layer LES/DNS.
- DNS: Central differencing (2nd or 4th order). Energy-conserving schemes are ideal
- LES: Bounded Central Differencing (Fluent), LUST (OpenFOAM), Central Differencing + small amount of upwind
- RANS: Second Order Upwind is sufficient
If you use upwind differencing, the vortices disappear, right?
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