混合層
Theory and Physics
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 Theory—The 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.
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 unstable and splashing, but after a while, the flow becomes steady, right? This "during the change" is described by the temporal term. The pulsation of blood flow from a heartbeat, the fluctuation of flow each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, solving first in steady-state 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. The warm air from a heater reaching the far end of a room is also because the "carrier" air transports heat by convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, 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 is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever left milk in coffee? Even without stirring, after a while they naturally mix, 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 is dominant. Conversely, in high Re number flows, convection overwhelms and diffusion plays a supporting 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. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. When switching to compressible analysis, if results become strange, confusion between absolute/gauge pressure might be the cause.
- Source Term $S_\phi$: Heated 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 generated by 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 the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (non-Newtonian fluids require viscosity models)
- Incompressibility Assumption (for Ma < 0.3): Treat density 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 (VOF/Level Set, etc. required)
Dimensional Analysis and Unit System
| 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 pressure 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
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?
Related Topics
なった
詳しく
報告