The Mixing Length Model in Computational Fluid Dynamics (CFD) is a fundamental method attempting to describe momentum transfer by turbulence Reynolds stresses within a Newtonian fluid boundary layer by means of an eddy viscosity. Developed by the renowned fluid dynamicist Ludwig Prandtl in the early 20th century, this model provides a simplified yet effective way to account for the effects of turbulence in fluid flows.
Understanding the Core Concept
At its heart, the mixing length model aims to bridge the gap between complex turbulent flows and their computational modeling. Turbulent flows are characterized by chaotic, fluctuating velocity components that significantly enhance momentum, heat, and mass transfer compared to laminar flows. Directly simulating these fluctuations (Direct Numerical Simulation - DNS) is computationally prohibitive for most engineering applications. Therefore, turbulence models are employed to represent the average effects of turbulence.
Reynolds Stresses and Eddy Viscosity
In turbulent flows, the time-averaged Navier-Stokes equations introduce additional terms known as Reynolds stresses. These stresses represent the apparent increase in stress due to momentum transfer by the fluctuating velocity components. The mixing length model provides a closure for these Reynolds stresses.
The model achieves this by introducing the concept of eddy viscosity (also known as turbulent viscosity, $\mu_t$). Just as molecular viscosity ($\mu$) describes momentum transfer due to random molecular motion in laminar flow, eddy viscosity describes the enhanced momentum transfer due to the macroscopic "mixing" or eddies in turbulent flow. Unlike molecular viscosity, which is a fluid property, eddy viscosity is a property of the flow itself and varies spatially.
The relationship can be expressed by Boussinesq's hypothesis:
$\tau_t = \mu_t \frac{dU}{dy}$
where $\tau_t$ is the turbulent shear stress, $\mu_t$ is the eddy viscosity, and $\frac{dU}{dy}$ is the mean velocity gradient.
The "Mixing Length" ($l_m$)
The model's name comes from its central idea: the mixing length ($l_m$). This concept is analogous to the mean free path in the kinetic theory of gases. It represents the average distance a parcel of fluid travels transverse to the mean flow direction before it loses its original momentum identity by mixing with the surrounding fluid.
Prandtl proposed that the eddy viscosity ($\mu_t$) could be related to the mixing length and the local mean velocity gradient:
$\mu_t = \rho l_m^2 \left| \frac{dU}{dy} \right|$
where $\rho$ is the fluid density.
The crucial challenge and limitation of the mixing length model lie in defining an appropriate value for $l_m$. It is not a constant but varies with position, especially within a boundary layer. For simple boundary layers, $l_m$ is often assumed to be proportional to the distance from the wall in the near-wall region and then to reach a constant value further away.
Applications in CFD
In CFD, the mixing length model is a simple yet foundational turbulence model. It is often used as a starting point for understanding more complex models or for flows where its assumptions are reasonably met.
- Boundary Layer Flows: It is particularly effective for simple, two-dimensional, incompressible boundary layer flows with a dominant shear direction, such as flow over a flat plate or through a pipe.
- Historical Significance: Many modern, more sophisticated turbulence models (like K-epsilon, K-omega) have roots in the concepts pioneered by Prandtl's mixing length theory.
- Computational Efficiency: Being an algebraic model (i.e., not involving transport equations for turbulence quantities), it is computationally very inexpensive, making it suitable for quick estimations or educational purposes.
Advantages and Limitations
While historically significant and computationally efficient, the mixing length model has distinct advantages and notable limitations:
| Aspect | Description |
|---|---|
| Advantages | - Simplicity and Efficiency: It's an algebraic model, meaning it doesn't require solving additional partial differential equations for turbulence quantities, making it computationally very cheap. - Good for Simple Flows: Provides reasonably accurate predictions for simple, fully-developed, one-dimensional shear flows (e.g., pipe flow, flat plate boundary layers). - Conceptual Basis: Offers an intuitive understanding of eddy viscosity and turbulent mixing. |
| Limitations | - Non-Universality of $l_m$: The mixing length $l_m$ is not a universal constant and must be prescribed empirically for each flow type. This makes it difficult to apply to complex geometries or flows without prior knowledge. - Flow History Independence: It's a "local" model, meaning the eddy viscosity at any point depends only on the mean flow properties at that point. It does not account for the upstream history of the flow, which is crucial for complex flows like those with separation, reattachment, or recirculation. - Anisotropy: It assumes an isotropic eddy viscosity, which is often not true in real turbulent flows where turbulent stresses are anisotropic. - Not Suitable for Complex Flows: Fails to accurately predict flows with strong pressure gradients, swirl, separation, recirculation, or three-dimensional effects. |
Historical Context
The mixing length model was a pioneering effort by Ludwig Prandtl in the early 20th century. His work laid the groundwork for modern turbulence modeling by providing a practical framework for incorporating turbulent effects into the momentum equations, long before the advent of powerful computers. It remains a foundational concept in the study of turbulent fluid mechanics and CFD.
