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Applying Reynolds Stress Equation Model to Turbulence Flow

Key Takeaways

  • Overview of Reynolds stress equation.

  • Applying the Reynolds stress equation model to turbulent fluid flow.

  • CFD tools for analyzing turbulent flow models.

Turbulence can be caused by a number of factors

Example of turbulent air flow

Both automobiles and aircraft are subject to a turbulent flow regime. Unfortunately, turbulence, which is caused by changes in the parameters of the airflow that passes over and around vehicles on the ground and planes in the air, is almost impossible to avoid. This is due to the fact that there are so many causes of turbulent fluid--or air--flow. For example, variation in wind velocity and/or pressure, which may be due to some physical obstruction, can easily be created. To mitigate these effects, it is best to rely on proven analysis techniques such as the Reynolds stress equation.

What Is the Reynolds Stress Equation?

When designing systems that are subject to airflow mechanics, being able to accurately model fluid flow, especially turbulence, is critical. Common turbulence models include the following:

  • k - 𝛆 model
  • k - 𝛚 model
  • Mentor’s shear stress transport (SST) model

Any of these methods may be broadly applied; however, they have limitations. The SST model is more applicable for aerospace applications. The k - 𝛆 and k - 𝛚 methods are 2-D modeling techniques, with the latter used as closure for the RANS equations. And, these models have difficulty when there are recirculating air flows--eddies--or flow separation at the boundary layer. Therefore, a more generally applicable model is needed.

The Reynolds stress equation and model, or RSM, are shown below. 

Reynolds stress equation components

Reynolds stress equation. From SlideToDoc.

The RSM fully utilizes the Reynolds stress transport equation as shown above. Moreover, as the rearrangement illustrates, the total stress tensor can be evaluated as individual components.

Applying the Reynolds Stress Equation Model to Turbulent Fluid Flow

In terms of accuracy for modeling energy cascades where there is significant eddy activity, the RSM outperforms other less complex methods. However, there are pros and cons to consider when opting to apply this method for your turbulence modeling.

Advantages of the Reynolds Stress Equation Model

  1. Only the initial and boundary conditions are needed. 
  2. All Reynolds transport equation components are solved.
  3. Most generally applicable method.
  4. Can be used to solve complex 3-D problems.
  5. Stress due to buoyancy and curvature effects can be reduced. 

Disadvantages of the Reynolds Stress Equation Model

  1. Complex and computationally costly.
  2. Can have divergence issues.

As shown above, the RSM model is comprehensive and can be widely applied. This makes the method a robust solution for turbulent flow modeling. However, computing resources and the establishment of a well-founded model are necessary to avoid or mitigate the negatives associated with utilizing the Reynolds stress equation model. Therefore, your choice of CFD tool is of significant importance. 

CFD Tools for Analyzing Turbulent Flow Models

When choosing a CFD solver, there are a number of considerations that should guide your selection.

  • ✅Are tool capabilities broad enough for my design requirement expectations?
    There are several fluid flow models and, depending upon your application, the functionality
    and capability needed may vary. Therefore, you need to ensure that your solver is robust enough to accommodate your immediate and future needs.
  • ✅Are there multiple solver options?  
    In conjunction with capabilities is the availability of different solution type options. The ability to apply the right solution method can improve efficiency and save development costs.
  • ✅Is support sufficient (i.e., training/example videos, online support, user community, etc.)?
    Implementing the Reynolds stress equation model can be a complex undertaking. Therefore, it is important that you have access to support to increase the slope of any learning curve involved.
  • ✅ Are CFD solutions easy to integrate into designs?
    Another important consideration is how easy it is to transform your CFD solutions into usable data or a format that can be integrated into your system design. 

Although it is the most robust solution for turbulence evaluation, the Reynolds stress equation model requires an advanced CFD tool like Cadence’s Omnis to fully leverage its advantages.  

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