# Assumptions and Insights of k-epsilon Low Re Yang-Shih Turbulence Model

**Key Takeaways**

- A new time scale-based k-epsilon (k-ɛ) model for near-wall turbulence is discussed. In this model, is chosen as the turbulent velocity scale.
- The Kolmogorov time scale bounds the time scale. When this time scale is used to reformulate the dissipation equation, there is no singularity at the wall, and introducing a pseudo-dissipation rate is avoided.
- A damping function, proposed as a function of instead of , as in Lam et al. 1981, allows the present model to be used for separated flows.

The k-ɛ model is one of the most widely used turbulence models. Although not very efficient in cases involving large adverse pressure gradients, it is a two-equation model, incorporating two additional transport equations to describe the turbulent properties of the flow (Wilcox et al. 1998). The number of equations denotes the number of additional partial differential equations (PDEs) being solved. This enables the model to account for historical effects such as the convection and diffusion of turbulent energy.

The first transported variable is the turbulent kinetic energy, k, which determines the energy in turbulence. The second transported variable is the turbulent dissipation, ε, which determines the scale of turbulence. There are two significant formulations of k-ɛ models (Jones et al. 1972 and Launder et al. 1974). The formulation by Launder et al. 1974 is typically called the standard k-ɛ model. The original motivation for developing the k-ɛ model was to improve the mixing length model and provide an alternative to algebraically describing turbulent length scales in moderate to highly complex flows. The k-ɛ model is useful for free shear layer flows with relatively small pressure gradients (Bardina et al. 1997). Similarly, the model produced good results for wall-bounded and internal flows only when the mean pressure gradients were small. Experimental evidence indicates that its accuracy diminishes for flows containing large adverse pressure gradients.

The standard k-ɛ model, designed for high Reynolds number turbulent flows, is traditionally used with a wall function for wall-bounded turbulent flows. However, no universal wall function exists for complex flows. Consequently, a low Reynolds number (LRN) Yang-Shih k-ɛ model was introduced to extend the model's applicability down to the wall.

## Approaches for the Efficient Implementation of the k-ɛ Turbulence Model

The transport equations for turbulent kinetic energy, k, and turbulent dissipation rate, ɛ, are modeled as follows:

where P is the turbulence production, T is the turbulent time scale, E is a term specific to the Yang-Shih, 1993, and is the Yap correction. The model constants are mostly the same as those in the standard k*-ɛ *model. Away from the wall, the present model reduces to the standard k-ɛ model. Thus, it is only necessary to assess the performance of the model for near-wall turbulence. Denoting the trace of a matrix product by , the production term is defined as follows:

where is the turbulent Reynolds stress tensor, and is the mean flow strain tensor.

NOTE: In the Cadence Fidelity Open Solver, the incompressible formulation of the production is considered, even when compressible flows are simulated. This assumption guarantees the production term never becomes negative since

A singularity would arise if the standard k-ɛ model were applied all the way to the wall due to the vanishing of turbulent kinetic energy, k, at the wall, which causes the time scale in the dissipation equation to become zero. To overcome this, Yang-Shih, 1993 used a modified time scale. The turbulent length scale is defined by the size of the energy-containing eddies, which, near the wall, would have a size of . Furthermore, the turbulent velocity field has the following expansions near the wall, as Hanjalic et al. 1976 demonstrated.

where are nonzero in general.

As the wall approaches, the turbulent length and velocity scales approach zero. However, the turbulent timescale, defined as the ratio of the length scale of energy-containing eddies to the turbulent velocity scale, remains non-zero and matches the Kolmogorov timescale due to dominant viscous dissipation near the wall. Therefore, the turbulent time scale is given by away from the wall and by the Kolmogorov time scale near the wall. Since is much larger than the Kolmogorov time scale away from the wall and vanishes near the wall due to the boundary condition on k.. The modified turbulent time scale is given by:

is the Kolmogorov time scale and is a constant of order one.

Now the time scale given by the modified turbulent time scale equation is bounded by the Kolmogorov time scale that is always positive. When this time scale is used in the dissipation equation, there will be no singularity at the wall. A Kolmogorov-Prandtl type formula defines the turbulent eddy viscosity .

where is the damping function that is used to account for the wall effect.

The damping function in the present model is taken to be a function of , which is defined as follows:

and takes the following form:

These constants are devised by comparing the performance of the model prediction and the direct numerical simulation (DNS) data for turbulent channel flow at

The damping function's use of instead of offers broader flow situation applicability, including separation and reattachment. However, this makes the model non-Galilean invariant. In complex geometries, like corner flows, y's meaning is unclear. As per Yang et al. 1993, in real-world scenarios, the non-Galilean invariance is not an issue when utilizing a body-fitted coordinate system where represents the wall-normal direction. For corner flows, the ambiguity only arises in very close proximity to the corner, where k is small, and y's precise definition is inconsequential.

In near-wall turbulence, the mean field's inhomogeneity creates a secondary source term in the dissipation equation, aside from the turbulence itself. The secondary source term E is given by the following equation:

Yap correction (Yap. C. J. 1987) modifies the epsilon equation by adding a source term, , to the right-hand side. The source term is written as:

The Yap correction, active in non-equilibrium flows, stabilizes the turbulence length scale, often enhancing predictions. It notably improves results with the k-ɛ in separated flows and stagnation regions. When implementing the Yap correction, it is only common to activate the correction when the source term is positive. Hence:

## References

- Wilcox, David C (1998). "Turbulence Modeling for CFD". Second edition. Anaheim: DCW Industries, 1998. pp. 174.
- Jones, W. P., and Launder, B. E. (1972), "The Prediction of Laminarization with a Two-Equation Model of Turbulence", International Journal of Heat and Mass Transfer, vol. 15, 1972, pp. 301-314.
- Launder, B. E., and Sharma, B. I. (1974), "Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disc", Letters in Heat and Mass Transfer, vol. 1, no. 2, pp. 131-138.
- Bardina, J.E., Huang, P.G., Coakley, T.J. (1997), "Turbulence Modeling Validation, Testing, and Development," NASA Technical Memorandum 110446.
- Lam, C. K. G., and Bremhorst, K., (1981)"A Modified Form of the k-e Model for Predicting Wall Turbulences," Journal of Fluids Engineering, Vol. 103, No. 3, pp. 456-460.
- Yang Z. and Shih T. H., (1993), "A k-epsilon model for turbulence and transitional boundary layer", Near-Wall Turbulent Flows, R.M.C. So., C.G. Speziale and B.E. Launder(Editors), Elsevier-Science Publishers B. V., pp. 165-175.
- Hanjalic, K., and Launder, B. E., (1976) "Contribution Towards a Reynolds-Stress Closure for Low-Reynolds-Number Turbulence," Journal of Fluid Mechanics, Vol. 74, Pt. 4, pp. 593-610.
- Yap, C. J. (1987),
*Turbulent Heat and Momentum Transfer in Recirculating and Impinging Flows*, PhD Thesis, Faculty of Technology, University of Manchester, United Kingdom.