Unlock Deeper Flow Insights with the V2-f Turbulence Model

Key Takeaways

• A ‘code-friendly’ variant of the model is proposed to enhance numerical stability.
• It has proven advantageous if an implicit segregated equation-by-equation approach is used. This modification alleviates the ‘stiffness problem’ associated with the original model of Durban caused by the boundary conditions at walls.
• This is particularly true when the Reynolds number is high, and the near-wall grid spacing is extremely small.

Unlocking the Potential of the V2 - f Turbulence Model for Complex Flows

In high-speed aerodynamics, studies utilizing second-moment closure for shock/boundary layer interaction are scarce. This scarcity stems from Reynolds stress equations lacking explicit second-order diffusion terms, leading to poor numerical stability. (e.g., Batten et al., 1997; Ha Minh and Vandromme, 1986; Lien and Leschziner, 1993). The issue worsens when all Reynolds stress components are stored at precise locations, resulting in velocity-stress decoupling and notorious checkerboard oscillations.

A significant challenge in implementing second-moment closure numerically comes from the complexity of setting boundary conditions on curved surfaces. Most low-Reynolds number (Re) second-moment closure models, such as those developed by Craft and Launder (1996), are highly sophisticated but have only been validated for simpler flow scenarios. Consequently, connecting high-Re models (Gibson and Launder 1978) with standard low-Re eddy viscosity models is expected when the y+ value is approximately 60, ensuring smoother transitions in turbulent flow simulations.

However, most low-Re      models reviewed by Patel et al. (1985) incorporate an ad-hoc viscous damping function  within the eddy-viscosity expression, i.e., 

The damping function corrects the improper asymptotic behavior of the eddy viscosity formulation when approaching a solid wall. It is often non-linear and causes numerical stiffness. Durbin (1991) suggested that the turbulence stress normal to streamlines, rather than k, should be used in the equation mentioned above to represent the kinematic blocking by the wall.

As a result, the model eliminates the need for a viscous damping function. The quantity is derived from a transport equation, simplified from the second-moment closure theory. The pressure-strain term in the  equation plays a crucial role by redistributing turbulent energy near the wall, ensuring the correct turbulence levels, and restoring isotropy. This is achieved through an auxiliary elliptic relaxation equation, which provides a more accurate representation of near-wall turbulence behavior.

Understanding the Core Principles of the V2-f Turbulence Model

Durbin’s 1995 Breakthrough: Unveiling the Original V2-f Model for Precision Flow Analysis

The Boussinesq approximation revolutionizes the stress-strain relationship for enhanced accuracy.

The eddy-viscosity is given by:

The turbulent time scale is given by:

The turbulent length scale is given by:

The turbulent time and length scale are determined from the standard k-ɛ equations:

The strain rate magnitude is defined as:

The production of turbulent kinetic energy is defined as:

Denoting with y the coordinate normal to the wall, on no-slip boundaries,  , yields

The above equation highlights how turbulence energy is redistributed from streamwise components, capturing non-local effects through an elliptic relaxation equation for the function f. This approach ensures a precise and realistic depiction of spatial interactions within the flow, offering deeper insights into complex turbulence dynamics.

The asymptotic behavior of the pressure-strain and the dissipation term  near a wall can be described as follows:

This yields the boundary condition for  in the original  model are:

The coefficients of the original model are as follows:

Where d is the distance to the wall.

Elliptic Relaxation Techniques: Code-Friendly Innovation in Turbulence by Lien and Durban, 1996; Lien et al., 1998

The boundary condition for  incorporates a  term raised to the fourth power, with  elegantly appearing in the denominator. This poses a challenge in the laminar and transitional regions, where the definition of  is ill-posed, giving rise to numerical oscillations. These oscillations become notably pronounced when a segregated numerical procedure is employed, as it prevents the implicit coupling between  and  near the wall. This inspired a reformulation of the terms in the transport equations:

Which redefine  , resulting in a modification of the elliptic relaxation equation to better capture the flow dynamics.

The boundary condition for  at the wall results in , which greatly enhances the numerical stability.

Note that such modifications retain the same asymptotic near-wall behavior of  as in the original model:

As , the kinematic blocking effect arising from elliptic relaxation disappears.

The constants of the resulting code-friendly model are denoted as:

References

Batten, P.E., Loyau, H., Leschziner, M.A., 1997. In: Proceedings of the Workshop on Shock/Boundary-Layer Interaction, UMIST, UK.

Ha Minh, H., Vandromme, D.D., 1986. Modeling of compressible turbulent flows: present possibilities and perspectives. In: Proceedings of the Shear Layer/Shock Wave Interaction, IUTAM Symposium, Palaiseau, Springer, Berlin, p. 13.

Lien, F.S., Leschziner, M.A., 1993. A pressure-velocity solution strategy for compressible flow and its application to shock/boundary-layer interaction using second-moment turbulence closure. ASME J. Fluid Eng. 115, 717.

Durbin, P.A., 1995. Separated flow computations with the k-ɛ-v2 model. AIAA J. 33, 659.

Durbin, P., 1996. On the k-ɛ stagnation point anomaly. Int. J. Heat Fluid Flow 17, 89.

Lien, F.S., Durbin, P.A., 1996. Non-linear k-ɛ-v2 modeling with application to high lift. In: Proceedings of the Summer Program 1996, Stanford University, p. 5. 

Lien, F.S., Kalitzin, G., Durbin, P.A., 1998. RANS modeling for compressible and transitional flows. In: Proceedings of the Summer Program 1998, Stanford University, p. 267.

Lien, F.S., Kalitzin, G., Computations of transonic flow with the v2 - f turbulence model.