The Predictions of Common Turbulence Models in a Mature Vortex

Abstract

A range of turbulence models is exercised in the problem of a mature two-dimensional isolated vortex with high circulation Reynolds number, mature meaning that the core size has grown to many times its initial value, and the vortex reaches a self-similar state in which all quantities scale with its outside circulation Γ and its age t. The reasoning closely follows the landmark papers of Govindaraju and Saffman (Phys. Fluids 14, 2074–2080, 1971) (GS) and Zeman (Phys. Fluids A 7, 135–143, 1995), and we call this the GSZ flow. The models produce radically different outcomes, and this simple problem clearly places each model into one of two classes, in terms of its predictions in vortices. Some reach a turbulent state, including the circulation overshoot first predicted by GS, whereas others reach a laminar state free of overshoot. We have no evidence of non-unique solutions or bifurcations for a given model. The overshoot consists in the circulation, as a function of radius r, rising from zero on the axis at r = 0 to values that exceed the outside circulation Γ, and then falling back down. This happens within the turbulent region, and it is associated with the creation by turbulence of mean vorticity of sign opposite to the core vorticity. We consider the second (i.e., the laminar) response as the correct one, the circulation overshoot appearing to us unphysical, a thought which was clear in both of the original papers and is supported by experimental and simulation results. The laminar state is given by eddy-viscosity models only when they include a rotation/curvature correction. A Reynolds-stress transport model and an Explicit Algebraic Reynolds-Stress Model return turbulent solutions but with weak turbulence, which we consider a positive result, although Zeman’s Reynolds-stress model prevented any overshoot and led to a laminar mature flow. We believe this flow can become a relevant, well-defined, standard test case for turbulence models.

Appendix

The analysis of the laminar-turbulent interface for the GSZ flow and the Spalart-Allmaras model is as follows. It involves the velocity u_𝜃(r, t) and the eddy viscosity ν_t(r, t), the momentum equation introduced above

$$r^{2}\frac{\partial u_{\theta } }{\partial t}=\frac{\partial }{\partial r}\left[ {r^{2}\nu_{t} \left( {\frac{\partial u_{\theta } }{\partial r}-\frac{u_{\theta } }{r}} \right)} \right] $$

and the model equation ([4], with the passive terms removed)

$$\frac{\partial \nu_{t} }{\partial t}=c_{b1} \left| \omega \right|\nu_{t} +\frac{1}{\sigma }\left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {r\nu_{t} \frac{\partial \nu_{t} }{\partial r}} \right)+c_{b2} \left( {\frac{\partial \nu_{t} }{\partial r}} \right)^{2}} \right] $$

The interface is propagating at a velocity c = dr₁/dt, so that to leading order a time derivative such as $\frac {\partial \nu _{t} }{\partial t}$ is equal to $-c\frac {\partial \nu _{t} }{\partial r}$. We assume that both the deviation from an irrotational vortex field, u_𝜃 −Γ/(2πr), and the eddy viscosity ν_t, approach zero at r = r₁ linearly (write u_𝜃 −Γ/(2πr) = A (r₁ − r) and ν_t = B (r₁ − r)). Thus, $\frac {\partial \nu _{t} }{\partial t}=cB$. The slopes A and B of these functions introduce two more unknowns besides c.

Once these distributions are introduced into the governing equations, the production (c_b1) term vanishes at r = r₁, and the model equation only involves the time derivative and diffusion terms much like in the original SA paper [11]. Then, both c and B drop out of the equations, which produces A as a function of Γ, r₁, and the model constants.

$$A=\frac{\sigma }{1+c_{b2} -\sigma }\frac{{\Gamma} }{\pi {r_{1}^{2}} } $$

The vorticity is actually –A.

$$\omega =\frac{1}{r}\frac{\partial \left( {r u_{\theta }} \right)}{\partial r} =-\frac{\sigma }{1+c_{b2} -\sigma }\frac{{\Gamma} }{\pi{r_{1}^{2}} } $$