RANS model overview

model categories

1. eddy viscosity models
   1. linear, algebraic
   2. linear, PDE-based
   3. non-linear, PDE-based
2. stress models
   1. algebraic (3 PDEs)
   2. Reynolds stresses (7 PDEs)

historical perspective III

• 1975: Launder-Reece-Rodi (LLR) RSM
  B. E. Launder, J. G. Reece, W. Rodi
• 1991: Speziale-Sarker-Gatski (SSG) RSM
  C. G. Speziale, S. Sarkar, T. B. Gatski
• 1996: Lien-Cubic $k$-$\varepsilon$ model
  F. S. Lien, W. L. Chen, M. A. Leschziner
• ...

Reynolds stress models (RSM)

approach to derive equations (Wilcox p. 41)

$$ \overline{u^\prime_i N(u_j) + u^\prime_j N(u_i)} = 0$$

$$N(u_i) = \partial_t u_j + \partial_i (u_iu_j) + \partial_j p - \partial_i (\nu (\partial_i u_j + \partial_j u_i)) = 0$$

favorable attributes

• improved performance in some flows
• most general RANS models

unfavorable attributes

• additional cost (7 PDEs)
• less validated than eddy viscosity models
• issues due to $\varepsilon$ equation
• numerical robustness

non-linear eddy viscosity models

viscoelastic analogy

$$\overline{u_i^\prime u_j^\prime} = f(\overline{S}_{ij}, k, \varepsilon, \dots)$$

$$\overline{u_i^\prime u_j^\prime} = f(\overline{S}_{ij}, D_t\overline{S}_{ij}, k, \varepsilon, \dots)$$

modeling guided by concepts of frame invariance and realizability

generalized eddy viscosity hypothesis

$$ \begin{aligned} -\overline{u_i^\prime u_j^\prime} &= 2\nu_t \overline{S}_{ij} - \frac{2}{3} k \delta_{ij}\\ &- C_1\nu_t \frac{k}{\varepsilon}(\overline{S}_{ik}\overline{S}_{kj}-\frac{1}{3}\overline{S}_{kl}\overline{S}_{kl}\delta_{ij})\\ &- ... \end{aligned} $$

quadratic and cubic expansions available

favorable attributes

• improved performance in some flows
• less expensive than RSM (2-3 PDEs)

unfavorable attributes

• less validated (potential calibration necessary)
• not always better than linear counterparts
• stability issues

summary RANS models

• zoo of linear, non-linear, Reynolds stress models
• model complexity tends to cost numerical stability
• standard for high $Re$ flows (think full aircraft)
• starting point for moderate $Re$ investigations