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 II

• 1974: $k$-$\varepsilon$ model
  B. E. Launder, B. I. Sharma
• 1988: $k$-$\omega$ model
  D. C. Wilcox
• 1992: Spalart-Allmaras model
  P. R. Splart, S. R. Allmaras
• 1994: $k$-$\omega$-SST model
  F. R. Menter

$k$-$\varepsilon$ model

transport equation of (specific) turbulence kinetic energy

1. trace of Reynolds stress transport equation
2. multiplication of Navier-Stokes and RANS equations and with velocity fluctuations; summation of components; subtraction; rearrangement

starting point

$$\nu_t \propto l_c u_c$$

velocity scale $u_c = k^{0.5}$

length scale $l_c = k^{1.5}/\varepsilon$

$\rightarrow \nu_t = C_\mu k^2/\varepsilon$

additional modeling

• $k$ equation terms
• $\varepsilon$ transport equations and terms

boundary conditions (high $Re$)

• $\nu_t$ - log-law, Spalding's function
• $k$ - zero gradient ($\nabla k \cdot \mathbf{n}=0$)
• $\varepsilon$ - fixed value ($\varepsilon_{log} = C_\mu k^{1.5}/\nu_t$)

inlet conditions

• ideally from experiments
• estimate $k = 1.5 (IU_{in})^2$
• estimate $\varepsilon = C_\mu^{0.75}k^{1.5}/l_c$

favorable attributes

• simple and numerically robust
• only free stream conditions needed
• good accuracy in many cases
• extensively validated

bad performance in certain flows

• flows in which wall functions are not applicable
• unconfined flows (jet)
• flows with large extra strain (curved boundaries, swirls - think tornados)
• rotating flows (fans, turbo machinery)
• flows driven by anisotropy of Reynolds stresses (fully developed flows in square duct)

$k$-$\omega$ model

starting point

$$\nu_t \propto l_c u_c$$

velocity scale $u_c = k^{0.5}$

turbulence frequency $\omega = \varepsilon / k$

length scale $l_c = k^{0.5}/\omega$

$\rightarrow \nu_t = k/\omega$

favorable attributes

• similar to $k$-$\varepsilon$ model
• integration of wall layer possible

unfavorable attributes

• similar weaknesses as $k$-$\varepsilon$ model
• sensitivity to free stream values

$k$-$\omega$-SST model

starting point $\nu_t = k/\omega$

observation: plugging definition of $\varepsilon = k\omega$ into transport equation of $\varepsilon$ yields $\omega$ transport equation with additional terms

idea: combine favorable attributes of both models

favorable attributes

• combines the best of $k$-$\varepsilon$ and $k$-$\omega$ models
• very well validated

$\rightarrow$ default model of most practitioners

unfavorable attributes in certain flows remain

summary

• based on mixing length idea
• mostly empirical modeling of transport equations
• many weaknesses related to assumption of isotropic eddy viscosity
• many variants exist $\rightarrow$ check implementation and/or documentation before usage