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 I

• 1877: Boussinesq hypothesis (eddy viscosity)
  Essay on the theory of running water
• 1894: Reynolds decomposition
  On the dynamical theory of incompressible viscous fluids and the determination of the criterion
• 1925: Prandtl's mixing length
  Report on the studies of developed turbulence

eddy viscosity hypothesis

RANS equation (momentum)

$$ \partial_t \overline{\mathbf{u}} + \nabla\cdot (\overline{\mathbf{u}}\overline{\mathbf{u}}) = -\nabla \overline{p} + \nabla\cdot \left(\nu \left(\nabla\overline{\mathbf{u}}+\nabla\overline{\mathbf{u}}^T \right) - \overline{\mathbf{u}^\prime\mathbf{u}^\prime}\right) $$

assumptions: incompressible, const. density, Newtonian fluid

mean strain rate tensor $$2\mathbf{S} = \nabla\overline{\mathbf{u}}+\nabla\overline{\mathbf{u}}^T$$

Reynolds stress tensor $$\mathbf{R} = - \overline{\mathbf{u}^\prime\mathbf{u}^\prime}$$

turbulence kinetic energy $$2k=-\mathrm{tr}(\mathbf{R})$$

observations

• increased loss/dissipation in turbulent flows
• turbulent stresses increase as the mean rate of deformation increases

$\rightarrow$ Boussinesq (1877): Reynolds stress might be proportional to mean rate of deformation

first trial

$$\mathbf{R}=-\overline{\mathbf{u}^\prime\mathbf{u}^\prime} = \nu_t 2\mathbf{S} = \nu_t \left(\nabla\overline{\mathbf{u}}+\nabla\overline{\mathbf{u}}^T\right)$$

$\nu_t$ - turbulent (eddy) viscosity

Issues?

$$ \mathbf{R} = \underbrace{-\overline{\mathbf{u}^\prime\mathbf{u}^\prime} + \frac{2}{3} k \mathbf{I}}_{\mathbf{R}_{dev}} - \underbrace{\frac{2}{3} k \mathbf{I}}_{\text{isotropic}} $$

$$ \mathbf{R}_{dev} = \nu_t 2 \mathbf{S} = \nu_t\left(\nabla\overline{\mathbf{u}}+\nabla\overline{\mathbf{u}}^T\right) $$

$$ \mathbf{R} = \nu_t 2 \mathbf{S} - \frac{2}{3} k \mathbf{I} $$

Which implicit assumption about $\nu_t$ have we made?

$$ D_t\overline{u} = -\nabla \overline{p} + \nabla\cdot \left(\nu 2 \mathbf{S} + \nu_t 2 \mathbf{S}-\frac{2}{3}k\mathbf{I}\right) $$ $$ D_t\overline{u} = -\nabla \tilde{\overline{p}} + \nabla\cdot \left(\nu_{eff} \left(\nabla\overline{\mathbf{u}}+\nabla\overline{\mathbf{u}}^T \right)\right) $$

with $\tilde{\overline{p}} = \overline{p}+\frac{2}{3}k$ and $\nu_{eff} = \nu + \nu_t$

passive scalar $\varphi$

$$ -\overline{u_i^\prime\varphi^\prime} = \Gamma_t \partial_i \overline{\varphi} $$

$\Gamma_t$ - turbulent/eddy diffusivity

$$ \partial_t \overline{\varphi} + \nabla\cdot (\overline{\mathbf{u}}\overline{\varphi}) = \nabla \cdot (\Gamma_{eff}\nabla \overline{\varphi}) $$

$\Gamma_{eff} = \Gamma + \Gamma_t$ - effective diffusivity

equivalence to molecular heat/mass transport

$Sc = \nu/D$ and $Pr = \nu/\alpha$

$$ \sigma_t = \nu_t / \Gamma_t $$

$\sigma_t$ - turbulent Schmidt/Prandtl number; around unity

How to determine $\nu_t$?

Prandtl's mixing length model

How do Reynolds stresses contribute to the mean momentum balance in a shear flow?

Prandtl's mixing length model (1925)

$$ \nu_t = l_m^2 |\partial_y \overline{u}_x | $$

Reynolds shear stress for shear layer

$$ R_{xy} = \nu_t \partial_y \overline{u}_x = l_m^2 |\partial_y \overline{u}_x | \partial_y \overline{u}_x $$

short comings

• not general (e.g., 2D, shear layer)
• case-dependent mixing length $l_m$

other algebraic eddy viscosity models

• Cebeci-Smith model (1967)
• Baldwin-Lomax model (1978)

distinction between inner and outer layer near walls; slightly more general; rarely used in practice nowadays; read more

summary

• Boussinesq: deviatoric part of turbulent stresses is proportional to mean strain rate
• eddy viscosity is isotropic (the same for all directions); often violated but not always critical
• Prandtl's mixing length approximation converts local unknown (eddy viscosity) into flow type dependent unknown (mixing length); basis for more advanced models