Reynolds Stress Models

The Reynolds stress model (RSM) determines the turbulent stresses by solving a transport equation for each stress component.

The RSM accounts for the effects of flow history and streamline curvature, as well as system rotation and stratification (Wilcox, 2000). Reynolds stress models are known to give superior results over one and two equation models when dealing with flows with streamline curvature, flows with sudden change in strain rate, and flows with secondary motions, all at the cost of an increased computing time (Bradshaw, 1997). There are many types of Reynolds stress models, the two most common being the RSM based on the dissipation rate (ε) and the RSM based on the eddy frequency (ω). In this section the dissipation rate (ε) model is discussed.

Transport Equations

Reynolds Stresses

$\bar{u'_{i} u'_{j}}$

(1)

$\frac{\partial (ρ \bar{u'_{i} u'_{j}})}{\partial t} + \frac{\partial (ρ \bar{u_{k}} \bar{u'_{i} u'_{j}})}{\partial x_{k}} = P_{R} + D_{R} + Π_{i j} + D_{R M} + D_{R T}$

Turbulent Dissipation Rate ε

(2)

$\frac{\partial (ρ ε)}{\partial t} + \frac{\partial (ρ \bar{u_{j}} ε)}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] + P_{ε} + D_{ε}$

Production Modeling

Reynolds Stresses

$\bar{u'_{i} u'_{j}}$

(3)

$P_{R} = - ρ (\bar{u'_{i} u'_{k}} \frac{\partial \bar{u_{j}}}{\partial x_{k}} + \bar{u'_{j} u'_{k}} \frac{\partial \bar{u_{i}}}{\partial x_{k}})$

Turbulent Dissipation Rate ε

(4)

$P_{ε} = C_{ε 1} \frac{ε}{k} μ_{t} S^{2} = C_{ε 1} \frac{ε}{k} P_{k}$

Dissipation Modeling

Reynolds Stresses

$\bar{u'_{i} u'_{j}}$

(5)

$D_{R} = - 2 μ (\frac{\partial \bar{u_{i}}}{\partial x_{k}} \frac{\partial \bar{u_{j}}}{\partial x_{k}})$

Turbulent Dissipation Rate ε

(6)

$D_{ε} = - C_{ε 2} ρ \frac{ε^{2}}{k}$

Molecular Diffusion Modeling

Reynolds Stresses

$\bar{u'_{i} u'_{j}}$

(7)

$D_{R M} = μ \frac{\partial}{\partial x_{k}} (\frac{\partial \bar{u'_{i} u'_{j}}}{\partial x_{k}})$

Turbulent Diffusion Modeling

Reynolds Stresses

$\bar{u'_{i} u'_{j}}$

(8)

$D_{R T} = \frac{μ_{t}}{ρ σ_{k}} \frac{\partial}{\partial x_{k}} (\frac{\partial \bar{u'_{i} u'_{j}}}{\partial x_{k}})$

Pressure-Strain Modeling

Reynolds Stresses

$\bar{u'_{i} u'_{j}}$

(9)

$Π_{i j} = Π_{i j, s} + Π_{i j, f} + Π_{i j, w}$

where $Π_{i j, s} = - C_{1} \frac{ρ ε}{k} (\bar{u'_{i} u'_{j}} - \frac{2}{3} k δ_{i j})$ , $Π_{i j, f} = Π_{i j, f 1} + Π_{i j, f 2}$ , $Π_{i j, f 1} = - C_{2} (- ρ (\bar{{u^{'}}_{i} {u^{'}}_{k}} \frac{\partial \bar{u_{j}}}{\partial x_{k}} + \bar{{u^{'}}_{j} {u^{'}}_{k}} \frac{\partial \bar{u_{i}}}{\partial x_{k}}) - \frac{\partial}{\partial x_{k}} (ρ \bar{u_{k}} \bar{{u^{'}}_{i} {u^{'}}_{j}}))$ , $Π_{i j, f 2} = - \frac{1}{3} δ_{i j} (- ρ (\bar{{u^{'}}_{k} {u^{'}}_{k}} \frac{\partial \bar{u_{k}}}{\partial x_{k}} + \bar{{u^{'}}_{k} {u^{'}}_{k}} \frac{\partial \bar{u_{k}}}{\partial x_{k}}) - \frac{\partial}{\partial x_{k}} (ρ \bar{u_{k}} \bar{{u^{'}}_{k} {u^{'}}_{k}}))$ , $Π_{i j, w} = Π_{i j, w 1} + Π_{i j, w 2}$ , $Π_{i j, w 1} = C_{3} \frac{ε_{n c}}{k_{n c}} (\bar{{u^{'}}_{k} {u^{'}}_{m}} η_{k} η_{m} δ_{i j} - \frac{3}{2} \bar{{u^{'}}_{i} {u^{'}}_{k}} η_{j} η_{k} - \frac{3}{2} \bar{{u^{'}}_{j} {u^{'}}_{k}} η_{i} η_{k}) \frac{k^{3 / 2}}{C_{w} ε χ}$ , $Π_{i j, w 2} = C_{4} (Π_{k m, f} η_{k} η_{m} δ_{i j} - \frac{3}{2} Π_{i k, f} η_{j} η_{k} - \frac{3}{2} Π_{j k, f} η_{i} η_{k}) \frac{k^{3 / 2}}{C_{w} ε χ}$

where $η_{k}$ is the $x_{k}$ component of the unit normal to the wall and $χ$ is the normal distance from the wall.

Modeling of Turbulent Viscosity $μ_{t}$

(10)

$μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}$

Model Coefficients

$C_{ε 1}$ = 1.44, $C_{ε 2}$ = 1.92, $C_{μ}$ = 0.09, $σ_{k}$ = 1.0, $σ_{ε}$ = 1.0, $C_{1}$ = 1.8, $C_{2}$ = 0.6, $C_{3}$ = 0.5, $C_{4}$ = 0.3, $C_{w}$ = $\frac{C_{μ}^{3 / 4}}{κ}$

Reynolds Stress Models

Transport Equations

Production Modeling

Dissipation Modeling

Molecular Diffusion Modeling

Turbulent Diffusion Modeling

Pressure-Strain Modeling

Modeling of Turbulent Viscosity μt<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>μ</mi> <mi>t</mi> </msub> </mrow> </math>

Model Coefficients

Modeling of Turbulent Viscosity $μ_{t}$