# Favre averaged Navier-Stokes equations

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+ The terms marked with $(1^*)-(8^*)$ are unknown, and have to be modeled in some way. + + Term $(1^*)$ and $(4^*)$ can be modeled using an eddy-viscosity assumption for the Reynolds stresses, $\tau_{ij}^{turb}$: + +
+ :$+ \tau_{ij}^{turb} \equiv + - \overline{\rho u''_i u''_j} \approx + 2 \mu_t \widetilde{S_{ij}^*} - + \frac{2}{3} \overline{\rho} k \delta_{ij} +$ + (24)
+ + Where $\mu_t$ is a turbulent viscosity, which is estimated with a turbulence model. The last term is included in order to ensure that the trace of the Reynolds stress tensor is equal to $-2 \rho k$, as it should be. + + Term $(2^*)$ and $(8^*)$ can be neglected if: + + +
+ :$+ \left| \widetilde{\tau_{ij}} \right| >> + \left| \overline{\tau''_{ij}} \right| +$ + (25)
+ + This is true for virtually all flows. + + Term $(3^*)$, corresponding to turbulent transport of heat, can be modeled using a gradient approximation for the turbulent heat-flux: + +
+ :$+ q_j^{turb} \defeq + C_p \overline{\rho u''_j T} \approx + - C_p \frac{\mu_t}{Pr_t} \frac{\partial \widetilde{T}}{\partial x_j} +$ + (26)
+ + Where $Pr_t$ is a turbulent Prandtl number. Often a constant $Pr_t$ is used ($\approx 0.9$. researchers have reported no significant improvement when using an + algebraic expression for the variation of $Pr_t$ \cite{Boyle:91}.

## Instantaneuos Equations

The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:

 $\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_j}\left[ \rho u_j \right] = 0$ (1) $\frac{\partial}{\partial t}\left( \rho u_i \right) + \frac{\partial}{\partial x_j} \left[ \rho u_i u_j + p \delta_{ij} - \tau_{ji} \right] = 0$ (2) $\frac{\partial}{\partial t}\left( \rho e_0 \right) + \frac{\partial}{\partial x_j} \left[ \rho u_j e_0 + u_j p + q_j - u_i \tau_{ij} \right] = 0$ (3)

For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:

 $\tau_{ij} = 2 \mu S_{ij}^*$ (4)

Where the trace-less viscous strain-rate is defined by:

 $S_{ij}^* \equiv \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij}$ (5)

The heat-flux, $q_j$, is given by Fourier's law:

 $q_j = -\lambda \frac{\partial T}{\partial x_j} \equiv -C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}$ (6)

Where the laminar Prandtl number $Pr$ is defined by:

 $Pr \equiv \frac{C_p \mu}{\lambda}$ (7)

To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:

 $\gamma \equiv \frac{C_p}{C_v} ~~,~~ p = \rho R T ~~,~~ e = C_v T ~~,~~ C_p - C_v = R$ (8)

Where $\gamma$, $C_p$, $C_v$ and $R$ are constant.

The total energy $e_0$ is defined by:

 $e_0 \equiv e + \frac{u_k u_k}{2}$ (9)

Note that the corresponding expression (15) for Favre averaged turbulent flows contains an extra term related to the turbulent energy.

Equations (1)-(9), supplemented with gas data for $\gamma$, $Pr$, $\mu$ and perhaps $R$, form a closed set of partial differential equations, and need only be complemented with boundary conditions.

## Favre Averaged Equations

It is not possible to solve the instantaneous equations directly for the applications of interest here. At the Reynolds numbers typically present in a turbine these equations have very chaotic turbulent solutions, and it is necessary to model the influence of the smallest scales. All turbulence models used in this work are based on one-point averaging of the instantaneous equations. The averaging procedure used is described in the next sections.

### Averaging

Let $\Phi$ be any dependent variable. It is convenient to define two different types of averaging of $\Phi$:

• Classical time average (Reynolds average):
 $\overline{\Phi} \equiv \frac{1}{T} \int_T \Phi(t) dt$ (10) $\Phi' \equiv \Phi - \overline{\Phi}$
• Density weighted time average (Favre average):
 $\widetilde{\Phi} \equiv \frac{\overline{\rho \Phi}}{\overline{\rho}}$ (11) $\Phi'' \equiv \Phi - \widetilde{\Phi}$

Note that with the above definitions $\overline{\Phi'} = 0$, but $\overline{\Phi''} \neq 0$.

### Open Turbulent Equations

In order to obtain an averaged form of the governing equations, the instantaneous continuity equation (1), momentum equation (2) and energy equation (3) are time-averaged. Introducing a density weighted time average decomposition (10) of $u_i$ and $e_0$, and a standard time average decomposition (11) of $\rho$ and $p$ gives the following exact open equations:

 $\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0$ (12)
 $\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) + \frac{\partial}{\partial x_j} \left[ \overline{\rho} \widetilde{u_i} \widetilde{u_j} + \overline{p} \delta_{ij} + \overline{\rho u''_i u''_j} - \overline{\tau_{ji}} \right] = 0$ (13)
 $\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{e_0} \right) + \frac{\partial}{\partial x_j} \left[ \overline{\rho} \widetilde{u_j} \widetilde{e_0} + \widetilde{u_j} \overline{p} + \overline{u''_j p} + \overline{\rho u''_j e''_0} + \overline{q_j} - \overline{u_i \tau_{ij}} \right] = 0$ (14)

The density averaged total energy $\widetilde{e_0}$ is given by:

 $\widetilde{e_0} \equiv \widetilde{e} + \frac{\widetilde{u_k} \widetilde{u_k}}{2} + k$ (15)

Where the turbulent energy, $k$, is defined by:

 $k \equiv \widetilde{\frac{u''_k u''_k}{2}}$ (16)

Equation (12), (13) and (14) are referred to as the Favre averaged Navier-Stokes equations. $\overline{\rho}$, $\widetilde{u_i}$ and $\widetilde{e_0}$ are the primary solution variables. Note that this is an open set of partial differential equations that contains several unkown correlation terms. In order to obtain a closed form of equations that can be solver it is neccessary to model these unknown correlation terms.

### Approximations

To analyze equation (12), (13) and (14) it is convenient to rewrite the unknown terms in the following way:

 $\overline{\tau_{ji}} = \widetilde{\tau_{ji}} + \overline{\tau''_{ji}}$ (17)
 $\overline{u''_j p} + \overline{\rho u''_j e''_0} = C_p \overline{\rho u''_j T} + u_i \overline{\rho u''_i u''_j} + \overline{\frac{\rho u''_j u''_i u''_i}{2}}$ (18)
 $\overline{q_j} = - \overline{C_p \frac{\mu}{Pr} \frac{\partial T}{\partial x_j}} = - C_p \frac{\mu}{Pr} \frac{\partial \widetilde{T}}{\partial x_j} - C_p \frac{\mu}{Pr} \frac{\partial \overline{T''}}{\partial x_j}$ (19)

 $\overline{u_i \tau_{ij}} = \widetilde{u_i} \widetilde{\tau_{ij}} + \overline{u''_i \tau_{ij}} + \widetilde{u_i} \overline{\tau''_{ij}}$ (20)

Where the perfect gas relations (8) and Fourier's law (6) have been used. Note also that fluctuations in the molecular viscosity, $\mu$, have been neglected.

Inserting (17)-(20) into (12), (13) and (14) gives:

 $\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0$ (21)
 $\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) + \frac{\partial}{\partial x_j} \left[ \overline{\rho} \widetilde{u_i} \widetilde{u_j} + \overline{p} \delta_{ij} + \underbrace{\overline{\rho u''_i u''_j}}_{(1^*)} - \widetilde{\tau_{ji}} - \underbrace{\overline{\tau''_{ji}}}_{(2^*)} \right] = 0$ (22)
 $\frac{\partial}{\partial t} \left(\overline{\rho} \widetilde{e_0} \right) + \frac{\partial}{\partial x_j} [ \overline{\rho} \widetilde{u_j} \widetilde{e_0} + \widetilde{u_j} \overline{p} + \underbrace{C_p \overline{\rho u''_j T}}_{(3^*)} + \underbrace{\widetilde{u_i} \overline{\rho u''_i u''_j}}_{(4^*)} + \underbrace{\overline{\frac{\rho u''_j u''_i u''_i}{2}}}_{(5^*)}$ $- C_p \frac{\mu}{Pr} \frac{\partial \widetilde{T}}{\partial x_j} - \underbrace{C_p \frac{\mu}{Pr} \frac{\partial \overline{T''}} {\partial x_j}}_{(6^*)}- \widetilde{u_i} \widetilde{\tau_{ij}} - \underbrace{\overline{u''_i \tau_{ij}}}_{(7^*)} - \underbrace{\widetilde{u_i} \overline{\tau''_{ij}}}_{(8^*)} ] = 0$ (23)

The terms marked with $(1^*)-(8^*)$ are unknown, and have to be modeled in some way.

Term $(1^*)$ and $(4^*)$ can be modeled using an eddy-viscosity assumption for the Reynolds stresses, $\tau_{ij}^{turb}$:

 $\tau_{ij}^{turb} \equiv - \overline{\rho u''_i u''_j} \approx 2 \mu_t \widetilde{S_{ij}^*} - \frac{2}{3} \overline{\rho} k \delta_{ij}$ (24)

Where $\mu_t$ is a turbulent viscosity, which is estimated with a turbulence model. The last term is included in order to ensure that the trace of the Reynolds stress tensor is equal to $-2 \rho k$, as it should be.

Term $(2^*)$ and $(8^*)$ can be neglected if:

 $\left| \widetilde{\tau_{ij}} \right| >> \left| \overline{\tau''_{ij}} \right|$ (25)

This is true for virtually all flows.

Term $(3^*)$, corresponding to turbulent transport of heat, can be modeled using a gradient approximation for the turbulent heat-flux:

 Failed to parse (unknown function\defeq): q_j^{turb} \defeq C_p \overline{\rho u''_j T} \approx - C_p \frac{\mu_t}{Pr_t} \frac{\partial \widetilde{T}}{\partial x_j} (26)

Where $Pr_t$ is a turbulent Prandtl number. Often a constant $Pr_t$ is used ($\approx 0.9$. researchers have reported no significant improvement when using an algebraic expression for the variation of $Pr_t$ \cite{Boyle:91}.

$\frac{\partial \overline{\rho}}{\partial t} + \frac{\partial}{\partial x_i}\left[ \overline{\rho} \widetilde{u_i} \right] = 0$

$\frac{\partial}{\partial t}\left( \overline{\rho} \widetilde{u_i} \right) + \frac{\partial}{\partial x_j} \left[ \overline{\rho} \widetilde{u_j} \widetilde{u_i} + \overline{p} \delta_{ij} - \widetilde{\tau_{ji}^{tot}} \right] = 0$