Non-dimensionalization of the Navier-Stokes equations

Question

In 2D simulations using Large Eddy Simulation (LES) methodology, Favre averaging is commonly applied to the variables involved in the Navier-Stokes equations, resulting in: \begin{align}\label{aq} \frac{\partial (\overline{\rho} \tilde{u}_1)}{\partial t} + \frac{\partial (\overline{\rho} \tilde{u}_1\tilde{u}_1)}{\partial x_1} + \frac{\partial (\overline{\rho} \tilde{u}_1 \tilde{u}_2)}{\partial x_2} = - \frac{\partial \overline{p}}{\partial x_1} + \underbrace{\frac{\partial}{\partial x_1} \left( \overline{\tau}_{11} - \overline{\rho} \widetilde{u_1''u_1''} \right) + \frac{\partial}{\partial x_2} \left( \overline{\tau}_{12} - \overline{\rho} \widetilde{u_1''u_2''} \right)}_{(i)} \\ \frac{\partial (\overline{\rho} \tilde{u}_2)}{\partial t} + \frac{\partial (\overline{\rho} \tilde{u}_2\tilde{u}_1)}{\partial x_1} + \frac{\partial (\overline{\rho} \tilde{u}_2 \tilde{u}_2)}{\partial x_2} = - \frac{\partial \overline{p}}{\partial x_2} + \underbrace{\frac{\partial}{\partial x_1} \left( \overline{\tau}_{21} - \overline{\rho} \widetilde{u_2''u_1''} \right) + \frac{\partial}{\partial x_2} \left( \overline{\tau}_{22} - \overline{\rho} \widetilde{u_2''u_2''} \right)}_{(ii)}, \end{align} where $\tilde{u}_i$ and $u_i''$ respectively denote the mean and fluctuating components of the velocity $u_i$. Due to their complexity, the terms $(i)$ and $(ii)$ can be approximated by simpler expressions.

I came across a document that exclusively presents the non-dimensional form of the Navier-Stokes equations, employed for simulating compressible flow at low Mach numbers: \begin{align} \frac{\partial(\overline{\rho}^*\tilde{u}_1^*)}{\partial t^*} + \frac{\partial (\overline{\rho}^*\tilde{u}_1^* \tilde{u}_1^*)}{\partial x_1^*} + \frac{\partial (\overline{\rho}^*\tilde{u}_1^* \tilde{u}_2^*)}{\partial x_2^*} &= - \frac{1}{\text{Ma}^2} \frac{\partial \overline{p}^*}{\partial x_1^*} + \underbrace{\left(1+ \frac{\mu_t}{\nu} \right)\frac{1}{\text{Re}} \left(\frac{\partial^2 \tilde{u}_1^*}{\partial (x_1^*)^2} + \frac{\partial^2 \tilde{u}_1^*}{\partial (x_2^*)^2} \right)}_{(iii)} \\ \frac{\partial(\overline{\rho}^*\tilde{u}_2^*)}{\partial t^*} + \frac{\partial (\overline{\rho}^*\tilde{u}_2^* \tilde{u}_1^*)}{\partial x_1^*} + \frac{\partial (\overline{\rho}\tilde{u}_2^* \tilde{u}_2^*)}{\partial x_2^*} &= - \frac{1}{\text{Ma}^2} \frac{\partial \overline{p}^*}{\partial x_2^*} + \underbrace{\left(1+ \frac{\mu_t}{\nu} \right)\frac{1}{\text{Re}} \left(\frac{\partial^2 \tilde{u}_2^*}{\partial (x_1^*)^2} + \frac{\partial^2 \tilde{u}_2^*}{\partial (x_2^*)^2} \right)}_{(iv)}, \end{align} where $\mu_t$ is the turbulent viscosity and $\nu$ the kinematic viscosity. I know that the nondimensional equations are obtained from the primitive equations through a substitution of variables: \begin{equation} x_1^* = \frac{x_1}{L}, \, \, x_2^{*} = \frac{x_2}{L}, \, \, t^{*} = \frac{U t}{L}; \,\, \tilde{u}_1^{*} = \frac{\tilde{u}_1}{U}, \, \, \tilde{u}_2^{*} = \frac{\tilde{u}_2}{U}; \, \, \bar{\rho}^{*} = \frac{\bar{\rho}}{{\rho}_{\infty}}, \, \, \bar{p}^{*} =\frac{\bar{p}}{p_{\infty}}, \, \, \text{Re} = \frac{ \rho_{\infty} U L}{\mu_{\infty}} = \frac{UL}{\nu}; \,\, \text{Ma} = \frac{U}{\sqrt{p_{\infty}/\rho_{\infty}}}. \end{equation} Is there a way to deduce the expression used to approximate the diffusive terms $(i)$ and $(ii)$ of the equations (in its primitive form), so that after the non-dimensionalization procedure these terms become respectively $(iii)$ and $(iv)$?

Answer 1

Focusing only on diffusion of the average flow and turbulent terms,

\begin{equation} \nabla \cdot \left[ \mathbb{S} + \rho \widetilde{\mathbf{u}' \mathbf{u}'} \right] \ . \end{equation} Writing

• the average viscous stress tensor in terms of the deformation velocity, \begin{equation} \mathbb{S} = 2 \mu \mathbb{D} = 2 \mu \frac{1}{2}(\nabla \mathbf{u} + \nabla^T \mathbf{u}) = \mu \left(\nabla \mathbf{u} + \nabla^T \mathbf{u} \right) = \dfrac{\rho \nu U}{L} \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) \end{equation}

• the turbulent stress with a turbulent viscosity model \begin{equation} \rho \widetilde{\mathbf{u}' \mathbf{u}'} = 2 \mu_t \mathbb{D} = \dots = \dfrac{\rho \nu_t U}{L} \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) = \dfrac{\nu_t}{\nu} \dfrac{\rho \nu U}{L} \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) \ . \end{equation}

Thus, performing the non-dimensionalization in the usual way and dividing by $\frac{\rho U}{T} = \frac{\rho U^2}{L}$ coming from the convective term, you get \begin{equation}\begin{aligned} \dots & = \dfrac{L}{\rho U^2} \dfrac{1}{L} \nabla^* \cdot \left[ \dfrac{\rho \nu U}{L} \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) + \dfrac{\nu_t}{\nu} \dfrac{\rho \nu U}{L} \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) \right] = \\ & = \dfrac{\nu}{U L} \left( 1 + \dfrac{\nu_t}{\nu} \right) \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) = \\ & = \dfrac{1}{\text{Re}} \left( 1 + \dfrac{\nu_t}{\nu} \right) \left(\nabla^* \mathbf{u}^* + \nabla^{*T} \mathbf{u}^* \right) \end{aligned}\end{equation}