The $\varphi-\overline{f}$ model.

The $\overline{v^2}-f$ model is designed to obtain a measure of non-local effects via elliptic relaxation. In order to account for non-local effects in the framework of an eddy viscosity model, the elliptic relaxation approach of the full RSM is simplified in the case of a plane channel flow. Taking into account that the appropriate velocity scale near the wall (Durbin, 1991) is $\overline{v^2}$, the turbulent viscosity can be expressed as:

 \begin{align*} \nu_t = C_{\mu}\overline{v^2}T \label{eq:nutDur} \end{align*}
With the time and length scales defined by
 \begin{align*} T = & \max \left(\frac{k}{\varepsilon},C_T\sqrt{\frac{\nu}{\varepsilon}}\right) \label {eq:TDur}\\[2mm] L = &C_L\max\left(\frac{k^{\tfrac{3}{2}}}{\varepsilon},C_{\eta}\left(\frac{\nu^3}{\varepsilon}\right)^{\tfrac{1}{4}} \right) \label{eq:LDur} \end{align*}

In order to make the $\overline{v^2}-f$ model more adaptable to an industrial code but without sacrificing its performance, another approach has been followed at the University of Manchester during the course of this work (See: Laurence et al. 2004).

By introducing a new variable, $\varphi$, defined as:

 \begin {equation*} \varphi = \frac{\overline{v^2}}{k} \label{eq:phidef} \end{equation*}

a transport equation can be solved for this new variable, which introduces some advantages. The resulting equations for $\varphi$ and $f$ are:

 \begin{gather*} \frac{\partial \varphi}{\partial t}+\left\langle U_j \right\rangle \frac{\partial \varphi}{\partial x_j} =f-P\frac{\varphi }{k}+\frac{2}{k}\left(\nu+\frac{\nu _{t}}{\sigma_{k}}\right)\frac{\partial \varphi}{\partial x_j}\frac{\partial k}{\partial x_j}+\frac{\partial }{\partial  x_{j}}\left[\left( \nu +\frac{\nu_{t}}{\sigma _{k}}\right) \frac{\partial\varphi}{\partial x_{j}}\right] \label{eq:phiOri}  \\[2mm] L^{2}\frac{\partial ^{2}f}{\partial x_{j}^{2}}-f =\frac{1}{T}(C_{1}-1)\left[\varphi-\frac{2}{3}\right] -C_{2}\frac{P_{k}}{k} \label{eq:fphiOri} \end{gather*}

In isotropic flow $\varphi \rightarrow 2/3$. A clear benefit is in the boundary condition for $f$:

 \begin{equation*} \underset{y\rightarrow 0}{\lim }\;f=-5\varepsilon \frac{\overline{v ^{2}% }}{k^{2}}=-5\varepsilon \frac{\varphi }{k} \end{equation*}
Although the resulting model has a boundary condition less stiff than the original one, it is non-zero at the wall but resulting from the ratio of three unknown variables, therefore still diminishing the robustness of the model. An interesting benefit is that the term $\varepsilon \overline{v^2}/k$ is no longer in the transport equation, which can be difficult to reproduce correctly in the near wall region since $\varepsilon$ becomes large and the ratio $\overline{v^2}/k$ tends to zero.

By using the limiting value of $\varepsilon$ near the wall and applying the $L'H\^opital$ theorem, the value of $f$ can be written in terms of $\varphi$ :

 \begin{eqnarray*} \underset{y\rightarrow 0}{\lim } \; \varepsilon &=&2\nu \frac{k}{y^{2}} \\[2mm] \underset{y\rightarrow 0}{\lim }\; f &=&-10\nu \frac{\varphi}{y^{2}}=-5\nu \frac{ \partial ^{2}\phi }{\partial x_{j}^{2}}=-5\nu \Delta \varphi \end{eqnarray*}
The singularity near the wall is now second order only (ratio of two discretised variables with a $y^{2}$ limit instead of $y^{4}$). It is essential because the lower the order the less the ''stiffness'' of the boundary condition and the less the sensitivity of the boundary condition to truncation error. By reformulating the limit of $f$ at the wall, it is possible to suggest a new change in variable that will lead to the zero boundary condition:
  \begin{equation*} f=\overline{f}-\dfrac{2\nu(\nabla\varphi\nabla k)}{k}\text{ \ }-\nu\nabla^{2}\varphi \label{eq:fbar_phi} \end{equation*} (1)

Considering the limit $y\rightarrow0$, it is possible to show that $\overline{f}\rightarrow0$. There are different possible substitutions to reach the homogeneous boundary condition for $\overline{f}$. Some of them, e.g. $f=\overline{f}$\ $-5\nu\nabla^{2}\varphi$, can lead to an ill-posed problem for $\varphi$ since in the right-hand side we obtain a negative diffusion term: $-4\nu\nabla^{2}\varphi$.

Introducing the definition for $\overline f$, the equations of the model can be found to be:

 \begin{align*} \frac{D\varphi}{Dt}& =\overline{f}-P_k\frac{\varphi }{k}+\frac{2}{k}\left(\frac{\nu _{t}}{\sigma _{k}}\right) \frac{\partial \varphi}{\partial x_j}\frac{\partial k}{\partial x_j} +\frac{\partial }{\partial  x_{j}}\left[\left(\nu+\frac{\nu_{t}}{\sigma _{k}}\right) \frac{\partial  \varphi}{\partial x_{j}}\right] \label{eq:phi}  \\[2mm] L^{2}\frac{\partial ^{2}\overline{f}}{\partial x_{j}^{2}}-\overline{f}& =\frac{1}{T}(C_{1}-1)\left[ \varphi-\frac{2}{3}\right]-C_{2}\frac{P_{k}}{k}- \frac{2 \nu}{k} \frac{\partial \varphi}{\partial x_j}\frac{\partial k}{\partial x_j} -\nu\frac{\partial^2\varphi}{\partial x_j^2}  \label{eq:4f} \end{align*}

Boundary conditions and closure constants.

The boundary conditions at the wall are as follows:
 \begin{align*} k(0) &  =0\text{, \ }\varepsilon(0)\rightarrow\dfrac{2\nu k}{y^{2}},\\ \varphi(0) &  =0\text{, \ }\overline{f}(0)=0. \end{align*}
The boundary conditions for both $\overline{f}$ and $\varphi$ are zero in the wall, which makes it possible to solve the system uncoupled.

The term $C_{\varepsilon1}$ is changed to:

 \begin{equation*} C_{\varepsilon1}=1.4\left(  1+A_1\sqrt{\frac{1}{\varphi}}\right) \label{eq:CE1phi}% \end{equation*}
Due to the introduction of the approximation in equation (1), the coefficients used in this formulation have been optimised using DNS data for a channel flow at the Reynolds $\operatorname{Re}_{\tau}=395$ as a target. The proposed coefficients are shown in the following table.
 \begin{table}[ht] \centering \begin{tabular}{||c|c|c|c|c|c|c|c|c||} \hline  $C_{\varepsilon 2}$ & $C_{\mu }$ & $\sigma _{\varepsilon }$ & $\sigma _{k}$ & $C1$ & $C2$ & $C_{L}$ & $C_{\eta}$& $A_1$ \\ \hline  1.9 & 0.22 & 1.3 & 1.0 & 1.4 & 0.3 & 0.3 & 100 & 0.06\\ \hline \end{tabular} \caption{Coefficients of the $\varphi$ model} \label{tab:PhiCnt} \end {table}


P. A. Durbin. Near-wall turbulent closure modelling without damping functions. Theoretical and computational fluid dynamics, pages 1–13, 1991.

D. Laurence, J.C. Uribe, and S. Utyuzhnikov. A robust formulation of the v2-f model. Flow, Turbulence and Combustion, 73, 2004.


These are the cases that have results using this model in the CfdTm database:

Test Case Case Results Author Model
Channel Flow TestCase001Res001 Juan Uribe Phi-f, SST, EBM (Elliptic Blending Model)
Channel Flow TestCase001Res004 J. Uribe
Vertical Heated Pipe TestCase005Res004 You et al [2003] DNS
Mixed convection in a vertical channel TestCase012Res000 Flavien Billard
Asymmetric plane diffuser TestCase013Res000 Juan Uribe, Flavien Billard SST, V2f models
Asymmetric plane diffuser TestCase013Res002 J .Uribe
Flow over 2D periodic hills TestCase014Res000 Juan Uribe
  TestCase015Res000 J. Uribe
  TestCase015Res001 E. Moreau
Parietal two-phase jet TestCase038Res000 M. Guillaud
Diurnal evolution of an atmospheric boundary layer TestCase055Res000 M. Milliez
Number of topics: 11

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Topic revision: r3 - 2011-07-07 - 10:32:20 - FlavienBillard
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