# Elliptic Blending Model

UNDER CONSTRUCTION

The model implemented here is the one described in Thielen et al. (Int Journal of Heat and Mass Transfer, 48, 2005, pp.1583-1598).

The model solves the full set Reynolds stresses %$\overline{u_i u_j}$%, a dissipation equation %$\varepsilon$% and an elliptic equation for %$\alpha$%.

The Reynolds stress equation can be written as:

%BEGINLATEX% \begin{align} \frac{D\overline{u_i u_j}}{Dt} = &P_{ij} + {\Phi^*}_{ij} - \varepsilon_{ij}+\frac{\partial}{\partial x_k}\left[ (\nu \delta_{kl} + C_s\overline{u_ku_l}\tau) \frac{\partial \overline{u_i u_j}}{\partial x_l} \right] \\ \frac{D\varepsilon}{Dt} = & \frac{C'_{\varepsilon 1}P - C_{\varepsilon 2} \varepsilon}{\tau}+\left[ (\nu \delta_{kl} + C_{\varepsilon}\overline{u_ku_l}\tau) \frac{\partial \varepsilon }{\partial x_l} \right] \end{align} %ENDLATEX%

The main idea of the model is to blend the near wall part with the farfiled one via %$\alpha$%, which is defined as:

%BEGINLATEX% $$\alpha-L^2\nabla^2\alpha = 1$$ %ENDLATEX%

With this parameter, the equation for the velocity-pressure-gradient correlation %$\Phi^*_{ij}$% can be written as:

%BEGINLATEX% $$\Phi^*_{ij} = (1-\alpha^2)\Phi^w_{ij}+\alpha^2\Phi^h_{ij}$$ %ENDLATEX%

The "homogeneous" part is taken from the SSG (Speziale et al.) %BEGINLATEX% \begin{align*} \Phi^h_{ij} =& -\left( C_1 +C_2\frac{P}{\varepsilon}\right) \varepsilon a_{ij} + C_3kS_{ij}\&+C_4k \left(a_{ik}S_{jk} +a_{jk}S_{ik} -\frac{2}{3}\delta_{ij}a_{lk}S_{kl} \right)\&+C_5k \left( a_{ik}\Omega_{jk} + a_{jk}\Omega_{ik} \right) \end{align*} %ENDLATEX%

And the near wall part is derived to satisfy the near wall asymptotic behaviour and stress budgets:

%BEGINLATEX% \begin{equation*} \Phi^w_{ij} = -5\frac{\varepsilon}{k} \left( \overline{u_i u_k} n_j n_k + \overline{u_j u_k} n_i n_k - \frac{1}{2} \overline{u_k u_l}n_k n_l (n_i n_j + \delta_{ij} ) \right) \end{equation*} %ENDLATEX%

with the following definitions:

%BEGINLATEX% \begin{align*} &a_{ij} = \frac{\overline{u_i u_j}}{k}-\frac{2}{3}\delta_{ij}; \quad S_{ij} = \frac{1}{2}\left(\frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right); \quad \Omega_{ij} = \frac{1}{2}\left(\frac{\partial U_i}{\partial x_j} - \frac{\partial U_j}{\partial x_i} \right) \\ &\mathbf{n} = \frac{\nabla \alpha}{\parallel\nabla \alpha\parallel}; \quad \tau = \max \left( \frac{k}{\varepsilon}; \, C_{\tau} \sqrt[2]{\frac{\nu}{\varepsilon}} \right); \quad L=C_L\max \left( \frac{k^{3/2}}{\varepsilon}; \, C_{\eta} \sqrt[4]{\frac{\nu^3}{\varepsilon}} \right) \end{align*} %ENDLATEX%

The constants used are:

%BEGINLATEX% \begin{tabular}{|c c c c c c c c c c c|} \hline $C_s$ & $C_{\varepsilon}$ & $C_{\varepsilon}^0$ & $C_{\varepsilon2}$ & $C_1$ & $C_2$ & $C_4$ & $C_5$ & $C_{\tau}$ & $C_L$ & $C_{\eta}$ \\ \hline 0.21&0.18&1.44&1.83&1.7&0.9&0.625&0.2&6.0&0.161&80 \\ \hline \end{tabular} \\ \centering $C_{\varepsilon 1} = C_{\varepsilon 1}^0 \left(1+0.03(1-\alpha^2)\sqrt{\frac{k}{\overline{u_i u_j} n_i n_j}}\right)$ %ENDLATEX%

-- JuanUribe - 2009-11-05

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gz ebm_channel.tar.gz manage 283.5 K 2009-11-05 - 12:41 JuanUribe File for the EBM for a channel flow calculation (v1.4.0)
Topic revision: r3 - 2009-11-06 - 10:17:22 - FlavienBillard
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