Entropy viscosity

The entropy viscosity method ([Kronbichler et al., 2012, Jean-Luc Guermond and Richard Pasquetti and Bojan Popov, 2011]) adds an artificial diffusion \(\nu_h\) to the weak form (28), where the diffusion term \(\left (k\nabla T, \nabla \varphi \right)\) is replaced by

\[\left(\max (k, \nu_h) \nabla T, \nabla \varphi \right).\]

The parameter \(\nu_h\) is chosen as a constant per cell as

\[v_h \vert_K = \min \left( v_h^\text{max} \vert_K, v_h^E \vert_K \right),\]

where \(v_h^\text{max}\) is the maximum dissipation defined as

\[v_h^\text{max} \vert_K = \alpha_\text{max} h \| \mathbf u \|_{\infty,K}\]

on each cell \(K\) with parameter \(\alpha_\text{max}\) (known as “beta” in the parameter files, see Parameter name: beta). By itself, this is commonly known as a first-order viscosity stabilization scheme, which is effective at stabilization, but too diffusive to be used by itself. In fact, one can show that this reduces the convergence order of smooth solutions to be only first order. This is avoided by taking the minimum with the entropy viscosity \(v_h^E|_K\) above. It is defined as

\[v_h^E \vert_K = \alpha_E \frac{h^2 \| r_E \|_{\infty, K}}{\| E - E_\text{avg} \|_{\infty, \Omega}}.\]

The constant \(\alpha_E\) is given by “cR” in the parameter files, see Parameter name: cR. In the denominator, the entropy viscosity above is scaled by the maximum deviation of the temperature entropy \(E=\frac{1}{2}(T-T_m)^2\) with \(T_m = \frac{1}{2}(T_\text{min}+T_\text{max})\) from the spatial average \(E_\text{avg} = \frac{1}{| \Omega |}\int E \;\text{d}x\). The residual \(r_E\) of the entropy equation for \(E\) is defined as

\[r_E = \frac{\partial E}{\partial t} + (T-T_m)(\mathbf{u}\cdot \nabla T - k\triangle T - F).\]

This residual is defined in such a way, that it is zero for the exact solution, large where the numerical approximation is poor (for example in areas with strong gradients), and small in areas where the numerical approximation is good.

The above definition assumes the entropy residual exponent (“alpha” in the parameter files, see Parameter name: alpha38]) is set to 2 (the default and recommended). For the choice of 1 for “alpha,” the entropy viscosity is defined as

\[v_h^E \vert_K = \alpha_E \frac{h |\Omega| \cdot \| \mathbf u \|_{\infty,K} \cdot \| r_E \|_{\infty, K}} {\| \mathbf u \|_{\infty,\Omega} \cdot (T_\text{max} - T_\text{min})}.\]

instead.

An additional parameter is the strain rate scaling factor “gamma” (see Parameter name: gamma), which changes the definition of the maximum dissipation \(\nu_h^\text{max}\) to

\[v_h^\text{max} \vert_K = \alpha_\text{max} h \|\lvert\mathbf u\rvert + \gamma h_K \lvert\varepsilon (\mathbf u)\rvert\|_{\infty,K},\]

where \(\gamma\geq 0\) is the aforementioned parameter in front of the strain rate.