Large eddy simulation of turbulent incompressible flows. Analytical and numerical results for a class of LES models.

*(English)*Zbl 1035.76001
Lecture Notes in Computational Science and Engineering 34. Berlin: Springer (ISBN 3-540-40643-3/pbk). xii, 261 p. (2004).

The author studies a number of turbulence models. Their derivation is mainly based on mathematical arguments. The unique existence of weak solutions is discussed. Numerical algorithms are performed and implemented into finite elements codes. To test the codes, turbulent flows in a mixing layer are considered.

Each function \(f(t,x)\) is mollified by passing to a function \(\overline{f}(t,x)= \) \(g_{\delta}\ast f\equiv\) \(\int g_{\delta}(x-z)f(t,z)dz\), with \(g_{\delta}(x)=\) \((\frac{6}{\delta^2 \pi })^{d/2}\) \(\exp (-6| x| ^2 /\delta ^2),\) \(x\in \mathbb{R}^d\). Starting from Navier-Stokes equations for velocity and pressure fields \(u(t,x)\) \(p(t,x)\), the derivation of equations for \(\overline{u}(t,x)\) and \(\overline{p}(t,x)\) cannot be performed without additional hypotheses on the Reynolds stress tensor \(\overline{u_i u_j}\). There is one more obstacle related to the fact that convolution and differentiation do not commute when \(u\) and \(p\) are extended by zero outside the bounded domain \(\Omega\). Thus, the commutation error should be estimated as \(\delta\to 0.\)

There are different ways to write Reynolds tensor as a function of \(\overline{u}\) and \(\overline{p}\). The Smagorinsky model is based on the constitutive law \(\mathbb{T}\)= \(c\delta ^2\) \(\sqrt{D(\overline{u}):D(\overline{u})}\) \(D(\overline{u})\), where \(\mathbb{T}\) is the deviator of the tensor \(\overline{u\otimes u}-\) \(\overline{u}\otimes\overline{u}\), and \(D(u)\) is the velocity strain tensor. The principal part of the book is devoted to the large eddy simulation (LES) approach. In this approach the decomposition \(u=\) \(\overline{u}+\) \(u'\) is used and the Reynolds tensor is modelled as follows. The terms \(L\equiv\) \(\overline{\overline{u}\otimes\overline{u}}+\) \(\overline{\overline{u}\otimes u'}+\) \(\overline{u'\otimes\overline{u}}\) are modified in five steps: 1) computing the Fourier transform \(\mathcal{F}(L)\); 2) replacing \(\mathcal{F}(u')\) by \(\mathcal{F}(\overline{u})\); 3) approximating the Fourier transform of Gaussian filter \(g_{\delta}\) by an appropriate simpler function; 4) neglecting all the terms which are in a certain sense of higher order in \(\delta\); 5) computing the inverse Fourier transform. As a result, one arrives at the representation formula \(L\simeq\) \(\overline{u}\otimes\overline{u}+\) \(\frac{\delta ^2}{2\gamma}\) \(\nabla\overline{u}\otimes\nabla\overline{u}\) or \(L\simeq\) \(\overline{u}\otimes\overline{u}+\) \(\frac{\delta ^2}{2\gamma}\) \((\mathbb{I}-\frac{\delta^2}{4\gamma}\Delta)^{-1}\) \(\nabla\overline{u}\otimes\nabla\overline{u}\). The first representation is due to a polynomial approximation of \(\mathcal{F}(g_{\delta})\), and the second is due to the approximation of \(\mathcal{F}(g_{\delta})\) by some rational function. In a similar manner, a model is derived for the subgrid scale term \(\overline{u' \otimes u'}\). After derivation of the LES models, the author studies the questions of unique solvability of a number of boundary value problems.

Discretization of LES models is performed according to the following strategy: 1) the time discretization leads at each discrete time step to a nonlinear system of equations; 2) the nonlinear system is linearized and reformulated as a variational problem; 3) the linear problem is discretized by finite elements. Then an efficient solution of the discrete system is performed, and numerical studies are presented.

Each function \(f(t,x)\) is mollified by passing to a function \(\overline{f}(t,x)= \) \(g_{\delta}\ast f\equiv\) \(\int g_{\delta}(x-z)f(t,z)dz\), with \(g_{\delta}(x)=\) \((\frac{6}{\delta^2 \pi })^{d/2}\) \(\exp (-6| x| ^2 /\delta ^2),\) \(x\in \mathbb{R}^d\). Starting from Navier-Stokes equations for velocity and pressure fields \(u(t,x)\) \(p(t,x)\), the derivation of equations for \(\overline{u}(t,x)\) and \(\overline{p}(t,x)\) cannot be performed without additional hypotheses on the Reynolds stress tensor \(\overline{u_i u_j}\). There is one more obstacle related to the fact that convolution and differentiation do not commute when \(u\) and \(p\) are extended by zero outside the bounded domain \(\Omega\). Thus, the commutation error should be estimated as \(\delta\to 0.\)

There are different ways to write Reynolds tensor as a function of \(\overline{u}\) and \(\overline{p}\). The Smagorinsky model is based on the constitutive law \(\mathbb{T}\)= \(c\delta ^2\) \(\sqrt{D(\overline{u}):D(\overline{u})}\) \(D(\overline{u})\), where \(\mathbb{T}\) is the deviator of the tensor \(\overline{u\otimes u}-\) \(\overline{u}\otimes\overline{u}\), and \(D(u)\) is the velocity strain tensor. The principal part of the book is devoted to the large eddy simulation (LES) approach. In this approach the decomposition \(u=\) \(\overline{u}+\) \(u'\) is used and the Reynolds tensor is modelled as follows. The terms \(L\equiv\) \(\overline{\overline{u}\otimes\overline{u}}+\) \(\overline{\overline{u}\otimes u'}+\) \(\overline{u'\otimes\overline{u}}\) are modified in five steps: 1) computing the Fourier transform \(\mathcal{F}(L)\); 2) replacing \(\mathcal{F}(u')\) by \(\mathcal{F}(\overline{u})\); 3) approximating the Fourier transform of Gaussian filter \(g_{\delta}\) by an appropriate simpler function; 4) neglecting all the terms which are in a certain sense of higher order in \(\delta\); 5) computing the inverse Fourier transform. As a result, one arrives at the representation formula \(L\simeq\) \(\overline{u}\otimes\overline{u}+\) \(\frac{\delta ^2}{2\gamma}\) \(\nabla\overline{u}\otimes\nabla\overline{u}\) or \(L\simeq\) \(\overline{u}\otimes\overline{u}+\) \(\frac{\delta ^2}{2\gamma}\) \((\mathbb{I}-\frac{\delta^2}{4\gamma}\Delta)^{-1}\) \(\nabla\overline{u}\otimes\nabla\overline{u}\). The first representation is due to a polynomial approximation of \(\mathcal{F}(g_{\delta})\), and the second is due to the approximation of \(\mathcal{F}(g_{\delta})\) by some rational function. In a similar manner, a model is derived for the subgrid scale term \(\overline{u' \otimes u'}\). After derivation of the LES models, the author studies the questions of unique solvability of a number of boundary value problems.

Discretization of LES models is performed according to the following strategy: 1) the time discretization leads at each discrete time step to a nonlinear system of equations; 2) the nonlinear system is linearized and reformulated as a variational problem; 3) the linear problem is discretized by finite elements. Then an efficient solution of the discrete system is performed, and numerical studies are presented.

Reviewer: Vladimir Shelukhin (Novosibirsk)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76F65 | Direct numerical and large eddy simulation of turbulence |

76M10 | Finite element methods applied to problems in fluid mechanics |

76F25 | Turbulent transport, mixing |