Abstract
Let us briefly reexamine the spatial scales present in turbulent flows before discussing how large-eddy simulation can be formulated. Consider the turbulent energy spectra for various turbulent flows across a wide range of Reynolds numbers. Shown in Fig. 8.1 are the energy spectra for various three-dimensional turbulent flows non-dimensionalized by the Kolmogorov scale [4]. Note that the vortices represented with low wave numbers are dependent on the problem. On the other hand, the small vortices represented by the high wave numbers exhibit a universal behavior (independent of the flow field). This is due to the isotropic nature of turbulence near the Kolmogorov scale. Based upon this observation, we can consider modeling the small-scale vortices that possess universality in their behavior and directly resolving the large scale vortices that are influenced by the setup of the flow field. Note however that it would not be appropriate to simply simulate only the large-scale vortices on a coarse grid with the Navier–Stokes equations, because there are interactions amongst vortices over wide range of scale due to the nonlinearity in turbulent flows. Large-eddy simulation (LES) resolves the large-scale vortices in the turbulent flow field and incorporates the influence of small-scale vortices that are not resolved by the grid through a model. This approach taken by LES has been shown to be successful in simulating a variety of complex turbulent flows.
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Notes
- 1.
A Gaussian profile decays fast but is nonzero over the entire space.
- 2.
For the turbulent stress \(\tau (u,v) = \overline{uv} - \overline{u} ~\overline{v}\) to be Galilean invariant, \(\tau (u+\alpha ,v+\beta ) = \tau (u,v)\) must hold for a system moving at a constant velocity of \((\alpha ,\beta )\). If we let velocity U to be the constant velocity \(\overline{U}\) for \(u^* = u + U\), then we have \(\overline{u^*}= \overline{u}+U\) and \({u^*}' = u'\). Thus the Leonard stress \(L_{ij}\) becomes
$$\begin{aligned} L_{ij} = \overline{\overline{u^*_i}~\overline{u^*_j}} - \overline{u^*_i}~\overline{u^*_j} - (\overline{\overline{u^*_i}}-\overline{u^*_i})U_j - (\overline{\overline{u^*_j}}-\overline{u^*_j})U_i, \end{aligned}$$which is not Galilean invariant since U remains in the expression. The modified \(L^m_{ij}\) on the other hand becomes
$$\begin{aligned} L^m_{ij} = \overline{\overline{u^*_i}~\overline{u^*_j}} - \overline{\overline{u^*_i}}~\overline{\overline{u^*_j}}, \end{aligned}$$which is Galilean invariant.
- 3.
It is often misinterpreted that the sum of the grid-scale and subgrid-scale energy \(k_{GS}+k_{SGS}\) is k. However it should be noted that this sum should be \(\overline{k}\). The kinetic energy distribution from experiments or DNS should be filtered when LES results are compared with such results.
- 4.
The modified pressure here is different from the pressure term \(\overline{P} = \overline{p} + \frac{2}{3}\rho k\) that appears in eddy-viscosity models for RANS but shares similarity in how it combines the isotropic stress components (see Sect. 7.3).
- 5.
Let us derive the interpolation equation for the kinematic (eddy) viscosity coefficient based on how the heat transfer rate is determined for a multispecies interface [23]. Figure 8.10 is provided for reference. Denoting the coefficient at the interface as \(\overline{\nu }\) and evaluating the flux with diffusive gradient, we have
$$\begin{aligned} \tau = - \overline{\nu } \frac{-u_M + u_P}{\Delta _m + \Delta _p}. \end{aligned}$$On the other hand, by assuming that the stress is constant over the segment M to P and the viscosity \(\nu \) is constant inside the cell, we can take the difference across the cell interface to express \(\tau \) as
$$\begin{aligned} \tau = - \nu _M \frac{- u_M + \overline{u}}{\Delta _m} = -\nu _P \frac{-\overline{u}+u_P}{\Delta _p}. \end{aligned}$$Eliminating \(\overline{u}\) in the above equation, we can derive
$$\begin{aligned} \frac{1}{\overline{\nu }} = \frac{1}{\Delta _m + \Delta _p}\left( \frac{\Delta _m}{\nu _M} + \frac{\Delta _p}{\nu _P} \right) , \end{aligned}$$which becomes the harmonic mean when the grid is uniform (\(\Delta _m = \Delta _p\)).
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Kajishima, T., Taira, K. (2017). Large-Eddy Simulation. In: Computational Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-45304-0_8
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