Abstract
This chapter deals with computation of turbulent flows. The nature of turbulence and three methods for its simulation are described: direct and large-eddy simulation and methods based on Reynolds-averaged Navier-Stokes equations. Some widely used models in the latter two approaches are described, including details related to boundary conditions. Examples of application of these approaches, including comparison of their performance, are presented.
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Notes
- 1.
There are also two-point closures which use equations for the correlation of the velocity components at two spatial points or, more often, the Fourier transform of these equations. These methods are not widely used in practice (Leschziner 2010) and most often are used in pure research, so we shall not consider them further. However, Lesieur (2010, 2011) presents a charming review with good insight to their history and state-of-the-art.
- 2.
- 3.
It could be worse! The NSF Report on Simulation-Based Engineering Science (2006) actually notes that the tyranny of scales dominates simulation efforts in many fields, including fluid mechanics. So, even for \(\mathrm{Re}_L \sim 10^7\), the length scale ratio is on the order of \(2\times 10^5\). However, they point out that for protein folding the time scale ratio is \(\sim 10^{12}\) while for advanced material design the spatial scale ratio is \(\sim 10^{10}\)!
- 4.
“Mesoscale” refers to weather systems with horizontal dimensions generally ranging from around 5 km to several hundred or perhaps \(10^3\) km.
- 5.
Subcritical flow past a sphere is a low Reynolds number regime in which the drag coefficient is not a function of Reynolds number and there is laminar boundary layer separation.
- 6.
Chen and Jaw (1998) illustrate the creation of an ensemble average in their Fig. 1.8.
- 7.
The reader may note that there is no mention of the grid here. The filter size is at least as large as the grid size and often significantly larger. In traditional LES, the filter width is assumed to be the grid size and for now we will follow that practice. In Sects. 10.3.3.4 and 10.3.3.7, we will explore the relationship between grid and filter widths and how that affects the modeling.
- 8.
As is the case for the time and ensemble averaged equations above, the filter operation is defined, but the equations are, in general, not explicitly filtered. The equations are essentially the result of an implicit and unknown filter in traditional LES. Bose et al. (2010) have shown the value of explicit filtering.
- 9.
The discerning reader will notice that we quietly extracted the coefficient \(C_S^2\), which is a function of space and time, from under the test-filter average in Eq. (10.17); this is equivalent to assuming that it is constant over the volume of the test filter. This is a convenient choice, but not the only possible one.
- 10.
Please note that Carati et al. (2001) and Chow et al. (2005), as well as many others, use a stress definition as the velocity product without including the density.
- 11.
Accordingly, level n has \(n+1\) terms from the expansion (10.24); see the Appendix of Shi et al. (2018a) for a discussion and an application.
- 12.
N.B.: A fine grid is needed also along the wall for DNS, but LES often has a rather high (horizontal to normal) grid aspect ratio near walls or the ground.
- 13.
For Coriolis-influenced flows, the velocity rotates with distance from the boundary and both components may be non-zero even if the flow is uni-directional far from the boundary; see Sect. 10.3.4.3.
- 14.
The potential temperature is used extensively in meteorology because it has properties that suit it to the study of stratified flow in a compressible medium (air); it is the temperature that a parcel of air at a height z would have after it was moved to the ground without exchanging heat with its surroundings (adiabatically); so \(\Theta (z)=T(z)[p_\mathrm{ground})/p(z)]^{0.286}\).
- 15.
Weather forecasters, in particular, use a finite set of simulations (which may not be statistically identical) and ensemble-average them to achieve improved predictions.
- 16.
For the unsteady case with persistent structures, the flow may not change monotonically and the result might look much like Fig. 10.6 if the LES line is imagined to represent the coherent structures in the ensemble-averaged flow. See Fig. 1.8 in Chen and Jaw (1998).
- 17.
Note the similarity to the subgrid-scale Reynolds stresses, Eq. 10.11.
- 18.
This relationship plays a large role in LES using a TKE-based SGS model; in that case, \(L=\Delta \) and a constant of proportionality of O(1) is used.
- 19.
The NASA Turbulence Modeling Resource (NASA TMR 2019) provides documentation for RANS turbulence models, including the latest (often corrected) versions of Spalart-Allmaras, Menter, Wilcox and other models and verification and validation test cases, grids, and databases.
- 20.
Note that the statements “high-Re” and “low-Re” have nothing to do with the actual Reynolds number for the particular flow problem—they are related to how close to wall the computational points reach.
- 21.
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Ferziger, J.H., Perić, M., Street, R.L. (2020). Turbulent Flows. In: Computational Methods for Fluid Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-99693-6_10
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DOI: https://doi.org/10.1007/978-3-319-99693-6_10
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