A systematic approach to explicit scheme

2.3.1 Overall concept

The main objective of this work is to develop an explicit LES scheme which fulfils LES principles and for which the numerical error converges to an acceptably small value. According to LES principles, filter widths need to lie within the inertial sub-range of the energy spectra and the scheme needs to accurately capture the energy containing eddies (Chow and Moin 2003). Kolmogorov’s universal theoretical slope in the inertial sub-range (proportional to wavenumbers

𝜅−5/3_{), provides a criterion for identifying this range and helps to demarcate the energy}

containing and dissipation ranges. However, in low Reynolds number flows the universal theoretical slope is scarcely observed, but this theoretical relationship becomes more apparent as the Reynolds number increases (Pope 2000).

A systematic and sequential approach to obtaining an appropriate LES solution using explicit filtering is proposed. Firstly, a fixed filter width is determined based on a physical parameter for which an analytical expression is available in the literature. Here, the BLT is considered as a suitable parameter as it characterizes the size of large scale eddies in wall bounded flows

37 Figure 2.6. Grid convergence of a hypothetical energy spectrum with progressively larger filter

to grid ratios (FGR) Δ/δx1 < Δ/δx2 < Δ/δx3. The dashed vertical line indicates the cut-off

wavenumber for the fixed filter width Δ

(Tennekes and Lumley 1972). BLT can be estimated from an analytical solution of idealized cases or empirical correlations. To reduce numerical errors (due to grid size) arising in the explicit schemes with respect to the fixed filter, the grid needs to be refined until the solution converges. A hypothetical example is shown by conducting a spectrum analysis (presented in Figures 2.6 and 2.7). In Figure 2.6 fixed filter width is chosen and the effect of reducing the grid size is investigated to obtain a grid-converged solution. The simulation is grid converged when further reductions in grid size lead to insignificant changes in the result, and the filter width remains constant. When grid-converged the example shown in Figure 2.6 effectively captures the energy containing range as the turbulence is resolved well into the inertial subrange denoted by

the region of the spectrum with constant slope. A larger explicit filter than the one used in Figure

2.6 may not capture the energy containing range even when grid-converged. The effect of introducing a progressively smaller explicit filter is shown schematically in Figure 2.7. Energy spectra for four different filter widths with Δ1 > Δ2 > Δ3 > Δ4 are shown, each being grid-

38 unable to fully capture the energy containing eddies. Each of the three progressively smaller filter widths capture the energy containing range effectively and thus fulfils the LES principle. In this hypothetical example the spectra with filter widths Δ3 and Δ4 are the same; however, the pursuit of such filter-convergence can be elusive, computationally expensive and of minor importance. Therefore, when implementing explicit filtering a filter width should be selected which lies within the inertial sub-range and its grid-converged solution should be able to capture

all energy containing scales. Although LES (Δ3) and LES (Δ4) may provide more accurate results

than LES (Δ2), the LES (Δ2) results can be deemed acceptable in that it conforms to LES

principles.

Figure 2.7. Hypothetical scenario of selection of filter width in an explicit scheme. Schematic shows the LES solution for four different filter widths with Δ1 > Δ2 > Δ3 > Δ4 (all grid

converged) and vertical lines are corresponding cut-off wavenumbers.

2.3.2 Proposed explicit filter width

In this study, the filter widths (∆) in the explicit scheme are taken as a function of the BLT (𝛿).

Two filter widths are selected as being 20% and 10% of the BLT, respectively. These values have been chosen according to the suggestions of Wilcox (1997) and Pope (2000) that at least ten grid points are required to accurately capture features of the boundary layer. The grid size

39

𝛿𝑥 = 0.1𝛿 is therefore taken as the reference point for the coarsest grid width and the base filter

width as twice that, ∆= 2𝛿𝑥, which is commonly adopted in contemporary literature (Gullbrand

2002, Gullbrand and Chow 2003, Radhakrishnan and Bellan 2012, 2013 and 2015). If necessary the filter width is refined and taken as 10% of the BLT to ensure the principles of LES (i.e. filter is within the inertial subrange and energy containing eddies are fully captured) are satisfied. Although not necessary in the present modelling, in general the filter could be further reduced by half (i.e. 5% of BLT) and so on. In the contemporary literature on implicitly filtered LES it is now common to see 20 or more grid points within the boundary layer; a situation that has been made possible by the growth in computing power. Such simulations can produce very detailed data sets of great value to the community. Here our approach is somewhat different. Our primary objective is to demonstrate a systematic approach to filter width selection in explicitly filtered LES. Accordance with LES principles is ensured but we do not attempt to produce super resolved LES, but acknowledge that higher resolution is possible and in some cases desirable. 2.3.3 Filter width to grid spacing ratio

As mentioned in Chapter 1, Gullbrand and Chow (2003) and Chow and Moin (2003) reported the importance of the FGR. They observed that higher order numerical schemes (fourth and sixth order), require two grids within the filter, while for second-order schemes at least four grids within the filter width are required. In the present work using a second-order code FGR values of 2, 4, 8 and 16 are used and the suitability of this choice is discussed.