Sensitivity Issues in Finite-Difference Large-Eddy Simulations of the Atmospheric Boundary Layer with Dynamic Subgrid-Scale Models

Abstract

The neutral atmospheric boundary layer (ABL) is simulated by finite-difference large-eddy simulations (LES) with various dynamic subgrid-scale (SGS) models. The goal is to understand the sensitivity of the results to several aspects of the simulation set-up: SGS model, numerical scheme for the convective term, resolution, and filter type. Three dynamic SGS models are tested: two scale-invariant models and the Lagrangian-averaged scale-dependent (LASD) model. The results show that the LASD model has the best performance in capturing the law-of-the-wall, because the scale invariance hypothesis is violated in finite-difference LES. Two forms of the convective term are tested, the skew-symmetric and the divergence forms. The choice of the convective term is more important when the LASD model is used and the skew-symmetric scheme leads to better simulations in general. However, at fine resolutions both in space and time, the sensitivity to the convective scheme is reduced. Increasing the resolution improves the performance in general, but does not better capture the law of the wall. The box and Gaussian filters are tested and it is found that, combined with the LASD model, the Gaussian filter is not sufficient to dissipate the small numerical noises, which in turn affects the large-scale motions. In conclusion, to get the most benefits of the LASD model within the finite-difference framework, the simulations need to be set up properly by choosing the right combination of numerical scheme, resolution, and filter type.

Appendix

1.1 Appendix 1: Planar-Averaged Scale-Invariant (PASI) SGS Model

Germano’s identity can be written as

$$\begin{aligned} L_{ij}=\overline{\widetilde{u}_{i}\widetilde{u}_{j}}-\overline{\widetilde{u}}_{i}\overline{\widetilde{u}}_{j}=T_{ij}-\overline{\tau }_{ij}, \end{aligned}$$

(20)

where $\overline{(\cdot )}$ denotes a test filtering with filter width of $\overline{\varDelta }=\alpha \varDelta$ and $\alpha$ is usually taken as 2, $L_{ij}$ is the resolved stress, and $T_{ij}=\overline{\widetilde{u_{i}u_{j}}}-\overline{\widetilde{u}}_{i}\overline{\widetilde{u}}_{j}$ is the SGS stress at the test filter scale. The Smagorinsky model is used for the deviatoric part of $T_{ij}$ as follows,

$$\begin{aligned} T_{ij}-\frac{1}{3}T_{kk}\delta _{ij}=-2\left( C_\mathrm{S}\alpha \varDelta \right) ^{2}|\overline{\widetilde{S}}|\overline{\widetilde{S}}_{ij}. \end{aligned}$$

(21)

Next Eqs. 3 and 21 are substituted into Eq. 20 to obtain the error

$$\begin{aligned} e_{ij}=L_{ij}-C_{\mathrm{S},\triangle }^{2}M_{ij}, \end{aligned}$$

(22)

where

$$\begin{aligned} M_{ij}=2\varDelta ^{2}\Big (\overline{|\widetilde{S}|\widetilde{S}_{ij}}-\alpha ^{2}\beta |\overline{\widetilde{S}}|\overline{\widetilde{S}}_{ij}\Big ), \end{aligned}$$

(23)

and

$$\begin{aligned} \beta =C_{\mathrm{S},\alpha \triangle }^{2}/C_{\mathrm{S},\triangle }^{2}. \end{aligned}$$

(24)

The parameter $\beta$ is the ratio between the coefficients at the test filter scale and at the filter scale. By minimization of the error using a least-square approach, and assuming that $\beta =1$ (i.e. $C_\mathrm{S}$ is scale-invariant) (Lilly 1992), the Smagorinsky coefficient at the test filter scale is obtained as

$$\begin{aligned} C_\mathrm{S}^{2}=\frac{\langle L_{ij}M_{ij} \rangle }{\langle M_{ij}M_{ij}\rangle }, \end{aligned}$$

(25)

where $\langle \cdot \rangle$ is a spatial average along the horizontal direction that eliminates numerical instability.

1.2 Appendix 2: Lagrangian-Averaged Scale-Invariant (LASI) SGS Model

On the basis of the PASI model, for general inhomogeneous turbulence where a spatial average is problematic, Meneveau et al. (1996) developed a weighted Lagrangian time average along the fluid trajectory as follows

$$\begin{aligned} C_\mathrm{S}^{2}=\frac{{\mathcal {J}}_{LM}}{{\mathcal {J}}_{LM}}, \end{aligned}$$

(26)

with

$$\begin{aligned} {\mathcal {J}}_{LM}=\int ^{t}_{-\infty }L_{ij}M_{ij}\Big (\varvec{x}\left( t'\right) ,t'\Big )W\left( t-t'\right) dt' \end{aligned}$$

(27)

and

$$\begin{aligned} {\mathcal {J}}_{LM}=\int ^{t}_{-\infty }M_{ij}M_{ij}\Big (\varvec{x}\left( t'\right) ,t'\Big )W\left( t-t'\right) dt', \end{aligned}$$

(28)

where $W(t-t')=(1/T){\text {exp}}((t-t')/T)$ is the weighting function and T is chosen as $T=1.5\varDelta ({\mathcal {J}}_{LM}{\mathcal {J}}_{MM})^{-1/8}$. The exponential form of $W(t-t')$ allows using forward relaxation-transport equations to replace the backward time integrals as follows

$$\begin{aligned} \frac{D{\mathcal {J}}_{LM}}{Dt}=\frac{\partial {\mathcal {J}}_{LM}}{\partial t}+\widetilde{\varvec{u}}\cdot {\mathcal {J}}_{LM}=\frac{1}{T_{\varDelta }}(L_{ij}M_{ij}-{\mathcal {J}}_{LM}) \end{aligned}$$

(29)

and

$$\begin{aligned} \frac{D{\mathcal {J}}_{MM}}{Dt}=\frac{\partial {\mathcal {J}}_{MM}}{\partial t}+\widetilde{\varvec{u}}\cdot {\mathcal {J}}_{MM}=\frac{1}{T_{\varDelta }}(M_{ij}M_{ij}-{\mathcal {J}}_{MM}). \end{aligned}$$

(30)

By using first-order numerical time and space schemes, Eqs. 29 and 30 can be solved easily and economically to update ${\mathcal {J}}_{LM}$ and ${\mathcal {J}}_{MM}$ at each timestep.

1.3 Appendix 3: Lagrangian-Averaged Scale-Dependent (LASD) SGS Model

The assumption of scale-invariance of $C_\mathrm{S}$, i.e. $\beta =1$, is questionable. Porté-Agel et al. (2000) and Bou-Zeid et al. (2005) introduced scale-dependent approaches by using a second test filter at scale $\widehat{\varDelta }=\alpha ^{2}\varDelta$ to calculate $\beta$ dynamically. Following the Bou-Zeid et al. approach, by applying the Germano identity and minimizing the error at the second test filter scale, the coefficient at this scale can be obtained as

$$\begin{aligned} C_{\mathrm{S},\alpha ^{2}\triangle }^{2}=\frac{{\mathcal {J}}_{QN}}{{\mathcal {J}}_{NN}}, \end{aligned}$$

(31)

where ${\mathcal {J}}_{QN}$ and ${\mathcal {J}}_{NN}$ are Lagrangian-averaged $Q_{ij}N_{ij}$ and $N_{ij}N_{ij}$, respectively, and $Q_{ij}=\widehat{\widetilde{u}_{i}\widetilde{u}_{j}}-\widehat{\widetilde{u}}_{i}\widehat{\widetilde{u}}_{j}$, $N_{ij}=2\varDelta ^{2}\Big (\widehat{|\widetilde{S}|\widetilde{S}_{ij}}-\alpha ^{4}\beta ^{2}|\widehat{\widetilde{S}}|\widehat{\widetilde{S}}_{ij}\Big )$. Assuming that $\beta$ is scale-invariant (this assumption is more reasonable than the scale-invariant assumption of $C_\mathrm{S}$), such that $\beta =C_{\mathrm{S},\alpha ^{2}\triangle }^{2}/C_{\mathrm{S},\alpha \triangle }^{2}=C_{\mathrm{S},\alpha \triangle }^{2}/C_{\mathrm{S},\triangle }^{2}$, implies that,

$$\begin{aligned} C_{\mathrm{S},\triangle }^{2}=C_{\mathrm{S},\alpha \triangle }^{2}/\beta =\frac{{\mathcal {J}}_{LM}/{\mathcal {J}}_{MM}}{\Big (\frac{{\mathcal {J}}_{QN}/{\mathcal {J}}_{MM}}{{\mathcal {J}}_{NN}/{\mathcal {J}}_{LM}}\Big )}. \end{aligned}$$

(32)

1.4 Appendix 4: Test and Second Test Filters in the Physical Space

The spatial filtering to a variable f at location $\varvec{x}$ is defined as the following convolution form

$$\begin{aligned} \widetilde{f}\left( \varvec{x}\right) =\int ^{+\infty }_{-\infty } \widetilde{G}\left( \varvec{x},\varvec{x}' \right) f\left( \varvec{x}'\right) \text {d}\varvec{x}', \end{aligned}$$

(33)

where $\widetilde{G}$ is the filter kernel satisfying the property of

$$\begin{aligned} \int ^{+\infty }_{-\infty }\widetilde{G}\left( \varvec{x},\varvec{x}'\right) =1. \end{aligned}$$

(34)

Here, in conjunction with the finite-difference methods, two filters, i.e. box (or top-hat) filter and Gaussian filter, are tested for their simplicity and wide use in applications. Specifically, for a filter width $\widetilde{\varDelta _{i}}$, the kernel of the 1D box filter is written as

$$\begin{aligned} \widetilde{G}(x_{i}-x'_{i})=\left\{ \begin{array}{ll} \frac{1}{\widetilde{\triangle _{i}}}, &{} {\, {\text {if}}\, |x_{i}-x'_{i}|\le \frac{\widetilde{\triangle _{i}}}{2};}\\ 0, &{} \,{\text {otherwise.}} \end{array}\right. \end{aligned}$$

(35)

Note that in the finite-difference discretization, the box filtering is implicitly applied at the filter width of the grid spacing (Najjar and Tafti 1996). For the 1D Gaussian filter, the kernel is

$$\begin{aligned} \widetilde{G}(x_{i}-x'_{i})=\Bigg (\frac{\gamma }{\pi \widetilde{\varDelta _{i}}^{2}}\Bigg )^{1/2}{\text {exp}} \Bigg (\frac{-\gamma |x_{i}-x'_{i}|^{2}}{\widetilde{\varDelta _{i}}^{2}}\Bigg ), \end{aligned}$$

(36)

where $\gamma =6$ is generally used (Pope 2000; Brasseur and Wei 2010). Here, the filtering is performed in a 2D manner along the horizontal directions in the physical space, i.e.

$$\begin{aligned} \widetilde{f}\left( \varvec{x}\right) =\int ^{+\infty }_{-\infty } \widetilde{G}\left( x_{1}-x'_{1}\right) \widetilde{G} \left( x_{2}-x'_{2}\right) f\left( \varvec{x}'\right) \text {d}\varvec{x}'. \end{aligned}$$

(37)

Following previous finite-difference LES, the trapezoidal rule is used to calculate the discrete integral (Zang et al. 1993; Balaras et al. 1995; Najjar and Tafti 1996).