Filtering¶

The previous page argued, on physical grounds, that LES should resolve the large eddies and model the small ones. To turn that argument into equations we need a precise, mathematical definition of “large” and “small”. That definition is the filter. Filtering is the operation that formally separates the resolved motion from the subgrid motion, and applying it to the governing equations is what produces the equations Nassu actually solves.

What a filter is¶

A filter is a local spatial average. It replaces the value of a field at a point with a weighted average of that field over a small surrounding region, the size of which is set by the filter width \(\Delta\). Formally, the filtered (resolved) part of any field \(\phi\) is the convolution

(1)¶\[ \bar{\phi}(\mathbf{x}) = \int G(\mathbf{x}-\mathbf{r};\,\Delta)\,\phi(\mathbf{r})\,\mathrm{d}\mathbf{r} \]

where \(G\) is the filter kernel, a function that integrates to one and is concentrated within a region of width \(\Delta\) [1][2]. The overbar \(\bar{\phi}\) denotes the resolved (filtered) field throughout this chapter. The effect is intuitive: fluctuations smaller than \(\Delta\) are smeared out and disappear from \(\bar{\phi}\), while structures larger than \(\Delta\) pass through almost unchanged. The filter is a low-pass operation in space, exactly the spectral cut introduced on the previous page.

The figure below (Rodi et al.[3]) shows this directly. The jagged curve is an instantaneous turbulent signal; the smooth curves are the same signal after filtering with width \(\Delta_{1}\) or the larger \(\Delta_{2}\). The wider the filter, the more small-scale detail is removed and the smoother the resolved field becomes.

Resolved and subgrid parts¶

Once the filter is defined, every flow variable splits into the part the filter keeps and the part it removes. A local quantity is written as a resolved component \(\bar{\phi}\) plus a subgrid fluctuation \(\phi^{'}\):

(2)¶\[ \phi = \bar{\phi} + \phi^{'} \]

This looks like the Reynolds decomposition of RANS, but it is not the same operation and the distinction is the heart of LES. The Reynolds average is a statistical mean, so re-averaging an averaged field changes nothing and the fluctuation averages to zero. A spatial filter has neither property in general: filtering a filtered field is not the same as filtering once, and the subgrid part \(\phi^{'}\) does not vanish under a second filtering [2][1]. The resolved field \(\bar{\phi}\) is therefore still a fluctuating, time-dependent, three-dimensional field, not a smooth mean. LES computes a turbulent flow; it just computes a blurred version of it.

Why this is not a Reynolds average

The filter looks like the Reynolds decomposition of RANS, but it separates scales in space, not statistics in time. The resolved field stays unsteady and three-dimensional, so LES delivers a time history of the large eddies. RANS returns only their mean. This is exactly why LES, and not RANS, recovers vortex shedding and gust loads.

The grid as an implicit filter¶

In Nassu the filter is never applied explicitly. The numerical grid does the filtering for free. A mesh of spacing \(\Delta x\) cannot represent any structure smaller than a cell, so eddies smaller than the grid are simply absent from the discrete solution: they fall through the mesh, while larger eddies are carried. The discretisation therefore acts as a low-pass filter whose width is the grid spacing itself, \(\Delta = \Delta x\) [3][1].

This implicit filtering is the standard and the most economical choice for engineering LES, and it has two practical consequences worth stating plainly. First, it ties the modelled physics to the mesh: refining the grid lowers \(\Delta\), moves the cut further up the spectrum, and resolves more of the turbulence directly. In the limit \(\Delta x \to \eta\), LES becomes DNS. Second, it means the cut-off is set wherever the engineer places grid points, so mesh design is not only a numerical-accuracy decision but a turbulence-modelling decision: the grid chooses which eddies are resolved and which are left to the subgrid model.

Mesh design is turbulence-model design

Because Nassu filters implicitly, the grid spacing \(\Delta x\) is the filter width \(\Delta\). There is no separate knob: wherever you place cells, you set the resolved/subgrid cut. Refine in a region of interest to resolve more of its turbulence directly; coarsen elsewhere to model it. A mesh that is too coarse where the loads are generated leaves energetic eddies to the subgrid model, and no model can recover what the grid never represented.

Filtered Navier-Stokes Equations¶

The filter is useful only because it can be applied to the equations of motion, not just to a signal. Since the filter is a linear operation that commutes with the space and time derivatives (exactly so for a uniform grid), passing it through the continuity and Navier-Stokes equations (NSE) yields a set of balance equations for the resolved field, shown here for incompressible flow [1][3]:

(3)¶\[\begin{split} \begin{aligned} \frac{\partial \bar{u}_{\alpha}}{\partial x_{\alpha}}&=0 \\ \frac{\partial \bar{u}_{\alpha}}{\partial t} + \frac{\partial \bar{u}_{\alpha}\bar{u}_{\beta}}{\partial x_{\beta}}&=-\frac{1}{\rho_{r}}\frac{\partial \bar{p}}{\partial x_{\alpha}}+\frac{\partial}{\partial x_{\beta}}\left(\nu\frac{\partial \bar{u}_{\alpha}}{\partial x_{\beta}}\right)-\frac{\partial \tau_{\alpha\beta}^{\mathrm{SGS}}}{\partial x_{\beta}}+g_{\alpha}\frac{\bar{\rho}-\rho_{r}}{\rho_{r}} \end{aligned} \end{split}\]

where \(\rho_{r}\) is the reference density and \(\tau^{\mathrm{SGS}}\) is the subgrid-scale (SGS) stress.

The closure problem¶

Equation (3) looks almost identical to the original Navier-Stokes equations, with the resolved velocity \(\bar{u}_{\alpha}\) in place of the full velocity and one extra term, \(\partial \tau_{\alpha\beta}^{\mathrm{SGS}} / \partial x_{\alpha}\). That single extra term is where all the difficulty lives. It arises because filtering does not commute with the nonlinear convective term: the filter of a product is not the product of the filters. What is left over is the subgrid stress

(4)¶\[ \tau_{\alpha\beta}^{\mathrm{SGS}}=\bar{u_{\alpha}u_{\beta}}-\bar{u}_{\alpha}\bar{u}_{\beta} \]

Read the right-hand side carefully. The first term, \(\bar{u_{\alpha}u_{\beta}}\), is the filter of the full (unfiltered) velocity product. Computing it would require the full velocity field, including the subgrid scales that the simulation, by construction, does not have. The second term uses only resolved quantities and can be computed. The difference, the subgrid stress, therefore depends on information that LES has thrown away. This is the closure problem: the equation for the resolved field contains a term that cannot be evaluated from the resolved field alone. Without a way to estimate \(\tau_{\alpha\beta}^{\mathrm{SGS}}\), Equation (3) is not a closed system and cannot be solved. A closure for the subgrid stress is therefore not an optional refinement; without one the filtered equations are not a closed system. Nassu supplies it with an explicit subgrid model (the alternative, implicit LES, is discussed in the note below) [2][1].

Physically, \(\tau_{\alpha\beta}^{\mathrm{SGS}}\) represents the effect of the unresolved fluctuations on the resolved motion. It is analogous to the Reynolds stress of RANS, but the analogy is partial: the Reynolds stress carries the effect of the entire turbulence spectrum on the mean flow, whereas \(\tau_{\alpha\beta}^{\mathrm{SGS}}\) carries only the effect of the small, subgrid scales on the resolved eddies. Because the subgrid scales lie in the universal part of the spectrum, modelling \(\tau_{\alpha\beta}^{\mathrm{SGS}}\) is a far less demanding task than modelling the full Reynolds stress, and this is precisely why LES is more reliable than RANS for the separated wakes of wind engineering.

The role of the subgrid stress is, on average, dissipative: it drains energy from the resolved scales at the rate the cascade would carry it down to the unresolved ones [2]. The most direct way to supply this drain is an explicit SGS model for \(\tau_{\alpha\beta}^{\mathrm{SGS}}\), which is the route Nassu takes.

Note

A second route is to let the truncation error of the numerical scheme supply the missing dissipation, with no explicit SGS term. This is implicit LES (ILES). It is not pursued here: Nassu uses an explicit subgrid model so that the dissipation is controlled by physics rather than by numerics.

From stress to eddy viscosity¶

To prepare the ground for the model, the subgrid stress is split into an isotropic and an anisotropic part:

(5)¶\[ \tau_{\alpha\beta}^{\mathrm{SGS}}=\underbrace{\left(\tau_{\alpha\beta}^{\mathrm{SGS}}-\frac{1}{3}\tau_{\gamma\gamma}^{\mathrm{SGS}}\delta_{\alpha\beta}\right)}_\text{anisotropic}+\underbrace{\frac{1}{3}\tau_{\gamma\gamma}^{\mathrm{SGS}}\delta_{\alpha\beta}}_\text{isotropic} \]

The trace \(\tau_{\gamma\gamma}^{\mathrm{SGS}}\) is twice the subgrid kinetic energy \(k_{\mathrm{SGS}}\), so the isotropic part \(\frac{1}{3}\tau_{\gamma\gamma}^{\mathrm{SGS}}\delta_{\alpha\beta}\) equals \(\frac{2}{3}k_{\mathrm{SGS}}\delta_{\alpha\beta}\). It acts like a pressure: it pushes equally in all directions and so can be folded into the pressure term as a modified pressure \(p^{*}=p+\frac{2}{3}k_{\mathrm{SGS}}\), removing it from the modelling problem. What remains to be modelled is the anisotropic part, the part that actually exchanges momentum between directions. The standard closure expresses it as an eddy-viscosity stress [2][1]:

(6)¶\[ \tau^{\mathrm{SGS}}_{\alpha\beta}-\frac{1}{3}\tau_{\gamma\gamma}^{\mathrm{SGS}}\delta_{\alpha\beta}=-2\nu_{\mathrm{SGS}}\bar{S}_{\alpha\beta} \]

where \(\nu_{\mathrm{SGS}}\) is the subgrid (eddy) viscosity and \(\bar{S}_{\alpha\beta}\) is the resolved rate-of-strain tensor. The reasoning behind this hypothesis, and how \(\nu_{SGS}\) is actually computed, is the subject of the next page. Substituting it back closes the resolved Navier-Stokes equations for quasi-incompressible flow:

(7)¶\[ \frac{\partial\bar{u}_{\alpha}}{\partial t}+\frac{\partial}{\partial x_{\beta}}\left(\bar{u}_{\alpha}\bar{u}_{\beta}\right)=-\frac{1}{\rho}\frac{\partial \bar{p}^{*}}{\partial x_{\alpha}}+2\frac{\partial}{\partial x_{\beta}}\left[\left(\nu+\nu_{\mathrm{SGS}}\right)\bar{S}_{\alpha\beta}\right] \]

The only change from the unfiltered equations is that the molecular viscosity \(\nu\) is augmented by the subgrid viscosity \(\nu_{\mathrm{SGS}}\). The whole effect of the unresolved turbulence has been reduced to an extra, position- and time-dependent viscosity. The remaining task is to model that one scalar field.