Spatial filtering of unsteady Navier---Stokes equations Filters are familiar separation devices in electronics and process applications

3.8.1 Spatial filtering of unsteady Navier---Stokes equations Filters are familiar separation devices in electronics and process applications

that are designed to split an input into a desirable, retained part and an un- desirable, rejected part. The details of the design of a filter – in particular its functional form and the cutoff width ∆ – determine precisely what is retained

and rejected.

3.8 LARGE EDDY SIMULATION 99

Filtering functions In LES we define a spatial filtering operation by means of a filter function

G(x, x ′, ∆) as follows:

2(x, t) ≡ G(x, x ′, ∆) φ(x′, t)dx 1 ′ dx 2 ′ dx 3 ′ (3.84)

where 2(x, t) = filtered function and φ(x, t) = original (unfiltered) function

and ∆ = filter cutoff width In this section the overbar indicates spatial filtering, not time-averaging.

Equation (3.84) shows that filtering is an integration, just like time-averaging in the development of the RANS equations, only in the LES the integration is not carried out in time but in three-dimensional space. It should be noted that filtering is a linear operation.

The commonest forms of the filtering function in three-dimensional LES computations are

• Top-hat or box filter:

(3.85a) @0

G(x, x ′, ∆) = ! 1/ ∆ 3 |x − x′| ≤ ∆ / 2

|x − x′| > ∆ / 2

• Gaussian filter:

A 3/2 γ D A |x − x′| 2 D

G(x, x ′, ∆) = B 2 E exp B − γ

2 E (3.85b)

C π∆ F C ∆

typical value for parameter γ=6 • Spectral cutoff:

3 sin[(x

G(x, x ′, ∆) = ∏

i −x i ′)/∆]

The top-hat filter is used in finite volume implementations of LES. The Gaussian and spectral cutoff filters are preferred in the research literature. The Gaussian filter was introduced for LES in finite differences by the Stanford group, which, over a period of more than three decades, has been the centre of research on LES and has established a rigorous basis for the technique as a turbulence modelling tool. Spectral methods (i.e. Fourier series to describe the flow variables) are also used in turbulence research, and the spectral filter gives a sharp cutoff in the energy spectrum at a wavelength of ∆/ π. The latter is attractive from the point of view of separation of the

large and small eddy scales, but the spectral method cannot be used in general-purpose CFD.

The cutoff width is intended as an indicative measure of the size of eddies that are retained in the computations and the eddies that are rejected. In principle, we can choose the cutoff width ∆ to be any size, but in CFD computations with the finite volume method it is pointless to select a cutoff width that is smaller than the grid size. In this type of computation only a single nodal value of each flow variable is retained on each grid cell, so all finer detail is lost anyway. The most common selection is to take the cutoff

CHAPTER 3 TURBULENCE AND ITS MODELLING

width to be of the same order as the grid size. In three-dimensional computa- tions with grid cells of different length ∆x, width ∆y and height ∆z the cutoff width is often taken to be the cube root of the grid cell volume:

∆= 3 ∆x∆y∆z

Filtered unsteady Navier---Stokes equations As before in section 3.3 we focus our attention on incompressible flows. As

usual we take Cartesian co-ordinates so that the velocity vector u has u-, v-, w-components. The unsteady Navier–Stokes equations for a fluid with con- stant viscosity µ are as follows:

+ div( ρu) = 0

∂t ∂(ρu)

∂p

+ div( ρuu) = − + µ div(grad(u)) + S u (2.37a)

+ div( ρvu) = − + µ div(grad(v)) + S v (2.37b)

+ div( ρwu) = −

+ µ div(grad(w)) + S w (2.37c)

∂t

∂z

If the flow is also incompressible we have div(u) = 0, and hence the viscous momentum source terms S u ,S v and S w are zero.

Considerable further simplification of the algebra is possible if we use the same filtering function G(x, x ′) = G(x − x′) throughout the computational domain, i.e. G is independent of position x. If we use such a uniform filter

function we can, by exploiting the linearity of the filtering operation, swap the order of the filtering and differentiation with respect to time, as well as the order of filtering and differentiation with respect to space co-ordinates. We have already seen this commutative property in action in section 3.3 when the time-averaged RANS flow equations were derived. Filtering of equation (2.4) yields the LES continuity equation:

∂t The overbar in this and all following equations in this section indicates a

filtered flow variable.

Repeating the process for equations (2.37a–c) gives

∂(ρR)

∂Q

+ div( ρ uu ) =− + µ div(grad(R)) (3.88a)

+ div( ρ vu ) =− + µ div(grad({)) (3.88b)

+ div( ρ wu ) =− + µ div(grad(S)) (3.88c)

∂t

∂z

3.8 LARGE EDDY SIMULATION 101

Equation set (3.87) and (3.88a–c) should be solved to yield the filtered veloc- ity field R, {, S and filtered pressure field Q. We now face the problem that we need to compute convective terms of the form div( ρ φu ) on the left hand

side, but we only have available the filtered velocity field R, {, S and pressure field Q. To make some progress we write

div( ρ φu ) = div(2=) + (div( ρ φu ) − div(2=)) The first term on the right hand side can be calculated from the filtered 2–

and R – fields and the second term is replaced by a model. Substitution into (3.88a–c) and some rearrangement yields the LES momentum equations:

∂(ρR) ∂Q + div( ρR=) = −

+ µ div(grad(R)) − (div(ρ uu ) − div( ρR=)) (3.89a) ∂t

∂x ∂(ρ{)

∂Q + div( ρ{=) = −

+ µ div(grad({)) − (div(ρ vu ) − div( ρ{=)) (3.89b) ∂t

∂y ∂(ρS)

∂Q + div( ρS=) = − + µ div(grad(S)) − (div(ρ wu ) − div( ρS=)) (3.89c) ∂t

∂z (I)

(V) The filtered momentum equations look very much like the RANS momen-

(II) (III)

(IV)

tum equations (3.26a–c) or (3.27a–c). Terms (I) are the rate of change of the filtered x-, y- and z-momentum. Terms (II) and (IV) are the convective and diffusive fluxes of filtered x-, y- and z-momentum. Terms (III) are the gradients in the x-, y- and z-directions of the filtered pressure field. The last terms (V) are caused by the filtering operation, just like the Reynolds stresses in the RANS momentum equations that arose as a consequence of time- averaging. They can be considered as a divergence of a set of stresses τ ij . In

suffix notation the i-component of these terms can be written as follows:

i u − div( ρu − ρ ) = i u) ∂(ρu i v − ρu v)

i w − ρu i w) ∂τ = ij (3.90a)

∂x j where τ ij = ρ u i u − ρ u i u = ρ u i u j − ρ u i u j

∂z

(3.90b) In recognition of the fact that a substantial portion of τ ij is attributable to

convective momentum transport due to interactions between the unresolved or SGS eddies, these stresses are commonly termed the sub-grid-scale stresses. However, unlike the Reynolds stresses in the RANS equations, the LES SGS stresses contain further contributions. The nature of these contri- butions can be determined with the aid of a decomposition of a flow variable φ(x, t) as the sum of (i) the filtered function 2(x, t) with spatial variations that are larger than the cutoff width and are resolved by the LES computation and (ii) φ′(x, t), which contains unresolved spatial variations at a length scale smaller than the filter cutoff width:

CHAPTER 3 TURBULENCE AND ITS MODELLING

(3.91) Using this decomposition in equation (3.90b) we can write the first term on

φ(x, t) = 2(x, t) + φ′(x, t)

the far right hand side as follows:

ρ u i u j = ρ ( R i +u ′)( i R j +u ′) j = ρ R i R j + ρ R i u j ′ + ρ u i ′ R j + ρ u i ′u j ′ = ρ u i u j +( ρ R i R j − ρ u i u j ) + ρ R i u j ′ + ρ u i ′ R j + ρ u i ′u ′ j

Now we can write the SGS stresses as follows:

τ ij = ρ u i u j − ρ u i u j =( ρ R i R j − ρ u i u j ) + ρ R i u j ′ + ρ u ′ i R j + ρ u i ′u ′ j (3.92)

(III) Thus, we find that the SGS stresses contain three groups of contributions: • Term (I), Leonard stresses L ij :

• Term (II), cross-stresses C ij :

C ij = ρ R i u j ′ + ρ u i ′ R j • Term (III), LES Reynolds stresses R ij :R ij = ρ u i ′u j ′ The Leonard stresses L ij are solely due to effects at resolved scale. They are

caused by the fact that a second filtering operation makes a change to a filtered flow variable, i.e. φ ≠ 2 for space-filtered variables, unlike in time- averaging, where ϕ(t) =+=Φ= ϕ(t) (compare equation (3.21)). These stress contributions were named after the American scientist A. Leonard, who first identified an approximate method to compute them from the filtered flow field (see Leonard (1974) for further details). The cross-stresses C ij are due to interactions between the SGS eddies and the resolved flow. An approx- imate expression for this term is given in Ferziger (1977). Finally, the LES Reynolds stresses R ij are caused by convective momentum transfer due to interactions of SGS eddies and are modelled with a so-called SGS turbu- lence model. Just like the Reynolds stresses in the RANS equations, the SGS stresses (3.92) must be modelled. Below we discuss the most prominent SGS models.