Multiple relaxation time

f_j−f_j⁽⁰⁾



. (4.53)

The collision matrix acts on the nonequilibrium distribution vector f_j− f_j⁽⁰⁾, resulting in a vector of changes to the distribution function. Other linear discrete collision operators, including BGK, can be found as a special case of this general linear discrete collision operator. In the BGK case,Ω_ij= −δ_ij/τ.

A number of alternative, more complex collision operators have been proposed to improve on the occasionally problematic BGK operator, though they generally build on the same principle of relaxing towards equilibrium. We will briefly go through the main ones as they all give a valuable additional understanding of the LB method.

4.3.1 Multiple relaxation time

Multiple relaxation time (MRT) operators evolved out of early experi-ments with general collision operators as described by (4.53) [8]. The distribution functions f_ican be represented as a q-dimensional vector f, which can be transformed to another basis. With MRT models, this basis is q hydrodynamic and nonhydrodynamic moments of f_i.

The hydrodynamic moments are typically Π0 = ρ, Πα = ρu, and Π_αβ, the moments which are relevant for the link to hydrodynamics. The nonhydrodynamic moments are not directly relevant for the hydrodynam-ics behaviour of the model, but must usually be present to fill out the moment basis.

The relaxation to equilibrium is performed in the moment basis instead of the f basis like the BGK operator. After relaxation, the moments are then transformed back to the f_i basis for streaming. The advantage of relaxing in the moment basis is that different moments can be relaxed at different rates.

98 Chapter 4 The lattice Boltzmann method

The transformation to moment basis is done using the moment trans-formation matrix M_ij, where

M f =m, M f⁽⁰⁾=m⁽⁰⁾. (4.54) m and m⁽⁰⁾ are the resulting moment vectors of the transformation.

The simplest possible practical example is the D1Q3 velocity set de-scribed in section 4.1.3. Slightly abusing the notation of the indices of f_i by using f+= f₁for the particles moving in the+x-direction and f₋= f₂ for those moving in the−x-direction, (4.54) becomes

⎡

D1Q3 can be handily transformed into a fully hydrodynamic basis.

However, other velocity sets also need some nonhydrodynamic moments to fill out the moment basis, as the number of hydrodynamic quantities is smaller than the number of velocities.

For instance, D2Q9 has six independent hydrodynamic variables: ˘Π0, Π˘x, ˘Πy, ˘Πxx, ˘Πxy, and ˘Πyy. The other three possible moments needed to fill out the basis cannot be entirely hydrodynamic, as they have to be linearly independent from the fully hydrodynamic moments which have been accounted for already. Choosing ˘Πxxy, ˘Πxyy, and ˘Πxxyyfor the other

The nonhydrodynamic moments are often called ghost modes. These

Ghost modes

Nonhydrodynamic behaviour that coexists with the hydrodynamic behaviour in LB simulations

moments are not relevant to the Chapman-Enskog expansion, and there-fore affect the fluid behaviour in LB simulations only indirectly. Using the discrete BGK collision operator, these moments decay to equilibrium with the relaxation timeτ, like the other moments.

4.3 Alternative collision operators 99

The q×q collision matrix Ω is assumed to be diagonalisable, or expressable as

Ω=M⁻¹T M (4.57)

where the relaxation matrixT is usually diagonal. Relaxation matrix The matrix in the MRT collision operator containing individual relaxation times for each moment

The generalised LBE (4.53), left-multiplied withM, results in a relaxa-tion equarelaxa-tion in moment space,

mout=m+T

m−m⁽⁰⁾

. (4.58)

mout can be seen as the post-collision moment vector, or the moment vector of the particles being streamed out of the node. We see that each element in the diagonal matrix T is a relaxation time for one of the moments in m. In the BGK special case, all relaxation times are equal, andT = −¹_τI so that

Ω= −_τ¹M⁻¹IM= −¹_τI;

the case mentioned previously.

More generally, the different moments can have different relaxation times. For the conserved moments, the relaxation times do not matter, as ρ = ρ⁽⁰⁾ and u = u⁽⁰⁾. For symmetry reasons, the non-conserved hydrodynamic momentsΠ_αβshould all have the same relaxation time;^* from section 4.1.2 we have that the relaxation of this second order moment determines the viscosity of the model.

The post-collision momentsmoutcannot be propagated directly, and must be converted back to the distribution function basis like f = M⁻¹m before streaming. Thus, in principle, the MRT algorithm works by stream-ing, conversion to moment basis, relaxation of the moments, conversion back to the distribution function basis, and streaming again. In practice, it is more efficient to computeΩ directly and perform the relaxation as in (4.53).

The usefulness of MRT lies in setting different relaxation times for the different nonhydrodynamic moments. For a given lattice and a given moment basis, an analysis can be carried out to find the optimal nonhydrodynamic relaxation times to optimise certain aspects of the behaviour of the LB method [83–86].

A downside is that such analyses do not give universal results. The results are specific to each velocity set, each choice of moment basis, and each desired optimal behaviour. These analyses can also be difficult both to perform and to comprehend.

However, one simple general-purpose choice that can vastly improve the accuracy and stability of the LBM is to choose a relaxation time of 1

*Note however that it is possible to use some special techniques when relaxingΠαβto allow setting the shear and bulk viscosity independently [83, Ch. 4].

100 Chapter 4 The lattice Boltzmann method

for the nonhydrodynamic moments [8, 78]. Thus, the nonhydrodynamic moments are always fully relaxed in each time step instead of oscillating analogously to the underrelaxation shown in Figure 4.4. In section 6.3 we shall look at a case where this choice allows accurate simulations with τ=1/2, i.e. with no viscosity at all.

One argument leveled against MRT is that it is a numerical technique with no corresponding physical basis in kinetic theory [87]. However, that does not in itself make the method less valuable.