It has been recently proposed to use the lat- tice Boltzmann approximation as an alternative to. Boolean lattice gas simulations [1-4]. In this ap- proach, the mean ...
Physica D 47 (1991) 231-232 North-Holland
RELATION BETWEEN THE LATTICE BOLTZMANN EQUATION AND THE NAVIER-STOKES EQUATIONS U. F R I S C H CNRS, Observatoire de Nice, B.P. 139, 06003 Nice Cedes, France Received 29 D e c e m b e r 1989
It is s h o w n t h a t the lattice gas B o l t z m a n n e q u a t i o n m a y be r e w r i t t e n in a form which brings out its close relation w i t h the N a v i e r - S t o k e s equations. Various consequences are pointed out.
It has been recently proposed to use the lattice Boltzmann approximation as an alternative to Boolean lattice gas simulations [1-4]. In this approach, the mean populations Ni (real numbers) are used and their evolution is computed exactly from the lattice B o l t z m a n n equation. One advantage is the absence of the noise inherent to Monte Carlo methods. The drawbacks have been clearly discussed in refs. [1,2], so t h a t we need not come back to them. Given that one then works with a limited number (say, 6 to 24) of real fields, defined on a spacetime grid, the question arises if there is any direct relation to the finite-difference discretized N a v i e r Stokes equations. The present investigation has been carried out so far only in a restricted framework: the Boltzm a n n approximation in differential form with a six-velocity two-dimensional model of the F H P - I type [5,7]. Generalizations of the present approach to F C H C - b a s e d models in two and three dimensions have been worked out recently [6]: In the following, notation is taken from ref. [7]. We start from the B o l t z m a n n equation in discrete form
OtNi + O~ci~Ni = Ai,
(1)
where A i is the collision term. We decompose the Ni's according to a suitably chosen orthogonal basis, such t h a t the first components correspond to the hydrodynamical fields and that the other ones should display as much invariance as possible:
Ni = l p + gpcie, 1 u,~ + Qia~S,~ + (--1)ia.
(2)
The tensor S~O is symmetric and traeeless (two independent components). We multiply (1) successively by 1, cia, Qiao, ( - 1 ) i and we sum over i. We then expand the collision term, treating Ni as a p e r t u r b a t i o n of gp; 1 this is justified at low Mach numbers. The first order of the expansion is expressible from the linearized collision m a t r i x Aij. It is necessary to go to second order in u to obtain the quadratic terms in the Navier-Stokes equations and their coefficient g(p) [1,8]. Higher than quadratic terms are irrelevant. Using the invariance properties, we obtain the following system (irrelevant terms omitted):
O,p + O~(puo ) = O,
O,(p o) + 1 oop + 3
(3) = 0,
+ l[Oo(pu ) + o (pu )
(4) -
- #(u,~u~ - ~ul 2 6 ~ ) + R~.rO.ra = - A S ~ , 60ta + R,~707S,~ ~ = - ~ a .
(5) (6)
The positive coefficients A, #, and ~ depend on the density and on the collision rules. T h e y are not given explicitly here. The third-order tensor Ra~7 is given by
R~Z7 = E
ci~c'~ciT(-1)i" i
It is G-invariant but not isotropic.
0167-2789/91/$03.50 @ 1991 - Elsevier Science Publishers B.V. (North-Holland)
(7)
232
U. Frisch / Lattice Boltzmann equation and the Navier-Stokes equations
In the usual hydrodynamical regime, where the variables p and u vary slowly, the variables S ~ and a are slaved because of the presence of negative eigenvalues (proportional to - A and - a ) in the linearized collision operator. S ~ is (up to a factor) the stress tensor, that is the m o m e n t u m flux tensor. It is slaved to a combination of the rate-of-strain and of the nonlinear flux. In the slaved regime, we exactly recover the equations of hydrodynamics (with the nonlinear term modified as usual by a coefficient g(p) = 3 # / A p ) . An implication is that the rate-of-strain tensor (symmetrical part of the velocity gradient) can be obtained directly from the mean populations Ni (via the tensor S ~ ) without taking any space-derivatives; this remark does unfortunately not extend to the vorticity (antisymmetrical part). As for the field a, it is irrelevant in the hydrodynamical limit. Its (tensor) generalization to the F C H C case (pseudo4D) can nevertheless become relevant for models with negative viscosities [9]: it is then necessary to include spatial derivatives up to fourth order in the hydrodynamical equations; also, it may not be legitimate to truncate the expansion of the collision term. The slaving of S ~ , that is the possibility to neglect O , S ~ , will hold so much more so as the eigenvalue A (roughly, the inverse collision viscosity) is larger. Thus, efforts to minimize the (collision) viscosity are also likely to improve the validity of the hydrodynamic approximation. With this view-point, the Boltzmann equation (1) can also be regarded as an approximation to the NavierStokes equations. This approximation is hyperbolic and introduces a slight delay (of order I/A) in the slaving of the stress tensor. We may now consider the lattice Boltzmann equations as a finite difference approximation to (1) and, thereby to the Navier-Stokes equations. The lattice Boltzmann equations read: N i ( t . + 1, r . + ci) - N i ( t . , r . ) ---- Ai.
(8)
It is known that existence-uniqueness-regularity problems are very hard at the Navier-Stokes level (in three dimensions), and even harder at the Boltzmann level. However, at the lattice Boltzmann level, we have a simple polynomia ! iteration which poses no existence-uniqueness-regularity problems. Furthermore, the H theorem (Hdnon's
version; see appendix F of ref. [7]) ensures "good" behaviour, even for large times, for those models satisfying semi-detailed balance. We may also consider the system (3)-(6) from the view-point of "large eddy simulations", that is for situations where the Reynolds number is too large for the resolution retained. Eq. (5) provides us then with an equation for the sub-grid-scale stresses. We have thus shown that the lattice Boltzmann technique may be regarded as a new finitedifference technique for the Navier-Stokes equations having the property of unconditional stability. One final remark: Grad [10] has shown that the Navier-Stokes equations can be obtained from the full (non-discrete) Boltzmann equation by an expansion procedure involving Hermite polynomials in the microscopic velocity. Our approach to the lattice Boltzmann equation can be viewed as a Grad-type expansion, but one which terminates after finitely many terms, because of the discreteness of velocity space. We are grateful to D. d'Humi~res and Y. Pomeau for useful remarks.
References [1] F.J. Higuera and J. Jimdnez, Europhys. Lett. 9 (1989) 663. [2] G.R. Mc Namara and G. Zanetti, Phys. Rev. Lett. 61 (1988) 2332. [3] F.J. Higuera, Lattice gas simulation based on the Boltzmann equation, in: Discrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrodynamics, Torino, September 20-24, ed. R. Monaco (World Scientific, Singapore, 1989) pp. 162-177. [4] F.J. Higuera and S. Succi, Europhys. Lett. 8 (1989) 517. [5] U. Frisch, B. Hasslacher and Y. Pomeau, Phys. Rev. Lett. 56 (1986) 1505. [6] M. Vergassola, R. Benzi and S. Succi, Europhys. Lett. 13 (1990) 411. [7] U. Frisch, D. d'Humi~res, B. Hasslacher, P. Lallemand, Y. Pomeau and J.P. Rivet, Complex Systems 1 (1987) 632.
[8] B. Dubrulle, Complex Systems 2 (1988) 577. [9] B. Dubrulle, U. Frisch, M. Hdnon and J.P. Rivet, Low viscosity lattice gases, J. Star. Phys. 59 (1990) 1187. [10] H. Grad, Commun. Pure Appl. Math. 2 (1949) 331; Phys. Fluids 6 (1963) 147.
