Modelling and numerical methods for the diffusion of impurities in a gas

Transcription

1 INERNAIONAL JOURNAL FOR NUMERICAL MEHODS IN FLUIDS Int. J. Numer. Meth. Fluids 6; : 6 [Version: /9/8 v.] Modelling and numerical methods for the diffusion of impurities in a gas E. Ferrari, L. Pareschi Department of Mathematics and Center for Modelling Computing and Statistics (CMCS) University of Ferrara, Via Machiavelli 35, 44 Ferrara, Italy. elisa.ferrari@unife.it, lorenzo.pareschi@unife.it. SUMMARY he dynamics of a mixture of impurities in a gas can be represented by a system of linear Bolzmann equations for hard spheres. We assume that the background is in thermodynamic equilibrium and that the polluting particles are sufficiently few (in comparison to the background molecules) to admit that there are no collisions among couples of them. In order to derive non trivial hydrodynamic models we close the Euler system around local Maxwellians which are not equilibrium states. he kinetic model is solved by using a Monte Carlo method, the hydrodynamic one by Implicit- Explicit (IMEX) Runge-Kutta schemes with WENO reconstruction []. Several numerical tests are then computed in order to compare the results obtained with the kinetic and the hydrodynamic models. Copyright c 6 John Wiley & Sons, Ltd. key words: Diffusion of impurities; Boltzmann equation; Hydrodynamic models; Monte Carlo methods; IMEX Runge Kutta schemes; WENO schemes. INRODUCION We start by considering a mixture of N species of impurities (treated as inelastic granular gases) constituted by particles with mass m i and diameter δ i (i =,,..., N), which interact with a background gas constituted by molecules with mass m b and diameter δ b. If we call respectively f i (x, v, t) and f b (x, v, t) the distribution functions of the particles, their dynamics can be described by a system of N + dissipative Boltzmann equations for hard spheres f i t + v xf i = N j= f b t + v xf b = ε b Q(f b, f b ) + Q(f i, f j ) + Q(f i, f b ) ε ij ε ib N j= ε jb Q(f j, f b ), () Correspondence to: Department of Mathematics, University of Ferrara, Via Machiavelli 35, 44 Ferrara, Italy. Copyright c 6 John Wiley & Sons, Ltd. Received Revised

2 E. FERRARI, L. PARESCHI where i =,..., N and ε ij = ε ji, ε ib = ε bi, ε b = ε bb are the Knudsen numbers. he collisions involving impurities are inelastic with coefficients of restitution < e ij, e ib respectively whereas the collision between background molecules are elastic e b =, so that the Boltzmann collision operators Q(f i, f j ), for i, j = b,,..., N, can be written as [ ] Q(f i, f j ) = δ ij π R 3 S (v w) n e ij f i (v )f j (w ) f i (v)f j (w) dwdn. () Here δ ij = δ i δ j is the distance between the centers of the colliding particles and n S is the unit vector of impact between the velocities v and w that collide returning v and w. hese pre-collisional velocities are given by ) [(v w) n] n v = v α ji ( + e ij w = w + α ij ( + e ij ) [(v w) n] n, m i with α ij =, so that α ji = α ij, for i, j = b,,..., N. m i + m j he collision operators conserve the particle number of each species of impurities and the total momentum N i,j= R 3 m i v Q(f i, f j ) dv + (3) R 3 Q(f i, f j ) dv = (4) N but part of the kinetic energy is dissipated according to the equation i= R 3 m i v Q(f i, f b ) dv =, (5) (v w ) n = e ij (v w) n. (6) From the relation (6) and the momentum conservation during the collision m i v + m j w = m i v + m j w, (7) we obtain the collisional mechanism { v = v α ji ( + e ij ) [(v w) n] n w = w + α ij ( + e ij ) [(v w) n] n. We introduce the parameters of inelasticity β ij = e ij, so that we can rewrite (8) as { v = v α ji ( β ij ) [(v w) n] n w = w + ( α ji )( β ij ) [(v w) n] n. he rest of the paper is organized as follows. Section is devoted to the introduction of the simplified model we consider in this work. Next we introduce the Monte Carlo method applied to the kinetic model in our simulations. hen we derive hydrodynamic models for Maxwellian molecules and hard spheres and describe the IMEX Runge-Kutta schemes used to solve them numerically. Finally we presents some numerical tests and we compare the results obtained with the two models and the corresponding numerical methods. (8) (9) Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

3 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS 3. A SIMPLIFIED KINEIC MODEL he complexity of the model describing the motion of impurities in a background gas, can be strongly reduced under the following simplifying assumptions. he particles are sufficiently few (i.e. the granular gases are sufficiently rarefied), in comparison to the number of background molecules, to admit that there are no collisions among couples of them. he background is in thermodynamic equilibrium, with temperature b and mean velocity u b, and its distribution function is the normalized Maxwellian ( ) 3/ mb M b (v) = exp { m b(v u b ) }, v R 3. π b b hus we have ε ij >> ε ib >> ε b, for i, j =,..., N, in system () so that we can neglect the terms ε ij Q(f i, f j ) and we can take f b = M b. his choice corresponds to take only the first term of the Chapman-Enskog expansion for f b. hus the moments of the background satisfy the closed Euler system b t + x ( b u b ) = b u b + x ( b u b u b + b b ) = () t E b + x (E b u b + b b u b ) =, t where the density b, the mean velocity u b, the mean temperature b and the energy E b are given by b = f b (v, t) dv, R 3 u b = v f b (v, t) dv, b R 3 b = () (v u b ) f b (v, t) dv, 3 b R 3 E b = v f b (v, t) dv. R 3 Finally system () can be written using N simplified dissipative linear Boltzmann equations f i t + v xf i = δ [ ] ib q n ε π R 3 S e f i (v )f b (w ) f i (v)f b (w) dwdn () ib f b = M b where q = v w and i =,..., N. For the system () it is possible to prove (see [9]) that the stationary equilibrium states of the collision operators are given by the Maxwellian distributions M i (v) = ( m i π i ) 3/ exp { m i(v u b ) i }, v R 3, (3) Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

4 4 E. FERRARI, L. PARESCHI having the same mean velocity of the background and temperatures i = ( α bi) ( β ib ) α bi ( β ib ) b (4) lower than the background one. For simplicity, from now on we are considering the case in which N =, so that the problem () is reduced to the dissipative Boltzmann equation f t + v xf = δ πε R 3 S q n [ e f(v )M b (w ) f(v)m b (w) ] dwdn, (5) where f = f is the distribution function of the impurities, δ = δ b is the distance between the centers of the colliding particles, e = e b is the restitution coefficient in the mixed collision and ε = ε b is the Knudsen number. 3. HYDRODYNAMIC MODELS In this section we summarize the derivation of hydrodynamic models from the kinetic one. We first consider the so called pseudo-maxwellian case, then the hard spheres case (see [4, 6]). he pseudo-maxwellian approximation consists in replacing the relative velocity q by a different vector S(x, t) q, where S is indipendent on the integration variables and q = q/ q. For simplicity we assume without loss of generality S(x, t) =. In both cases considered, the existence of a Maxwellian equilibrium at non-zero temperature (3) allows to construct hydrodynamic models for the granular flow. However, here only the mass of the inelastic particles is preserved. hus the mass is the unique hydrodynamic variable and the Euler system is reduced to the single advection equation t + x (u) =. (6) o obtain more detailed hydrodynamic models, we perform a closure for the moment equations by replacing f with the local Maxwellian M(, u, ) were, u and are the the density, the mean velocity and the temperature of f at the time t. Off course the validity of this assumption may be questionable and one of the aim of the present paper is in fact to verify numerically the results obtained under this closure. In order to avoid the term in the right hand side of (5) and to obtain the equations for e the moments, it is useful to consider the weak form of the Boltzmann equation. Given any test-function ϕ(v), we have < ϕ, Q(f, M b ) >= δ πε R 3 w R3 v S q n (ϕ(v ) ϕ(v))f(v)m b (w)dwdvdn, (7) where <, > represents the inner product in L (R 3 ) and the post-collisional velocity v is defined as v = v α( β)(q n)n. (8) As usual, we choose as test function respectively ϕ = (that is the unique collision invariant), ϕ = v and ϕ = v. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

5 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS 5 As shown in [4, 6], the weak form of the Boltzmann equation for pseudo-maxwellian molecules leads to the following dissipative Euler system t + (u) = t u + (u u ) + ( R ) α( β) = b (u u b ) (9) ( ε ( t u + 3 ) ) ( ( R + u u + 5 ) ) R = b D(x, t) ε with D(x, t) = α( β) (u u b ) u (u u b ) 3R [ α( β)] ( ). he constant R = k B /m, where k B is the Boltzmann constant. If we apply the same procedure to the hard spheres case, we obtain where + (u) =, t t u + (u u ) + ( R ) = t B(x, t) = (u u b) 4 ( ( u + 3 R ) ) + 5 ( u α( β) ε ( u + 5 ) R = A(x, t) = 4 [ R 3 ψ( ) + (u u b) 5 [ α( β) + ψ( ) 3 ] + 4 R ψ( ) (u u b) ψ( ) b A(x, t) π R α( β) ε ] (u u b ), () ψ( ) b B(x, t) π R [ α( β) ψ( ) ] u (u u b) 3 8R 3ψ( ) u (u u b) + 8 R [α( β) ψ( ) ], ψ( ) and ψ( ) = α ( α) b + α. 4. HYBRID MONE CARLO MEHODS FOR HE KINEIC MODEL In order to solve numerically the kinetic system we consider the development of an hybrid scheme in which we simulate the evolution of the particles accordingly to the following method. he background moments b, u b, b are computed using a deterministic method (finite volume / finite differences / finite elements). he particles evolution is computed with a probabilistic method (Direct Simulation Monte Carlo). Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

6 6 E. FERRARI, L. PARESCHI he two steps are connected by sampling the background velocities from the Maxwellian distribution M b = M( b, u b, b ). his can be done in a conservative as in [, 3, 4]. During the interaction the velocity of a particle of impurity changes following the relation where α = α b and β = β b. Using the identity we can rewrite () as v = v α ( β) [(v w) n] n, () (q n) + φ ((q n) n) dn = q S 4 S φ ( q q n ) dn, v = v α ( β) [(v w) v w n]. () he unit vector n is chosen uniformly in the unit sphere, so it is cos φ sin θ n = sin φ sin θ cos θ, (3) where θ = arccos( ξ ) and φ = π ξ, with ξ, ξ uniformly distributed random variables in [, ]. he classical fictious collision method [] for variable hard spheres with Acceptance- Rejection strategy is then applied, so that the collision between a particle of impurity, having velocity v i, and one of background, with velocity w j, happens only if Σ ξ < v i w j, where ξ [, ] is a uniformly distributed random variable and the upper bound Σ should be given by Σ = max v i w j. i,j he computation of the upper bound Σ, if N is the number of colliding couples, requires O(N ) operations. However taking the larger bound Σ = max i v i ū + max w j ū b + ū ū b, j we can reduce the computational cost at O(N). Here we denoted with ū b and ū the mean velocities respectively of the background and of the impurities. We refer to [] and references therein for further details on the method for the Boltzmann equation. 5. IMEX-WENO SCHEMES FOR HE FLUID MODEL o solve numerically the hydrodynamic models (9) and () we apply the implicit-explicit (IMEX) Runge-Kutta schemes up to the third order []. hese schemes are strong-stabilitypreserving (SSP) for the limiting system of conservation laws in the limit of vanishing Knudsen Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

7 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS 7 number. hese schemes assure the asymptotic preserving property, i.e. the consistency of the scheme with the equilibrium system and the asymptotic accuracy, so that the order of accuracy is maintained also in the zero relaxation limit. he high accuracy in space is obtained by using a Weighted Essentially Non Oscillatory (WENO) reconstruction [5]. Let us summarize shortly the construction of such schemes. In one space dimension an hyperbolic system with relaxation for conservative variables has the form t U + x F (U) = R(U), x R. (4) ε he Jacobian matrix F (U) has real eigenvalues and admits a basis of eigenvectors U R n. ε is the relaxation parameter. An IMEX Runge-Kutta scheme consists of applying an implicit discretization to the source terms and an explicit one to the non stiff term. When applied to the previous system it takes the form U (i) = U n+ = i U n t ã ij x F (U (j) ) + t U n t j= ν w i x F (U (i) ) + t i= ν j= ν i= a ij ε R(U (j) ) w i ε R(U (i) ) he matrices à = (ã ij ), ã ij = for j i and A = (a ij ) are ν ν matrices such that the resulting scheme is explicit in F, and implicit in R. An IMEX Runge-Kutta scheme is characterized by these two matrices and the coefficient vectors w = ( w,..., w ν ), w = (w,..., w ν ). Since the simplicity and efficiency of solving the algebraic equations corresponding to the implicit part of the discretization at each step is of paramount importance, it is natural to consider diagonally implicit Runge-Kutta (DIRK) schemes for the source terms (a ij =, for j > i). IMEX Runge-Kutta schemes can be represented by a double tableau in the usual Butcher notation, c à c A w w where the coefficients c and c used for the treatment of non autonomous systems, are given by the usual relation i i c i = ã ij, c i = a ij. (6) j= he use of a DIRK scheme for R is a sufficient condition to guarantee that F is always evaluated explicitly. In all the schemes we have considered the implicit tableau corresponds to an L-stable scheme, that is w A e =, e being a vector whose components are all equal to, whereas the explicit tableau is SSPk, where k denotes the order of the SSP scheme. We shall use the notation SSPk(s, σ, p), where the triplet (s, σ, p) characterizes the number s of stages of the implicit scheme, the number σ of stages of the explicit scheme and the order p of the IMEX scheme. In particular, we will make use of the asymptotically SSP schemes of second and third order, reported in ables I and II. j= (5) Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

8 8 E. FERRARI, L. PARESCHI / / γ γ γ γ γ / / γ = able I. ableau for the explicit (left) implicit (right) IMEX-SSP(,,) L-stable scheme / /4 /4 /6 /6 /3 α α α α α α / β η / β η α α /6 /6 /3 α = , β = η = able II. ableau for the explicit (left) implicit (right) IMEX-SSP3(4,3,3) L-stable scheme In all the computations described in this paper we used finite difference WENO schemes with Lax-Friedrichs flux and conservative variables. his choice, together with the monotonicity condition, will ensure that the overall scheme will not produce spurious numerical oscillations []. Finally let us remark that because of the non-linearity of the terms A(x, t) and B(x, t) in (), the corresponding implicit solver has been solved by using a Newton s method. 6. NUMERICAL ESS We are interested in comparing the results obtained with the kinetic model and the hydrodynamic model () in the case of hard spheres. he tests presented in this section differ for the initial condition of the impurities, whereas the background for simplicity is assumed to be constant, with mass, mean velocity and temperature given by b = u b = b =. In all the graphics presented in this section we plot the initial condition for the together with the solution obtained by and the solution with IMEX-SSP(,,) and/or IMEX- SSP3(4,3,3). 6.. Inert concentration of impurities In the first test case we consider a concentration of impurities, transported by a constant wind in -d. he evolution is studied for different values of the Knudsen number, namely ε =.,.5,.. he mass density of the impurities consists in a square wave profile characterized by = for. x and = elsewhere. he mean velocity is assumed to be zero (u = ) and the temperature to be small ( =.5). he boundary conditions are periodic. As shown in Figs. -6, there is a good agreement between the results obtained with the different models for all range of Knudsen numbers. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

9 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS 9 When the value of ε decreases, the equilibrium condition u = u b and = is achieved faster. In particular, when ε = 4 the equilibrium is reached in few time steps and the impurities are just transported with velocity u b and temperature. In this case (Figs. 4-6) we report also the results of the third order IMEX-SSP3(4,3,3) schemes which clearly shows the better resolution of the discontinuity. 6.. Concentrations of impurities with different mean velocities In this test we consider two concentrations of impurities characterized respectively by = for. x.3 and =.6 for.5 x.7 ( = elsewhere), the first one having zero mean velocity, the second one having negative mean velocity. he initial condition for velocity and temperature correspond to a shock profile, with the following left-hand side and right-hand side values u l =, l =. and u r =.5, r =.. he boundary conditions are characterized by a zero flux and the evolution is studied for ε =.,.. We present in Figs. 7-9 the results for ε =. at the instants t = and t =, and in Figs. - the results for t =. when ε =.. In this last case we show the results obtained with IMEX-SSP3(4,3,3) instead of IMEX-SSP(,,). Again there is a good agreement between the two numerical solutions even for large values of the relaxation rate ɛ Shock waves profiles We finally consider a Riemann problem. In particular we have chosen the classical Lax data l = 45, u l =.698, p l = 3.58 and r =.5, u r =, p r =.57, where p = R, and here R =. We still consider zero flux conditions on the boundary and present in Figs. 3-8 the pictures of the solutions at t = 5 and t = 3 for ε = and at t = and t = for ε =.. A good agreement of the two models is observed also in this case. 7. CONCLUSIONS In this paper we have presented some recent results on the modelling of the diffusion of impurities in a gas. he main goal of using hydrodynamic models is the possibility to strongly reduce the computational cost of the full kinetic model. his is especially true when the mean free path of particles is small. We have introduced a Monte Carlo scheme which is suitable for the solution of the kinetic model and an IMEX-WENO method for the solution of the hydrodynamic one. For such model the time step is independent of the mean free path between particles. For the test cases considered here, the results obtained with the hydrodynamic model () are very closed to the ones of the kinetic model. his is very promising for future application based on the fluid model. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

10 E. FERRARI, L. PARESCHI he hydrodynamic model presented in this paper is derived by using a logarithmic entropy functional. As described in [] this is not the unique possibility. It is under investigation the choice of a quadratic entropy functional and the comparison of the corresponding model with the kinetic one. Finally, we hope to extend the schemes to the case of more than one kind of impurities and including some simple chemical reactions. REFERENCES. Bisi, M., Spiga, G., Fluid-dynamic equations for granular particles in a host medium, J. Math. Phys. 46, pp. 33: (5).. Bobylev, A.V., Carrillo, J.A., Gamba, I.M., On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys., 98, pp (). 3. Brey, J.J., Dufty, J.W., Santos, A., Kinetic Models for Granular Flow, J. Statist. Phys., 97, pp. 8 3 (999). 4. Brull, S., Pareschi, L., Dissipative hydrodynamic models for the diffusion of impurities in a gas, Appl. Math. Lett., 9(6), pp (6). 5. Ferrari, E., Pareschi, L., Hybrid Monte Carlo methods for the diffusion of impurities in a gas, Modelling and Numerics of Kinetic Dissipative Systems, L. Pareschi, G. Russo, G. oscani editors, Nova-Science, New York, pp. 9 4 (6). 6. Garzò, V., Dufty, J.W., Homogeneous cooling state for a granular mixture, Phys. Rev. E, 6 (999). 7. Garzò, V., Montanero, J. M., Diffusion of impurities in a granular gas, Phys. Rev. E, 69 (4). 8. LeVeque, R.J., Numerical Methods for Conservation Laws, Lectures in Mathematics, Birkhäuser Verlag, Basel (99). 9. Lods, B., oscani, G., he dissipative linear Boltzmann equation for hard spheres, J. Statist. Phys., 7 (3 4), pp (4).. Pareschi, L., Russo, G., An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM Proc.,, pp ().. Pareschi, L., Russo, G., Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput. 5, no. -, 9 55 (5).. Pareschi, L., Russo, G., ime Relaxed Monte Carlo methods for the Boltzmann equation, SIAM J. Sci. Comput., 3, pp (). 3. Pareschi, L., razzi, S., Numerical solution of the Boltzmann equation by ime Relaxed Monte Carlo (RMC) methods, Int. J. Num. Meth. Fluids, 48, pp (5). 4. Pullin, D.I., Generation of normal variates with given sample, J. Stat. Comput. Simul., 9, pp (979). 5. Shu, C.-W., Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Math. 697, Springer, Berlin, pp (998). 6. Spiga, G., oscani, G., he dissipative linear Boltzmann equation, Appl. Math. Lett., 7 (3), pp (4). Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

11 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS.4 time =.8.4 time = ε = ε =.5 Figure. est - Evolution of the mass density with ε =. (left) and ε =.5 (right). 5 time =.8. time = ε = ε =.5 Figure. est - Evolution of the velocity with ε =. (left) and ε =.5 (right). time =.8.4 time = ε = ε =.5 Figure 3. est - Evolution of the temperature with ε =. (left) and ε =.5 (right). Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

12 E. FERRARI, L. PARESCHI.4. SSP SSP3 time = ε =. Figure 4. est - Evolution of the mass density with ε =... time = SSP SSP ε =. Figure 5. est - Evolution of the velocity with ε =...4. SSP SSP3 time = ε =. Figure 6. est - Evolution of the temperature with ε =.. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

13 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS 3.4 time =..4 time = ε = ε =. Figure 7. est - Evolution of the mass density with ε =. at t = and t =..5 time =..5 time = ε = ε =. Figure 8. est - Evolution of the velocity with ε =. at t = and t =..4 time =..4 time = ε = ε =. Figure 9. est - Evolution of the temperature with ε =. at t = and t =. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

14 4 E. FERRARI, L. PARESCHI.4. time = ε =. Figure. est - Evolution of the mass density with ε =. at t =... time = ε =. Figure. est - Evolution of the velocity with ε =. at t =...4. time = ε =. Figure. est - Evolution of the temperature with ε =. at t =.. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

15 MODELLING AND NUMERICS FOR HE DIFFUSION OF IMPURIIES IN A GAS 5 time = 5. time = ε = ε = Figure 3. est 3 - Evolution of the mass density with ε = at t = 5 and t = 3..6 time = 5..7 time = ε = ε = Figure 4. est 3 - Evolution of the velocity with ε = at t = 5 and t = 3. time = 5. time = ε = ε = Figure 5. est 3 - Evolution of the temperature with ε = at t = 5 and t = 3. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6

16 6 E. FERRARI, L. PARESCHI time =. time = ε =. ε =. Figure 6. est 3 - Evolution of the mass density with ε =. at t = and t =.. time =.. time = ε = ε =. Figure 7. est 3 - Evolution of the velocity with ε =. at t = and t =. time =. time = ε = ε =. Figure 8. est 3 - Evolution of the temperature with ε =. at t = and t =. Copyright c 6 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 6; : 6