A generalized method for advective-diffusive computations in engineering

Fluid Structure Interaction and Moving Boundary Problems 563 A generalized method for advective-diffusive computations in engineering H. Gómez, I. Colominas, F. Navarrina & M. Casteleiro Department of Mathematical Methods, University of A Coruña, Spain Abstract Besides the computational cost of solving advective-diffusive problems in convection dominated situations, the standard statement for this phenomenon leads to the result that mass can propagate at an infinite speed. This paradoxical result occurs as a consequence of using Fick s law [1] and it is related to the appearance of boundary layers on outflow borders when convection dominates diffusion. For these reasons we propose to use Cattaneo s law [2] instead of Fick s law as the constitutive equation of the problem. This approach leads to a totally hyperbolic system of partial differential equations. A finite diffusive velocity can be defined by using this approach. Several problems have been solved to show that the proposed formulation leads to stable results in convection dominated situations. Keywords: advection-diffusion, Taylor Galerkin, Cattaneo s equation. 1 Introduction Transport problems involving convective and diffusive processes in fluid media have a great applicability in engineering. This kind of phenomena can be modeled by using the so-called advective-diffusive equation. Unfortunately, it is very complicated to obtain an accurate and stable solution for this equation when the convective term becomes important. To overcome this problem, many interesting numerical methods have been introduced, but completely satisfactory results have not been achieved (for a detailed presentation of most of these methods see reference [3]). In this paper we review the formulation of the advective-diffusive equations applied to the spillage of a pollutant into a fluid medium. In particular, we notice that Fick s law leads to the result that mass can propagate at an infinite speed. This fact is related to the appearance of spurious oscillations in the numerical

564 Fluid Structure Interaction and Moving Boundary Problems solution of this equation [4, 5]. To overcome the infinite speed paradox we use a generalized constitutive equation proposed by Cattaneo in [2]. Cattaneo s law has been used in practical applications for pure-diffusive problems instead of Fick s law, but, up to the authors knowledge, it has not been used for problems with convective term. In this study the advective-diffusive problem has been formulated by using Cattaneo s law as the constitutive equation. This approach leads to a totally hyperbolic system of conservation laws. Therefore, a finite diffusive velocity can be defined. Finally, several practical examples have been solved. 2 Standard formulation of the advective-diffusive problem 2.1 Problem statement The classic description for the advective-diffusive problem under the assumption of incompressibility is based on the equations t + a x (u)+ x (q) =0 (1) q = K x (u) (2) where Eqn (1) is the pollutant mass conservation law and Eqn (2) is the constitutive equation known as Fick s law. In the above, u is the pollutant concentration, a is the velocity vector which satisfies the hydrodynamic equations of an incompressible fluid, q is the diffusive flux per unit fluid density and K is the diffusivity tensor which is assumed to be positive definite. Clearly, the system of Eqns (1) (2) is fully decoupled as we can introduce Eqn (2) into Eqn (1) and solve t + a x (u) x (K x (u)) = 0 (3) which is a scalar equation. It is well known that the Eqn (3) is a parabolic one. Therefore, boundary conditions must be imposed everywhere on the boundary of the domain. 2.2 A pure-diffusive example to analyze the infinite speed paradox In what follows we will show that the above formulation leads to mass propagation at an infinite speed. Let us consider an (incompressible) homogeneous, isotropic (hence, if I is the identity tensor, K = ki for a certain k > 0) and one-dimensional medium. We consider a pure-diffusive situation. We suppose the domain to be long enough to be approximated as infinitely long. Finally, we assume that the pollutant is added to the medium as a rapid pulse. In this case we should solve the following problem: find u: R [0, ) R such that t k 2 u =0 x2 x R t>0 (4) u(x, 0) = δ(x) x R (5)

Fluid Structure Interaction and Moving Boundary Problems 565 where δ is the Dirac distribution. The solution of Eqns (4) (5) is u(x, t) = 1 4πkt e x2 4kt, x R, t > 0. (6) which is the Gauss distribution function at each time t. So, it is straightforward (see [5] for further details) that the mean velocity of the pollutant is not bounded. 3 Proposed formulation of the advective-diffusive problem 3.1 Problem statement We will derive this formulation by substituting Eqn (2) by Cattaneo s law which involves a tensorial function τ. This mapping transforms each point (x,t) of the domain into that point relaxation tensor. The coordinates of the relaxation tensor are specific diffusion process times. Up to the authors knowledge, Cattaneo s equation has been only used in non-advective thermal problems. Thus, we had to find Cattaneo s equation with convective term [5]. Hence, basic equations for the transport problem described by using Cattaneo s law are t + a x (u)+ x (q) =0 (7) ( ) q q + τ t + x (q) a = K x (u) (8) where Eqn (8) is Cattaneo s law with convective term. It should be noted that we are using a generalized constitutive equation as we recover the classic formulation when τ is the zero tensor. 3.2 A pure-diffusive example with a finite propagation velocity In order to compare the solution of the classic formulation with the solution of the generalized formulation we now solve the Cattaneo counterpart of Eqns (4) (5). In this simple case the system of Eqns (7) (8) can be reduced to a second order partial differential equation (for further details, see [5]). Now we need two initial conditions because this problem involves second order derivatives with respect to the time. Then, we consider an (incompressible) homogeneous, isotropic, onedimensional and non-convective medium. With the above assumptions we can state this problem as [5]: find u: R [0, ) R such that τ 2 u t 2 + t k 2 u =0 x2 x R t>0 (9) u(x, 0) = δ(x) x R (10) (x, 0) = 0 t x R (11)

566 Fluid Structure Interaction and Moving Boundary Problems If we call c = k/τ mass wave celerity and we denote s = c 2 t 2 x 2,the solution of Eqns (9) (11) is [ 1 2 e c 2 2k t δ( x ct)+ c ( ) ( )] c 2k I 0 2k s + c2 c 2ks ti 1 2k s, x ct u(x, t) = 0, x >ct (12) where I 0 and I 1 are the modified Bessel functions of the first kind of order 0 and 1. We compare in Figure 1 solutions of Eqns (4) (5) and Eqns (9) (11) at t =4and t =10respectively. Clearly, if we use Cattaneo s equation a wave front exists which advances with a celerity c. 3.3 Study of the proposed model as a system of conservation laws The system of Eqns (7) (8) can not be reduced to a second order partial differential equation in multidimensional problems with a non-constant velocity field. In this case we must solve a coupled system of first order partial differential equations. If we suppose the medium to be (incompressible) homogeneous and isotropic, the system of Eqns (7) (8) can be written as a system of conservation laws. As we consider two-dimensional problems, we will use the notation q =(q 1,q 2 ) T, a = (a 1,a 2 ) T. Then, the system of Eqns (7) (8) can be written as U t + x (F )=S (13) where u ua 1 + q 1 ua 2 + q 2 0 U = τq 1 ; F = τq 1 a 1 + ku τq 1 a 2 ; S = q 1 (14) τq 2 τq 2 a 1 τq 2 a 2 + ku q 2 It can be shown that the system of Eqns (13) is totally hyperbolic because its associated characteristic equation [6] has 3 different real solutions. At this point, it is very useful to introduce the dimensionless number H = a c (15) which plays a similar role to Mach number in gas dynamics or Froude number in shallow water problems. Then, we define supercritical flow as one characterized by a local H number such that H > 1. In the same way, we define subcritical flow as one characterized by H<1 and critical flow as the flow which satisfies H =1. It should be noted that in supercritical flow it is not possible a pollutant transport towards upstream as the circular (diffusive) mode of propagation travels with a velocity lesser than the convective one.

15 10 5 0 5 10 15 Fluid Structure Interaction and Moving Boundary Problems 567 0.16 0.14 Cattaneo s law Fick s law 0.16 0.14 Cattaneo s law Fick s law 0.12 0.12 Concentration 0.1 0.08 0.06 Concentration 0.1 0.08 0.06 0.04 0.04 0.02 0.02 0 Spatial coordinate 0 15 10 5 0 5 10 15 Spatial coordinate Figure 1: Comparison at t =4(left) and at t =10(right) between the solution of Eqns (4) (5) (dashed line) and the solution of Eqns (9) (11) (solid line). Parameters k and τ have a value of one. The question of boundary conditions which can be prescribed for the system of Eqns (13) is not trivial. A very frequent situation is to have to impose a boundary condition at a fixed solid wall. In this situation only the normal component of flux q must be prescribed. Then, boundary condition in this case will be q n =0, being n the unit outward normal vector to that boundary. However, computational domains are usually limited by inflow and outflow boundaries, as well. We call inflow boundary the part of the boundary in which a n < 0 and outflow boundary the part in which a n > 0. In these types of boundaries the issue of boundary conditions is more complicated. However, by means of a Riemann analysis [6] in the direction of the outward normal we conclude the following: on a supercritical inflow boundary all components of U must be prescribed while no components should be imposed on a supercritical outflow boundary. In contrast, on a subcritical outflow boundary only one component of U will be prescribed while 2 components of U must be imposed on a subcritical inflow boundary. Independently of boundary conditions, an initial condition must be imposed to Eqns (13). 4 Numerical examples This section is devoted to the presentation of the results obtained by using the second order Taylor Galerkin method proposed by Donea in [7]. We will present two practical examples which consist of a pollutant transport in a rectangular channel. The unstructured meshes were generated by using the code GEN4U based on the algorithm by Sarrate and Huerta [8]. 4.1 Subcritical flow in a rectangular channel We consider a rectangular channel over the domain [0, 10] [0, 2]. A typical mesh of quadrilaterals has been used. Grid steps size are x = y =0.25 m.

568 Fluid Structure Interaction and Moving Boundary Problems The values of the parameters are k = 1 and τ = 1 which implies c = 1. The velocity field employed in the calculations is a(x 1,x 2 )=(0.8x 2 (2 x 2 ), 0) T (16) which is a divergence free field. By using Eqn (15) and Eqn (16) it is straightforward that the considered flow is subcritical at each point of the domain. As initial conditions we use u(x 1,x 2 )=e (x2 1 +x2 2 ) ; q(x 1,x 2 )=0. (17) As boundary conditions, we prescribe concentration and q 2 on the inflow boundary; q 1 on the outflow boundary and q 2 = 0 at the solid wall (in all cases we impose the values which result of initial conditions). A time step of t =0.1s has been used in the calculations. We step in time until the steady state has been reached. The solution is considered to be steady when Residual = U n+1 U n U n+1 10 5 (18) We show in Figure 2 concentration solutions at t =3s, t =6s, t =9sandt =12 s Further, we plot the steady state solution. 4.2 Transcritical flow in a rectangular channel We consider a rectangular channel over the domain [0, 10] [0, 2]. The computational mesh (1375 elements) is plotted in Figure 3. The values of the parameters are k =10 2 and τ =1. Therefore, the mass wave celerity is c =0.1. We will solve this problem by using the velocity field a(x 1,x 2 )=(2x 2 (2 x 2 ), 0) T (19) which is a solenoidal one. By using Eqn (15) we find the flow to be supercritical except near the walls of the channel. In fact, this is a high velocity problem as it satisfies H = 20 at the straight line x 2 = 1. We use Eqn (17) as initial conditions. As boundary conditions, we prescribe all unknowns on the supercritical inflow boundary (we impose the values which result of the initial conditions). At the solid wall we impose q 2 =0which is also consistent with initial conditions. A time step of t =2. 10 2 s has been used in the calculations. We march in time until the Eqn (18) is satisfied. We show in Figure 4 the initial condition involving concentration and concentration solutions at t =1.5s, t =3s, t =4.5s, t =6s and t =12sFurther, we plot the steady state solution. 5 Conclusions In this paper we propose to use Cattaneo s law as the constitutive equation of the advective-diffusive problem. This approach leads to a wave-like solution and it

Fluid Structure Interaction and Moving Boundary Problems 569 Figure 2: Subcritical channel flow. Initial condition and concentration solutions at t =3s, t =6s, t =9sandt =12s in downward direction. We plot the steady state solution in the lower graphic. Figure 3: Transcritical channel flow. Computational mesh (1375 elements). avoids the infinite speed paradox which is reached by using the standard formulation. The proposed formulation constitutes a generalized approach for advectivediffusive phenomena as the standard formulation can be considered as a subcase

570 Fluid Structure Interaction and Moving Boundary Problems Figure 4: Transcritical channel flow. Initial condition and concentration solutions at t =1.5 s, t =3s, t =4.5 s, t =6sandt =12sindownward direction. We plot the steady state solution in the lower graphic. of the proposed one. When Cattaneo s law is used two kinds of flow can occur: subcritical flow (pollutant can propagate towards any direction) and supercritical flow (propagation towards upstream is not possible). From a numerical point of view, two problems have been solved to show that the Cattaneo-type model leads, on the whole, to stable solutions even if we use the standard Galerkin method for the spatial discretization.

Acknowledgements This work has been partially supported by Grant # PGDIT01PXI11802PR of the SXID of the Xunta de Galicia, Grant # DPI200200297 of the SGPICT of the Ministerio de Ciencia y Tecnología ofthe Spanish Government, research fellowships of the Universidad de A Coruña and research fellowships of the Fundación de la Ingeniería Civil de Galicia. References Fluid Structure Interaction and Moving Boundary Problems 571 [1] Fick, A., Uber diffusion, Poggendorff s Annalen der Physik und Chemie, 94, pp. 59 86, 1855. [2] Cattaneo, M.C., Sur une forme de l equation de la chaleur éliminant le paradoxe d une propagation instantaneé, Comptes Rendus de L Academie des Sciences: Series I-Mathematics, 247, pp. 431 433, 1958. [3] Donea, J. and Huerta, A., Finite element methods for flow problems, John Wiley & Sons: Chichester, 2003. [4] Gómez, H., Colominas, I., Navarrina, F. and Casteleiro, M., On the intrinsic instability of the advection diffusion equation, Proc. of the 4 th European Congress on Computational Methods in Applied Sciences and Engineering (CDROM), eds. P. Neittaanmäki,T.Rossi,S.Korotov,E.Oñate,J.Périaux y D. Knörzer, Jyväskylä, Finland, 2004. [5] Gómez, H., A new formulation for the advective-diffusive transport problem, Technical Report (in Spanish), University of A Coruña, 2003. [6] Courant R. and Hilbert D. Methods of mathematical physics. Vol II., John Wiley & Sons: New York, 1989. [7] Donea, J., A Taylor Galerkin method for convective transport problems, International Journal for Numerical Methods in Engineering, 20, pp. 101 120, 1984. [8] Sarrate, J. and Huerta, A., Efficient unstructured quadrilateral mesh generation, International Journal for Numerical Methods in Engineering, 49, pp. 1327 1350, 2000.