summary:In this paper, we are concerned with the numerical solution of 2D/3D flows through a branching channel where viscous incompressible laminar fluid flow is considered. The mathematical model in this case can be described by the system of the incompressible Navier-Stokes equations and the continuity equation. In order to obtain the steady state solution the artificial compressibility method is applied. The finite volume method is used
for spatial discretization. The arising system of ordinary differential equations (ODE) is solved by a multistage Runge-Kutta method. Numerical results for both 2D and 3D cases are presented

Keslerová, Radka

Kozel, Karel

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Numerical solution of 2D and 3D incompressible laminar flows through a branching channel

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PANM 12Radka Keslerová; Karel KozelNumerical solution of 2D and 3D incompressible laminar flows through a branching channelIn: Jan Chleboun and Petr Přikryl and Karel Segeth (eds.): Programs and Algorithms of Numerical Mathematics,Proceedings of Seminar. Dolní Maxov, June 6-11, 2004. Institute of Mathematics AS CR, Prague, 2004. pp. 94–101.Persistent URL: http://dml.cz/dmlcz/702781Terms of use:© Institute of Mathematics AS CR, 2004Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly forpersonal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://dml.czNUMERICAL SOLUTION OF 2D AND 3D INCOMPRESSIBLELAMINAR FLOWS THROUGH A BRANCHING CHANNEL ∗Radka Keslerová, Karel KozelAbstractIn this paper, we are concerned with the numerical solution of 2D/3D flows througha branching channel where viscous incompressible laminar fluid flow is considered.The mathematical model in this case can be described by the system of the incom-pressible Navier-Stokes equations and the continuity equation. In order to obtainthe steady state solution the artificial compressibility method is applied. The finitevolume method is used for spatial discretization. The arising system of ordinary dif-ferential equations (ODE) is solved by a multistage Runge-Kutta method. Numericalresults for both 2D and 3D cases are presented.1. Mathematical modelThe motivation for numerical solution of the fluid flow in branching channelsarises in many applications, e.g., in biomedicine, the solution of the blood flow incardiovascular system is of interest. The study of the blood flow in large and mediumarteries is a very complex task because of the heterogeneous nature of the problemand the extreme complexity of blood and arterial wall dynamics. Mathematicaland numerical investigation of the blood circulatory system is one of the majorchallenges of the coming decades. During the 1970s, in vitro experiments were themain mode of cardiovascular investigations. Recently, the advances in computationalfluid dynamics have lead to a significant breakthrough in vascular research. Physicalquantities that are hard to measure in vivo can be computed for real geometries now(for details see, e.g., [4])In this paper the system of Navier-Stokes equations for incompressible laminarflow is treated, which can be considered as a simplified mathematical model of theblood flow in a cardiovascular problem. The system of 2D Navier-Stokes equationsand the continuity equation in two dimensional case written in the conservative formreadsR̃Wt + Fx + Gy =R̃Re∆W, R̃ = diag‖0, 1, 1‖, (1)whereW = (p, u, v)T , F = (u, u2 + p, uv)T , G = (v, uv, v2 + p)T . (2)The following notation is used:∗This work was partly supported by grant No. 201/05/0005 and Research PlanMSM 68 4077000394Fig. 1: Considered forms of channel for 2D case.(u, v)T - velocity vector for 2D in dimensionless formp - kinetic pressureRe - Reynolds number defined as Re = q∞lνν - the kinematic viscosityl - the height of the entranceq∞ - reference velocity valueAt the inlet, the Dirichlet boundary condition for the velocity vector (u, v)T is pre-scribed, at the outlet the pressure value is given. On the wall, the zero Dirichletboundary conditions for the components of velocity are used.2. Numerical solution by the finite volume methodIn what follows we are interested in steady state solutions. In such a case, theartificial compressibility method can be applied for the solution of the system (1),i.e.,Wt + F̃x + G̃y = 0, (3)whereF̃ = F − 1ReF v, G̃ = G− 1ReGv,and F, G are inviscid fluxes defined by (2), whereas F v, Gv are viscous fluxes,F v = (0, ux, vx)T , Gv = (0, uy, vy)T .Eq. (3) is integrated over Dij (Dij is a finite volume cell, µij =∫∫Dijdxdy),∫∫DijWtdxdy = −∫∫Dij(F̃x + G̃y)dxdy, (4)the mean value theorem is applied to the left-hand side of (4), and Green’s theoremon the right-hand side of (4), so that95Wt |ij= − 1µij∮∂DijF̃dy − G̃dx. (5)Next, we numerically approximate the integrals on the right hand side of (5) byWt |ij= − 1µij4∑k=1F̃ij,k∆yk − G̃ij,k∆xk, (6)where viscous fluxes in F̃ , G̃ are conjucted using dual volumes.DDDDi−1,j Diji,j+1i+1,ji,j−1k=2k=3k=4k=1∆x∆y22Fig. 2: Finite volume cell.The ODE system (6) is time-discretized with the aid of a multistage Runge-Kuttamethod, i.e.,W nij = W(0)ijW(r)ij = W(0)ij − αr∆tRW (r−1)ij (7)W n+1ij = W(m)ij r = 1, . . . , m,where m = 3, α1 = α2 = 0.5, α3 = 1, RWnij = RWnij −DW nij, and the steadyresidual RWij is defined byRWij =1µij4∑k=1[(F ik −1ReF vk)∆yk −(Gik −1ReGvk)∆xk]. (8)The added artificial viscosity term DWij of Jameson’s type (for details see, e.g., [3])is defined asDWij = E [γi (Wi+1,j − 2Wij + Wi−1,j) + γj (Wi,j+1 − 2Wij + Wi,j−1)] (9)E = diag‖0, ε1, ε2‖, ε1, ε2 ∈ <, γi = max(γi1, γi2), γj = max(γj1, γj2),96γi1 =| pi+1,j − 2pij + pi−1,j || pi+1,j + 2pij + pi−1,j | , γi2 =| pij − 2pi−1,j + pi−2,j || pij + 2pi−1,j + pi−2,j | ,γj1 =| pi,j+1 − 2pij + pi,j−1 || pi,j+1 + 2pij + pi,j−1 | , γj2 =| pij − 2pi,j−1 + pi,j−2 || pij + 2pi,j−1 + pi,j−2 | .In order to satisfy the stability condition, the time step is chosen as (for details see,e.g., [1]):∆t = mini,j,kCFL µijρA∆yk + ρB∆xk +2Re((∆xk)2+(∆yk)2µij) , (10)ρA =| û | +√û2 + 1 ρB =| v̂ | +√v̂2 + 1,and | û |, | v̂ | are the maximal values of the components of velocity inside thecomputational domain; the definition of ∆xk, ∆yk is shown in Fig. 2.The computation is performed until the value of the L2-norm of residual satisfiesRez W nij ≤ εERR with εERR small enough (MN denotes the number of grid cells inthe computational domain), whereRez W nij =√√√√ 1MN∑ij(W n+1ij −W nij∆t)2. (11)3. Numerical resultsIn this paper we present the numerical results for channels with one entrance andtwo exit parts. The computation of both two-dimensional and three-dimensionalcases was performed. First, in Figs. 3 and 4 the numerical results for the two-dimensional case are shown. Fig. 3 shows the fluid velocity distribution for Reynold’snumber 1000 in the channel of the form of reverse T and the convergence of the resid-uals of the vector W = (p, u, v)T . Fig. 4 shows the velocity isolines for Reynold’snumber 1500 for the symmetric branching channel of the form Y. By the symbol q,the velocity magnitude is denoted, i.e. q =√u2 + v2.Next, the computation for fully three-dimensional fluid flow in two distinct caseswas performed. Figs. 5, 6 show the velocity isolines in the cross-sections of thebranching channels for Reynold’s number 300. Figs. 7, 8 show the velocity iso-lines in the cross-sections of the symmetric branching channel for Reynold’s num-ber 300. In Fig. 7, the convergence of the residuals of the components of the vectorW = (p, u, v, w)T is shown. Symbol q denotes the velocity magnitude for the three-dimensional case, i.e., q =√u2 + v2 + w2.970 25000 50000 75000 100000-7-6-5-4-3rez prez urez v0.991.071.141.070.990.080.150.380.910.23 0.690.380.530.610.61 0.840.690.080.61q1.14191.06570.98960.91350.83740.76120.68510.60900.53290.45670.38060.30450.22840.15220.0761Fig. 3: Velocity magnitude distribution in two dimensional channel of reverse T form(Reynolds number Re = 1000).1.011.010.290.471.110.831.011.201.381.381.010.291.380.920.200.831.291.01q1.37731.28661.19601.10531.01460.92400.83330.74270.65200.56130.47070.38000.28940.19870.1080Fig. 4: Velocity magnitude distribution in two dimensional symmetric channel of Y form(Reynolds number Re = 1500).981.021.131.23 1.33 1.43 1.541.330.200.92 0.721.230.410.31 0.61 0.820.200.100.200.101.130.82q1.53531.43301.33061.22831.12601.02360.92130.81900.71660.61430.51200.40960.30730.20500.1026Fig. 5: Velocity magnitude distribution in the central cut of the three dimensional channel(Reynolds number Re = 300).0.200.921.431.330.61 0.410.721.230.311.541.430.720.101.020.410.920.311.021.430.200.820.200.100.100.200.200.100.200.100.100.310.310.200.101.131.331.130.410.200.920.51Fig. 6: Velocity magnitude distribution in the cuts of the 3D channel from Figure 5.990 2500 5000 7500 10000-7.5-7-6.5-6-5.5-5-4.5-4-3.5-3-2.5rez prez vrez urez w0.26 0.130.130.390.260.260.130.390.130.130.130.26q1.93161.80321.67481.54641.41791.28951.16111.03270.90420.77580.64740.51900.39050.26210.1337Fig. 7: Velocity magnitude distribution in the central cut of the three dimensional channel(Reynolds number Re = 300).0.261.161.421.801.42 0.651.160.520.781.931.290.391.801.931.671.550.780.520.391.161.160.261.161.671.801.420.900.131.161.420.901.551.161.291.551.551.030.261.670.780.900.520.261.291.421.42 1.030.901.291.421.290.390.39Fig. 8: Velocity magnitude distribution in the cuts of the 3D channel from Figure 7.4. ConclusionA numerical model for the simulation of fluid flow in a branching channel withone entrance and two exit parts for two-dimensional and three-dimensional cases wasdeveloped. However, the method was applied for several different types of channelconfigurations. The presented results can be useful as a blood flow approximation butseveral significant simplifications made in the model should be mentioned. First, the100general blood flow can be characterized rather by the non-Newtonian fluid flow model(see [4]) so that the use of the Newtonian fluid flow model is questionable. Second, thevessel’s walls cannot be considered as fixed walls and thus the computational domaindeformations needs to be taken into account. This phenomenon can be treated, e.g.,by the arbitrary Lagrangian-Eulerian (ALE) method (for ALE description see, e.g.,[7], for practical applications see, e.g., [5], [6]). Moreover, the wall dynamics needsto be modelled and coupled with the fluid flow model. However, the presentedresults can be still used as an approximation of the problem, which can provideuseful information, e.g., the fluid flow character, the appearance of separation zones,pressure distribution, etc. The applied method serves as a starting point and willbe further developed. From this point of view it is necessary to test the methodon some standard problems and compare the results with the other available data(see e.g. [8]). Moreover, the presented method can be extended and applied to morerealistic blood flow model, which is the subject of our future work.References[1] R. Dvořák, K. Kozel. Mathematical Modelling in Aerodynamics. CTU Prague1996. (In Czech.)[2] K. Kozel, R. Keslerová, V. Prokop. Numerical Solution of Icompressible LaminarFlows through a Branching Channel. In Proceedings of International Conferenceof Applied Mathematics and Informatics at Universities 2002, Trnava, SlovakRepublic[3] R. C. Swanson, E. Turkel. Multistage Schemes With Multigrid for Euler andNavier-Stokes Equations. NASA Technical Paper 3631, 1997.[4] A. Quarteroni, M. Tuveri, A. Veneziani. Computational vascular fluid dynam-ics: Problems, models and methods. Computing and Visualization in Science 2,2000, pp. 163–197.[5] P. Sváček, M. Feistauer, J. Horáček. Numerical Simulation of a Flow InducedAirfoil Vibrations. In Flow Induced Vibrations, volume 2, pp. 57–62. Ecole Poly-technique, 2004.[6] P. Sváček, M. Feistauer, J. Horáček. Numerical simulation of a flow inducedairfoil vibrations with large amplitudes. Preprint No. MATH-knm-2005/1, pp.29, Faculty of Mathematics and Physics, Charles University Prague 2005.[7] T. Nomura, T. J. R. Hughes. An arbitrary Lagrangian-Eulerian finite elementmethod for interaction of fluid and a rigid body. Computer Methods in AppliedMechanics and Engineering 95, 1992, pp. 115–138[8] K. Kozel, V. Prokop. Numerical Simulation of Some Cardiovascular Problems.In Proceedings of Topical Problems of Fluid Mechanics, 2003, pp. 57-60.101

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Numerical solution of 2D and 3D incompressible laminar flows through a branching channel

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