A new numerical approach is presented for simulating the movement of test particles suspended in an incompressible fluid flowing through a porous matrix. This two-phase particle-laden flow is based on the Navier-Stokes equations for incompressible fluid flow and equations of motion for the individual particles in which Stokes drag is dominant. The Immersed Boundary method is applied to incorporate the geometric complexity of the porous medium. A symmetry-preserving finite volume discretization method in combination with a volume penalization method resolves the flow within the porous material. The new Lagrangian particle tracking is such that for mass-less test particles no (numerical) collision with the coarsely represented porous medium occurs at any spatial resolution

Geurts, Bernard J.

Ghazaryan, Lilya

Lopez Penha, David J.

Stolz, Steffen

English

Ghazaryan, L.

Lopez Penha, D.J.

Geurts, Bernardus J.

Stolz, S.

NARCIS 

Physically consistent simulation of transport of inertial particles in porous media.

University of Twente Research Information

MAMERN11: 4th International Conference on Approximation Methodsand Numerical Modelling in Environment and Natural ResourcesSaidia (Morocco), May 23-26, 2011Physically consistent simulation of transport of inertialparticles in porous mediaLilya Ghazaryan1, David J. Lopez Penha1, Bernard J.Geurts1,2and Steffen Stolz3,11Multiscale Modeling and Simulation, Department of AppliedMathematics,Faculty EEMCSUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands{l.ghazaryan,d.j.lopezpenha,b.j.geurts}@ewi.utwente.nl2 Anisotropic Turbulence, Department of Applied Physics, EindhovenUniversity of TechnologyP.O. Box 513, 5600 MB Eindhoven, The Netherlands3 Philip Morris Products S.A., PMI Research & DevelopmentQuai Jeanrenaud 5,2000 Neuchâtel, SwitzerlandSteffen.Stolz@pmintl.comKeywords: Lagrangian transport model, Particle filtration, Slow flow,PorousmediaAbstract. A new numerical approach is presented for simulating the move-ment of test particles suspended in an incompressible fluid flowing througha porous matrix. This two-phase particle-laden flow is basedon the Navier-Stokes equations for incompressible fluid flow and equationsof motion for theindividual particles in which Stokes drag is dominant. The Immersed Bound-ary method is applied to incorporate the geometric complexity of the porousmedium. A symmetry-preserving finite volume discretization method in combi-nation with a volume penalization method resolves the flow within the porousmaterial. The new Lagrangian particle tracking is such thatfor mass-lesstest particles no (numerical) collision with the coarsely rep esented porousmedium occurs at any spatial resolution.12 L. GHAZARYAN , D.J. LOPEZPENHA, B.J. GEURTS AND S. STOLZ1 IntroductionIn a wide range of natural processes and industrial applications the motionof small particles suspended in a fluid flowing through a porous medium isone of the major phenomena of interest. Understanding the beavior of parti-cles can help to control and improve processes like pollution percolation intounderground areas, oil recovery and various refinement processes.In this work the emphasis is on capturing the fully detailed motion of par-ticles in a flow through complex porous media. We describe a computationalapproach for simulating the underlying flow of fluid and the motion of embed-ded particles in the presence of a porous medium. The employed computa-tional algorithms provide a particle tracking method whichcan be used bothfor treating test particles as tracers, that follow precisely the fluid, and as amethod for quantifying filtering properties of porous filters. For describing thefluid phase the Eulerian approach is used, while the particlephase is modeledusing the Lagrangian transport model. To allow study of complex geometriesof various porous materials we employ the Immersed Boundary(IB) method[1].In the following sections we will shortly describe the mathematical modeland the applied numerical methods. Finally, some illustrations based on oursimulations for a test geometry will be presented.2 Mathematical modeling and numerical treatment of two-phaseflow in porous mediaThe system of governing equations is as follows:∇ · uf = 0 (1)∂uf∂t+ uf · ∇uf = −1ρf∇p+ ν∇2uf + f (2)dxpdt= up(t) (3)dupdt=1τ(uf (xp, t)− up(t)) (4)with ∇ ≡ (∂/∂x, ∂/∂y, ∂/∂z)T representing the vector differential operatorand∇2 ≡ ∇ · ∇ the Laplace operator. Here,uf = (uf , vf , wf )T is the fluidvelocity,up = (up, vp, wp)T is the particle velocity,xp = (xp, yp, zp)T is theparticle position,ρf is the fluid mass density,p is the pressure andν is thekinematic viscosity.The set of equations (1-4) is composed of the Navier-Stokes equations forincompressible fluids describing the fluid phase, Eqs. (1) and (2), and equa-INERTIAL PARTICLES IN POROUS MEDIA 3tions of motion restricted to the dominant drag-force to describe the particlephase [2], Eqs. (3) and (4). One-way coupling of the two phases is considered,implying that the particle phase has no influence on the fluid phase [3]. Theterm f is a body force per unit mass, used to approximate no-slip boundaryconditions at all solid-fluid interfaces. In this contribution we use a forcingfunction based on a volume-penalization strategy.The essential parameter in the equations of motion is the velocity responsetime or relaxation timeτ [4]. Considering the Stokes drag, the particle re-sponse time depends on its diameterDp, the density of the particleρp and themolecular viscosity of the carrier fluidµ: τ = (ρpD2p)/(18µ). The smallerτ ,the more sensitive is the particle motion to temporal and spatial variations inthe flow field. By making the equation of motion non-dimensional, based on acharacteristic length scale and velocity, we obtain a non-dimensional parame-terSt, denoting the Stokes number, which expresses the inertia ofhe particle.Another important dimensionless parameterRe (Reynolds number) is intro-duced in the non-dimensional formulation of Eq.(2), which characterizes theflow regime.The solution of the governing system of equations has various properties,which represent important physical features of the problem. In order to pre-serve these properties on the level of the numerical solution, suitable numericalmethods need to be developed. For resolving the gas flow we usesymmetry-preserving finite-volume discretization on a staggered gri[5]. The IB reso-lution of the gas flow on the staggered grid results in non-zero v locity fieldsat the locations at which the solid-liquid interfaces are defined. This plays animportant role in the tracking of particles, implying the possibility of a small,unphysical deposition of mass-less test particles on the solid urface. This isnot acceptable, since mass-less particles are supposed to follow the stream-lines, which are consistent with the no-slip condition on all so id-liquid inter-faces. We restrict the velocities to all solid-liquid interfaces in such a way thatmass-less particles do not hit the surface of the solid. Thisis essential whenparticles are considered as tracers. The new developed approach of interpo-lating the velocities close to the solid-liquid interface is also important for anaccurate estimation of deposition due to impaction for verysmall particles.3 Test particles in a structured porous mediumIn order to verify and validate the described approach we simulate the mo-tion of an ensemble of particles in a test geometry composed of a staggeredarrangement of 3D square rods (Figure 1(a)).We present the dynamics of particles withSt = 0.05 for a creeping flow4 L. GHAZARYAN , D.J. LOPEZPENHA, B.J. GEURTS AND S. STOLZ0 0.20.4 0.60.8 11.2 1.41.6 1.8200.5100.20.40.60.81xzy(a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.50.60.70.80.91xy(b)Figure 1: Staggered arrangement of square rods in 3D (a). Particle t ajectories ema-nating from a line of initial particle positions (b). Each curve corresponds to a par-ticular moment in time. The particle positions originate from a straight line of initialconditions on the left side of the figure. We simulated the motion of particles withSt = 0.05 and for a creeping flow ofRe = 1.of Reynolds numberRe = 1. Computed trajectories of the particles smoothlyfollow the flow streamlines (Figure 1(b)). Our simulations confirm for testparticles with no inertia (St → 0, i.e. τ → ∞) physically consistent behavior:particles move with the flow and never hit the solid walls. Forrelatively largevalues ofSt we observe strong inertial effect on the trajectories of theparticles.By following the trajectories for different values ofτ andRe, one can quantifythe filtering properties of a given geometry.REFERENCES[1] R. Mittal and G. Iaccarino, Immersed boundary methods,Annu. Rev. FluidMech., 37, 239–261 (2005).[2] M.R. Maxey and J.J. Riley, Equation of motion for a small rigid sphere ina nonuniform flow,Phys. Fluids, Vol. 26, No. 4 (1988).[3] S. Elghobashi and G.C. Truesdell, On the two-way interaction betweenhomogeneous turbulence and dispersed solid particles. I: Turbulence mod-ification,Phys. Fluids A5, 1790-1801 (1993).[4] W.C. Hinds, Aerosol Technology: properties, behavior,and measurementof airborne particles, Second Edition,John Wiley & Sons Inc.(1999).[5] R.W.C.P Verstappen and A.E.P. Veldman, Symmetry preserving dis-cretization of turbulent flow,Journal of Computational Physics, 187, 343-368 (2003).

Aerosol Technology: properties, behavior, and measurement of airborne particles, Second Edition,

Equation of motion for a small rigid sphere in a nonuniform ﬂow,

Immersed boundary methods,

On the two-way interaction between homogeneous turbulence and dispersed solid particles. I: Turbulence modiﬁcation,

Symmetry preserving discretization of turbulent ﬂow,

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Physically consistent simulation of transport of inertial particles in porous media

Physically consistent simulation of transport of inertial particles in porous media

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