Simulation of Multiphase Flow using Lattice Boltzmann Method

Submitted by: Saba Mushtaq MSPE-17 Supervised by: Mrs. Romana BasitDepartment of Chemical EngineeringPakistan Institute Of Engineering And AppliedSciences Nilore, Islamabad PakistanTable of ContentsTable of Contents ...........................................................................................................................................2 Abstract..........................................................................................................................................................3 1 Introduction .................................................................................................................................................4 2 Literature Survey........................................................................................................................................5 3 Lattice Boltzmann Method .........................................................................................................................7 4 Single Component Multiphase LBM........................................................................................................10 5 Multicomponent Multiphase LBM ...........................................................................................................13 6 Boundary Conditions................................................................................................................................14 7 Results And Discussions ...........................................................................................................................17 References....................................................................................................................................................28 Table of Figures2Figure 1: D2Q9 model ..................................................................................................................................8Figure 2: Streaming of particles ....................................................................................................................9Figure 3: Effect of density on interaction potentials....................................................................................11Figure 4: Effect of pressure reduction on density for G = -48, -72, -92.4, -120, -144................................12Figure 5: Comparison of velocity profile by LBM and Analytical.............................................................18Figure 6: Velocity profiles for different values of ...................................................................................19Figure 7: Density variation with number of iteration for = 0.5,1.75,1.85,1.99........................................19Figure 8: Obstacle in flow path at x=10......................................................................................................20Figure 9: Velocity contours for Re = 50......................................................................................................21Figure 10: Pressure contours for Re = 50 with obstacle present at x=10....................................................21Figure 11: Stream traces (a) Re =10, (b) Re=20, (c) Re=50, (d) Re=100, (e) Re=150 (f) Re=162............22Figure 12: Time series plot of vapor phase separation. Results for various time steps a) = 1, b) = 100,c) = 500,d) = 1000, e) = 2000, f) = 4000, g) = 8000, h) =15000, i) 25600 ts.....................................................24Figure 13: Density profile of a liquid droplet at time=1000 (ts).................................................................25Figure 14: Effect of interaction strength|G| on phase separation...............................................................26Figure 15: Pressure difference inside and outside of droplets as function of 1/r.........................................27 AbstractThe objective of this report is to demonstrate the capability of Lattice Boltzmann method to simulate single and multiphase flow. Simulations have been performed for two dimensional Poisuelle flow using D2Q9 model. The results obtained are compared with analytical resultsandtheyshowgood agreement. Then, boundary layer separation has been studied for various Reynolds number. Phase transition has also been demonstrated usingShanandChensinglecomponent multiphasemodel. Theresults wereusedto compute the surface tension using Laplace law. The value of surface tension computed is 14.322 lu mu/ts2. This is very close to literature value (14.332 lu mu/ts2). The error is only0.06977%. After comparisonwithliterature, it has beenconcludedthat lattice Boltzmann method is a valid numerical scheme emerging as an innovative computational fluid dynamics technique.1 IntroductionThe lattice Boltzmann method (LBM) is an innovative computational fluid dynamics (CFD)techniquefor simulatingfluidflows andheat transfer. It has emerged as a powerful competitor for traditional CFD methods, especially for simulations of multiphase flows and flows involving complex boundaries.It has been used extensively to simulate flow through porous mediaand blood flow simulation.Compared with the conventional CFD approach, the LBM is easy for programming, intrinsically parallel, and it is easy to incorporate complicated boundary conditions such as those in porous media. Itpossesses several distinct features than the conventional CFDtechniques.The convection operator is linear in phase space unlike Navier Stokes (NS) equations. Also, 4pressure is obtained through an equation of state rather than solving poisson equation.LBMandLatticegascellular automata(LGCA) arebasedonmesoscopickinetic equationrelating themicroscopic dynamicsofparticlesto macroscopic properties. Historically, LBM evolves from LGCA. LGCA describes the physical system by particles present on nodes having only two states (Boolean variables: 0 and 1).The evolution of lattice gas models proceeds in two steps that take place during each time step. The first step is propagation or streaming in which the particles move to new sites according to their previous positions and their velocities. Then particles collide and scatter according to the collision rules . Momentum and energy remain conserved during both steps. The major problem in LGCA is the presence of noise which causes a problem in achieving highaccuracy. InLBM, it hasbeen removed bydescribingthephysicalsystem using continuousdistribution functionrather than using Boolean variables. Distribution functionisdefinedastheprobabilityof findingaparticular moleculewithacertain position and momentum at certain time.The physical system is again comprises of the same steps: streaming and collision.1.1 Research ObjectivesIn the 3rd and 4th semester, poisuelle and single component multiphase (SCMP) flows have been simulated using D2Q9 LBM model. Poisuelle flow is the simplest flow that can be simulated using LBM. Boundary layer separation has also been observed for various Reynold numbers. SCMP flow has been simulated using the most popular Shan and Chen model used. Surface tension in this two phase mixture has also been computed.In the research semester, simulation of multicomponent multiphase flow will be performed and its surface tension will be computed.2 Literature SurveyShanand Chen(1994)hasdescribeda modeltosimulate flows involvingmultiple phases andmultiplecomponents. However,the focus was on single component which obeynonideal gas equationandundergophasetransitionprocess. Theyhavealso obtaineddensityprofileintheliquid-gasinterfaceasfunctionofthetemperature-like parameter .Multicomponent multiphaseflowisof tremendousimportanceinpetroleumindustry. Bulk-flow simulations of reservoir-flow performance are an important ingredient in the decisionofwhether toinvest inmajorrecoveryprojects. Buckleset al (1994)have carried out the simulations of an oil recovery project using the lattice Boltzmann model. Multiple components involved in the system are oil and water. They have plotted relative permeabilities for the wetting fluid (oil) and the nonwetting fluid (water) as a function of the wetting-fluid saturation for different values of the capillary number.The concluded that as saturation of the wetting fluid increases (the fraction of pore space it occupies) its relativepermeabilityincreases, that is, thefluidflowsmoreeasily. Similarly, asthe saturation of the nonwetting fluid decreases its relative permeability decreases.The lattice Boltzmann equation (LBE) has been directly derived from the continuous Boltzmann equation byXiaoyi and Luo (1997). They have demonstrated that the LBE is a special discretized form of the Boltzmann equation. A general procedure to derive the latticeBoltzmannmodel fromtheBoltzmannequationhavealsobeendescribed. The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models MartysandDouglas(2001)havecalculatedtheequilibriumpropertiesofLBfluid mixtures to characterize the critical phenomenon occurring in these processes . They have also studied the phase separation under quiescent and shearing conditions. Ladd and Verberg (2001) have applied LBM to particle fluid suspensions. They have summarizedtheavailablemethods along with applications for simulations of colloidal suspensions.Finally, they have also derived the equations for LBMthat includes transverse and longitudinal fluctuations in momentum .Baochang(2007)hasdirectlyappliedthelatticeBoltzmannmethodforconvection diffusion equation with source termto solve some importantnonlinear complex equations, including nonlinear Schrodinger(NLS) equation, Klein-Gordon equation etc. by using complex-valueddistribution function and relaxation time. He carried out their detailed numerical simulation and his results agree well with the analytical solution .Physical behavior of anumber of mesoscopic models for multiphase flow have been analyzed by Sbragaglia et al (2007). They have developed an extended pseudopotential model which permits to tune the equation of state and surface tension independently of each other . Their model has also enhanced the flexibility of LBM to high density ratios.Huanget al(2008)havestudiedtheviscouscouplingeffectsforthesimulationof immiscible two phase flow through porous media using Shan and Chen type multiphase LBM.They have investigatedcoupling effects due to capillary number, viscosity ratio and wetting angle.They have observed that when the wetting phase is less viscous and covers the solid surface, the relative permeability of the non-wetting phase may be greater than unity .6Gokaltun and McDaniel (2010) have studied the feasibility of multiple relaxation time LBM for multiphase flow simulations The LBM becomes unstable when simulating high density ratioandlow viscositymultiphase flows. This instability has been avoided by using multiple relaxation time rather than single relaxation time.Nwatchoket al (2010) have developed an LBM based model to solve conservation equationsfor momentumfor PoisuelleRayeighBernard(PRB) mixedflow. InPRB mixed flow, there exists longitudinal pressure gradient in a horizontal rectangular channel which is heated from below and cooled from above. PRB mixed flow simulation is an active and vast topic of research Myre et al (2010)have examined the performance of single component single phase and multicomponent multiphase flows on Graphics processing unit (GPU) clusters. They have concluded that all LBM simulations depend on effects corresponding to simulation geometry and not on the architectural aspects of GPU .3 Lattice Boltzmann MethodLBMisbasedonBoltzmannequationwhichdescribes thesystembycontinuous distribution function. It discretizes continuous Boltzmann equation to reduce the number of possible particle spatial positions, microscopic momenta and time. Particle positions are confined to the nodes of a lattice. Variations in momenta that could have been due to a continuum of velocity directions,magnitudes and varying particle mass are reduced to eight directions, 3 magnitudes and a single particle mass in a D2Q9 model . D2Q9 model means 2 dimensional and having 9 velocity vectors as shown in Figure 1. Point (0,0) represents the coordinates of particles which are at rest. Points (1,0), (0,1), (-1,0), (0,-1) shows the position of particles which are present on coordinate axis, whereas (1,1), (-1,1), (-1,-1), (1,-1) shows the particles present along diagonals. The fundamental measure of length is lattice unit (lu), unitof time is time steps (ts) and of mass is mass unit (mu). Mass of particleshas uniformly taken as 1 mu throughout the domain.The particles present on (0,0) coordinates are stationary. The velocity magnitude of 1 through 4velocityvectorsis1 lu/tsandthat ofindex 5to8is2lu/ts.These velocitiesare exceptionally convenient in that all of their x and y-components are either 0 or 1 .Figure 1: D2Q9 model The Boltzmann equation without external force term(which has been neglected because of its lesser magnitude in comparison with intermolecular force) is given by fv f Qt+ r (1)Where,fis the distribution function,vris the velocity vector and Qis the collision integral. Applying the BGK approximation to the collision integral( )1 eqfv f f ft + r (2)Whereis known as the collision time andeqf is the equilibriumdistribution function. The distribution function depends on space, momenta and time. Since mass unit has beentakenas unity, somomentais equivalent tothevelocity. Thevelocityis discretized to 9 directions in a D2Q9 resulting in a set of 9 distribution functions at any point at a given time. The discretized equation is( )1 eq ii i ifv f f ft + r(3)Two steps involved in LBM are:1) Streaming 2) CollisionInstreamingdirectionspecificdensities(distributionfunctions)aremovedtothe nearest neighbor lattice nodes.The streaming of distribution functions from one node to the other is shown in Figure 2. 8Figure 2: Streaming of particles In collision, distribution functions collide with each other to get new distribution while conserving momentum. Both steps can be represented by following mathematical expression( ) ( ) ( )1, , , ( , )StreamingCollisioneqi i i if t t t f t f t f t1 + + ]ix e x x x1 4 4 4 442 4 4 4 4 431 4 4 4 2 4 4 4 3(4)Where ieindicatesthemicroscopicvelocityofithparticle. Collisionofthefluid particles isconsideredasrelaxationtowardslocal equilibrium. Thiscollisionstepis skipped at solid nodes, because bounceback boundary condition which is applied on solid nodes, is also a type of collision. The equilibrium distribution function eqifis defined as,( )( )22( )2 4 2. . 9 31 32 2i eq ii if wc c c1 + + 1 1 ]e u e u ux(5)Where the weights iware4 9 for the rest particles ( 0 i ),1 9 for { } 1, 2, 3, 4 i and 1 36 for { } 5, 6, 7, 8 i .crepresentsthebasicspeedonlattice(inthesimpleLBM models1 c 1luts) and stands for macroscopic density.The macroscopic variables are calculated by using the following formulas:Macroscopic density80iif (6)Macroscopic velocity u801i iifu e (7)Pressure P P RT (8)Kinematic viscosity 1 13 2 _ ,(9)Relaxation time should be greater than 1/ 2 to get physically realistic results so, the relaxation parameter which is defined as the inverse of should be less than 2. The values of distribution functions and equilibrium distribution functions should be positive tomeettheconsistencyrequirements. Unlike traditional CFD techniques, pressure has beencomputedbyusingideal gaslawratherthanusingPoissonequation. Moreover, pressure does not depend explicitly on velocity.4 Single Component Multiphase LBMLBMha capability tosimulate single componentsingle phase (SCSP),single component multiphase (SCMP)and multicomponent multiphase (MCMP) flows. There are many LBM models to simulate multiphase flow, most of them are very difficult to apply. Here, the method proposed by Shan and Chen has been described because of its relative ease when compared with other methods. Also, in this model, separation of two phasestakes placespontaneouslywhenever themagnitudeof interactionstrength|G| exceedsa certainthreshold value.Since,it is easy to use so it is very popular among programmers . The other LBM models are relatively complex and difficult to apply.InSCMPflow, anattractive forceis introducedbetween the particles through interaction potential and interaction strength. This force leads to a reduction in pressure in accordance with the van der Waals Equation of state. The reduced pressure causes the separation. Onsubatomiclevel, itcan be justified as the repulsion caused by electron cloud when they are very close to each other under the influence of external attractive force.In this model, equilibrium distribution function is computed as( )( )22( )2 4 2.. 9 3, 1 32 2eqeq eqieq ii if t wc c c 1 1 + + 1 ]e ue u ux (10)Where, the equilibrium velocity equis calculated as10eq +Fu u (11)The velocity u is defined as80i iif eu(12)The interparticle forceFis given by( )80( , ) ( , ) ,i i iit G t w t t + F x x x e e(13)Where G is the interaction strength, being negative for interparticle attraction.iw is 1/9 for{ } 1, 2, 3, 4 i , for{ } 1, 2, 3, 4 i is 1/36 for and is the interaction potential: ( )00exp _ , (14)0and0Beingthearbitraryconstants. Here, thevalues of0and0hasbeen arbitrarily chosen as 4 and 200 respectively, because the model behavior with these values has been explored thoroughly. According to equation (14) the attractive force is stronger whendensityishigher,andviceversa. Hence,theliquidregionsexperiencestronger cohesive force than the vapor leading to the surface tension phenomenon . The variation of with is shown in Figure 3.Figure 3: Effect of density on interaction potentialsThe pressure is calculated by using the following non-linear and non-ideal equation of state:( )22GRTP RT +1 ](15)The value ofRT is fixed for both SCSP and SCMP models:13RT (16) The second term on the right hand side of is non-ideal part that stands for attractive force between the molecules. Since G is negative, it leads to the reduction of pressure. WhenGis adequatelynegative, theEOSis non-monotonicthat leads tothephase separation. Using (16), the equation for pressure becomes( )26RTP RT + (17)The pressure density relationship is demonstrated in Figure 4 for G = -48, -72, -92.4 (critical), -120, -144. It has been observed that when |G| exceeds a critical value of 92.4, there corresponding to each pressure there exists two values of densities; the higher one showing the density of liquid and lower indicating vapor density.Figure 4: Effect of pressure reduction on density for G = -48, -72, -92.4, -120, -144Here, only attractive forces has been considered and repulsive forces has been ignored which become dominant when a gas is present in compressed form.125 Multicomponent Multiphase LBMIn Multicomponent multiphase (MCMP) system, more than one species are combined throughinteractionforces(repulsive) toformimmisciblefluids. MCMPflowsareof tremendous economic and environmental importance. Their economic importance is due to the fact that petroleum is found with water forming a MCMP system. Non aqueous phase liquids present in the subsurface and acting as a long lived source of ground water contamination contribute to the environmental importance of MCMP. Tosimplifythesystem, onlytwo components have beenconsideredhere. In LBM algorithm, the addition of a second component is incorporated by the the addition of a new array and a new loop. The equilibrium distribution function is calculated using the following composite macroscopic velocityu11a aaafuf e (18)The composite macroscopic velocity represents the flow of the bulk of fluid and differs from the macroscopic uncoupled velocities of the individual components u801a aau f e (19)Where,denotes the fluid components and = {1,2} for two component mixture. The macroscopic density of each fluid component is given by 80aaf (20)The density of both components cannot be zero at any point in the calculation domain, otherwise physically unreakistic values ofvelocities will be obtained.To obtain equilibrium velocitiesequ , the gravity (body force) and interaction forcs are added tou. The repulsive interaction forc on the fluid component is given by( ) ( , ) ( , )a a aaF x G x t w x t t + e e(21)Where, G is the interaction strength, is the one fluid component and denotes the other component.and are usually taken as the fluid densities such as and . The addition of this interaction forc termF touis equ u F +(22)equ Fu +(23)Thevelocityincrement depends onthemagnitudeof Gandfluiddensities.The magnitude of the velocity increment in the above Equation should be kept small. 6 Boundary ConditionsThe dynamics of fluid flow depends on the surrounding environment. This influence is described via boundaryconditions. So, Boundaryconditions play a crucial role in computing any meaningful results. Periodic, Bounce back and Dirichlet boundaries have been described below.6.1 Periodic boundary ConditionsPeriodicboundaryconditionsarethesimplest boundaryconditionsbecausesystem becomes closed by the edges as if they are attached with each other. They are sufficient wheresurfaceeffects arenegligible. Inflowsimulationinaslit, periodicboundary condition is applied at the open end of the slit and bounce back boundary condition is applied at the slit walls. Periodic boundary condition is implemented in streamwise x direction by treating the nodes on the inflow and outflow faces as the neighbors if they share common y and z coordinates. For boundarynodes, theneighboringnodes arepresent ontheopposite boundary. Here are some conditionals that check if the neighboring nodes lie outside the computational domain and then assign the appropriate node on the opposite boundary in that case to achieve periodicity. Pseudo code to implement Periodic boundary condition in x direction is described below (1 is the first node in both x and y direction and lx and ly are the last nodes in x and in y direction respectively).( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )do y=1,lyf 1,1,y f 1,lx,yf 5,1,y f 5,lx,yf 8,1,y f 1,lx,yf 3,1x,y f 3,l,yf 6,1x,y f 6,l,yf 3,1x,y f 7,l,yenddo (24)Using this convention, the populations on all incoming links at inflow and outflow is defined automatically by streaming process. Periodic boundary condition is imposed as 14thenatural part of streamingsothat outgoingpopulations at oneendbecomes the incoming populations at the other end. Fully periodic boundary conditions are also applied in some cases e.g. the simulation of multiphase flow in an infinite domain.6.2 Bounce Back Boundary ConditionThenext simplest boundaryconditionisthebouncebackboundarycondition, also knownasno-slipboundarycondition. It playsamajorroleinmakingLBMpopular becauseit canconvenientlyhandlethecomplexboundariessuchasthosepresent in porous media. It specifies that velocity of fluid at the wall is same as that of wall. In programming, one just has to declare a particular node as the solid object and no special treatment is necessary. Thus it is easy to incorporate images of porous media and immediately compute flow in them. Solids nodes are separated into two categories:1. Boundary solids2. Isolated solidsBoundary solids lie at the solid-fluid interface and isolated solids are those that has no contact with fluid. By using this division it is possible to eliminate unnecessary computationsat inactivenodes; thisisveryimportant insimulatingflowinfractured media where fraction of total space occupied by open space in contact with fluids, which needs to program is very small.Ithasbeenassumed that solid surface is aligned with the grid. Two types of implementations will be discussed:1. On-grid or Full way bounceback 2. Mid grid or Half way bounceback On-grid means that physical boundary lies exactly on grid line, whereas, in mid-grid conditiontheboundaryliesinbetweentwogridlines. Thefirstone iseasy asit just reverse all the populations sitting on the boundary node. It simply implies that for any domain ( ) ( )in outf N =f N( ) ( )in outf S =f S(25)whereN andSrepresent north and south rows of the domain. For a D2Q9 model, it is implemented as:( ) ( )( ) ( )( ) ( )f 6,x,y =f 2,x,yf 7,x,y =f 3,x,yf 8,x,y =f 4,x,y (26)The on-grid bounceback possess only first order accuracy because of of sided character of streaming process at the boundary. Second order accuracy can be achieved by using the mid grid or half way bounceback condition, which is modestly more complex than Full way bounce back.It is implemented as:( ) ( )in outf N =f NF( ) ( )in outf S =f SF(27)WhereNand Srepresent ageneric site on the topand bottomwalls, respectively, whereas NF,SF is the set of fluid sites connected to N,S respectively.For a D2Q9 model it is implemented as:( ) ( )( ) ( )( ) ( )f 6,x,y= f 2,x-1,y-1f 7,x,y= f 3,x,y-1f 8,x,y= f 4,x+1,y-1(28)6.3 Dirichlet Boundary ConditionInreal flowexperiment, someformofregulationat inlet andoutlet isneeded. A common type involves the constant pressure difference between the inlet and the outlet. TheDirichlet boundaryconditionconstrainsthepressure/densityat aninlet oroutlet boundary. To impose it, it is assumed that inflow face is oriented normal to an axis of the lattice.It is also assumed that velocity tangent to boundary is zero and the normal to the boundary component of velocity is evaluated. First of all, a density ois specified, using it then the velocity is computed. Density specification is equivalent to pressure specification because both are linked by equation of state (Which for single component D2Q9 model isP RT , whereas RT=1/3). In addition to the macroscopic velocity, the proper distribution function at the boundarynodesneedsto bedetermined.After the streaming thereare threeunknown directional densities at each lattice node that point from boundary into the fluid. These three unknowns can be solved in a way to maintain the specified pressure/density o at their lattice nodes. 166.4 Von Neumann Boundary ConditionVon Neumann boundary condition specifies the flux at the boundary. A velocity vector containingxand ycomponents [ ]0 0= u v uis specified from which density/pressure is computed. Macroscopicdensity/pressureisonlypartofwhat needstobecomputed. After streaming there there are three unknown directional densities4 7 8,andf f fat each latticenode (pointinginto thedomain). These three unknowns are solved in a wayto maintain the specified velocity at lattice nodes. Then density is computed by the summation of all distribution functions.7 Results And DiscussionsThe simplest LBMthat has been implemented is for SCSP and SCMP. The programming language used is FORTRAN and lattice model is D2Q9. TECPLOT 360 has been used as post processing tool. Here the simulation results have been presented for poisuelle flow and multiphase flow. 7.1 Plane poisuelle FlowTheviscousflowthroughachannel under theaction ofpressure gradientiscalled poisuelle flow. The flow is one dimensional incompressible and laminar. The analytical solution gives a parabolic velocity profile with maximum velocity in the center and zero at the walls of the channel. In LBM, poisuelle flow has been simulated using a uniform grid, which contains thirty lattice units (lu) in x-direction and twenty in y direction. Bounceback boundary condition has been applied along the walls. In the flow direction Periodic boundaries has been used, which means the fluid leaving the domain reenters in the opposite end of the channel. The system is effectively infinite in the flow direction and there are no end effects . The time lengthis 3000timesteps. Thefluidis restinginthebeginningandis thenslowly accelerated. The value of relaxation parameter (the inverse of collision time) is 1.85. Thevalueofinitial densitydistributionfunctionchosenis 0.1. Figure5shows the comparison of LBM and analytical results. Parabolic velocity profile obtained show excellent agreement with analytical results of poisuelle flow. Only slight difference has been observed at the wall due to the use of full waybounce backboundarycondition whichshows first order accuracy. It can be improved usinghalf waybounce backboundarycondition, beingthe secondorder accurate. In the full way bounce back boundary conditionthe physical boundary (wall) lies exactly on a grid line, whereas in the half way bounce back boundary condition, the boundary lies between two grid lines.Figure 5: Comparison of velocity profile by LBM and AnalyticalTo see the effect of which is a convergence parameter, velocity profiles have been plotted for various values of , as shown in Figure 6. 18Figure 6: Velocity profiles for different values of It is observed that when the value of is closet to zero the magnitude of velocity obtained is very less than actual whereas, when approaches two, the flow becomes unstable. The value of very close to 0 shows infinite viscosity beyond the limit simulated by LBM. Thevalueofgreater thantworepresentsnegativeviscosity. Bothextremesleadto stability issues in LBM model. So, the appropriate choice of lies between one and two.The variation of average density with time steps has been shown in Figure 7 for 4 values of . It is observed that when is close zero, the average velocity obtained is negligible, due to viscous effects. When is near two, the flow becomes unstable and no steady state is achieved. When lies between one and two ( = 1.75, 1,85), a steady state is achieved after 3000 iterations.Fluid flowing past an object tends to drag the object in the direction of fluid flow.If an object is moving through a stationary fluid the drag tends to slow down it and if body is stationary in flowing fluid the drag tends to move it in the direction of flow. Boundary layer separation occurs when fluid strikes with a blunt body such as flat plate perpendicular to the flow.-2.00E-020.00E+002.00E-024.00E-026.00E-028.00E-021.00E-011.20E-011.40E-011.60E-010 500 1000 1500 2000 2500 3000 3500Avg velocity(lu/ts)No. of iterations(ts)w=0.5w=1.75w=1.85w=1.99Figure 7: Density variation with number of iteration for = 0.5,1.75,1.85,1.99To observe the boundary layer separation, the obstacle has been introduced in the flow, perpendicular to the flow at x=10 lu and the grid size has been changed to 5020 lattice to minimize the flow disturbance as shown in Figure 8. X (lu)Y(lu)10 20 30 40 505101520OBST0.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.20.150.10.05Figure 8: Obstacle in flow path at x=10Thecontours ofvelocityforRe = 50, with obstacle present at x=10, are shown in Figure 9. X (lu)Y(lu)10 20 30 40 505101520VX0.140.130.120.110.100.090.080.070.060.050.040.030.020.010.0020Figure 9: Velocity contours for Re = 50The velocity of fluid when it strikes with obstacle becomes zero and is converted into pressure. Hence, pressure is maximum at x = 10. Along the sides of the obstacle, since flowarea has been reduced now, topreserve continuityequation, thevelocityhas increased to maximum. Just behind the obstacle, a low pressure zone is formed where boundary layer has been separated along with formation of vortices. The kinetic energy of the fluid has been used in vortex formation. The pressure difference between the front and rear edge of obstacle has been illustrated in Figure 10. When the fluid strikes the obstacle, the pressure is highest at that point, whereas just behind the obstacle a low pressure zone is formed.X (lu)Y(lu)10 20 30 40 505101520PRESS0.03430.03420.03410.0340.03390.03380.03370.03360.03350.03340.03330.03320.03310.0330.03290.0328Figure 10: Pressure contours for Re = 50 with obstacle present at x=10The boundary layer separation has been studied for various Reynolds numbers. The streamtraces areshowninFigure11forRe=10,20,50,100,150,162. At Re=10, streamlinesseparate and recombine after the obstacle ends with no vortex formation. At Re = 20, little vortex formation has been observed on downstream side. As Re is further increased, vortex grows in size at the rear end. Finally, when Re has been increased from 162, instabilityinstreamlineflowhasbeenobserved.Thisinstabilityisduetothe presence of large size of the obstacle.Figure 11: Stream traces (a) Re =10, (b) Re=20, (c) Re=50, (d) Re=100, (e) Re=150 (f) Re=16222a)X (lu)Y(lu)10 20 30 40 505101520d)X (lu)Y(lu)10 20 30 40 505101520b)X (lu)Y(lu)10 20 30 40 505101520e)X (lu)Y(lu)10 20 30 40 505101520c)X (lu)Y(lu)10 20 30 40 505101520f)X (lu)Y(lu)10 20 30 40 5051015207.2 SCMP FlowSimulation of multiphase flow is an interesting and challenging task. Uniform grid has beenusedhaving200X200latticenodes. Initial densityintroducedis200lu/ts2plus perturbationinducedbyarandomnumberbetween0and1. Thevalueofinteraction strength G chosen is -120, which leads to the phase separation process. The relaxation factor chosen is 1 (ts). Periodicboundaries have been used in all directions. The time length is 25600 (ts). The phase separation process for various time intervals is shown in Figure 12.a) b)c)d) e) f)g) h)i)The red color is indicating the denser phase (liquid) and blue phase is showing the lighter phase (vapor). The molecules sitting on the interface experience a net force driven by the intermolecular distancebetweenthetwofluids, andasaresult likemolecules start moving towardseach other. Itcan be observed that the smaller droplets are relatively moreunstablethanthelargeones. Thisisduetothesurfacetensioneffectsonthe interfacewithalarger curvature. Alargepressuredifferenceoccurs inthesmaller droplets causing them to evaporate . Phase separation ultimately leads to a single droplet in vapor atmosphere. Whether liquid drops or vapor bubbles are formed depends on the 24Figure 12: Time series plot of vapor phase separation. Results for various time steps a) = 1, b) = 100,c) = 500,d) = 1000, e) = 2000, f) = 4000, g) = 8000, h) =15000, i) 25600 tstotal mass in the domain and consequently on the initial density selected. When phase separation occurs then the interfaces try to minimize their total area thus achieving the shape of a circle. Fig. 13 illustrates the density profile of a droplet which is immersed in itsownvapors. At thevapor liquidinterfacethedensitychanges fromminimumto maximum. Figure 13: Density profile of a liquid droplet at time=1000 (ts)

Effect of interaction strength on phase separation process has been shown in Figure 14. Below the critical value of |G| there is only one phase present. As the value of |G| exceeds this threshold value, in this case, it is 92.4, two phases form having different densities. The one having minimum density is the vapor phase and with maximum density is the liquid phase. Figure 14: Effect of interaction strength|G| on phase separationSurfacetensionis veryimportant for multiphaseflowsystem. Here, it has been calculated using Laplace law, which states that, if a liquid droplet of radius r is immersed in its own vapors, pressure difference (outside insideP P P ) across the droplet is related tothe radiusofthedropletbythe following relationship wheredenotes the surface tension. Pr (29)According to Laplace law, plot of P and 1/r should be linear and its slop represents the surfacetension. Thesurfacetensionhasbeencomputedbymeasuringtheradii of droplets of various sizes, immersed in their own vapors and their pressure. The plot of P versus 1/r is shown in Figure 15, which is linear, as stated by Laplace law. The surface tension which is the slope of the straight line comes out to be 14.322 lu mu/ts2.The value of surface tension found in literature is 14.332 lu mu/ts2. The error is only 0.06977%.26Figure 15: Pressure difference inside and outside of droplets as function of 1/r8 CONCLUSIONSIn this report, LBM has been introduced as a competitive CFD toolto simulate the poisuelleflowandphasetransitionprocess.For poisuelleflow, simulationhasbeen performed using a 3020 lattice. The velocity profiles has been studied for vaious values of and optimum value of has been found to be lying between 1 and two. The results of poisuelle flow are in excellent agreement with analytical results for =1.85. The simulation results for SCMP have been depicted by using Shan and Chen SCMP model, are also in good agreement with results present in literature. Phenomenon of phase separation has been shownfor various values of interaction strength whichcauses reduction in pressure and hence leading to phase separation phenomenon. Density profile inadroplet presentinthevaporatmospherehasalsobeenshown. Densityshowsan abrubpt change at the interface. Surface tension computed also match quite well with the literature values. The value of surface tension computed is 14.322 lu mu/ts2.This is very close to literature value (14.332 lu mu/ts2). The error is only 0.06977%. Although SCSP and SCMP flow have been simulated, LBM has capability to simulate MCMP.The simulation of MCMP flow using LBM will be performed in the research semester.References[1]. Michael C.Sukop, Daniel T.Thorne, Jr.,Lattice Boltzmann Modeling An IntroductionforGeoscientistsandEngineers,SpringerVerlagBerlinHeidelberg, 2006.[2]. 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