Research Article Lattice Boltzmann Model of 3D Multiphase...

Research ArticleLattice Boltzmann Model of 3D Multiphase Flow inArtery Bifurcation Aneurysm Problem

Aizat Abas1 N Hafizah Mokhtar1 M H H Ishak1 M Z Abdullah2 and Ang Ho Tian1

1School of Mechanical Engineering Universiti Sains Malaysia Engineering Campus 14300 Nibong Tebal Penang Malaysia2School of Aerospace Engineering Universiti Sains Malaysia Engineering Campus 14300 Nibong Tebal Penang Malaysia

Correspondence should be addressed to Aizat Abas aizatabasusmmy

Received 6 January 2016 Revised 5 March 2016 Accepted 31 March 2016

Academic Editor Giuseppe Pontrelli

Copyright copy 2016 Aizat Abas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper simulates and predicts the laminar flow inside the 3D aneurysm geometry since the hemodynamic situation in the bloodvessels is difficult to determine and visualize using standard imaging techniques for example magnetic resonance imaging (MRI)Three different types of Lattice Boltzmann (LB)models are computed namely single relaxation time (SRT)multiple relaxation time(MRT) and regularized BGKmodelsThe results obtained using these different versions of the LB-based code will then be validatedwith ANSYS FLUENT a commercially available finite volume- (FV-) based CFD solver The simulated flow profiles that includevelocity pressure and wall shear stress (WSS) are then compared between the two solvers The predicted outcomes show that allthe LB models are comparable and in good agreement with the FVM solver for complex blood flow simulation The findings alsoshow minor differences in their WSS profiles The performance of the parallel implementation for each solver is also included anddiscussed in this paper In terms of parallelization it was shown that LBM-based code performed better in terms of the computationtime required

1 Introduction

Aneurysm is a condition in which a blood vessel wall ispathologically dilated It normally happens at the vicinity ofthe circle of Willis the main cerebral arteries and has alsobeen found to develop in the iliac and intracranial arteries[1] Aneurysm could lead to artery rupture resulting in strokeor even death The arterial wall may rupture when the wallitself is not strong enough to withstand the stresses exertedon it during the blood flow As the treatment methods foraneurysm carry a high degree of risk [2] understanding thevesselrsquos internal fluid flow is important It can be utilizedto improve aneurysm diagnosis and treatment methodsHashimoto et al [3] pointed out that the aneurysm growthcan be associated with the intra-aneurysmal hemodynamicsHence good understanding and knowledge of hemodynamicparameters are crucial Various computational fluid dynamics(CFD) studies [4ndash9] have been carried out to investigate therelationship between hemodynamic factors and the risk ofan artery wallrsquos rupture These factors were the impingementforce the WSS the pressure and the blood velocity

For the past two decades the simplicity and the effec-tiveness of the Lattice Boltzmann method (LBM) in solvingflow related problem has created awareness in the CFDcommunity LBM-based computations have been applied tosimulate different complex and irregular geometries pre-sented in flow problems Some of these experiments demon-strated that LBM performed better than the classical CFDtools in certain applications [10ndash12] The LB simulation isbased on the Boltzmann theory instead of the expensiveand time-consuming Navier-Stokes equation LBM did haveits limitations despite the increasing popularity in solvingflow problems For example though LBM is capable ofsolving high-Mach number flow in aerodynamics it still lacksa consistent thermohydrodynamic scheme However someefforts were concerted to sort out these limitations [13]

The past few years have seen many researchers resortingto LBM in an attempt biomedical problem In the work byLiu biviscosity constitutive relations and control dynamicsbased on a D2Q9 lattice are used to solve a stenosis bloodflow problem [14] Sun and Munn predicted the blood flowby developing an LBM model that considers the blood

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 6143126 17 pageshttpdxdoiorg10115520166143126

2 Computational and Mathematical Methods in Medicine

as a suspension of particles in the plasma [15] In addi-tion Leitner et al introduced new LBM boundary condi-tions for the cardiovascular domain using a D2Q9 latticemodel [16] The application of LBM to three-dimensional(3D) problems is rather limited and noticeable applica-tions can be found in the work by [17] featuring a 3Dsimulation of intracranial aneurysm geometry and in thework by [18] on wall orientation and shear stress in theartery

In general the application of LBM is not restricted tothe fluid flow simulation It has been found to be an effi-cient solver in sound absorption applications [19] Anotherinteresting use is in the simulation of floating rigid bodieson a free surface [20] The LBM and the FVM solver canbe combined together to solve the rigid body problemFollowing this development various other simulations wereconducted utilizing this coupling method [4 21ndash23]

Vast amount of performance analysis and optimizationtechniques has been found conducted to utilize the com-putation efficiency of LBM software Kopta et al [24] andTian et al [25] try to evaluate the parallel performance ofLBM in terms of its scalability and efficiency In additiona number of optimization steps can be constructed on theLB code to improve the simulation efficiency [26] NormallyOpenMPI and OpenMP are the two principle programmingmodels often used in high-performance computing (HPC)The parallelization usually includes a pure message-passing-interface (MPI) algorithm that can be used along with LBMsolver A study has been found that combines both of thesemodels in an effort to reduce the computation resources[26] especially involving complex geometries Parallel per-formance can be further improved by replacing the CPUwith Graphics Processing Units (GPU) algorithms duringnumerical calculation GPU was originally developed forcomputer games and will provide better computation powerfor scientific applications compared to the conventional CPUunit [27]

In the current work the prediction and the investigationof the complex unsteady blood flow inside aneurysm geom-etry is explored by using LBM-based software Palabos Tooptimize the results obtained three different collision mod-els namely the incompressible BGK model the regularizedBGK model and the multiple relaxation time (MRT) modelare compared to investigate the efficacy of each methodin modelling blood flow problem [28] The accuracy ofthese different LB-based scheme in solving aneurysm blowflow simulations will be examined by comparing the hemo-dynamic parameters with the conventional finite volumemethod based solver To the best of the authorsrsquo knowledgethere is no study conducted to compare the capability ofboth FVM and LBMmodels typically in the simulation bloodflow problems in terms of the solution accuracy and parallelperformance The current study attempts to compare thedistribution of mesh for both LBM and FVM models in thedetection of highwall shear stress (WSS) region in the arterialwall The parallel speedup with increasing number of cores isalso compared between both LBM and FVM solvers in termsof its parallel efficacy

Table 1 Weighting functions for D3Q19

Model 1198882

119904Node no Weight

D3Q19 131198910

131198911ndash1198916

1181198917ndash11989118

136

0

1

2

3

4

5

6

78

9

10

11

1213

14

15

16

17

18

Figure 1 3D lattice arrangements for D3Q19

2 Lattice Boltzmann Models

The results shown in this section are formulated using D3Q19lattice model The LBM equation can be summarized in

119891 (119903 + 119888119889119905 119888 + 119865119889119905 119905 + 119889119905) minus 119891 (119903 119888 119905) = Ω119891 (119903 119888 119905) (1)

in which the right-hand side represents the streaming stepThe left-hand term denotes the collision term which can berepresented using the well-known Bhatnagar-Gross-Krook(BGK) model as given in

Ω = 120596 (119891eqminus 119891) =

1

120591

(119891eqminus 119891) (2)

120596 and 120591 denote the relaxation frequency and time 119891eq

represents the equilibrium function that relates to the latticearrangement

The equilibrium function 119891eq can be described as

119891eq(120588 119906)

= 120588119908[1 +

1

1198882

119904

(119888 sdot 119906) +

1

21198884

119904

(119888 sdot 119906)2minus

1

21198882

119904

(119906 sdot 119906)]

(3)

in which 119908 represents weighting function across differentlattice links For the case ofD3Q19 latticemodel as depicted inFigure 1 the weighting functions can be described in Table 1

Computational and Mathematical Methods in Medicine 3

Microscopic velocities for aD3Q19 latticemodel are givenas

1198900= (0 0 0)

11989012

= (plusmn1 0 0)

11989034

= (0 plusmn1 0)

11989056

= (0 0 plusmn1)

1198907ndash10 = (plusmn1 plusmn1 0)

11989011ndash14 = (plusmn1 0 plusmn1)


(4)

21 Conventional BGK Collision Model The underlying the-ory of LBM is based on the discrete Boltzmann equationDue to the complicated collision term that exists in the RHSof the Boltzmann equation it is difficult and burdensome tosolve To ease the computational effort Bhatnagar et al [32]proposed a simplified version of the collision operator Thecollision operator can be replaced as

Ω = 120596 (119891eqminus 119891) =

1

120591

(119891eqminus 119891) (5)

where 120596 = 1120591 The coefficient 120596 denotes the collisionfrequency and 120591 is the relaxation factor 119891eq is the Maxwell-Boltzmann equilibriumdistribution function By substitutingthe approximate collision operator the discrete Boltzmannequation can be shown as

119891119894( +

997888rarr119890119894 119905 + 1) = 119891

119894( 119905) + Ω (6)

where 119894 is the index selected between the possible discretevelocity directions and 119890

119894is the direction of the selected

velocityThe fluid density and macroscopic velocity can be found

from the moment of the distribution function as shownbelow

120588 = sum

119894

119891119894

=

1

120588

sum

119894

119891119894

997888rarr119890119894

(7)

Subsequently the equilibrium distribution 119891eq can be

arranged according to the Maxwell-Boltzmann distributionform as

119891eq119894( 119905) = 119908

119894120588 [1 + 3 (

997888rarr119890119894) +

9

2

(997888rarr119890119894)

2

minus

3

2

2] (8)

Equation (5) will replace the commonly used Navier-Stokesequation in CFD simulations It is also possible to derive N-S equation from Boltzmann equation using the Chapman-Enskog model

22 Incompressible BGK Model In the fundamental form ofBGK the equilibrium term is multiplied by the fluid density

as shown in (5) In incompressiblemodel the density term 120588will be formulated as 120588 = 120588

0+ 120575120588 where 120588

0is a constant value

and 120575120588 is the infinitesimal changes in densityThe equilibriumdistribution function then becomes [16]

119891eq119894( 119905) = 119908

119894[120588 + 120588

03 (

997888rarr119890119894) +

9

2

(997888rarr119890119894)

2

minus

3

2

2] (9)

inwhich all terms containing aremultiplied by the constant1205880 Since the velocity is defined to be proportional to the

momentum the incompressible model is computationallycheaper than the standard BGK as it avoids the division by120588 in (3) It also reduces the compressibility errors introducedby the quasi-incompressible nature of LBM

23 Regularized BGK Model The regularized model can beextended from the standard BGK dynamics with relativelysimpler precollision steps based on the study by Latt andChopard [33] The use of this model can increase numericalstability with less consumption of the computational over-head The equilibrium term 119891

eq reads

119891eq119894

= 120588119905119894[1 +

V119894120572119906prop

1198882

119904

+

1

21198884

119904

119876119894prop120573

119906prop119906120573] (10)

where a repeated Greek index implies a summation over thisindexThe tensors119876

119894prop120573are defined to119876

119894prop120573= V119894120572119906119894120573minus 1198882

119904120575prop120573

and 119905119894rsquos as well as 119888

119904are the coefficients specific to the lattice

topology

24 Multiple Relaxation Time Model (MRT) The linear colli-sion operator is recast into the space of velocity moments andtheir corresponding relaxation parameters are individuallyadjusted to improve numerical stability or to modify thephysics of the model In the MRT model the local discreteMaxwellian function [34] can be approximated as

119891eq119872prop

(120588 )

= 119908prop

1 +

997888rarr119890propsdot 120588

RT+

(997888rarr119890propsdot 120588)

2

2 (RT)2minus

1

2

sdot 120588

RT

(11)

where119908propis the weighting factor In this representation of the

local Maxwellian and the factor RT is related to the speed ofsound 119888

119904 of the model through RT = 119888

2

119904 where 119888

119904= 1radic3119888

25 Finite Volume Method FVM simulation has servedas a viable alternative to experimental methods and ismore superior in terms of its ability to determine flowvariables which are difficult to obtain in experiments Thebasic concept of CFD problems mostly involved solvingthe Navier-Stokes (N-S) equation FVM is one of the mostcommonly used approaches in CFD and recent progressin computational capability has led to the use of FVM insolving various critical medical problems These medicalcases typically for blood flow problems often require complexsimulation that demands substantial memory usage andhigh calculation speed Many researchers have utilized N-S equation in solving flow related problems of aneurysm


Figure 2Themeaning of pseudopotential force Each lattice popu-lation experiences an attractive force from color-mates neighboursand a repulsive one from the opposite color

cases with highly reliable simulation results produced usingfinite volume based numerical method [35ndash50] In FVMthe N-S equations comprise continuity mass and energyconservation equations The simplified version of the finitevolume equation for fluid flow yields the governing equationsas described in (12) and (13)

Continuity Equation Consider the following

120597120588

120597119905

+ nabla sdot 120588 = 0 (12)

Momentum Equation Consider the following

120597

120597119905

120588 + nabla sdot (120588) = minusnabla119875 + nabla sdot 120591 + 120588 (13)

3 Parallelization Message PassingInterface (MPI)

Parallelization is performed via the message-passing-interface (MPI) paradigm of the MPI library It is astandardized portable MPI system that works on a widevariety of parallel computer platforms MPI helps tosubdivide large problem into smaller and more manageableones that are equally distributed for each processor in theparallel system Each processor will then solve the problemconcurrently and the results are combined at the end of theparallelization The main function of MPI is to speedup theexecution of application by subdividing the job among allmultiple processing elements in parallel device Exploitationof all processors or cores in parallel machines is necessary toensure significant speedup in calculation

4 Multiphase Formulation

Shan et al introduced pseudopotential model for multi-component [51] and multiphase flow [52 53] as depicted inFigure 2 It was a popular model based on collision operatorthat has been improved by long-range interaction force that

imitates the pairwise interaction potential between dissimilarphases or components and their findings have shown thatimpressive results are obtainable with fairly minimal effortThe Shan-Chen model is therefore highly popular despitepotential numerical deficiencies in some areas [52]The Shan-Chen model represents the source term as follows

119881(119909 1199091015840) = 119866 (119909 119909

1015840) 120595 (119909) 120595 (119909

1015840) (14)

where 119866 is a Greenrsquos function and 120595 = 120595(119901) is the ldquoeffectivemassrdquo Given the potential the interaction force is written as

119865 = minus119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) 120595 (119909)sum

120572

120595 (119909 + 119890120572) 119890120572 (15)

The strength of interaction is controlled by the amplitudeparameter 119866 Note that positive 119866 means attraction mean-while negative 119866 means a repulsion As a result phaseseparation is determined by positive entries in the diagonalinteraction matrix 119866 and negative (repulsion) on the off-diagonal

For most practical purposes this interaction is describedin its ldquocorralrdquo form as

119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) =

119866 if 1003816100381610038161003816119890120572

1003816100381610038161003816= 119888

0 if 1003816100381610038161003816119890120572

1003816100381610038161003816gt 119888

(16)

where 119888 is the lattice spacing The velocity 119906(eq) is changedto 119906 + 120591119865120588 when the equilibrium distribution functionis computed The averaged momentum before and aftercollision is 120588119906 + 1198652 The nonideal gas equation of state(EOS) was specified by120595(120588) = 120588

119900[1minusexp(minus120588120588

119900)] Analytical

calculation can be computed to find the value of 119866 and 120588where the value of 119866 is determined by the temperature Forthe same fluid the coexistence of two densities presents at arange of 120588 where 119889119901119889120588 is negative for the case that is belowthe critical valuesThe advantage of using this model in phasesegregation is generated automaticallyThe trade-off howeveris that the collision operator does not conserve momentumlocally and instead momentum is conserved globally if thereis no momentum exchanged occurring at the boundaries

5 Methodology

51 Simulation Setup An artery bifurcation aneurysm geom-etry located at the branch point of the common internaland external carotid arteries is created using the CADsoftware as shown in Figure 3 The carotid arteries functionas blood vessel that carries blood from the neck to thebrain region El-Sabrout and Cooley found out that theoccurrence of aneurysm in the extracranial carotid arteryis found to be rare [54] Their research also revealed thatin most cases aneurysm in the external portion is seldomreported and limited research has been conducted to studythe treatment procedure since it involves diverse etiologiesand present diagnostic and therapeutic challenges Howeverthis aneurysm affected carotid artery if left untreated maycause neurologic symptom from embolization or seriouscomplications that include rupture and thrombosis [55 56]


Table 2 Differences between iliac and cerebral arteries dimensions[29ndash31]

Artery Vessel caliber(cm)

Wall thickness(cm)

Iliac artery 104 0076Cerebralintracranialartery 016ndash077 0026

Figure 3 Artery bifurcation aneurysms geometry

Various treatment methods are recommended for patientswith unruptured intracranial aneurysm as early precautionto future aneurysm rupture with varying risk depending onage smoking and size of the aneurysm [29] Johnston et alfound that post detection of intracranial aneurysm includesrupture or subarachnoid hemorrhage associated with 32ndash67 fatality and 10ndash20 long term effect at the specifiedbrain area [57] Their study also shows that for rupturedaneurysm early detection or study on the intracranial regionwithin 24 to 72 hours can help to reduce the chance ofsubsequent subarachnoid hemorrhage In a separate study itwas reported that 50 of rupture also occurs in the aneurysmaffected iliac artery region with 50 to 70 mortality rate[58]The common iliac artery originates from the abdominalaorta that is the main blood vessel in the abdominal areaGiven the fatality rate associated with aneurysm cases it isimperative that the aneurysm be detected at the onset of itsdevelopment and intervention steps begin [30]

The carotid artery aneurysm is chosen in this paperfor future development of stent design studies in the treat-ment of aneurysm that may cause serious complications tothe wall artery The bifurcation region of the intracranialartery comprising common internal and external carotidarteries is chosen with the range of diameters given as043ndash077 cm 0394ndash053 cm and 036ndash056 cm respectively[59ndash61] Table 2 summarizes the differences between theintracranialcerebral and iliac arteries with respect to thevessel caliber and wall thickness Incidentally these differentwall dimensions will incur different stress levels on the arterywall that can be detected using numerical simulation Thetreatment procedure can then be proposed depending on the

Figure 4 Regular mesh in LBMmodel at particular plane

distribution of wall stresses that is highly dependent on thebranching angle and flow type

Norma et al study the development factor of aneurysmthat includes role of bifurcation age gender and envi-ronmental factor It was shown that for study conductedon elderly age older than 65 years old the prevalence ofaneurysm is approximately 5 to 6 in men while 1 to2 for women The study has also shown the importance ofbranching angle study with acute angle branch commonlyattributed with smokers and aging factors could give riseto turbulence and increased in wall shear stress [62] Byreferring to the research by Rossitti and Lofgren theirmathematical formulation has shown that the blood flowrate influences the wall shear stress proportional to the thirdpower of the vessel radius These changes in blood flowrate will continuously change the vessel radius resulting ininduction of high stress region at certain parts of the arterywall [63] In Figure 3 the carotid artery branch model con-sisting of one inlet (common artery) and two outlets (externaland internal arteries) with different diameters is createdusing CAD model [64] The inlet diameter is approximately063 cm while the first and the second outlet diameters areabout 045 cm and 03 cm respectively

Mesh generation is one of the most critical parts influid simulation The quality of the mesh will determine theaccuracy and the stability of the numerical calculation Thetriangularmesh setupwas generated using both of the LB andFVmethods The mesh creation process of Palabos is writtenin the source code and the mesh is automatically generatedwhen the Palabos code is compiled and runThe constructedmesh using the LBM approach is all regular and no additionalcomplexity is required to create suchmesh Figure 4 shows anexample of blood flow model that has been discretized intoregular mesh The upper model has a higher resolution ascompared to the lower model The interfaces and the fluidproperties in each cell are calculated and solved using theLattice Boltzmann equation

In comparison FLUENT needs external meshing soft-ware to produce the volume mesh This volume meshgeneration process is structured carefully to ensure reli-able simulation results Moreover the simulation results arestrongly dependent on the final mesh as well Figure 5 depicts


Table 3 Flow parameters

Parameters ValueDensity of blood 1060 kgm3

Kinematic viscosity 37037 times 10minus6m2s

(m)001000750005000250

Figure 5 Mesh created by external software in ANSYS FLUENT

an unstructured tetrahedral mesh generated using ANSYSFLUENT

To optimize the LBM simulation three different types ofsolvers were chosen to compare their accuracy against thatof the FVM solver They are the incompressible BGK modelthe regularized BGKmodel and the multiple relaxation time(MRT) model In FLUENT the pressure-based type solveris selected The pressure-based approach was developed forlow-speed incompressible flows and is compatible with thecurrent blood flow problem In each simulation the blood ismodelled as a Newtonian fluid to ensure the physical prop-erties remain consistent throughout the experiment Table 3shows the blood flow parameters used in this simulation

A constant parabolic velocity profile is imposed on theinlet with zero pressure set at both outlets The wall bound-aries are also imposed using nonslip boundary conditionTheaverage velocity is set at 01ms in this research accordingto blood flow velocity in artery used to investigate aneurysmcases based on the experiment and simulation from researchoutcome by Gabe et al [65] Since the inlet velocity profilein FLUENT is constant a user-defined file (UDF) of theparabolic profile is created to standardize both simulationsThe UDF parabolic profile is created according to the PowerLaw Laminar formula = 2119906avg[1 minus (119903119877)

2] where 119877 is the

inlet radius and 119906avg denotes the average velocity over thecross section In a recent study [53] the parabolic formula-tion produced highly similar results compared to the CFDsimulation for cases involving the effect of the inlet velocityprofile using real-patience-specific inlet velocity profile Bycombining all the parameters to the inlet diameter the flowrsquosReynolds number was estimated at around 170

The physical properties of a fluid must be converted intothe LB discrete system during LBM implementation Thereare constraints that must be adhered to in determining thechoice of the lattice variables The first principle is that thediscrete system must be equivalent to the physical systemfor example Reynolds number of incompressible flow mustgive the same value Another constraint is that the param-eters should be fine-tuned in order to reach and maintain

the required accuracy and stability during simulation runFor example the grid should be sufficiently resolved orthe discrete time step should be small enough to ensuresmooth simulation computation process In this simulationthe discrete space interval of 120575119909 can be obtained from theaneurysm geometry and the discrete time step 120575119905 is chosenas another fixed variable The discrete space interval can bedefined as the reference length divided by the number of cells119873 used to discretize the length The resolution 119873 is set at350 with the reference length set at 005589m correspondingto 120575119909 = 0000156563 Since the physical time step is set as001 s 120575119905 is discretized as 000005 The same physical timestep is also applied in the FLUENT solver With two knownvariables 120575119909 and 120575119905 other variables such as discrete averagevelocity 119906LB discrete kinematic viscosity VLB and relaxationtime 120591 can now be easily transformed between the physicaland the LB system as shown in

119906LB = 119906physical times120597119905

120597119909

VLB = Vphysical times120597119905

1205971199092

120591 = 3VLB + 05

(17)

Variables 119906LB and VLB were calculated as 00319361 and000755492 respectively with 120591 = 052266 This relaxationthat is approaching 05 yields a stable simulation using doubleprecision arithmetic during the computations The final stepis conducted to validate the first principle through Reynoldsnumber in lattice unitsTheReynolds numbers in LB unit canbe calculated via

Re =119906LB119873inlet

VLB (18)

where 119873inlet is the lattice nodes of the inlet with value givenby 119873inlet asymp 1936477 lattices nodes The calculated Reynoldsnumber is then a value of around 170

52 Parallel Computation A comparative study is conductedto compare the efficacy of parallel implementation scheme inthe LBM and the FVM scheme in amultiprocessor computerQuad Cores Intel Xeon CPU W3520267GHz with a 12-GB installed memory (12-GB ram) The system type is 64-bit Microsoft Windows XP operating system MPI was usedas the message passing library to provide a straightforwardinterface to write a software program that can utilize multiplecores of the computer A program written using Palabos canbe automatically parallelized through the use of the MPIlibrary Major improvement in the speed up could be seenwhen all of the cores in a parallel system are being exploitedhighlighting the importance of parallel architecture In con-trast in FLUENT a built-inMPI system is included and readyfor use To ensure similar setup between LBM and FVMthe number of lattice elements of meshing was set to almostsimilar value for the two geometries The mesh informationof the fluid zone domain in FLUENT comprised 1866320elements and 455001 nodes whereas in LBM it comprised


Figure 6 Coordinates of 15 points of interest

Figure 7 Orientation of 3 planes (119883119884 plane is red119884119885 plane is blueand 119885119883 plane is green)

1728430 elements and 487359 nodes For comparison onlythe incompressible BGK model was selected as the LBMcollision method to be compared to the FLUENT simulation

6 Results and Discussion

61 Comparisons between Methods The solutions wereobserved and analyzed after running the aneurysm caseon both solvers The wall artery of aneurysm geometryitself along with additional 15 points of interest and threeplanes has been used to explore the blood flow reaction Thedissimilarity of these two solvers is also investigated Figure 6shows the fifteen coordinate points distributed throughoutthe geometry and Figure 7 depicts the orientation of theplanes The position of points and orientation of planesis chosen in this way so that every part of the flow canbe well understood and studied The monitoring processstarted with the pressure profile when the steady state of theflow is reached Figure 8 compares the pressure contours offour different methods in wall artery The pressure contours

of incompressible BGK model (IncBGK) regularized BGKmodel (RLBM) and multiple relaxation time model (MRT)are almost identical compared to the FVM model All fourpressure contours show that pressures are high at the vicinityof the inlet region and the pressure gradually decreases untilit reaches zero value at the two outlets

The blood flow problem is related to the Poiseuille flowand there must exist some pressure difference as the bloodflows through the artery Another likeliness of the four mod-els is the high pressure spot found in the bifurcation regionThis is probably due to the collision of the bloodstream andwall as the stream separates into two branches resulting inhigh pressure spot The pressure of each appointed pointis compared among the solvers and graphed in Figure 9Figure 10 displays the percentage difference of pressure andthese differences are computed from

Percentage Diff =

10038161003816100381610038161003816100381610038161003816

119875LBM minus 119875FVM119875FVM

10038161003816100381610038161003816100381610038161003816

times 100 (19)

All LBM models give high degree of accuracy relative to theFVM model In terms of the pressure profile the maximumpercentage difference is of just merely 5 for comparisonat point 8 Furthermore Figure 7 shows that all lines areoverlapping with each other Thus it can be concluded thatLBMmodel is able to produce comparable results with FVMmodel in terms of the pressure profile

For the velocity profile the velocity magnitude contoursfor different simulations were placed side by side and com-pared according to their respective cross section Figures 1112 and 13 show the velocity contours in 119883119884 119884119885 and 119885119883

planeFrom Figures 11 to 13 it was revealed that the characteris-

tics of the flow field are similar for different simulationmeth-ods at the given cross section All LBM models have almostidentical velocity contours between each other although someminor deviation can be found for the solutions obtainedusing FVM As depicted in Figure 11 the stagnation flow isobserved inside the aneurysm for all simulation since thevelocity is almost zeroThere are high velocities of flow streamdetected near the bifurcation point for every simulation Thehigh velocity flow is believed to have hit the wall causinghigh pressure spot as previously demonstrated in Figure 8The same characteristic of flow can be predicted from 119885119883

plane in which the high velocity can be observed near thebifurcation region while zero velocity region is visible insidethe aneurysm Similarly for the pressure profile the velocitymagnitude at each interest point location will be compared todetermine the accuracy between these twomethodsThe plotof velocity magnitude and percentage differences is compiledin Figures 12 and 13

Referring to Figure 14 the velocity magnitude at eachlocation for FLUENT incompressible BGK LBM modelmultiple relaxing time LBM model and regularized BGKLBMmodels shows small variation and almost overlaps witheach other Figure 15 shows the ldquoabsolute changerdquo of velocityfor all LBM models relative to the FLUENT FVM model


Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

Figure 8 Pressure contours in the different simulations at the wall artery of aneurysm geometry

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT

Figure 9 Comparison of the pressure of points of interest betweenLBM and FVMmodels

The absolute change in velocities for all LBM models relativeto the FVMmodel is computed from

Absolute Change = 1003816100381610038161003816119881FVM minus 119881LBM

1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

Figure 10 Percentage difference (pressure) at each point of interestfor all LBMmodels relative to FVM

Note that at point 3 RLBM and MRT have similar values ofvelocities relative to the FVM models In comparison for allpoints the velocity values of all LBM models show minor


X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow

Stagnation flow inside aneurysm

FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

Figure 11 Velocity magnitude contours at119883119884 plane for different simulations

differences relative to the FVMmodelThis demonstrates thecapability of all LBM models compared to the conventionalestablished FVMmodel

Wall shear stress is a flow-induced stress that can bedescribed as the frictional force of viscous blood It hasalways been thought of as the most important parameter forclinical diagnosis of artery disease It is also one of the mostreliable indicators for discriminating the rupture point of theaneurysm geometry Two theories can be used to explain themechanism of aneurysms rupture which are high-flow andlow-flow theories Consistent with these theories some CFDstudies came out with different findings regarding the rupturestatus of aneurysm In a study by Bai-Nan et al [62] andCebral et al [66] it was reported that the ruptured aneurysmsare associated with higher wall shear stress but recent studiespointed out that the rupture point of aneurysm is inducedby remarkably low shear stress [8 67] In our simulationwe found that both highest and lowest wall shear stress existin the geometry The highest wall shear stress (WSS) occursnear to the bifurcation region whereas the lowest wall shearstress is observed in the aneurysm wall These two areasmay possess the highest risk of rupture since there exist tworegions of highly concentrated stresses These high stressregions may pose problem if they exceed the allowable stresslevel causing the wall to deform Subsequently it will lead toaneurysm deformation or in worst case scenario can lead tothe rupture at the artery wall From Figure 14 both LBM andFVM solutions can precisely indicate the area of highest andlowestWSSThe FVMmodel tends to have higherWSS at thevicinity of the bifurcation point As demonstrated previouslyin Figures 8 and 16 the highest WSS area also corresponds torelatively high pressure and velocity This is the region where

the bloodstreampersistently hit the walls thereby causing thishigh WSS values

In order to compare the capability of LBM and FVMsolver the histogramofWSS profile at wall artery of geometryis studied and analyzed Only FLUENT and incompressibleLBMmodel were compared since the remaining LBMmodelsgave comparatively similar results The WSS histogramsfor both FVM and LBM are shown in Figures 17 and 18respectively

In comparison the WSS profile for LBM shows relativelybetter distribution of mesh compared to FVMmodel This isdue to the distribution of mesh in LBM as shown in Figure 18which is decreasing more uniformly from one peak valueto the other indicating highly continuous distribution ofnumerical approximation across all of themesh For the FVMmodel as depicted in Figure 17 sudden peak particularly atthe points withWSS value of 14ndash15 Pa and low concentrationof mesh distribution for WSS value in the range of 18ndash28 Pa demonstrates the lack of continuity from on meshto the other To complement this statement two box plotswere constructed as depicted in Figure 19 based on thepercentage of mesh number between WSS values of 0ndash28 Pato compare the variability and uniformity of the currentmeshdistribution in LBM and FVM models From Figure 19 twooutlier values of velocities can be seen in the FVMmodel withthe box plot having minimal variation of data between thefirst and third quartile indicating less uniform percentage ofmesh distribution In contrast it was shown that no outlierswere observed in the LBM with considerably better variationof velocity data ranging from 05 to 75 Pa between the firstand the third quartiles This spread of data shows good


X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

Figure 12 Velocity magnitude contours at 119884119885 plane for different simulations

uniformity for LBMmethod mesh distribution moving fromone WSS value to the other

Since the treatment method is dependent on the distri-bution of WSS highly uniform mesh typically at the highWSS region as depicted in Figure 18 for the LBM model canaccurately identify the critical WSS region to determine thebest possible treatment options Early studies have pointedout that morphological changes in the vessel wall may resultin growth of aneurysm over time [68ndash71] Aneurysm rupturecan occur if the arterial wall is unable to withstand theincrease in hemodynamic forces [70] It is believed that thelow-WSS segments can weaken the arterial wall structure andhigh WSS can lead to aneurysm wall destructive modelling[72] Therefore treatment strategies of aneurysm could beconducted to reduce risk of rupture For small unrupturedand asymptomatic aneurysm observation of the wall arteryusing image scanning can be conducted every year with nosurgical treatment advised However for high risk aneurysmthat may lead to rupture it could be treated by surgicalclipping or endovascular treatment such as stent placementcoil embolization and flow diversion [73 74]

62 Multiphase Blood Flow Simulation In this section theimpact of high velocity 3D flow model towards the arterywall is observed The 3D flows with velocity higher than005ms will be visualized to examine its impact on the arterywall The multiphase simulation profile of blood flow withvelocity more than 005ms at various time steps is shownin Figure 20

The blood will stream and get separated into two mainbranches Noticeable nonvisible region can be observed onthe left side of the branch This region with velocity less than005ms will exhibit the highest wall shear stress (WSS)WSSis a dynamic frictional force induced by viscous fluid movingalong a surface of solid material It is highly important thatthis region of high WSS is diagnosed earlier before it burstsas a result of significant buildup of WSS

Figure 21 depicts the 2D multiphase profile of velocity atvarious time steps In Figure 21(b) there exists low velocitydomain in the aneurysm region as previously described inFigure 20(b) Figure 21(b) shows that higher velocity flowregion occurs near the bottom parts of the bifurcation regionwhere high WSS pressure is induced


Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z

Stagnation flowinside aneurysm


Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT

Figure 14 Comparison of the velocity of 15 points of interestbetween LBM and FVMmodels

63 Parallel Implementation In Table 4 the LBM solver isshown to perform more efficiently than the conventionalFVM solver LBM takes considerably smaller time than FVMto execute the simulation with 5879 smaller computationtime when applied on single core platform As seen in

Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018

Figure 15 Absolute change (velocity) at each point of interest for allLBMmodels

Figure 22 the simulation time obtained using FLUENT isrelatively similar compared to Palabos as the numbers ofcores is increased Even so it still imposes around 3240


Table 4 The tabulated form of simulation time and percentage difference

Number of cores PALABOS time (s) FLUENT time (s)Difference between

FLUENT time (s) andPALABOS time (s)

Percentage difference ()

1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

Figure 16 Wall shear stress in the different simulations at the wall artery of aneurysm geometry

00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)

Figure 17 Histogram of percentage of mesh number at differentWSS values for FVM

20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)

Figure 18 Histogram of percentage of mesh number at differentWSS values for LBM


LBM box plotFVM box plot

Two outlierrsquosvelocity values

012345678910111213141516171819

Figure 19 Box plot of LBM and FVMmodels for percentage of mesh number between WSS range of 0ndash28 Pa

(a) Simulation profile at time step 119905 = 004 s (b) Simulation profile at time step 119905 = 007 s

(c) Simulation profile at time step 119905 = 022 s (d) Simulation profile at time step 119905 = 028 s

(e) Simulation profile at time step 119905 = 060 s

Figure 20 3D multiphase blood flow for velocity more than 005ms at different time steps


(a) (b)

(c) (d)

Figure 21 2D Multiphase velocity of blood flow for different time steps

The simulation time versus number of cores

PalabosFLUENT

1 2 3 4 50Number of cores used

0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)

Figure 22 Comparisons of the simulation time versus number ofcores for Palabos and FLUENT

and 3164 differences when 3 and 4 cores are used Thespeed-up curve shows smaller gradient as the number ofcores is increased from single core up to 4 cores Similar

behavior can be observed for both Palabos and FLUENTMany reasons could have attributed to this issue but it isprimarily due to the interconnection and communicationtime between the different cores used As the number ofcores is increased the tasks will be subdivided into muchsmaller task resulting to smaller computation time to solveIn the same time as the numerical of core increases theinterconnection and communication time between the coreswill increase substantially Therefore it is not worth to addmore cores at some point when the computation time becamesaturated This is consistent with the findings by Abas andAbdul-Rahman in which it was found that as the number ofcores increases for similar applied problem at some point itwill reach saturation point where most of the computationtime is spent on the interconnection and communicationbetween individual cores [75]

7 Conclusions

In this study the Lattice Boltzmann method (LBM) for3D transient blood flow in aneurysm geometry has been


validated by comparing the simulation results with thoseobtained by conventional commercial FVM solver The LBMsolver can provide nearly identical solutions to those of theFV method Several versions of the LBM were tested and allLBMmodels showed the same results The pressure velocityand the WSS profile from both methods were compared anddemonstrated with good agreement between the differentsolvers The maximum percentage difference of pressure forthe 15 points of interest come out to a mere 5 differenceand the absolute change for the velocity values ranges from0 to 0017msminus1 It was also found that the WSS contours forthe LBM are observed to be more acceptable compared to asignificantly high WSS induced by FVM in the bifurcationregion In addition this paper evaluates the performance ofparallel implementation in eachmethod From the findings itwas shown that LBM is more efficient than the FVM in termsof the parallel computing setup

The main purpose of this paper is to demonstrate thatLBM presents a viable alternative for computation of hemo-dynamics problems It is self-evident that the validation ofthis new method against existing established methodologieshas provided an additional platform for CFD analysis inbiomedical application Nevertheless this study shows thatLBM can be considered an alternative solver for compu-tational hemodynamics that can produce results of similarquality compared to those obtained from the Navier-Stokessolvers

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work was partly supported by the Short Term Grant60313020 from the Division of Research and InnovationUniversiti Sains Malaysia and FRGS Grant 6071322 from theMinistry of Higher Education Malaysia

References

[1] D Scheinert M Schroder H Steinkamp J Ludwig and GBiamino ldquoTreatment of iliac artery aneurysms by percutaneousimplantation of stent graftsrdquo Circulation vol 102 no 19supplement 3 pp Iii-253ndashIii-258 2000

[2] D O Wiebers J P Whisnant J Huston et al ldquoUnrupturedintracranial aneurysms natural history clinical outcome andrisks of surgical and endovascular treatmentrdquo The Lancet vol362 no 9378 pp 103ndash110 2003

[3] T Hashimoto H Meng and W L Young ldquoIntracranialaneurysms links among inflammation hemodynamics andvascular remodelingrdquo Neurological Research vol 28 no 4 pp372ndash380 2006

[4] S Zou X-F Yuan X Yang W Yi and X Xu ldquoAn integratedlattice Boltzmann and finite volume method for the simulationof viscoelastic fluid flowsrdquo Journal of Non-Newtonian FluidMechanics vol 211 pp 99ndash113 2014

[5] J R Cebral F Mut J Weir and C M Putman ldquoAssociation ofhemodynamic characteristics and cerebral aneurysm rupturerdquo

American Journal of Neuroradiology vol 32 no 2 pp 264ndash2702011

[6] J Xiang S K Natarajan M Tremmel et al ldquoHemodynamic-morphologic discriminants for intracranial aneurysm rupturerdquoStroke vol 42 no 1 pp 144ndash152 2011

[7] J-D Muller M Jitsumura and N H F Muller-KronastldquoSensitivity of flow simulations in a cerebral aneurysmrdquo Journalof Biomechanics vol 45 no 15 pp 2539ndash2548 2012

[8] Y Miura F Ishida Y Umeda et al ldquoLow wall shear stressis independently associated with the rupture status of middlecerebral artery aneurysmsrdquo Stroke vol 44 no 2 pp 519ndash5212013

[9] A J Geers I Larrabide H G Morales and A F FrangildquoApproximating hemodynamics of cerebral aneurysms withsteady flow simulationsrdquo Journal of Biomechanics vol 47 no 1pp 178ndash185 2014

[10] S Alapati S Kang and Y K Suh ldquoParallel computation oftwo-phase flow in a microchannel using the lattice Boltzmannmethodrdquo Journal of Mechanical Science and Technology vol 23no 9 pp 2492ndash2501 2009

[11] K R Tubbs and F T-C Tsai ldquoMultilayer shallow waterflow using lattice Boltzmann method with high performancecomputingrdquo Advances in Water Resources vol 32 no 12 pp1767ndash1776 2009

[12] D Beugre S Calvo M Crine D Toye and P Marchot ldquoGasflow simulations in a structured packing by lattice BoltzmannmethodrdquoChemical Engineering Science vol 66 no 17 pp 3742ndash3752 2011

[13] D Yu R Mei L-S Luo and W Shyy ldquoViscous flow computa-tions with the method of lattice Boltzmann equationrdquo Progressin Aerospace Sciences vol 39 no 5 pp 329ndash367 2003

[14] Y Liu ldquoA lattice Boltzmann model for blood flowsrdquo AppliedMathematical Modelling vol 36 pp 2890ndash2899 2012

[15] C Sun and L LMunn ldquoLattice-Boltzmann simulation of bloodflow in digitized vessel networksrdquo Computers amp Mathematicswith Applications vol 55 no 7 pp 1594ndash1600 2008

[16] D Leitner S Wassertheurer M Hessinger and A HolzingerldquoA Lattice Boltzmann Model for pulsative blood flow in elasticvesselsrdquo Elektrotechnik und Informationstechnik vol 123 no 4pp 152ndash155 2006

[17] G Zavodszky and G Paal ldquoValidation of a lattice Boltzmannmethod implementation for a 3D transient fluid flow in anintracranial aneurysm geometryrdquo International Journal of Heatand Fluid Flow vol 44 pp 276ndash283 2013

[18] M Matyka Z Koza and Ł Mirosław ldquoWall orientation andshear stress in the lattice Boltzmann modelrdquo Computers andFluids vol 73 pp 115ndash123 2013

[19] K Habibi and L Mongeau ldquoPrediction of sound absorption bya circular orifice termination in a turbulent pipe flow using theLattice-Boltzmann methodrdquo Applied Acoustics vol 87 pp 153ndash161 2015

[20] S Bogner and U Rude ldquoSimulation of floating bodies withthe lattice Boltzmann methodrdquo Computers ampMathematics withApplications vol 65 no 6 pp 901ndash913 2013

[21] J R Clausen D A Reasor Jr and C K Aidun ldquoParallel perfor-mance of a lattice-Boltzmannfinite element cellular blood flowsolver on the IBM Blue GeneP architecturerdquo Computer PhysicsCommunications vol 181 no 6 pp 1013ndash1020 2010

[22] Z Li M Yang and Y Zhang ldquoA coupled lattice Boltzmannand finite volume method for natural convection simulationrdquoInternational Journal of Heat and Mass Transfer vol 70 pp864ndash874 2014


[23] ACVelivelli andKMBryden ldquoDomain decomposition basedcoupling between the lattice Boltzmannmethod and traditionalCFD methodsmdashpart I formulation and application to the 2-DBurgersrsquo equationrdquo Advances in Engineering Software vol 70pp 104ndash112 2014

[24] P KoptaMKulczewski K Kurowski et al ldquoParallel applicationbenchmarks and performance evaluation of the Intel Xeon 7500family processorsrdquo Procedia Computer Science vol 4 pp 372ndash381 2011

[25] M Tian W Gu J Pan and M Guo ldquoPerformance analysis andoptimization of palabos on petascale sunway BlueLight MPPsupercomputerrdquo in Parallel Computational Fluid Dynamics KLi Z Xiao Y Wang J Du and K Li Eds vol 405 ofCommunications in Computer and Information Science pp 311ndash320 Springer Berlin Germany 2014

[26] L Biferale F Mantovani M Pivanti et al ldquoOptimization ofmulti-phase compressible lattice Boltzmann codes onmassivelyparallel multi-core systemsrdquo Procedia Computer Science vol 4pp 994ndash1003 2011

[27] I Tanno T Hashimoto T Yasuda Y Tanaka K Morinishi andN Satofuka ldquoSimulation of turbulent flow by lattice Boltzmannmethod and conventional method on a GPUrdquo Computers ampFluids vol 80 pp 453ndash458 2013

[28] D A Perumal and A K Dass ldquoApplication of lattice Boltzmannmethod for incompressible viscous flowsrdquo Applied Mathemati-cal Modelling vol 37 no 6 pp 4075ndash4092 2013

[29] The International Study of Unruptured Intracranial AneurysmsInvestigators ldquoUnruptured intracranial aneurysmsmdashrisk ofrupture and risks of surgical interventionrdquo The New EnglandJournal of Medicine vol 339 pp 1725ndash1733 1998

[30] F-J H TimoKrings DMMandell T-R Kiehl et al ldquoIntracra-nial aneurysms from vessel wall pathology to therapeuticapproachrdquoNature Reviews Neurology vol 7 no 10 pp 547ndash5592011

[31] A P Avolio ldquoMulti-branched model of the human arterialsystemrdquoMedical and Biological Engineering and Computing vol18 no 6 pp 709ndash718 1980

[32] P L Bhatnagar E P Gross andMKrook ldquoAmodel for collisionprocesses in gases I Small amplitude processes in charged andneutral one-component systemsrdquo Physical Review vol 94 no3 pp 511ndash525 1954

[33] J Latt and B Chopard ldquoLattice Boltzmann method withregularized pre-collision distribution functionsrdquo MathematicsandComputers in Simulation vol 72 no 2ndash6 pp 165ndash168 2006

[34] K N Premnath and J Abraham ldquoThree-dimensional multi-relaxation time (MRT) lattice-Boltzmann models for multi-phase flowrdquo Journal of Computational Physics vol 224 no 2pp 539ndash559 2007

[35] P Berg G Janiga and D Thevenin ldquoDetailed comparisonof numerical flow predictions in cerebral aneurysms usingdifferent CFD softwarerdquo in Proceedings of the Conference onModelling Fluid Flow J Vad Ed pp 128ndash135 Budapest Hun-gary September 2012

[36] S Ahmed I D Sutalo H Kavnoudias and AMadan ldquoNumer-ical investigation of hemodynamics of lateral cerebral aneurysmfollowing coil embolizationrdquo Engineering Applications of Com-putational Fluid Mechanics vol 5 no 3 pp 329ndash340 2011

[37] Y Qian H Takao M Umezu and Y Murayama ldquoRisk analysisof unruptured aneurysms using computational fluid dynamicstechnology preliminary resultsrdquo American Journal of Neurora-diology vol 32 no 10 pp 1948ndash1955 2011

[38] G Janiga P Berg O Beuing et al ldquoRecommendations for accu-rate numerical blood flow simulations of stented intracranialaneurysmsrdquo Biomedizinische Technik vol 58 no 3 pp 303ndash3142013

[39] R Torii M Oshima T Kobayashi K Takagi and T E Tez-duyar ldquoFluid-structure interaction modeling of blood flow andcerebral aneurysm significance of artery and aneurysm shapesrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 45-46 pp 3613ndash3621 2009

[40] A Valencia H Morales R Rivera E Bravo and MGalvez ldquoBlood flow dynamics in patient-specific cerebralaneurysm models the relationship between wall shear stressand aneurysm area indexrdquoMedical Engineering and Physics vol30 no 3 pp 329ndash340 2008

[41] I Chatziprodromou A Tricoli D Poulikakos and Y VentikosldquoHaemodynamics and wall remodelling of a growing cerebralaneurysm a computational modelrdquo Journal of Biomechanicsvol 40 no 2 pp 412ndash426 2007

[42] K Yokoi F Xiao H Liu and K Fukasaku ldquoThree-dimensionalnumerical simulation of flows with complex geometries in aregular Cartesian grid and its application to blood flow in cere-bral artery with multiple aneurysmsrdquo Journal of ComputationalPhysics vol 202 no 1 pp 1ndash19 2005

[43] HGirija Bai K BNaidu andGVasanthKumar ldquoCFD analysisof aortic aneurysms on the basis of mathematical simulationrdquoIndian Journal of Science amp Technology vol 7 no 12 pp 2020ndash2032 2014

[44] R M Shabestari K Hassani and F Izadi ldquoModeling ofabdominal aorta aneurysm and study of the pathology usingcomputational fluid dynamics methodrdquo Biomedical Engineer-ing Applications Basis and Communications vol 23 no 4 pp295ndash305 2011

[45] T Hassan E V Timofeev M Ezura et al ldquoHemodynamicanalysis of an adult vein of Galen aneurysm malformationby use of 3D image-based computational fluid dynamicsrdquoAmerican Journal of Neuroradiology vol 24 no 6 pp 1075ndash1082 2003

[46] W Jeong and K Rhee ldquoHemodynamics of cerebral aneurysmscomputational analyses of aneurysm progress and treatmentrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 782801 11 pages 2012

[47] B V Rathish Kumar ldquoA space-time analysis of blood flow in a3D vessel with multiple aneurysmsrdquo Computational Mechanicsvol 32 no 1-2 pp 16ndash28 2003

[48] S Kelly and M OrsquoRourke ldquoFluid solid and fluid-structureinteraction simulations on patient-based abdominal aorticaneurysm modelsrdquo Proceedings of the Institution of MechanicalEngineers Part H Journal of Engineering in Medicine vol 226no 4 pp 288ndash304 2012

[49] I Szikora G Paal AUgron et al ldquoImpact of aneurysmal geom-etry on intraaneurysmal flow a computerized flow simulationstudyrdquo Neuroradiology vol 50 no 5 pp 411ndash421 2008

[50] I C S Cito J Pallares and A Vernet ldquoCFD challengegiant internal carotid artery aneurysm simulation using thecommercial finite volume solver fluentrdquo in Proceedings of theASME Summer Bioengineering Conference pp 123ndash124 FajardoPuerto Rico June 2012

[51] X Shan and G Doolen ldquoMulticomponent lattice-Boltzmannmodel with interparticle interactionrdquo Journal of StatisticalPhysics vol 81 no 1-2 pp 379ndash393 1995


[52] X Shan and H Chen ldquoLattice Boltzmannmodel for simulatingflows withmultiple phases and componentsrdquo Physical Review Evol 47 no 3 pp 1815ndash1819 1993

[53] X Shan and H Chen ldquoSimulation of nonideal gases andliquid-gas phase transitions by the lattice Boltzmann equationrdquoPhysical Review E vol 49 no 4 pp 2941ndash2948 1994

[54] R El-Sabrout and D A Cooley ldquoExtracranial carotid arteryaneurysms Texas Heart Institute experiencerdquo Journal of Vascu-lar Surgery vol 31 no 4 pp 702ndash712 2000

[55] G Roche-Nagle and G Oreopolous ldquoExtracranial carotidartery aneurysmsrdquoTheAmerican Journal of Surgery vol 197 no6 pp e64ndashe65 2009

[56] R EWelling A Taha T Goel et al ldquoExtracranial carotid arteryaneurysmsrdquo Surgery vol 93 no 2 pp 319ndash323 1983

[57] S C Johnston R T Higashida D L Barrow et al ldquoRec-ommendations for the endovascular treatment of intracranialaneurysms a statement for healthcare professionals from theCommittee on Cerebrovascular Imaging of the AmericanHeartAssociation Council on Cardiovascular Radiologyrdquo Stroke vol33 no 10 pp 2536ndash2544 2002

[58] M Nagarajan P Chandrasekar E Krishnan and S Muralidha-ran ldquoRepair of iliac artery aneurysms by endoluminal graftingthe systematic approach of one institutionrdquoTexasHeart InstituteJournal vol 27 no 3 pp 250ndash252 2000

[59] J Krejza M Arkuszewski S E Kasner et al ldquoCarotid arterydiameter in men and women and the relation to body and necksizerdquo Stroke vol 37 no 4 pp 1103ndash1105 2006

[60] M A Williams and A N Nicolaides ldquoPredicting the normaldimensions of the internal and external carotid arteries from thediameter of the common carotidrdquo European Journal of VascularSurgery vol 1 no 2 pp 91ndash96 1987

[61] Y R Limbu G Gurung R Malla R Rajbhandari and S RRegmi ldquoAssessment of carotid artery dimensions by ultrasoundin non-smoker healthy adults of both sexesrdquo Nepal MedicalCollege Journal vol 8 no 3 pp 200ndash203 2006

[62] X Bai-Nan W Fu-Yu L Lei Z Xiao-Jun and J Hai-YueldquoHemodynamics model of fluid-solid interaction in internalcarotid artery aneurysmsrdquo Neurosurgical Review vol 34 no 1pp 39ndash47 2011

[63] S Rossitti and J Lofgren ldquoOptimality principles and floworderliness at the branching points of cerebral arteriesrdquo Strokevol 24 no 7 pp 1029ndash1032 1993

[64] F H Netter J Craig and J Perkins Netter Atlas of Neu-roanatomy and Neurophysiology The Netter Collection of Med-ical Illustrations 2002

[65] I T Gabe J H Gault J Ross Jr et al ldquoMeasurement ofinstantaneous blood flow velocity and pressure in consciousman with a catheter-tip velocity proberdquo Circulation vol 40 no5 pp 603ndash614 1969

[66] J R Cebral F Mut J Weir and C Putman ldquoQuantitativecharacterization of the hemodynamic environment in rupturedand unruptured brain aneurysmsrdquo American Journal of Neuro-radiology vol 32 no 1 pp 145ndash151 2011

[67] K Fukazawa F Ishida Y Umeda et al ldquoUsing computationalfluid dynamics analysis to characterize local hemodynamicfeatures of middle cerebral artery aneurysm rupture pointsrdquoWorld Neurosurgery vol 83 no 1 pp 80ndash86 2015

[68] A Kamiya J Ando M Shibata and H Masuda ldquoRoles of fluidshear stress in physiological regulation of vascular structure andfunctionrdquo Biorheology vol 25 no 1-2 pp 271ndash278 1988

[69] T F Luscher and F C Tanner ldquoEndothelial regulation ofvascular tone and growthrdquo American Journal of Hypertensionvol 6 no 7 pp 283Sndash293S 1993

[70] D M Sforza C M Putman and J R Cebral ldquoHemodynamicsof cerebral aneurysmsrdquo Annual Review of Fluid Mechanics vol41 pp 91ndash107 2009

[71] W E Stehbens ldquoEtiology of intracranial berry aneurysmsrdquoJournal of Neurosurgery vol 70 no 6 pp 823ndash831 1989

[72] G Paal A Ugron I Szikora and I Bojtar ldquoFlow in simplifiedand real models of intracranial aneurysmsrdquo International Jour-nal of Heat and Fluid Flow vol 28 no 4 pp 653ndash664 2007

[73] A J P Goddard P P J Raju and A Gholkar ldquoDoes the methodof treatment of acutely ruptured intracranial aneurysms influ-ence the incidence and duration of cerebral vasospasm andclinical outcomerdquo Journal of Neurology Neurosurgery andPsychiatry vol 75 no 6 pp 868ndash872 2004

[74] J R Vanzin D G Abud M T S Rezende and J Moret ldquoNum-ber of coils necessary to treat cerebral aneurysms according toeach size group a study based on a series of 952 embolizedaneurysmsrdquo Arquivos de Neuro-Psiquiatria vol 70 no 7 pp520ndash523 2012

[75] A Abas and R Abdul-Rahman ldquoAdaptive FEM with domaindecomposition method for partitioned-based fluidndashstructureinteractionrdquo Arabian Journal for Science and Engineering vol41 no 2 pp 611ndash622 2016

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


as a suspension of particles in the plasma [15] In addi-tion Leitner et al introduced new LBM boundary condi-tions for the cardiovascular domain using a D2Q9 latticemodel [16] The application of LBM to three-dimensional(3D) problems is rather limited and noticeable applica-tions can be found in the work by [17] featuring a 3Dsimulation of intracranial aneurysm geometry and in thework by [18] on wall orientation and shear stress in theartery

In general the application of LBM is not restricted tothe fluid flow simulation It has been found to be an effi-cient solver in sound absorption applications [19] Anotherinteresting use is in the simulation of floating rigid bodieson a free surface [20] The LBM and the FVM solver canbe combined together to solve the rigid body problemFollowing this development various other simulations wereconducted utilizing this coupling method [4 21ndash23]

Vast amount of performance analysis and optimizationtechniques has been found conducted to utilize the com-putation efficiency of LBM software Kopta et al [24] andTian et al [25] try to evaluate the parallel performance ofLBM in terms of its scalability and efficiency In additiona number of optimization steps can be constructed on theLB code to improve the simulation efficiency [26] NormallyOpenMPI and OpenMP are the two principle programmingmodels often used in high-performance computing (HPC)The parallelization usually includes a pure message-passing-interface (MPI) algorithm that can be used along with LBMsolver A study has been found that combines both of thesemodels in an effort to reduce the computation resources[26] especially involving complex geometries Parallel per-formance can be further improved by replacing the CPUwith Graphics Processing Units (GPU) algorithms duringnumerical calculation GPU was originally developed forcomputer games and will provide better computation powerfor scientific applications compared to the conventional CPUunit [27]

In the current work the prediction and the investigationof the complex unsteady blood flow inside aneurysm geom-etry is explored by using LBM-based software Palabos Tooptimize the results obtained three different collision mod-els namely the incompressible BGK model the regularizedBGK model and the multiple relaxation time (MRT) modelare compared to investigate the efficacy of each methodin modelling blood flow problem [28] The accuracy ofthese different LB-based scheme in solving aneurysm blowflow simulations will be examined by comparing the hemo-dynamic parameters with the conventional finite volumemethod based solver To the best of the authorsrsquo knowledgethere is no study conducted to compare the capability ofboth FVM and LBMmodels typically in the simulation bloodflow problems in terms of the solution accuracy and parallelperformance The current study attempts to compare thedistribution of mesh for both LBM and FVM models in thedetection of highwall shear stress (WSS) region in the arterialwall The parallel speedup with increasing number of cores isalso compared between both LBM and FVM solvers in termsof its parallel efficacy

Table 1 Weighting functions for D3Q19

Model 1198882

119904Node no Weight

D3Q19 131198910

131198911ndash1198916

1181198917ndash11989118

136

0

1

2

3

4

5

6

78

9

10

11

1213

14

15

16

17

18

Figure 1 3D lattice arrangements for D3Q19

2 Lattice Boltzmann Models

The results shown in this section are formulated using D3Q19lattice model The LBM equation can be summarized in

119891 (119903 + 119888119889119905 119888 + 119865119889119905 119905 + 119889119905) minus 119891 (119903 119888 119905) = Ω119891 (119903 119888 119905) (1)

in which the right-hand side represents the streaming stepThe left-hand term denotes the collision term which can berepresented using the well-known Bhatnagar-Gross-Krook(BGK) model as given in

Ω = 120596 (119891eqminus 119891) =

1

120591

(119891eqminus 119891) (2)

120596 and 120591 denote the relaxation frequency and time 119891eq

represents the equilibrium function that relates to the latticearrangement

The equilibrium function 119891eq can be described as

119891eq(120588 119906)

= 120588119908[1 +

1

1198882

119904

(119888 sdot 119906) +

1

21198884

119904

(119888 sdot 119906)2minus

1

21198882

119904

(119906 sdot 119906)]

(3)

in which 119908 represents weighting function across differentlattice links For the case ofD3Q19 latticemodel as depicted inFigure 1 the weighting functions can be described in Table 1



1198900= (0 0 0)

11989012

= (plusmn1 0 0)

11989034

= (0 plusmn1 0)

11989056

= (0 0 plusmn1)




(4)


Ω = 120596 (119891eqminus 119891) =

1

120591

(119891eqminus 119891) (5)


119891119894( +

997888rarr119890119894 119905 + 1) = 119891

119894( 119905) + Ω (6)





120588 = sum

119894

119891119894

=

1

120588

sum

119894

119891119894

997888rarr119890119894

(7)



119891eq119894( 119905) = 119908

119894120588 [1 + 3 (

997888rarr119890119894) +

9

2

(997888rarr119890119894)

2

minus

3

2

2] (8)




0+ 120575120588 where 120588



119891eq119894( 119905) = 119908

119894[120588 + 120588

03 (

997888rarr119890119894) +

9

2

(997888rarr119890119894)

2

minus

3

2

2] (9)




eq reads

119891eq119894

= 120588119905119894[1 +

V119894120572119906prop

1198882

119904

+

1

21198884

119904

119876119894prop120573

119906prop119906120573] (10)



119894prop120573= V119894120572119906119894120573minus 1198882

119904120575prop120573



topology


119891eq119872prop

(120588 )

= 119908prop

1 +

997888rarr119890propsdot 120588

RT+

(997888rarr119890propsdot 120588)

2

2 (RT)2minus

1

2

sdot 120588

RT

(11)




2

119904 where 119888

119904= 1radic3119888






120597120588

120597119905

+ nabla sdot 120588 = 0 (12)


120597

120597119905







119881(119909 1199091015840) = 119866 (119909 119909

1015840) 120595 (119909) 120595 (119909

1015840) (14)


119865 = minus119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) 120595 (119909)sum

120572

120595 (119909 + 119890120572) 119890120572 (15)



119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) =

119866 if 1003816100381610038161003816119890120572

1003816100381610038161003816= 119888

0 if 1003816100381610038161003816119890120572

1003816100381610038161003816gt 119888

(16)


119900[1minusexp(minus120588120588

119900)] Analytical


5 Methodology





Wall thickness(cm)














(m)001000750005000250










120597119909


1205971199092

120591 = 3VLB + 05

(17)


Re =119906LB119873inlet

VLB (18)











Percentage Diff =

10038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816

times 100 (19)








Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of


















1198900= (0 0 0)

11989012

= (plusmn1 0 0)

11989034

= (0 plusmn1 0)

11989056

= (0 0 plusmn1)




(4)


Ω = 120596 (119891eqminus 119891) =

1

120591

(119891eqminus 119891) (5)


119891119894( +

997888rarr119890119894 119905 + 1) = 119891

119894( 119905) + Ω (6)





120588 = sum

119894

119891119894

=

1

120588

sum

119894

119891119894

997888rarr119890119894

(7)



119891eq119894( 119905) = 119908

119894120588 [1 + 3 (

997888rarr119890119894) +

9

2

(997888rarr119890119894)

2

minus

3

2

2] (8)




0+ 120575120588 where 120588



119891eq119894( 119905) = 119908

119894[120588 + 120588

03 (

997888rarr119890119894) +

9

2

(997888rarr119890119894)

2

minus

3

2

2] (9)




eq reads

119891eq119894

= 120588119905119894[1 +

V119894120572119906prop

1198882

119904

+

1

21198884

119904

119876119894prop120573

119906prop119906120573] (10)



119894prop120573= V119894120572119906119894120573minus 1198882

119904120575prop120573



topology


119891eq119872prop

(120588 )

= 119908prop

1 +

997888rarr119890propsdot 120588

RT+

(997888rarr119890propsdot 120588)

2

2 (RT)2minus

1

2

sdot 120588

RT

(11)




2

119904 where 119888

119904= 1radic3119888






120597120588

120597119905

+ nabla sdot 120588 = 0 (12)


120597

120597119905







119881(119909 1199091015840) = 119866 (119909 119909

1015840) 120595 (119909) 120595 (119909

1015840) (14)


119865 = minus119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) 120595 (119909)sum

120572

120595 (119909 + 119890120572) 119890120572 (15)



119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) =

119866 if 1003816100381610038161003816119890120572

1003816100381610038161003816= 119888

0 if 1003816100381610038161003816119890120572

1003816100381610038161003816gt 119888

(16)


119900[1minusexp(minus120588120588

119900)] Analytical


5 Methodology





Wall thickness(cm)














(m)001000750005000250










120597119909


1205971199092

120591 = 3VLB + 05

(17)


Re =119906LB119873inlet

VLB (18)











Percentage Diff =

10038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816

times 100 (19)








Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















120597120588

120597119905

+ nabla sdot 120588 = 0 (12)


120597

120597119905







119881(119909 1199091015840) = 119866 (119909 119909

1015840) 120595 (119909) 120595 (119909

1015840) (14)


119865 = minus119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) 120595 (119909)sum

120572

120595 (119909 + 119890120572) 119890120572 (15)



119866 (1003816100381610038161003816119890prop

1003816100381610038161003816) =

119866 if 1003816100381610038161003816119890120572

1003816100381610038161003816= 119888

0 if 1003816100381610038161003816119890120572

1003816100381610038161003816gt 119888

(16)


119900[1minusexp(minus120588120588

119900)] Analytical


5 Methodology





Wall thickness(cm)














(m)001000750005000250










120597119909


1205971199092

120591 = 3VLB + 05

(17)


Re =119906LB119873inlet

VLB (18)











Percentage Diff =

10038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816

times 100 (19)








Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















Wall thickness(cm)














(m)001000750005000250










120597119909


1205971199092

120591 = 3VLB + 05

(17)


Re =119906LB119873inlet

VLB (18)











Percentage Diff =

10038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816

times 100 (19)








Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















(m)001000750005000250










120597119909


1205971199092

120591 = 3VLB + 05

(17)


Re =119906LB119873inlet

VLB (18)











Percentage Diff =

10038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816

times 100 (19)








Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
























Percentage Diff =

10038161003816100381610038161003816100381610038161003816


10038161003816100381610038161003816100381610038161003816

times 100 (19)








Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Pres

sure

(Pa)

FLUENT IncBGK

MRTRLBM

X Y

Z

X Y

Z

X Y

Z

X Y

Z

0000e + 000

9777e + 000

1955e + 001

2933e + 001

3911e + 001

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)

0000e + 00

8

16

24

32

3911e + 01

Pres

sure

(Pa)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points

10

15

20

25

30

35

Pres

sure

(Pa)

IncBGKMRT

RLBMFLUENT




1003816100381610038161003816 (20)

Perc

enta

ge d

iffer

ence

()

IncBGKMRTRLBM

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160Points




X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Velo

city

pla

ne (m

sminus1 )

High velocity flow


FLUENT IncBGK

MRTRLBM

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u








X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















X Y

Z

Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z








Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Velo

city

pla

ne (m

sminus1 )

FLUENT IncBGK

MRTRLBM

0000e + 000

4875e minus 002

9750e minus 002

1462e minus 001

1950e minus 001

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

0000e + 00

004

008

012

016

1950e minus 01

u

X Y

Z

X Y

Z

X Y

Z

X Y

Z



Velo

city

(ms

)

0005

01015

02025

2 3 4 5 6 7 8 9 10 11 12 13 14 151Points

FLUENTIncBGK

RLBMMRT



Abso

lute

chan

ge(m

s)

IncBGKRLBMMRT

2 3 4 5 6 7 8 9 10 11 12 13 14 151Point

00002000400060008

0010012001400160018








1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















1 136402 21660 80198 58795325582 78388 11640 38012 48492116143 59366 7860 19234 32399016274 49224 6480 15576 3164310093

FLUENT IncBGK

MRTRLBM

Wal

l she

ar W

SS (P

a)

X

YZ

X

YZ

X

YZ

X

YZ

2690e + 00

0000e + 000

6725e minus 001

1345e + 000

2017e + 000

2690e + 000

1837e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS

2690e + 00

1637e minus 03

05

1

15

2

25

WSS


00000002000000400000060000008000000

100000001200000014000000160000001800000020000000

Perc

enta

ge (

)

25

02

18

13

09

28

00

07

10

27

05

06

14

15

03

04

19

20

24

22

23

01

08

26

16

17

12

11

21

Wall shear (Pa)


20

03

18

07

16

22

25

10

12

23

21

27

07

05

04

01

08

00

13

26

09

28

14

Wall shear (Pa)

000200400600800

100012001400160018002000

Perc

enta

ge (

)





012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















012345678910111213141516171819







(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b)

(c) (d)



PalabosFLUENT


0

5000

10000

15000

20000

25000

Sim

ulat

ion

time (

s)




7 Conclusions





Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















Competing Interests


Acknowledgments


References




















































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of












































































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Research Article Lattice Boltzmann Model of 3D Multiphase...

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Transcript of Research Article Lattice Boltzmann Model of 3D Multiphase...