An Improved WCSPH Method to Simulate the Non-Newtonian Power Law Fluid Flow Induced by Motion of a Square Cylinder

R.Shamsoddini,N.Aminizadehand M.Se fid

An Improved WCSPH Method to Simulate the Non-Newtonian Power Law Fluid Flow Induced by Motion of a Square Cylinder

R.Shamsoddini1,N.Aminizadeh1and M.Se fid2

In this study,an improved weakly compressible Smoothed Particle Hydrodynamics method is introduced and applied for investigation of the non-Newtonian power-law fluid flow which is induced by motion of a square cylinder.The method is based on a predictor-corrector scheme and pressure velocity coupling to overcome the non-physical fluctuations of WCSPH.The numerical method is also supported by the corrective tensors and shifting algorithm.The results are validated against the well known test cases and benchmark data.The square motion is tested in various Reynolds numbers for various power law indices.The results show that the drag coef ficient increases by increasing the power-law index although there are some irregular behaviors for the lower power-law indices at higher Reynolds numbers.

SPH,moving boundary,power-law,square cylinder.

1 Inrtoduction

There are several meshless methods such as Element Free Galerkin(EFG),Reproducing Kernel Particle Method(RKPM),Partition of Unity Finite Element Method(PUFEM),Natural Element Method(NEM),Meshless Local Petrov-Galerkin(ML PG)and Smoothed Particle Hydrodynamics(SPH)[Atluri and Shen(2002);Atluri(2004)].SPH is the oldest one.It was first proposed by Lucy(1998)and Gingold and Monaghan(1997)for astrophysical applications.It was gradually extended to model a wide range of research and engineering applications including elasticity[Libersky,Petschek(1991)],multi-material and multiphase- flows[Cleary(1998)]and blood simulation[Muller,Schirm,Teschner,(2004)].Since it has the Lagrangian nature and is based on particles,the Smoothed Particle Hydrodynamics is a proper method to model the complex problems such as multi phase andmaterial[Cleary(1998)],free surface flow[Dong,Liu,Jiang,Gu,Xiao,Yu and Liu(2013);Jiang,Tang and Ren(2014)],Surface tension[Qiang,Chen and Gao(2011)]and moving boundary and fluid solid intraction[Ra fiee and Thiagarajan(2009)]and special engineering problems such as metal forging[Cleary,Prakash,Das and Ha(2012)]and high pressure die casting process[Cleary,Ha,Prakash and Nguyen(2006)].Some of these problems have their own complexity for other numerical methods especially for the Eulerian and grid-based methods.For example,moving boundary and fluid solid interaction modeling are intricate numerical problems for grid-based methods.Arbitrary Eulerian-Lagrangian(ALE)[Wang and Yan(2010)]and Immersed boundary methods[FadlunVerzicco Orlandi and Mohd-Yusof(2000)]are some of the methods which have been applied for gridbased methods to model moving boundary problems.Each of the above methods has their own complexities,restrictions and applications.Lagrangian and meshfree properties facilitate the complex simulations.Some investigations of the moving boundary and FSI problems has been recently done using the SPH method;Ra fiee and Thiagarajan(2009)proposed an incompressible Smoothed Particle Hydrodynamics for simulation of fluid–structure interaction problems,deploying the pressure Poisson equation to satisfy incompressibility constraints.Kajtar and Monaghan(2012)described how the swimming of linked-rigid bodies can be simulated using SPH.To simulate two way coupled fluid structure interaction,Cohen,Cleary,and Mason(2012)simulated the dolphin kick swimming by SPH to evaluate variants of this swimming stroke technique.Hashemi,Fatehi,and Manzari[2011;2012]simulated two way coupled fluid solid interaction using WCSPH method for the Newtonian and non-Newtonian fluid.SPH can also successfully simulate the non-Newtonian fluid flows.A signi ficant part of the numerical analysis which has been carried out using SPH is related to non-Newtonian fluid flows;Zhou,Ge and Li(2010)studied the rheological behavior in a two-dimensional Couette flow simulated using smoothed particle hydrodynamics.Zhu,Martys,Ferraris and Kee(2010)applied a Lagrangian formulation of the Navier–Stokes equations,based on the Smoothed Particle Hydrodynamics(SPH)approach to determine how well rheological parameters such as plastic viscosity can be determined from vane rheometer measurements.Ra fiee,Manzari and Hosseini(2007)presented an incompressible smoothed particle hydrodynamics(SPH)method to solve unsteady free-surface flows for Newtonian and viscoelastic fluids.Jiang,Ouyang,Li,Ren and Yang(2011)proposed and extended a corrected smoothed particle hydrodynamics(CSPH)method to the numerical simulation of transient viscoelastic fluid flows.Fan,Tanner and Zheng,(2010)developed an implicit SPH for non-Newtonian flow,which is completely matrix free,to solve the equation system iteratively and robustly.They introduced the arti ficial pressure between particles to stabilize the SPH system avoiding the tensile instability.

Although the studies of the non-Newtonian fluid flows over the stationary bluff bodies,especially square cylinders,are considerable[Sahu,Chhabra and Eswaran(2010);Bouaziz,Kessentini and Turki,(2010);Sahu and Chhabra(2009)],the shortage of the studies on the moving body is felt.

In the present study,an improved weakly compressible Smoothed Particle hydrodynamics scheme is applied to model the flow induced by motion of a square cylinder in the non-Newtonian fluids.Most of the fluids in the oil industry possess a pseudoplastic behavior[Escobar,Bonilla and Cicery,(2012)].Motion of pipeline pigs and cleaning devices inside the oil pipelines are examples of application of moving boundary in the non-Newtonian fluids.Pseudoplastic fluids obey the power law with the power law index less than one.Motion of the balloon catheter inside an artery is included in the motion of the rigid bodies in the pseudoplastic fluids.However,some past studies ignored the non-Newtonian effects for simpli fication[Torii,Wood,Hughes,Thom,Aguado-Sierra,Davies,Francis,Parker and Xu(2007)].Present results show the effect of power law index on the flow field and drag coefficient of the motion of a square cylinder in the initially stationary non-Newtonian fluid for various Reynolds numbers.In the followings, first,the governing equations and numerical algorithm are discussed.Next,the validation and veri fication of the computational algorithm is considered.Then,the results and discussions are presented.

2 Governing equations and numerical procedure

2.1 Formulation

The governing equations for the fluid flow are respectively mass,momentum and the state equations:

whereρ,,pandCare respectively the fluid’s density,velocity,pressure,and speed of sound andτis the shear stress tensor which is as follow:

wheredfor two dimensional flow is equal to 2.For incompressible flow,∇is zero.However for the standard WCSPH∇=0.Appropriate selection for the speed of sound can lead to the incompressible flow.The standard WCSPH has a number of shortcomings[Vignjevic,Reveles,Campbell(2006)].It suffers from the non-Physical fluctuations.In the present study,using a predictive corrective scheme,the divergence of the velocity is related to the laplacian of the pressure.This correction causes to remove the non-physical fluctuation.So the fluctuation of the density is removed and the divergence of Eq.(4)can be approximated by

The Smoothed Particles Hydrodynamics(SPH)method is applied to solve the above equations.As a Lagrangian method,the formulation of SPH is based on an integral form which indicates each continuous de fined function f,over an interest domain Ω can be stated as:

where∀jis the volume of j-th particle.In the present study the fifth-order Wendland(1995)kernel function is used.Dehnen and Aly(2012)showed that the use of this Kernel function causes to increase the accuracy.

whereW0is respectivelyfor 1,2 and 3 dimensional cases.

To descritze the second order derivative terms such asfor i-th particle,the new method introduced by Fatehi and Manzari(2011a)is used:

where eeeijis the unit vector in the inter-particle direction.Becauseµpis not constant,it is approximated byµpijwhich is given by:

whereµpis apparent viscosity which is given by:

where k is consistency index,n is power law index andis

where α and β are the Cartesian indices.The velocity gradient apparent in the Eq.(9)is calculated by

in which I is the second-order unitary tensor.

The SPH descritization of the pressure gradient divided by density for each particle is calculated by:

Standard WCSPH solves the governing equations(Eq.(1)to Eq.(3))as follows:

2.2 Predictive corrective scheme

The standard WCSPH suffers from the pressure and density fluctuations[Lee,Moulinec,Xu,Violeau,Laurence and Stansby(2008)].Fatehi and Manzari(2011b;2011c)showed that the connection between the divergences of the velocity and the Laplacian of the pressure considerably decreases the non-physical fluctuation.So,according to the pressure velocity decoupling,a pseudo constant density method is introduced to remove the density and pressure fluctuations to improve the WCSPH.The present method has been based on a predictor-corrector scheme which is as follow:

1-Predictor step:

Each particle has constant mass so the conservation of mass equation(1)leads to

where α is a coef ficient which can vary between 0 and 1.Dividing Eq.(2)by ρ leads to:

Applying the divergence operator on the Eq.(24)and substituting in the Eq.(21)leads to

Tensile instability,defects and clustering distribution are complications in the SPH simulations.To avoid these unfavorable phenomena,a shifting algorithm similar to the particle shifting strategy of Xu,Stansby,and Laurence(2009),has been applied in the present study.The direction and amount of shifting are determined from the arrangement of neighboring particles;theis de fined as shifting particle vector which is calculated by:

whereεcan vary between 0 and 0.1 and¯riis equal to

If the particles are homogeneously distributed around the particlei,thenwill be zero.Otherwise this vector shows that the distribution of neighboring particles around of the particle is not balanced.Then the particle is slightly shifted byUltimately it is necessary to interpolate the flow field variables in the new position.These modi fications according to the first order Taylor series expansion are

This process is repeated once for each time-step.

The dummy particles are used to de fine the wall boundary condition;the dummy particles are distributed similar to those applied by Lee,Moulinec,Xu,Violeau,Laurence and Stansby(2008);three layers of dummy particles with the same velocity of the wall particles(no slip condition)are arranged nearby each wall boundary condition.These particles have also the same pressure of the wall in the normal direction to satisfy the Neumann boundary condition.A pseudo-code of the present solution algorithm is shown in Tab.1.

If Eq.(28)is compared with Eq.(18),the main difference of the present algorithm with the standard WCSPH is indicated;the present algorithm uses the divergence of the intermediate velocity and pressure Laplacian while the standard WCSPH uses only the divergence of velocity of previous time-step.This modi fication creates a velocity pressure coupling which reduces the non-physical fluctuations of WCSPH.

The present method also uses the advanced SPH discretization and modi fication tensors(and).The third main improvement is the shifting algorithm which causes to homogenize the particles distribution.

Table 1:pseudo-code of the present solution algorithm.

3 Results and Discussion

3.1 Veri fication and validation

Before applying present code for the typical problem of non-Newtonian fluid flow,it has been validated against a number of well-known benchmark results.It may be noted that non-Newtonian fluid flow induced by motion of a square cylinder in a rectangular cavity containing the fluid which is initially at rest has not been reported.However,the same cases using Newtonian fluids are available[Lee,Violeau,Laurence,Stansby,Moulinec,(2007)].Present code is validated by these results in Fig.1 and Fig.2.Contours of velocity magnitude for the present simulation and the reference solution in the SPHERIC test case 6 forRe=ρUsLs?µ=150 and t=8s is plotted in Fig.1 whereUsandLsare respectively velocity and side length of the square cylinder.

Figure1:Contourofthevelocitymagnitudeoftheforcemotionofasquarecylinder for Re=150 in the initially stationary flow for SPHERIC test case 6[Lee,Violeau,Laurence,Stansby,Moulinec,(2007)](top)and present SPH algorithm simulation(bottom).

In Fig.2,the time variations of the drag coef ficient is plotted which is calculated by

whwehreer eρnixsi st hxe-d iflr eucitdio dne unnsiit tvye,cUtomr,aAxiiss athreea manadxTim isu smtre svse tleoncsiotyr: of the square,Lis the squTa=reσs-ipdIe, andbis the transversal length of the prismatic surface[Lee,Vio(3l6e)au,Laurence,Stansby,Moulinec,(2007)]andFxis

wherenn nxis x-direction unit vector,Ais area andTis stress tensor:

whereis viscous stress tensor,pis the pressure acting on the wall square.

In Fig.2,the results of Finite Volume Particle Method(FVPM)reported by Nestor,Basa and Quinlan,(2008)and standard WCSPH(solving Eq.(17)to Eq.(20))

Figure 2:Time variations of drag coef ficient for Reynolds numbers 50,100 and 150 for SPHERIC test case 6 solution[Lee,Violeau,Laurence,Stansby,Moulinec,(2007)],FVPM[Nestor,Basa and Quinlan,(2008)],present WCSPH and standard WCSPH results.

are also shown.As indicated,the present algorithm considerably reduces the nonphysical fluctuations in comparison with the FVPM and standard WCSPH results.To validate the method proposed for calculations of the non-Newtonian power law behaviors,the present code is compared with three traditional problems;the first is the flow in the lid driven cavity and the second is the flow between two parallel plates.In fig.3,the results of present code are compared by those reported by Bell and Surana(1994)for lid driven cavity flow.The Reynolds number of power law fluid is calculated by

whereUwis the velocity of top lid,Lis the cavity length andnis power law index.The results shown in fig.3 are achieved at RePL=100.

Figure 3:The Comparison of the vertical and horizontal middle sections velocity pro files of the lid driven cavity between present study and Bell and Sourana results(1994)for RePL=100 and two values of power law index.

The flow vector fields of these cases are shown in Fig.4.

Figure 4:Flow vector fields for two values of power law index(left n=0.5 and right n=1.5)in the Re=100 case.

Another traditional problem is the simulation of the non-Newtonian power law flow in a 2D channel( flow between two parallel plates).The non-dimensional velocity pro file of the power-law flow for a straight channel with width of 2his given by:

The results of present code forn=0.5,1.0 and 1.5 are compared by the analytical results in Fig.5.

Figure 5:Non-dimensional velocity pro files of the present SPH simulation for power law indices n=0.5,1.0&1.5 in comparison with the analytical solutions.

The results of the SPH simulations are extracted from the RePL100 case although the Reynolds number has no effect on the non-dimensional velocitypro file.Thedummyparticlesareappliedtode finenoslipboundarycondition for the channel walls and the periodic boundary condition is de fined for the flow boundaries.As shown in Fig.5,there is well agreement between the present SPH and analytical results.

The last test case which is selected to examine the proposed algorithm is the numerical investigation of the power-law fluid flow behaviors past a circular cylinder con fined in a channel(Fig.6).For this problem,no slip condition is applied for the wall boundaries.For the flow boundaries,a similar approach with that of Federico(2010)is applied.He de fined inlet and outlet regions to exert the inlet and outlet flow boundary conditions.

For the present flow boundary condition,the velocity of the particles which are in the inlet region is renormalized to the power-law Poiseuille velocity pro file for each time-step.For the case:RePL=and D/H=0.25,time averaged drag coef ficient for different power-law indices in comparison with the Bijjam and Dhiman(2012)results are shown in Tab.2.

Figure 6:Schematic of flow across a con fined circular cylinder.

Table 2:Comparison of present results with Bijjam and Dhiman(2012)results for Re=100.

As indicated in Tab.2,drag coef ficient increases by increasing the power-law index.For the three cases of Tab.2,the SPH simulations for contours of vorticity are shown in Fig.7.The vortices are stretched more longitudinally by increasing the power-law index.Fig.7 also shows that the vortices are formed and separated easily for the lower power-law indices.

3.2 Particle study

To study the flow induced by motion of a square cylinder in the fluid which is initially at rest,a reasonable particles number is needed to satisfy both of the accuracy and calculations time.

Fig.8 shows the drag coef ficient for thecase for three cases with 60000,125000 and 250000 particles to demonstrate the convergence and particle independency.The differences between the cases with 125000 and 250000 particles can be neglected.So the case with 125000 particles is appropriate for this study.

Figure 7:Contours of vorticity for the three cases of Tab.2.

Figure 8:The effects of particles number on the time variations of drag coef ficient.

3.3 Effects of power law index on the induced flow

To investigate the effects of power law index on the flows fiesl d induced by the square cylinder motion,four Reynolds numbers150)are considered for five power law indices(n=0.5,0.75,1.0,1.25&1.5).Time variations of drag coef ficient for these cases are plotted in Fig.9.

TheSPHmethodsuffersfromsomeinherentproblems[Liu,M.B.Liu,G.R(2006)].

Figure 9:Time variations of drag coef ficient for various power law indices in the Re=10,50,100&150 cases.

Although the present method uses advanced corrective tensors and a predictive corrective scheme to reduce the non-Physical fluctuations,there are still some fluctuations in the drag coef ficients time variations as shown in Fig.1.These fluctuations come from the pressure waves returning from the rectangular cavity walls.The pressure waves are frequently repeated between the cylinder and rectangular walls.So for each curve,an average regression curve(a polynomial regression curve with order 10)is replaced instead of the former style for better comparison.In the Re=10 case,the effects of power law index on the drag coef ficient can be neglected forn＞0.75.Otherwise forn=0.5 or fewer,the effects of shear thinning behaviors may be considerable.For Re=50,increasing the power law index causes to increase the drag coef ficient regularly.In the Re=100 and Re=150 cases,increasing the power law index also causes to increase the drag coef ficient forn＞1.0.However there are some irregularities forn＜1.0 for these two Reynolds numbers.In Fig.10,contours of velocity magnitude for various power law indices for the Re=100 case are plotted.The maximum velocity magnitude is occurred past the square cylinder.The maximum velocity magnitude decreases by increasing the power law index.

Also the region of the fluid which is affected by motion of the square cylinder is broader for the larger power law indices.

Figure 10:Contours of velocity magnitude for varius power law indices for the Re=100 case.

4 Conclusion

In the present study a robust modi fied weakly compressible Smoothed Particles Hydrodynamic method based on a renormalized predictor-corrector scheme was introduced and applied to study the flow induced by motion of a square cylinder in non-Newtonian power law fluids.The results had good agreement with the benchmark test cases and previous data.The method also reduces considerably the nonphysical fluctuations in comparison with the other particle method.To study the effects of power law index on the induced fluid flow field, five power law indices at four Reynolds numbers were considered.The results showed that:

1.For Re=10 case,the effects of power-law can be neglected reasonably.However forn=0.5 or fewer the differences may be considerable.

2.ForRe=50case,thereisaregularincreaseinthedragcoef ficientwithincreasing the power law index.

3.For the Re=100 and 150 cases,increasing the power law index also causes to increase the drag coef ficient although there are irregularities in some cases withn＜1.

4.The maximum velocity occurred past the square cylinder increases with decrease the power law index.Also the region of fluid in fluenced by motion of the square cylinder is broader in the larger power law indices.

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1Sirjan University of Technology,Sirjan,Iran.

2Yazd University,Yazd,Iran.