Comparison on Different Schemes of Direct Numerical Simulation for Low/medium Reynolds Flow

WANG Jian-chun,WU Cheng-sheng,WANG Xing

(China Ship Scientific Research Center,Wuxi 214082,China)

Comparison on Different Schemes of Direct Numerical Simulation for Low/medium Reynolds Flow

WANG Jian-chun,WU Cheng-sheng,WANG Xing

(China Ship Scientific Research Center,Wuxi 214082,China)

Direct numerical simulation(DNS)for low/medium Reynolds lid-driven cavity flow with different schemes are presented.The semi-implicit method for pressure-linked equations(SIMPLE), pressure-implicit splitting of operations(PISO)and pseudo-compressibility schemes are used.The N-S equations are all discretised by the Finite Volume Method for the three schemes,with the same staggered grid arrangement,the fully implicit time-stepping scheme and the QUICK scheme for the discretization of the temporal items and convective transport terms,the results are compared with the benchmark solution reported by Ghia[1].Under the same convergence criteria condition,difference in the stability,accuracy and convergence rate are analyzed.The PISO scheme is the most accurate scheme for low Reynolds number of Re=400 and 1 000 flow.The pseudo-compressibility scheme is found to be the most accurate for Re=5 000 flow.Besides,pseudo-compressibility scheme cost the minimum time to achieve convergence for all the cases,which shows it is one of the best choice for DNS of the low/medium flows.

SIMPLE;PISO;pseudo-compressibility;DNS;accuracy;convergence rate; stability

0 Introduction

In recent years,along with the continuous improvement of computer performance,deepening of the high performance parallel computing study and the fervent need for the industry to the research of the turbulent meticulous flow field,direct numerical simulation(DNS)of the turbulence are constantly studied.Incompressible fluids are the main objects of the DNS research,their governing equations are the incompressible N-S equations.However,a lot of prerequisites need to be met before carrying out the DNS research,for example,a good solver of the governing equations,massively parallel cluster,high precision scheme for the temporal and space discretization,high resolution to catch the minimum scale vortices,and so on.Nowadays,the DNS research is main focused on the low/medium flows since there are so many restrict prerequisites.Among these prerequisites,a good solvers is the fundamental conditions that you can choose to accelerate the DNS program.The main solvers for the incompressible fluid are solving the primitive variable N-S equations,vorticity-stream function method andother methods.SIMPLE and PISO method belong to the first one,the pseudo-compressibility method belongs to other methods.

The SIMPLE algorithm was relatively straightforward and soon became the main solver of the incompressible flow since it was first put forward in 1972 and it had been successfully implemented in numerous CFD procedures recent years.The PISO scheme was first put forward by Issa in 1986 and was early designed to solve the unsteady N-S equations.The PISO was considered as an extension for the SIMPLE scheme.At the same time,the pseudo-compressibility scheme attracted lots of researchers’attention because the continuous equation and the momentum equations were solved synchronously and the scheme itself had high efficiency.In recent years,these schemes were applied to perform the DNS research by some researchers. Wang et al[2]applied GPU accelerated DNS with the SIMPLE[3-4]scheme to the Re=1 000 and Re=10 000 lid-driven cavity flow and the results agreed with the literature well.Dousset and Pothérat[5]carried out the DNS of low Reynolds Re=100 and Re=200 flows past a truncated square cylinder through the PISO[6]scheme for both steady and unsteady flows and analyzed the formation mechanism of hairpin vortices in the wake of the truncated square cylinder in a duct.Skovorodko[7]used the pseudo-compressibility[8]scheme to complete the DNS of compressible fully developed turbulent Couette flow between two parallel plates and analyzed the slip effects in compressible turbulent channel flow.However,rare articles are found about using the pseudo-compressibility scheme to the DNS of incompressible flows.

Since the better solver of the N-S equations chosen,the easier the DNS be performed, being aware of the difference among the different kinds of solvers to the governing equations is very important for the DNS research,since you can choose the best solver to your problems. Convergence,accuracy and stability are the three important features used to judge whether a scheme is good or bad for the studied problems.Difference of these important features for the three different schemes is presented in this paper.Same conditions as shown latter are implemented for these different schemes to avoid the influence of other factors in addition to these schemes theirselves.

1 Numerical methods

The integral form of the dimensionless incompressible N-S equations is:

Finite-volume discretion momentum equation in the staggered grid is:

Mass conservation equation is:

where the subscripts e,n,w and s represent the control-volume faces and E,N,W and S represent the grid points,nb represents the neighbor grid point as shown in the staggered grid[4,9]system(Fig.1)below.

Fig.1 Staggered grid system

1.1 SIMPLE scheme

There is evidently no equation for the pressure in the incompressible N-S equations,coupling between pressure and velocity is hidden in the continuity equation,bring the problem that how to solve the pressure alone?The Semi-Implicit Method for Pressure-Linked Equation (SIMPLE)was originally put forward by Patankar and Spalding in 1972,resolved the pressurevelocity coupled problem well.Staggered grid arrangement is used in this paper and the main calculation procedures of the SIMPLE algorithm are as follows:

(2)Solve the discretised momentum Eqs.(4)and(5)to get the u*,v*,using the estimated pressure or the pressure calculated on the last level remarked as P*.

(3)Calculate the pressure correction P′,ensure the（u*+u′）,（v*+v′）corresponding to（P*+P′）satisfy the continuity equation.The pressure correction is then obtained by substituting the corrected velocity into discretised continuity Eq.(6),using the relationship between u′, v′and P′.

(4)Calculate the velocity corrections u′,v′,ensure the（ue*+ue′ ）and（vn*+vn′）still satisfy the linearized momentum equation.

(5)Set the（u*+u′）,（v*+v′）and（P*+aPP′）as the answer of this level and start the calculation for the next level,aPis the under-relax factor,repeat step one to five until the flow field converges,namely the velocities can satisfy both the momentum equations and the continuity equation.The so-called‘level’is the solving process of the algebraic equation set consisted of the fixed coefficient and source.SIMPLE method is essentially a guess-and-correct scheme,the guess steps are 1～3 and correct steps are 4～5.Details refer to the Ref.[6]or[7].

1.2 PISO scheme

The PISO method consists of one guess step and two correct steps,the guess step and the first correct step are almost the same of SIMPLE scheme,the main procedure is:

(1)Guess step-same as the SIMPLE scheme

Solving discretised momentum equation implicitly based on the estimated or the last level pressure value,remarked as P（）k:

u*,v*and P（）kin this step satisfy the momentum equation but may not satisfy the continuity equation.

(2)First correct step-same as the SIMPLE scheme

(3)The second correct step-special feature for the PISO scheme

Search for the second corrected pressure P**and velocities u***,v***based on the calculated pressure P*and velocities u**,v**in the first correct step,make them satisfy the continuity equation and momentum equations better.

Then,P**,u***,v***are calculated，the second correct step is complemented.Set the P**, u***,v***as the initial value for the next level and continue the procedure above until it satifies the the convergence condition.Details refer to the Ref.[5].

1.3 Pseudo-compressibility scheme

Pseudo-compressibility scheme was first put forward by Chorin and Vladimirova separately.The wind tunnel test was started,the wind speed was gradual changed from zero to the stability value,this accelerate process was unsteady.This change from the unsteady to the steady process was essentially the change of the type of governing equations.The idea for the pseudo-compressibility scheme is:If the steady incompressible N-S equations are added an time derivative term?/?t,then they are translated to a virtual unsteady compressible N-S equations.The continuity equation is added pressure derivative term?P/?t,momentum equations are added velocities derivative terms?v→/?t and the N-S equations for the pseudo-compressibility scheme are:

The procedure for the pseudo-compressibility is very simple:

(1)Calculate the velocities u,v from the momentum equations(10)based on the estimated pressure or the last level pressure.

(2)Using the velocities u,v calculated above and the equation(9)to calculate the pressure P.

(3)Repeat the steps(1-2)until the u,v and P satisfy the convergence condition.

2 Numerical results

The convergence criterion[10-11]is:

where rpis the residual reduction factor,its value range from 0.05 to 0.25.0.1 is used in this paper.

Accuracy,stability and rapid convergence of the three scheme are compared for three different Reynolds and the corresponding grid number.The results are presented in the Tabs.1-2 and Figs.2-4.As showed in Tab.1,under the same discretised scheme and convergence criterion,different time and iterate number are needed to achieve convergence for different scheme. The pseudo-compressibility cost the minimum time to converge,followed by SIMPLE scheme and then PISO scheme.For the case of Re=400,Re=1 000 and Re=5 000,the corresponding time consumed for SIMPLE scheme is 20.4,18.4 and 44.7 times as the pseudo-compressibility scheme,the time consumed for PISO scheme is 26.2,23.1 and 48.2 times as the pseudocompressibility scheme.Much time are saved for the pseudo-compressibility scheme since it is a non-iterative scheme,while 80%of the time costed in calculation is to solve the pressurecorrect equation.One more pressure correct equation is needed for PISO compare with SIMPLE,which shows the PISO scheme cost maximum time to simulate the steady lid-driven flow.

Tab.1 Fast convergence for different method under different conditions

Fig.2 Iterative process or the residual monitor for different scheme at Re=1 000,40*40 grids: UP(SIMPLE);RIGHT(PISO);LEFT(pseudo-compressibility)

For the stability during iteration,SIMPLE and PISO are better than pseudo-compressibility as the corresponding residual monitor curve shown in Fig.2 for the case Re=1 000.

It can be seen from Fig.3.that PISO is the most accurate scheme for simulating the velocities in the central line in the case Re=400 and Re=1 000,followed by pseudo-compressibility and then SIMPLE scheme.While for Re=5 000,the pseudo-compressibility scheme is themost accurate scheme,also the streamline for different methods in the case Re=5 000 in Fig.4 shows that the pseudo-compressibility scheme can simulate the secondary vortices that occur in the bottom right corner better and the curve of streamline matches the benchmark results by Ghia better than the SIMPLE and PISO.for simulating the lid-driven flow,followed by pseudo-compressibility and then SIMPLE.In the case of mid Re such as Re=5 000,pseudo-compressibility is thought to be the most accurate scheme for simulating the lid-driven flow.PISO is more accurate than SIMPLE scheme for simulating the lid-driven flow for all the Re number,since PISO has one more pressure correct step than SIMPLE.

Fig.3 Non-dimensional horizontal(U)and vertical(V)velocity component profiles along the vertical(y)and horizontal(x)centerlines of a wall-driven square enclosure flow

Fig.4 Streamline for different methods at Re=5 000Thus,in the case of low Re such as Re=400 and 1 000,PISO is the most accurate scheme

Tab.2 The location of primary and the secondary vortices

Tab.2 shows that the location of primary and the secondary vortices,when Re=400,the position deviation for the location of primary vortices compared with Ghia’s results for SIMPLE is(+0.003 5,+0.001 9),PISO is(+0.001 5,+0.002 1)and pseudo-compressibility is(+0.003 1, +0.002 5),which shows that the PISO is the most accurate scheme,followed by pseudo-compressibility and then SIMPLE in this case.Similar,when Re=5 000,the position deviation for SIMPLE is(+0.004 4,-0.012 9),for PISO is(+0.008 8,-0.001 7),for pseudo-compressibility is(+0.003 5,-0.000 2),which shows that the pseudo-compressibility is the most accuratescheme in this case.Similarly comparison can be performed to approve the conclusion reached last paragraph.

3 Conclusions

Three different numerical schemes applied to numerically simulate the low-medium Reynolds lid-driven cavity flow are presented in this paper,results are compared with benchmark solution reported by Ghia in 1982,difference among these schemes are analyzed and the results are as follows:

(1)The convergence rate is the best for pseudo-compressibility in different cases,also the accuracy of this scheme is the best in the case of medium and high Reynolds.The iterate residual curve of SIMPLE and PISO schemes is smoother than pseudo-compressibility scheme.

(2)In the cases of low Reynolds flow,PISO is found to be the most accurate scheme.However,the convergence rate of PISO is the worst among these schemes.It is always more accurate than SIMPLE in all cases.

In this paper,for medium or high Reynolds like Re=5 000,the pseudo-compressibility is more accurate than the other schemes and for all Reynolds studied in the paper,the convergence rate of pseudo-compressibility is the best.The pseudo-compressibility scheme is considered to be the best choice for direct numerical simulation of low/medium Reynolds steady laminar or turbulence in the future.

However,there are still some shortcomings for pseudo-compressibility scheme need to be overcome,for example,when come to the unsteady flow,the scheme needs to be re-designed. Besides,the key parameter c for different situations is always different and not easy to design.

[1]Ghia U,Ghia K N,Shin C T.High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method[J].Journal of Computational Physics,1982,48(3):387-411.

[2]Wang J,Xu M,Ge W,et al.GPU accelerated direct numerical simulation with SIMPLE arithmetic for single-phase flow [J].Chin Sci Bull,2010,55:1979-1986.

[3]Patankar S V,Spalding D B.A calculation procedure for heat,mass and momentum transfer in three-dimensional parabolic flows[J].International Journal of Heat and Mass Transfer,1972,15(10):1787-1806.

[4]Patankar S.Numerical heat transfer and fluid flow[M].CRC Press,1980.

[5]Dousset V,Pothérat A.Formation mechanism of hairpin vortices in the wake of a truncated square cylinder in a duct[J]. Journal of Fluid Mechanics,2010,653:519-536.

[6]Issa R I.Solution of the implicitly discretised fluid flow equations by operator-splitting[J].Journal of Computational Physics, 1986,62(1):40-65.

[7]Skovorodko P A.Slip effects in compressible turbulent channel flow[J].arXiv preprint arXiv:1210.2152,2012.

[8]Chorin A J.A numerical method for solving incompressible viscous flow problems[J].Journal of Computational Physics, 1967,2(1):12-26.

[9]Harlow F H,Welch J E.Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface [J].Physics of Fluids,1965,8(12):2182.

[10]Patankar S V.A calculation procedure for two-dimensional elliptic situations[J].Numerical Heat Transfer,1981,4(4): 409-425.

[11]Van Doormaal J P,Raithby G D.Enhancements of the SIMPLE method for predicting incompressible fluid flows[J].Numerical Heat Transfer,1984,7(2):147-163.

中低雷諾數(shù)流動(dòng)直接數(shù)值模擬的算法比較

王建春，吳乘勝，王星
（中國(guó)船舶科學(xué)研究中心，江蘇無(wú)錫214082）

文章采用不同的算法對(duì)中低雷諾數(shù)方腔驅(qū)動(dòng)流動(dòng)進(jìn)行了直接數(shù)值模擬，所用算法分別是人工壓縮方法、SIMPLE算法以及PISO算法。三種算法均采用有限體積法基于交錯(cuò)網(wǎng)格技術(shù)離散N-S方程，時(shí)間項(xiàng)采用全隱格式離散，對(duì)流項(xiàng)采用QUICK格式離散，并將它們得到結(jié)果與Ghia發(fā)表的基準(zhǔn)解進(jìn)行了比對(duì)。文中分析了在同樣的收斂條件下，不同算法之間的穩(wěn)定性，收斂速率以及準(zhǔn)確性的差異，發(fā)現(xiàn)PISO算法在較低雷諾數(shù)Re=400和Re=1 000情況下最準(zhǔn)確，而人工壓縮算法在雷諾數(shù)為5 000時(shí)最準(zhǔn)確，在所有計(jì)算的不同Re數(shù)條件下，發(fā)現(xiàn)人工壓縮法達(dá)到收斂所需時(shí)間都是最少的，這可以使它成為中低雷諾數(shù)下研究直接數(shù)值模擬最好的算法之一。

SIMPLE；PISO；人工壓縮方法；DNS；準(zhǔn)確性；收斂速度；穩(wěn)定性

O35

:A

王建春（1989-），男，中國(guó)船舶科學(xué)研究中心碩士研究生；

O35

:A

10.3969/j.issn.1007-7294.2017.06.001

1007-7294（2017）06-0651-10

吳乘勝（1976-），男，中國(guó)船舶科學(xué)研究中心研究員；

date：2017-01-12

Biography：WANG Jian-chun(1989-),male,master student,E-mail:664148138@qq.com; WU Cheng-sheng(1976-),male,researcher.

王星（1988-），男，中國(guó)船舶科學(xué)研究中心工程師。