FOURTH ORDER COMPACT FORMULATION OF STEADY NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS

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1 International Journal of Mechanical Engineering and Technology (IJMET Volume 9 Issue 10 October 2018 pp Article ID: IJMET_09_10_11 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed FOURTH ORDER COMPACT FORMULATION OF STEADY NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS Ercan Erturk Bahcesehir University Mechatronics Engineering Department Besiktas Istanbul Turkey Orhan Gokcol Bahcesehir University Computer Education and Instructional Technologies Department Besiktas Istanbul Turkey ABSTRACT In the present work the fourth order compact formulation introduced by Erturk and Gokcol (Int. J. Numer. Methods Fluids will be extended to non-uniform grids. The advantage of this formulation is that the final form of the compact formulation is in the same form with the non-uniform grid formulation of the Navier- Stokes equations such that any existing second order code for Navier-Stokes equations on non-uniform grids can be easily altered to provide fourth order solutions by just adding some coefficients into the existing code. The efficiency of the formulation is demonstrated on the lid driven cavity flow benchmark problem. Key words: High Order Compact Scheme Fourth Order Spatial Accuracy Non- Uniform Grids Driven Cavity Flow Cite this Article: Ercan Erturk and Orhan Gokcol Fourth Order Compact Formulation of Steady Navier-Stokes Equations on Non-Uniform Grids International Journal of Mechanical Engineering and Technology 9( pp INTRODUCTION In computational fluid dynamics (CFD field of study fourth order compact formulations provide fourth order spatial accuracy in a point compact stencil. A standard five point wide discretization would also provide fourth order spatial accuracy however this type of discretization requires special treatment near boundaries. Fourth order compact formulations do not have this complexity near the boundaries. Erturk [1] have discussed the advantages and disadvantages of wide and compact formulation and also compared the numerical performances of both formulations in terms of stability and convergence characteristics editor@iaeme.com

2 Fourth Order Compact Formulation of Steady Navier-Stokes Equations on Non-Uniform Grids For steady 2-D incompressible Navier-Stokes equations Dennis and Hudson [2] MacKinnon and Johnson [] Gupta Manohar and Stephenson [] Spotz and Carey [5] and Li Tang and Fornberg [6] have demonstrated the efficiency of high order compact schemes. Similar to these Erturk and Gokcol [7] have presented a new fourth order compact formulation for the Navier-Stokes equations. The uniqueness of their [7] formulation is that the presented Fourth Order Navier-Stokes (FONS equations are in the same form with the Navier-Stokes (NS equations with additional coefficients. Therefore any numerical method used for the NS equations can be easily applied to the introduced FONS equations. Moreover since the NS equations are actually a subset of the FONS equations any existing second order code can be easily modified just by adding the coefficients of the FONS equations into the existing code and then the code can provide fourth order accurate solutions. Erturk [8] have showed that the extra CPU work depends on the numerical method used for the solution and for FONS equations the CPU time needed for convergence is in the order of times of the CPU time needed for convergence for NS equations depending on the numerical method. In all of these studies [2-8] the presented fourth order compact formulations are strictly for uniform grid meshes. Unfortunately in most of the CFD applications non-uniform grids are used to capture the flow physics for example more points are used near the solid wall boundaries and etc. In the literature Spotz and Carey [9] and Ge and Zhang [10] have demonstrated the efficiency of fourth order compact formulations for convection diffusion equation on non-uniform meshes. Also Pandit Kalita and Dalal [11] and Wang Zhong and Zhang [12] have applied the fourth order compact formulation to the Navier-Stokes equations for non-uniform grids. In this study the fourth order compact formulation introduced by Erturk and Gokcol [7] will be extended to non-uniform grids and we will present fourth order compact formulation for steady incompressible 2-D Navier-Stokes equations for non-uniform grids. Such a formulation is needed for practical applications since in most of the cases a graded mesh is needed in CFD studies. Apart from similar studies found in the literature the uniqueness of the present formulation is that it allows both second order and fourth order accurate solutions in a single equation for non-uniform grids. The efficiency of the presented fourth order compact formulation will be demonstrated on the lid driven cavity flow benchmark problem with using several graded meshes. 2. FOURTH ORDER COMPACT FORMULATION In non-dimensional form steady 2-D incompressible Navier-Stokes equations in streamfunction ( and vorticity ( formulation are given as (1 where and are the Cartesian coordinates and is the Reynolds number. The grid points in the Cartesian coordinates ( are non-uniform. These grid points are mapped on to the computational domain ( where the grid spacing is uniform as shown in Figure 1 and we obtain the governing equations as the following!! 2 (2 ( editor@iaeme.com

3 Ercan Erturk and Orhan Gokcol!! ( Figure 1 Transformation of the physical domain to computational domain Equations ( and ( are the Navier-Stokes equations in the computational domain where the grid spacing is uniform. We will apply the compact fourth order formulation defined in Erturk and Gokcol [7] to these equations. Unfortunately we cannot obtain a fourth order compact formulation for these equations in this form. As stated briefly in Spotz and Carey [9] in order to obtain a fourth order compact formulation the grid transformation has to satisfy the following condition 0 (5 This restriction states that the grid mesh should be orthogonal. Imposing this restriction the governing equations become the following & ' ( (6 where & 0 (7 ' ( -. ' - and / ( - In order to obtain the fourth order equations we follow the procedure defined in Erturk and Gokcol [7]. Using the Taylor Series one can obtain the first and second derivatives in -direction as and where 1 is any differentiable quantity and 1 and 1 denote standard three point second order central difference approximation such that :0 7;9 and :0 7 <0 7;9 where subscript is 2 the grid index. Substituting these into equations (6 and (7 we obtain editor@iaeme.com

4 Fourth Order Compact Formulation of Steady Navier-Stokes Equations on Non-Uniform Grids 2 6 & 6 & ' ' 2 ( ( ( > > 2 2?? (9 In these equations (8 and (9 if we can approximate the terms with third and fourth order derivatives with respect to - and -directions ( then these equations (8 and (9 are fourth order accurate In order to approximate these derivatives we use equations (6 and (7. For example in order to find an expression for we differentiate the whole equation (6 with respect to such that & ' ( (10 and obtain & ' > (? (11 We note that using three point central differencing every term on the right hand side of equation (11 can be approximated with second order accuracy as the following > (? & ' (12 Similarly following the same procedure in order to find an expression for 6 6 we differentiate equation (6 twice with respect to and we obtain & 6 2 ' 2> > ( 2?? (1 Again similarly using three point central differencing every term on the right hand side of equation (1 can be approximated with second order accuracy as the following & 2 ' 2 > > ( 2?? (1 We note that in equations (8 and (9 the terms with third and fourth order derivatives with respect to - and -directions ( are multiplied either with 5 or 5. Therefore when the expressions of and 6 in equations (12 and (1 are multiplied by 5 6 or 5 then the error on right hand side of equations (12 and (1 will become fourth order accurate editor@iaeme.com

5 Ercan Erturk and Orhan Gokcol Following the same procedure described above after a long tedious derivation following Erturk and Gokcol [7] we write the final form of the fourth order accurate expressions of the streamfunction and vorticity equations as the following. && ' ' ( ( A (15 < < < < 0 (16 where the coefficients & ' ( & and are defined in the Appendix. Here we would like to note that the numerical solutions of equations (15 and (16 satisfy the streamfunction and vorticity equations (6 and (7 with fourth order spatial accuracy. Moreover in these equations (15 and (16 if we set the values of the coefficients & ' ( & and as zero then the numerical solution of equations (15 and (16 will satisfy the streamfunction and vorticity equations (6 and (7 with second order spatial accuracy. Equations (15 and (16 offer both fourth and second order accurate solutions depending on the choice of the values of the coefficients. Therefore a single code for the solutions of equations (15 and (16 would provide both fourth and second order accurate numerical solutions. Moreover with the help of equations (15 and (16 if one already has a second order code for the streamfunction and vorticity equations for non-uniform grids by adding the coefficients & ' ( & and into the existing code and the code could turn into a fourth order accurate code.. GRID STRETCHING In this study we numerically solve equations (15 and (16 for the simulation of the benchmark driven cavity flow using non-uniform grids. We use grid clustering both in - and -directions near the wall as shown in Figure 2 using the following stretching function C D FG2H C I FG2H (17 E E where [01][01] are the coordinates of the grids in the non-uniform physical domain [01][01] are the coordinates of the grids in the uniform computational domain and also M and M are the stretching parameter in - and -directions respectively where 0 M < 1. We note that when M and M are chosen as zero then there is no grid clustering. We also note that from equation (17 we can analytically calculate the derivatives 6 6 easily. For the considered driven cavity flow problem since 6 6 and then the coefficients & ' ( and - reduce to & ' ( - For the coefficients & ' ( & and defined in the Appendix we need to calculate the derivatives of the coefficients & ' ( and - (i.e. and so on. Using the stretching function given in equation (17 we can easily calculate these derivatives of the coefficients. For example the derivative of can be calculated as the following : 2 ( editor@iaeme.com

6 Fourth Order Compact Formulation of Steady Navier-Stokes Equations on Non-Uniform Grids In the derivatives of these coefficients expressions for the inverse metrics (i.e. and so on are needed. We note that in almost all of the cases the grid stretching is done using an analytical stretching function therefore the transformation metrics can be calculated analytically. However depending on the stretching function used sometimes the inverse transformation function is not easy to obtain analytically and hence the inverse metrics cannot be calculated analytically. This is the case in this study. For the stretching function defined in equation (17 it is not easy to obtain an analytical expression for the inverse transformation (i.e. and. However using the transformation metrics we can easily calculate the inverse transformation metrics as the following SD ST similarly we find : :V : : 10 : (19 : (20 (21 : :V 6 6 (22 Following the same procedure we can also calculate 6 6 easily.. RESULTS AND DISCUSSIONS In order to demonstrate the efficiency of the compact fourth order accurate formulation of the streamfunction and vorticity equations we presented in equations (15 and (16 we simulate the benchmark driven cavity flow problem with fourth order accuracy using several different graded mesh. We use Alternating Direction Implicit (ADI method and solve equations (15 and (16 up to very low residuals. We note that one can use a mesh with small number of grid points and can still obtain accurate solutions using the fourth order accurate equations. In many practical CFD applications in order to resolve the flow grid clustering is needed. In this study we will demonstrate that it is possible to obtain very accurate numerical solutions by solving fourth order spatial accurate equations on a graded mesh with small number of grid points. For this purpose in this study we will use only a 66 number of grid points for the solution of the benchmark driven cavity flow. Figure 2 shows several graded mesh we use in this study. In this figure one every two grid points is shown with M M M. Figure 2 Grid refinement near the boundaries 18 editor@iaeme.com

7 Ercan Erturk and Orhan Gokcol Figure Streamfunction contours for 1000 for M0.0 Figure Streamfunction contours for 1000 for M0.9 We have solved the driven cavity flow for 1000 first using a stretching parameter ranging between 0.0 M 0.90 on a grid mesh with 66 grid points. Figure and show the streamfunction contours of the cavity flow obtained using a stretching parameter M0.0 where there is no stretching (i.e. uniform grids and M0.9 respectively. From Figure we can clearly see that when there is no grid stretching even though the spatial accuracy is fourth order accurate the flow is not resolved enough with using this many grid points especially near the Figure 5 Streamfunction contours for 2500 for M0.0 Figure 6 Streamfunction contours for 2500 for M editor@iaeme.com

8 Fourth Order Compact Formulation of Steady Navier-Stokes Equations on Non-Uniform Grids solid wall boundaries. However from Figure we can see that with fourth order spatial accuracy even 66 graded mesh is enough to resolve the flow accurately. Similarly we then have solved the cavity flow for 2500 using the same number of grid points and Figures 5 and 6 show the streamfunction contours for 2500 for M0.0 and M0.9 respectively. From Figures 5 and 6 we again can see that for 2500 using a 66 grid points while a uniform mesh is not enough to resolve the flow a graded mesh with M0.9 resolve the flow accurately and the streamfunction contours are very smooth. Botella and Peyret [1] have used a Chebyshev collocation method and obtained highly accurate spectral solutions for the driven cavity flow with a maximum of grid mesh of [160 (polynomial degree. They [1] have tabulated highly accurate vorticity data from inside the cavity along a vertical line and along a horizontal line passing through the center of the cavity Figure 7 Fourth order vorticity values along a horizontal line passing through the center of the cavity 1000 Figure 8 Fourth order vorticity values along a vertical line passing through the center of the cavity 1000 Figure 9 Second order vorticity values along a horizontal line passing through the center of the cavity 1000 Figure 10 Second order vorticity values along a vertical line passing through the center of the cavity editor@iaeme.com

9 Ercan Erturk and Orhan Gokcol for In Figures 7 and 8 we have plotted the fourth order accurate vorticity values for different grid stretching parameters. Since the equations (15 and (16 offer either second or fourth order solutions depending on the values of the coefficients & ' ( & and we decided to solve the same flow problem with second order accuracy using different grid stretching parameter also and compare the second order solutions with the fourth order solutions for a given non-uniform stretched mesh. For this in equations (15 and (16 we simply set the values of the coefficients & ' ( & and to zero and run the simulation again. For comparison we have plotted the second order accurate vorticity values along a vertical line and along a horizontal line passing through the center of the cavity similarly for different grid stretching parameters in Figures 9 and 10. One can clearly see the difference between fourth and second order accurate solutions by comparing Figures 7 and 8 with Figures 9 and 10. Expectedly the fourth order solutions provide much more accurate results than the second order accurate solutions. From Figures and 10 we can also see that grid stretching provides better results for both second and fourth order accurate solutions. 5. CONCLUSIONS In this study the fourth order compact formulation introduced by Erturk and Gokcol [7] is extended to non-uniform grids and forth order accurate compact formulation for 2-D steady incompressible Navier-Stokes equations for non-uniform grids are presented. The uniqueness of the presented formulation is that it provides both second order and fourth order accurate solutions according to the chosen coefficients. Moreover using the presented formulation any second order code for non-uniform grids can easily be altered to provide fourth order accurate solutions just by adding some coefficients in to the existing code. In order to demonstrate the efficiency of the presented fourth order compact formulation we have solved the driven cavity flow using a 6 6 grid mesh. Our solutions show that fourth order accurate equations provide very accurate results even with 6 6 grid points when grid stretching is used for the model problem. The presented fourth order accurate compact formulation for 2-D steady incompressible Navier-Stokes equations for non-uniform grids is proved to be very efficient. APPENDIX The coefficients & ' ( and A defined in equation (15 are defined as the following 2 & 2 & ' 2 ' ( 2 ( T > > T > T T >> T T? T A 2 >? ( (? T ' 2 & T > T >? T > T T? 2 '? \? > \? \ > \?? \ ( >? \? \? \ \ \ > ' \? \ \ \ ( > T & ' & The & and defined in equation (16 are defined as the following editor@iaeme.com

10 Fourth Order Compact Formulation of Steady Navier-Stokes Equations on Non-Uniform 2 > ' T > T \? \ \ \] T ] - ]> - ] 2 & T > T T? ( & \? \ \] - ]? 2 ' >> T ] T > T ' T ] > \? \ > \ > T ] >] T - T ] T? \] \] -?] \ \] \ \] - ]? >] T ] ] - ]> ]] T ]] \ ] 2 ( >? T T? T (?? \ > T ] - >] T T ] T? \] 2 T ] - >]?] \] 2 \? \ ]?] \ - \] \ T ] -?] ] - ]] \ ( 2 ]] T T \ & 2 > ]? & >? T? >]?] ] - '2 ] ] ( 2 ] ] ] - >? \ > ' T ] - >] \ ] -?] ] editor@iaeme.com

11 Ercan Erturk and Orhan Gokcol REFERENCES [1] Erturk E. Comparison of Wide and Compact Fourth Order Formulations of the Navier-Stokes Equations International Journal for Numerical Methods in Fluids 2009; 60: [2] Dennis SC Hudson JD. Compact h Finite Difference Approximations to Operators of Navier-Stokes Type Journal of Computational Physics 1989; 85: [] MacKinnon RJ Johnson RW. Differential-Equation-Based Representation of Truncation Errors for Accurate Numerical Simulation International Journal for Numerical Methods in Fluids 1991; 1: [] Gupta MM Manohar RP Stephenson JW. A Single Cell High Order Scheme for the Convection-Diffusion Equation with Variable Coefficients International Journal for Numerical Methods in Fluids 198; : [5] Spotz WF Carey GF. Formulation and Experiments with High-Order Compact Schemes for Nonuniform Grids International Journal of Numerical Methods for Heat Fluid Flow 1998; 8: [6] Li M Tang T Fornberg B. A Compact Forth-Order Finite Difference Scheme for the Steady Incompressible Navier-Stokes Equations International Journal for Numerical Methods in Fluids 1995; 20: [7] Erturk E Gokcol C. Fourth Order Compact Formulation of Navier-Stokes Equations and Driven Cavity Flow at High Reynolds Numbers International Journal for Numerical Methods in Fluids 2006; 50: [8] Erturk E. Numerical Performance of Compact Fourth-Order Formulation of the Navier-Stokes Equations Communications in Numerical Methods in Engineering 2008; 2: [9] Spotz WF Carey GF. High-Order Compact Scheme for the Steady Streamfunction Vorticity Equations International Journal for Numerical Methods in Engineering 1995; 8: [10] Ge L Zhang J. High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Nonuniform Grids Journal of Computational Physics 2001; 171: [11] Pandit SK Kalita JC Dalal DC. A Fourth-Order Accurate Compact Scheme for the Solution of Steady Navier-Stokes Equations on Non-Uniform Grids Computers Fluids 2008; 7: [12] Wang J Zhong W Zhang J. High Order Compact Computation and Nonuniform Grids for Streamfunction Vorticity Equations Applied Mathematics and Computation 2006; 179: [1] Botella O Peyret R. Benchmark spectral results on the lid-driven cavity flow Computers and Fluids 1998; 27: editor@iaeme.com