# American Institute of Mathematical Sciences

January  2019, 24(1): 127-147. doi: 10.3934/dcdsb.2018111

## A dimension splitting and characteristic projection method for three-dimensional incompressible flow

 1 School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China 2 Department of Civil and Mechanical Engineering, University of Missouri-Kansas City, Kansas City, MO, 64110, USA 3 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China 4 School of Mechanical Engineering, Dongguan University of Technology, Dongguan 523000, China 5 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 64509, USA

* Corresponding author: Yuchuan Chu(ychuan.chu@hit.edu.cn)

Received  May 2017 Revised  September 2017 Published  January 2019 Early access  March 2018

Fund Project: The first author is supported by the Fundamental Research Funds for the Central Universities of China, grant 2682015CX044.

A dimension splitting and characteristic projection method is proposed for three-dimensional incompressible flow. First, the characteristics method is adopted to obtain temporal semi-discretization scheme. For the remaining Stokes equations we present a projection method to deal with the incompressibility constraint. In conclusion only independent linear elliptic equations need to be calculated at each step. Furthermore on account of splitting property of dimension splitting method, all the computations are carried out on two-dimensional manifolds, which greatly reduces the difficulty and the computational cost in the mesh generation. And a coarse-grained parallel algorithm can be also constructed, in which the two-dimensional manifold is considered as the computation unit.

Citation: Hao Chen, Kaitai Li, Yuchuan Chu, Zhiqiang Chen, Yiren Yang. A dimension splitting and characteristic projection method for three-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 127-147. doi: 10.3934/dcdsb.2018111
##### References:
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Wang, A dimension split method for the incompressible Navier-Stokes equations in three dimensions, Int. J. Numer. Meth. Fluids, 73 (2013), 409-435.  doi: 10.1002/fld.3803. [7] H. Chen, J. Su, K. Li and S. Wang, A characteristic projection method for incompressible thermal flow, Numer. Heat Tr. B-Fund., 65 (2014), 554-590.  doi: 10.1080/10407790.2013.836052. [8] Z. Chen, Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems, Comput. Meth. Appl. Mech. Eng., 191 (2002), 2509-2538.  doi: 10.1016/S0045-7825(01)00411-X. [9] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2. [10] A. J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comput., 23 (1969), 341-353.  doi: 10.1090/S0025-5718-1969-0242393-5. [11] J. Douglas Jr and T. F. Russell, Numerical method for convection-dominated diffusion problem based on combining the method of characteristics with finite element of finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.  doi: 10.1137/0719063. [12] C. J. Freitas, R. L. Street, A. N. Findikakis and J. R. Koseff, Numerical simulation of three-dimensional flow in a cavity, Int. J. Numer. Meth. Fluids, 5 (1985), 561-575.  doi: 10.1002/fld.1650050606. [13] C. J. Freitas and R. L. Street, Non-linear transient phenomena in a complex recirculating flow: A numerical investigation, Int. J. Numer. Meth. Fluids, 8 (1988), 769-802.  doi: 10.1002/fld.1650080704. [14] U. Ghia, K. N. Ghia and C. T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), 387-411.  doi: 10.1016/0021-9991(82)90058-4. [15] J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Meth. Appl. Mech. Eng., 195 (2006), 6011-6045.  doi: 10.1016/j.cma.2005.10.010. [16] M. Hermanns, Parallel programming in Fortran 95 using OpenMP, 2002. Available from: http://www.openmp.org/wp-content/uploads/F95_OpenMPv1_v2.pdf. [17] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987. [18] J. R. Koseff and R. L. Street, Visualization studies of a shear driven three-dimensional recirculating flow, J. Fluids Eng., 106 (1984), 21-27.  doi: 10.1115/1.3242393. [19] J. R. Koseff and R. L. Street, The lid-driven cavity flow: A synthesis of qualitative and quantitative observations, J. Fluids Eng., 106 (1984), 390-398.  doi: 10.1115/1.3243136. [20] O. A. Ladyzhenskaya, the Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, 1969. [21] K. Li, A. Huang and W. Zhang, A dimension split method for the 3-d compressible Navier-Stokes equations in turbomachine, Commun. Numer. Meth. Eng., 18 (2002), 1-14. [22] K. W. Morton, A. Priestley and E. Süli, Convergence Analysis of the Lagrange-Galerkin Method with Non-Exact Integration, Technical report, Oxford University Computing Laboratory. Rept. N86/14, Oxford, 1986. [23] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994. [24] C. Shu, X. D. Niu and Y. T. Chew, Taylor series expansion and least squares-based lattice boltzmann method: three-dimensional formulation and its applications, Int. J. Mod. Phys. C, 14 (2003), 925-944.  doi: 10.1142/S0129183103005133. [25] E. Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53 (1988), 459-483.  doi: 10.1007/BF01396329. [26] R. Temam, Sur l'approximation de la solution des equations de Navier-Stokes par la ḿethode des fractionnarires Ⅱ, Arch. Rational Mech. Anal., 33 (1969), 377-385. [27] C. Wu, A general theory of three-dimensional flow in subsonic and supersonic turbomachines of axial-, radial-, and mixed-flow types, Tech. Notes. Nat. Adv. Comm. Aeronaut., 1952 (1952), ⅱ+93 pp. [28] P. X. Yu and Z. F. Tian, A compact streamfunction-velocity scheme on nonuniform grids for the 2D steady incompressible Navier-Stokes equations, Comput. Math. Appl., 66 (2013), 1192-1212.  doi: 10.1016/j.camwa.2013.07.013. [29] O. C. Zienkiewicz, P. Nithiarasu and R. L. Taylor, the Finite Element Method for Fluid Dynamics, seventh ed., Elsevier/Butterworth Heinemann, Amsterdam, 2014.

show all references

##### References:
 [1] A. Allievi and R. Bermejo, Finite element modified of characteristics for the Navier-Stokes equations, Int. J. Numer. Meth. Fluids, 32 (2000), 439-463.  doi: 10.1002/(SICI)1097-0363(20000229)32:4<439::AID-FLD946>3.0.CO;2-Y. [2] V. Babu and S. Korpela, Numerical solution of the incompressible three-dimensional Navier-Stokes equations, Comput. Fluids, 22 (1994), 675-691.  doi: 10.1016/0045-7930(94)90009-4. [3] O. Botella and R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids, 27 (1998), 421-433.  doi: 10.1016/S0045-7930(98)00002-4. [4] R. Bouffanais, M. O. Deville and E. Leriche, Large-eddy simulation of the flow in a lid-driven cubical cavity, Phys. Fluids, 19 (2007), 055108. doi: 10.1063/1.2723153. [5] J. Chan, J. A. Evans and W. Qiu, A dual Petrov-Galerkin finite element method for the convection-diffusion equation, Comput. Math. Appl., 68 (2014), 1513-1529.  doi: 10.1016/j.camwa.2014.07.008. [6] H. Chen, K. Li and S. Wang, A dimension split method for the incompressible Navier-Stokes equations in three dimensions, Int. J. Numer. Meth. Fluids, 73 (2013), 409-435.  doi: 10.1002/fld.3803. [7] H. Chen, J. Su, K. Li and S. Wang, A characteristic projection method for incompressible thermal flow, Numer. Heat Tr. B-Fund., 65 (2014), 554-590.  doi: 10.1080/10407790.2013.836052. [8] Z. Chen, Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems, Comput. Meth. Appl. Mech. Eng., 191 (2002), 2509-2538.  doi: 10.1016/S0045-7825(01)00411-X. [9] A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2. [10] A. J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comput., 23 (1969), 341-353.  doi: 10.1090/S0025-5718-1969-0242393-5. [11] J. Douglas Jr and T. F. Russell, Numerical method for convection-dominated diffusion problem based on combining the method of characteristics with finite element of finite difference procedures, SIAM J. Numer. Anal., 19 (1982), 871-885.  doi: 10.1137/0719063. [12] C. J. Freitas, R. L. Street, A. N. Findikakis and J. R. Koseff, Numerical simulation of three-dimensional flow in a cavity, Int. J. Numer. Meth. Fluids, 5 (1985), 561-575.  doi: 10.1002/fld.1650050606. [13] C. J. Freitas and R. L. Street, Non-linear transient phenomena in a complex recirculating flow: A numerical investigation, Int. J. Numer. Meth. Fluids, 8 (1988), 769-802.  doi: 10.1002/fld.1650080704. [14] U. Ghia, K. N. Ghia and C. T. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), 387-411.  doi: 10.1016/0021-9991(82)90058-4. [15] J. L. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows, Comput. Meth. Appl. Mech. Eng., 195 (2006), 6011-6045.  doi: 10.1016/j.cma.2005.10.010. [16] M. Hermanns, Parallel programming in Fortran 95 using OpenMP, 2002. Available from: http://www.openmp.org/wp-content/uploads/F95_OpenMPv1_v2.pdf. [17] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987. [18] J. R. Koseff and R. L. Street, Visualization studies of a shear driven three-dimensional recirculating flow, J. Fluids Eng., 106 (1984), 21-27.  doi: 10.1115/1.3242393. [19] J. R. Koseff and R. L. Street, The lid-driven cavity flow: A synthesis of qualitative and quantitative observations, J. Fluids Eng., 106 (1984), 390-398.  doi: 10.1115/1.3243136. [20] O. A. Ladyzhenskaya, the Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Science Publishers, New York, 1969. [21] K. Li, A. Huang and W. Zhang, A dimension split method for the 3-d compressible Navier-Stokes equations in turbomachine, Commun. Numer. Meth. Eng., 18 (2002), 1-14. [22] K. W. Morton, A. Priestley and E. Süli, Convergence Analysis of the Lagrange-Galerkin Method with Non-Exact Integration, Technical report, Oxford University Computing Laboratory. Rept. N86/14, Oxford, 1986. [23] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994. [24] C. Shu, X. D. Niu and Y. T. Chew, Taylor series expansion and least squares-based lattice boltzmann method: three-dimensional formulation and its applications, Int. J. Mod. Phys. C, 14 (2003), 925-944.  doi: 10.1142/S0129183103005133. [25] E. Süli, Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53 (1988), 459-483.  doi: 10.1007/BF01396329. [26] R. Temam, Sur l'approximation de la solution des equations de Navier-Stokes par la ḿethode des fractionnarires Ⅱ, Arch. Rational Mech. Anal., 33 (1969), 377-385. [27] C. Wu, A general theory of three-dimensional flow in subsonic and supersonic turbomachines of axial-, radial-, and mixed-flow types, Tech. Notes. Nat. Adv. Comm. Aeronaut., 1952 (1952), ⅱ+93 pp. [28] P. X. Yu and Z. F. Tian, A compact streamfunction-velocity scheme on nonuniform grids for the 2D steady incompressible Navier-Stokes equations, Comput. Math. Appl., 66 (2013), 1192-1212.  doi: 10.1016/j.camwa.2013.07.013. [29] O. C. Zienkiewicz, P. Nithiarasu and R. L. Taylor, the Finite Element Method for Fluid Dynamics, seventh ed., Elsevier/Butterworth Heinemann, Amsterdam, 2014.
Splitting the flow domain $\Omega$
Grid structure of two-dimensional manifold $D$
Sketch of three-dimensional lid-driven cavity flow
Velocity profiles on middle plane z = 0.5 for Re = 100
Velocity profiles on middle plane z = 0.5 for Re = 400
Velocity profiles on middle plane z = 0.5 for Re = 1000
Streamline profile for various Reynolds numbers: Re = 100(A, B, C); Re = 400(D, E, F); x = 0.5(A, D); z = 0.5(B, E); y = 0.5(C, F)
Streamline profile for various Reynolds numbers: Re = 1000(A, B, C); Re = 2000(D, E, F); x = 0.5(A, D); z = 0.5(B, E); y = 0.5(C, F)
Three dimensional streamline for different Reynolds numbers: Re = 100(A, B, C); Re = 400(D, E, F)
Three dimensional streamline for different Reynolds numbers: Re = 1000(A, B, C); Re = 2000(D, E, F)
Error of numerical solution with different mesh sizes
 $\frac{1}{h}$ $\|\vec u-\vec u_h\|_{L^2}$ $\alpha$ $\|p-p_h\|_{L^2}$ $\alpha$ $\kappa_{div}$ $4$ 1.149E-002 - 2.931E-001 - 4.741E-002 $8$ 3.513E-003 1.710 1.394E-001 1.072 6.953E-003 $16$ 8.765E-004 1.856 6.172E-002 1.124 2.304E-003 $32$ 1.927E-004 1.966 1.961E-002 1.301 3.826E-004
 $\frac{1}{h}$ $\|\vec u-\vec u_h\|_{L^2}$ $\alpha$ $\|p-p_h\|_{L^2}$ $\alpha$ $\kappa_{div}$ $4$ 1.149E-002 - 2.931E-001 - 4.741E-002 $8$ 3.513E-003 1.710 1.394E-001 1.072 6.953E-003 $16$ 8.765E-004 1.856 6.172E-002 1.124 2.304E-003 $32$ 1.927E-004 1.966 1.961E-002 1.301 3.826E-004
Convergence rate with different mesh sizes
 $\frac{1}{h}$ DSM-C DSM-D $U_{L^2}$ rate $P_{L^2}$ rate CPU(s) $U_{L^2}$ rate $P_{L^2}$ rate CPU(s) 4 - - 44.5 - - 43.7 8 1.710 1.072 138.7 1.42 0.876 162.4 16 1.856 1.124 206.3 1.49 1.075 383.2 32 1.966 1.301 957.6 1.53 0.971 1996.3
 $\frac{1}{h}$ DSM-C DSM-D $U_{L^2}$ rate $P_{L^2}$ rate CPU(s) $U_{L^2}$ rate $P_{L^2}$ rate CPU(s) 4 - - 44.5 - - 43.7 8 1.710 1.072 138.7 1.42 0.876 162.4 16 1.856 1.124 206.3 1.49 1.075 383.2 32 1.966 1.301 957.6 1.53 0.971 1996.3
Parallel performance of DSM-C at $1/h = 8, 16$
 $p$ $1/h=8$ $1/h=16$ $T_p$ $S_{p}$ $E_{p}$ $T_p$ $S_{p}$ $E_{p}$ 1 52.35 - - 451.69 - - 2 37.93 1.38 0.69 303.14 1.49 0.75 4 21.63 2.42 0.61 170.44 2.65 0.66 6 16.72 3.13 0.52 123.07 3.67 0.61 8 15.17 3.45 0.43 102.42 4.41 0.55 10 15.31 3.42 0.34 91.81 4.92 0.49 12 22.76 2.30 0.19 94.10 4.80 0.40
 $p$ $1/h=8$ $1/h=16$ $T_p$ $S_{p}$ $E_{p}$ $T_p$ $S_{p}$ $E_{p}$ 1 52.35 - - 451.69 - - 2 37.93 1.38 0.69 303.14 1.49 0.75 4 21.63 2.42 0.61 170.44 2.65 0.66 6 16.72 3.13 0.52 123.07 3.67 0.61 8 15.17 3.45 0.43 102.42 4.41 0.55 10 15.31 3.42 0.34 91.81 4.92 0.49 12 22.76 2.30 0.19 94.10 4.80 0.40
Parallel performance of DSM-C at $1/h = 32, 64$
 $p$ $1/h=32$ $1/h=64$ $T_p$ $S_{p}$ $E_{p}$ $T_P$ $S_{p}$ $E_{p}$ 1 3847.32 - - 32861.04 - - 2 2171.17 1.77 0.89 17077.32 1.92 0.96 4 1183.79 3.25 0.81 9225.06 3.56 0.89 6 875.91 4.39 0.73 6744.55 4.87 0.81 8 717.34 5.36 0.67 5402.69 6.08 0.76 10 648.78 5.93 0.59 4627.37 7.10 0.71 12 616.41 6.24 0.52 4147.19 7.92 0.66
 $p$ $1/h=32$ $1/h=64$ $T_p$ $S_{p}$ $E_{p}$ $T_P$ $S_{p}$ $E_{p}$ 1 3847.32 - - 32861.04 - - 2 2171.17 1.77 0.89 17077.32 1.92 0.96 4 1183.79 3.25 0.81 9225.06 3.56 0.89 6 875.91 4.39 0.73 6744.55 4.87 0.81 8 717.34 5.36 0.67 5402.69 6.08 0.76 10 648.78 5.93 0.59 4627.37 7.10 0.71 12 616.41 6.24 0.52 4147.19 7.92 0.66
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