# 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  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 & Continuous Dynamical Systems - B, 2019, 24 (1) : 127-147. doi: 10.3934/dcdsb.2018111
##### References:

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##### References:
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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