Numerical solution of the time-fractional Navier–Stokes equations for incompressible flow in a lid-driven cavity

Abstract

This paper addresses a computational technique for solving 2D unsteady Navier–Stokes equations (NSEs) with time-fractional order in the Caputo sense in the formulation of stream function-vorticity. The finite difference-based method of lines is used to discretize the time-fractional NSEs on a collocated grid that construct a fractional differential algebraic equations system. After solving the discretized complementary Poisson’s equation, this system is reduced to a system of fractional differential equations (FDEs). The resulting FDEs are solved by fractional backward differentiation formulas. The flow in a square lid-driven cavity is considered as the model problem.

Appendix A

If $v_{ij}\ge 0$ and $w_{ij}\ge 0$, then

$$\begin{aligned}&A_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&B_{i,j}=\frac{u_{i,j}}{\bigtriangleup x}+\dfrac{1}{Re \bigtriangleup x^{2}}\nonumber \\&C_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&D_{i,j}=\frac{v_{i,j}}{\bigtriangleup x}+\dfrac{1}{Re \bigtriangleup y^{2}}\nonumber \\&E_{i,j}=-\frac{u_{i,j}}{\bigtriangleup x}-\frac{v_{i,j}}{\bigtriangleup x}-\dfrac{2}{Re \bigtriangleup y^{2}}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}} \end{aligned}$$

(A1)

If $v_{ij}\ge 0$ and $w_{ij}< 0$, then

$$\begin{aligned}&A_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&B_{i,j}=\frac{u_{i,j}}{\bigtriangleup x}+\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&C_{i,j}=-\frac{v_{i,j+1}}{\bigtriangleup y}+\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&D_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&E_{i,j}=-\frac{u_{i,j}}{\bigtriangleup x}+\frac{v_{i,j+1}}{\bigtriangleup x}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}}. \end{aligned}$$

(A2)

If $v_{ij}< 0$ and $w_{ij}\ge 0$, then

$$\begin{aligned}&A_{i,j}=-\frac{u_{i+1,j}}{\bigtriangleup x}+\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&B_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&C_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&D_{i,j}=\frac{v_{i,j}}{\bigtriangleup y}+\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&E_{i,j}=\frac{u_{i+1,j}}{\bigtriangleup x}-\frac{v_{i,j+1}}{\bigtriangleup x}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}}. \end{aligned}$$

(A3)

If $v_{ij}< 0$ and $w_{ij}< 0$, then

$$\begin{aligned}&A_{i,j}=-\frac{u_{i+1,j}}{\bigtriangleup x}+\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&B_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup x^{2}}\nonumber \\&C_{i,j}=-\frac{v_{i,j}}{\bigtriangleup y}+\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&D_{i,j}=\dfrac{1}{\mathrm{Re} \bigtriangleup y^{2}}\nonumber \\&E_{i,j}=\frac{u_{i+1,j}}{\bigtriangleup x}-\frac{v_{i,j+1}}{\bigtriangleup x}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}}-\dfrac{2}{\mathrm{Re} \bigtriangleup y^{2}}. \end{aligned}$$

(A4)