An Efficient Implicit Compact Streamfunction Velocity Formulation of Two Dimensional Flows

Abstract

In this paper, we develop a class of implicit, unconditionally stable compact schemes for solving coupled fourth order and second order semilinear streamfunction and temperature equations respectively representing unsteady Navier–Stokes (N–S) equations for incompressible viscous fluid flows. The governing equations are presented in a generic form from which specific equations are obtained for several two dimensional (2D) incompressible viscous fluid flow problems. The streamfunction and the temperature equations are all solved iteratively in a coupled system of equations for the four field variables consisting of streamfunction, two velocities and temperature. The discretized form of biharmonic equation in streamfunction consists on 5-point stencil of streamfunction and is found to be second-order accurate both for spatially and temporally. In addition, we have considered two strategies for the discretization of the energy equation, which are second order accurate 5-point and fourth order accurate 9-point compact scheme. The formulation is made efficient by decreasing the computational complexity and is used to solve several 2D time dependent well studied benchmark problems. It is seen to efficiently capture both time wise and steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. The results obtained using our proposed scheme are in excellent agreement with the results available in the literature and they clearly establish the efficiency and the accuracy of the proposed scheme.

Appendices

Appendix 1

The coefficients of Eq. (27) are listed as follows:

$$\begin{aligned}&A_{i-1,j}= \displaystyle k^{4}(lh^{2}-12a\mu {\Delta t}+gh^2\mu \Delta t);\quad A_{i,j-1}= \displaystyle h^{4}(lk^{2}-12c\mu {\Delta t}+gk^2\mu \Delta t);\\&A_{i,j}= \displaystyle {-2lh^{2}k^{2}(h^{2}+k^{2})+24\mu {\Delta t}(ak^{4}+ch^{4})-2gh^2k^2(h^2+k^2)\mu \Delta t};\\&A_{i+1,j}= \displaystyle k^{4}(lh^{2}-12a\mu {\Delta t}+gh^2\mu \Delta t);\quad A_{i,j+1}= \displaystyle h^{4}(lk^{2}-12c\mu {\Delta t}+gk^2\mu \Delta t);\\&A_{i-1,j}^{'}= \displaystyle k^{4}(lh^{2}-12a(\mu -1){\Delta t}+gh^2(\mu -1)\Delta t);\\&A_{i,j-1}^{'}= \displaystyle h^{4}(lk^{2}-12c(\mu -1){\Delta t}+gk^2(\mu -1)\Delta t);\\&A_{i,j}^{'}= \displaystyle {-2lh^{2}k^{2}(h^{2}+k^{2})-2gh^2k^2(h^2+k^2)(\mu -1) \Delta t+24(\mu -1){\Delta t}(ak^{4}+ch^{4})};\\&A_{i+1,j}^{'}= \displaystyle k^{4}(lh^{2}-12a(\mu -1){\Delta t}+gh^2(\mu -1)\Delta t);\\&A_{i,j+1}^{'}= \displaystyle h^{4}(lk^{2}-12c(\mu -1){\Delta t}+gh^2(\mu -1)\Delta t);\\&R_{i+k_1,j+k_2}=\mu \displaystyle {\Delta t}r_{i+k_1,j+k_2};\quad R_{i+k_1,j+k_2}^{'}=(1-\mu )\displaystyle {\Delta t}r_{i+k_1,j+k_2};\\&P_{i+k_1,j+k_2}=\mu \displaystyle {\Delta t}p_{i+k_1,j+k_2};\quad P_{i+k_1,j+k_2}^{'}=(1-\mu )\displaystyle {\Delta t}p_{i+k_1,j+k_2}; \end{aligned}$$

with

$$\begin{aligned}&r_{i-1,j-1}=\displaystyle {\frac{b}{4}h^{2}k^{3}};\quad r_{i,j-1}=\displaystyle h^{2}k(12ch^{2}-bk^{2}-2ekh^{2});\\&r_{i+1,j-1}=\displaystyle {\frac{b}{4}h^{2}k^{3}};\quad r_{i-1,j}=\displaystyle {-eh^{2}k^{4}};\\&r_{i,j}=\displaystyle {-2eh^{2}k^{2}(h^{2}+k^{2})};\quad r_{i+1,j}=\displaystyle {-eh^{2}k^{4}};\\&r_{i-1,j+1}=\displaystyle {-\frac{b}{4}h^{2}k^{3}};\quad r_{i,j+1}=\displaystyle h^{2}k(-12ch^{2}+bk^{2}-2ekh^{2});\\&r_{i+1,j+1}=\displaystyle {-\frac{b}{4}h^{2}k^{3}}; \end{aligned}$$

and

$$\begin{aligned}&p_{i-1,j-1}=\displaystyle {-\frac{b}{4}h^{3}k^{2}}; \quad p_{i,j-1}=\displaystyle {\frac{d}{4}h^{4}k^{2}};\\&p_{i+1,j-1}=\displaystyle {\frac{b}{4}h^{3}k^{2}}; \quad p_{i-1,j}=\displaystyle {hk^{2}(-12ak^{2}+bh^{2}+2dhk^{2})};\\&p_{i,j}= \displaystyle {-2dh^{2}k^{2}(h^{2}+k^{2})};\quad p_{i+1,j}=\displaystyle {hk^{2}(12ak^{2}-bh^{2}+2dhk^{2})};\\&p_{i-1,j+1}=\displaystyle {-\frac{b}{4}h^{3}k^{2}};\quad p_{i,j+1}=\displaystyle {dh^{4}k^{2}};\\&p_{i+1,j+1}=\displaystyle {\frac{b}{4}h^{3}k^{2}}. \end{aligned}$$

Appendix 2

The coefficients of Eq. (30) are listed as follows:

$$\begin{aligned} \gamma ^{*}_{i,j}= & {} \frac{\gamma _{i,j}-2\delta _{x}\alpha _{i,j}}{\alpha _{i,j}},\nonumber \\ \nu ^{*}_{i,j}= & {} \frac{\nu _{i,j}-2\delta _{y}\beta _{i,j}}{\beta _{i,j}},\nonumber \\ L_{i,j}= & {} \alpha _{i,j}+\frac{h^{2}}{12}\bigg [\bigg (\delta ^{2}_{x}\alpha _{i,j} +2\delta _{x}\gamma _{i,j}\bigg )+\gamma ^{*}_{i,j}\bigg (\delta _{x}\alpha _{i,j}+\gamma _{i,j}\bigg )\bigg ] +\frac{k^{2}}{12} \bigg [\nu ^{*}_{i,j}\delta _{y}\alpha _{i,j}+\delta ^{2}_{y}\alpha _{i,j}\bigg ], \nonumber \\ M_{i,j}= & {} \beta _{i,j}+\frac{h^{2}}{12} \bigg [\gamma ^{*}_{i,j}\delta _{\xi }\beta _{i,j}+\delta ^{2}_{\xi }\beta _{i,j}\bigg ]+\frac{k^{2}}{12} \bigg [\bigg (\delta ^{2}_{\eta }\beta _{i,j} +2\delta _{\eta }\nu _{i,j}\bigg )+\nu ^{*}_{i,j}\bigg (\delta _{\eta }\beta _{i,j}+\nu _{i,j}\bigg )\bigg ], \nonumber \\ N_{i,j}= & {} \gamma _{i,j}+ \frac{h^{2}}{12}\bigg [\delta ^{2}_{\xi }\gamma _{i,j} +\gamma ^{*}_{i,j}\delta _{\xi }\gamma _{i,j}\bigg ]+\frac{k^{2}}{12}\bigg [\delta ^{2}_{\eta }\gamma _{i,j}+ \nu ^{*}_{i,j}\delta _{\eta }\gamma _{i,j}\bigg ],\nonumber \\ O_{i,j}= & {} \nu _{i,j} +\frac{h^{2}}{12}\bigg [\delta ^{2}_{\xi }\nu _{i,j}+ \gamma ^{*}_{i,j}\delta _{\xi }\nu _{i,j}\bigg ]+ \frac{k^{2}}{12}\bigg [\delta ^{2}_{\eta }\nu _{i,j} +\nu ^{*}_{i,j}\delta _{\eta }\nu _{i,j}\bigg ],\nonumber \\ G_{i,j}= & {} \frac{h^{2}}{12}\bigg [2\delta _{x}\nu _{i,j}+\gamma ^{*}_{i,j}\nu _{i,j}\bigg ] +\frac{k^{2}}{12}\bigg [2\delta _{y}\gamma _{i,j}+ \nu ^{*}_{i,j}\gamma _{i,j}\bigg ],\nonumber \\ X_{i,j}= & {} h^{2}\beta _{i,j}+k^{2}\alpha _{i,j},\\ Y_{i,j}= & {} h^{2}\nu _{i,j}+k^{2}(2\delta _{y}\alpha _{i,j}\nu ^{*}_{i,j}\alpha _{i,j}),\\ Z_{i,j}= & {} h^{2}(2\delta _{x}\beta _{i,j}+\gamma ^{*}_{i,j}\beta _{i,j})+k^{2}\gamma _{i,j}. \end{aligned}$$