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<p>This work presents a flux approximation scheme for the finite volume method (FVM) applied to a semi-discretized system of the time-fractional Navier–Stokes equations (TFNSE) on a staggered grid. The cell-face fluxes are evaluated by solving appropriate local nonlinear inhomogeneous boundary value problems (BVPs), in which the effects of cross-fluxes and pressure gradients are incorporated into the source term. Consequently, the cell-face flux is decomposed into homogeneous and inhomogeneous components. The homogeneous component represents the effects of convection and viscous forces, while the inhomogeneous component accounts for the contributions from gradients of cross-flux and pressure terms. Therefore, the resulting numerical fluxes incorporate the collective influence of the forces governing the flow field. The fully discrete TFNSE system is then constructed using a finite difference scheme (FDS) in time, with the Caputo time-fractional derivative approximated through piecewise linear interpolation. To test the efficacy of the proposed scheme, we solve some benchmark problems. First, we numerically implemented the proposed scheme for the Taylor–Green vortex (TGV) problem in a time-fractional framework for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0&lt; \gamma &lt; 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and the result demonstrates that the scheme achieves a second-order convergence in space. Next, we numerically simulate the flow in a unit square cavity driven by the motion of the upper lid, while the remaining walls are kept stationary and the fluid is initially at rest. We present contours of the velocity field and stream function for various values of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> at different Reynolds numbers <i>Re</i>.</p>

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A finite volume numerical scheme for time-fractional Navier–Stokes equation

  • 期刊论文

  • 发表日期：

    2026年6月18日

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  • Rinki Rawat,
  • Chitranjan Pandey,
  • B. V. Rathish Kumar

摘要

This work presents a flux approximation scheme for the finite volume method (FVM) applied to a semi-discretized system of the time-fractional Navier–Stokes equations (TFNSE) on a staggered grid. The cell-face fluxes are evaluated by solving appropriate local nonlinear inhomogeneous boundary value problems (BVPs), in which the effects of cross-fluxes and pressure gradients are incorporated into the source term. Consequently, the cell-face flux is decomposed into homogeneous and inhomogeneous components. The homogeneous component represents the effects of convection and viscous forces, while the inhomogeneous component accounts for the contributions from gradients of cross-flux and pressure terms. Therefore, the resulting numerical fluxes incorporate the collective influence of the forces governing the flow field. The fully discrete TFNSE system is then constructed using a finite difference scheme (FDS) in time, with the Caputo time-fractional derivative approximated through piecewise linear interpolation. To test the efficacy of the proposed scheme, we solve some benchmark problems. First, we numerically implemented the proposed scheme for the Taylor–Green vortex (TGV) problem in a time-fractional framework for \(0< \gamma < 1\) 0 < γ < 1 , and the result demonstrates that the scheme achieves a second-order convergence in space. Next, we numerically simulate the flow in a unit square cavity driven by the motion of the upper lid, while the remaining walls are kept stationary and the fluid is initially at rest. We present contours of the velocity field and stream function for various values of \(\gamma \) γ at different Reynolds numbers Re.