A Novel Gaussian filter-based Pressure Correction Technique with Super Compact Scheme for Unsteady 3D Incompressible, Viscous Flows

TL;DR

The paper advances computational fluid dynamics by coupling a Gaussian filter-based pressure correction with a super-compact, high-order finite-difference scheme for unsteady 3D incompressible viscous flows. It introduces a $(19,7)$-point stencil and a Gaussian smoothing step to accelerate pressure-iteration convergence within a modified compressibility framework, demonstrated on a 3D Burgers equation and two lid-driven cavity variants. Validation shows fourth-order spatial accuracy and second-order temporal accuracy, along with significant reductions in pressure-iteration counts and overall computational cost, without sacrificing accuracy. The approach yields robust, efficient simulations for complex 3D flows and holds promise for larger-scale CFD problems. All mathematical notation is presented within $...$ delimiters.

Abstract

This work deals with a novel Gaussian filter-based pressure correction technique with a super compact higher order finite difference scheme for solving unsteady three-dimensional (3D) incompressible, viscous flows. This pressure correction technique offers significant advantages in terms of optimizing computational time by taking minimum iterations to reach the required accuracy, making it highly efficient and cost-effective. Pressure fields often exhibit highly nonlinear behavior, and employing the Gaussian filter can help to enhance their reliability by reducing noise and uncertainties. On the other hand, the super compact scheme uses minimum grid points to produce second-order accuracy in time and fourth-order accuracy in space variables. The main focus of this study is to enhance the accuracy and efficiency and minimize the computational cost of solving complex fluid flow problems. The super compact scheme utilizes 19 grid points at the known time level (i.e., $n^{th}$ time level) and only seven grid points from the unknown time level (i.e., $n + 1$ time level). By employing the above strategies, it becomes possible to notably decrease computational expenses while maintaining the accuracy of the computational scheme for solving complex fluid flow problems. We have implemented our methodology across three distinct scenarios: the 3D Burger's equation having analytical solution, and two variations of the lid-driven cavity problem. The outcomes of our numerical simulations exhibit a remarkable concordance, aligning exceptionally well with both the analytical benchmarks and previously validated numerical findings for the cavity problems.

Figures (18)

• Figure 1: The super-compact unsteady stencil
• Figure 2: Comparison of surface Plots: Numerical vs. Analytical Solution for $Re = 10$ at $t = 1.0$ and $x=0.4$ plane (a) Numerical solution using $11\times 11\times11$ grid size (b) Numerical solution using $41\times 41\times41$ grid size (c) Analytical solution
• Figure 3: Comparison of Surface Plots: Numerical vs. Analytical Solution for $Re = 10$ at $t = 1.0$ and $x=0.4$ plane (a) Numerical solution using $11\times 11\times11$ grid size (b) Numerical solution using $41\times 41\times 41$ grid size (c) Analytical solution
• Figure 4: (a) Illustration of the configuration in the 3D lid-driven cavity scenario and (b) View of the grids at a resolution of $91\times91\times91$.
• Figure 5: Velocity values along the different grid sizes at an observing point (0.75, 0.75, 0.75) (a) $u$ and (b) $w$