International Journal of Aerospace and Lightweight Structures

doi: 10.3850/S2010428612000359

A Matrix-Free Implicit Gradient Smoothing Method (GSM) for Compressible Flows

Sheng Wang^1,a, George Xiangguo Xu², G. R. Liu³ and Boo Cheong Khoo^{1,b 1}Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore. ^ag0800348@nus.edu.sg ^bmpekbc@nus.edu.sg ²Parsons Brinckerhoff Pte Ltd, 300 Beach Road 05-00, the Concourse, Singapore 199555, Singapore. xuxiangguo@gmail.com ³University of Cincinnati, Cincinnati, OH, 45221-0070, USA. liugr@ucmail.uc.edu

ABSTRACT

The explicit gradient smoothing method (GSM) implemented directly on the strong form of Navier-Stokes equations has already been successfully established for solving both incompressible and compressible fluid flows based on unstructured triangular mesh. This study intends to develop an implicit GSM in which the implicit dual time integration scheme is applied in the time domain instead of the explicit time integration scheme so as to find a more efficient way for solving the compressible flows. The 1st- and 2ndorder spatial derivatives of field variables are numerically approximated by consistent and successive application of gradient smoothing operation over the node-associated and midpoint associated smoothing domains. The 2nd-order Roe flux differencing splitting upwind scheme is employed for approximating the inviscid flux in order to ensure the stability of the calculation. The implicit Lower-Upper Symmetric Gauss-Seidel (LUSGS) scheme is applied in the sub-iteration of the dual time stepping process. The explicit GSM with five-stage Runge-Kutta time integration scheme is also employed in the present paper for the comparison purpose. Benchmarks including both steady and transient fluid flows are tested to demonstrate the accuracy, efficiency and stability of the proposed implicit GSM, comparing with the explicit GSM. Numerical results from the proposed implicit GSM can match quite well with the experimental and referential data. The implicit GSM can also give remarkably computational efficiency without any loss in accuracy comparing with the explicit GSM.

Keywords: GSM, GSD, Implicit, Numerical method, Fluid dynamics, Stencil.

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