SUPG finite element method for adiabatic flows

The purpose of this paper is to present the SUPG finite element method for adiabatic flows and to compare the results with those obtained via various computational methods. The incompressibility assumption is often used to solve transient viscous flows. On the other hand, compressible flow analyses are also popular because natural flows include compressibility even if it is a negligible amount. It is well known that incompressibility is a limit state of compressibility. To solve compressible flows, three kinds of governing equations are needed: the conservation of mass, momentum, and energy, in which density, velocity, and internal energy are independent variables. If we assume that there is no heat transfer in or out of a system, the energy equation can be eliminated from the governing equations. Density and velocity can be considered independent variables. These kinds of flows are referred to as adiabatic flows in this paper. The SUPG formulation is one of the most widely used methods in the finite element analyses of fluid flows. In this paper, the SUPG method for adiabatic flows is presented. The polytropic law is introduced for the equation of state. Moreover, the computational results obtained by four models of fluid flows are compared: adiabatic flows, compressible flows, constant acoustic velocity flows, and incompressible flows. Verifications are carried out using a lid-driven cavity flow. Boundary effects are estimated using a wide computational domain. Drag forces are compared with those computed using incompressible flows. The pressure coefficients over a surface of a circular cylinder located in fluid flows computed by the present method show good correspondence with the experimental measurement. On the other hand, significant discrepancies are observed between the pressure coefficients computed assuming constant density and the experimental results.

[1]  Mutsuto Kawahara,et al.  A finite element method for high Reynolds number viscous fluid flow using two step explicit scheme , 1983 .

[2]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[3]  U. Ghia,et al.  High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method , 1982 .

[4]  S. Mittal,et al.  Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements , 1992 .

[5]  Hiroshi Ishiyama,et al.  Shape optimization of body located in incompressible viscous flow , 2008, Int. J. Comput. Math..

[6]  M. Kawahara,et al.  Two dimensional shape optimization using partial control and finite element method for compressible flows , 2010 .

[7]  M. Kawahara,et al.  A three-dimensional two-step explicit finite element method for high Reynolds number viscous fluid flow , 1983 .

[8]  H. Okumura,et al.  A New Stable Bubble Element for Incompressible Fluid Flow Based on a Mixed Petrov–Galerkin Finite Element Formulation , 2003 .

[9]  Olivier Pironneau,et al.  Numerical analysis for compressible viscous isentropic stationary flows , 1989, IMPACT Comput. Sci. Eng..

[10]  M. Kawahara,et al.  Shape Optimization of Body Located in Incompressible Navier--Stokes Flow Based on Optimal Control Theory , 2000 .

[11]  Leszek Demkowicz,et al.  A new finite element method for solving compressible Navier-Stokes equations based on an operator splitting method and h-p adaptivity , 1990 .

[12]  E. Erturk,et al.  Discussions on driven cavity flow , 2004, cs/0411048.

[13]  Shape optimization of a body located in incompressible flow using adjoint method and partial control algorithm , 2011 .

[14]  Jacques Periaux,et al.  Active Control and Drag Optimization for Flow Past a Circular Cylinder , 2000 .

[15]  Tayfun E. Tezduyar,et al.  Stabilization and shock-capturing parameters in SUPG formulation of compressible flows , 2004 .

[16]  Thomas J. R. Hughes,et al.  Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations , 1984 .

[17]  S. Mittal,et al.  Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries , 1995 .

[18]  Shape Optimization of a Body Located in Viscous Flows Using Acoustic Velocity Method , 2012 .

[19]  Shape optimization of a body in compressible inviscid flows , 2008 .

[20]  M. Rumpfkeil,et al.  The optimal control of unsteady flows with a discrete adjoint method , 2010 .

[21]  Ionel M. Navon,et al.  Suppression of vortex shedding for flow around a circular cylinder using optimal control , 2002 .

[22]  Thomas J. R. Hughes,et al.  Stabilized Methods for Compressible Flows , 2010, J. Sci. Comput..

[23]  Mutsuto Kawahara,et al.  Shape Identification for Fluid-Structure Interaction Problem Using Improved Bubble Element , 2001 .

[24]  Thomas J. R. Hughes,et al.  Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations†‡ , 1987 .

[25]  B. Kirk.  Adiabatic shock capturing in perfect gas hypersonic flows , 2009 .