In order for design of thermal-fluid systems to be based solely on computational fluid dynamics (CFD), error bounds must be provided for each CFD solution.  Only when there is confidence in each CFD solution can design decisions be based on solid foundations.  There are four main sources of errors in CFD solutions.  The first is inadequate modeling of physics that are not resolved by first principles such as turbulence and combustion.  The second is inadequate information on boundary conditions (e.g., those at inflow and outflow boundaries).  The third is non-physical effects such as numerical diffusion, dispersion, and other spurious modes that result when the governing partial differential equations (PDEs) are discretized into algebraic equations on a discrete domain by finite-difference, finite-volume, or finite-element methods.  The fourth source of error is from the mesh.

For most users of CFD, especially those who do not have access to the source code, the mesh and the time-step size are the only parts of the solution procedure in which the user has full control.  The importance of the mesh cannot be over emphasized.  The mesh must represent the geometry with sufficient detail and enable the algebraic analog of the governing PDEs to resolve the relevant flow physics.  For complicated steady and unsteady, three-dimensional problems, the number of grid points or cells that can be used in a mesh is restricted by either the available computer resource or a need to have a practical turn-around time in computing a solution.  With a constraint on the number of grid points or cells, accuracy demands grid points to be placed in regions where they are most needed to resolve the geometry and flow physics (e.g., by r- or h-refinement).  Unfortunately, this non-uniform distribution can create what are referred to as poor-quality cells, which can induce considerable errors in the computed solutions.  For most problems of engineering interest, the situation is made worse in that it is generally not feasible to generate grid-independent solutions so that there are errors from poor quality cells and from inadequate resolution.

Thus, the questions are (1) what can one believe in the computed solutions?  (2) What is the error bound on the solutions from a poor-quality or an insufficiently fine grid?  (3) How can the grid or mesh be improved to increase accuracy?  This talk describes recent work that attempts to answer these questions by providing methods for estimating grid-induced errors.

 

 

Tom I-P. Shih joined Purdue as professor and head of the School of Aeronautics and Astronautics in August 2009.  Previously, he was professor and chair of the Department of Aerospace Engineering at Iowa State University (2003-09).  He has also held faculty positions at Michigan State University (1998-2003), Carnegie Mellon University (1988-98), and the University of Florida (1983-88) and was a mechanical engineer at NASA – Lewis (now Glenn) Research Center (1981-82).  He started his undergraduate education at West Virginia University but completed his B.S. degree at the National Cheng Kung University in Taiwan in 1976.  He received his M.S.E. and Ph.D. degrees from The University of Michigan at Ann Arbor in 1977 and 1981, respectively.  Shih’s research centers on computational fluid dynamics (CFD).  He and his students have developed a number of algorithms and codes for studying reacting and non-reacting, compressible and incompressible flows.  Algorithms and codes have also been developed for automatic/knowledge-based grid generation, error estimation of CFD solutions, and direct numerical simulation of particle-particle interactions that resolve the flow around each particle.  In using CFD, Shih and his students have studied a wide range of problems in energy, power, and propulsion systems, including piston and Wankel rotary engines, automotive torque converters, control of shock-wave/boundary-layer interactions by bleed, aerodynamics of iced airfoils and wings, and internal and film cooling of gas turbine components.  In these endeavors, Shih has authored and co-authored more than 200 technical papers in journals and conferences; presented over 160 invited seminars, lectures, workshops, and keynotes at conferences; and served as advisor and co-advisor to 21 Ph.D. and 44 M.S. students.  Shih is a Fellow of ASME and AIAA.