Development of Triangular Equilibrium Flux Method Using Graphics Processing Unit Acceleration for Solving Euler and Navier-Stokes Equations

• Main Authors: Yen-ChihChen, 陳彥志
• Other Authors: Matthew Ross Smith
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/21974715382338895824

碩士 === 國立成功大學 === 機械工程學系 === 102 === This study presents two new methods, named the “Uniform Equilibrium Flux Method” (UEFM) and “Triangular Equilibrium Flux Method” (TEFM) for solving flows governed by the Euler and Navier-Stokes equations. In the original Equilibrium Flux Method (EFM), the EFM fluxes are calculated by integrating the Maxwell-Boltzmann equilibrium velocity probability distribution function over velocity space. However, the equilibrium velocity probability distribution function and its integral contain both exponential and error functions. These exponential and error functions are complex – without closed form – and thus their solution is highly time-consuming. Accordingly, the proposed UEFM and TEFM methods use a series of uniform and triangular probability distribution functions designed to approximate the equilibrium velocity probability distribution function. This simplification results in a significant reduction in the computational cost. Furthermore, the extension of UEFM and TEFM method to the second order spatial accuracy is realized by casting of the present flux expressions as a direction-decoupled surface flux. Based on the Taylor series expansion in space, the second order split surface fluxes are able to be reconstructed at the cell interfaces. The MINMOD and MC flux limiters are employed to avoid the non-physical oscillations related to the integration of gradient terms over the velocity and physical space. The analysis of the numerical dissipation for the first order TEFM scheme predicts that the diffusion coefficient for TEFM mass flux is associated with the local Mach number and results in a significant decrease in diffusion with increasing Mach numbers. This does not occur with UEFM mass flux, and hence the majority of the results focus on the superior TEFM method. The higher order TEFM fluxes are then applied to large scale parallel computation using Nvidia’s Graphics Processing Units, or GPUs, through the CUDA API. Due to the vector split nature of the TEFM fluxes, the GPU parallel computation is relatively straight forward. The GPU computation is implemented through the creation of several key CUDA kernels for the evaluation of TEFM split fluxes, gradients of split fluxes, Navier-Stokes viscous fluxes and the state computations. All computations are executed entirely on the GPU device, leaving the CPU (host) idle during the computation stage. Multiple numerical benchmarks, including Sod’s one-dimensional shock tube problem, the two-dimensional shock bubble interaction, Euler two-dimensional four shocks and four contacts problems, are solved based on the Euler equation for an inviscid, ideal gas to verify both the first order and second order TEFM schemes. The one-dimensional shock tube problem indicates that the first order TEFM scheme demonstrates results appropriately equivalent to the original EFM method. In addition, the second order extension of TEFM method improves the resolution of contact surface and shock wave regions for the shock tube problem. The two-dimensional numerical implementations with various numbers of computational cells and second order extension provide sharp resolution in regions involving large gradients of flow properties. The two-dimensional backward facing step flow is further used to demonstrate the TEFM method for solving the Navier-Stokes equations with viscous, ideal flow. As the speedup results indicate, hundreds of times speedup can be achieved using the Nvidia’s GTX Titan, Tesla C2075 and GTX 670 GPU when compared with a single core of an Intel i3-2120 CPU.