Implicit Least Squares Kinetic Upwind Method (LSKUM) And Implicit LSKUM Based On Entropy Variables (q-LSKUM)

Abstract

With increasing demand for computational solutions of fluid dynamical problems, researchers around the world are working on the development of highly robust numerical schemes capable of solving flow problems around complex geometries arising in Aerospace engineering. Also considerable time and effort are devoted to development of convergence acceleration devices, for reducing the computational time required for such numerical solutions. Reduction in run times is very vital for production codes which are used many times in design cycle. In this present work, we consider a numerical scheme called LSKUM capable of operating on any arbitrary distribution of points. LSKUM is being used in CFD center (IIsc) and DRDL (Hyderabad) to compute flows around practical geometries and presently these LSKUM based codes are explicit- It has been observed already by the earlier researchers that the explicit schemes for these methods are robust. Therefore, it is absolutely essential to consider the possibility of accelerating explicit LSKUM by making it LSKUM-Implicit. The present thesis focuses on such a study. We start with two kinetic schemes namely Least Squares Kinetic Upwind Method (LSKUM) and LSKUM based on entropy variables (q-LSKUM). We have developed the following two implicit schemes using LSKUM and q-LSKUM. They are

(i)Non-Linear Iterative Implicit Scheme called LSKUM-NII. (ii)Linearized Beam and Warming implicit Scheme, called LSKUM-BW. For the purpose of demonstration of efficiency of the newly developed above implicit schemes, we have considered flow past NACA0012 airfoil as a test example. In this regard we have tested these implicit schemes for flow regimes mentioned below •Subsonic Case: M∞ = 0.63, a.o.a = 2.0° •Transonic Case: M∞ = 0.85, a.o.a = 1.0° The speedup of the above two implicit schemes has been studied in this thesis by operating them on different grid sizes given below •Coarse Grid: 4074 points •Medium Grid: 8088 points •Fine Grid: 16594 points

The results obtained by running these implicit schemes are found to be very much encouraging. It has been observed that these newly developed implicit schemes give as much as 2.8 times speedup compared to their corresponding explicit versions. Further improvement is possible by combining LKSUM-Implicit with modern iterative methods of solving resultant algebraic equations. The present work is a first step towards this objective.