Application of least squares kinetic upwind method to strongly rotating viscous flows

Abstract

The Least Squares Kinetic Upwind Method (LSKUM) is a node-based method. Ghosh and Deshpande, as well as Ramesh & Deshpande, have demonstrated that LSKUM works on any type of grid—structured, unstructured, Cartesian, Chimera, hybrid, etc.—or even on an arbitrary distribution of points. LSKUM can be considered a generalization of the finite difference method that does not require a regular grid. The method combines least squares discretization with the Kinetic Flux Vector Splitting (KFVS) approach, which is based on the Boltzmann equation from the kinetic theory of gases. The Navier–Stokes equations can be solved using kinetic methods in two ways:

Standard KFVS approach: Discretize the convective terms using KFVS and apply central differencing or least squares with a full stencil for viscous terms. Extended KFVS approach: Extend KFVS based on the Chapman–Enskog distribution so that both inviscid and viscous terms are treated in an upwind manner through splitting of viscous as well as inviscid fluxes.

We have developed the LSKUM-NS code for computing 2D and 3D laminar viscous flows. The code uses least squares discretization of spatial derivatives and employs first-order as well as higher-order Runge–Kutta time-marching schemes. Flux splitting is performed using the extended KFVS method. Anandhanarayanan and Deshpande carried out extensive studies on various 2D problems using KFVS-based Navier–Stokes solvers. However, so far, LSKUM has primarily been applied to high-speed rotating viscous flows, with emphasis on aerospace problems in external aerodynamics.

Long-Term Objective The ultimate long-term aim of this thesis is to develop an LSKUM-based code for 3D viscous rotating flow with multi-body configurations. The development of such a 3D solver is best accomplished through a series of steps:

Step 1: Develop a 2D viscous solver for rotating flows. Step 2: Create tools for pre-processing the cloud of points and domain decomposition for parallelization. Step 3: Develop a 3D LSKUM-NS solver for viscous rotating flow. Step 4: Build tools for pre-processing the 3D cloud of points and domain decomposition to enable fast numerical simulation of multi-body configurations under strong rotation on parallel computers.

In the present thesis, we have executed the first three steps:

A pre-processor for generating connectivity. Development of the LSKUM solver for 2D viscous rotating flow. Parallelization of the solver.

Additionally, work was carried out to extend the 2D LSKUM for rotating flows to 3D viscous rotating flows for relatively simple geometric configurations.

Organization of the Thesis

Chapter 1: Introduction to KFVS and basics of 2D LSKUM. Chapter 2: Generation of cloud of points and connectivity issues. Chapter 3: LSKUM applied to viscous non-rotating flows, including boundary conditions and test cases. Chapter 4: LSKUM applied to viscous rotating flows, including rotating viscous flows, associated boundary layers, and test cases. Chapter 5: Parallelization issues and implementation for the 2D LSKUM-NS solver. Includes an application using the parallel 2D LSKUM-NS code for viscous rotating flow. Chapter 6: Extension to viscous 3D LSKUM for rotating flow, including 3D geometry, boundary conditions, and test cases. Chapter 7: Concluding remarks.