### Least Squares Spectral Element Method for Laminar and Turbulent Flows, –Continuous and Discontinuous Approaches–

*A Dissertation Presented for the Doctor of Philosophy in Computational Engineering, The
University of Tennessee at Chattanooga*

#### Jaber Javanshir Hasbestan, August 2016

Abstract:

In this research, application of a least squares spectral element method for compressible
laminar and turbulent flow problems is investigated. For the turbulent Reynolds Averaged
Navier-Stokes (RANS), a modified Spalart-Allmaras (SA) turbulence model is employed
and integrated with the mean flow equations in a segregated fashion. Two different
approaches are presented for solving the SA model using the least squares method.
The first method represents a simple rearrangement of the equation. However, proper
arrangement of the SAmodel is required in order to produce a stable scheme for the
least squares methodology. The second approach is into divide the SA equation to two
hyperbolic and elliptic partial differential equations. To improve and condition number
of the Jacobian matrix, as well as the convergence of the nonlinear system, a weighting
for the least squares spectral method is introduced. This modification is essential
for systems that include different scales in the formulation. Least squares formulations
require the Navier-Stokes equations be re-cast in first order form. Therefore, additional
independent variables, are introduced which in turn increases the memory requirements.
Fortunately due to symmetry only half of the Jacobian matrix needs to be stored. To
further reduce memory requirements, an assembly free pleasingly parallel discontinuous
methodology is developed by modifying the cost function. This approach eliminates
the storage of the Jacobian matrix and its preconditioner at the expense of adding
an extra Newton iterative-loop. As such, the system can be solved at the element level
using a Cholesky factorization algorithm. This formulation is ideally suited for shared
memory paradigms such as OpenMp or CUDA, as it does not need blocking communication.
H-refinement is implemented for both steady and unsteady test cases using the discontinuous
formulation. Feature based adaptation is utilized in the adaptive refinement process.
Least squares method is known to be conditionally stable. In this study, an unconditionally
stable method is derived by modifying the weighting function and the results are presented
for the method of manufactured solution for Euler equations only for steady state
case. To demonstrate the versatility of the assembly free least squares approach,
this methodology is additionally applied to incompressible flows. For all simulations
presented P5 quadrilateral elements are utilized. A simple method is presented for
generating smooth higher order meshes from given P1 meshes for two-dimensional problems.

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