**Least squares spectral element method for laminar and turbulent flows
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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