The strand-Cartesian grid approach is a unique method of generating and computing fluid dynamic simulations. The strand-Cartesian approach provides highly desirable qual- ities of fully-automatic grid generation and high-order accuracy. This thesis focuses on the implementation of the Spalart-Allmaras turbulence model to the strand-Cartesian grid framework. Verification and validation is required to ensure correct implementation of the turbulence model.Mathematical code verification is used to ensure correct implementation of new algo- rithms within the code framework. The Spalart-Allmaras model is verified with the Method of Manufactured Solutions (MMS). MMS shows second-order convergence, which implies that the new algorithms are correctly implemented.Validation of the strand-Cartesian solver is completed by simulating certain cases for comparison against the results of two independent compressible codes; CFL3D and FUN3D. The NASA-Langley turbulence resource provided the inputs and conditions required to run the cases, as well as the case results for these two codes. The strand solver showed excellent agreement with both NASA resource codes for a zero-pressure gradient flat plate and bump- in-channel. The treatment of the sharp corner on a NACA 0012 airfoil is investigated, resulting in an optimal external sharp corner configuration of strand vector smoothing with a base Cartesian grid and telescoping Cartesian refinement around the trailing edge. Results from the case agree well with those from CFL3D and FUN3D. Additionally, a NACA 4412 airfoil case is examined, and shows good agreement with CFL3D and FUN3D, resulting in validation for this case.

Foram feitas simulações de um jato plano livre e incompressível usando o modelo de uma equação Spalart-Allmaras padrão e um modelo Spalart-Allmaras modificado através da alteração da escala de comprimento turbulenta. Sabe-se da literatura que no caso de jatos livres o modelo Spalart-Allmaras não consegue predizer adequadamente os resultados observados experimentalmente. Os resultados das simulações foram comparados com experimentos da literatura através de perfis de velocidade e da taxa de expansão do jato. Como esperado, os resultados obtidos das simulações utilizando o modelo Spalart-Allmaras padrão foram considerados insatisfatórios, porém o modelo Spalart-Allmaras modificado teve uma melhor concordância com os resultados experimentais. Além disso, o modelo Spalart-Allmaras modificado foi usado para simular os casos do escoamento sobre uma placa plana sem gradiente de pressão e o escoamento em um degrau com separação e gradiente adverso de pressão. Quando comparados com resultados experimentais da literatura e com resultados obtidos usando o modelo padrão, os resultados do modelo modificado obtidos para ambos os casos foram muito satisfatórios, concluindo-se que a modificação da escala de comprimento permite obter uma maior generalidade para o modelo Spalart-Allmaras. / Simulations of a plane and incompressible free jet using the standard Spalart-Allmaras model and a Spalart-Allmaras model modified by changing the turbulent length scale were carried out. It is known from literature that, in the case of the free jet, the Spalart-Allmaras model fails to adequately predict the experimentally observed results. The results of our simulations were compared with published experiments using the velocity profiles and the jet spreading rate. As expected, the results of simulations using the standard Spalart-Allmaras model were considered unsatisfactory while the modified Spalart-Allmaras model had a better agreement with the experimental results. Furthermore, the modified Spalart-Allmaras model was used to simulate the cases of flow over a flat plate with no pressure gradient and flow through a backward facing step with separation and adverse pressure gradient. When compared with experimental results from the literature and with results obtained using the standard model, the results of the modified model for both cases were very satisfactory, allowing the conclusion that the change in the length scale provided a greater generality for the Spalart-Allmaras model.

Foram feitas simulações de um jato plano livre e incompressível usando o modelo de uma equação Spalart-Allmaras padrão e um modelo Spalart-Allmaras modificado através da alteração da escala de comprimento turbulenta. Sabe-se da literatura que no caso de jatos livres o modelo Spalart-Allmaras não consegue predizer adequadamente os resultados observados experimentalmente. Os resultados das simulações foram comparados com experimentos da literatura através de perfis de velocidade e da taxa de expansão do jato. Como esperado, os resultados obtidos das simulações utilizando o modelo Spalart-Allmaras padrão foram considerados insatisfatórios, porém o modelo Spalart-Allmaras modificado teve uma melhor concordância com os resultados experimentais. Além disso, o modelo Spalart-Allmaras modificado foi usado para simular os casos do escoamento sobre uma placa plana sem gradiente de pressão e o escoamento em um degrau com separação e gradiente adverso de pressão. Quando comparados com resultados experimentais da literatura e com resultados obtidos usando o modelo padrão, os resultados do modelo modificado obtidos para ambos os casos foram muito satisfatórios, concluindo-se que a modificação da escala de comprimento permite obter uma maior generalidade para o modelo Spalart-Allmaras. / Simulations of a plane and incompressible free jet using the standard Spalart-Allmaras model and a Spalart-Allmaras model modified by changing the turbulent length scale were carried out. It is known from literature that, in the case of the free jet, the Spalart-Allmaras model fails to adequately predict the experimentally observed results. The results of our simulations were compared with published experiments using the velocity profiles and the jet spreading rate. As expected, the results of simulations using the standard Spalart-Allmaras model were considered unsatisfactory while the modified Spalart-Allmaras model had a better agreement with the experimental results. Furthermore, the modified Spalart-Allmaras model was used to simulate the cases of flow over a flat plate with no pressure gradient and flow through a backward facing step with separation and adverse pressure gradient. When compared with experimental results from the literature and with results obtained using the standard model, the results of the modified model for both cases were very satisfactory, allowing the conclusion that the change in the length scale provided a greater generality for the Spalart-Allmaras model.

A two-dimensional thin-layer Navier-Stokes cascade flow solver for turbomachinery is developed. A second-order finite-difference scheme and a second and fourth-difference dissipation scheme are used. Periodic and non-reflecting inlet and outlet boundary conditions are implemented into the approximate-factorization numerical method. Turbulence is modeled through the one-equation Spalart-Allmaras model. A two-dimensional turbomachinery cascade structured grid generator is developed to produce six-block H-type grids. The validity of this work is tested in various ways. A grid convergence study is performed showing the effect of grid density. The non-reflecting inlet and outlet boundary conditions are tested for boundary placement influence. Comparisons of the flow solver numerical results are performed against experimental results. A Mach number sweep and angle of attack sweep are performed on two similar transonic turbine cascades.

A two-dimensional thin-layer Navier-Stokes cascade flow solver for turbomachinery is developed. A second-order finite-difference scheme and a second and fourth-difference dissipation scheme are used. Periodic and non-reflecting inlet and outlet boundary conditions are implemented into the approximate-factorization numerical method. Turbulence is modeled through the one-equation Spalart-Allmaras model. A two-dimensional turbomachinery cascade structured grid generator is developed to produce six-block H-type grids. The validity of this work is tested in various ways. A grid convergence study is performed showing the effect of grid density. The non-reflecting inlet and outlet boundary conditions are tested for boundary placement influence. Comparisons of the flow solver numerical results are performed against experimental results. A Mach number sweep and angle of attack sweep are performed on two similar transonic turbine cascades.

In this thesis study, Detached Eddy Simulation turbulence model is studied in two dimension mainly for flow over single element airfoils in high Reynolds numbers to gain experience with model before applying it to a three dimensional simulations. For this aim, Spalart-Allmaras and standard DES ,DES97, turbulence models are implemented to parallel, viscous, hybrid grid flow solver. The flow solver ,Set2d, is written in FORTRAN language. The Navier-Stokes equations are discretized by first order accurately cell centered finite volume method and solved explicitly by using Runge-Kutta dual time integration technique. Inviscid fluxes are computed using Roe flux difference splitting method. The numerical simulations are performed in parallel environment using domain decomposition and PVM library routines for inter-process communications. To take into account the effect of unsteadyness after the convergence is ensured by local time stepping technique for four order magnitude drop in density residual, global time stepping is applied for 20000 iterations. The solution algorithm is validated aganist the numerical and experimental studies for single element airfoils in subsonic and transonic flows. It is seen that Spalart-Allmaras and DES97 turbulence models give the same results in the non-seperated flows. Grey area is investigated by changing $C_{DES}$ coefficient. Modeled Stress Depletion which cause reduction of eddy viscosity is observed.

An unstructured explicit, Reynolds averaged Navier-Stokes solver is developed to operate on inviscid flows, laminar flows and turbulent flows and one equation Spalart-Allmaras turbulence modeling is implemented to the solver. A finite volume formulation, which is cell-center based, is used for numerical discretization of Navier-Stokes equations in conservative form. This formulation is combined with one-step, explicit time marching upwind numerical scheme that is the first order accurate in space. Turbulent viscosity is calculated by using one equation Spalart-Allmaras turbulence transport equation. In order to increase the convergence of the solver local time stepping technique is applied. Eight test cases are used to validate the developed solver,for inviscid flows, laminar flows and turbulent flows. All flow regimes are tested on NACA-0012 airfoil. The results of NACA-0012 are compared with the numerical and experimental data.

A three-dimensional, parallel, finite volume solver which uses Roe&#039 / s upwind flux differencing scheme for spatial and Runge-Kutta explicit multistage time stepping scheme for temporal discretization on unstructured meshes is developed for the unsteady solution of external viscous flow around rotating bodies. The main aim of this study is to evaluate the aerodynamic dynamic stability derivative coefficients for rotating missile configurations. Arbitrary Lagrangian Eulerian (ALE) formulation is adapted to the solver for the simulation of the rotation of the body. Eigenvalues of the Euler equations in ALE form has been derived. Body rotation is simply performed by rotating the entire computational domain including the body of the projectile by means of rotation matrices. Spalart-Allmaras one-euqation turbulence model is implemented to the solver. The solver developed is first verified in 3-D for inviscid flow over two missile configurations. Then inviscid flow over a rotating missile is tested. Viscous flux computation algorithms and Spalarat-Allmaras turbulence model implementation are validated in 2-D by performing calculations for viscous flow over flat plate, NACA0012 airfoil and NLR 7301 airfoil with trailing edge flap. The ALE formulation is validated in 2-D on a rapidly pitching NACA0012 airfoil. Afterwards three-dimensional validation studies for viscous, laminar and turbulent flow calculations are performed on 3-D flat plate problem. At last, as a validation test case, unsteady laminar and turbulent viscous flow calculations over a spinning M910 projectile configuration are performed. Results are qualitatively in agreement with the analytical solutions, experimental measurements and previous studies for steady and unsteady flow calculations.

In this thesis study, implementation, numerical stability and convergence rate issues of turbulence modeling are explored. For this purpose, a one equation turbulence model, Spalart-Allmaras, and a two-equation turbulence model, SST k-w, are adapted to an explicit, cell centered, finite volume method based, structured / hybrid multi grid flow solver, SENSE2D, developed at TUBITAK-SAGE. Governing equations for both the flow and the turbulence are solved in a loosely coupled manner, however, each set of equations are solved using a coupled, semi-implicit solution algorithm. In multigrid solutions, the semi-implicit solution algorithm and the turbulence model equations are employed only in the finest level grid. As a result, stable and convergent numerical solutions are obtained. In order to validate the turbulence models and the semi-implicit solution algorithm implemented, turbulent flow solutions over a flat plate, RAE2822 airfoil and NLR7301 multi element airfoil are performed. The results are compared with the experimental data and the numerical results of the commercial CFD package FLUENT. It is shown that the numerical results obtained by SENSE2D are in good agreement with the experimental data and the FLUENT results. In addition to the turbulence modeling studies, convergence rate studies are also performed by multigrid and semi-implicit solution methods. It is shown that, the convergence rates of the semi-implicit solutions are increased about 5 times for single grid and 35% for multigrid solutions in comparison to the explicit solutions.

The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.