The objective of this research is to develop an efficient and accurate methodology to resolve flow non-linearity of fluid-structural interaction. To achieve this purpose, a numerical strategy to apply the detached-eddy simulation (DES) with a fully coupled fluid-structural interaction model is established for the first time. The following novel numerical algorithms are also created: a general sub-domain boundary mapping procedure for parallel computation to reduce wall clock simulation time, an efficient and low diffusion E-CUSP (LDE) scheme used as a Riemann solver to resolve discontinuities with minimal numerical dissipation, and an implicit high order accuracy weighted essentially non-oscillatory (WENO) scheme to capture shock waves. The Detached-Eddy Simulation is based on the model proposed by Spalart in 1997. Near solid walls within wall boundary layers, the Reynolds averaged Navier-Stokes (RANS) equations are solved. Outside of the wall boundary layers, the 3D filtered compressible Navier-Stokes equations are solved based on large eddy simulation(LES). The Spalart-Allmaras one equation turbulence model is solved to provide the Reynolds stresses in the RANS region and the subgrid scale stresses in the LES region. An improved 5th order finite differencing weighted essentially non-oscillatory (WENO) scheme with an optimized epsilon value is employed for the inviscid fluxes. The new LDE scheme used with the WENO scheme is able to capture crisp shock profiles and exact contact surfaces. A set of fully conservative 4th order finite central differencing schemes are used for the viscous terms. The 3D Navier-Stokes equations are discretized based on a conservative finite differencing scheme, which is implemented by shifting the solution points half grid interval in each direction on the computational domain. The solution points are hence located in the center of the grid cells in the computational domain (not physical domain). This makes it possible to use the same code structure as a 2nd order finite volume method. A finite differencing high order WENO scheme is used since a finite differencing WENO scheme is much more efficient than a finite volume WENO scheme. The unfactored line Gauss-Seidel relaxation iteration is employed for time marching. For the time accurate unsteady simulation, the temporal terms are discretized using the 2nd order accuracy backward differencing. A pseudo temporal term is introduced for the unsteady calculation following Jameson's method. Within each physical time step, the solution is iterated until converged based on pseudo time step. A general sub-domain boundary mapping procedure is developed for arbitrary topology multi-block structured grids with grid points matched on sub-domain boundaries. The interface of two adjacent blocks is uniquely defined according to each local mesh index system (MIS) which is specified independently. A pack/unpack procedure based on the definition of the interface is developed to exchange the data in a 1D array to minimize data communication. A secure send/receive procedure is employed to remove the possibility of blocked communication and achieve optimum parallel computation efficiency. Two terms, "Order" and "Orientation", are introduced as the logics defining the relationship of adjacent blocks. The domain partitioning treatment of the implicit matrices is to simply discard the corner matrices so that the implicit Gauss-Seidel iteration can be implemented within each subdomain. This general sub-domain boundary mapping procedure is demonstrated to have high scalability. Extensive numerical experiments are conducted to test the performance of the numerical algorithms. The LDE scheme is compared with the Roe scheme for their behavior with RANS simulation. Both the LDE and the Roe scheme can use high CFL numbers and achieve high convergence rates for the algebraic Baldwin-Lomax turbulence model. For the Spalart-Allmaras one equation turbulence model, the extra equation changes the Jacobian of the Roe scheme and weakens the diagonal dominance. It reduces the maximum CFL number permitted by the Roe scheme and hence decreases the convergence rate. The LDE scheme is only slightly affected by the extra equation and maintains high CFL number and convergence rate. The high stability and convergence rate using the Spalart-Allmaras one equation turbulence model is important since the DES uses the same transport equation for the turbulence stresses closure. The RANS simulation with the Spalart-Allmaras one equation turbulence model is the foundation for DES and is hence validated with other transonic flows including a 2D subsonic flat plate turbulent boundary layer, 2D transonic inlet-diffuser, 2D RAE2822 airfoil, 3D ONERA M6 wing, and a 3D transonic duct with shock boundary layer interaction. The predicted results agree very well with the experiments. The RANS code is then further used to study the slot size effect of a co-flow jet (CFJ) airfoil. The DES solver with fully coupled fluid-structural interaction methodology is validated with vortex induced vibration of a cylinder and a transonic forced pitching airfoil. For the cylinder, the laminar Navier-Stokes equations are solved due to the low Reynolds number. The 3D effects are observed in both stationary and oscillating cylinder simulation because of the flow separations behind the cylinder. For the transonic forced pitching airfoil DES computation, there is no flow separation in the flow field. The DES results agree well with the RANS results. These two cases indicate that the DES is more effective on predicting flow separation. The DES code is used to simulate the limited cycle oscillation of NLR7301 airfoil. For the cases computed in this research, the predicted LCO frequency, amplitudes, averaged lift and moment, all agree excellently with the experiment. The solutions appear to have bifurcation and are dependent on the initial perturbation. The developed methodology is able to capture the LCO with very small amplitudes measured in the experiment. This is attributed to the high order low diffusion schemes, fully coupled FSI model, and the turbulence model used. This research appears to be the first time that a numerical simulation of LCO matches the experiment. The DES code is also used to simulate the CFJ airfoil jet mixing at high angle of attack. In conclusion, the numerical strategy of the high order DES with fully coupled FSI model and parallel computing developed in this research is demonstrated to have high accuracy, robustness, and efficiency. Future work to further maturate the methodology is suggested.

In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems. The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case. The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach. Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks. Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability. Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated.

A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic and elliptic systems of conservation laws on body- fitted multi-block mesh. The spatial discretization of the hyperbolic (inviscid) terms is based on a hybrid solution reconstruction procedure that combines an unlimited high-order k-exact least-squares reconstruction technique following from a fixed central stencil with a monotonicity preserving limited piecewise linear reconstruction algorithm. The limited reconstruction is applied to computational cells with under-resolved solution content and the unlimited k-exact reconstruction procedure is used for cells in which the solution is fully resolved. Switching in the hybrid procedure is determined by a solution smoothness indicator. The hybrid approach avoids the complexity associated with other ENO schemes that require reconstruction on multiple stencils and therefore, would seem very well suited for extension to unstructured meshes. The high-order elliptic (viscous) fluxes are computed based on a k-order accurate average gradient derived from a (k+1)-order accurate reconstruction. A novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady refinement of the AMR mesh. The predictive capabilities of the proposed high-order AMR scheme are demonstrated for the Euler and Navier-Stokes equations governing two-dimensional compressible gaseous flows as well as for advection-diffusion problems characterized by the full range of Peclet numbers, Pe. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown for a range of problems. Moreover, the ability to perform mesh refinement in regions of smooth but under-resolved and/or non-smooth solution content to achieve the desired resolution is also demonstrated.