Klaus A. Hoffmann

Detached-Eddy Simulation With Compressibility Corrections Applied to a Supersonic Axisymmetric Base Flow

Detached-eddy simulation is applied to an axisymmetric base flow at supersonic conditions. Detached-eddy simulation is a hybrid approach to modeling turbulence that combines the best features of the Reynolds-averaged Navier-Stokes and large-eddy simulation approaches. In the Reynolds-averaged mode, the model is currently based on either the Spalart-Allmaras turbulence model or Menter’s shear stress transport model; in the largeeddy simulation mode, it is based on the Smagorinski subgrid scale model. The intended application of detached-eddy simulation is the treatment of massively separated, highReynolds number flows over complex configurations (entire aircraft, automobiles, etc.). Because of the intented future application of the methods to complex configurations, Cobalt, an unstructured grid Navier-Stokes solver, is used. The current work incorporates compressible shear layer corrections in both the Spalart-Allmaras and shear stress transport-based detached-eddy simulation models. The effect of these corrections on both detached-eddy simulation and Reynolds-averaged Navier-Stokes models is examined, and comparisons are made to the experiments of Herrin and Dutton. Solutions are obtained on several grids—both structured and unstructured—to test the sensitivity of the models and code to grid refinement and grid type. The results show that predictions of base flows using detached-eddy simulation compare very well with available experimental data, including turbulence quantities in the wake of the axisymmetric body. @DOI: 10.1115/1.1517572#

Numerical analysis of effects of leading-edge protuberances on aircraft wing performance

The effects of biologically inspired leading-edge protuberances on aircraft wings were investigated by using a numerical scheme. Simulations were performed at a Reynolds number of 5:7 10 on wings with leading-edge sinusoidal protuberances and baseline configuration. All wings had the same cross section ofNACA2412. In all cases of the modified wings, a decrease in lift and an increase in drag at low angles of attack were observed. However, at high angles of attack ( 16 deg), the lift of themodifiedwingswas up to 48%greater than that of thebaselinewing, with 40% less drag or no drag penalty. The amplitude of protuberances significantly affects wing performance. Although the maximum lift generated by modified wings was lower than baseline, protuberances along the leading edge of the wing proved to have a profound advantage in obtaining higher lift at high angles of attack.

Numerical Simulations of Magnetic Flow Control in Hypersonic Chemically Reacting Flows

Hypersonic flows over blunt bodies subject to magnetic fields are numerically investigated. The magnetogasdynamic equations in the high magnetic Reynolds number formulation form an eight-equation system, with density, momentum, magnetic field, and total energy as unknowns. In the low magnetic Reynolds number approximation, the magnetic field induction is ignored, which leads to a five-equation system, where the magnetic interaction is represented by source terms in the momentum and energy equations. A four-stage modified Runge-Kutta scheme with the Davis-Yee symmetric total variation diminishing model as a postprocessing stage is used to solve the magnetogasdynamic equations. High-temperature effects are simulated by equilibrium and nonequilibrium chemistry models. The equilibrium model computes thermodynamic properties by interpolation from experimental data. The nonequilibrium model is a 1-temperature, 5-species, 17-reaction model solved by an implicit flux-vector splitting scheme. A loosely coupled approach is implemented to communicate between the magnetogasdynamic equations and the chemistry models

Numerical investigation of high Reynolds number flows over square and circular cylinders

Numerical solutions of flows at high Reynolds numbers are investigated by detached-eddy simulation (DES). Two cylinders in crossflow are selected as the test cases; flow around a circular cylinder is simulated at Reynolds numbers of 1.4 × 10 5 and 3.6 × 10 6 and simulation for a square cylinder is performed at a Reynolds number of 2.2 × 10 4 . These simple geometries produce complex flow phenomena such as recirculation, vortex shedding, and unsteady turbulent separation, which are very common flows associated with complex geometries. However, most numerical simulations have been performed at low Reynolds numbers, and only a few are reported at high Reynolds numbers ranges. DES with the Spalart‐Allmaras turbulence model is used for turbulent treatment. It functions as a Reynolds averaged approach in the near-wall region and transfers to large eddy simulation (LES) far from the wall. This procedure requires fewer grid points compared to LES. To assess the quality of solutions, the results are evaluated by comparison with experimental data and other numerical results. In addition, laminar solutions and trip functions are also investigated in the circular cylinder cases. Even though fewer grid points are used, most of the results compare well with experimental data and other numerical solutions.

Development of a modified Runge-Kutta scheme with TVD limiters for the ideal two-dimensional MHD equations

A fourth-order modified Runge-Kutta scheme augmented with TVD models have been developed to solve the ideal two-dimensional MHD equations. The numerical scheme is applied to supersonic flow within a channel including compression and expansion corners in the current investigation. The flowfield includes oblique shocks, expansion waves, shock reflections, shock/expansion interactions, and secondary wave patterns due to magnetic field. The solutions are compared to analytical solutions when available. The numerical scheme is shown to be accurate with ability to capture physical phenomena occurring in the flowfield without any or with minimum oscillations in the solution.

Development of a modified Runge-Kutta scheme with TVD limiters for the ideal 1-D MHD equations

A fourth-order modified Runge-Kutta scheme augmented with TVD models have been developed to solve the one-dimensional MHD equations. The numerical scheme is applied to the benchmark magnetic shock tube problem. Several limiters for each TVD model have been extensively investigated, and the solutions are compared to each other and analytical solutions when available. The numerical scheme is proven to be accurate with ability to capture strong shocks, thin contact surfaces, and other physical phenomena occurring in the flowfield without any or with minimum oscillations in the solution.

Numerical Solution of the Ideal Magnetohydrodynamic Equations for a Supersonic Channel Flow

A fourth-order modified Runge-Kutta scheme augmented with total variation diminishing (TVD) models have been developed to solve the two-dimensional magnetohydrodynamic equations for supersonic flows. Several limiters for each TVD model have been extensively investigated. The numerical scheme is proven to be accurate with the ability to capture strong shocks, expansion waves, reflective waves, and other physical phenomena occurring in the flowfield without any or with minimum oscillations in the solution. The numerical scheme has been formulated in generalized coordinates and is applied to a supersonic flow within a channel including compression and expansion corners. The effect of magnetic field on the flowfield has been investigated with the application of several externally applied magnetic fields. The solutions indicate changes in the wave strength and the formation of secondary waves.

Numerical Solutions of the Eight-Wave Structure Ideal MHD Equations by Modified Runge-Kutta Scheme with TVD

The one-dimensional eight-wave ideal MHD equations are used to investigate the accuracy of the proposed Runge-Kutta scheme with added TVD acting as a dissipation mechanism. This investigation is performed to establish the procedure for extension to multidimensio nal problems in MHD. A set of eigenvalues and a set of new associated eigenvectors have been developed. Three well known TVD models and several limiters for each model are investigated for their accuracy and robustness. Solutions are obtained for diffferent cases, each under the influence of a different magnetic field in order to identify the formation of wave patterns in the flow field. The numerical solutions are compared to each other and analytical solutions when available.

Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves

Abstract Improvements in the numerical algorithm for the dynamics of flows that involve discontinuities and broadband fluctuations simultaneously are proposed. These two flow features suggest numerical strategies of a paradoxical nature because the discontinuities demand dissipation, and the small-scale smooth features require the opposite. There may be several ways to approach such a complicated issue, but the natural choice is a numerical technique that can adjust adaptively with flow regimes. The weighted essentially non-oscillatory (WENO) scheme may be this choice. However, there are two sources of dissipation associated with the WENO procedure: the upwind optimal stencil and the nonlinear adaption mechanism. The current work suggests a robust and comprehensive treatment for the minimization of dissipation error from these two sources. The optimization technique, which is guided by restriction on the linear optimal weights derived from stability and consistency requirement, is used to delay the dissipation of the upwind optimal stencil to those wavenumbers for which the dispersion error is large. The parallel advantage of this technique is the improvement of the dispersion property. Nevertheless, optimization decreases the formal order of accuracy of the optimal stencil from fifth order to third order. This loss of accuracy is derived by Taylor series expansion. Using Taylor-series expansion and WENO procedure, the third-order accuracy is verified in the smooth region, except at the critical point of order two, where the order of accuracy reduces to at least second order. This possible loss of accuracy at the second-order critical point is restored in an attempt to reduce the dissipation induced by the nonlinear adaptive weights. Modification of the nonlinear weights to reduce the dissipation is introduced by redefining them with an additional smoothness indicator. Other suggestions to minimize the dissipation of the nonlinear weights are also reviewed. The numerical approximation of the spatial derivative is performed by means of a conservative and consistent finite difference method based on monotone local Lax–Friedrichs Riemann solver. The resulting scheme is then integrated by the optimal third-order TVD Runge–Kutta method to ensure the nonlinear stability of the overall numerical method. A variety of benchmark problems, ranging from non-broadband to broadband, are solved using the proposed schemes and compared with the existing ones. Most test problems are validated against exact or reference data. The numerical results with bandwidth optimization and modification of the nonlinear weights are consistently superior.

Numerical solutions of ideal MHD equations for a symmetric blunt body at hypersonic speeds

The eight-wave ideal magnetohydrodynamics (MHD) equations are extended to two-dimensional problems. Several developments are reported in this study. (1) The two-dimensional governing equations of ideal MHD equations are derived in the generalized coordinate system, (2) a set of eigenvectors associated with the system of equations are developed for the implementation of the numerical scheme, (3) solutions are obtained for a blunt body at hypersonic speed, and (4) the effect of the magnetic field on the flowfield structure is investigated. The numerical scheme used is the modified Runge-Kutta scheme with the TVD model augmented as a postprocessor approach at each time level.