Frederick G. Blottner

Accurate Navier-Stokes results for the hypersonic flow over a spherical nosetip

The unsteady, thin-layer Navier-Stokes equations for a perfect gas are solved with a linearized block, alternating direction implicit finite-difference solution procedure. Solution errors due to numerical dissipation added to the governing equations are evaluated. Errors in the numerical predictions on three-differe nt grids are determined where Richardson extrapolation is used to estimate the exact solution. Accurate computational results are tabulated for the hypersonic laminar flow over a spherical body, which can be used as a benchmark test case. Predictions obtained from the code are in good agreement with inviscid numerical results and experimental data.

Assessment of One- and Two-Equation Turbulence Models for Hypersonic Transitional Flows

Hypersonic transitional flows over a flat plate and a sharp cone are studied using four turbulence models: the one-equation eddy viscosity transport model of SpalartAllmaras, a low Reynolds number k-ε model, the Menter k-ωmodel, and the Wilcox k-ωmodel. A framework is presented for the assessment of turbulence models that includes documentation procedures, solution accuracy, model sensitivity, and model validation. The accuracy of the simulations is addressed, and the sensitivities of the models to grid refinement, freestream turbulence levels, and wall y+ spacing are presented. The flat plate skin friction results are compared to the well-established laminar and turbulent correlations of Van Driest. Correlations for the sharp cone are discussed in detail. These correlations, along with recent experimental data, are used to judge the validity of the simulation results for skin friction and surface heating on the sharp cone. The Spalart-Allmaras performs the best with regards to model sensitivity and model accuracy, while the Menter k-ω model also performs well for these zero pressure gradient boundary layer flows. Nomenclature Cf skin friction coefficient Cμ turbulence modeling constant (= 0.09) f generic solution variable H total enthalpy, J/kg h specific enthalpy, J/kg k specific turbulent kinetic energy, m2/s2  1 † Senior Member of Technical Staff, MS 0835, E-mail: cjroy@sandia.gov, Member AIAA ‡ Distinguished Member of Technical Staff, MS 0825, E-mail: fgblott@sandia.gov, AIAA Fellow * Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. This paper is declared a work of the U. S. Government and is not subject to copyright protection in the United States. M Mach number m Van Driest correlation parameter Pr Prandtl number (= 0.71) PrT turbulent Prandtl number (= 1.0) p pressure, N/m2 q heat flux, W/m2 Re Reynolds number rf recovery factor St Stanton number s surface distance, m T temperature, K Tu freestream turbulence intensity percent t time, s u axial velocity, m/s V velocity magnitude, m/s x axial coordinate, m y wall normal direction, m y+ wall normal direction in wall coordinates ratio of specific heats specific dissipation rate, m2/s3 θ momentum thickness, m absolute viscosity, Ns/m2 kinematic viscosity, m2/s density, kg/m3 τ wall shear stress, N/m2 specific turbulent frequency, 1/s Subscripts aw adiabatic wall value c cone value e edge condition fp flat plate value k mesh level RE Richardson Extrapolated value t turbulent quantity w wall value ∞ freestream value γ ε

Development of a One-Equation Transition/Turbulence Model

This paper reports on the development of a unified one-equation model for the prediction of transitional and turbulent flows. An eddy viscosity - transport equation for non-turbulent fluctuation growth based on that proposed by Warren and Hassan (Journal of Aircraft, Vol. 35, No. 5) is combined with the Spalart-Allmaras one-equation model for turbulent fluctuation growth. Blending of the two equations is accomplished through a multidimensional intermittence function based on the work of Dhawan and Narasimha (Journal of Fluid Mechanics, Vol. 3, No. 4). The model predicts both the onset and extent of transition. Low-speed test cases include transitional flow over a flat plate, a single element airfoil, and a multi-element airfoil in landing configuration. High-speed test cases include transitional Mach 3.5 flow over a 5{degree} cone and Mach 6 flow over a flared-cone configuration. Results are compared with experimental data, and the spatial accuracy of selected predictions is analyzed.

Review and Assessment of Turbulence Models for Hypersonic Flows: 2D/Axisymmetric Cases

Turbulence modeling remains a major source of uncertainty in the computational prediction of aerodynamic forces and heating for hypersonic vehicles. The first goal of this paper is to update the previous comprehensive review published in 1991 by Settles and Dodson (G. S. Settles and L. J. Dodson, “Hypersonic Shock/Boundary-Layer Interaction Database,” NASA CR 177577, April 1991). In their review, Settles and Dodson developed a methodology for assessing experiments appropriate for turbulence model validation and critically surveyed the existing hypersonic experimental database. We limit the scope of our current effort by considering only twodimensional/axisymmetric flows in the hypersonic speed regime where calorically perfect gas models are appropriate. We extend the prior database of recommended hypersonic experiments by adding three new cases. The first two cases, the flat plate/cylinder and the sharp cone, are canonical test cases which are amenable to theory-based correlations, and these correlations are discussed in detail. The third case added is the two-dimensional shock impinging on a flat plate boundary layer. The second goal is to review and assess the validation usage of various turbulence models on the existing experimental database. Here we limit the scope to one- and two-equation turbulence models where integration to the wall is used (i.e., we omit studies involving wall functions). In order to preserve a models prior validation history, we omitted corrections to the standard turbulence models in cases where the impact of such corrections on low-speed flows had not been adequately addressed (either through a re-validation of the models on a wide range of low-speed test cases or theoretical arguments). A methodology for validating turbulence models is given, and turbulence model comparisons from various authors are compiled and presented in graphical form. Conclusions are drawn for those models which have been applied to a sufficiently wide range of two-dimensional/axisymmetric hypersonic flows, and recommendations for future experimental and modeling efforts are given.

Finite-difference solution of the incompressible three-dimensional boundary layer equations for a blunt body

Abstract The governing equations for a laminar flow are solved in terms of an orthogonal surface coordinate system. One of the coordinate is determined by the intersection with the body surface of meridional planes which pass through an axis containing the stagnation point. The other coordinate is obtained numerically from the orthogonality condition. The momentum equations have been written in a standard from which allows additional equations of this form to be added with a small modification of the computer code. This equation is replaced with a nonlinear finite-difference equation which is solved as an iterative solution of linear tridiagonal equations. The special form of the governing equations at the stagnation point and the plane of symmetry is determined and the solution of these equations is obtained to provide a unified code. Numerical solutions have been obtained for several special cases and compared to results of other authors. New results are presented for an ellipsoid ar angle of attack and an elliptic-paraboloid at zero incidence.

Bluff-Body Flow Simulations using Hybrid RANS/LES

The Detached Eddy Simulation (DES) and steady-state Reynolds-Averaged Navier-Stokes (RANS) turbulence modeling approaches are examined for the incompressible flow over a square cross-section cylinder at a Reynolds number of 21,400. A compressible flow code is used which employes a second-order Roe upwind spatial discretization. Efforts are made to assess the numerical accuracy of the DES predictions with regards to statistical convergence, iterative convergence, and temporal and spatial discretization error. Three-dimensional DES simulations compared well with two-dimensional DES simulations, suggesting that the dominant vortex shedding mechanism is effectively two-dimensional. The two-dimensional simulations are validated via comparison to experimental data for mean and RMS velocities as well as Reynolds stress in the cylinder wake. The steady-state RANS models significantly overpredict the size of the recirculation zone, thus underpredicting the drag coefficient relative to the experimental value. The DES model is found to give good agreement with the experimental velocity data in the wake, drag coefficient, and recirculation zone length.

A spatial marching technique for the inviscid blunt body problem

A technique has been developed for obtaining approximate solutions of the inviscid, hypersonic flow on a blunt body with a spatial marching scheme. The scheme introduces the Vigneron pressure gradient approximation into the momentum equation in the direction along the body surface. The resulting governing equations are hyperbolic. With a specified shock wave these equations are solved at the stagnation streamline with an iteration procedure and are solved in the downstream direction with a marching scheme. The complete Euler equations are solved with the numerical scheme when the flow is supersonic. A global iteration procedure is required to obtain the shock wave location. The approximate results from the spatial marching technique are compared with the complete solution of the Euler equations for flow over a sphere. The two results are shown to be in approximate agreement and the spatial marching technique provides useful engineering predictions while requiring considerably less computational time.

Memoirs: Obtaining the right solution to thermophysics problems

In many cases experiments can provide and are required to obtain needed information to engineering problems. However this paper is concerned with using numerical solution techniques to obtain the desired results where the governing equations provide an adequate simulation. With the development of improved numerical schemes for solving the governing equations of computational fluid dynamics and thermophysics and the development of larger and faster computers, significant improvement in prediction capabilities have evolved. The level and complexity of the governing equations required, the numerical solution scheme employed, and the availability of computer power (speed and memory size) have determined what problems are reasonable to solve at any given time. The emphasis in this paper is on the authors contributions in the development of an implicit finite-difference scheme for solving boundary layer flows and in modeling chemically reacting boundary layer flows. This work is used to illustrate the development of prediction capabilities with time. An approach to determine the accuracy or discretization error of the numerical solution and verification of numerical solutions is discussed. These issues have been a concern of the author since the initial boundary layer solutions were obtained.

A domain decomposition study of massively parallel computing in compressible gas dynamics

The appropriate utilization of massively parallel computers for solving the Navier-Stokes equations is investigated and determined from an engineering perspective. The issues investigated are: (1) Should strip or patch domain decomposition of the spatial mesh be used to reduce computer time? (2) How many computer nodes should be used for a problem with a given sized mesh to reduce computer time? (3) Is the convergence of the Navier-Stokes solution procedure (LU-SGS) adversely influenced by the domain decomposition approach? The results of the paper show that the present Navier-Stokes solution technique has good performance on a massively parallel computer for transient flow problems. For steady-state problems with a large number of mesh cells, the solution procedure will require significant computer time due to an increased number of iterations to achieve a converged solution. There is an optimum number of computer nodes to use for a problem with a given global mesh size.

Numerical solution of diffusion-convection equations☆

Abstract A finite-difference scheme is described for solving an ordinary differential equation with a thin diffusion region. The approach takes advantage of the change in character of the governing equation from the inner to the outer region where the convection dominates. The resulting difference scheme is pseudo-second-order accurate and when reasonably accurate results are desired, the stability restrictions are not important. The difference scheme is compared to other methods and is used to solve two example problems.