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Vortex flows caused by a swept wing are described in some detail. The thin-shear-layer Navier-Stokes equations with an algebraic eddy-viscosity turbulence model are solved. An implicit, factored numerical scheme is used. On the upper surface of the wing, flow is primarily along the free-stream and spanwise direction, respectively, at small and large angles of attack.
The numerical method developed by Schiff and Sturek (1980) on the basis of the thin-layer parabolized Navier-Stokes equations of Schiff and Steger (1980) is extended to the case of turbulent supersonic flows on pointed bodies at high angles of attack. The governing equations, the numerical scheme, and modifications to the algebraic eddy-viscosity turbulence model are described; and results for three cones and one ogive-cylinder body (obtained using grids of 50 nonuniformly spaced points in the radial direction between the body and the outer boundary) are presented graphically and compared with published experimental data. The grids employed are found to provide sufficient spatial resolution of the leeward-side vortices; when combined with the modified turbulence model, they are shown to permit accurate treatment of flows with large regions of crossflow separation.
An upwind-biased implicit approximate factorization Navier-Stokes algorithm is applied to a variety of steady transonic airfoil cases, using the NACA 0012, RAE 2822, and Jones supercritical airfoils. The thin-layer form of the compressible Navier-Stokes equations is used. Both the CYBER 205 and CRAY 2 supercomputers are utilized, with average computational speeds of about 18 and 16 microsec/gridpoint/iteration, respectively. Lift curves, drag polars, and variations in drag coefficient with Mach number are determined for the NACA 0012 and Jones supercritical airfoils. Also, several cases are computed for comparison with experiment. The effect of grid density and grid extent on a typical turbulent airfoil solution is shown. An algebraic eddy-viscosity turbulence model is used for all of the computations.
Essential aspects are presented for computing flow past an oblique wing with the thin-layer Navier-Stokes equations. A new method is developed for generating a grid system around a realistic wing. This method utilizes a series of conformal transformations. The thin-shear-layer approximation and an algebraic eddy-viscosity turbulence model are used to simplify the Reynolds-averaged Navier-Stokes equations. An implicit, factored numerical scheme and the concept of pencil data structure are utilized. For the first time, some flow fields caused by the oblique wing in a supersonic free stream are discussed, emphasizing the separated vortex flows associated with such a wing.
The unsteady, compressible Reynolds-averaged thin-layer Navier-Stokes equations are used to solve for symmetric and asymmetric vortical flows around slender pointed bodies of revolution. The modified Baldwin and Lomax algebraic two-layer turbulent model is used for the eddy viscosity calculation in prescribed turbulent regions. The implicit, upwind flux-difference splitting finite-volume scheme is used to obtain the solutions. Computational results are presented for a low-speed symmetrical vortical flow around a 3.5-caliber tangent-ogive cylinder for fully laminar flow. Computed results are presented for a high-speed asymmetric vortex-shedding flow around a cone at 3.286 relative incidence, using a fine computational grid. The asymmetric vortex-shedding flow has been produced through a small asymmetric transitional perturbation in the cross-flow plane.
The physical aspects governing accurate numerical simulation of turbulent flows having large regions of crossflow separation are re-examined. Time-accurate, three-dimensional fine-grid Navier-Stokes solutions were obtained for turbulent subsonic and supersonic flows over a slender ogive-cylinder body of revolution at large angles of attack. These flowfields are complex and contain regions of crossflow separation and an organized leeward-side vortex structure. An algebraic eddy-viscosity turbulence model has been modified to correctly account for the effects of the vortices on the underlying viscous layers. The numerical results show that the vortical flow structure of both subsonic and supersonic high incidence flows are similar, and can be accurately treated by the turbulence model. As the angle of attack is increased, the effectiveness of the model increases since the influence of the vortices on the underlying boundary layer decreases.
A comparative study was made with four different codes for solving the compressible Navier-Stokes equations using three different test problems. The first of these cases was hypersonic flow through the P8 inlet, which represents inlet configurations typical of a hypersonic airbreathing vehicle. The free-stream Mach number in this case was 7.4. This 2-D inlet was designed to provide an internal compression ratio of 8. Initial calculations were made using two state-of-the-art finite-volume upwind codes, CFL3D and USA-PG2, as well as NASCRIN, a code which uses the unsplit finite-difference technique of MacCormack. All of these codes used the same algebraic eddy-viscosity turbulence model. In the experiment, the cowl lip was slightly blunted; however, for the computations, a sharp cowl leading edge was used to simplify the construction of the grid. The second test problem was the supersonic (Mach 3.0) flow in a three-dimensional corner formed by the intersection of two wedges with equal wedge angles of 9.48 degrees. The flow in such a corner is representative of the flow in the corners of a scramjet inlet. Calculations were made for both laminar and turbulent flow and compared with experimental data. The three-dimensional versions of the three codes used for the inlet study (CFL3D, USA-PG3, and SCRAMIN, respectively) were used for this case. For the laminar corner flow, a fourth code, LAURA, which also uses recently-developed upwind technology, was also utilized. The final test case is the two-dimensional hypersonic flow over a compression ramp. The flow is laminar with a free-stream Mach number of 14.1. In the experiment, the ramp angle was varied to change the strength of the ramp shock and the extent of the viscous-inviscid interaction. Calculations were made for the 24-degree ramp configuration which produces a large separated-flow region that extends upstream of the corner.
A finite difference, three dimensional incompressible Navier-Stokes formulation to calculate the flow through turbopump components is utilized. The solution method is based on the pseudo compressibility approach and uses an implicit upwind differencing scheme together with the Gauss-Seidel line relaxation method. Both steady and unsteady flow calculations can be performed using the current algorithm. Here, equations are solved in steadily rotating reference frames by using the steady state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using an algebraic mixing-length turbulence model. Numerical results are compared with experimental measurements and a good agreement is found between the two.
A finite-difference, three-dimensional incompressible Navier-Stokes formulation to calculate the flow through turbopump components is utilized. The solution method is based on the pseudocompressibility approach and uses an implicit-upwind differencing scheme together with the Gauss-Seidel line relaxation method. Both steady and unsteady flow calculations can be performed using the current algorithm. In this work, the equations are solved in steadily rotating reference frames by using the steady-state formulation in order to simulate the flow through a turbopump inducer. Eddy viscosity is computed by using an algebraic mixing-length turbulence model. Numerical results are compared with experimental measurements and a good agreement is found between the two. Included in the appendix is a paper on incompressible viscous flow through artificial heart devices with moving boundaries. Time-accurate calculations, such as impeller and diffusor interaction, will be reported in future work.
Procedures and methods for veri.cation of coding algebra and for validations of models and calculations used in the aerospace computational fluid dynamics (CFD) community would be ef.cacious if used by the glacier dynamics modeling community. This paper presents some of those methods, and how they might be applied to uncertainty management supporting code veri.cation and model validation for glacier dynamics. The similarities and differences between their use in CFD analysis and the proposed application of these methods to glacier modeling are discussed. After establishing sources of uncertainty and methods for code veri.cation, the paper looks at a representative sampling of veri.cation and validation efforts that are underway in the glacier modeling community, and establishes a context for these within an overall solution quality assessment. Finally, a vision of a new information architecture and interactive scienti.c interface is introduced and advocated.
Structural dynamics is one of the critical disciplines for the successful design, development, and testing of space launch vehicles. It is applied from the smallest component (turbine blades), all the way to the entire vehicle, and has to be calculated for every phase of a mission, from ascent and orbit to landing. Successful application of structural dynamics requires extensive knowledge of Fourier techniques, linear algebra, random variables, finite element modeling, and essentials of SDOF and MDOF vibration theory. Working knowledge of fluid dynamics, statistics, and data analysis also extremely useful.
A comparison is made of the performance of two locally nonequilibrium and one equilibrium algebraic stress closures in calculating combustor flows. Effects of four different pressure-strain models on these closure models are also analyzed. The results show that the pressure-strain models have a much greater influence on the calculated mean velocity and turbulence field than the algebraic stress closures, and that the best mean strain model for the pressure-strain terms is that proposed by Launder, Reece and Rodi (1975). However, the equilibrium algebraic stress closure with the Rotta return-to-isotropy model (1951) for the pressure-strain terms gives as good a correlation with measurements as when the Launder et al. mean strain model is included in the pressure-strain model. Finally, comparison of the calculations with the standard k-epsilon closure results show that the algebraic stress closures are better suited for simple turbulent flow calculations.
The considered investigation is concerned with periodic solutions in the context of a forced, dissipative, barotropic spectral model truncated to three complex coefficients with constant forcing on only the intermediate scale. It is found that determining a periodic solution of this three-coefficient model also reduces to finding the algebraic roots of a real polynomial. In the derivation of this polynomial, a class of hydrodynamic spectral systems is described for which a periodic solution might be similarly specified. The existence of periodic solutions of the three-coefficient model is controlled by the roots of the stability polynomial of the basic stationary solution, which represents the simplest response to the constant forcing. When the forcing exceeds a critical value, the basic solution becomes unstable. Owing to the nature of the roots of the stability polynomial at critical forcing, bifurcation theory guarantees the existence of a periodic solution.
A numerical model of the Ares I upper stage main propulsion system is formulated based on first principles. Equation's are written as non-linear ordinary differential equations. The GASP fortran code is used to compute thermophysical properties of the working fluids. Complicated algebraic constraints are numerically solved. The model is implemented in Simulink and provides a rudimentary simulation of the time history of important pressures and temperatures during re-pressurization, boost and upper stage firing. The model is validated against an existing reliable code, and typical results are shown.
A novel algorithm is presented for adaptive learning of an unknown function that separates two regions of a domain.In the context of reliability analysis these two regions represent the failure domain, where a set of constraints or requirements are violated, and a safe domain where they are satisfied. The Limit State Function (LSF) separates these two regions. Evaluating the constraints for a given parameter point requires the evaluation of a computational model that may well be expensive. For this reason we wish to construct a meta-model that can estimate the LSFas accurately as possible, using only a limited amount of training data. This work presents an adaptive strategy employing a Support Vector Machine (SVM) as a meta-model to provide a semi-algebraic approximation of the LSF.We describe an optimization process that is used to select informative parameter points to add to training data at each iteration to improve the accuracy of this approximation. A formulation is introduced for bounding the predictions of the meta-model; in this way we seek to incorporate this aspect of Gaussian Process Models (GPMs) within anSVM meta-model. Finally, we apply our algorithm to two benchmark test cases, demonstrating performance that is comparable with, if not superior, to a standard technique for reliability analysis that employs GPMs
We describe an algebra of Edge-Valued Decision Diagrams (EVMDDs) to encode arithmetic functions and its implementation in a model checking library. We provide efficient algorithms for manipulating EVMDDs and review the theoretical time complexity of these algorithms for all basic arithmetic and relational operators. We also demonstrate that the time complexity of the generic recursive algorithm for applying a binary operator on EVMDDs is no worse than that of Multi- Terminal Decision Diagrams. We have implemented a new symbolic model checker with the intention to represent in one formalism the best techniques available at the moment across a spectrum of existing tools. Compared to the CUDD package, our tool is several orders of magnitude faster
We describe an algebra of Edge-Valued Decision Diagrams (EVMDDs) to encode arithmetic functions and its implementation in a model checking library along with state-of-the-art algorithms for building the transition relation and the state space of discrete state systems. We provide efficient algorithms for manipulating EVMDDs and give upper bounds of the theoretical time complexity of these algorithms for all basic arithmetic and relational operators. We also demonstrate that the time complexity of the generic recursive algorithm for applying a binary operator on EVMDDs is no worse than that of Multi-Terminal Decision Diagrams. We have implemented a new symbolic model checker with the intention to represent in one formalism the best techniques available at the moment across a spectrum of existing tools: EVMDDs for encoding arithmetic expressions, identity-reduced MDDs for representing the transition relation, and the saturation algorithm for reachability analysis. We compare our new symbolic model checking EVMDD library with the widely used CUDD package and show that, in many cases, our tool is several orders of magnitude faster than CUDD.
Direct numerical simulations (DNS) of flat-plate, zero-pressure-gradient turbulent boundary layers are presented for nominal freestream Mach numbers of 11 and 14 and a highly cooled wall (wall-to-recovery temperature of approximately 0.2). The flow conditions of the DNS are representative of the experimental data for a Mach 11.1 turbulent boundary layer on a flat plate that was tested at Calspan–University of Buffalo Research Center (CUBRC) and the operational conditions of the AEDC Hypervelocity Tunnel No. 9 at Mach 14. The wall shear stress and turbulent heat flux predicted by DNS show good comparisons with those measured at CUBRC and those modeled with a Baldwin-Lomax model. The generated DNS database extends the range of Reynolds numbers from those typical of previous DNS studies for hypersonic boundary-layer flows and confirms the validity of Morkovin’s hypothesis within the relatively broader range of Reynolds numbers considered in this study. A priori assessment of turbulent heat flux models with DNS data shows that the algebraic energy flux (AEF) formulation gives better predictions of the streamwise turbulent heat flux than the commonly used constant turbulent Prandtl number model while both models give similarly good predictions of the wall-normal turbulent heat flux. A posteriori assessment shows that the AEF model improves the prediction of the temperature profiles while keeping a good prediction of the velocity field.