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Set of computer codes solves three-dimensional Navier-Stokes equations for flow over nonaxisymmetric nozzles. Codes compute internal and external viscous flowfield about isolated nozzle, so flow characteristics and performance of three-dimensional jet engine exhaust nozzles can be predicted. Programs written in FORTRAN IV and ASSEMBLER.
The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic, and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6 deg and 20 deg, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching step sizes with the implicit shock fitting procedure. A correlation is found between the marching step size, Reynolds number, and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.
The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6 deg and 20 deg, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching-step sizes with the implicit shock fitting procedure. A correlation is found between the marching-step size, Reynolds number and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.
A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the hypersonic, viscous, chemically reacting flow around two-dimensional or axisymmetric bodies. The new code is an extension of the upwind (perfect gas) PNS code of Lawrence et al. (1986). The upwind algorithm is based on Roe's flux-difference splitting scheme which has been modified to account for real gas effects. The algorithm solves the gas dynamic and species continuity equations in a 'loosely' coupled manner. The new code has been validated by computing the laminar flow (at free stream Mach number 25) of chemically reacting air over a wedge and a cone. The results of these computations are compared with the results from a centrally-differenced, fully coupled, nonequilibrium PNS code. The agreement is excellent, except in the vicinity of the shock wave where the present code exhibits superior shock capturing capabilities.
An implicit Navier-Stokes analysis using a single deforming mesh has been developed for the unsteady rotor-stator interaction problem. The technique has been used to simulate the flow through a turbine stator-rotor stage. Periodic two-dimensional solutions have been obtained using 1000 time steps per cycle without iteration at each time step. Computed surface pressure distributions compare favorably with experimental data available for this configuration.
Unsteady, thin-layer Navier-Stokes equations written in rotor coordinates are solved using a partially flux-split, implicit numerical algorithm to calculate the flowfields of a hovering rotor blade at subsonic and transonic conditions. Numerical results are in good agreement with experimental data for both nonlifting and lifting rectangular blades. For the lifting case, the rotor wake effects are modeled by applying a correction to the geometric angle of attack of the blades. Alternate methods of calculating the hovering rotor flowfields in blade-fixed mode that have the same circulation distribution as a hovering blade are explored. All of the results presented in this paper were computed on a Cray-2 supercomputer.
An incompressible Navier-Stokes solver using an upwind finite-difference algorithm is employed to investigate low-speed, three-dimensional, laminar, leading-edge vortex flows over three round-edged low-aspect-ratio wings. The effects of grid density, Reynolds number, and wing planform on the flowfield structures and integral values are studied. Computed results show good qualitative and quantitative agreement with the available experimental data.
A new upwind, parabolized Navier-Stokes (PNS) code has been developed to compute the three-dimensional flow of chemically reacting air around hypersonic vehicles. The code is a modification of the perfect gas, three-dimensional UPS code of Lawrence et al. (1986) which has been extended in the present study to permit the calculation of hypersonic, viscous flows in chemical nonequilibrium. The algorithm solves the PNS equations using a finite-volume, upwind TVD method based on Roe's approximate Riemann solver that has been modified to account for real gas effects. The present code solves the fluid dynamic and species continuity equations in a loosely-coupled manner. The fluid medium is assumed to be a chemically reacting mixture of thermally perfect (but calorically imperfect) gases in thermal equilibrium. Results are presented for the hypersonic laminar flow over a cone at 0- and 10-deg angles of attack and for a generic hypersonic vehicle. Calculations are performed assuming either perfect gas, equilibrium air, or finite-rate chemistry.
The PARC2D Navier-Stokes Code was modified to compute the performance parameters for rocket engine nozzles. The perfect gas code was applied to the SSME engine nozzle for inviscid, laminar and turbulent flow. Inviscid computations compare well with Rocketdyne computations. Performance degradation due to the boundary layers is very reasonable. Application of the code to nontraditional nozzle geometries and to low Reynolds nozzles is demonstrated. Modification of the code for equilibrium H2/O2 chemistry is described. Thermodynamic and equilibrium constants are determined from statistical mechanics and the transport properties from exact kinetic theory, using collison integrals determined from appropriate intermolecular potentials. The equilibrium code was used to compute the SSME flowfield. Modifications of the flowfield due to the change of composition are described.
The Proteus Navier-Stokes Code is evaluated for 2-D/axisymmetric, viscous, incompressible, internal, and external flows. The particular cases to be discussed are laminar and turbulent flows over a flat plate, laminar and turbulent developing pipe flows, and turbulent pipe flow with swirl. Results are compared with exact solutions, empirical correlations, and experimental data. A detailed description of the code set-up, including boundary conditions, initial conditions, grid size, and grid packing is given for each case.
An extension of the continuum model beyond the Navier-Stokes (NS) level and related issues on problem formulation are examined for a hypersonic shock layer on the basis of Grad's thirteen-moment equations for a Maxwell gas. The 13-moment system, simplified consistently with a fully viscous version of the thin shock-layer approximation, permits correlation with the corresponding NS-based solution. With the exception of pressure and density, several flow properties including normal stress, shear stress and normal heat flux along a streamline are unaffected by translational nonequilibrium and are therefore predicted correctly by the NS solution to the leading order, even in a domain far from translational equilibrium where molecular-transport processes rank equally with the convection.
An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.
The Proteus Navier-Stokes Code is evaluated for two-dimensional/axisymmetric, viscous, incompressible, internal and external flows. The particular cases to be discussed are laminar and turbulent flows over a flat plate, laminar and turbulent dveloping pipe flows and turbulent pipe flow with swirl. Results are compared with exact solutions, empirical correlations and experimental data. A detailed description of the code set-up, including boundary conditions, intitial conditions, grid size and grid packing is given for each case.
Navier-Stokes simulations of high Reynolds number flow around an axisymmetric body supported in a water tunnel were made. The numerical method is based on a finite-differencing high resolution second-order accurate implicit upwind scheme. Four different configurations were investigated, these are: (1) barebody; (2) body with an operating propeller; (3) body with a ring wing; and (4) body with a ring wing and an operating propeller. Pressure and velocity components near the stern region were obtained computationally and are shown to compare favorably with the experimental data. The method correctly predicts the existence and extent of stern flow separation for the barebody and the absence of flow separation for the three other configurations with ring wing and/or propeller.
Navier-Stokes computations at high angles of attack over aerodynamic configurations are presented using an implicit finite-volume algorithm. The spatial differencing is upwind-biased for the convective and pressure terms and central for the shear stress and heat transfer terms. The equations are relaxed to steady state with a spatially factored implicit algorithm. In order to treat general geometries, a multiblock patched-grid framework is implemented. Applications and detailed comparisons with experimental data are made for two simple but representative geometric shapes: (1) a highly swept delta wing and (2) a prolate spheroid of 6:1 length-to-diameter. Recent extensions of the algorithm to compute the flow over an F-18 forebody-strake configuration are shown, including comparisons with wind tunnel and flight test results. Comparisons across the range of Reynolds number for all cases indicate that either low Reynolds number or high Reynolds number flows are simulated well, but that flows at intermediate Reynolds number cannot be simulated accurately without a detailed knowledge of the transitional zone between laminar and turbulent flow.
Numerical solutions of the time-dependent compressible Navier-Stokes equations are employed to analyze the unsteady features of jet flows which contribute to jet noise. The turbulent-mass flux spectra are examined in detail in the numerical analysis, and MacCormack's explicit finite difference algorithm is used to solve the governing equations. The computational grid is based on an axisymmetric jet with simple exponential stretching used to optimize the resolution of critical unsteady flow features. The time-varying numerical data are analyzed, and the time averages of the velocity and turbulence-intensity profiles are shown to agree with experimental data. Grid refinement and longer run times improved the prediction of the experimental mass-flux spectra. The numerical procedure can be extended to 3D jet-flow configurations to study the related 3D unsteady flow features.
A new optimization procedure, in which a parabolized Navier-Stokes solver is coupled with a non-linear least-squares optimization algorithm, is applied to the design of a Mach 14, laminar two-dimensional hypersonic subscale flight inlet with an internal contraction ratio of 15:1 and a length-to-throat half-height ratio of 150:1. An automated numerical search of multiple geometric wall contours, which are defined by polynomical splines, results in an optimal geometry that yields the maximum total-pressure recovery for the compression process. Optimal inlet geometry is obtained for both inviscid and viscous flows, with the assumption that the gas is either calorically or thermally perfect. The analysis with a calorically perfect gas results in an optimized inviscid inlet design that is defined by two cubic splines and yields a mass-weighted total-pressure recovery of 0.787, which is a 23% improvement compared with the optimized shock-canceled two-ramp inlet design. Similarly, the design procedure obtains the optimized contour for a viscous calorically perfect gas to yield a mass-weighted total-pressure recovery value of 0.749. Additionally, an optimized contour for a viscous thermally perfect gas is obtained to yield a mass-weighted total-pressure recovery value of 0.768. The design methodology incorporates both complex fluid dynamic physics and optimal search techniques without an excessive compromise of computational speed; hence, this methodology is a practical technique that is applicable to optimal inlet design procedures.
An approach for solving the compressible Euler and Navier-Stokes equations upon meshes composed of nearly arbitrary polyhedra is described. Each polyhedron is constructed from an arbitrary number of triangular and quadrilateral face elements, allowing the unified treatment of tetrahedral, prismatic, pyramidal, and hexahedral cells, as well the general cut cells produced by Cartesian mesh approaches. The basics behind the numerical approach and the resulting data structures are described. The accuracy of the mixed volume grid approach is assessed by performing a grid refinement study upon a series of hexahedral, tetrahedral, prismatic, and Cartesian meshes for an analytic inviscid problem. A series of laminar validation cases are made, comparing the results upon differing grid topologies to each other, to theory, and experimental data. A computation upon a prismatic/tetrahedral mesh is made simulating the laminar flow over a wall/cylinder combination.