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NASA NTRS1993

Ice accretion and performance degradation calculations with LEWICE/NS

The LEWICE ice accretion computer code has been extended to include the solution of the two-dimensional Navier-Stokes equations. The code is modular and contains separate stand-alone program elements that create a grid, calculate the flow field parameters, calculate the droplet trajectory paths, determine the amount of ice growth, calculate aeroperformance changes, and plot results. The new elements of the code are described. Calculated results are compared to experiment for several cases, including both ice shape and drag rise.

Potapczuk, Mark G.↗

NASA NTRS1996

Preconditioned Mixed Spectral Element Methods for Elasticity and Stokes Problems

Preconditioned iterative methods for the indefinite systems obtained by discretizing the linear elasticity and Stokes problems with mixed spectral elements in three dimensions are introduced and analyzed. The resulting stiffness matrices have the structure of saddle point problems with a penalty term, which is associated with the Poisson ratio for elasticity problems or with stabilization techniques for Stokes problems. The main results of this paper show that the convergence rate of the resulting algorithms is independent of the penalty parameter, the number of spectral elements Nu and mildly dependent on the spectral degree eta via the inf-sup constant. The preconditioners proposed for the whole indefinite system are block-diagonal and block-triangular. Numerical experiments presented in the final section show that these algorithms are a practical and efficient strategy for the iterative solution of the indefinite problems arising from mixed spectral element discretizations of elliptic systems.

Pavarino, Luca F.↗

NASA NTRS1988

Scaling laws for homogeneous turbulent shear flows in a rotating frame

The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri=-2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.

Speziale, Charles G.↗

NASA NTRS1989

Scaling laws for homogeneous turbulent shear flows in a rotating frame

The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri = -2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.

Speziale, Charles G.↗

NASA NTRS1968

Study and development of a mathematical analysis for the performance assessment of space communication system parameters

Electronic computer program user manual for optimum design of space communication system

Brinkman, K. L.↗

NASA NTRS1976

Solutions of the Navier-Stokes equations for vortex breakdown

A numerical investigation of vortex breakdown has been undertaken in an attempt to understand its properties, and the mechanisms responsible for it. Solutions of the full steady axisymmetric Navier-Stokes equations for breakdown in an unconfined viscous vortex have been obtained for core Reynolds numbers up to 200, for a two-parameter family of assumed upstream velocity distributions. Diffusion and convection of vorticity away from the vortex core, and the strong coupling between the circumferential and axial velocity fields in highly-swirling flows, are shown to lead to stagnation and reversal of the axial flow near the axis. The various theories of vortex breakdown are considered in light of the present numerical solutions.

Grabowski, W. J.↗

NASA NTRS1987

A numerical study of three-dimensional vortex breakdown

A numerical simulation of bubble-type vortex breakdown using a unique discrete form of the full 3-D, unsteady incompressible Navier-Stokes equations was performed. The Navier-Stokes equations were written in a vorticity-velocity form and the physical problem was not restricted to axisymmetric flow. The problem was parametized on a Rossby- Reynolds-number basis. Utilization of this parameter duo was shown to dictate the form of the free-field boundary condition specification and allowed control of axial breakdown location within the computational domain. The structure of the breakdown bubble was studied through time evolution plots of planar projected velocity vectors as well as through plots of particle traces and vortex lines. These results compared favorably with previous experimental studies. In addition, profiles of all three velocity components are presented at various axial stations and a Fourier analysis was performed to identify the dominant circumferential modes. The dynamics of the breakdown process were studied through plots of axial variation of rate of change of integrated total energy and rate of change of integrated enstrophy, as well as through contour plots of velocity, vorticity and pressure.

Spall, Robert E.↗

NASA NTRS1986

Numerical simulation of boundary layers. Part 3: Turbulence and relaminarization in sink flows

Direct numerical simulations of sink-flow boundary layers, with acceleration parameters K between 1.5 and 3.0 x 10 to the -6 power, are presented. The three-dimensional, time-dependent Navier-Stokes equations are solved numerically using a spectral method, with about 10 to the -6 degrees of freedom. The flow is assumed to be statistically steady, and self-similar. A multiple-scale approximation and periodic conditions are applied to the fluctuations. The turbulence is studied using instantaneous and statistical results. Good agreement with the experiments of Jones and Launder is observed. Two effects of the favorable pressure gradient are to extend the logarithmic layer, and to alter the energy balance of the turbulence near the edge of the boundary layer. At very low Reynolds number the logarithmic layer is shortened and slightly displaced, but wall-layer streaks are present even at the lowest values of the Reynolds number for which turbulence can be sustained. Large quiescent patches appear in the flow. Relaminarization occurs at K = 3.0 x 10 to the -6 power, corresponding to a Reynolds number, based on momentum thickness, of about 330.

Spalart, Philippe R.↗

NASA NTRS1992

On the drag of model dendrite fragments at low Reynolds number

An experimental study of low Reynolds number drag on laboratory models of dendrite fragments has been conducted. The terminal velocities of the dendrites undergoing free fall along their axis of symmetry were measured in a large Stokes flow facility. Corrections for wall interference give nearly linear drag vs Reynolds number curves. Corrections for both wall interference and inertia effects show that the dendrite Stokes settling velocities are always less than that of a sphere of equal mass and volume. In the Stokes limit, the settling speed ratio is found to correlate well with the primary dendrite arm aspect ratio and a second dimensionless shape parameter which serves as a measure of the fractal-like nature of the dendrite models. These results can be used to estimate equiaxed grain velocities and distance of travel in metal castings. The drag measurements may be used in numerical codes to calculate the movement of grains in a convecting melt in an effort to determine macrosegregation patterns caused by the sink/float mechanism.

Zakhem, R.↗

NASA NTRS1993

Collisional excitation of CO by H2O - An astrophysicist's guide to obtaining rate constants from coherent anti-Stokes Raman line shape data

Rate constants for excitation of CO by collisions with H2O are needed to understand recent observations of comet spectra. These collision rates are closely related to spectral line shape parameters, especially those for Raman Q-branch spectra. Because such spectra have become quite important for thermometry applications, much effort has been invested in understanding this process. Although it is not generally possible to extract state-to-state rate constants directly from the data as there are too many unknowns, if the matrix of state-to-state rates can be expressed in terms of a rate-law model which depends only on rotational quantum numbers plus a few parameters, the parameters can be determined from the data; this has been done with some success for many systems, especially those relevant to combustion processes. Although such an analysis has not yet been done for CO-H2O, this system is expected to behave similarly to N2-H2O which has been well studies; modifications of parameters for the latter system are suggested which should provide a reasonable description of rate constants for the former.

Green, Sheldon↗

NASA NTRS1983

A study on the spectral models for waves in finite water depth

From an extension of the Wallops Spectrum (Huang et al., 1981) for the deep water waves, spectral models for waves in finite water depths are developed. Stokes wave expansions are found to offer a good approximation for intermediate water depth. The spectral function in this case is controlled by three parameters: the significant slope, the nondimensional depth, and the peak frequency. It is pointed out that solitary and cnoidal wave models must be used for the shallow water waves. The controlling parameters now reduce to the Urell number and the peak frequency. Even though the resulting spectral models place special emphasis on the energy-containing range of the spectrum, they are not limited to this range and they are not limited to any particular sea state. They are seen as offering a possible explanation of the variations in the special slope observed by previous investigators.

Huang, N. E.↗

NASA NTRS1989

Progress on a Taylor weak statement finite element algorithm for high-speed aerodynamic flows

A new finite element numerical Computational Fluid Dynamics (CFD) algorithm has matured to the point of efficiently solving two-dimensional high speed real-gas compressible flow problems in generalized coordinates on modern vector computer systems. The algorithm employs a Taylor Weak Statement classical Galerkin formulation, a variably implicit Newton iteration, and a tensor matrix product factorization of the linear algebra Jacobian under a generalized coordinate transformation. Allowing for a general two-dimensional conservation law system, the algorithm has been exercised on the Euler and laminar forms of the Navier-Stokes equations. Real-gas fluid properties are admitted, and numerical results verify solution accuracy, efficiency, and stability over a range of test problem parameters.

Baker, A. J.↗

NASA NTRS1990

Patterns of vortex shedding from an oscillating circular cylinder

Vortex shedding from an oscillating circular cylinder was numerically investigated at Re = 100 with the Navier-Stokes equations and the new boundary conditions. The detailed shedding patterns are characterized by means of streakline plotting and lift-coefficient curves. A parameter map is presented which distinguishes the synchronized shedding from the asynchronous and the double vortices shedding from the single vortex shedding. The computational result is in good agreement with earlier experimental results.

Chang, Keun-Shik↗

NASA NTRS1992

Computational study of transition front on a swept wing leading-edge model

The e(N) method is employed with Navier-Stokes calculations to study 3D supersonic boundary layers at the leading-edge region of swept wings with attention given to the conditions for linear stability. Mean-flow profiles are computed as solutions to the Navier-Stokes equations, and the e(N) method is used to identify the onset of the transition. Specific treatment is given to the identification of parameters which affect the transition such as free-stream Reynolds number, wall cooling, and boundary-layer suction levels. The boundary layer can be stabilized to below N=10 levels in all locations by utilizing distributed suction, and enhanced stabilization can also be achieved by employing wall cooling and reduced free-stream Reynolds numbers. This study presents key calculations for corresponding experimental investigations by shedding light on requirements for test conditions and measurement requirements.

Iyer, Venkit↗

NASA NTRSDate not supplied

Optimization of Plug Nozzles Using CFD Informed Design Space Querying

This paper presents an axisymmetric CFD approach based on Reynolds-averaged Navier-Stokes (RANS) calculations for the shape optimization of plug nozzle contours for both supersonic cruise and landing/takeoff (LTO) conditions. Three key external plug design parameters were defined as independent variables in the optimization scheme, while nozzle performance was evaluated using thrust coefficient (Cfg) and discharge coefficient (Cd) as the primary dependent variables. By varying these parameters, a wide range of axisymmetric conic section plug profiles can be achieved, simplifying the problem. The paper also discusses the supersonic plug nozzle optimization approach, design space seeding, meshing, and CFD simulations for the optimization process. The results demonstrate that the approach is effective in exploring the parameter space and reducing computational expenses, making it a viable option for optimizing nozzle plug contours. Using this optimization toolchain, the study yielded preliminary designs for LTO, cruise, and LTO-cruise balanced plug nozzles.

nozzles↗

NASA NTRSDate not supplied

Optimization of Plug Nozzles Using CFD Informed Design Space Querying

CFD↗

NASA NTRS1985

On the penetration of a hot diapir through a strongly temperature-dependent viscosity medium

The ascent of a hot spherical body through a fluid with a strongly temperature-dependent viscosity has been studied using an axisymmetric finite element method. Numerical solutions range over Peclet numbers of 0.1 - 1000 from constant viscosity up to viscosity variations of 100,000. Both rigid and stress-free boundary conditions were applied at the surface of the sphere. The dependence of drag on viscosity variation was shown to have no dependence on the stress boundary condition except for a Stokes flow scaling factor. A Nusselt number parameterization based on the stress-free constant viscosity functional dependence on the Peclet number scaled by a parameter depending on the viscosity structure fits both stress-free and rigid boundary condition data above viscosity variations of 100. The temperature scale height was determined as a function of sphere radius. For the simple physical model studied in this paper pre-heating is required to reduce the ambient viscosity of the country rock to less than 10 to the 22nd sq cm/s in order for a 10 km diapir to penetrate a distance of several radii.

Daly, S. F.↗

NASA NTRS1990

Computational analysis of the flowfield of a two-dimensional ejector nozzle

A time-iterative full Navier-Stokes code, PARC, is used to analyze the flowfield of a two-dimensional ejector nozzle system. A parametric study was performed for two controlling parameters, duct to nozzle area ratio and nozzle pressure ratio. Results show that there is an optimum area ratio for the efficient pumping of secondary flow. At high area ratios, a freestream flow passes directly through the mixing duct without giving adequate pumping. At low area ratios, the jet boundary blocks the incoming flow. The nozzle pressure ratio variation shows that the pumping rate increases as the pressure ratio increases, provided there is no interaction between the shroud wall and the shock cell structure.

Choi, Y. H.↗