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NASA NTRS1992

STME Hydrogen Mixer Study

The hydrogen mixer for the Space Transportation Main Engine is used to mix cold hydrogen bypass flow with warm hydrogen coolant chamber gas, which is then fed to the injectors. It is very important to have a uniform fuel temperature at the injectors in order to minimize mixture ratio problems due to the fuel density variations. In addition, the fuel at the injector has certain total pressure requirements. In order to achieve these objectives, the hydrogen mixer must provide a thoroughly mixed fluid with a minimum pressure loss. The AEROVISC computational fluid dynamics (CFD) code was used to analyze the STME hydrogen mixer, and proved to be an effective tool in optimizing the mixer design. AEROVISC, which solves the Reynolds Stress-Averaged Navier-Stokes equations in primitive variable form, was used to assess the effectiveness of different mixer designs. Through a parametric study of mixer design variables, an optimal design was selected which minimized mixed fuel temperature variation and fuel mixer pressure loss. The use of CFD in the design process of the STME hydrogen mixer was effective in achieving an optimal mixer design while reducing the amount of hardware testing.

Blumenthal, Rob↗

NASA NTRS1993

On solving the compressible Navier-Stokes equations for unsteady flows at very low Mach numbers

The properties of a preconditioned, coupled, strongly implicit finite-difference scheme for solving the compressible Navier-Stokes equations in primitive variables are investigated for two unsteady flows at low speeds, namely the impulsively started driven cavity and the startup of pipe flow. For the shear-driven cavity flow, the computational effort was observed to be nearly independent of Mach number, especially at the low end of the range considered. This Mach number independence was also observed for steady pipe flow calculations; however, rather different conclusions were drawn for the unsteady calculations. In the pressure-driven pipe startup problem, the compressibility of the fluid began to significantly influence the physics of the flow development at quite low Mach numbers. The present scheme was observed to produce the expected characteristics of completely incompressible flow when the Mach number was set at very low values. Good agreement with incompressible results available in the literature was observed.

Pletcher, R. H.↗

NASA NTRS1993

On solving the compressible Navier-Stokes equations for unsteady flows at very low Mach numbers

The properties of a preconditioned, coupled, strongly implicit finite difference scheme for solving the compressible Navier-Stokes equations in primitive variables are investigated for two unsteady flows at low speeds, namely the impulsively started driven cavity and the startup of pipe flow. For the shear-driven cavity flow, the computational effort was observed to be nearly independent of Mach number, especially at the low end of the range considered. This Mach number independence was also observed for steady pipe flow calculations; however, rather different conclusions were drawn for the unsteady calculations. In the pressure-driven pipe startup problem, the compressibility of the fluid began to significantly influence the physics of the flow development at quite low Mach numbers. The present scheme was observed to produce the expected characteristics of completely incompressible flow when the Mach number was set at very low values. Good agreement with incompressible results available in the literature was observed.

Pletcher, R. H.↗

NASA NTRS1996

The Multigrid-Mask Numerical Method for Solution of Incompressible Navier-Stokes Equations

A multigrid-mask method for solution of incompressible Navier-Stokes equations in primitive variable form has been developed. The main objective is to apply this method in conjunction with the pseudospectral element method solving flow past multiple objects. There are two key steps involved in calculating flow past multiple objects. The first step utilizes only Cartesian grid points. This homogeneous or mask method step permits flow into the interior rectangular elements contained in objects, but with the restriction that the velocity for those Cartesian elements within and on the surface of an object should be small or zero. This step easily produces an approximate flow field on Cartesian grid points covering the entire flow field. The second or heterogeneous step corrects the approximate flow field to account for the actual shape of the objects by solving the flow field based on the local coordinates surrounding each object and adapted to it. The noise occurring in data communication between the global (low frequency) coordinates and the local (high frequency) coordinates is eliminated by the multigrid method when the Schwarz Alternating Procedure (SAP) is implemented. Two dimensional flow past circular and elliptic cylinders will be presented to demonstrate the versatility of the proposed method. An interesting phenomenon is found that when the second elliptic cylinder is placed in the wake of the first elliptic cylinder a traction force results in a negative drag coefficient.

Ku, Hwar-Ching↗

NASA NTRS1973

Numerical models of the primitive solar nebula.

The equations giving force balance in the radial and vertical directions in the primitive solar nebula are developed. The techniques used to compute the gravitational potential and to compute the opacity of the gas for energy transport in the vertical direction are described, as well as the properties of the primitive solar nebula models thus constructed. The rate at which gaseous dissipation should occur within the solar nebula, producing the sun by inward flow of matter, is discussed.

Cameron, A. G. W.↗

NASA NTRS1987

Time-marching solution of incompressible Navier-Stokes equations for internal flow

Primitive variables with central differencing on a staggered grid are used in the present, factored ADI finite-difference scheme for artificial compressibility method solution of the incompressible Navier-Stokes equations, leading to a close coupling between velocity and pressure that both enhances stability and eliminates the need for artificial damping. Computational efficiency is enhanced through the use of a spatially variable, fixed Courant number-based time-step. The numerical results obtained for a driven cavity at Re of 10,000, with local cell Re as high as 100, exhibits no flow variable spatial oscillations on a 40 x 40 stretched grid solution.

Soh, W. Y.↗

NASA NTRS1995

Entropy jump across an inviscid shock wave

The shock jump conditions for the Euler equations in their primitive form are derived by using generalized functions. The shock profiles for specific volume, speed, and pressure are shown to be the same, however density has a different shock profile. Careful study of the equations that govern the entropy shows that the inviscid entropy profile has a local maximum within the shock layer. We demonstrate that because of this phenomenon, the entropy, propagation equation cannot be used as a conservation law.

Salas, Manuel D.↗

NASA NTRS1991

A Variational Assimilation Method for Satellite and Conventional Data: a Revised Basic Model 2B

A variational objective analysis technique that modifies observations of temperature, height, and wind on the cyclone scale to satisfy the five 'primitive' model forecast equations is presented. This analysis method overcomes all of the problems that hindered previous versions, such as over-determination, time consistency, solution method, and constraint decoupling. A preliminary evaluation of the method shows that it converges rapidly, the divergent part of the wind is strongly coupled in the solution, fields of height and temperature are well-preserved, and derivative quantities such as vorticity and divergence are improved. Problem areas are systematic increases in the horizontal velocity components, and large magnitudes of the local tendencies of the horizontal velocity components. The preliminary evaluation makes note of these problems but detailed evaluations required to determine the origin of these problems await future research.

Achtemeier, Gary L.↗

NASA NTRS1992

A time-accurate algorithm for chemical non-equilibrium viscous flows at all speeds

A time-accurate, coupled solution procedure is described for the chemical nonequilibrium Navier-Stokes equations over a wide range of Mach numbers. This method employs the strong conservation form of the governing equations, but uses primitive variables as unknowns. Real gas properties and equilibrium chemistry are considered. Numerical tests include steady convergent-divergent nozzle flows with air dissociation/recombination chemistry, dump combustor flows with n-pentane-air chemistry, nonreacting flow in a model double annular combustor, and nonreacting unsteady driven cavity flows. Numerical results for both the steady and unsteady flows demonstrate the efficiency and robustness of the present algorithm for Mach numbers ranging from the incompressible limit to supersonic speeds.

Shuen, J.-S.↗

NASA NTRS1993

A coupled implicit method for chemical non-equilibrium flows at all speeds

The present time-accurate coupled-solution procedure addresses the chemical nonequilibrium Navier-Stokes equations over a wide Mach-number range uses, in conjunction with the strong conservation form of the governing equations, five unknown primitive variables. The numerical tests undertaken address steady convergent-divergent nozzle flows with air dissociation/recombination, dump combustor flows with n-pentane/air chemistry, and unsteady nonreacting cavity flows.

Shuen, Jian-Shun↗

NASA NTRS1993

The NASA/GISS Mars general circulation model: Preliminary experiments

The NASA/GISS Mars General Circulation Model (GCM) is an adapted version of the GISS Global Climate/Middle Atmosphere Model, specifically developed for the diagnostic validation and objective analysis of measured atmospheric temperatures from the Mars Observer Pressure Modulator Infrared Radiometer (PMIRR) experiment. The GISS Mars GCM has 23 vertical layers extending from the surface to approximately 80 km altitude, representing a vertical resolution of about 0.3 scale heights. The primitive (vertically hydrostatic) equations are solved in finite difference form on the Krakawa B grid, with a horizontal resolution of 8 deg x 10 deg (latitude-longitude). The model includes a diurnal solar cycle, heat transport within a two-layer ground, and a high-order 'slopes-scheme' for the advection of heat in the upper atmosphere. The radiative transfer scheme is based on the correlated k distribution method for the treatment of nongray gaseous absorption thermal emission, and multiple scattering, including options for suspended dust. A special feature of the model of particular importance for Mars is a parameterization of gravity-wave-induced drag incorporating orographic forcing, wind shear, convection, and radiative damping. The implementation of the GISS Mars model includes global maps of topography, roughness, and albedo.

Allison, Michael↗

NASA NTRS1994

An implicit numerical scheme for the simulation of internal viscous flows on unstructured grids

The Navier-Stokes equations are solved numerically for two-dimensional steady viscous laminar flows. The grids are generated based on the method of Delaunay triangulation. A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. A preconditioning matrix is added to the equations so that low Mach number flows can be solved economically. The equations are time marched using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient like method. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speed up in the matrix solver is typical for the viscous equations. Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a four-port valve. Comparisons are made with available results obtained by other investigators.

Jorgenson, Philip C. E.↗

NASA NTRS1987

A numerical method for the solution of three dimensional, incompressible, viscous flows over slender bodies

A marching iterative method for the solution of the three dimensional, incompressibhle, steady and parabolized Navier-Stokes equations is described. The equations are written in primitive variables and discretized in general axisymmetric orthogonal coordinate systems. The coupled set of finite-difference equations are solved without any splitting or factorization errors. Moreover, the continuity equation and the two crossflow momentum equations are exactly satisfied at every step of the iterative process. The solution scheme is equivalent to the solution of one Poisson equation by the Successive Plane Over Relaxation method and has good convergence properties. Other existing solution methods resemble a Jacobi-type iterative scheme and therefore are less efficient. Numerical experiments include the laminar, incompressible flow over prolate spheroids at incidence.

Rosenfeld, Moshe↗

NASA NTRS1994

Numerical Wave Propagation 211 Based on Wave Primitives

Compact higher order finite difference equations are applied to a sequence of problems in wave propagation and aeroacoustics. Systems of PDE's are reduced to a sequence of simple wave primitives using a local eigenvector decomposition. The wave primitives are first order PDE's in two independent variable and allow natural boundary conditions to be imposed for both single- and multidimensional problems. The method uses a "discrete dispersion relation" approach to obtain high order approximations to the wave primitives on a 3 spatial point / 2 time level computational molecule. The scheme is fourth order accurate for the class of system with constant coefficients, e.g., those that support exponential solutions. Weakly non-linear PDE's are solved in a similar manner using a variant of the "method of frozen coefficients." Experience with the new algorithm for linear, non-linear, and multi-dimensional test problems will be described.

Davis, Sanford S.↗

NASA NTRS1978

Development of a three-dimensional Navier-Stokes code on CDC star-100 computer

A three-dimensional code in body-fitted coordinates was developed using MacCormack's algorithm. The code is structured to be compatible with any general configuration, provided that the metric coefficients for the transformation are available. The governing equations are developed in primitive variables in order to facilitate the incorporation of physical boundary conditions and turbulence-closure models. MacCormack's two-step, unsplit, time-marching algorithm is used to solve the unsteady Navier-Stokes equations until steady-state solution is achieved. Cases discussed include (1) flat plate in supersonic free stream; (2) supersonic flow along an axial corner; (3) subsonic flow in an axial corner at M infinity = 0.95; and (4) supersonic flow in an axial corner at M infinity 1.5.

Vatsa, V. N.↗

NASA NTRS1976

Finite difference procedure for boundary layers including effects of longitudinal and transverse curvatures

A second order viscous layer solution procedure has been developed that does in a consistent way include curvature effects and the corresponding normal pressure gradients. In the present system, the normal momentum equation is retained. The parabolic system of nonlinear partial differential equations is converted by linear finite differencing procedures to a system of linear algebraic equations and solved in primitive coordinates. The solutions have been shown to give smooth stable distributions for all the variables, most particularly the normal velocity which plays an important role in the interaction procedure. An algorithm for matching the viscous layer solution with a rotational characteristics outer solution has been developed.

Tassa, Y.↗

NASA NTRS1994

Three-dimensional fully spectral numerical method for mantle convection with depth-dependent properties

A semi-implicit fully spectral collocation method for the simulation of three-dimensional mantle convection with depth-dependent thermo-dynamic and transport properties is presented. The variable property Navier-Stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations. The periodic horizontal boundary conditions allow a Fourier expansion for the two horizontal directions. The stress-free, impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a Chebyshev expansion in the vertical direction. In the limit of an infinite Prandtl number appropriate to mantle convection, the inertial terms in the momentum equation are unimportant. In this case an explicit solution of a Poisson equation for pressure can be avoided; instead a fourth-order equation for vertical velocity can be solved. Simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed. The pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique. The numerical method employed here avoids time-splitting errors and enforces velocity boundary conditions and continuity over the entire domain, including the boundaries, to machine accuracy. Strongly time-dependent three-dimensional solutions up to a surface Rayleigh number of 1 x 10(exp 7) have been obtained. Strong upwellings, pulsating chaotically, are formed by the collective merging of cylindrical plumes.

Balachandar, S.↗

NASA NTRS1982

A simple finite difference procedure for the vortex controlled diffuser

A simple prediction procedure for sudden expansion incompressible flows is developed and applied to the vortex controlled diffuser. Transient Navier-Stokes equations of an incompressible fluid are solved by means of their associated finite difference equations in terms of the primitive pressure velocity variables. A computer code is developed using a laminar flow simulation with free slip or no slip wall boundary conditions. In addition, predicted results confirm that effectiveness increases with increases in duct length and bleed flow rate

Busnaina, A. A.↗