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NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

A computer code called Proteus 3D has been developed to solve the three dimensional, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort has been to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation have been emphasized. The governing equations are solved in generalized non-orthogonal body-fitted coordinates by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This is the Analysis Description, and presents the equations and solution procedure. It describes in detail the governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models.

Towne, Charles E.↗

NASA NTRS1981

Conservation form of the Navier-Stokes equations in general nonsteady coordinates

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Warsi, Z. U. A.↗

NASA NTRS1983

Implicit conservative characteristic modeling schemes for the Euler equations - A new approach

An implicit characteristic-modeling solution scheme for the Euler equations is presented. The scheme does not require the governing equations to be written in characteristic variables or the flux terms to be split into positive and negative contributions. For the two-dimensional problem of a shock wave reflecting from a flat plate, this feature and the simple solution algorithm combine to reduce the computational work per mesh point by 40 percent from that required by a standard, central-difference, implicit, solution algorithm. Application of the method to the quasi-one-dimensional nozzle flow equations for subsonic and supersonic flows without shocks shows the method to be well-conditioned for large time steps.

Wornom, S. F.↗

NASA NTRS1984

A rule for selecting analytical boundary conditions for the conservative quasi-one-dimensional nozzle flow equations

For the one-dimensional nozzle flow equations, the number of analytical boundary conditions that may be applied corresponds to the number of exterior characteristics reaching the boundary points. Since the slopes of the characteristics are the characteristic speeds u, u+c, and u-c, one concludes that two analytical boundary conditions are required at a subsonic inflow boundary and one at a subsonic outflow boundary. However, using this guideline alone can lead to nonunique solutions in subsonic cases. This paper presents a rule for selecting analytical boundary conditions for the dependent variables which will yield unique solutions if they exist.

Wornom, S. F.↗

NASA NTRS1972

Conservation of quasiparticles in weakly turbulent plasmas.

An expansion making use of the eikonal is shown to yield a solution to the equations of a general weak turbulent plasma which is weakly dependent on space and time. The method is used to derive a quasi-particle conservation equation for quasi-static perturbations of Vlasov plasmas with general equilibrium field configurations.

Bloomberg, H. W.↗

NASA NTRS1987

A finite difference scheme for three-dimensional steady laminar incompressible flow

A finite difference scheme for three-dimensional steady laminar incompressible flows is presented. The Navier-Stokes equations are expressed conservatively in terms of velocity and pressure increments (delta form). First order upwind differences are used for first order partial derivatives of velocity increments resulting in a diagonally dominant matrix system. Central differences are applied to all other terms for second order accuracy. The SIMPLE pressure correction algorithm is used to satisfy the continuity equation. Numerical results are presented for cubic cavity flow problems for Reynolds numbers up to 2000 and are in good agreement with other numerical results.

Hwang, Danny P.↗

NASA NTRS1987

A finite difference scheme for three-dimensional steady laminar incompressible flow

Hwang, Danny P.↗

NASA NTRS1979

Description and preliminary results of the 9-level UCLA general circulation model

The 9-level general circulation model, with the upper boundary of 50 mb, is a modification of the earlier 6- and 12-level versions used at UCLA. Two major design changes have been made in an effort to improve the model: (1) a potential enstrophy conserving scheme in the equation of motion has replaced the previous scheme which only conserved enstrophy for nondivergent flow, and (2) the treatment of the planetary boundary layer has been modified by making the predicted planetary boundary layer top a coordinate surface. Experiments with idealized steep mountains have been made to test each of the major modifications, and the model is now ready for general circulation simulation experiments.

Suarez, M. J.↗

NASA NTRS1991

Sensing and actuation in a liquid propellant rocket engine

Equations describing the conservation of mass, momentum, and energy, and multicomponent transport and chemical reaction in a liquid propellant rocket (LPR) engine are given, along with appropriate boundary conditions. The formulation includes a detailed description of the two-phase (gas/liquid) flow in terms of a moving boundary problem, with droplets and vapor occupying time-dependent subdomains of the LPR chamber. Several sensor and actuator types are considered in detail, and their mathematical descriptions are incorporated into the differential equations and boundary conditions.

Bentsman, Joseph↗

NASA NTRS1994

Particle Distribution in Suspension Shear Flow

An earlier proposed constitutive relation for normal stresses originated by random particle fluctuations is used to describe a joint effect of thermal and shear-induced fluctuations on concentrational distributions in suspension flow. Averaged products of components of the fluctuation velocity are evaluated on a basis of the rational mechanics approach combined with a simple kinematic consideration. The equation of momentum conservation of the dispersed phase of a suspension closed with this constitutive relation is applied to unidirectional shear flow in the gravity field and to rotational Couette flow. Coupling of the thermal and shear-induced fluctuations results in that the ability of shear flow to suspend particles has a minimum at a certain particle size, all other things being equal. The developed model provides also for a reasonable explanation of particle distributions observed in Couette flow. The approach based on the consideration of momentum balance for the dispersed phase is proved to lead to an effective equation of convective diffusion of the suspended particles. Coefficients of mutual diffusion due to both thermal and shear-induced fluctuations are drastically different from corresponding self-diffusivities as regards both their scaling and their concentrational dependence.

Buyevich, Yu A.↗

NASA NTRS1979

Cropping management using color and color infrared aerial photographs

The Universal Soil Loss Equation (USLE) is a widely accepted tool for erosion prediction and conservation planning. Solving this equation yields the long-term average annual soil loss that can be expected from rill and inter-rill erosion. In this study, manual interpretation of color and color infrared 70 mm photography at the scale of 1:60,000 is used to determine the cropping management factor in the USLE. Accurate information was collected about plowing practices and crop residue cover (unharvested vegetation) for the winter season on agricultural land in Pheasant Branch Creek watershed in Dane County, Wisconsin.

Morgan, K. M.↗

NASA NTRS2009

Hall-Effect Thruster Simulations with 2-D Electron Transport and Hydrodynamic Ions

A computational approach that has been used extensively in the last two decades for Hall thruster simulations is to solve a diffusion equation and energy conservation law for the electrons in a direction that is perpendicular to the magnetic field, and use discrete-particle methods for the heavy species. This "hybrid" approach has allowed for the capture of bulk plasma phenomena inside these thrusters within reasonable computational times. Regions of the thruster with complex magnetic field arrangements (such as those near eroded walls and magnets) and/or reduced Hall parameter (such as those near the anode and the cathode plume) challenge the validity of the quasi-one-dimensional assumption for the electrons. This paper reports on the development of a computer code that solves numerically the 2-D axisymmetric vector form of Ohm's law, with no assumptions regarding the rate of electron transport in the parallel and perpendicular directions. The numerical challenges related to the large disparity of the transport coefficients in the two directions are met by solving the equations in a computational mesh that is aligned with the magnetic field. The fully-2D approach allows for a large physical domain that extends more than five times the thruster channel length in the axial direction, and encompasses the cathode boundary. Ions are treated as an isothermal, cold (relative to the electrons) fluid, accounting for charge-exchange and multiple-ionization collisions in the momentum equations. A first series of simulations of two Hall thrusters, namely the BPT-4000 and a 6-kW laboratory thruster, quantifies the significance of ion diffusion in the anode region and the importance of the extended physical domain on studies related to the impact of the transport coefficients on the electron flow field.

plasma↗

NASA NTRS1993

A time-accurate finite volume method valid at all flow velocities

A finite volume method to solve the Navier-Stokes equations at all flow velocities (e.g., incompressible, subsonic, transonic, supersonic and hypersonic flows) is presented. The numerical method is based on a finite volume method that incorporates a pressure-staggered mesh and an incremental pressure equation for the conservation of mass. Comparison of three generally accepted time-advancing schemes, i.e., Simplified Marker-and-Cell (SMAC), Pressure-Implicit-Splitting of Operators (PISO), and Iterative-Time-Advancing (ITA) scheme, are made by solving a lid-driven polar cavity flow and self-sustained oscillatory flows over circular and square cylinders. Calculated results show that the ITA is the most stable numerically and yields the most accurate results. The SMAC is the most efficient computationally and is as stable as the ITA. It is shown that the PISO is the most weakly convergent and it exhibits an undesirable strong dependence on the time-step size. The degenerated numerical results obtained using the PISO are attributed to its second corrector step that cause the numerical results to deviate further from a divergence free velocity field. The accurate numerical results obtained using the ITA is attributed to its capability to resolve the nonlinearity of the Navier-Stokes equations. The present numerical method that incorporates the ITA is used to solve an unsteady transitional flow over an oscillating airfoil and a chemically reacting flow of hydrogen in a vitiated supersonic airstream. The turbulence fields in these flow cases are described using multiple-time-scale turbulence equations. For the unsteady transitional over an oscillating airfoil, the fluid flow is described using ensemble-averaged Navier-Stokes equations defined on the Lagrangian-Eulerian coordinates. It is shown that the numerical method successfully predicts the large dynamic stall vortex (DSV) and the trailing edge vortex (TEV) that are periodically generated by the oscillating airfoil. The calculated streaklines are in very good comparison with the experimentally obtained smoke picture. The calculated turbulent viscosity contours show that the transition from laminar to turbulent state and the relaminarization occur widely in space as well as in time. The ensemble-averaged velocity profiles are also in good agreement with the measured data and the good comparison indicates that the numerical method as well as the multipletime-scale turbulence equations successfully predict the unsteady transitional turbulence field. The chemical reactions for the hydrogen in the vitiated supersonic airstream are described using 9 chemical species and 48 reaction-steps. Consider that a fast chemistry can not be used to describe the fine details (such as the instability) of chemically reacting flows while a reduced chemical kinetics can not be used confidently due to the uncertainty contained in the reaction mechanisms. However, the use of a detailed finite rate chemistry may make it difficult to obtain a fully converged solution due to the coupling between the large number of flow, turbulence, and chemical equations. The numerical results obtained in the present study are in good agreement with the measured data. The good comparison is attributed to the numerical method that can yield strongly converged results for the reacting flow and to the use of the multiple-time-scale turbulence equations that can accurately describe the mixing of the fuel and the oxidant.

Kim, S.-W.↗

NASA NTRS1963

CONVECTION IN PLANETARY INTERIORS

Formulation of linearized equations safeguarding the conservation of mass, energy and momentum of viscous flow inside a fluid sphere, and application to the problem of convection in planetary interiors

VISCOUS FLOW↗

NASA NTRS1972

Numerical computation of multishocked, three-dimensional supersonic flow fields with real gas effects.

A computational procedure is presented which is capable of determining the supersonic flow field surrounding three-dimensional wing-body configurations such as a delta-wing space shuttle. The governing equations in conservation-law form are solved by a finite difference method using a second-order noncentered algorithm between the body and the outermost shock wave, which is treated as a sharp discontinuity. Secondary shocks which form between these boundaries are captured automatically, and the intersection of these shocks with the bow shock posed no difficulty. Resulting flow fields about typical blunt nose shuttle-like configurations at angle of attack are presented. The differences between perfect and real gas effects for high Mach number flows are shown.

Kutler, P.↗

NASA NTRS1974

A comparison of numerical solutions of the advective equation

Second- and third-order finite-difference methods recently applied to problems in high-speed fluid flow are applied to the model advection equation cast in conservative form. The differencing methods considered use forward time differencing with both centered and preferential space differences employed in the predictor-corrector sequences. The free parameter required for stability in the third-order method is adjusted to cause the solution to be either minimum dispersive or minimum dissipative in nature. Results indicate that the third-order method using minimum dispersion is the most accurate method tested. Computer time requirements are approximately twice those needed for second-order techniques.

Anderson, D.↗

NASA NTRS1975

Three-dimensional, shock-on-shock interaction problem

The unsteady, three-dimensional flowfield resulting from the interaction of a plane shock with a cone-shaped vehicle traveling supersonically is determined, using a second-order, shock-capturing, finite-difference approach. The time-dependent, inviscid gasdynamic equations are transformed to include the self-similar property of the flow, to align various coordinate surfaces with known shock waves, and to cluster points in the vicinity of the intersection of the transmitted incident shock and the surface of the vehicle. The governing partial differential equations in conservation-law form are then solved iteratively using MacCormack's (1969) algorithm.

Kutler, P.↗

NASA NTRS1976

A BBGKY framework for fluid turbulence

A framework is presented for a systematic kinetic theory of turbulence originating from the Liouville equation for the Fourier coefficients of fluid variables. The real and imaginary parts of these Fourier coefficients play the role that particle coordinates (positions and momenta) play in the BBGKY theory. The basic relations of the problem are the incompressible Navier-Stokes equations in two dimensions with zero viscosity, with the probability distributions of Fourier coefficients rather than moments being the basic variables of the theory. A kinetic equation is derived and shown to possess a number of requirements that any reasonable kinetic equation must have: conservation laws, positive-definite spectral densities, and an H-theorem. The major lack in the theory is any reliable information on the relaxation predicted by the complicated linear operator H. Closure of the hierarchy is achieved by the hypothesis that the five-coefficient correlation function is negligible. Problems associated with inclusion of viscosity and external driving forces are discussed.

Montgomery, D.↗