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NASA NTRS1982

Flow aerodynamics modeling of an MHD swirl combustor - Calculations and experimental verification

This paper describes a computer code for calculating the flow dynamics of constant density flow in the second stage trumpet shaped nozzle section of a two stage MHD swirl combustor for application to a disk generator. The primitive pressure-velocity variable, finite difference computer code has been developed to allow the computation of inert nonreacting turbulent swirling flows in an axisymmetric MHD model swirl combustor. The method and program involve a staggered grid system for axial and radial velocities, and a line relaxation technique for efficient solution of the equations. Turbulence simulation is by way of a two-equation Kappa-epsilon model. The code produces as output the flowfield map of the nondimensional stream function, axial, and swirl velocity. Good agreement was obtained between the theoretical predictions and the qualitative experimental results. The best seed injector location for uniform seed distribution at combustor exit is with injector located centrally on the combustor axis at entrance to the second stage combustor.

Gupta, A. K.↗

NASA NTRS1982

ADI on staggered mesh - A method for the calculation of compressible convection

An alternating direction implicit (ADI) method has been applied to a staggered grid for the computation of convection in a highly stratified fluid. Since artificial viscosity is not needed, subtle effects like the onset of convection can be studied. These computations compare well with the 2-D results by Graham (1975) and also agree with standard Boussinesq results when taken to that limit. Good efficiency has been achieved with a time step hundreds of times larger than the stability limit imposed by the explicit treatment of diffusion and the Courant number is not restricted to be below 1. The Navier-Stokes equation contains cross spatial derivatives which are treated explicitly in most ADI schemes. The destabilizing effect of such a practice on a 2-D model system with second-order spatial derivative terms only was analyzed and found to be not excessive.

Chan, K. L.↗

NASA NTRS1984

Variational objective analyses for cyclone studies

The basic analysis equations, i.e., the two horizontal momentum equations, the hydrostatic equation, and the integrated continuity equation were derived for the nonlinear vertical coordinate, nondimensionalized, and expressed in finite differences on a staggered grid. Special care was taken to transform the hydrostatic equation and the pressure gradient terms of the horizontal momentum equations to nearly eliminate truncation error over steeply sloping terrain. This formulation also eliminated explicit reference to orographically induced variations in the thermodynamic variables so that the variational adjustments are on the scale of the meteorological perturbations. The analysis equations were subjected to the Euler-Lagrange operations as expressed for finite differences and an additional set of five partial differential equations was derived, bringing to nine the number of equations in Model I. Higher order terms, terms containing observed quantities, and terms containing none of the variables to be adjusted were grouped into forcing functions and the equations were solved for the zero order terms. Zero order variables were eliminated between these equations and there resulted two diagnostic equations which take the form of general linear second order partial differential equations with nonconstant coefficients.

Achtemeier, G. L.↗

NASA NTRS1985

On spectral multigrid methods for the time-dependent Navier-Stokes equations

A splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.

Zang, T. A.↗

NASA NTRS1986

On spectral multigrid methods for the time-dependent Navier-Stokes equations

Zang, T. A.↗

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 NTRS1986

Numerical prediction of cold turbulent flow in combustor configurations with different centerbody flame holders

A numerical study of cold turbulent flow in a combustor containing a centerbody flame holder is presented. The axisymmetric Navier-Stokes equations incorporating a k-epsilon turbulence model were solved in a nonorthogonal curvilinear coordinate system. The finite volume method, applied to a staggered grid system, was used to discretize the differential equations. These finite differenced equations were then solved iteratively utilizing the SIMPLE algorithm described by Patankar (1980). Solutions were obtained for two combustor duct geometries with various centerbody flame holders which included a disk, a cone, and a sphere. The extent of mixing due to these bodies was evaluated. A comparison with previously obtained experimental data yielded moderately good agreement.

Yung, C. N.↗

NASA NTRS1988

Developing fluid flow in a curved duct of square cross-section and its fully developed dual solutions

Developing fluid flow in a curved duct of square cross-section is studied numerically by a factored ADI finite-difference method on a staggered grid. A central-difference scheme with primitive variables is used inside the computational domain to reduce numerical diffusion. Two Reynolds numbers, 574 and 790, based upon a bulk velocity and hydraulic diameter are chosen for curvature ratios of 1/6.45 and 1/2.3, respectively. It is found that the secondary flow is far more complicated than expected, with the appearance of at least two pairs of vortices. Main-flow separation is also observed for the higher curvature ratio. Furthermore, it is observed that the flow develops into two quite different states downstream, depending upon the inlet conditions. Solutions of the fully developed Navier-Stokes equations is shown to be not unique beyond a certain critical Reynolds number. Developing flow seems to evolve into the fully developed state along a particular branch into which the fully developed solution bifurcates.

Soh, W. Y.↗

NASA NTRS1988

Numerical modeling of chemical vapor deposition (CVD) in a horizontal reactor

In the present numerical prediction of the deposition rate of silicon from silane in a CVD process, the conservation equations for mass, momentum, energy, and chemical species are solved on a staggered grid using the SIMPLE algorithm, while the rate of chemical reactions in the gas phase and on the susceptor surface is obtained from an Arrhenius rate equation. Predicted deposition rates as a function of position along the susceptor with and without the gas phase chemical reaction are compared with the available experimental and numerical data; agreement is excellent except at the leading edge of the susceptor, where the deposition rate is overpredicted.

Sheikholeslami, M. Z.↗

NASA NTRS1989

Control-volume based Navier-Stokes equation solver valid at all flow velocities

A control-volume based finite difference method to solve the Reynolds averaged Navier-Stokes equations is presented. A pressure correction equation valid at all flow velocities and a pressure staggered grid layout are used in the method. Example problems presented herein include: a developing laminar channel flow, developing laminar pipe flow, a lid-driven square cavity flow, a laminar flow through a 90-degree bent channel, a laminar polar cavity flow, and a turbulent supersonic flow over a compression ramp. A k-epsilon turbulence model supplemented with a near-wall turbulence model was used to solve the turbulent flow. It is shown that the method yields accurate computational results even when highly skewed, unequally spaced, curved grids are used. It is also shown that the method is strongly convergent for high Reynolds number flows.

Kim, S.-W.↗

NASA NTRS1989

Calculations of rotational flows using stream function

The stream function equation is solved for steady two-dimensional (and axisymmetric) rotational flows. Both finite differences and finite volumes discretization techniques are studied, using generalized body fitted coordinates and unstructured staggered grids, respectively. For inviscid transonic flows, a new artificial viscosity scheme which does not produce any artificial vorticity is introduced, for the stability of the mixed flow calculations and for capturing shocks. The solution of Euler equations, in primitive variables, are also considered. The effects of the artificial viscosity and numerical boundary conditions on the total enthalpy and the vorticity distributions are demonstrated.

Hafez, M.↗

NASA NTRS1991

Choice of velocity variables for complex flow computation

The issue of adopting the velocity components as dependent velocity variables for the Navier-Stokes flow computations is investigated. The viewpoint advocated is that a numerical algorithm should preferably honor both the physical conservation law in differential form and the geometric conservation law in discrete form. With the use of Cartesian velocity vector, the momentum equations in curvilinear coordinates can retain the full conservation-law form and satisfy the physical conservation laws. With the curvilinear velocity components, source terms appear in differential equations and hence the full conservation law form can not be retained. In discrete expressions, algorithms based on the Cartesian components can satisfy the geometric conservation-law form for convection terms but not for viscous terms; those based on the curvilinear components, on the other hand, cannot satisfy the geometric conservation-law form for either convection or viscous terms. Several flow solutions for domain with 90 and 360 degree turnings are presented to illustrate the issues of using the Cartesian velocity components and the staggered grid arrangement.

Shyy, W.↗

NASA NTRS1991

Computational modelling of dump combustors flowfield

A computational model aimed at predicting the flowfield of dump combustors is presented. The turbulent combustion model is based on the conserved scalar approach and on a convenient specification of its probability density function, which reduces the computation of the mean density to a closed form. Turbulence is modeled by means of the k-epsilon model. The averaged conservation equations are solved by a technique based on a staggered grid and on the SIMPLE solver. The computational model is applied to a simple dump combustor to assess the computer time requirements and accuracy. The turbulent combustion model is shown to reduce the computer time by an order of magnitude when compared to evaluating the mean density by numerical quadrature.

Lentini, D.↗

NASA NTRS1990

Prediction of the Thrust Performance and the Flowfield of Liquid Rocket Engines

In an effort to improve the current solutions in the design and analysis of liquid propulsive engines, a computational fluid dynamics (CFD) model capable of calculating the reacting flows from the combustion chamber, through the nozzle to the external plume, was developed. The Space Shuttle Main Engine (SSME) fired at sea level, was investigated as a sample case. The CFD model, FDNS, is a pressure based, non-staggered grid, viscous/inviscid, ideal gas/real gas, reactive code. An adaptive upwinding differencing scheme is employed for the spatial discretization. The upwind scheme is based on fourth order central differencing with fourth order damping for smooth regions, and second order central differencing with second order damping for shock capturing. It is equipped with a CHMQGM equilibrium chemistry algorithm and a PARASOL finite rate chemistry algorithm using the point implicit method. The computed flow results and performance compared well with those of other standard codes and engine hot fire test data. In addition, the transient nozzle flowfield calculation was also performed to demonstrate the ability of FDNS in capturing the flow separation during the startup process.

Wang, T.-S.↗

NASA NTRS1991

A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations: Validation and model problems

An algorithm is described for the solution of the laminar, incompressible Navier-Stokes equations. The basic algorithm is a multigrid based on a robust, box-based smoothing step. Its most important feature is the incorporation of automatic, dynamic mesh refinement. This algorithm supports generalized simple domains. The program is based on a standard staggered-grid formulation of the Navier-Stokes equations for robustness and efficiency. Special grid transfer operators were introduced at grid interfaces in the multigrid algorithm to ensure discrete mass conservation. Results are presented for three models: the driven-cavity, a backward-facing step, and a sudden expansion/contraction.

Thompson, C. P.↗

NASA NTRS1992

An analysis of the flow field in the region of the ASRM field joints

The flow field in the region of a solid rocket motor field joint is very important since fluid dynamic and mechanical propellant stresses can couple to cause a motor failure at a joint. Presented here is an examination of the flow field in the region of the Advanced Solid Rocket Motor (ASRM) field joints. The analyses were performed as a first step in assessing the design of the ASRM forward and aft field joints in order to assure the proper operation of the motor prior to further development of test firing. The analyses presented here were performed by employing a two-dimensional axisymmetric assumption. Fluent/BFC, a three dimensional full Navier-Stokes flow field code, was used to make the numerical calculations. This code utilizes a staggered grid formulation along with the SIMPLER numerical algorithm. Wall functions are used to determine the character of the laminar sublayer, and a standard kappa-epsilon turbulence model is used to close the fluid dynamic equations. The analyses performed to this date verify that the ASRM field joint design operates properly. The fluid dynamic stresses at the field joints are small due to the inherent design of the field joints. A problem observed in some other solid rocket motors is that large fluid dynamic stresses are generated at the motor joint on the downstream propellant grain due to forward facing step geometries. The design of the ASRM field joints are such that this is not a problem as shown by the analyses. Also, the analyses of the inhibitor stub left protruding into the port flow from normal propellant burn back show that more information is necessary to complete these analyses. These analyses were performed as parametric analyses in relation to the height of the inhibitor stub left protruding into the motor port. A better estimate of the amount of the inhibitor stub remaining at later burn times must be determined since the height which the inhibitor stub protrudes into the port flow drastically affects the fluid dynamics induced stresses on the propellant grain at the field joints.

Dill, Richard A.↗

NASA NTRS1992

Linear-stability theory of thermocapillary convection in a model of float-zone crystal growth

Linear-stability theory has been applied to a basic state of thermocapillary convection in a model half-zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half-zone is of finite, O(1) aspect ratio with two-dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities will remain partial differential equations. The disturbance equations are treated by a staggered-grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases.

Neitzel, G. P.↗

NASA NTRS1992

Adaptive multigrid domain decomposition solutions for viscous interacting flows

Several viscous incompressible flows with strong pressure interaction and/or axial flow reversal are considered with an adaptive multigrid domain decomposition procedure. Specific examples include the triple deck structure surrounding the trailing edge of a flat plate, the flow recirculation in a trough geometry, and the flow in a rearward facing step channel. For the latter case, there are multiple recirculation zones, of different character, for laminar and turbulent flow conditions. A pressure-based form of flux-vector splitting is applied to the Navier-Stokes equations, which are represented by an implicit lowest-order reduced Navier-Stokes (RNS) system and a purely diffusive, higher-order, deferred-corrector. A trapezoidal or box-like form of discretization insures that all mass conservation properties are satisfied at interfacial and outflow boundaries, even for this primitive-variable, non-staggered grid computation.

Rubin, Stanley G.↗