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NASA NTRS1990

Estimation of time- and state-dependent delays and other parameters in functional differential equations

A parameter estimation algorithm is developed which can be used to estimate unknown time- or state-dependent delays and other parameters (e.g., initial condition) appearing within a nonlinear nonautonomous functional differential equation. The original infinite dimensional differential equation is approximated using linear splines, which are allowed to move with the variable delay. The variable delays are approximated using linear splines as well. The approximation scheme produces a system of ordinary differential equations with nice computational properties. The unknown parameters are estimated within the approximating systems by minimizing a least-squares fit-to-data criterion. Convergence theorems are proved for time-dependent delays and state-dependent delays within two classes, which say essentially that fitting the data by using approximations will, in the limit, provide a fit to the data using the original system. Numerical test examples are presented which illustrate the method for all types of delay.

Murphy, K. A.↗

NASA NTRS1990

Development of a time-dependent incompressible Navier-Stokes solver based on a fractional-step method

The development, validation and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems are discussed. A solution method that combines a finite-volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries was previously developed for fixed-grids. In the present research effort, this solution method is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality.

Rosenfeld, Moshe↗

NASA NTRS1990

Development of a time-dependent incompressible Navier-Stokes solver based on a fractional-step method

The main goals are the development, validation, and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems. A solution method that combines a finite volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality.

Rosenfeld, Moshe↗

NASA NTRS1990

Finite element thermo-viscoplastic analysis of aerospace structures

The time-dependent thermo-viscoplastic response of aerospace structures subjected to intense aerothermal loads is predicted using the finite-element method. The finite-element analysis uses the Bodner-Partom unified viscoplastic constitutive relations to determine rate-dependent nonlinear material behavior. The methodology is verified by comparison with experimental data and other numerical results for a uniaxially-loaded bar. The method is then used (1) to predict the structural response of a rectangular plate subjected to line heating along a centerline, and (2) to predict the thermal-structural response of a convectively-cooled engine cowl leading edge subjected to aerodynamic shock-shock interference heating. Compared to linear elastic analysis, the viscoplastic analysis results in lower peak stresses and regions of plastic deformations.

Pandey, Ajay K.↗

NASA NTRS1990

Finite-element thermo-viscoplastic analysis of aerospace structures

Pandey, Ajay↗

NASA NTRS1991

Numerical simulation of transition in wall-bounded shear flows

The current status of numerical simulation techniques for the transition to turbulence in incompressible channel and boundary-layer flows is surveyed, and typical results are presented graphically. The focus is on direct numerical simulations based on the full nonlinear time-dependent Navier-Stokes equations without empirical closure assumptions for prescribed initial and boundary conditions. Topics addressed include the vibrating ribbon problem, space and time discretization, initial and boundary conditions, alternative methods based on the triple-deck approximation, two-dimensional channel and boundary-layer flows, three-dimensional boundary layers, wave packets and turbulent spots, compressible flows, transition control, and transition modeling.

Kleiser, Leonhard↗

NASA NTRS1991

A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

The time-dependent, three-dimensional incompressible Navier-Stokes equations are presently solved in generalized coordinate systems by means of a fractional-step method whose primitive variable formulation uses as dependent variables, in place of the Cartesian components of the velocity: (1) pressure (defined at the center of the computational cell), and (2) volume fluxes across the faces of the cells. The momentum equations are solved by means of an approximate factorization method. A novel 'ZEBRA' scheme incorporating four-color ordering efficiently solves the Poisson equation. Illustrative two- and three-dimensional laminar flow test cases are computed and evaluated relative to extant numerical and experimental results, and good agreement is obtained.

Rosenfeld, Moshe↗

NASA NTRS1991

Multigrid for hypersonic viscous two- and three-dimensional flows

The use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows is considered. The time-dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that remove the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock-capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are given for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional viscous flow over a blunt biconic.

Turkel, E.↗

NASA NTRS1991

A time-dependent incompressible viscous BEM for moderate Reynolds numbers

The boundary element method is applied to transient viscous incompressible flow. The time-domain formulation allows a boundary-only solution for linear Stokes flow. For higher speed flows in which the nonlinear convective effects cannot be ignored, a volume integral must be retained. However, the introduction of reference velocities often limits the nonlinear region to the vicinity of obstacles or boundary surfaces. Additionally, the volume terms are rewritten to eliminate the need for the calculation of velocity gradients. A general purpose numerical implementation of this new formulation then produces a very attractive tool for engineering analysis. This implementation includes a Newton-Raphson algorithm, permitting accurate solutions up to the moderate Reynolds number range. Several numerical examples are provided to validate the present approach.

Dargush, G. F.↗

NASA NTRS1992

Fractional-Step, Finite-Volume Computation Of Flow

Method of solving Navier-Stokes equations of incompressible flow in general nonorthogonal curvilinear coordinates incorporates fractional-step and finite-volume approaches. Developed in continuing effort to achieve accuracy without need for excessive computation time in numerical simulation of time-dependent, three-dimensional flows bounded by surfaces of complicated shape.

Kwak, Dochan↗

NASA NTRS1992

A numerical study of two-dimensional vortex shedding from rectangular cylinders

An efficient time-marching, non-iterative calculation method is used to analyze time-dependent flows around rectangular cylinders. The turbulent flow in the wake region of a square section cylinder is analyzed using an anisotropic k-epsilon model. Initiation and subsequent development of the vortex shedding phenomenon is naturally captured once a perturbation is introduced in the flow. Transient calculations using standard eddy-viscosity and an anisotropic k-epsilon model averaged over an integral number of cycles to get the fluctuating energy (organized and turbulent) are compared with experimental data. It is shown that the anisotropic k-epsilon model resolves the anisotropy of the Reynolds stresses and gives mean energy distribution closer to the experiment than the standard k-epsilon model.

Hadid, A. H.↗

NASA NTRS1991

Time-dependent solutions of viscous incompressible flows in moving co-ordinates

A time-accurate solution method for the incompressible Navier-Stokes equations in generalized moving coordinates is presented. A finite-volume discretization method that satisfies the geometric conservation laws for time-varying computational cells is used. The discrete equations are solved by a fractional-step solution procedure. The solution is second-order-accurate in space and first-order-accurate in time. The pressure and the volume fluxes are chosen as the unknowns to facilitate the formulation of a consistent Poisson equation and thus to obtain a robust Poisson solver with favorable convergence properties. The method is validated by comparing the solutions with other numerical and experimental results. Good agreement is obtained in all cases.

Rosenfeld, Moshe↗

NASA NTRS1992

Calculation of two-dimensional shear-driven cavity flows at high Reynolds numbers

The time-dependent Navier-Stokes equations are numerically integrated for two-dimensional incompressible viscous flow in a shear-driven square cavity. Using a time-splitting method and finite differences on a staggered mesh, the momentum and pressure equations are directly solved by a tensor product method where one finite difference direction is diagonalized by eigenvalue decomposition. The effects of increasing Reynolds number are studied and the developing boundary layer is captured by using a finely clustered mesh. At Re = 30,000 the flow is in a continuously developing unsteady regime. Power spectrum plots indicate that the unsteady flow oscillates with one fundamental frequency and exhibits some characteristics of transition between laminar and turbulent states.

Huser, A.↗

NASA NTRS1992

A finite-volume numerical method to calculate fluid forces and rotordynamic coefficients in seals

A numerical method to calculate rotordynamic coefficients of seals is presented. The flow in a seal is solved by using a finite-volume formulation of the full Navier-Stokes equations with appropriate turbulence models. The seal rotor is perturbed along a diameter such that the position of the rotor is a sinusoidal function of time. The resulting flow domain changes with time, and the time-dependent flow in the seal is solved using a space conserving moving grid formulation. The time-varying fluid pressure reaction forces are then linked with the rotor center displacement, velocity and acceleration to yield the rotordynamic coefficients. Results for an annular seal are presented, and compared with experimental data and other more simplified numerical methods.

Athavale, M. M.↗

NASA NTRS1992

Development of a fractional-step method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems

A fractional step method is developed for solving the time-dependent three-dimensional incompressible Navier-Stokes equations in generalized coordinate systems. The primitive variable formulation uses the pressure, defined at the center of the computational cell, and the volume fluxes across the faces of the cells as the dependent variables, instead of the Cartesian components of the velocity. This choice is equivalent to using the contravariant velocity components in a staggered grid multiplied by the volume of the computational cell. The governing equations are discretized by finite volumes using a staggered mesh system. The solution of the continuity equation is decoupled from the momentum equations by a fractional step method which enforces mass conservation by solving a Poisson equation. This procedure, combined with the consistent approximations of the geometric quantities, is done to satisfy the discretized mass conservation equation to machine accuracy, as well as to gain the favorable convergence properties of the Poisson solver. The momentum equations are solved by an approximate factorization method, and a novel ZEBRA scheme with four-color ordering is devised for the efficient solution of the Poisson equation. Several two- and three-dimensional laminar test cases are computed and compared with other numerical and experimental results to validate the solution method. Good agreement is obtained in all cases.

Rosenfeld, Moshe↗

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 NTRS1995

Finite Element Analysis of Poroelastic Composites Undergoing Thermal and Gas Diffusion

A theory for time-dependent thermal and gas diffusion in mechanically time-rate-independent anisotropic poroelastic composites has been developed. This theory advances previous work by the latter two authors by providing for critical transverse shear through a three-dimensional axisymmetric formulation and using it in a new hypothesis for determining the Biot fluid pressure-solid stress coupling factor. The derived governing equations couple material deformation with temperature and internal pore pressure and more strongly couple gas diffusion and heat transfer than the previous theory. Hence the theory accounts for the interactions between conductive heat transfer in the porous body and convective heat carried by the mass flux through the pores. The Bubnov Galerkin finite element method is applied to the governing equations to transform them into a semidiscrete finite element system. A numerical procedure is developed to solve the coupled equations in the space and time domains. The method is used to simulate two high temperature tests involving thermal-chemical decomposition of carbon-phenolic composites. In comparison with measured data, the results are accurate. Moreover unlike previous work, for a single set of poroelastic parameters, they are consistent with two measurements in a restrained thermal growth test.

Salamon, N. J.↗

NASA NTRS1997

An Adaptive Flow Solver for Air-Borne Vehicles Undergoing Time-Dependent Motions/Deformations

This report describes a concurrent Euler flow solver for flows around complex 3-D bodies. The solver is based on a cell-centered finite volume methodology on 3-D unstructured tetrahedral grids. In this algorithm, spatial discretization for the inviscid convective term is accomplished using an upwind scheme. A localized reconstruction is done for flow variables which is second order accurate. Evolution in time is accomplished using an explicit three-stage Runge-Kutta method which has second order temporal accuracy. This is adapted for concurrent execution using another proven methodology based on concurrent graph abstraction. This solver operates on heterogeneous network architectures. These architectures may include a broad variety of UNIX workstations and PCs running Windows NT, symmetric multiprocessors and distributed-memory multi-computers. The unstructured grid is generated using commercial grid generation tools. The grid is automatically partitioned using a concurrent algorithm based on heat diffusion. This results in memory requirements that are inversely proportional to the number of processors. The solver uses automatic granularity control and resource management techniques both to balance load and communication requirements, and deal with differing memory constraints. These ideas are again based on heat diffusion. Results are subsequently combined for visualization and analysis using commercial CFD tools. Flow simulation results are demonstrated for a constant section wing at subsonic, transonic, and a supersonic case. These results are compared with experimental data and numerical results of other researchers. Performance results are under way for a variety of network topologies.

Singh, Jatinder↗