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NASA NTRS1990

The calculation of rotor/fuselage interaction for two-dimensional bodies

Unsteady rotor wake interactions with the empennage, tail boom, and other aerodynamic surfaces have a significant influence on the aerodynamic performance of the helicopter, ride quality, and vibration. A Computational Fluid Dynamic (CFD) method for computing the aerodynamic interaction between an interacting vortex wake and the viscous flow about arbitrary 2-D bodies was developed to address this helicopter problem. The vorticity and flow field velocities are calculated on a body-fitted computational mesh using an uncoupled iterative solution. The interacting vortex wake is represented by an array of discrete vortices which, in turn, are represented by a finite core model. The evolution of the interacting vortex wake is calculated by Lagrangian techniques. The flow around circular and elliptic cylinders in the absence of an interacting vortex wake was calculated. These results compare very well with other numerical results and with results obtained from experiment and thereby demonstrate the accuracy of the viscous solution. The interaction of a simulated rotor wake with the flow about 2-D bodies, representing cross sections of fuselage components, was calculated to address the vortex interaction problem. The vortex interaction was calculated for the flow about a circular and an elliptic cylinder at 45 and 90 degrees incidence. The results demonstrate the significant variation in lift and drag on the 2-D bodies during the vortex interaction.

Stremel, Paul M.↗

NASA NTRS1976

Digital controllers for VTOL aircraft

Using linear-optimal estimation and control techniques, digital-adaptive control laws have been designed for a tandem-rotor helicopter which is equipped for fully automatic flight in terminal area operations. Two distinct discrete-time control laws are designed to interface with velocity-command and attitude-command guidance logic, and each incorporates proportional-integral compensation for non-zero-set-point regulation, as well as reduced-order Kalman filters for sensor blending and noise rejection. Adaptation to flight condition is achieved with a novel gain-scheduling method based on correlation and regression analysis. The linear-optimal design approach is found to be a valuable tool in the development of practical multivariable control laws for vehicles which evidence significant coupling and insufficient natural stability.

Stengel, R. F.↗

NASA NTRS1987

Position Control for Non-linear, Multiple-Link Robots

Several approaches to complicated control problem examined. Report surveys methods for controlling motion of robot manipulators. Applies to coupled, highly nonlinear multiple-link robots. Presents adaptive control technique based on discrete linear model of robot arm, obtained by linearizing nonlinear dynamical equations about nominal trajectory. Parameters of model calculated with adaptive parameter-identification algorithm. Based on system model, optimal control input computed using position and velocity feedback.

Seraji, H.↗

NASA NTRS1991

Post-glacial relaxation of a viscously stratified compressible mantle

The postglacial relaxation phenomenon of a viscously compressible nonself-gravitating spherical-shell model is investigated. Analytical solutions to an exponentially depth-dependent viscosity are developed with various types of compressible density models such as those with an exponential dependence of the radius and an algebraic-root dependence of the radius. The solutions of a viscously compressible multilayered model with constant thermodynamic properties and viscosity are developed by the propagator matrix method. The results show that inferences of deep mantle viscosity from postglacial rebound would be hampered by mantle compressibility for long-wavelength harmonics because of the smaller excitation of compressible eigenfunctions in the lower mantle. The relaxation times and velocity fields are more sensitive to higher viscosity contrast for the exponentially varying viscosity than for models with discrete jumps in the viscosity structure.

Wu, J.↗

NASA NTRS1956

The Near Noise Field of Static Jets and Some Model Studies of Devices for Noise Reduction

An experimental study is presented of the pressure fluctuations near jet exhaust streams made during unchoked operation of a turbojet engine and a 1-inch-diameter high-temperature model jet and during choked operation of various sizes of model jets with unheated air. The tests for unchoked operation indicate a random spectrum of rather narrow band width which varies in frequency content with axial position along the jet. Pressure surveys from the model tests along lines parallel to the 15 degree jet boundary indicate that the station of greatest pressure fluctuations is determined by the jet velocity and the radial distance, with a tendency of the maximum to shift downstream as either parameter is increased. From model tests the magnitude of the fluctuations appears to increase as about the second power of jet velocity at points just outside the jet boundary and as increasingly higher powers of jet velocity as distance from the boundary is increased. A laboratory method of noise reduction with model jets was found to produce large decreases in the magnitude of the lower-frequency components of the spectra and thereby also to reduce the total radiated energy. Choked operation of model jets with unheated air indicates the appearance of a discrete-frequency component of very large magnitude. Shadowgraph records of the flow show that this condition is associated with the appearance of flow formations suggestive of partly formed toroidal vortices in the vicinity of the shocks. Elimination of these formations is found to eliminate the discrete component and thereby to reduce the overall noise level.

Lassiter, Leslie W↗

NASA NTRS2006

Simulations of Sea-Ice Dynamics Using the Material-Point Method

In recent years, the availability of large volumes of recorded ice motion derived from high-resolution SAR data has provided an amazingly detailed look at the deformation of the ice cover. The deformation is dominated by the appearance of linear kinematic features that have been associated with the presence of leads. These remarkable data put us in a position to begin detailed evaluation of current coupled mechanical and thermodynamic models of sea ice. This presentation will describe the material point method (MPM) for solving these model equations. MPM is a numerical method for continuum mechanics that combines the best aspects of Lagrangian and Eulerian discretizations. The material points provide a Lagrangian description of the ice that models convection naturally. Thus, properties such as ice thickness and compactness are computed in a Lagrangian frame and do not suffer from errors associated with Eulerian advection schemes, such as artificial diffusion, dispersion, or oscillations near discontinuities. This desirable property is illustrated by solving transport of ice in uniform, rotational and convergent velocity fields. Moreover, the ice geometry is represented by unconnected material points rather than a grid. This representation facilitates modeling the large deformations observed in the Arctic, as well as localized deformation along leads, and admits a sharp representation of the ice edge. MPM also easily allows the use of any ice constitutive model. The versatility of MPM is demonstrated by using two constitutive models for simulations of wind-driven ice. The first model is a standard viscous-plastic model with two thickness categories. The MPM solution to the viscous-plastic model agrees with previously published results using finite elements. The second model is a new elastic-decohesive model that explicitly represents leads. The model includes a mechanism to initiate leads, and to predict their orientation and width. The elastic-decohesion model can provide similar overall deformation as the viscous-plastic model; however, explicit regions of opening and shear are predicted. Furthermore, the efficiency of MPM with the elastic-decohesive model is competitive with the current best methods for sea ice dynamics. Simulations will also be presented for an area of the Beaufort Sea, where predictions can be validated against satellite observations of the Arctic.

Sulsky, D.↗

NASA NTRS2018

A Linear-Elasticity Solver for Higher-Order Space-Time Mesh Deformation

A linear-elasticity approach is presented for the generation of meshes appropriate for a higher-order space-time discontinuous finite-element method. The equations of linear-elasticity are discretized using a higher-order, spatially-continuous, finite-element method. Given an initial finite-element mesh, and a specified boundary displacement, we solve for the mesh displacements to obtain a higher-order curvilinear mesh. Alternatively, for moving-domain problems we use the linear-elasticity approach to solve for a temporally discontinuous mesh velocity on each time-slab and recover a continuous mesh deformation by integrating the velocity. The applicability of this methodology is presented for several benchmark test cases.

higher-order↗

NASA NTRS1996

A Mixed Finite Volume Element Method for Flow Calculations in Porous Media

A key ingredient in the simulation of flow in porous media is the accurate determination of the velocities that drive the flow. The large scale irregularities of the geology, such as faults, fractures, and layers suggest the use of irregular grids in the simulation. Work has been done in applying the finite volume element (FVE) methodology as developed by McCormick in conjunction with mixed methods which were developed by Raviart and Thomas. The resulting mixed finite volume element discretization scheme has the potential to generate more accurate solutions than standard approaches. The focus of this paper is on a multilevel algorithm for solving the discrete mixed FVE equations. The algorithm uses a standard cell centered finite difference scheme as the 'coarse' level and the more accurate mixed FVE scheme as the 'fine' level. The algorithm appears to have potential as a fast solver for large size simulations of flow in porous media.

Jim E Jones↗

NASA NTRS1993

Large-scale computation of incompressible viscous flow by least-squares finite element method

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations and results in symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli function-vorticity formulation for incompressible viscous flows is also tested. For three-dimensional cases, an additional compatibility equation, i.e., the divergence of the vorticity vector should be zero, is included to make the first-order system elliptic. The simple substitution of the Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this scheme for large-scale computation, we give numerical results for 2D driven cavity problem at Re = 10000 with 408 x 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1,000 with 50 x 50 x 50 trilinear elements. The Taylor-Goertler-like vortices are observed for Re = 1,000.

Jiang, Bo-Nan↗

NASA NTRS1973

Fundamental Consideration of Noise radiation by rotary wings

An historical review of progress in understanding of rotor noise is presented. Initial work was principally on propellers, but has many obvious applications to noise from rotary wings. Current understanding of rotor noise radiation is then reviewed in some detail. The principal noise sources appear to be: (1) discrete frequency due to distorted inflow; (2) low frequency broadband due to turbulent inflow, and (3) high frequency broadband due to tip effects. On a helicopter rotor each of these sources seems to be intimately connected with the shed vortex wakes. Tip modifications offer one method for controlling the effects. The implications for the designer are discussed. Rotor subjective noise levels appear to obey a velocity to the eighth power law, independent of thrust. Experiments to rectify some of the present deficiencies in knowledge are suggested.

Lowson, M. V.↗

NASA NTRS1990

Simulation Of Unsteady, Viscous, Incompressible Flow

Method for numerical solution of Navier-Stokes equations of viscous, incompressible flow developed based on use of fractional-step procedure. Accurate to second order in both space and time. Attempt made to minimize Poisson-equation difficulties by choosing pressures at centers and volume fluxes across faces of computational cells as dependent variables instead of familiar Cartesian components of velocity. Choice ensures satisfaction of discrete equation of conservation of mass to within round-off errors in any coordinate system and has favorable effects on convergence properties.

Rosenfeld, Moshe↗

NASA NTRS1995

Sensitivity analysis for the dynamic response of thermoviscoplastic shells of revolution

A computational procedure is presented for evaluating the sensitivity coefficients of the dynamic axisymmetric, fully-coupled, thermoviscoplastic response of shells of revolution. The analytical formulation is based on Reissner's large deformation shell theory with the effects of large-strain, transverse shear deformation, rotatory inertia and moments turning around the normal to the middle surface included. The material model is chosen to be viscoplasticity with strain hardening and thermal hardening, and an associated flow rule is used with a von Mises effective stress. A mixed formulation is used for the shell equations with the fundamental unknowns consisting of six stress resultants, three generalized displacements and three velocity components. The energy-balance equation is solved using a Galerkin procedure, with the temperature as the fundamental unknown. Spatial discretization is performed in one dimension (meridional direction) for the momentum and constitutive equations of the shell, and in two dimensions (meridional and thickness directions) for the energy-balance equation. The temporal integration is performed by using an explicit central difference scheme (leap-frog method) for the momentum equation; a predictor-corrector version of the trapezoidal rule is used for the energy-balance equation; and an explicit scheme consistent with the central difference method is used to integrate the constitutive equations. The sensitivity coefficients are evaluated by using a direct differentiation approach. Numerical results are presented for a spherical cap subjected to step loading.

Kulkarni, Makarand↗

NASA NTRS1988

Calculation of symmetric and asymmetric vortex seperation on cones and tangent ogives based on discrete vortex models

An inviscid discrete vortex model, with newly derived expressions for the tangential velocity imposed at the separation points, is used to investigate the symmetric and asymmetric vortex separation on cones and tangent ogives. The circumferential locations of separation are taken from experimental data. Based on a slender body theory, the resulting simultaneous nonlinear algebraic equations in a cross-flow plane are solved with Broyden's modified Newton-Raphson method. Total force coefficients are obtained through momentum principle with new expressions for nonconical flow. It is shown through the method of function deflation that multiple solutions exist at large enough angles of attack, even with symmetric separation points. These additional solutions are asymmetric in vortex separation and produce side force coefficients which agree well with data for cones and tangent ogives.

Chin, S.↗

NASA NTRS1998

A Posteriori Bounds for Linear-Functional Outputs of Crouzeix-Raviart Finite Element Discretizations of the Incompressible Stokes Problem

We present a finite element technique for the efficient generation of lower and upper bounds to outputs which are linear functionals of the solutions to the incompressible Stokes equations in two space dimensions; the finite element discretization is effected by Crouzeix-Raviart elements, the discontinuous pressure approximation of which is central to our approach. The bounds are based upon the construction of an augmented Lagrangian: the objective is a quadratic "energy" reformulation of the desired output; the constraints are the finite element equilibrium equations (including the incompressibility constraint), and the intersubdomain continuity conditions on velocity. Appeal to the dual max-min problem for appropriately chosen candidate Lagrange multipliers then yields inexpensive bounds for the output associated with a fine-mesh discretization; the Lagrange multipliers are generated by exploiting an associated coarse-mesh approximation. In addition to the requisite coarse-mesh calculations, the bound technique requires solution only of local subdomain Stokes problems on the fine-mesh. The method is illustrated for the Stokes equations, in which the outputs of interest are the flowrate past, and the lift force on, a body immersed in a channel.

Patera, Anthony T.↗

NASA NTRS1990

Effects of streamwise curvature variations on Goertler vortices

The influence of wall curvature oscillations on Goertler vortices is studied using an asymptotic method. A large number of oscillations leads to a stabilization of the Goertler vortices. In addition, basic velocity profile of a flow over a wall with a large number of oscillations is similar to that over a wall without oscillation. The curvature stays positive while oscillating and the wall is concave. On a wall with curvature decreasing to zero, the Goertler vortices move outside of the boundary layer; no discrete eigenvalue of Goertler instability is found on a wall with zero curvature. In this case the discrete eigenvalues merge into a continuous spectrum.

Jallade, Sophie↗

NASA NTRS1982

A vortex wake capturing method for potential flow calculations

A method is presented for modifying finite difference solutions of the potential equation to include the calculation of non-planar vortex wake features. The approach is an adaptation of Baker's 'cloud in cell' algorithm developed for the stream function-vorticity equations. The vortex wake is tracked in a Lagrangian frame of reference as a group of discrete vortex filaments. These are distributed to the Eulerian mesh system on which the velocity is calculated by a finite difference solution of the potential equation. An artificial viscosity introduced by the finite difference equations removes the singular nature of the vortex filaments. Computed examples are given for the two-dimensional time dependent roll-up of vortex wakes generated by wings with different spanwise loading distributions.

Murman, E. M.↗

NASA NTRS1993

A least-squares finite element method for 3D incompressible Navier-Stokes equations

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Jiang, Bo-Nan↗

NASA NTRS1996

Three-Dimensional Simulations of Marangoni-Benard Convection in Small Containers by the Least-Squares Finite Element Method

This paper reports a numerical study of the Marangoni-Benard (MB) convection in a planar fluid layer. The least-squares finite element method (LSFEM) is employed to solve the three-dimensional Stokes equations and the energy equation. First, the governing equations are reduced to be first-order by introducing variables such as vorticity and heat fluxes. The resultant first-order system is then cast into a div-curl-grad formulation, and its ellipticity and permissible boundary conditions are readily proved. This numerical approach provides an equal-order discretization for velocity, pressure, vorticity, temperature, and heat conduction fluxes, and therefore can provide high fidelity solutions for the complex flow physics of the MB convection. Numerical results reported include the critical Marangoni numbers (M(sub ac)) for the onset of the convection in containers with various aspect ratios, and the planforms of supercritical MB flows. The numerical solutions compared favorably with the experimental results reported by Koschmieder et al..

Yu, Sheng-Tao↗