Results for “linear potential flow” — Page 5

NASA NTRS1974

A finite difference method for the solution of the transonic flow around harmonically oscillating wings

A finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented. The partial differential equation for the unsteady transonic flow was linearized by dividing the flow into separate steady and unsteady perturbation velocity potentials and by assuming small amplitudes of harmonic oscillation. The resulting linear differential equation is of mixed type, being elliptic or hyperbolic whereever the steady flow equation is elliptic or hyperbolic. Central differences were used for all derivatives except at supersonic points where backward differencing was used for the streamwise direction. Detailed formulas and procedures are described in sufficient detail for programming on high speed computers. To test the method, the problem of the oscillating flap on a NACA 64A006 airfoil was programmed. The numerical procedure was found to be stable and convergent even in regions of local supersonic flow with shocks.

Ehlers, E. F.↗

NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for unsteady and steady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux-vector-splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flows results are made as well as with experimental data where available.

Anderson, W. Kyle↗

NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for steady and unsteady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux vector splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flow results are made, as well as with experimental data where available.

Anderson, W. Kyle↗

NASA NTRS1993

Turbine blade forced response prediction using FREPS

This paper describes a software system called FREPS (Forced REsponse Prediction System) that integrates structural dynamic, steady and unsteady aerodynamic analyses to efficiently predict the forced response dynamic stresses in axial flow turbomachinery blades due to aerodynamic and mechanical excitations. A flutter analysis capability is also incorporated into the system. The FREPS system performs aeroelastic analysis by modeling the motion of the blade in terms of its normal modes. The structural dynamic analysis is performed by a finite element code such as MSC/NASTRAN. The steady aerodynamic analysis is based on nonlinear potential theory and the unsteady aerodynamic analyses is based on the linearization of the non-uniform potential flow mean. The program description and presentation of the capabilities are reported herein. The effectiveness of the FREPS package is demonstrated on the High Pressure Oxygen Turbopump turbine of the Space Shuttle Main Engine. Both flutter and forced response analyses are performed and typical results are illustrated.

Murthy, Durbha, V.↗

NASA NTRS1991

Development of a steady potential solver for use with linearized, unsteady aerodynamic analyses

A full potential steady flow solver (SFLOW) developed explicitly for use with an inviscid unsteady aerodynamic analysis (LINFLO) is described. The steady solver uses the nonconservative form of the nonlinear potential flow equations together with an implicit, least squares, finite difference approximation to solve for the steady flow field. The difference equations were developed on a composite mesh which consists of a C grid embedded in a rectilinear (H grid) cascade mesh. The composite mesh is capable of resolving blade to blade and far field phenomena on the H grid, while accurately resolving local phenomena on the C grid. The resulting system of algebraic equations is arranged in matrix form using a sparse matrix package and solved by Newton's method. Steady and unsteady results are presented for two cascade configurations: a high speed compressor and a turbine with high exit Mach number.

Hoyniak, Daniel↗

NASA NTRS1984

A linear aerodynamic analysis for unsteady transonic cascades

A potential flow analysis to predict unsteady airloads produced by the vibrations of turbomachinery blades operating at transonic Mach numbers is presented. The unsteady aerodynamic model includes the effects of blade geometry, finite mean pressure variation across the blade row, high frequency blade motion, and shock motion within the framework of a linearized, frequency domain formulation. The unsteady equations are solved implicit, least squares, finite difference approximation which is applicable on arbitrary grids. A numerical solution for the entire unsteady field is determined by matching a solution determined on a rectilinear type cascade mesh, which covers an extended blade passage region, to a solution determined on a detailed polar type local mesh, which covers and extends well beyond the supersonic region(s) adjacent to a blade surface. Cascades of double circular arc and flat plate blades demonstrate the unsteady analysis, and partially illustrate the effects of blade geometry, inlet Mach number, blade vibration frequency and shock motion on unsteady response.

Verdon, J. M.↗

NASA NTRS1995

Sensitivity analysis for aeroacoustic and aeroelastic design of turbomachinery blades

A new method for computing the effect that small changes in the airfoil shape and cascade geometry have on the aeroacoustic and aeroelastic behavior of turbomachinery cascades is presented. The nonlinear unsteady flow is assumed to be composed of a nonlinear steady flow plus a small perturbation unsteady flow that is harmonic in time. First, the full potential equation is used to describe the behavior of the nonlinear mean (steady) flow through a two-dimensional cascade. The small disturbance unsteady flow through the cascade is described by the linearized Euler equations. Using rapid distortion theory, the unsteady velocity is split into a rotational part that contains the vorticity and an irrotational part described by a scalar potential. The unsteady vorticity transport is described analytically in terms of the drift and stream functions computed from the steady flow. Hence, the solution of the linearized Euler equations may be reduced to a single inhomogeneous equation for the unsteady potential. The steady flow and small disturbance unsteady flow equations are discretized using bilinear quadrilateral isoparametric finite elements. The nonlinear mean flow solution and streamline computational grid are computed simultaneously using Newton iteration. At each step of the Newton iteration, LU decomposition is used to solve the resulting set of linear equations. The unsteady flow problem is linear, and is also solved using LU decomposition. Next, a sensitivity analysis is performed to determine the effect small changes in cascade and airfoil geometry have on the mean and unsteady flow fields. The sensitivity analysis makes use of the nominal steady and unsteady flow LU decompositions so that no additional matrices need to be factored. Hence, the present method is computationally very efficient. To demonstrate how the sensitivity analysis may be used to redesign cascades, a compressor is redesigned for improved aeroelastic stability and two different fan exit guide vanes are redesigned for reduced downstream radiated noise. In addition, a framework detailing how the two-dimensional version of the method may be used to redesign three-dimensional geometries is presented.

Lorence, Christopher B.↗

NASA NTRS1978

Computational wing optimization and comparisons with experiment for a semi-span wing model

A computational wing optimization procedure was developed and verified by an experimental investigation of a semi-span variable camber wing model in the NASA Ames Research Center 14 foot transonic wind tunnel. The Bailey-Ballhaus transonic potential flow analysis and Woodward-Carmichael linear theory codes were linked to Vanderplaats constrained minimization routine to optimize model configurations at several subsonic and transonic design points. The 35 deg swept wing is characterized by multi-segmented leading and trailing edge flaps whose hinge lines are swept relative to the leading and trailing edges of the wing. By varying deflection angles of the flap segments, camber and twist distribution can be optimized for different design conditions. Results indicate that numerical optimization can be both an effective and efficient design tool. The optimized configurations had as good or better lift to drag ratios at the design points as the best designs previously tested during an extensive parametric study.

Waggoner, E. G.↗

NASA NTRS1981

Three-dimensional periodic disturbances acting upon airfoils in cascade

A theory is developed for three-dimensional periodic gusts interacting with loaded airfoils. The unsteady disturbances are linearized with respect to the mean potential flow of the airfoils. The vorticity transport equation is then integrated analytically in a Lagrangian form. The vorticity vector shows strong variations in its magnitude and wavelength as it interacts with the flowfield of a lifting airfoil. The streamwise component of the vorticity increases significantly with flow acceleration and flow turning. For simplicity the rectangular fan approximation is used and a single Helmholtz-like equation is derived to characterize the unsteady 3D flowfield. This is a new and significant result. The present theory is first applied to symmetric airfoils.

Atassi, H.↗

NASA NTRS1981

Three-dimensional periodic disturbances acting upon airfoils in cascade

A theory for three dimensional periodic gusts interacting with loaded airfoils was developed. The unsteady disturbances are linearized with respect to the mean potential flow of the airfoils. The vorticity transport equation is integrated analytically in a Lagrangian form. The vorticity vector shows strong variations in its magnitude and wavelength as it interacts with the flow field of a lifting airfoil. The streamwise component of vorticity increases significantly with flow acceleration and flow turning. The rectangular fan approximation is used and a single Helmholtz equation is derived that characterizes the unsteady three dimensional flow field.

Atassi, H.↗

NASA NTRS1988

Irreversible wave-mean flow interactions in a mechanistic model of the stratosphere

A wave-1 minor warming is simulated in a mechanistic, global, primitive equation model. The modification of the zonal flow by the rearrangement of potential vorticity on a middle stratopheric isentrope is compared in a fully nonlinear model and in a model with only one wave and the zonal flow (a quasi-linear model). The permanent rearrangement of potential vorticity during the wave episode is more intense and more localized meridionally in the fully nonlinear model, which is able to capture the process of planetary wave breaking in some detail. Additional experiments reveal that the differences between the quasi-linear and nonlinear models persist for a broad range of wave amplitudes, and that the quasi-linear model can qualitatively reproduce the modification of the zonal flow by the wave when the diffusive dissipation of the wave is enhanced. These results are discussed in the context of the theory of barotropic Rossby waves in shear flows, and in comparison with recent numerical simulations of the middle atmosphere.

Robinson, Walter A.↗

NASA NTRSDate not supplied

Enhancements to Linear Stability-Based, CFD-integrated Transition Prediction for High-Speed Flows

Combining linear stability calculations with computational fluid dynamics (CFD) simulations has great potential for the automated modeling of high-speed flows, especially when adequate information about the configuration and the disturbance environment is available. However, a significant impediment to the applicability of this technique is the lack of an efficient method to calculate the crucial amplification ratio corresponding to the onset of transition in hypersonic flows. This ratio, also known as the "transition N-factor," is dependent upon the freestream disturbance environment as well as the surface properties of the test article. In response to the need for an engineering solution to predict the transition N-factor within conventional hypersonic wind tunnels, this paper presents a data-driven correlation that expands the existing correlations from straight circular cones with a narrow range of half angles to a broader array of axisymmetric configurations. Furthermore, when tested against a chosen dataset that was not used in its calibration, the suggested correlation shows good predictive accuracy with an RMS error of only 6.9%. Although similar accuracy may also be achieved via existing correlations based on similar datasets, predictions based on the proposed correlation have the advantage of not requiring an extensive amount of configuration-specific data. Practical applications often have access to the input parameters needed for this correlation, such as the freestream disturbance intensity, Mach number, and body-based slenderness Reynolds number. Additionally, this correlation outperforms the traditional assumption of a constant N-factor, particularly for configurations with blunted nose geometries. The development of this correlation is grounded in an extensive dataset encompassing conical models with body half-angles varying between 5 degrees and 16 degrees, Mach numbers ranging from 5 to 14, and nosetip-based Reynolds numbers approaching the transition reversal limit for blunt-nosed cones.

CFD↗

NASA NTRSDate not supplied

Enhancements to Linear Stability-Based, CFD-integrated Transition Prediction for High-Speed Flows

Combining linear stability calculations with computational fluid dynamics (CFD) simulations has great potential for the automated modeling of high-speed flows, especially when adequate information about the configuration and the disturbance environment is available. However, a significant impediment to the applicability of this technique is the lack of an efficient method to calculate the crucial amplification ratio corresponding to the onset of transition in hypersonic flows. This ratio, also known as the "transition N-factor," is dependent upon the freestream disturbance environment as well as the surface properties of the test article. In response to the need for an engineering solution to predict the transition N-factor within conventional hypersonic wind tunnels, this paper presents a data-driven correlation that expands the existing correlations from straight circular cones with a narrow range of half angles to a broader array of axisymmetric configurations. Furthermore, when tested against a chosen dataset that was not used in its calibration, the suggested correlation shows good predictive accuracy with an RMS error of only 6.9%. Although similar accuracy may also be achieved via existing correlations based on similar datasets, predictions based on the proposed correlation have the advantage of not requiring an extensive amount of configuration-specific data. Practical applications often have access to the input parameters needed for this correlation, such as the freestream disturbance intensity, Mach number, and body-based slenderness Reynolds number. Additionally, this correlation outperforms the traditional assumption of a constant N-factor, particularly for configurations with blunted nose geometries. The development of this correlation is grounded in an extensive dataset encompassing conical models with body half-angles varying between 5 degrees and 16 degrees, Mach numbers ranging from 5 to 14, and nosetip-based Reynolds numbers approaching the transition reversal limit for blunt-nosed cones.

CFD↗

NASA NTRS1983

Spectral multigrid methods with applications to transonic potential flow

Spectral multigrid methods are demonstrated to be a competitive technique for solving the transonic potential flow equation. The spectral discretization, the relaxation scheme, and the multigrid techniques are described in detail. Significant departures from current approaches are first illustrated on several linear problems. The principal applications and examples, however, are for compressible potential flow. These examples include the relatively challenging case of supercritical flow over a lifting airfoil.

Streett, C. L.↗

NASA NTRS1985

Spectral multigrid methods with applications to transonic potential flow

Spectral multigrid methods are demonstrated to be a competitive technique for solving the transonic potential flow equation. The spectral discretization, the relaxation scheme, and the multigrid techniques are described in detail. Significant departures from current approaches are first illustrated on several linear problems. The principal applications and examples, however, are for compressible potential flow. These examples include the relatively challenging case of supercritical flow over a lifting airfoil.

Streett, C. L.↗

NASA NTRS1991

A modal aeroelastic analysis scheme for turbomachinery blading

An aeroelastic analysis is developed which has general application to all types of axial-flow turbomachinery blades. The approach is based on linear modal analysis, where the blade's dynamic response is represented as a linear combination of contributions from each of its in-vacuum free vibrational modes. A compressible linearized unsteady potential theory is used to model the flow over the oscillating blades. The two-dimensional unsteady flow is evaluated along several stacked axisymmetric strips along the span of the airfoil. The unsteady pressures at the blade surface are integrated to result in the generalized force acting on the blade due to simple harmonic motions. The unsteady aerodynamic forces are coupled to the blade normal modes in the frequency domain using modal analysis. An iterative eigenvalue problem is solved to determine the stability of the blade when the unsteady aerodynamic forces are included in the analysis. The approach is demonstrated by applying it to a high-energy subsonic turbine blade from a rocket engine turbopump power turbine. The results indicate that this turbine could undergo flutter in an edgewise mode of vibration.

Smith, Todd E.↗

NASA NTRS1978

The practical application of a finite difference method for analyzing transonic flow over oscillating airfoils and wings

Separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady equations for small disturbances was performed. The steady velocity potential was obtained first from the well known nonlinear equation for steady transonic flow. The unsteady velocity potential was then obtained from a linear differential equation in complex form with spatially varying coefficients. Since sinusoidal motion is assumed, the unsteady equation is independent of time. The results of an investigation into the relaxation-solution-instability problem was discussed. Concepts examined include variations in outer boundary conditions, a coordinate transformation so that the boundary condition at infinity may be applied to the outer boundaries of the finite difference region, and overlapping subregions. The general conclusion was that only a full direct solution in which all unknowns are obtained at the same time will avoid the solution instabilities of relaxation. An analysis of the one-dimensional form of the unsteady transonic equation was studied to evaluate errors between exact and finite difference solutions. Pressure distributions were presented for a low-aspect-ratio clipped delta wing at Mach number of 0.9 and for a moderate-aspect-ratio rectangular wing at a Mach number of 0.875.

Weatherill, W. H.↗

NASA NTRS1994

Simulations of bypass transition for spatially evolving disturbances

The spatial evolution of disturbances in plane Poiseuille flow and zero pressure gradient boundary layer flow is considered. For disturbances governed by the linearized equations, potential for significant transient growth of the amplitude is demonstrated. The maximum amplification occurs for disturbances with zero or near zero frequencies. Spatial numerical simulations of the transition scenario involving a pair of oblique waves has been conducted for both flows. A fully spectral solver using a simple but efficient fringe region technique allowed the flows to be computed with high resolution into the fully turbulent domain. A modal decomposition of the simulation results indicates that non-linear excitation of the transient growth is responsible for the rapid emergence of low-frequency structures. Physically, this corresponds to streaky flow structures, as seen from the results of a numerical amplitude expansion. Thus, this spatial transition scenario has been found to be similar to the corresponding temporal one. In the boundary layer simulations the streaks are seen to break down from what appears to be a secondary instability.

Lundbladh, A.↗