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NASA NTRS1976

Transonic flow calculations

The development of relaxation methods to calculate transonic flows is discussed. Rather accurate predictions can be made for a number of flows of interest using the transonic potential flow equation, which may be derived from the Euler equations for inviscid compressible flow by introducing the assumption that the flow is irrotational. Once the choice of a mathematical model has been settled, the numerical procedure for actually computing a solution contains two main elements: the construction of a discrete approximation which converges to the solution of the continuous problem in the limit as the mesh width is reduced to zero, and the solution of the resulting set of nonlinear difference equations by a convergent iterative scheme. The choice of an appropriate coordinate system and its influence on the accuracy of the discrete approximation is discussed. Several applications of the general method are described.

Jameson, A.↗

NASA NTRS1980

Advanced stability theory analyses for laminar flow control

Recent developments of the SALLY computer code for stability analysis of laminar flow control wings are summarized. Extensions of SALLY to study three dimensional compressible flows, nonparallel and nonlinear effects are discussed.

Orszag, S. A.↗

NASA NTRS1989

Aerodynamics of engine-airframe interaction

The development of the variational approach for the solution of inviscid aerodynamic problems using solution adaptive grids is discussed. The formulation of a new, directional weighting, functional has been shown to have desirable properties. The scheme has been applied to compute the transonic flow past two-dimensional airfoils using the Euler equations of inviscid, compressible flow. Transonic flows in quasi-one-dimensional nozzles and over the two dimensional airfoils are solved on the various solution-adaptive-grids to demonstrate the applicability of the proposed directional-concentration functional and the grid adaptation process from the stand point of improving the solution accuracy and demonstrating the overall convergence.

Caughey, David A.↗

NASA NTRS1995

Investigation of Convection and Pressure Treatment with Splitting Techniques

Treatment of convective and pressure fluxes in the Euler and Navier-Stokes equations using splitting formulas for convective velocity and pressure is investigated. Two schemes - controlled variation scheme (CVS) and advection upstream splitting method (AUSM) - are explored for their accuracy in resolving sharp gradients in flows involving moving or reflecting shock waves as well as a one-dimensional combusting flow with a strong heat release source term. For two-dimensional compressible flow computations, these two schemes are implemented in one of the pressure-based algorithms, whose very basis is the separate treatment of convective and pressure fluxes. For the convective fluxes in the momentum equations as well as the estimation of mass fluxes in the pressure correction equation (which is derived from the momentum and continuity equations) of the present algorithm, both first- and second-order (with minmod limiter) flux estimations are employed. Some issues resulting from the conventional use in pressure-based methods of a staggered grid, for the location of velocity components and pressure, are also addressed. Using the second-order fluxes, both CVS and AUSM type schemes exhibit sharp resolution. Overall, the combination of upwinding and splitting for the convective and pressure fluxes separately exhibits robust performance for a variety of flows and is particularly amenable for adoption in pressure-based methods.

Thakur, Siddharth↗

NASA NTRS1997

Preconditioning for Numerical Simulation of Low Mach Number Three-Dimensional Viscous Turbomachinery Flows

A preconditioning scheme has been implemented into a three-dimensional viscous computational fluid dynamics code for turbomachine blade rows. The preconditioning allows the code, originally developed for simulating compressible flow fields, to be applied to nearly-incompressible, low Mach number flows. A brief description is given of the compressible Navier-Stokes equations for a rotating coordinate system, along with the preconditioning method employed. Details about the conservative formulation of artificial dissipation are provided, and different artificial dissipation schemes are discussed and compared. The preconditioned code was applied to a well-documented case involving the NASA large low-speed centrifugal compressor for which detailed experimental data are available for comparison. Performance and flow field data are compared for the near-design operating point of the compressor, with generally good agreement between computation and experiment. Further, significant differences between computational results for the different numerical implementations, revealing different levels of solution accuracy, are discussed.

Tweedt, Daniel L.↗

NASA NTRS1992

Wall functions for the kappa-epsilon turbulence model in generalized nonorthogonal curvilinear coordinates

A k-epsilon turbulence model suitable for compressible flow, including the new wall function formulation, has been incorporated into an existing compressible Reynolds-averaged Navier-Stokes code, F3D. The low Reynolds number k-epsilon model of Chien (1982) was added for comparison with the present method. A number of features were added to the F3D code including improved far-field boundary conditions and viscous terms in the streamwise direction. A series of computations of increasing complexity was run to test the effectiveness of the new formulation. Flow over a flat plate was computed by using both orthogonal and nonorthogonal grids, and the friction coefficients and velocity profiles compared with a semi-empirical equation. Flow over a body of revolution at zero angle of attack was then computed to test the method's ability to handle flow over a curved surface. Friction coefficients and velocity profiles were compared to test data. All models gave good results on a relatively fine grid, but only the wall function formulation was effective with coarser grids. Finally, in order to demonstrate the method's ability to handle complex flow fields, separated flow over a prolate spheroid at angle of attack was computed, and results were compared to test data. The results were also compared to a k-epsilon model by Kim and Patel (1991), in which one equation model patched in at the wall was employed. Both models gave reasonable solutions, but improvement is required for accurate prediction of friction coefficients in the separated regions.

Sondak, D. L.↗

NASA NTRS2011

Numerical Study Comparing RANS and LES Approaches on a Circulation Control Airfoil

A numerical study over a nominally two-dimensional circulation control airfoil is performed using a large-eddy simulation code and two Reynolds-averaged Navier-Stokes codes. Different Coanda jet blowing conditions are investigated. In addition to investigating the influence of grid density, a comparison is made between incompressible and compressible flow solvers. The incompressible equations are found to yield negligible differences from the compressible equations up to at least a jet exit Mach number of 0.64. The effects of different turbulence models are also studied. Models that do not account for streamline curvature effects tend to predict jet separation from the Coanda surface too late, and can produce non-physical solutions at high blowing rates. Three different turbulence models that account for streamline curvature are compared with each other and with large eddy simulation solutions. All three models are found to predict the Coanda jet separation location reasonably well, but one of the models predicts specific flow field details near the Coanda surface prior to separation much better than the other two. All Reynolds-averaged Navier-Stokes computations produce higher circulation than large eddy simulation computations, with different stagnation point location and greater flow acceleration around the nose onto the upper surface. The precise reasons for the higher circulation are not clear, although it is not solely a function of predicting the jet separation location correctly.

Rumsey, Christopher L.↗

NASA NTRS1992

Vibrational relaxation of anharmonic oscillators in expanding flows

Although the Landau-Teller vibrational model accurately predicts vibrational excitation process in post-shock and compressing flows, it under-predicts the rate of de-excitation in cooling and expanding flows. In the present paper, detailed calculations of the vibrational relaxation process of N2 and CO in cooling flows are conducted. A coupled set of vibrational transition rate equations and quasi-one-dimensional fluid dynamic equations is solved. Multiple quantum level transition rates are computed using SSH theory. The SSH transition rate results are compared with available experimental data and other theoretical models. Vibration-vibration exchange collisions are responsible for some vibrational relaxation acceleration in situations of high vibrational temperature and low translational temperature. The present results support the relaxation mechanisms proposed by Bray and by Treanor Rich and Rehm.

Ruffin, Stephen M.↗

NASA NTRS1989

The upper-branch stability of compressible boundary layer flows

The upper-branch linear and nonlinear stability of compressible boundary layer flows is studied using the approach of Smith and Bodonyi (1982) for a similar incompressible problem. Both pressure gradient boundary layers and Blasius flow are considered with and without heat transfer, and the neutral eigenrelations incorporating compressibility effects are obtained explicitly. The compressible nonlinear viscous critical layer equations are derived and solved numerically and the results indicate some solutions with positive phase shift across the critical layer. Various limiting cases are investigated including the case of much larger disturbance amplitudes and this indicates the structure for the strongly nonlinear critical layer of the Benney-Bergeon (1969) type. It is also shown how a match with the inviscid neutral inflexional modes arising from the generalized inflexion point criterion, is achieved.

Gajjar, J. S. B.↗

NASA NTRS1991

Computations of the three-dimensional flow and heat transfer within a coolant passage of a radial turbine blade

A numerical code is developed for computing three-dimensional, turbulent, compressible flow within coolant passages of turbine blades. The code is based on a formulation of the compressible Navier-Stokes equations in a rotating frame of reference in which the velocity dependent variable is specified with respect to the rotating frame instead of the inertial frame. The algorithm employed to obtain solutions to the governing equation is a finite-volume LU algorithm that allows convection, source, as well as diffusion terms to be treated implicitly. In this study, all convection terms are upwind differenced by using flux-vector splitting, and all diffusion terms are centrally differenced. This paper describes the formulation and algorithm employed in the code. Some computed solutions for the flow within a coolant passage of a radial turbine are also presented.

Shih, T. I.-P.↗

NASA NTRS1986

Transonic flow solutions using a composite velocity procedure for potential, Euler and RNS equations

Solutions for transonic viscous and inviscid flows using a composite velocity procedure are presented. The velocity components of the compressible flow equations are written in terms of a multiplicative composite consisting of a viscous or rotational velocity and an inviscid, irrotational, potential-like function. This provides for an efficient solution procedure that is locally representative of both asymptotic inviscid and boundary layer theories. A modified conservative form of the axial momentum equation that is required to obtain rotational solutions in the inviscid region is presented and a combined conservation/nonconservation form is applied for evaluation of the reduced Navier-Stokes (RNS), Euler and potential equations. A variety of results is presented and the effects of the approximations on entropy production, shock capturing, and viscous interaction are discussed.

Gordnier, R. E.↗

NASA NTRS1989

Transonic flow solutions using a composite velocity procedure for potential, Euler and RNS equations

Gordnier, R. E.↗

NASA NTRS2005

CFD - Mature Technology?

Over the past 30 years, numerical methods and simulation tools for fluid dynamic problems have advanced as a new discipline, namely, computational fluid dynamics (CFD). Although a wide spectrum of flow regimes are encountered in many areas of science and engineering, simulation of compressible flow has been the major driver for developing computational algorithms and tools. This is probably due to a large demand for predicting the aerodynamic performance characteristics of flight vehicles, such as commercial, military, and space vehicles. As flow analysis is required to be more accurate and computationally efficient for both commercial and mission-oriented applications (such as those encountered in meteorology, aerospace vehicle development, general fluid engineering and biofluid analysis) CFD tools for engineering become increasingly important for predicting safety, performance and cost. This paper presents the author's perspective on the maturity of CFD, especially from an aerospace engineering point of view.

Kwak, Dochan↗

NASA NTRS2014

Development of Dynamic Sub-Grid Models for Variational Multiscale Methods

A dynamic Variational Multiscale Method (Hughes et al. 1998) is developed by leveraging the Germano procedure from classical Large-eddy Simulations (LES). The similarity between the classical and variational approaches is analyzed in the context of incompressible flow. This analysis leads to a consistent modeling approach for both incompressible and compressible flows, the latter being demonstrated in a priori testing for low-speed attached and separated boundary layers. Similar to the classical LES procedure from which it is derived, the variational dynamic procedure does not guarantee a positive semi-definite coefficient in the general case. However, reproducing the behavior of the classical LES dynamic approach is seen as a necessary first step to develop a VMM that automatically adjusts to the local resolution and flow physics.

Variational multiscale↗

NASA NTRS1975

Numerical calculation of the internal flow field in a centrifugal compressor impeller

An iterative numerical method has been developed for the calculation of steady, three-dimensional, viscous, compressible flow fields in centrifugal compressor impellers. The computer code, which embodies the method, solves the steady three dimensional, compressible Navier-Stokes equations in rotating, curvilinear coordinates. The solution takes place on blade-to-blade surfaces of revolution which move from the hub to the shroud during each iteration.

Walitt, L.↗

NASA NTRS1971

A numerical method for computing three dimensional viscous supersonic flow fields about slender bodies

A numerical method is presented for the calculation of steady, three-dimensional, viscous, compressible flow fields about slender bodies at angle of attack and at supersonic speeds. Approximations are introduced in modeling the flow in the longitudinal direction. Accordingly, the flow fields calculated with the program were computed with a model that permits viscous crossflow together with inviscid axial flow. An analysis of the errors introduced by such a treatment is presented. Numerical calculations were made and compared with experimental results for an ogive-cylinder and an airplane fuselage configuration. Generally, good agreement with experiment was obtained. However, boundary layer separation and body vortex positions differed from experimental locations on the ogive-cylinder, and the shock induced by the fuselage canopy was predicted at a slightly different location.

Walitt, L.↗

NASA NTRS1980

General design method for three-dimensional potential flow fields. 1: Theory

A general design method was developed for steady, three dimensional, potential, incompressible or subsonic-compressible flow. In this design method, the flow field, including the shape of its boundary, was determined for arbitrarily specified, continuous distributions of velocity as a function of arc length along the boundary streamlines. The method applied to the design of both internal and external flow fields, including, in both cases, fields with planar symmetry. The analytic problems associated with stagnation points, closure of bodies in external flow fields, and prediction of turning angles in three dimensional ducts were reviewed.

Stanitz, J. D.↗

NASA NTRS1984

Three-dimensional viscous flows with large secondary velocity

A new system of approximation equations is derived for three-dimensional steady viscous compressible flows in which a primary-flow direction is present, but in which both transverse velocity components can be large. Previous approaches which address simplification of the steady Navier-Stokes equations are discussed, and a new approach is proposed. The transverse velocity vector which corrects a given potential flow has been decomposed into potential and rotational components. It is found that the potential-velocity vector may be assumed small, whereas the rotational-velocity vector may be assumed small, whereas the rotational velocity vector and hence the composite secondary flow can be of order unity. This assumption leads to a system of governing equations whose characteristic polynomial has a non-elliptic form for arbitrary Mach numbers. The resulting non-elliptic approximation equations can be solved as an initial/boundary-value problem. Computed results confirm the small scalar-potential approximation.

Briley, W. R.↗