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NASA NTRS1986

Transition and mixing in axisymmetric jets and vortex rings

A class of impulsively started, axisymmetric, laminar jets produced by a time dependent joint source of momentum are considered. These jets are different flows, each initially at rest in an unbounded fluid. The study is conducted at three levels of detail. First, a generalized set of analytic creeping flow solutions are derived with a method of flow classification. Second, from this set, three specific creeping flow solutions are studied in detail: the vortex ring, the round jet, and the ramp jet. This study involves derivation of vorticity, stream function, entrainment diagrams, and evolution of time lines through computer animation. From entrainment diagrams, critical points are derived and analyzed. The flow geometry is dictated by the properties and location of critical points which undergo bifurcation and topological transformation (a form of transition) with changing Reynolds number. Transition Reynolds numbers were calculated. A state space trajectory was derived describing the topological behavior of these critical points. This state space derivation yielded three states of motion which are universal for all axisymmetric jets. Third, the axisymmetric round jet is solved numerically using the unsteady laminar Navier Stokes equations. These equations were shown to be self similar for the round jet. Numerical calculations were performed up to a Reynolds number of 30 for a 60x60 point mesh. Animations generated from numerical solution showed each of the three states of motion for the round jet, including the Re = 30 case.

Allen, G. A., Jr.↗

NASA NTRS1986

Efficient solutions of two-dimensional incompressible steady viscous flows

A simple, efficient, and robust numerical technique is provided for solving two dimensional incompressible steady viscous flows at moderate to high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations. Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established.

Morrison, J. H.↗

NASA NTRS1986

Numerical studies of interacting vortices

To get a basic understanding of the physics of flowfields modeled by vortex filaments with finite vortical cores, systematic numerical studies of the interactions of two vortices and pairs of coaxial axisymmetric circular vortex rings were made. Finite difference solutions of the unsteady incompressible Navier-Stokes equations were carried out using vorticity and stream function as primary variables. Special emphasis was placed on the formulation of appropriate boundary conditions necessary for the calculations in a finite computational domain. Numerical results illustrate the interaction of vortex filaments, demonstrate when and how they merge with each other, and establish the region of validity for an asymptotic analysis.

Liu, G. C.↗

NASA NTRS1986

Application of the method of lines for solution of the Navier-Stokes equations using nonuniform grid distributions

The method of lines is investigated for the numerical solution of the stream function and velocity form of the Navier-Stokes equations on nonuniform grids. Stiffness characteristics of a linear one-dimensional model equation are examined to establish the feasibility of applying the method to the volicity equation in two dimensions. It is observed that the transformed governing equations in computational domain become stiffer with increasing the number of grid points and the differencing technique affects the stiffness characteristics. The results demonstrate that the method of lines under certain circumstances is highly suitable for the numerical solution of physical problems on domains covered with variable grids.

Tiwari, S. N.↗

NASA NTRS1986

Interactions of coupled acoustic and vortical instability

In the past, the acoustic combustion instability was studied independently of the hydrodynamic instability induced by vortex motions. This paper is intended to combine the two different sources of energy everywhere within the spatial domain and determine the effect of one upon the other. This can be achieved by calculating the mean flow velocities and vorticities and their fluctuating parts of velocities and vortices, as well as the fluctuating pressure. The Orr-Sommerfeld equation is utilized to determine the wavenumbers and unsteady stream functions from which vortically coupled acoustic instability growth constants are calculated. This process demonstrates that there are two different frequencies, acoustic and hydrodynamic, various combinations of which contribute to either damping or amplification. It is found that stability boundaries for coupled acoustic and vortical oscillations are somewhat similar to the classical hydrodynamic stability boundaries, but they occur in the form of multiple islands.

Chung, T. J.↗

NASA NTRS1986

Another look at North Sea pole tide dynamics

The mechanism proposed by Wunsch (1974) to explain pole tide observations in the North Sea is evaluated. Wunsch's equations governing pole tide in the North Sea are presented, and solutions for correcting the depth, stream function, and deviation of the tidal height from the equilibrium values are described. The similarity between the Stokes paradox and the tidal equations of the North Sea, and the need for inclusion of inertial terms in the tidal equations are discussed.

Dickman, S. R.↗

NASA NTRS1987

Computation of viscous transonic flow over porous airfoils

The viscous effects on transonic flow past an airfoil which contains a shallow cavity beneath a porous surface are studied numerically. The porous region occupies a small portion of the total airfoil surface, and is located near the shock. Both an interactive boundary layer (IBL) algorithm and a thin-layer Navier-Stokes (TLNS) algorithm have been modified for use in studying the outer flow, whereas a stream-function formulation has been used to model the inner flow in the small cavity. The coupling procedure at the porous surface is based on Darcy's law and on the assumption of a constant total presusre in the cavity. In addition, a modified Baldwin-Lomax turbulence model is used to consider the transpired turbulent boundary layer in the TLNS approach, and the Cebeci-Smith turbulence model is used in the IBL approach. According to the present analysis, a porous surface can reduce the wave drag appreciably, but it can also increase viscous losses. As has been observed experimentally, the numerical results indicate that the total drag is reduced at higher Mach numbers and increased at lower Mach numbers when the angles of attack are small. Furthermore, the streamline patterns of passive-shock and boundary-layer interaction are revealed in this study.

Chow, Chuen-Yen↗

NASA NTRS1986

Vertical structure of the wind field during the Special Observing Period I of the Global Weather Experiment

The vertical structure of the global atmosphere is analyzed for selected periods of the Special Observing Period I (SOP-I) for the Global Weather Experiment (GWE). The analysis consists of projection of the stream-function and velocity potential at 200 and 850 mb on spherical harmonics and of the wind and height fields on the normal modes of a linearized form of the primitive equations for a basic state at rest. The kinematic vertical structure is discussed in terms of correlation coefficients of the 200 mb and 850 mb winds and analysis of the internal and external normal modes of the primitive equations. The reliability of the results is checked by applying the same analysis methods to data sets obtained from three different institutions: Geophysical Fluid Dynamics Laboratory (GFDL), European Center for Medium Range Weather Forecasting (ECMWF), and Goddard Laboratory for the Atmospheres (GLA). It is found that, on a global basis, vertically reversing circulations are as important as the equivalent barotropic structures. For the verticaly reversing components, the gravity and mixed Rossby-gravity modes have contributions of the same order of magnitude as those of the Rossby modes in tropical latitudes.

Paegle, J. N.↗

NASA NTRS1987

The stratosphere - Climatologies of the radiative heating and cooling rates and the diabatically diagnosed net circulation fields

Stratospheric solar, IR, and net radiative heating are calculated on a monthly basis using solar and IR radiative codes and satellite derived distributions of ozone, water vapor, and temperature. Divergence-free, zonally averaged, advective fields are diagnosed using the calculated diabatic heating; associated stream functions are derived. The stratospheric transport of inert tracers is studied. Analysis of the diagnosed advective fields reveal that: (1) entry into the mid- to upper stratosphere of tropospheric air is mainly from altitude regions of + or - 10 deg at the equatorial tropopause; (2) at latitudes poleward of + or - 15 deg, tracers transported from the troposphere into the stratosphere are transported toward the pole and then downward and out of the stratosphere; and (3) the presence of net cooling cells in the lower stratospheric polar regions is important. The interannual variability of the diabatic circulation is estimated using heating and advection fields derived from LIMS data.

Callis, Linwood B.↗

NASA NTRS1986

Characteristics of precipitation regimes during the global weather experiment

Rainfall data were used in conjunction with heating rate values obtained from GLA/NASA to describe large scale heating changes and concurrent circulation fluctuations for weekly averages during the Special Observing Period I (January-March 1979) of the Global Weather Experiment. The GLA diabatic heating rates were obtained as residues of the thermodynamic energy equation during the four-dimensional data assimilation cycle (as described by Kalnay and Baker, 1984). In week one, the heating and rainfall rate profiles display a positive anomaly on the West Pacific in both fields. The anomaly appears to propagate eastward into the East Pacific and South America by the fifth week. By the week four, the negative anomaly over the Indian Ocean becomes positive. These heating patterns are reflected on positive divergent anomalies at 200 mb analyzed for the same periods. The stream-function anomalies display anticyclonic circulations over the heating anomalies.

Ahrens, Glen D.↗

NASA NTRS1988

Computer Program Predicts Turbine-Stage Performance

MTSBL updated version of flow-analysis programs MERIDL and TSONIC coupled to boundary-layer program BLAYER. Method uses quasi-three-dimensional, inviscid, stream-function flow analysis iteratively coupled to calculated losses so changes in losses result in changes in flow distribution. Manner effects both configuration on flow distribution and flow distribution on losses taken into account in prediction of performance of stage. Written in FORTRAN IV.

Boyle, Robert J.↗

NASA NTRS1987

Nonlinear magnetohydrodynamic waves in a steady zonal circulation for a shallow fluid shell on the surface of a rotating sphere

This paper considers two-dimensional nonlinear MHD waves of large horizontal spatial scales for a thin magnetofluid layer on the surface of a rotating sphere. The 'shallow fluid' hydrodynamic equations are generalized to include the effects of magnetic fields, and it is shown that the resulting MHD equations can be reduced to a single scalar equation for a stream function involving several free functions. For special choices of these free functions, two kinds of finite-amplitude MHD waves are obtained, propagating in the azimuthal direction relative to the uniformly rotating background atmosphere in the presence of a background zonal magnetic field and a steady differential zonal flow. These two kinds of MHD waves are fundamentally due to the joint effects of the uniform rotation of the background atmosphere and background magnetic field; the first is an inertial wave of the Rossby (1939) and Haurwitz (1940) type, modified by the presence of the background zonal magnetic field, while the second is a magnetic Alfven-like wave which is modified by the uniform rotation of the background atmosphere.

Lou, Y. Q.↗

NASA NTRS1988

Entropy and vorticity corrections for transonic flows

Different models for inviscid transonic flows are examined. The common assumptions that the flow is isentropic and irrotational are critically evaluated. Entropy and vorticity correction procedures for potential and stream function formulations are presented, together with the details of the treatment of shocks and wakes, and drag and lift calculations. The non-uniqueness problem of the potential formulation is studied using different artificial viscosity forms. Numerical results are compared with Euler solutions.

Hafez, M.↗

NASA NTRS1988

Efficient solutions of two-dimensional incompressible steady viscous flows

Morrison, J. H.↗

NASA NTRS1987

Simulation of separated flow past a bluff body using Navier-Stokes equations

Two-dimensional flow past a bluff body is presently simulated on the basis of an analysis that employs the incompressible, unsteady Navier-Stokes equations in terms of vorticity and stream function. The fully implicit, time-marching, alternating-direction, implicit-block Gaussian elimination used is a direct method with second-order spatial accuracy; this allows it to avoid the introduction of any artificial viscosity. Attention is given to the simulation of flow past a circular cylinder with and without symmetry, requiring the use of either the half or the full cylinder, respectively.

Ghia, K. N.↗

NASA NTRS1988

The field lines of an axisymmetric magnetic field

The equations of Willis and Young (1987) for the field lines of an arbitrary axisymmetric multipole are generalized to an arbitrary linear combination of multipoles, i.e., to an arbitrary axisymmetric magnetic field B outside a sphere of radius a, S(a), centered on the origin, and containing all the sources of B. For this field, axisymmetric Stokes stream function is expressed in terms of the Gauss coefficients. It is shown that if only one Gauss coefficient is nonzero, the field line equations are identical to those obtained by Willis and Young.

Backus, George E.↗

NASA NTRS1988

Computational unsteady aerodynamics for lifting surfaces

Two dimensional problems are solved using numerical techniques. Navier-Stokes equations are studied both in the vorticity-stream function formulation which appears to be the optimal choice for two dimensional problems, using a storage approach, and in the velocity pressure formulation which minimizes the number of unknowns in three dimensional problems. Analysis shows that compact centered conservative second order schemes for the vorticity equation are the most robust for high Reynolds number flows. Serious difficulties remain in the choice of turbulent models, to keep reasonable CPU efficiency.

Edwards, John W.↗

NASA NTRS1989

A skin friction law for compressible turbulent flow

An algebraic skin friction law is derived for adiabatic, compressible, equilibrium, turbulent boundary layer flow. An outer solution in terms of the Clauser defect stream function is matched to an inner empirical expression composed of compressible laws of the wall and wake. The modified Crocco temperature-velocity relationship and the Clauser eddy viscousity model are used in the outer solution. The skin friction law pertains for all pressure gradients in the incompressible through supersonic range and for small pressure gradients in the hypersonic range. Excellent comparisons with experiment are obtained in the appropriate parameter ranges. The application to numerical computation is discussed.

Barnwell, Richard W.↗