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NASA NTRS1990

Reynolds stress closure in jet flows using wave models

A collection of papers is presented. The outline of this report is as follows. Chapter three contains a description of a weakly nonlinear turbulence model that was developed. An essential part of the application of such a closure scheme to general geometry jets is the solution of the local hydrodynamic stability equation for a given jet cross-section. Chapter four describes the conformal mapping schemes used to map such geometries onto a simple computational domain. Chapter five describes a solution of a stability problem for circular, elliptic, and rectangular geometries. In chapter six linear models for the shock shell structure in non-circular jets is given. The appendices contain reprints of papers also published during this study including the following topics: (1) instability of elliptic jets; (2) a technique for predicting the shock cell structure in non-circular jets using a vortex sheet model; and (3) the resonant interaction between twin supersonic jets.

Morris, Philip J.↗

NASA NTRS1990

Zero wavenumber modes of a compressible supersonic mixing layer

It is shown that there exists a family of supersonic neutral modes for a compressible mixing layer in an unbounded domain. These modes have zero wavenumber and frequency with nonzero phase speed. They are analogous to the supersonic neutral modes of the compressible vortex sheet found by Miles. The results presented give a more complete picture of the spectrum of the disturbances in this flow.

Jackson, T. L.↗

NASA NTRS1990

Instability waves in twin supersonic jets

Calculations are presented for the characteristics of instability waves in the initial mixing region of twin circular supersonic jets. Two models for the basic jet flow are used. In the first, the jets are modeled as two circular vortex sheets. In the second, realistic velocity and density profiles are used. It is shown that the unsteady flow fields of the two jets interact before the time-averaged jets flows have merged. The normal modes or instability waves are classified by their symmetry properties in the twin-jet case and their asymptotic behavior for large jet separations. Calculations of the growth rates and phase velocities are made for these modes as a function of jet separation and mixing-layer thickness. The associated pressure distributions are also presented. In the realistic jet profile calculations the effect of jet separation is found to be relatively weak. For modes that are even about the symmetry plane between the two jets the pressure levels are found to increase near this plane as the jet separation decreases.

Morris, Philip J.↗

NASA NTRS1991

The optical analogy for vector fields

This paper develops the optical analogy for a general vector field. The optical analogy allows the examination of certain aspects of a vector field that are not otherwise readily accessible. In particular, in the cases of a stationary Eulerian flow v of an ideal fluid and a magnetostatic field B, the vectors v and B have surface loci in common with their curls. The intrinsic discontinuities around local maxima in absolute values of v and B take the form of vortex sheets and current sheets, respectively, the former playing a fundamental role in the development of hydrodyamic turbulence and the latter playing a major role in heating the X-ray coronas of stars and galaxies.

Parker, E. N.↗

NASA NTRS1991

Variational formulation of hybrid problems for fully 3-D transonic flow with shocks in rotor

Based on previous research, the unified variable domain variational theory of hybrid problems for rotor flow is extended to fully 3-D transonic rotor flow with shocks, unifying and generalizing the direct and inverse problems. Three variational principles (VP) families were established. All unknown boundaries and flow discontinuities (such as shocks, free trailing vortex sheets) are successfully handled via functional variations with variable domain, converting almost all boundary and interface conditions, including the Rankine Hugoniot shock relations, into natural ones. This theory provides a series of novel ways for blade design or modification and a rigorous theoretical basis for finite element applications and also constitutes an important part of the optimal design theory of rotor bladings. Numerical solutions to subsonic flow by finite elements with self-adapting nodes given in Refs., show good agreement with experimental results.

Liu, Gao-Lian↗

NASA NTRS1991

Broadband shock-associated noise from supersonic jets in flight

The stochastic model theory of broadband shock-associated noise from supersonic jets, developed earlier (Tam, 1987, 1990), is extended to jets in flight. Forward flight affects broadband shock-associated noise generation through noise source modification. It also affects the intensity and characteristics of the noise radiated in a given direction through the mean flow convection effect. Noise source modification arises when the external flow associated with forward flight alters the mean velocity profile and shock cell structure of the jet. The mean flow convection effect arises because the radiated sound waves now have to propagate through a moving medium to reach the observation point. The present theory suggests that both the noise source modification effect and the mean flow convection effect would produce narrower broadband shock-associated noise spectral peaks with lower peak frequencies. The theory is further extended to the case of a supersonic jet embedded in an open wind tunnel bounded by a vortex sheet. Detalied comparisons between the calculated results and the measurements of Norum and Shearin (1984, 1986) are reported here. Very good agreements are found over a broad range of jet Mach numbers and wind tunnel Mach numbers.

Tam, C. K. W.↗

NASA NTRS1993

Three-dimensional numerical simulation of gradual opening in a wave rotor passage

The evolution of the contact interface and the propagation of compression waves inside a single wave rotor passage gradually opening to and traversing an inlet port is studied numerically using an inviscid formulation of the governing equations. Insights into the response of the interface and kinematics of the flow field to various opening times are given. Since the opening time is inversely proportional to the rotational speed of the rotor, the effects of passage rotation such as centripetal and Coriolis accelerations are intrinsically coupled to the gradual opening process. Certain three-dimensional features associated with the gradual opening process as a result of centripetal and Coriolis accelerations are illustrated. For the range of opening times or rotational speeds considered, a portion of the interface behaves like a vortex sheet that can degenerate into a complex interfacial structure. The vortices produced along the interface can serve as a stirring mechanism to promote local mixing. Coriolis and centripetal accelerations can introduce three dimensional effects such as interfacial distortions in meridional planes and spanwise migration of fluid elements.

Larosiliere, Louis M.↗

NASA NTRS1992

Instability of rectangular jets

The instability of rectangular jets is investigated using a vortex sheet model. It is shown that such jets support four linearly independent families of instability waves. Within each family there are infinitely many modes. A way to classify these modes according to the characteristics of their mode shapes or eigenfunctions is proposed. A parametric study of the instability wave characteristics has been carried out. A sample of the numerical results is reported here. It is found that the first and third modes of each instability wave family are corner modes. The pressure fluctuations associated with these instability waves are localized near the corners of the jet. The second mode, however, is a center mode with maximum fluctuations concentrated in the central portion of the jet flow. The center mode has the largest spatial growth rate. It is anticipated that as the instability waves propagate downstream the center mode would emerge as the dominant instability of the jet.

Tam, Christopher K. W.↗

NASA NTRS1993

The rollup of a vortex layer near a wall

The behavior of an inviscid vortex layer of non-zero thickness near a wall is studied, both through direct numerical simulation of the two-dimensional vorticity equation at high Reynolds numbers, and using an approximate ordinary nonlinear integro-differential equation which is satisfied in the limit of a thin layer under long-wavelength perturbations. For appropriate initial conditions the layer rolls up and breaks into compact vortices which move along the wall at constant speed. Because of the effect of the wall, they correspond to equilibrium counter-rotating vortex dipoles. This breakup can be related to the disintegration of the initial conditions of the approximate nonlinear dispersive equation into solitary waves. The study is motivated by the formation of longitudinal vortices from vortex sheets in the wall region of a turbulent channel.

Jimenez, Javier↗

NASA NTRS1993

Instability of rectangular jets

The instability of rectangular jets is investigated using a vortex-sheet model. It is shown that such jets support four linearly independent families of instability waves. Within each family there are infinitely many modes. A way to classify these modes according to the characteristics of their mode shapes or eigenfunctions is proposed. It is demonstrated that the boundary element method can be used to calculate the dispersion relations and eigenfunctions of these instability wave modes. The method is robust and efficient. A parametric study of the instability wave characteristics has been carried out. A sample of the numerical results is reported here. It is found that the first and third modes of each instability wave family are corner modes. The pressure fluctuations associated with these instability waves are localized near the corners of the jet. The second mode, however, is a center mode with maximum fluctuations concentrated in the central portion of the jet flow. The center mode has the largest spatial growth rate. It is anticipated that as the instability waves propagate downstream the center mode would emerge as the dominant instability of the jet.

Tam, Christopher K. W.↗

NASA NTRS1993

Three-dimensional numerical simulation of gradual opening in a wave rotor passage

Larosiliere, Louis M.↗

NASA NTRS1993

Quasi-periodic transverse plasma flow associated with an evolving MHD vortex street in the outer heliosphere

We study a transverse plasma flow induced by the evolution of a Karman vortex street using a Chebyshev-Fourier spectral algorithm to solve both the compressible Navier-Stokes and MHD equations. The evolving vortex street is formed by the nonlinear interaction of two vortex sheets initially in equilibrium. We study spatial profiles of the total plasma velocity, the density, the meridional flow angle and the location of sector boundaries and find generally good agreement with Voyager 2 measurements of quasi-periodic transverse flow in the outer heliosphere. The pressure pulses associated with the meridional flows in the simulation are too small, although they are correctly located, and this may be due to the lack of any 'warp' in the current sheet in this model. A strong, flow-aligned magnetic field, such as would occur in the inner heliosphere, is shown to lead to weak effects that would be masked by the background interplanetary turbulence. We also study the plasma and magnetic transport resulting from the meridional flow and find that deficits of magnetic quantities do occur near the ecliptic. While the effect is relatively small, it is in general agreement with the most recent analysis of 'flux deficit' in the outer heliosphere.

Siregar, Edouard↗

NASA NTRS1987

Design of Turbomachinery Blading in Three-Dimensional Flow By the Circulation Method: A Progress Report

In this paper, we report on the progress of the further development and application of an analytical theory for the design of turbomachinery blading in three-dimensional flow. After a section describing the recent development of the theory, we discuss the various applications including the investigations of the design of radial inflow turbines. Numerical examples are also presented to illustrate the application of the theory to the design of a free vortex fan, an axial turbine rotor, a free vortex compressor stage for low and high Mach numbers, a non-free vortex fan with trailing vortex sheets, and a counter-rotating propeller in the limit of a large number of blades. Further possible applications of the theory to the design of other aero- and hydrodynamic devices are discussed.

W R Hawthorne↗

NASA NTRS1993

Pinching solutions of slender cylindrical jets

Simplified equations for slender jets are derived for a circular jet of one fluid flowing into an ambient second fluid, the flow being confined in a circular tank. Inviscid flows are studied which include both surface tension effects and Kelvin-Helmholtz instability. For slender jets a coupled nonlinear system of equations is found for the jet shape and the axial velocity jump across it. The equations can break down after a finite time and similarity solutions are constructed, and studied analytically and numerically. The break-ups found pertain to the jet pinching after a finite time, without violation of the slender jet ansatz. The system is conservative and admissible singular solutions are those which conserve the total energy, mass, and momentum. Such solutions are constructed analytically and numerically, and in the case of vortex sheets with no surface tension certain solutions are given in closed form.

Papageorgiou, Demetrios T.↗

NASA NTRS1993

The acoustic and instability waves of jets confined inside an acoustically lined rectangular duct

An analysis of linear wave modes associated with supersonic jets confined inside an acoustically lined rectangular duct is presented. Mathematical formulations are given for the vortex-sheet model and continuous mean flow model of the jet flow profiles. Detailed dispersion relations of these waves in a two-dimensional confined jet as well as an unconfined free jet are computed. Effects of the confining duct and the liners on the jet instability and acoustic waves are studied numerically. It is found that the effect of the liners is to attenuate waves that have supersonic phase velocities relative to the ambient flow. Numerical results also show that the growth rates of the instability waves could be reduced significantly by the use of liners. In addition, it is found that the upstream propagating neutral waves of an unconfined jet could become attenuated when the jet is confined.

Hu, Fang Q.↗

NASA NTRS1994

The effects of profiles on supersonic jet noise

The effect of velocity profiles on supersonic jet noise are studied by using stability calculations made for a shock-free coannular jet, with both the inner and outer flows supersonic. The Mach wave emission process is modeled as the noise generated by the large scale turbulent structures or the instability waves in the mixing region. Both the vortex-sheet and the realistic finite thickness shear layer models are considered. The stability calculations were performed for both inverted and normal velocity profiles. Comparisons are made with the results for an equivalent single jet, based on equal thrust, mass flow rate and exit area to that of the coannular jet. The advantages and disadvantages of these velocity profiles as far as noise radiation is concerned are discussed. It is shown that the Rayleigh's model prediction of the merits and demerits of different velocity profiles are in good agreement with the experimental data.

Tiwari, S. N.↗

NASA NTRS1993

The effect of velocity profiles on supersonic jet noise

This paper presents stability calculations made for a shock-free coannular jet, with both the inner and outer flows supersonic, using the instability wave model. The Mach wave emission process is modeled as the noise generated by the large scale turbulent structures or the instability waves in the mixing region. Both the vortex-sheet and the realistic finite thickness shear layer models are considered. The stability calculations have been performed for both inverted and normal velocity profiles. Comparisons are made with the results for an equivalent single jet, based on equal thrust, mass flow rate and exit area to that of the coannular jet. The advantages and disadvantages of these velocity profiles as far as noise radiation is concerned are discussed.

Bhat, T. R. S.↗

NASA NTRS1995

Nonlinear interaction between a pair of oblique modes in a supersonic mixing layer: Long-wave limit

The nonlinear interaction between a pair of symmetric, oblique, and spatial instability modes is studied in the long-wave limit using asymptotic methods. The base flow is taken to be a supersonic mixing layer whose Mach number is such that the corresponding vortex sheet is marginally stable according to Miles' criterion. It is shown that the amplitude of the mode obeys a nonlinear integro-differential equation. Numerical solutions of this equation show that, when the obliqueness angle is less than pi/4, the effect of the nonlinearity is to enhance the growth rate of the instability. The solution terminates in a singularity at a finite streamwise location. This result is reminiscent of that obtained in the vicinity of the neutral point by other authors in several different types of flows. On the other hand, when the obliqueness angle is more than pi/4, the streamwise development of the amplitude is characterized by a series of modulations. This arises from the fact that the nonlinear term in the amplitude equation may be either stabilizing or destabilizing, depending on the value of the streamwise coordinate. However, even in this case the amplitude of the disturbance increases, though not as rapidly as in the case for which the angle is less than pi/4. Quite generally then, the nonlinear interaction between two oblique modes in a supersonic mixing layer enhances the growth of the disturbance.

Balsa, Thomas F.↗