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NASA NTRS1992

Three dimensional particle simulation of high altitude rocket plumes

The interaction of two nozzles exhausting into vacuum generates a complex three-dimensional shock structure. The shock structure and resulting plume flowfield is characterized by the nozzle separation distance. For the appropriate range of penetration Knudsen numbers, the analysis of this shock structure can be suitably accomplished through a Monte Carlo simulation. This paper describes the application of a general three-dimensional Monte Carlo simulation on the Connection Machine CM-2 to the analysis of the plume self-interaction shock in the near field. Results are presented for two cases, corresponding to a small and a large nozzle separation distance. The results correctly reproduce the expected flow features and demonstrate the ability of this method to properly simulate the start of the plume self-interaction shock. This has significance not only for allowing analysis of the self-interacting plume in the near field, but also for allowing the subsequent simulation of the far field flow through the use of a continuation downstream exit boundary.

Dagum, Leonardo↗

NASA NTRS1981

Spectral energy transfer for inhomogeneous turbulence

It is noted that several terms in the two-point spectral equation for homogeneous turbulence can be interpreted as spectral-transfer terms; that is, they represent the net rate of energy transfer into a wavenumber region from all other wavenumbers. This holds for terms associated with both turbulence and self-interaction and interaction between turbulence and mean gradients. It is not seen as obvious, however, that similar interpretations apply when the turbulence is not homogeneous. In particular, one might question the interpretation for the terms associated with turbulence self-interaction because the condition of homegeneity is generally used in making the interpretation. It is the purpose here to consider whether terms interpretable as transfer terms exist in the equations for inhomogeneous turbulence. It is found that certain terms in the two-point spectral equation can be interpreted as transfer terms.

Deissler, R. G.↗

NASA NTRS1975

Comparison of theory and experiment for homogeneous turbulence with shear

Solutions for uniformly-sheared turbulence, in which the interaction of the turbulence with the mean shear dominates the turbulent self-interaction, are compared with experiment. An anisotropic spectral tensor, which appears general enough to represent the initial experimental turbulence, is used for the initial condition in the calculations. The evolution of one-point turbulence components and microscales, as well as two-point velocity correlations, are considered. In most cases the agreement with experiment is good. The theory correctly predicts the presence of a negative region for two-point longitudinal-velocity correlations only for point separations in the direction normal to the flow and the mean gradient.

Deissler, R. G.↗

NASA NTRS1980

Gravitational radiation quadrupole formula is valid for gravitationally interacting systems

An argument is presented for the validity of the quadrupole formula for gravitational radiation energy loss in the far field of nearly Newtonian (e.g., binary stellar) systems. This argument differs from earlier ones in that it determines beforehand the formal accuracy of approximation required to describe gravitationally self-interacting systems, uses the corresponding approximate equation of motion explicitly, and evaluates the appropriate asymptotic quantities by matching along the correct space-time light cones.

Walker, M.↗

NASA NTRS1987

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. The spectral transfers are found to be dominated by two dynamically distinct components, one due to nonlinear self-interactions within the transient flow and the other arising from coupled interactions between the stationary and transient parts of the flow. The former seems to be attributable to two-dimensional homogeneous turbulence, while the second cannot be understood within that theory. It is argued that the mixed stationary-transient interactions can be understood physically as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. The transients act mainly to strengthen the stationary flow, and this 'forcing' operates on a timescale comparable to presumed viscous timescales.

Shepherd, Theodore G.↗

NASA NTRS1989

A new class of non-topological solitons

A class of non-topological solitons was constructed in renormalizable scalar field theories with nonlinear self-interactions. For large charge Q, the soliton mass increases linearly with Q, i.e., the soliton mass density is approximately independent of charge. Such objects could be naturally produced in a phase transition in the early universe or in the decay of superconducting cosmic strings.

Frieman, Joshua A.↗

NASA NTRS1989

On the Nature of Navier-stokes Turbulence

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, largest Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The solutions, separate exponentially with time, having a positive Liapunov exponent. Thus the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number is reduced. Several qualitatively different flows are noted. These are, fully chaotic, complex period, weakly chaotic, simple periodic, and fixed-point. Of these, only the fully chaotic flows are classified as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, is both time-dependent and aperiodic. Turbulent solutions are obtained in which energy cascades from large to small-scale motions. In general, the spectral energy transfer takes place between wavenumber bands that are considerably separated. The special transfer can occur either as a result of nonlinear turbulence self-interaction or by interaction of turbulence with mean gradients. Turbulent systems are compared with those studied in kinetic theory. The two types of systems are fundamentally different (continuous and dissipative as opposed to discrete and conservative), but there are similarities. For instance, both are nonlinear and show sensitive dependence on initial conditions. Also, the turbulent and molecular stress tensors are identical if the macroscopic velocities for the turbulent stress are replaced by molecular velocities.

Deissler, Robert G.↗

NASA NTRS1991

Resonant triad in boundary-layer stability. Part 1: Fully nonlinear interaction

A first principles theory is developed to study the nonlinear spatial evolution of a near-resonance triad of instability waves in boundary layer transition. This triad consists of a plane wave at fundamental frequency and a pair of symmetrical, oblique waves at the subharmonic frequency. A low frequency, high Reynolds number asymptotic scaling leads to a distinct critical layer where nonlinearity first becomes important; the development of the triad's waves is determined by the critical layer's nonlinear, viscous dynamics. The resulting theory is fully nonlinear in that all nonlinearly generated oscillatory and nonoscillatory components are accounted for. The presence of the plane wave initially causes exponential of exponential growth of the oblique waves. However, the plane wave continues to follow the linear theory, even when the oblique waves' amplitude attains the same order of magnitude as that of the plane wave. A fully interactive stage then comes into effect when the oblique waves exceed a certain level compared to that of the plane wave. The oblique waves react back on the fundamental, slowing its growth rate. The oblique waves' saturation results from their self-interaction - a mechanism that does not require the presence of the plane wave. The oblique waves' saturation level is independent of their initial level, but decreases as the obliqueness angle increases.

Mankbadi, Reda R.↗

NASA NTRS1991

Resonant Triad in Boundary-Layer Stability: Composite Solution and Comparison with Observations - Part 2

Here, numerical results are computed from an asymptotic near-resonance triad analysis. The analysis considers a resonant triad of instability waves consisting of a plane fundamental wave and a pair of symmetrical oblique subharmonic waves. The relevant scaling ensures that nonlinearity is confined to a distinct critical layer. The analysis is first used to form a composite solution that accounts for both the flow divergence and nonlinear effects. It is shown that the backreaction on the plane Tollmien Schlichting (TS) fundamental wave, although fully accounted for, is of little significance. The observed enhancement at the fundamental frequency disturbance is not in the plane TS wave, but is caused by nonlinearly generated waves at the fundamental frequency that result from nonlinear interactions in the critical layer. The saturation of the oblique waves is caused by their self-interaction. The nonlinear phase-locking phenomenon, the location of resonance with respect to the neutral stability curve, low frequency effects, detuning in the streamwise wave numbers, and nonlinear distortion of the mode shapes are discussed. Nonlinearity modifies the initially two dimensional Blasius profile into a fuller one with spanwise periodicity. The interactions at a wide range of unstable spanwise wave numbers are considered, and the existence of a preferred spanwise wave number is explained by means of the vorticity distribution in the critical layer. Besides presenting novel features of the phenomena and explaining the delicate mechanisms of the interactions, the results of the theory are in excellent agreement with experimental and numerical observations for all stages of the development and for various input parameters.

Mankbadi, Reda R.↗

NASA NTRS1992

On the nonlinear development of three-dimensional instability waves in natural transition

The nonlinear development of a pair of symmetrical three-dimensional oblique instability waves in transitioning boundary layers is investigated based on critical-layer nonlinearity (CLN). Particular emphasis is placed on understanding the mechanisms leading to the naturally occurring transition in which the amplitudes of the plane and oblique modes can be of the same order. Results indicate that the self-interaction of the oblique waves reduces their growth rate. A relatively large oblique mode can either suppress or enhance the growth of the plane mode, depending on the initial phase difference. If the plane mode amplitude is not negligible with respect to that of the oblique modes, it causes a strong amplification of the oblique mode at the subharmonic frequency. The comparison with observations are quite encouraging and several novel features of the natural transition process are revealed.

Mankbadi, Reda R.↗

NASA NTRS1991

On the origin of streaks in turbulent shear flows

The paper substantiates the notion that selective amplification and direct resonance, based on linear theory, does not provide a selection mechanism for the well-defined streak spacing of about 100 wall units observed in wall-bounded turbulent shear flows. For the direct resonance theory, it is shown that the streaks are created by the nonlinear self-interaction of the vertical velocity rather than that of the directly forced vertical vorticity. It is proposed that the selection mechanism must be inherently nonlinear and correspond to a self-sustaining process. For the case of plane Poiseuille flow the 100-wall-unit criterion corresponds to a critical Reynolds number of 1250, based on the centerline velocity and the channel half-width, which is close to the usually quoted value of about 1000. In plane Couette flow, it corresponds to a critical Reynolds number of 625, based on the half-velocity difference and the half-width.

Waleffe, Fabian↗

NASA NTRS1991

On the origin of the streak spacing in turbulent shear flows

It is shown that the ideas of selective amplification and direct resonance, based on linear theory, can not provide an explanation for the well-defined streak spacing of about 100 wall units (referred to as 100(+) hereafter) in wall-bounded turbulent shear flows. In addition, for the direct resonance theory, the streaks are created by the non-linear self-interaction of the vertical velocity rather than of the directly forced vertical vorticity. In view of the failure of these approaches, it is then proposed that the selection mechanism must be inherently non-linear and correspond to a self-sustaining mechanism. The 100(+) value should thus be considered as a critical Reynolds number for that mechanism. Indeed, in the case of Poiseuille flow, this 100(+) criterion for transition to turbulence corresponds to the usually quoted value of 1000 based on the half-width and the centerline velocity. In Couette flow, it corresponds to a critical Reynolds number of about 400 based on the half width and half velocity difference.

Waleffe, Fabian A.↗

NASA NTRS1994

Vortex rings and the solar granulation

Observations indicate that solar granules have the flow topology of updraft vortex loops. We interpret granule behavior in terms of the mutual and self-interactions of such loops. In particular, the expansion phase that granules commonly undergo is explained by the self-expansion of a vortex ring in a stratified fluid. For a range of granular parameters, we find that the expansion velocity of a vortex ring varies from 0.7 to 1.5 times the maximum surface flow velocity, in agreement with granule observations. We also present speculation on the nature of granule fragmentation.

Arendt, Steve↗

NASA NTRS1995

Pressure correlations at a fluid/structure interface

The structure of pressure-pressure correlations at the interface of an incompressible steady-state turbulent flow with a rigid boundary was investigated. For the sake of completeness, the absolute value of the correlation between two random varying functions is herein defined as a number greater than or equal to zero and less than or equal to unity which is a measure of that fraction of one of the functions that 'follows' the second function (or vice versa). It was found that the soughtafter correlations can be determined by consideration of the high Re Navier-Stokes equation, but that the complexity of boundary layer turbulence, in particular the inhomogeneity perpendicular to the boundary and the anisotropy due to convective flow gradients, makes the structure of said correlations extremely difficult to assess. One of the earlier researchers in this field described the quantity under present consideration as 'a quantity which is beyond assessment.' Nonetheless, it was found that under some rather simplifying assumptions the determination of the required structure necessitates the formulation of the related structure of second order two-point correlations of turbulent velocity gradients, as well as third order two-point correlations of velocity gradients. The presence of these latter gradients is due to the nonlinearity in the turbulence ('turbulence self-interaction'). Both of these correlations are scaled, although not similarly, by factors dependent upon the magnitude of the convective flow, which can be modeled using a log law approximation. Fourth order correlations, although present, can be ignored, since they constitute 'higher order terms.' In a slightly more complex situation, it was found that convective flow gradients also have to be incorporated. At the moment, no definitive algebraic information peculiar to pressure-pressure correlations is available in the most highly idealized cases.

Trevino, George↗

NASA NTRS1995

The role of nonlinear critical layers in boundary layer transition

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called 'critical layer', with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

Goldstein, M.E.↗

NASA NTRS1995

The Effect of Nonlinear Critical Layers on Boundary Layer Transition

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear and spatially growing instability waves evolve downstream in nominally two-dimensional and spanwise periodic laminar boundary layers. The first nonlinear reaction takes place locally within a so-called 'critical layer' with the flow outside this layer consisting of a locally parallel mean flow plus an appropriate superposition of linear instability waves. The amplitudes of these waves are determined by either a single integro-differential equation or by a pair of integro-differential equations with quadratic to quartic-type nonlinearities.

Goldstein, Marvin E.↗

NASA NTRS1998

Nonlinear Interaction of Detuned Instability Waves in Boundary-Layer Transition: Amplitude Equations

The non-equilibrium critical-layer analysis of a system of frequency-detuned resonant-triads is presented. In this part of the analysis, the system of partial differential critical-layer equations derived in Part I is solved analytically to yield the amplitude equations which are analyzed using a combination of asymptotic and numerical methods. Numerical solutions of the inviscid non-equilibrium oblique-mode amplitude equations show that the frequency-detuned self-interaction enhances the growth of the lower-frequency oblique modes more than the higher-frequency ones. All amplitudes become singular at the same finite downstream position. The frequency detuning delays the occurrence of the singularity. The spanwise-periodic mean-flow distortion and low-frequency nonlinear modes are generated by the critical-layer interaction between frequency-detuned oblique modes. The nonlinear mean flow and higher harmonics as well as the primary instabilities become as large as the base mean flow in the inviscid wall layer in the downstream region where the distance from the singularity is of the order of the wavelength scale.

Lee, Sang Soo↗

NASA NTRS2003

Computation of Large-Scale Structure Jet Noise Sources With Weak Nonlinear Effects Using Linear Euler

An approximate technique is presented for the prediction of the large-scale turbulent structure sound source in a supersonic jet. A linearized Euler equations code is used to solve for the flow disturbances within and near a jet with a given mean flow. Assuming a normal mode composition for the wave-like disturbances, the linear radial profiles are used in an integration of the Navier-Stokes equations. This results in a set of ordinary differential equations representing the weakly nonlinear self-interactions of the modes along with their interaction with the mean flow. Solutions are then used to correct the amplitude of the disturbances that represent the source of large-scale turbulent structure sound in the jet.

Dahl, Milo D.↗