Spectral broadening and flow randomization in free shear layers

2012 ◽  
Vol 706 ◽  
pp. 431-469 ◽  
Author(s):  
Xuesong Wu ◽  
Feng Tian
Keyword(s):  
Phase Modulation ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Present Theory ◽  
Shear Layers ◽  
Evolution System ◽  
Free Shear Layer ◽  
Spectral Broadening ◽  
Free Shear Layers ◽  
Combination Frequencies

AbstractIt has been observed experimentally that when a free shear layer is perturbed by a disturbance consisting of two waves with frequencies ${\omega }_{0} $ and ${\omega }_{1} $, components with the combination frequencies $(m{\omega }_{0} \pm n{\omega }_{1} )$ ($m$ and $n$ being integers) develop to a significant level thereby causing flow randomization. This spectral broadening process is investigated theoretically for the case where the frequency difference $({\omega }_{0} \ensuremath{-} {\omega }_{1} )$ is small, so that the perturbation can be treated as a modulated wavetrain. A nonlinear evolution system governing the spectral dynamics is derived by using the non-equilibrium nonlinear critical layer approach. The formulation provides an appropriate mathematical description of the physical concepts of sideband instability and amplitude–phase modulation, which were suggested by experimentalists. Numerical solutions of the nonlinear evolution system indicate that the present theory captures measurements and observations rather well.

On the spatial instability of piecewise linear free shear layers

1987 ◽  
Vol 174 ◽  
pp. 553-563 ◽  
Author(s):  
T. F. Balsa
Keyword(s):  
Velocity Profile ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Reynolds Numbers ◽  
Shear Layers ◽  
Spatial Instability ◽  
Free Shear Layers ◽  
Linear Shear ◽  
High Reynolds ◽  
Sommerfeld Equation

The main goal of this paper is to clarify the spatial instability of a piecewise linear free shear flow. We do this by obtaining numerical solutions to the Orr–Sommerfeld equation at high Reynolds numbers. The velocity profile chosen is very much like a piecewise linear one, with the exception that the corners have been rounded so that the entire profile is infinitely differentiable. We find that the (viscous) spatial instability of this modified profile is virtually identical to the inviscid spatial instability of the piecewise linear profile and agrees qualitatively with the inviscid results for the tanh profile when the shear layers are convectively unstable. The unphysical features, previously identified for the piecewise linear velocity profile, arise only when the flow is absolutely unstable. In a nutshell, we see nothing wrong with the inviscid spatial instability of piecewise linear shear flows.

Higher-order approximation for free shear layers in almost rigid rotations

1975 ◽  
Vol 69 (1) ◽  
pp. 191-195 ◽  
Author(s):  
L. Pamela Cook ◽  
G. S. S. Ludford
Keyword(s):  
Angular Velocity ◽  
Order Approximation ◽  
Shear Layers ◽  
Higher Order ◽  
Rigid Rotation ◽  
Free Shear Layer ◽  
Curvature Effects ◽  
Parallel Disks ◽  
Free Shear Layers ◽  
Theory And Experiment

The free shear layer stemming from a discontinuity in angular velocity at either of two parallel disks in almost rigid rotation with fluid between is re-examined. The sole discrepancy between theory and experiment is unaffected by higher-order approximation unless curvature effects are included, when it is reduced.

The effect of initial conditions on the development of a free shear layer

1966 ◽  
Vol 26 (2) ◽  
pp. 225-236 ◽  
Author(s):  
P. Bradshaw
Keyword(s):  
Turbulent Mixing ◽  
Initial Conditions ◽  
Mixing Layer ◽  
Initial Boundary ◽  
Shear Layers ◽  
Free Shear Layer ◽  
Turbulent Mixing Layer ◽  
Free Shear Layers ◽  
Final Approach ◽  
Reynolds Number Effects

The distance between the separation point and the final approach to a fully developed turbulent mixing layer is found to be of the order of a thousand times the momentum-deficit thickness of the initial boundary layer, whether the latter be laminar or turbulent. There are correspondingly large shifts in the virtual origin of the mixing layer, resulting in spurious Reynolds-number effects which cause considerable difficulties in tests of model jets or blunt-based bodies, and which are probably responsible for the disagreements over the influence of Mach number on the development of free shear layers. These effects are explained.

scholarly journals Nonlinear evolution of interacting oblique waves on two-dimensional shear layers

1989 ◽  
Vol 207 ◽  
pp. 97-120 ◽  
Author(s):  
M. E. Goldstein ◽  
S.-W. Choi
Keyword(s):  
Wave Amplitude ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Shear Layers ◽  
Instability Wave ◽  
Two Dimensional ◽  
Instability Waves ◽  
Downstream Distance ◽  
Parallel Streams

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that they always end in a singularity at a finite downstream distance.

scholarly journals Nonlinear dynamics of large-scale coherent structures in turbulent free shear layers

2015 ◽  
Vol 787 ◽  
pp. 396-439 ◽  
Author(s):  
Xuesong Wu ◽  
Xiuling Zhuang
Keyword(s):  
Nonlinear Dynamics ◽  
Coherent Structures ◽  
Large Scale ◽  
Shear Layers ◽  
Small Scale ◽  
Evolution System ◽  
Mean Flow ◽  
Free Shear Layers ◽  
The Mean ◽  
Scale Turbulence

Fully developed turbulent free shear layers exhibit a high degree of order, characterized by large-scale coherent structures in the form of spanwise vortex rollers. Extensive experimental investigations show that such organized motions bear remarkable resemblance to instability waves, and their main characteristics, including the length scales, propagation speeds and transverse structures, are reasonably well predicted by linear stability analysis of the mean flow. In this paper, we present a mathematical theory to describe the nonlinear dynamics of coherent structures. The formulation is based on the triple decomposition of the instantaneous flow into a mean field, coherent fluctuations and small-scale turbulence but with the mean-flow distortion induced by nonlinear interactions of coherent fluctuations being treated as part of the organized motion. The system is closed by employing a gradient type of model for the time- and phase-averaged Reynolds stresses of fine-scale turbulence. In the high-Reynolds-number limit, the nonlinear non-equilibrium critical-layer theory for laminar-flow instabilities is adapted to turbulent shear layers by accounting for (1) the enhanced non-parallelism associated with fast spreading of the mean flow, and (2) the influence of small-scale turbulence on coherent structures. The combination of these factors with nonlinearity leads to an interesting evolution system, consisting of coupled amplitude and vorticity equations, in which non-parallelism contributes the so-called translating critical-layer effect. Numerical solutions of the evolution system capture vortex roll-up, which is the hallmark of a turbulent mixing layer, and the predicted amplitude development mimics the qualitative feature of oscillatory saturation that has been observed in a number of experiments. A fair degree of quantitative agreement is obtained with one set of experimental data.

Nonlinear evolution of oblique waves on compressible shear layers

1989 ◽  
Vol 207 ◽  
pp. 73-96 ◽  
Author(s):  
M. E. Goldstein ◽  
S. J. Leib
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Landau Equation ◽  
Shear Layers ◽  
Instability Wave ◽  
Instability Waves ◽  
Downstream Distance ◽  
Rate Effects ◽  
Parallel Streams ◽  
Parameter Ranges

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on compressible shear layers between two parallel streams. The analysis shows that mean temperature non-uniformities cause nonlinearity to occur at much smaller amplitudes than it does when the flow is isothermal. The nonlinear instability wave growth rate effects are described by an integro-differential equation which bears some resemblance, to the Landau equation in that it involves a cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that inviscid solutions always end in a singularity at a finite downstream distance but that viscosity can eliminate this singularity for certain parameter ranges.

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