Evolution of localized disturbances in the elliptic instability

2014 ◽  
Vol 755 ◽  
pp. 603-627 ◽  
Author(s):  
Yuji Hattori ◽  
Mohd Syafiq bin Marzuki
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Energy Spectrum ◽  
Short Wavelength ◽  
Isotropic Turbulence ◽  
Inertial Range ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Elliptical Flow ◽  
The Mean

AbstractThe time evolution of localized disturbances in an elliptical flow confined in an elliptical cylinder is studied by direct numerical simulation (DNS). The base flow is subject to the elliptic instability. The unstable growth of localized disturbances predicted by the short-wavelength stability analysis is captured. The time evolution can be divided into four stages: linear, weakly nonlinear, nonlinear and turbulent. In the linear stage a single wavepacket grows exponentially without changing its shape. The exponential growth is accompanied by large oscillations which have time period half that of the fluid particles in the elliptical flow. An averaged wavepacket, which is a train of bending waves that has a finite spatial extent, also grows exponentially, while the oscillations of the growth rate are small. The averaged growth rate increases as the kinematic viscosity decreases; the inviscid limit is close to the value predicted by the short-wavelength stability analysis. In the weakly nonlinear stage the energy stops growing. The vortical structure of the initial disturbances is deformed into wavy patterns. The energy spectrum loses the peak at the initial wavenumber, developing a broad spectrum, and the flow goes into the next stage. In the nonlinear stage weak vorticity is scattered in the whole domain although strong vorticity is still localized. The probability density functions (p.d.f.) of a velocity component and its longitudinal derivative are similar to those of isotropic turbulence; however, the energy spectrum does not have an inertial range showing the Kolmogorov spectrum. Finally in the turbulent stage fine-scale structures appear in the vorticity field. The p.d.f. of the longitudinal derivative of velocity shows the strong intermittency known for isotropic turbulence. The energy spectrum attains an inertial range showing the Kolmogorov spectrum. The turbulence is not symmetric because of rotation and strain; the component of vorticity in the compressing direction is smaller than the other two components. The energy of the mean flow as well as the total energy decreases. The ratio of the lost energy to the initial energy of the mean flow is large in the core region.

scholarly journals The Growth and Saturation of Submesoscale Instabilities in the Presence of a Barotropic Jet

2018 ◽  
Vol 48 (11) ◽  
pp. 2779-2797 ◽  
Author(s):  
Megan A. Stamper ◽  
John R. Taylor ◽  
Baylor Fox-Kemper
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Initial Conditions ◽  
Boussinesq Equations ◽  
Small Region ◽  
Computational Domain ◽  
Mean Flow ◽  
The Mean

AbstractMotivated by recent observations of submesoscales in the Southern Ocean, we use nonlinear numerical simulations and a linear stability analysis to examine the influence of a barotropic jet on submesoscale instabilities at an isolated front. Simulations of the nonhydrostatic Boussinesq equations with a strong barotropic jet (approximately matching the observed conditions) show that submesoscale disturbances and strong vertical velocities are confined to a small region near the initial frontal location. In contrast, without a barotropic jet, submesoscale eddies propagate to the edges of the computational domain and smear the mean frontal structure. Several intermediate jet strengths are also considered. A linear stability analysis reveals that the barotropic jet has a modest influence on the growth rate of linear disturbances to the initial conditions, with at most a ~20% reduction in the growth rate of the most unstable mode. On the other hand, a basic state formed by averaging the flow at the end of the simulation with a strong barotropic jet is linearly stable, suggesting that nonlinear processes modify the mean flow and stabilize the front.

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

scholarly journals WKB theory for rapid distortion of inhomogeneous turbulence

1999 ◽  
Vol 390 ◽  
pp. 325-348 ◽  
Author(s):  
S. NAZARENKO ◽  
N. K.-R. KEVLAHAN ◽  
B. DUBRULLE
Keyword(s):  
Large Scale ◽  
Isotropic Turbulence ◽  
Two Dimensional ◽  
Turbulent Spot ◽  
Mean Flow ◽  
Mean Velocity ◽  
Inhomogeneous Turbulence ◽  
Large Scale Vortex ◽  
The Mean ◽  
Mean Strain

A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

Generation mechanism of a hierarchy of vortices in a turbulent boundary layer

2019 ◽  
Vol 865 ◽  
pp. 1085-1109 ◽  
Author(s):  
Yutaro Motoori ◽  
Susumu Goto
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Energy Spectrum ◽  
Turbulent Boundary Layer ◽  
Large Scale ◽  
Generation Mechanism ◽  
Small Scale ◽  
Generation Process ◽  
Mean Flow ◽  
The Mean

To understand the generation mechanism of a hierarchy of multiscale vortices in a high-Reynolds-number turbulent boundary layer, we conduct direct numerical simulations and educe the hierarchy of vortices by applying a coarse-graining method to the simulated turbulent velocity field. When the Reynolds number is high enough for the premultiplied energy spectrum of the streamwise velocity component to show the second peak and for the energy spectrum to obey the$-5/3$power law, small-scale vortices, that is, vortices sufficiently smaller than the height from the wall, in the log layer are generated predominantly by the stretching in strain-rate fields at larger scales rather than by the mean-flow stretching. In such a case, the twice-larger scale contributes most to the stretching of smaller-scale vortices. This generation mechanism of small-scale vortices is similar to the one observed in fully developed turbulence in a periodic cube and consistent with the picture of the energy cascade. On the other hand, large-scale vortices, that is, vortices as large as the height, are stretched and amplified directly by the mean flow. We show quantitative evidence of these scale-dependent generation mechanisms of vortices on the basis of numerical analyses of the scale-dependent enstrophy production rate. We also demonstrate concrete examples of the generation process of the hierarchy of multiscale vortices.

scholarly journals Transitional Flow Dynamics Behind a Micro-Ramp

2019 ◽  
Vol 104 (2-3) ◽  
pp. 533-552
Author(s):  
J. Casacuberta ◽  
K. J. Groot ◽  
Q. Ye ◽  
S. Hickel
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Large Scale ◽  
Vortex Pair ◽  
Base Flow ◽  
Transitional Flow ◽  
Mean Flow ◽  
Primary Vortex ◽  
The Mean ◽  
Near Wall

AbstractMicro-ramps are popular passive flow control devices which can delay flow separation by re-energising the lower portion of the boundary layer. We compute the laminar base flow, the instantaneous transitional flow, and the mean flow around a micro-ramp immersed in a quasi-incompressible boundary layer at supercritical roughness Reynolds number. Results of our Direct Numerical Simulations (DNS) are compared with results of BiLocal stability analysis on the DNS base flow and independent tomographic Particle Image Velocimetry (tomo-PIV) experiments. We analyse relevant flow structures developing in the micro-ramp wake and assess their role in the micro-ramp functionality, i.e., in increasing the near-wall momentum. The main flow feature of the base flow is a pair of streamwise counter-rotating vortices induced by the micro-ramp, the so-called primary vortex pair. In the instantaneous transitional flow, the primary vortex pair breaks up into large-scale hairpin vortices, which arise due to linear varicose instability of the base flow, and unsteady secondary vortices develop. Instantaneous vortical structures obtained by DNS and experiments are in good agreement. Matching linear disturbance growth rates from DNS and linear stability analysis are obtained until eight micro-ramp heights downstream of the micro-ramp. For the setup considered in this article, we show that the working principle of the micro-ramp is different from that of classical vortex generators; we find that transitional perturbations are more efficient in increasing the near-wall momentum in the mean flow than the laminar primary vortices in the base flow.

Similarity and self-preservation in isotropic turbulence

1951 ◽  
Vol 243 (867) ◽  
pp. 359-386 ◽  
Keyword(s):  
Energy Spectrum ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Wave Number ◽  
Inertial Subrange ◽  
Local Similarity ◽  
Mean Flow ◽  
Spectrum Function ◽  
Energy Spectrum Function

Measurements of the double and triple velocity correlation functions and of the energy spectrum function have been made in the uniform mean flow behind turbulence-producing grids of several shapes at mesh Reynolds numbers between 2000 and 100000. These results have been used to assess the validity of the various theories which postulate greater or less degrees of similarity or self-preservation between decaying fields of isotropic turbulence. It is shown that the conditions for the existence of the local similarity considered by Kolmogoroff and others are only fulfilled for extremely small eddies at ordinary Reynolds numbers, and that the inertial subrange in which the spectrum function varies as k -35 ( k is the wave-number) is non-existent under laboratory conditions. Within the range of local similarity, the spectrum function is best represented by an empirical function such as k -a log k , and it is concluded that all suggested forms for the inertial transfer term in the spectrum equation are in error. Similarity of the large scale structure of flows of differing Reynolds numbers at corresponding times of decay has been confirmed, and approximate measurements of the Loitsianski invariant in the initial period have been made. Its value, expressed non-dimensionally, decreases slowly with grid Reynolds number within the range of observation. Turbulence-producing grids of widely different shapes are found to produce flows identical in energy decay and in structure of the smaller eddies. The largest eddies depend markedly on the grid shape and are, in general, significantly anisotropic. Within the initial period of decay, the greater part of the energy spectrum function is self-preserving, and this part has a shape independent of the shape of the turbulence-producing grid. The part that is not self-preserving contains at least one-third of the total energy, and it is concluded that theories postulating quasi-equilibrium during decay must be considered with great caution.

Fully coupled resonant-triad interactions in a free shear layer

1994 ◽  
Vol 278 ◽  
pp. 101-121 ◽  
Author(s):  
R. Mallier ◽  
S. A. Maslowe
Keyword(s):  
Flow Direction ◽  
Mixing Layer ◽  
Adverse Pressure Gradient ◽  
Critical Layer ◽  
Free Shear Layer ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Initial Interaction ◽  
The Mean ◽  
Fully Coupled

We report the results of an investigation of the weakly nonlinear evolution of a triad of waves, each slightly amplified on a linear basis, that are superimposed on a tanh y mixing layer. The triad consists of a plane wave and a pair of oblique modes that act as a subharmonic of order 1/2. The oblique modes are inclined at approximately ±60°. to the mean flow direction and because the resonance conditions are satisfied exactly the analysis is entirely self-consistent as an asymptotic theory. The nonlinearity first occurs within the critical layer and the initial interaction is of the parametric resonance type. This produces faster than exponential growth of the oblique waves, behaviour observed recently in the experiments of Corke & Kusek (1993). The critical-layer dynamics lead subsequently to coupled integro-differential equations governing the amplitude evolution and, as first shown in related work by Goldstein & Lee (1992) on boundary layers in an adverse pressure gradient, these equations develop singularities in a finite time.

scholarly journals Moisture Modes and the Eastward Propagation of the MJO

2013 ◽  
Vol 70 (1) ◽  
pp. 187-192 ◽  
Author(s):  
Adam Sobel ◽  
Eric Maloney
Keyword(s):  
Growth Rate ◽  
Moisture Gradient ◽  
Moist Static Energy ◽  
Mean Flow ◽  
Considerable Uncertainty ◽  
Low Level ◽  
Zonal Advection ◽  
The Mean ◽  
Frictional Convergence ◽  
Static Energy

Abstract The authors discuss modifications to a simple linear model of intraseasonal moisture modes. Wind–evaporation feedbacks were shown in an earlier study to induce westward propagation in an eastward mean low-level flow in this model. Here additional processes, which provide effective sources of moist static energy to the disturbances and which also depend on the low-level wind, are considered. Several processes can act as positive sources in perturbation easterlies: zonal advection (if the mean zonal moisture gradient is eastward), modulation of synoptic eddy drying by the MJO-scale wind perturbations, and frictional convergence. If the sum of these is stronger than the wind–evaporation feedback—as observations suggest may be the case, though with considerable uncertainty—the model produces unstable modes that propagate weakly eastward relative to the mean flow. With a small amount of horizontal diffusion or other scale-selective damping, the growth rate is greatest at the largest horizontal scales and decreases monotonically with wavenumber.

scholarly journals The theory of axisymmetric turbulence

1946 ◽  
Vol 186 (1007) ◽  
pp. 480-502 ◽  
Keyword(s):  
Viscous Dissipation ◽  
Invariant Theory ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Practical Interest ◽  
Mean Flow ◽  
Velocity Correlation ◽  
Rates Of Change ◽  
The Mean ◽  
Velocity Pressure

This paper discusses a type of turbulence in a uniform stream which is next to isotropic turbulence in order of simplicity. Instead of spherical symmetry, or isotropy, axially symmetical turbulence possesses symmetry about an axis which in practice is usually the direction of mean flow. The analysis is developed with the aid of invariant theory, as suggested by a previous paper by Robertson. The form of the fundamental velocity correlation is obtained, and scales of axisymmetric turbulence are defined. The results of greatest practical interest concern the time rates of change of the mean squares of the lateral and longitudinal velocity components. The rates of change involve two terms, the first representing viscous dissipation, and the second representing a transfer of energy from one component to the other due to the finite correlation between the velocity and pressure at neighbouring points. The effect of the velocity-pressure correlation is to bring the two velocity components towards equality, while the effect of the viscous dissipation will only be towards equality if an inequality between the curvatures at the origin of two particular velocity correlation coefficient curves, both of which are measurable, is obeyed. The rates of change of the mean squares of the vorticity components are also obtained.

DNS of the thermal effects of laser energy deposition in isotropic turbulence

2010 ◽  
Vol 654 ◽  
pp. 387-416
Author(s):  
SHANKAR GHOSH ◽  
KRISHNAN MAHESH
Keyword(s):  
Shock Wave ◽  
Time Scale ◽  
Energy Deposition ◽  
Isotropic Turbulence ◽  
Radial Distance ◽  
Working Fluid ◽  
Shock Propagation ◽  
Mean Flow ◽  
Turbulent Reynolds Number ◽  
The Mean

The interaction of a laser-induced plasma with isotropic turbulence is studied using numerical simulations. The simulations use air as the working fluid and assume local thermodynamic equilibrium. The numerical method is fully spectral and uses a shock-capturing scheme in a corrector step. A model problem involving the effect of energy deposition on an isolated vortex is studied as a first step towards plasma/turbulence interaction. Turbulent Reynolds number Reλ = 30 and fluctuation Mach numbers Mt = 0.001 and 0.3 are considered. A tear-drop-shaped shock wave is observed to propagate into the background, and progressively become spherical in time. The turbulence experiences strong compression due to the shock wave and strong expansion in the core. This behaviour is spatially inhomogeneous and non-stationary in time. Statistics are computed as functions of radial distance from the plasma axis and angular distance across the surface of the shock wave. For Mt = 0.001, the shock wave propagates on a much faster time scale compared to the turbulence evolution. At Mt of 0.3, the time scale of the shock wave is comparable to that of the background. For both cases the mean flow is classified into shock formation, shock propagation and subsequent collapse of the plasma core, and the effect of turbulence on each of these phases is studied in detail. The effect of mean vorticity production on the turbulent vorticity field is also discussed. Turbulent kinetic energy budgets are presented to explain the mechanism underlying the transfer of energy between the mean flow and background turbulence.

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