Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability

2018 ◽  
Vol 838 ◽  
pp. 478-500 ◽  
Author(s):  
Mathieu Marant ◽  
Carlo Cossu
Keyword(s):  
Initial Conditions ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Basic Flow ◽  
Sensitivity Analyses ◽  
Mean Flow ◽  
Helmholtz Instability ◽  
Flow Distortion ◽  
Streamwise Streaks ◽  
The Stability

The optimal energy amplifications of streamwise-uniform and spanwise-periodic perturbations of the hyperbolic-tangent mixing layer are computed and found to be very large, with maximum amplifications increasing with the Reynolds number and with the spanwise wavelength of the perturbations. The optimal initial conditions are streamwise vortices and the most amplified structures are streamwise streaks with sinuous symmetry in the cross-stream plane. The leading suboptimal perturbations have opposite (varicose) symmetry. When forced with finite amplitudes these perturbations modify the characteristics of the Kelvin–Helmholtz instability. Maximum temporal growth rates are reduced by optimal sinuous perturbations and are slightly increased by varicose suboptimal ones. In contrast, the onset of absolute instability is delayed by varicose suboptimal perturbations and is slightly promoted by sinuous optimal ones. We show that if, instead of the computed fully nonlinear basic-flow distortions, the stability analysis is based on a shape assumption for the flow distortions, then opposite effects on the flow stability are predicted in most of the considered cases. These strong differences are attributed to the spanwise-uniform component of the nonlinear basic-flow distortion which, we conclude, should be systematically included in sensitivity analyses of the stability of two-dimensional basic flows to three-dimensional basic-flow perturbations. We finally show that the leading-order quadratic sensitivity of the eigenvalues to the amplitude of the streaks is preserved if the effects of the mean flow distortion are included in the sensitivity analysis.

Inviscid spatial stability of a three-dimensional compressible mixing layer

1991 ◽  
Vol 231 ◽  
pp. 35-50 ◽  
Author(s):  
C. E. Grosch ◽  
T. L. Jackson
Keyword(s):  
Dimensional Stability ◽  
Stability Problem ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Compressible Mixing Layer ◽  
Spatial Stability ◽  
Family Of Curves ◽  
The Stability

We present the results of a study of the inviscid spatial stability of a parallel three-dimensional compressible mixing layer. The parameters of this study are the Mach number of the fast stream, the ratio of the speed of the slow stream to that of the fast stream, the ratio of the temperature of the slow stream to that of the fast stream, the direction of the crossflow in the fast stream, the frequency, and the direction of propagation of the disturbance wave. Stability characteristics of the flow as a function of these parameters are given. Certain theoretical results are presented which show the interrelations between these parameters and their effects on the stability characteristics. In particular, the three-dimensional stability problem for a three-dimensional mixing layer at Mach zero can be transformed to a two-dimensional stability problem for an equivalent two-dimensional mean flow. There exists a one-parameter family of curves such that for any given direction of mean flow and of wave propagation one can apply this transformation and obtain the growth rate from the universal curves. For supersonic couvective Mach numbers, certain combinations of crossflow angle and propagation angle of the disturbance can increase the growth rates by a factor of about two. and thus enhance mixing.

Mach number effects on the global mixing modes induced by ramp injectors in supersonic flows

2014 ◽  
Vol 757 ◽  
pp. 403-431 ◽  
Author(s):  
Luca Massa
Keyword(s):  
Mach Number ◽  
Free Stream ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Base Flow ◽  
Recirculation Region ◽  
Mean Flow ◽  
Mixing Rate ◽  
Flow Distortion ◽  
The Mean

AbstractModern injectors for supersonic combustors (hypermixers) augment the fuel–air mixing rate by energizing the perturbation in the mixing layer. From an instability point of view, the increased perturbation growth is linked to the increased complexity of the equilibrium base flow when compared to the axisymmetric mixing layer. Common added features are streamwise vortex streaks, oblique recompression shocks and Prandtl–Meyer expansions. One of the main effects of such distortions of the mean flow is to transform the instability responsible for the creation of fine scales from a local amplified mode to a global self-sustained fluctuation. The focus of the present research is on the flow distortion induced by flushed ramps for free-stream Mach numbers in the range 2.5–3.5. The principal mean flow features are the recirculation region due to the recompression of the flow after the ramp, the shear layer over the recirculation region and the vortex streaks propagating from the ramp corners. A global three-dimensional stability analysis and three-dimensional direct numerical simulations of small perturbations of the mean flow are performed. The growth and energy distribution of the dominant and subdominant fluctuations supported by the three-dimensional steady laminar base flow are computed. The main results are the growth rates of the self-sustained varicose and sinuous modes and their correlation to the variation in the free-stream Mach number. The complex three-dimensional wavemaker is investigated by evaluating the three-dimensional eigenfunctions of the direct and adjoint modes, and the effects of the axial vorticity generated by the ramp corners are discussed.

Experimental and Theoretical Investigation of the Supersonic Boundary Layer Stability on the Swept Wing

2008 ◽  
Vol 3 (3) ◽  
pp. 34-38
Author(s):  
Sergey A. Gaponov ◽  
Yuri G. Yermolaev ◽  
Aleksandr D. Kosinov ◽  
Nikolay V. Semionov ◽  
Boris V. Smorodsky
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Supersonic Boundary Layer ◽  
Mean Flow ◽  
Swept Wing ◽  
Wing Model ◽  
Dimensional Boundary ◽  
The Stability ◽  
Good Agreement

Theoretical and an experimental research results of the disturbances development in a swept wing boundary layer are presented at Mach number М = 2. In experiments development of natural and small amplitude controllable disturbances downstream was studied. Experiments were carried out on a swept wing model with a lenticular profile at a zero attack angle. The swept angle of a leading edge was 40°. Wave parameters of moving disturbances were determined. In frames of the linear theory and an approach of the local self-similar mean flow the stability of a compressible three-dimensional boundary layer is studied. Good agreement of the theory with experimental results for transversal scales of unstable vertices of the secondary flow was obtained. However the calculated amplification rates differ from measured values considerably. This disagreement is explained by the nonlinear processes observed in experiment

scholarly journals Inviscid spatial stability of a compressible mixing layer. Part 3. Effect of thermodynamics

1991 ◽  
Vol 224 ◽  
pp. 159-175 ◽  
Author(s):  
T. L. Jackson ◽  
C. E. Grosch
Keyword(s):  
Prandtl Number ◽  
Mixing Layer ◽  
Mean Flow ◽  
Temperature Relation ◽  
Compressible Mixing Layer ◽  
Spatial Stability ◽  
Mean Temperature ◽  
The Mean ◽  
The Difference ◽  
The Stability

We report the results of a comprehensive comparative study of the inviscid spatial stability of a parallel compressible mixing layer using various models for the mean flow. The models are (i) the hyperbolic tangent profile for the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity–temperature relation and a Prandtl number of one; (ii) the Lock profile for the mean speed and the Crocco relation for the mean temperature, with the Chapman viscosity-temperature relation and a Prandtl number of one; and (iii) the similarity solution for the coupled velocity and temperature equations using the Sutherland viscosity–temperature relation and arbitrary but constant Prandtl number. The purpose of this study was to determine the sensitivity of the stability characteristics of the compressible mixing layer to the assumed thermodynamic properties of the fluid. It is shown that the qualitative features of the stability characteristics are quite similar for all models but that there are quantitative differences resulting from the difference in the thermodynamic models. In particular, we show that the stability characteristics are sensitive to the value of the Prandtl number and to a particular value of the temperature ratio across the mixing layer.

Turret Mooring Design Based on Analytical Expressions of Catastrophes of Slow-Motion Dynamics

1998 ◽  
Vol 120 (3) ◽  
pp. 154-164 ◽  
Author(s):  
M. M. Bernitsas ◽  
L. O. Garza-Rios
Keyword(s):  
Parametric Design ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Slow Motion ◽  
Sensitivity Analyses ◽  
Design Parameters ◽  
Mooring Lines ◽  
The Stability ◽  
Fifth Order ◽  
Analytical Expressions

Analytical expressions of the bifurcation boundaries exhibited by turret mooring systems (TMS), and expressions that define the morphogeneses occurring across boundaries are developed. These expressions provide the necessary means for evaluating the stability of a TMS around an equilibrium position, and constructing catastrophe sets in two or three-dimensional parametric design spaces. Sensitivity analyses of the bifurcation boundaries define the effect of any parameter or group of parameters on the dynamical behavior of the system. These expressions allow the designer to select appropriate values for TMS design parameters without resorting to trial and error. A four-line TMS is used to demonstrate this design methodology. The mathematical model consists of the nonlinear, fifth-order, low-speed, large-drift maneuvering equations. Mooring lines are modeled with submerged catenaries, and include nonlinear drag. External excitation consists of time-independent current, wind, and mean wave drift.

scholarly journals A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

Effect of an Azimuthal Mean Flow On the Structure and Stability of Thermoacoustic Modes in an Annular Combustor Model with Electroacoustic Feedback

2020 ◽  
Author(s):  
Sylvain C. Humbert ◽  
Jonas Moeck ◽  
Alessandro Orchini ◽  
Christian Oliver Paschereit
Keyword(s):  
Mach Number ◽  
Initial Conditions ◽  
Mean Flow ◽  
Final State ◽  
Mean Velocity ◽  
Acoustic Damping ◽  
Annular Combustor ◽  
The Mean ◽  
High Mach ◽  
The Stability

Abstract Thermoacoustic oscillations in axisymmetric annular combustors are generally coupled by degenerate azimuthal modes, which can be of standing or spinning nature. Symmetry breaking due to the presence of a mean azimuthal flow splits the degenerate thermoacoustic eigenvalues, resulting in pairs of counter-spinning modes with close but distinct frequencies and growth rates. In this study, experiments have been performed using an annular system where the thermoacoustic feedback due to the flames is mimicked by twelve identical electroacoustic feedback loops. The mean azimuthal flow is generated by fans. We investigate the standing/spinning nature of the oscillations as a function of the Mach number for two types of initial states, and how the stability of the system is affected by the mean azimuthal flow. It is found that spinning, standing or mixed modes can be encountered at very low Mach number, but increasing the mean velocity promotes one spinning direction. At sufficiently high Mach number, spinning modes are observed in the limit cycle oscillations. In some cases, the initial conditions have a significant impact on the final state of the system. It is found that the presence of a mean azimuthal flow increases the acoustic damping. This has a beneficial effect on stability: it often reduces the amplitude of the self-sustained oscillations, and can even suppress them in some cases. However, we observe that the suppression of a mode due to the mean flow may destabilize another one. We discuss our findings in relation with an existing low-order model.

The development of three-dimensional wave-packets in unbounded parallel flows

1968 ◽  
Vol 32 (4) ◽  
pp. 801-808 ◽  
Author(s):  
M. Gaster ◽  
A. Davey
Keyword(s):  
Wave Packet ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Mean Flow ◽  
Physical Plane ◽  
Parallel Flows ◽  
Flow Velocity Profile ◽  
The Stability ◽  
Special Case ◽  
Dimensional Wave

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

Secondary instability of a temporally growing mixing layer

1987 ◽  
Vol 184 ◽  
pp. 207-243 ◽  
Author(s):  
Ralph W. Metcalfe ◽  
Steven A. Orszag ◽  
Marc E. Brachet ◽  
Suresh Menon ◽  
James J. Riley
Keyword(s):  
Growth Rates ◽  
Stokes Equations ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Mixing Layers ◽  
Navier Stokes ◽  
Small Scale ◽  
Two Dimensional ◽  
Mean Flow ◽  
Direct Numerical Integration

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.

Three-dimensional destabilization of Stuart vortices: the influence of rotation and ellipticity

1999 ◽  
Vol 387 ◽  
pp. 205-226 ◽  
Author(s):  
P. G. POTYLITSIN ◽  
W. R. PELTIER
Keyword(s):  
Cross Sections ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Steady State Solution ◽  
Vorticity Distribution ◽  
Columnar Vortex ◽  
Elliptical Instability ◽  
The Stability ◽  
Dimensional Instability ◽  
Stuart Vortices

We investigate the influence of the ellipticity of a columnar vortex in a rotating environment on its linear stability to three-dimensional perturbations. As a model of the basic-state vorticity distribution, we employ the Stuart steady-state solution of the Euler equations. In the presence of background rotation, an anticyclonic vortex column is shown to be strongly destabilized to three-dimensional perturbations when background rotation is weak, while rapid rotation strongly stabilizes both anticyclonic and cyclonic columns, as might be expected on the basis of the Taylor–Proudman theorem. We demonstrate that there exist three distinct forms of three-dimensional instability to which strong anticyclonic vortices are subject. One form consists of a Coriolis force modified form of the ‘elliptical’ instability, which is dominant for vortex columns whose cross-sections are strongly elliptical. This mode was recently discussed by Potylitsin & Peltier (1998) and Leblanc & Cambon (1998). The second form of instability may be understood to constitute a three-dimensional inertial (centrifugal) mode, which becomes the dominant mechanism of instability as the ellipticity of the vortex column decreases. Also evident in the Stuart model of the vorticity distribution is a third ‘hyperbolic’ mode of instability that is focused on the stagnation point that exists between adjacent vortex cores. Although this short-wavelength cross-stream mode is much less important in the spectrum of the Stuart model than it is in the case of a true homogeneous mixing layer, it nevertheless does exist even though its presence has remained undetected in most previous analyses of the stability of the Stuart solution.

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