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.

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.

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.

Rotating free-shear flows. Part 2. Numerical simulations

1995 ◽  
Vol 293 ◽  
pp. 47-80 ◽  
Author(s):  
Olivier Métais ◽  
Carlos Flores ◽  
Shinichiro Yanase ◽  
James J. Riley ◽  
Marcel Lesieur
Keyword(s):  
Solid Body ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Rossby Number ◽  
Two Dimensional ◽  
Body Rotation ◽  
Mean Flow ◽  
Hairpin Vortices ◽  
Solid Body Rotation ◽  
Rotation Rates

The three-dimensional dynamics of the coherent vortices in periodic planar mixing layers and in wakes subjected to solid-body rotation of axis parallel to the basic vorticity are investigated through direct (DNS) and large-eddy simulations (LES). Initially, the flow is forced by a weak random perturbation superposed on the basic shear, the perturbation being either quasi-two-dimensional (forced transition) or three-dimensional (natural transition). For an initial Rossby number Ro(i), based on the vorticity at the inflexion point, of small modulus, the effect of rotation is to always make the flow more two-dimensional, whatever the sense of rotation (cyclonic or anticyclonic). This is in agreement with the Taylor–Proudman theorem. In this case, the longitudinal vortices found in forced transition without rotation are suppressed.It is shown that, in a cyclonic mixing layer, rotation inhibits the growth of three-dimensional perturbations, whatever the value of the Rossby number. This inhibition exists also in the anticyclonic case for |Ro(i)| ≤ 1. At moderate anticyclonic rotation rates (Ro(i) < −1), the flow is strongly destabilized. Maximum destabilization is achieved for |Ro(i) ≈ 2.5, in good agreement with the linear-stability analysis performed by Yanase et al. (1993). The layer is then composed of strong longitudinal alternate absolute vortex tubes which are stretched by the flow and slightly inclined with respect to the streamwise direction. The vorticity thus generated is larger than in the nonrotating case. The Kelvin–Helmholtz vortices have been suppressed. The background velocity profile exhibits a long range of nearly constant shear whose vorticity exactly compensates the solid-body rotation vorticity. This is in agreement with the phenomenological theory proposed by Lesieur, Yanase & Métais (1991). As expected, the stretching is more efficient in the LES than in the DNS.A rotating wake has one side cyclonic and the other anticyclonic. For |Ro(i)| ≤ 1, the effect of rotation is to make the wake more two-dimensional. At moderate rotation rates (|Ro(i)| > 1), the cyclonic side is composed of Kármán vortices without longitudinal hairpin vortices. Karman vortices have disappeared from the anticyclonic side, which behaves like the mixing layer, with intense longitudinal absolute hairpin vortices. Thus, a moderate rotation has produced a dramatic symmetry breaking in the wake topology. Maximum destabilization is still observed for |Ro(i)| ≈ 2.5, as in the linear theory.The paper also analyses the effect of rotation on the energy transfers between the mean flow and the two-dimensional and three-dimensional components of the field.

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 compressible mixing layer

1989 ◽  
Vol 208 ◽  
pp. 609-637 ◽  
Author(s):  
T. L. Jackson ◽  
C. E. Grosch
Keyword(s):  
Mach Number ◽  
Growth Rates ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Value ◽  
Propagation Direction ◽  
General Behaviour ◽  
Compressible Mixing Layer ◽  
Spatial Stability ◽  
Wave Stability

We present the results of the inviscid spatial stability of a parallel compressible mixing layer. The parameters of this study are the Mach number of the moving stream, the ratio of the temperature of the stationary stream to that of the moving 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. It is shown that if the Mach number exceeds a critical value there are always two groups of unstable waves. One of these groups is fast with phase speeds greater than ½ and is supersonic with respect to the stationary stream. The other is slow with phase speeds less than ½ and supersonic with respect to the moving stream. Phase speeds for the neutral and unstable modes are given, as well as growth rates for the unstable modes. Finally, we show that three-dimensional modes have the same general behaviour as the two-dimensional modes but with higher growth rates over some range of propagation direction.

Three-dimensional simulations of large eddies in the compressible mixing layer

1991 ◽  
Vol 224 ◽  
pp. 133-158 ◽  
Author(s):  
N. D. Sandham ◽  
W. C. Reynolds
Keyword(s):  
Mach Number ◽  
Large Scale ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Large Scale Structure ◽  
Two Dimensional ◽  
Instability Waves ◽  
Compressible Mixing Layer ◽  
Mach Numbers ◽  
High Mach

The effect of Mach number on the evolution of instabilities in the compressible mixing layer is investigated. The full time-dependent compressible Navier–Stokes equations are solved numerically for a temporally evolving mixing layer using a mixed spectral and high-order finite difference method. The convective Mach number Mc (the ratio of the velocity difference to the sum of the free-stream sound speeds) is used as the compressibility parameter. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers (Mc > 0.6) oblique waves grow more rapidly than two-dimensional waves. Simulations are then presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves. A change in the developed large-scale structure is observed as the Mach number is increased, with vortical regions oriented in a more oblique manner at the higher Mach numbers. At convective Mach numbers above unity the two-dimensional instability is found to have little effect on the flow development, which is dominated by the oblique instability waves. The nonlinear structure which develops from a pair of equal and opposite oblique instability waves is found to resemble a pair of inclined A-vortices which are staggered in the streamwise direction. A fully nonlinear computation with a random initial condition shows the development of large-scale structure similar to the simulations with forcing. It is concluded that there are strong compressibility effects on the structure of the mixing layer and that highly three-dimensional structures develop from the primary inflexional instability of the flow at high Mach numbers.

Three-dimensional disturbances in flow between parallel planes

1960 ◽  
Vol 254 (1279) ◽  
pp. 562-569 ◽  
Keyword(s):  
Reynolds Number ◽  
Dimensional Stability ◽  
Flow Direction ◽  
Simple Procedure ◽  
Three Dimensional ◽  
Stability Diagram ◽  
Wave Number ◽  
Two Dimensional ◽  
General Answer ◽  
The Stability

The close connexion between the stability of three-dimensional and two-dimensional disturbances in flow between parallel walls has been examined and this has led to the formation of a three-dimensional stability diagram where ‘stability surfaces’ replace stability curves. The problem which has been investigated is whether the most highly amplifying disturbance at any given Reynolds number above the minimum critical Reynolds number is a two-dimensional or a three-dimensional disturbance. It has been shown that the most unstable disturbance is a two-dimensional one for a certain definite range of Reynolds number above the critical. For Reynolds numbers greater than this no definite general answer has been found; each basic undisturbed flow must be treated separately and a simple procedure has been given which, in principle, determines the type of disturbance which is most unstable. Difficulty arises in following this procedure because it requires knowledge of the two-dimensional stability curves in a certain region where this knowledge is very scanty at the moment. Althoughth is difficulty arises, in Poiseuille flow the calculations available indicate very strongly that the most unstable disturbance at any given Reynolds number above the critical is two-dimensional. Further, it is believed that this result holds for all other basic flows. A second result is that if the wave number (a) in the flow direction is specified, as well as the Reynolds number, then for a in a certain range, the most unstable disturbance is three-dimensional.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

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

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