Nonlinear evolution of secondary instabilities in boundary-layer transition

Methods of nonlinear stability theory are applied to analyze the evolution of disturbances in the three-dimensional stage immediately preceding the breakdown of a laminar boundary layer. A perturbation scheme is used to solve the nonlinear equations and to develop a dynamical model for the interaction of primary and secondary instabilities. The first step solves for the two-dimensional primary wave in the absence of secondary disturbances. Once this finite-amplitude wave is calculated, it is decomposed into a basic-flow component and an interaction component. The basic-flow component acts as a parametric excitation for the three-dimensional secondary wave, while the interaction component captures the resonance between the secondary and primary waves. Results are presented in two principal forms: amplitude growth curves and velocity profiles. Our results agree with experimental data and the few available results of transition simulations and, moreover, reveal the origin of the observed phenomena. The method described establishes the basis for physical transition criteria in a given disturbance environment.This work has been supported by the Air Force Office of Scientific Research under Contract F46920-87-K-0005 and Grant AFOSR-88-0186 (TH) and by an ONT Postdoctoral Fellowship (JDC).     

The effects of resonance interactions between wave perturbations in a boundary layer

The transition to turbulence in a boundary layer can be induced by perturbations of low intensity and is accompanied by a growth in their energy, the development of three-dimensional structures, and a change in the spectral composition of the field. A number of important properties of the process admit interpretation in the framework of nonlinear stability theory and can be due to a resonance interaction. Experiments [1, 2] have revealed a transition accompanied by an appreciable enhancement of pulsations whose period is twice that of the driving vibrating tape. Theoretical investigations [3–9] have revealed the existence of a resonance mechanism capable of strong excitation of three-dimensional Tollmien-Schlichting waves at the frequency of a subharmonic. It has been suggested [4] that the observed transition regime is the result of evolution of triplets of resonantly coupled oscillations forming symmetric triplets [10]. In contrast to the type of transition considered by Craik et al. [10, 11], the leading role is played by subharmonics distinguished parametrically in the background. Experimental confirmations have been obtained [12, 13] of the coupling of the resonances in symmetric triplets with the subharmonic regime. Further investigation of the resonance mechanism is an important topical problem. This paper presents a study on the formation and special characteristics of the initial stage in the nonlinear development of triplets; the collective interaction of a two-dimensional Tollmien-Schlichting wave with a packet of three-dimensional waves is examined; the behavior of the system is analyzed, taking into account the resonance coupling with the harmonic of the main wave. A comparison is made between Craik's model and experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 23–30, July–August, 1984.The auothors wish too express their gratitude to A. G. Volodin for useful discussions and V. Ya, Levchenko for his interest in the work.     

Evolution of resonance disturbances in a hartmann flow

A rapid increase of energy of fluctuation motion is observed after a severe loss of stability of laminar regimes. This phenomenon does not find explanation in the scope of the linear theory of stability, which, though it predicts an exponential increase of disturbances in the supercritical region, gives quite small values of the increments. The explosionlike turbulence is due to a nonlinear mechanism. The simplest collective interaction of disturbances is illustrated by a set of three harmonic oscillations whose parameters are associated by resonance relations. Such triplets, being an elementary but sufficiently meaningful model of the nonlinear theory of hydrodynamic stability, have become in recent years the object of interesting investigations [1–4]. In [5–7] branching of stationary triplets of small amplitude from laminar regimes was investigated and it was shown that, beginning with certain Reynolds numbers, the triplet can be composed of neutral waves and Tolman-Schlichting waves increasing according to the linear theory. It is shown in the article that a quite rich example in this case is Hartmann flow, where the existence of triplets of disturbances having a different symmetry relative to the axis of the channel is admitted. The evolution of triplets is studied for near-critical values of the parameters in the framework of amplitude equations obtained on the basis of the Galerkin method with the use of eigenfunctions of the linear theory of stability as the basis [8]. Regimes stationary in the mean are calculated in the supercritical region: limiting cycles and strange attractors; in the latter case a spectral analysis is carried out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 33–39, September–October, 1978.The authors thank M. A. Gol'dshtik and M. I. Rabinovich for discussing the work.     

Occurrence of Tolman-Schlichting waves in the boundary layer under the effect of external perturbations

Under small external perturbations, the initial stage of the laminar into turbulent flow transition process in boundary layers is the development of natural oscillations, Tolman-Schlichting waves, which are described by the linear theory of hydrodynamic stability. Subsequent nonlinear processes start to appear in a sufficiently narrow band of relative values of the perturbation amplitudes (1–2% of the external flow velocity) and progress quite stormily. Hence, the initial linear stage of relatively slow development of perturbations is governing, in a known sense, in the complete transition process. In particular, the location of the transition point depends, to a large extent, on the spectrum composition and intensity of the perturbations in the boundary layer, which start to develop according to linear theory laws, resulting in the long run in destruction of the laminar flow mode. In its turn, the initial intensity and spectrum composition of the Tolman-Schlichting waves evidently depend on the corresponding characteristics of the different external perturbations generating these waves. The significant discrepancy in the data of different authors on the transition Reynolds number in the boundary layer on a flat plate [1–4] is probably explained by the difference in the composition of the small perturbing factors (which have not, unfortunately, been fully checked out by far). Moreover, it is impossible to expect that all kinds of external perturbations will be transformed identically into the natural boundary-layer oscillations. The relative role of external perturbations of different nature is apparently not identical in the Tolman-Schlichting wave generation process. However, how the boundary layer reacts to small external perturbations, under what conditions and in what way do external perturbations excite Tolman-Schlichting waves in the boundary layer have practically not been investigated. The importance of these questions in the solution of the problem of the passage to turbulence and in practical applications has been emphasized repeatedly recently [5, 6], Only the first steps towards their solution have been taken at this time [4, 7–10], Out of all the small perturbing factors under the real conditions of the majority of experiments to investigate the flow stability and transition in the case of smooth polished walls, three are apparently most essential, viz.: the turbulence of the external flow, acoustic perturbations, and model vibrations. In principle, all possible mechanisms for converting the energy of these perturbations into Tolman-Schlichting waves can be subdivided into two classes (excluding the nonlinear interactions which are not examined here): 1) distributed wave generation in the boundary layer; and 2) localized wave generation at the leading edge of the streamlined model. Among the first class is both the possibility of the direct transformation of the external flow perturbations into Tolman-Schlichting waves through the boundary-layer boundary, and wave excitation because of the active vibrations of the model wall. Among the second class are all possible mechanisms for the conversion of acoustic or vortical perturbations, as well as the vibrations of the streamlined surface, into Tolman-Schlichting waves, which occurs in the area of the model leading edge.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–94, September–October, 1978.     

Development of three-dimensional disturbances in a boundary layer

In the region of transition from a two-dimensional laminar boundary layer to a turbulent one, three-dimensional flow occurs [1–3]. It has been proposed that this flow is formed as the result of nonlinear interaction of two-dimensional and three-dimensional disturbances predicted by linear hydrodynamic stability theory. Using many simplifications, [4, 5] performed a calculation of this interaction for a free boundary layer and a boundary layer on a wall with a very coarse approximation of the velocity profile. The results showed some argreement with experiment. On the other hand, it is known that disturbances of the Tollmin—Schlichting wave type can be observed at sufficiently high amplitude. This present study will use the method of successive linearization to calculate the primary two- and three-dimensional disturbances, and also the average secondary flow occurring because of nonlinear interaction of the primary disturbances. The method of calculation used is close to that of [4, 5], the disturbance parameters being calculated on the basis of a Blazius velocity profile. A detailed comparison of results with experimental data [1] is made. It developed that at large disturbance amplitude the amplitude growth rate differs from that of linear theory, while the spatial distribution of disturbances agree s well with the distribution given by the natural functions and their nonlinear interaction. In calculating the secondary flow an experimental correction was made to the amplitude growth rate.     

Nonlinear Group Interactions of the Taylor–Görtler Disturbances in Supersonic Axisymmetric Jets

The nonlinear group interaction of the Taylor-Görtler disturbances (streamwise vortices) at the initial section of a supersonic axisymmetric jet is numerically studied within the framework of the weakly nonlinear theory of stability. The experimentally observed spectrum of disturbances is considered. The regular and specific features of the streamwise dynamics of various wave components for a turbulent jet are studied. It is shown that such an interaction in the coupled mode in resonant group triplets allows one to describe the experimentally observed elevated growth of background components of the real spectrum.     

Nonlinear Mechanisms of the Initial Stage of the Laminar–Turbulent Transition at Hypersonic Velocities

Weakly nonlinear development of waves in an axisymmetric hypersonic boundary layer is studied by the method of bispectral analysis. The type of nonlinear interaction that was not observed previously in such flows is found. The possibility of subharmonic resonance of the second mode at the nonlinear stage of transition is demonstrated. The previously discovered nonlinear generation of the harmonic of the fundamental wave of the second mode of disturbances is observed.     

Spatial structure of two-dimensional wave disturbances in a boundary layer

In [1] on the basis of a numerical integration of the Navier-Stokes equations the authors investigated the nonlinear evolution of two-dimensional disturbances of the traveling wave type in the boundary layer on a flat plate. The process of interaction of two waves with different wave numbers and initial amplitudes was examined. In this article the study of these interactions is continued. Special attention is paid to the spatial structure of the disturbances with respect to the cross-flow coordinate (with respect to the longitudinal coordinate the disturbances are assumed to be periodic) at various moments of time. It is shown that if the initial amplitude of one of the waves is sufficiently large, i.e., exceeds a certain threshold value, an undamped quasisteady regime is established during the interaction process. At lower amplitudes the process degenerates and the waves develop independently. In these two cases the evolution of the spatial distribution of the perturbation amplitudes is qualitatively different. In the first case the shape of the amplitude distribution varies only slightly with time, while in the second it depends importantly on the parameters of the wave numbers and the Reynolds number. When the parameters are such that one of the finite-amplitude waves is damped, its amplitude distribution rapidly evolves into the form characteristic of disturbances of the continuous spectrum of the linear stability problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–24, September–October, 1990.     

Resonance properties of two-dimensional wave disturbances in a boundary layer

The nonlinear development of disturbances of the traveling wave type in the boundary layer on a flat plate is examined. The investigation is restricted to two-dimensional disturbances periodic with respect to the longitudinal space coordinate and evolving in time. Attention is concentrated on the interactions of two waves of finite amplitude with multiple wave numbers. The problem is solved by numerically integrating the Navier-Stokes equations for an incompressible fluid. The pseudospectral method used in the calculations is an extension to the multidimensional case of a method previously developed by the authors [1, 2] in connection with the study of nonlinear wave processes in one-dimensional systems. Its use makes it possible to obtain reliable results even at very large amplitudes of the velocity perturbations (up to 20% of the free-stream velocity). The time dependence of the amplitudes of the disturbances and their phase velocities is determined. It is shown that for a fairly large amplitude of the harmonic and a particular choice of wave number and Reynolds number the interacting waves are synchronized. In this case the amplitude of the subharmonic grows strongly and quickly reaches a value comparable with that for the harmonic. As distinct from the resonance effects reported in [3, 4], which are typical only of the three-dimensional problem, the effect described is essentially two-dimensional.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 37–44, March–April, 1990.     

Instability due to three-dimensional disturbances

In the linear theory of the stability of parallel flows of a viscous fluid, most attention is usually given to plane-wave disturbances. The reason is the validity in many cases of the Squire theorem, which states that the critical Reynolds number R is determined by two-dimensional disturbances [1]. It is shown in the present paper that for large R the region generating the turbulence in the initial stage of its development is formed by three-dimensional disturbances. This feature applies both to the generating range of wave numbers and the dimension of the wall layer, where the fluctuating energy is produced. The consequences of the Squire transformations for parallel flows are analyzed. The contribution of resonant nonlinear triad coupling to the rapid growth of fluctuating energy is studied for the case of an explosive instability in an extended laminar mode. It is shown that the rate of turbulent energy production is not governed by the small derivatives of linear theory, but by nonlinear triad coupling of neutral and growing disturbances, with their three-dimensional nature playing an important role.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 29–34, September–October, 1976.The author thanks M. A. Gol'dshtik for his interest in the work and for discussion of the results.     

Resonant interaction of wave packets in a boundary layer

The space-time evolution of resonance-coupled triads of wave packets in a Blasius boundary layer is studied within the framework of weakly nonlinear stability theory. The amplitude behavior of the packet envelopes is determined in relation to their initial shape, the carrier frequency and the region of propagation. As in the case of triads with a discrete spectrum, interaction leads to parametric pumping of the low-frequency fluctuations and explosive nonlinear growth of the packet maxima. The space-time evolution characteristics are expressed in the deformation of the shape and the spectra of the disturbance. Parts of the envelopes are amplified, depending on the local values of the parameters. This leads to sharp discrimination of the peaks and the equalization of their propagation velocities. These effects make it possible to explain the broadening of the spectrum, the stable distribution of the visualization pattern, and the appearance of irregularities in the oscillograms observed in the S transition. In order to analyze the nonlinear evolution of a disturbance initiated by an instantaneous point source, the interaction of a two-dimensional wave train with variable carrier frequency and pairs of three-dimensional low-frequency packets is examined. (The train frequency corresponds to the local maximum of the linear growth rate with respect to R.) The possibility of the progressive parametric excitation of fluctuations over the entire band of frequency parameters is established. This may explain the acceleration of the transition process in the presence of an impulsive disturbance of the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 67–71, November–December, 1988.The authors are grateful to I. I. Maslennikov for useful discussions.     

Coupled magnetoacoustic waves in a conducting paramagnetic fluid

We consider the propagation of small disturbances in a paramagnetic conducting fluid in a uniform constant magnetic field. Because of coupling of the mechanical and magnetic effects, coupled magnetoacoustic oscillations of a wave nature develop in a certain (resonant) frequency region. The usual MHD waves and uniform magnetization oscillations occur far from resonance. Dissipative processes are accounted for.The equations of motion for a conducting paramagnetic fluid in which interaction of the hydrodynamic velocity with the magnetization and the magnetic field was taken into account phenomenologically were obtained in [1], One of the consequences of this interaction is the propagation of coupled magnetoelastic waves in the fluid; this phenomenon is examined in the present paper.     

Three-Wave Interactions of Disturbances in a Supersonic Boundary Layer

Within the framework of the weakly nonlinear stability theory, group interaction of disturbances in a supersonic boundary layer is considered. The disturbances are represented by two spatial packets of traveling instability waves (wave trains) with multiple frequencies. The possibility of energy redistribution in such wave systems in the case of three-wave resonant interactions of packet constituents is considered. The model is used to test the dynamics of unstable waves arising due to introduction of controlled high-intensity disturbances into a supersonic boundary layer. It is found that this mechanism is not the main one for the features of streamwise dynamics of such nonlinear waves being observed.     

Relaxation of overdriven detonation waves with finite reaction rate

A numerical study is made of the interaction of a detonation wave having finite reaction velocity with a rarefaction wave of different intensity which approaches it from the rear, for the Zeldovich-Neumann-Doring (ZND) model with a single irreversible reaction A B. It is found that, for a fixed value of the parameter characterizing the initial supercompression (depending on the activation energy and the heating value of the mixture), the considered interaction leads either to a gradual relaxation of the detonation wave and its transition to the Chapman-Jouguet (CJ) regime, or to the development of undamped oscillations.Interest in the problems of detonation and supersonic combustion has increased in recent years. This is associated with the appearance and development of new experimental and theoretical techniques; it is also associated with the further development of air-breathing reaction engines, and other practical requirements. The present state of detonation theory is reflected in the survey [1].It has been established [2] that the detonation wave in gases nearly always has a complex nonuniform structure. Transverse disturbances are observed under a wide range of conditions and differ both in amplitude and wavelength. At the same time, behind the detonation leading front there is a region of uncompletely burned gas corresponding to the effective ignition induction period [3]. In spinning detonation the induction period is significantly longer than the heat release period and transverse detonation waves traveling in the induction zone of the head wave appear [3, 4]. Such a secondary detonation wave is free of transverse disturbances. The same is true of the detonation waves observed in the wake behind a body moving at high speed in a combustible medium [5] or in a gas which has been preheated by a shock wave [6].Although it is possible, under favorable conditions, to study in detail the system of discontinuities accompanying detonation, information on the extensive zones in which heat release takes place is scarce, the mechanism of detonation wave autonomy (in particular, the role of the rarefaction zone behind the wave) is not entirely clear, and the fact that, in spite of the complex structure, an autonomous detonation propagates with the CJ velocity calculated on the basis of one-dimensional theory has not yet been explained.In studying the nonlinear phenomena associated with the finite reaction rate it is quite acceptable to investigate only the simple one-dimensional detonation model, with which it is convenient to restrict ourselves to a single effective chemical reaction. This model is particularly reasonable since, in certain cases, the real detonation is virtually one-dimensional.The question of the stability of the one-dimensional detonation wave to disturbances of its structure has been examined by several authors [7–13]. The use of computers makes possible the direct computation of flows with heat release and the study of their properties. This method has been used in [11–13] to study the stability problem for a detonation wave with respect to finite disturbances.In the present paper we present a numerical study of the interaction of a detonation wave having finite chemical reaction rate with a rarefaction wave of different intensity approaching it from the rear for the ZND model with a single irreversible reaction A B. It is found that for a fixed value of the parameter characterizing the difference between detonation and the CJ waves, depending on the activation energy E and the mixture heating value Q_m, the interaction in question leads either to a gradual relaxation of the detonation wave and its transition to the CJ regime (this relaxation may be accompanied by decaying oscillations) or to the appearance of undamped oscillations (the unstable regime). The parameters E and Q_m affect the wave stability differently: with increase of Q_m, the wave is stabilized; with increase of E, it is destabilized. The boundary between the stable and unstable detonation wave propagation regimes is found. This boundary has a weak dependence on the rarefaction wave intensity. Estimates and calculated examples show that the amplitude of the unstable wave oscillations is finite and that the average detonation propagation velocity is close to the CJ velocity computed for the given heating value Qm.The author wishes to thank G. G. Chernyi for his guidance and L. A. Chudov for advice on computational questions.     

On nonlinear evolution equations of a three-dimensional capillary-gravity wave packet at the interface of two superposed fluids

Assuming amplitudes as slowly varying functions of space and time and using a perturbation method, two coupled nonlinear partial differential equations are derived that give the nonlinear evolution of the amplitude of a three-dimensional capillary-gravity wave packet at the interface of two superposed incompressible fluid layers of finite depths, including the effect of its interaction with a long gravity wave. Starting from these two coupled equations, a balanced set of modulation equations, both at nonresonance and at resonance, is derived. The balanced set of modulation equations, at nonresonance, reduces to a single nonlinear Schrödinger equation, if it is assumed that space variation of the amplitudes depends only on variation along an arbitrary fixed horizontal direction. Modulational instability conditions, both at resonance and at nonresonance, are also deduced. The advantage of the perturbation method adopted in the present problem, over the reductive perturbation method, is noticed.     

Development of finite-amplitude two-dimensional and three-dimensional disturbances in jet flows

The laminar-turbulent transition zone is investigated for a broad class of jet flows. The problem is considered in terms of the inviscid model. The solution of the initial-boundary value problem for three-dimensional unsteady Euler equations is found by the Bubnov-Galerkin method using the generalized Rayleigh approach [1–4]. The occurrence, subsequent nonlinear evolution and interaction of two-dimensional wave disturbances are studied, together with their secondary instability with respect to three-dimensional disturbances.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 8–19, September–October, 1985.     

Nonstationary one-dimensional gas flows in channels in the presence of periodic perturbations of the boundary conditions

A study is made of the propagation in channels of forced oscillations generated by harmonic variation of the boundary conditions at the entrance and exit sections. Linear theory is used to find classes of boundary conditions and frequencies of the forced oscillations corresponding to the greatest gain or attenuation of high-frequency oscillations in a channel of variable section and of oscillations of arbitrary frequency in a channel of constant section. The resonance phenomenon that arises in channels when the frequencies of the forced oscillations and the fundamental oscillations are equal is studied. The wave process in a channel of variable section is investigated numerically, its characteristics found, and a comparison made with the linear theory. It is shown that the results of the calculations and the data of the linear analysis agree well.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–135, September–October, 1980.We thank A. B. Vatazhin for helpful discussions.     

Wave generation on an interface by vortex disturbances in a shear flow

A linear problem of oscillations of an interface in a two-layer system, in which the upper layer is at rest and the lower layer has a constant velocity shear, is considered. The dynamic perturbations in the lower layer are represented as the sum of vortex and wave disturbances (disturbances with zero vorticity). It is shown that in the shear flow the evolution of the vortex disturbances with a nonsmooth or a singular initial vorticity distribution can result in the resonant excitation of waves on the interface. The occurrence of the resonance corresponds to the coincidence of the oscillation frequencies of the perturbations of both classes. In the absence of hydrodynamic instability of the shear flow, the resonant excitation can be one of the main mechanisms of wave generation in two-layer systems.     

Development of three-dimensional disturbances in a boundary layer with pressure gradient

The results of an experimental investigation of the three-dimensional stability of a boundary layer with a pressure gradient are presented. A low-turbulence subsonic wind tunnel was employed. The development of a three-dimensional wave packet of oscillations harmonic in time in the boundary layer on a model wing is studied. The amplitudephase distributions of the pulsations in the wave packet are subjected to a Fourier analysis. Spectral (with respect to the wave numbers) decomposition of the oscillations enables the flow stability with respect to plane waves with different directions of propagation to be examined. The results are compared with the corresponding data obtained in flat plate experiments. The effect of the pressure gradient on the development of the three-dimensional spectral components of the disturbances and the dispersion properties of the flow is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 85–91, May–June, 1988.     

Secondary instability modes generated by a Tollmien-Schlichting wave scattering from a bump

The plane finite-amplitude Tollmien-Schlichting wave interaction with a three-dimensional bump on a wall is considered for plane channel flow. The scattering of this wave leads to the production of unsteady three-dimensional disturbances which transform into growing secondary instability modes. The generation of such modes is studied assuming the three-dimensional disturbances to be small in comparison with the primary plane instability wave. The solution predicts that secondary disturbance amplification takes place only within a narrow wedge downstream of the bump. The qualitative comparison of results with experiments on turbulent wedge origination at an isolated roughness in a boundary layer is presented.