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.     

Stability of a spanwise nonuniform boundary layer flow

The linear stability of a boundary layer flow with a spanwise-periodic nonuniformity in the velocity profile is investigated. This flow can be considered as a model of a streaky structure occurring in the boundary layer at a high freestream turbulence level. It is shown that for a small nonuniformity amplitude symmetric modes similar to Tollmien-Schlichting waves are the most unstable. At higher nonuniformity amplitudes, antisymmetric modes, qualitatively different from Tollmien-Schlichting waves and having a larger phase velocity, are the most amplified. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 54–63, November–December, 1998. The study was carried out with the support of the International Scientific and Technical Center (project No. 199-95) and the Russian Foundation for Basic Research (project No. 95-01-01201a).     

Nonlinear equation for amplitude of Tollmien-Schlichting waves in a boundary layer

The nonlinear evolution of a Tollmien-Schlichting wave is analyzed with allowance for the flow being nonparallel in a boundary layer. In contrast to the early work of Zel'man [19], strict allowance is made for the fact that the extent to which the flow is nonparallel is not independent of the Reynolds number — the departure from parallel flow in a boundary layer is small only at large Reynolds numbers. Therefore, an asymptotic theory of Tollmien-Schlichting waves is constructed under the assumption that the Reynolds number tends to infinity.     

Propagation of wave packets in the boundary layer on a curved surface

The development of wave packets excited in a boundary layer by means of a local deformation of the surface in the longitudinal-transverse interaction regime is considered. A solution of the linearized system of equations of interaction theory is constructed using a Laplace transformation with respect to time and a Fourier transformation with respect to the space variables. Two problems are separately examined. In the first, the disturbances are induced by a surface deformation sinusoidal in the transverse direction. It is shown that the center of the wave packet with the greatest oscillation amplitude moves in a direction opposite to that of the flow in the boundary layer. At the same time the wave packet expands, so that in the course of time any fixed point will enter the region of growing oscillations. In the second problem the source of the disturbances is isolated. In this case the wave packet acquires a horseshoe shape. Expanding, it carries the disturbances away from the source in all directions, including upstream relative to the flow in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 59–68, March–April, 1990.     

Simplified system of equations for calculating the stability of the boundary layer gas flow on a swept wing

A composite system of equations for calculating the stability of a compressible flow in a three-dimensional boundary layer is developed. A comparison with the calculations in accordance with the complete system of equations shows that the system proposed almost accurately predicts the growth rates of both the Tollmien-Schlichting mode and the transverse flow instability mode over a wide range of frequencies and transverse wavenumbers at moderate Mach numbers. At the same time, the computation time reduces by an order.     

Wave packet formation in an incompressible boundary layer under the influence of impulsive vibration of the wall

The formation of a Tollmien-Schlichting wave packet excited by the impulsive motion of a localized area of the boundary surface is investigated. A solution of the mixed problem for the linearized system of Navier-Stokes equations is constructed by the Fourier method for a parallel main flow. The initial stage of the disturbed motion is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 56–60, May–June, 1989.     

Nonlinear interaction of a large vortex with a boundary layer

The nonlinear problem of boundary layer instability under the influence of a plane vortex is investigated for high Reynolds numbers. The vortex occupies the entire thickness of the boundary layer and has a longitudinal dimension of the order of the Tollmien-Schlichting wavelength. The initial vortex is rapidly swept away by the flow, inducing a Stokes layer near the surface of the plate. Expanding, this layer reaches the dimensions of the viscous sublayer of free interaction theory, where wave packet generation takes place. In the case in question a feature of the nonlinear stage of development of the disturbances is the formation of a concentrated vortex, which arises in the Stokes layer and grows rapidly, whereas the wave packet propagated ahead of it remains linear. From the calculations there emerges a tendency for the new vortex to be formed above the wail, whereas the maximum vorticity of the vortex generated in the Stokes layer corresponds to the wall itself.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 70–77, January–February, 1993.The authors are grateful to V. V. Kozlov for his interest in their work.     

A DIRECT NUMERICAL SIMULATION OF CHANNEL FLOW TRANSITION UNDER AN EXTERNAL BODY FORCE

This research seeks to increase our understanding on the laminar-turbulent transition under an external body force. Direct numerical simulation by the spectral method with a weak formulation is used to solve the transient 3-D Navier-Stokes equations. Initial disturbances consist of the finite-amplitude 2-D Tollmien-Schlichting wave and two 3-D oblique waves. Competitions among different modes were computed during transition for different Richardson numbers.

It is found that the body force can modify the transition mechanism of flows between two vertical plates. The body force was found to hasten the formation of three-dimensional flow. A non-laminar flow induced by the body force may present when the background flow is still laminar.     

Excitation of Tollmien-Schlichting waves py a vibrating section of a surface with flow round it

In the context of the problem of describing the transition of a laminar boundary layer to a turbulent, great interest attaches to the study of susceptibility, i.e., of the reaction of the flow to various external influences, such as acoustic perturbations, surface roughness, vibration of the wall, turbulence of the unperturbed flow, etc. A general property of the effect of the factors mentioned above on the flow in a laminar boundary layer was discovered in experimental and numerical studies and is noted in [1]: in all cases an external forcing perturbation leads to the excitation of normal modes of oscillation in the boundary layer which propagate downstream, namely, Tollmien-Schlichting waves. There is an analytical calculation in [2, 3] of the amplitude of a wave excited by harmonic oscillations of a narrow band on the surface of a plane plate, the Reynolds number having been assumed to be infinitely large, and the frequency of the vibrator corresponding to the neighborhood of the lower branch of the neutral cuirve [4], In [5] the amplitude of the wave of instability generated is calculated by the method of expansion of the solution in a biorthogonal system of eigenfunctions. The amplitudes of the Tollmien-Schlichting waves are calculated below by means of a generalization of the method of [2] for the whole range of Reynolds numbers and frequencies of the vibrator corresponding to the region of instability: for moderate Reynolds numbers the problem is solved numerically, while for large Reynolds numbers an asymptotic solution is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 46–51, July–August, 1987.The author is grateful to M. N. Kogan and V. V. Mikhailov for useful discussions of the results of the study.     

Spatial direct numerical simulation of boundary-layer transition mechanisms: Validation of PSE theory

A study of instabilities in incompressible boundary-layer flow on a flat plate is conducted by spatial direct numerical simulation (DNS) of the Navier-Stokes equations. Here, the DNS results are used to evaluate critically the results obtained using parabolized stability equations (PSE) theory and to study mechanisms associated with breakdown from laminar to turbulent flow. Three test cases are considered: two-dimensional Tollmien-Schlichting wave propagation, subharmonic instability breakdown, and oblique-wave breakdown. The instability modes predicted by PSE theory are in good quantitative agreement with the DNS results, except a small discrepancy is evident in the mean-flow distortion component of the two-dimensional test problem. This discrepancy is attributed to far-field boundary-condition differences. Both DNS and PSE theory results show several modal discrepancies when compared with the experiments of subharmonic breakdown. Computations that allow for a small adverse pressure gradient in the basic flow and a variation of the disturbance frequency result in better agreement with the experiments.     

Numerical modeling of three-dimensional vortex structures in hypersonic transverse flow past a cylinder

A new mechanism of the formation of spatially periodic structures on the nose surfaces of cylindrically blunted bodies in a hypersonic transverse flow is investigated. According to this mechanism, a curved shock wave produces a vortex flow, while the vortex, which is conserved in the presence of weak dissipation, acts on the shock and maintains its curved shape. The realizability of this vortex formation mechanism is verified by direct numerical simulation using the FLUENT software package. It is confirmed that in the case of uniform hypersonic freestream both plane and three-dimensional modes of the steady flow past the cylinder nose can exist. The three-dimensional mode is characterized by periodic-in-span vortex structures and considerable heat flux peaks on the nose surface. The calculated results are compared with the experimental data.     

Stability of a Liquid Film Flowing Down an Oscillating Inclined Surface

The stability of flow of a liquid film along an inclined plate subject to periodic oscillations under the action of the gravity force is investigated with allowance for the surface tension. An equation of the Orr-Sommerfeld type with time-periodic coefficients is used. A method for determining the eigenvalues of the linear stability problem is developed on the basis of Floquet theory, spectral representation of the variables, and multistep methods of integration of ordinary differential equations. The bifurcation spectrum of the resonance modes is investigated, and the amplification coefficients and phase velocities are calculated for the surface waves, Tollmien-Schlichting waves, and resonance waves. The influence of external parameters, namely, the inclination, the surface tension, and the layer thickness, on the resonance modes and the steady-state flow modes is studied.     

An experimental study of oblique transition in a Blasius boundary layer flow

Transition initiated by a pair of oblique waves was investigated experimentally in a Blasius boundary layer flow by using hot-wire measurements and flow visualisation. The oblique waves were generated by periodic blowing and suction through an array of pipes connecting to the flow through a transverse slit in the flat plate model. The structure of the flow field is described and the amplitude of individual frequency-spanwise wave number modes was determined from Fourier transforms of the disturbance velocity. In contrast to results from investigations of oblique transition at subcritical flow conditions, the transition process at the present conditions suggests the combined effect of non-modal growth of streaks and a second stage with exponential growth of oblique waves to initiate the final breakdown stage.     

Determination of the shape of the shocks ahead of a transverse jet injected into a supersonic flow

The three-dimensional shape of the shock wave formed ahead of a sonic jet flowing out into a supersonic flow through the surface of a sharp cone is determined. The shape of the wave in the longitudinal and transverse cross-sections of the model is constructed using schlieren photographs taken for various angles of rotation and freestream Mach numbers M=1.75–3. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 41–44, March–April, 1998. This research was carried out with financial support from the Russian Foundation for Basic Research (project No. 95-01-00709a).     

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.     

Shear polarisation of elastic waves by a structured interface

The article addresses the elastodynamic behaviour of a ‘shear-type structured’ interface separating two elastic regions. The structure includes a system of horizontal elastic bars connected by transverse massless elastic links. For a plane wave transmission problem, the analytical results obtained here show that such an interface acts as a ‘shear polariser’. A special attention is paid to the analysis of the defect modes (also referred to as resonance modes) within the interface. It is shown that these vibration modes induce enhanced transmission within the elastic system.     

Numerical study of three-dimensional detonation structure transformations in a narrow square tube: from rectangular and diagonal modes into spinning modes

Three-dimensional (3-D) detonation structure transformations from rectangular and diagonal modes into spinning modes in a narrow square tube are investigated by high-resolution simulation. Numerical simulations are performed with a Riemann solver of the HLLC-type, new cell-based structured adaptive mesh refinement data structure, high-order, parallel adaptive mesh refinement reactive flow code. A simplified one-step kinetic reaction model is used to reveal the 3-D detonation structure. The four different types of initial disturbances applied in the ZND profiles lead to the structures of rectangular in phase, rectangular out of phase, rectangular partial out of phase and diagonal, respectively, during the initial stages of detonation propagation. Eventually, all these detonation structures evolve into the self-sustained spinning detonations. The asymmetric disturbance leads to a stable spinning detonation much faster than the rest. The important features in the formation of spinning detonation are revealed using a 3-D visualization, and a remarkable qualitative agreement with experimental and numerical results is obtained with respect to the transverse wave dynamics and detonation front structures. The transverse wave collisions produce the unburnt gas pockets and the energy to sustain the detonation front propagation and distortion. The periodic pressure oscillation of front plays a complex role as it shifts the reaction zone structure with an accompanying change in the driving energy of transition and the detonation parameters which result in the more distorted front and the unstable detonation. Eventually, the unstable distorted detonation evolves into a spinning detonation.     

Acoustic Field Effect on Laminar Turbulent Transition on a Swept Wing in the Favorable Pressure Gradient Region

The development of disturbances in a three-dimensional boundary layer on a swept wing model is studied both under natural conditions and for artificial excitation of traveling waves by an acoustic field. It is found that steady-state streamwise structures are formed in the three-dimensional boundary layer; under natural conditions a wave packet leading to turbulence is detected. When the flow is exposed to the action of an acoustic field at a frequency from the wave packet, disturbances whose velocity along the streamwise structures is equal to 0.55 of the oncoming flow velocity are formed, while the laminar-turbulent transition is displaced upstream.     

THE APPLICATION OF MULTI－SCALE PERTURBATION METHOD TO THE STABILITY ANALYSIS OF PLANE COUETTE FLOW

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Some exact expressions for the temporal evolution of long Rossby waves on a beta-plane

Some exact expressions are derived to describe the temporal evolution of forced Rossby waves in a two-dimensional beta-plane configuration where the background flow has constant zonal-mean velocity. The meridional length scale of the problem is assumed to be small relative to the zonal length scale and so the long-wave limit of zero aspect ratio is taken. In the case where the background flow velocity is zero, an exact solution is obtained in terms of generalized hypergeometric functions. A late-time asymptotic approximation is obtained and it shows that the solution oscillates with time and its amplitude goes to zero in the limit of infinite time. In the case of a non-zero background flow velocity, the solution is evaluated using two different procedures which give two equivalent expressions in terms of different generalized hypergeometric functions. The late-time asymptotic behaviour is investigated and it is found that the solution approaches a steady state in the limit of infinite time.We also derive a solution in the form of an asymptotic series expansion for the more general situation where a Rossby wave packet is generated by a zonally-localized boundary condition comprising a continuous spectrum of wavenumbers or Fourier modes. The exact solutions found here can be used as leading-order solutions in weakly-nonlinear analyses and other studies involving more realistic configurations for time-dependent Rossby waves or wave packets.