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.     

Thermal excitation of waves in a boundary layer

An expression is obtained for the effective (reduced to the neutral surface) intensity of Tollmien-Schlichting waves induced by thermal motion in a boundary layer with Blasius velocity profile. On this basis, an expression is found for the amplitude of a quasisinusoidal wave at Reynolds numbers corresponding to the stabilization of the transition point in Wells's study [7].     

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.     

Effect of nonisothermality of the surface of a thin profile on the stability of a laminar boundary layer

Investigations of the stability of a subsonic laminar boundary layer have shown that, other things being equal, the stability of the laminar flow is considerably improved by cooling the entire surface of the body to a constant temperature T_w=const lower than the temperature of the free stream [1–3]. This is attributable to an increase in the critical Reynolds number of loss of stability and a decrease in the range of unstable perturbations of the Tollmien-Schlichting wave type when the surface is cooled. Recently, in the course of investigating the stability of laminar flow over a flat plate it was found [4, 5] that a similar improvement in flow stability can be achieved by raising the temperature of a small part of the surface near the leading edge of the plate. In this study we examine the possibility of delaying the transition to turbulent flow by creating a nonuniform temperature distribution along the surface of thin profiles, where the development of an adverse pressure gradient in the outer flow has a destabilizing effect on the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 36–42, September–October, 1986.In conclusion, the authors wish to thank M. N. Kogan for useful discussions of their results.     

Viscous Effects Can Destabilize Linear and Nonlinear Water Waves

Long waves on a running stream in shallow water are shown theoretically to be susceptible, in some circumstances, to a viscous instability, which can lead to rapid linear and nonlinear growth. The theory is based on high Reynolds numbers and involves viscous-inviscid interplay, leading in effect to a viscosity-modified version of the classical nonlinear K dV equation. This is with a pre-existing mean flow present. The modification is due to a Stokes wall layer and it can cause severe linear and nonlinear instability. A model profile for the original mean flow is studied first, followed by a smooth realistic profile, the latter provoking a nonlinear critical layer in addition. The theory is linked with interactive-boundary-layer analysis and linear and nonlinear Tollmien-Schlichting waves and there is some analogy with the recent findings (in work by the authors) of nonlinear break-ups occurring in any unsteady interactive boundary layer, including the external boundary layer and internal channel or pipe flows.     

Localized Coherent Structures in the Boundary Layer

A Blasius laminar boundary layer and a steady turbulent boundary layer on a flat plate in an incompressible fluid are considered. The spectral characteristics of the Tollmien—Schlichting (TS) and Squire waves are numerically determined in a wide range of Reynolds numbers. Based on the spectral characteristics, relations determining the three–wave resonance of TS waves are studied. It is shown that the three–wave resonance is responsible for the appearance of a continuous low–frequency spectrum in the laminar region of the boundary layer. The spectral characteristics allow one to obtain quantities that enter the equations of dynamics of localized perturbations. By analogy with the laminar boundary layer, the three–wave resonance of TS waves in a turbulent boundary layer is considered.     

Stability of a flow in a circular tube

Many studies, both theoretical and experimental, have been dedicated to the stability of flow in a circular tube (see, for example, review [1]). In every case mathematical investigation has not succeeded in obtaining an expression for hydrodynamic instability of such a flow for disturbances of sufficiently low amplitude. (An exception is [2].) Experiment also indicates the stability of such a flow [3], with a laminar mode being extended to Reynolds numbers of the order of tens of thousands. These facts are the basis for the assumption that the flow of a viscous incompressible liquid in a circular tube is stable for small perturbations. However, there is no analytical or even numerical proof of this hypothesis. Moreover, some studies, for example [2], indicate the instability of such a flow in relation to three-dimensional nonaxiosymmetric perturbations. The analysis of hydrodynamic stability with respect to three-dimensional disturbances of flow within a circular tube conducted in this study showed the stability of the flow over a wide range of wave numbers and Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 20–24, January–February, 1973.     

Suppression of unstable oscillations in a boundary layer

The study considers emission of Tollmien—Schlichting waves by a vibrator mounted on a plate with a viscous incompressible fluid flowing round it. It is shown that by changing the shape of a membrane working at a supercritical frequency, it is possible not only to reduce greatly the amplitude of the forced oscillations, but also to achieve their complete degeneration. This possibility opens the door to the suppression of an already formed Tollmien—Schlichting wave by a vibrator with specially chosen parameters. This type of equipment makes it possible to suppress perturbations in a laminar boundary layer and delay its transition to the turbulent state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 20–26, March–April, 1987.The authors are grateful to the referee V. A. Buchin for a useful observation expressed in the course of preparation of the article for the press.     

The origin and subsequent development in space of Tollmien-Schlichting waves in a boundary layer

Two kinds of small-parameter expansions are used to derive a series of ordinary differential equations governing spatially growing or decaying disturbances of the travelling-wave type in non-parallel laminar boundary layers. Eigensolutions of the lowest-order equation are shown to tend to resemble eigensolutions of the unsteady boundary-layer equation as the Reynolds number Re decreases, and to the Tollmien-Schlichting wave solutions of the Orr-Sommerfeld equation as Re increases. Examination of the next-order approximation reveals that the unsteady boundary-layer equation has eigensolutions only if Re exceeds a critical value depending on the frequency of disturbances, while it remains a partial differential equation for lower Reynolds numbers. Thus the Tollmien-Schlichting wave may be considered to originate at the above critical value of Re. The subsequent development of the wave with fixed frequency can be followed by integrating the spatial growth rate of the eigensolutions with respect to the downstream distance. The amplitude is at first damped but grows even higher than its original value farther downstream, provided the frequency is less than a certain value.     

Large eddy simulation of boundary layer flow under cnoidal waves

Water waves in coastal areas are generally nonlinear, exhibiting asymmetric velocity profiles with different amplitudes of crest and trough. The behaviors of the boundary layer under asymmetric waves are of great significance for sediment transport in natural circumstances. While previous studies have mainly focused on linear or symmetric waves, asymmetric wave-induced flows remain unclear, particularly in the flow regime with high Reynolds numbers.Taking cnoidal wave as a typical example of asymmetric waves, we propose to use an infinite immersed plate oscillating cnoidally in its own plane in quiescent water to simulate asymmetric wave boundary layer. A large eddy simulation approach with Smagorinsky subgrid model is adopted to investigate the flow characteristics of the boundary layer. It is verified that the model well reproduces experimental and theoretical results. Then a series of numerical experiments are carried out to study the boundary layer beneath cnoidal waves from laminar to fully developed turbulent regimes at high Reynolds numbers, larger than ever studied before.Results of velocity profile, wall shear stress, friction coefficient, phase lead between velocity and wall shear stress, and the boundary layer thickness are obtained. The dependencies of these boundary layer properties on the asymmetric degree and Reynolds number are discussed in detail.     

Development of a Shear Instability in Nodal Zones of a Standing Internal Wave

The results of experimentally investigating the initial stage of development of shear instability of the interface between two immiscible fluids relatively oscillating during the parametric excitation of standing internal waves are presented. Three stages of distortion of the sinusoidal wave profile are distinguished: the formation of short secondary waves, their breaking, and transition to large-scale vortex formations. It is shown that in the nodal zones of a standing wave quasi-stationary wave perturbations start to develop at wave steepnesses Γ ～ 0.08–0.13 and critical Reynolds numbers of the laminar boundary layer R ～ 90–300. The experimental data are compared with the classical Kelvin-Helmholtz theory.     

Effect of transverse stream velocity in an incompressible boundary layer on the stability of the laminar flow regime

The stability of the laminar flow regime in the boundary layer developed on a wall is increased considerably by the relatively slight extraction of fluid from the wall [1–4]. In the theoretical study of this phenomenon, all the investigators known to the present authors have taken into account only the increase in the fullness of the velocity profile in the boundary layer with suction. Computations of the stability characteristics have been made on the assumption that there are no transverse velocities in the laminar boundary layer.We present below an analysis of the stability of the laminar boundary layer in the presence of a constant transverse velocity in the near-wall region (suction). The calculations made show the existence for a given velocity profile in the boundary layer of a relative suction velocity v=v such that with suction velocities greater than v the flow remains stable at all Reynolds numbers, while the method used in the cited references gives a definite finite critical Reynolds number, equal in our notation to the Reynolds number at v=0, for each relative suction velocity.It was found that with suction of fluid from the boundary layer the region of instability has finite dimensions, i.e., there exist lower and upper critical Reynolds numbers. The flow is stable if its Reynolds number is less than the lower, or greater than the upper values of the critical Reynolds number.The instability region diminishes with increase in the relative suction velocity, and at a value of this velocity which is specific for each value of the velocity profile the instability region degenerates into a point-the flow becomes absolutely stable. Thus, with distributed suction it is advisable to increase the relative suction velocity only to a definite magnitude corresponding to disappearance of the instability region. The computational results presented make it possible to estimate this velocity for velocity profiles ranging from a Blasius profile to an asymptotic profile. Specific calculations were made for a family of Wuest profiles, since under actual conditions with suction there always exists a starting segment of the boundary layer [1, 2].     

Effect of surface temperature on the stability of the swept attachment line boundary layer

At fairly high Reynolds numbers instability may develop on the line of attachment of the potential flow to the leading edge of a swept wing and lead to a transition to boundary layer turbulence directly at the leading edge [1, 2]. Although the first results relating to the stability and transition of laminar flow at the leading edge of swept wings were obtained almost 30 years ago, the problem remains topical. The stability of the swept attachment line boundary layer was recently investigated numerically with allowance for compressibility effects [3, 4]. Below we examine the effect of surface temperature on the stability characteristics of the laminar viscous heat-conducting gas flow at the leading edge of a side slipping wing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 78–82, November–December, 1990.     

Hypersonic viscous flow around extended bodies

The characteristic feature of flow around an extended body is the interaction of the thickened boundary layer with the external nonviscous flow. This phenomenon becomes more significant at low Reynolds numbers and high Mach numbers. Theoretical investigation of this interaction is difficult because of the presence of shock waves, which are characteristic of hypersonic velocities; the position and curvature of these shock waves depend on the state of the boundary layer developing in conditions of pronounced vorticity of the external flow. With increasing rarefaction of the flow, the problem begins to take on an elliptic character, and this necessitates the use of methods of investigation of more general form than the classical boundary-layer theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 164–166, March–April, 1976.The authors thank V. G. Farafonov and V. N. Arkhipov for guidance and assistance in the work.     

Investigation of boundary layer stability in a gradient flow with a high degree of free-stream turbulence

The results of an experimental investigation of boundary layer stability in a gradient flow with a high degree of free-stream turbulence are presented. The question of the possible artificial generation, the further development and the effect on laminar-turbulent transition of instability waves (Tollmien-Schlichting waves) in the boundary layer on a wing profile is considered for a level of free-stream turbulence =1.75% of the free-stream velocity; the sensitivity of the flow to the disturbances and their control by means of boundary layer suction are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 52–58, March–April, 1990.     

The near wake structure of a square cylinder

The near wake structure of a square cross section cylinder in flow perpendicular to its length was investigated experimentally over a Reynolds number (based on cylinder width) range of 6700–43,000. The wake structure and the characteristics of the instability wave, scaling on θ at separation, were strongly dependent on the incidence angle () of the freestream velocity. The nondimensional frequency (St_θ) of the instability wave varied within the range predicted for laminar instability frequencies for flat plate wakes, jets and shear layers. For = 22.5°, the freestream velocity was accelerated over the side walls and the deflection of the streamlines (from both sides of the cylinder) towards the center line was higher compared to the streamlines for = 0°. This caused the vortices from both sides of the cylinder to merge by x/d 2, giving the mean velocity distribution typical of a wake profile. For = 0°, the vortices shed from both sides of the cylinder did not merge until x/d 4.5. The separation boundary layer for all cases was either transitional or turbulent, yet the results showed good qualitative, and for some cases even quantitative, agreement with linearized stability results for small amplitude disturbances waves in laminar separation layers.     

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.     

Effect of periodic action on boundary layer stability

The effect of a wave traveling over the surface and suction-blowing in the form of a traveling wave on boundary layer stability and laminarturbulent transition is investigated. The perturbation parameters are assumed not to be related to the parameters of the Tollmien-Schlichting wave. The parameters corresponding to an increase in the critical Reynolds number by a factor of 2–2.5 are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 109–115, May–June, 1988.The author is grateful to V. A. Kuparev for supplying the program for calculating the stability of the boundary layer.     

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.