Laminar boundary layer stability on membrane type deformable surfaces

In connection with the successful experiments of Kramer [l, 2] on models sheathed by flexible coverings, attempts have been made to explain theoretically the effect of boundary deformation on the position of the point of stability loss in the boundary layer. Korotkin [3] examined the stability of a plane laminar boundary layer on an elastic surface under the assumption of a linear connection between the pressure perturbation and the normal deformation of the surface. Benjamin [4] and Landahl [5] investigated the stability of the laminar boundary layer on a membrane type surface under the assumption that the physical characteristics of the surface depend on the perturbing flow wavelength. In the following we examine stability of Blasius flow on a membrane type surface whose physical characteristics are constant along the length.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, vol. 10, no. 6, pp. 52–56, November–December, 1969.     

Stability of the laminar boundary layer flow encountering a row of roughness elements: Biglobal stability approach and DNS

The stability of the laminar boundary layer developing on a flat plate in the presence of a periodic row of roughness elements is investigated. A Direct Numerical Simulation is performed to compute the steady flow downstream of the roughness elements, which contains a pair of two counter-rotating streamwise vortices per element, which can be considered as a “pre-streaky” structure. The linear stability of this base flow is analyzed by means of the so-called “biglobal” stability approach. Three-dimensional eigenmodes are found, which are shown to be the continuation of the Tollmien–Schlichting waves present in the case of an unperturbed boundary layer. Moreover, a stabilizing effect due to the roughness-induced vortices is found. A Direct Numerical Simulation of the interaction between a two-dimensional Tollmien–Schlichting wave and the roughness array is also performed. The computed perturbation traveling downstream of the roughness elements is shown to be a linear combination of the biglobal eigenmodes.     

Small perturbations spectrum in boundary-layer flow on a flat plate

The small perturbation spectrum of a number of flows has recently been analyzed carefully [1–3]. At the same time, investigations for the boundary layer have been limited within the framework of linear perturbation theory to the neighborhood of the neutral curve although a spectrum analysis is of indubitable interest not only to find the stability criterion of a laminar stream, but also to solve a problem with initial data about the time development of an arbitrary small perturbation. In particular, the possibility of representing an arbitrary perturbation in terms of a system of basis functions is related to the question of the completeness of the system. The finiteness was proved [4] and an estimate was obtained of the domain of eigenvalue existence in an investigation of the boundary-layer stability and a deduction has been made about the finiteness of the small perturbations spectrum for boundary-layer flow on this basis. A sufficiently complete survey of the investigation of the neutral stability of a laminar boundary layer can be found in the monograph [5]. The small perturbations spectrum in a boundary layer flow is obtained in this paper by methods of the linear theory of hydrodynamic stability by using the complete boundary conditions on the outer boundary. It is shown that the small perturbations spectrum is finite for each fixed value of the wave number . Singularities in the spectrum behavior are investigated for sufficiently small .Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 112–115, July–August, 1975.The author is grateful to M. A. Gol'dshtik and V. N. Shtern for useful discussions of the results of the research.     

The effect of non-linear interaction between gas and particle velocities on the hydrodynamic stability in the Blasius boundary layer

A weakly non-linear stability analysis of two phase flow in the Blasius boundary layer has been carried out. Two mathematical models have been established based on the perturbation shape preserved assumption and linear stability model of two phase flow proposed by Stuart [On the non-linear mechanics of hydrodynamic stability, J. Fluid Mech. 4 (1958) 1-21] and Saffman [On the stability of laminar flow of dusty gas, J. Fluid Mech. 13 (1962) 120-128], respectively. The perturbation model and the perturbation energy balance equation are solved numerically with Chebyshev spectral method and artificial boundary condition. The numerical program adopted in the present study is verified by comparison with former works. The results show that the non-linear interaction between mean flow and perturbation reduces the growth rate of perturbation, while the non-linear interaction between particle phase and gas phase increases the growth rate of perturbation amplitude. The distortion of the mean flow caused by the Reynolds stress modifies the rate of transfer of energy from the mean flow to disturbance. The existence of particle alleviates the distortedness. The result also indicates that the weakly non-linear stability theory is consistent to linear stability theory, and the addition of fine and coarse particles reduces and increases the critical Reynolds number.     

Effect of Thermal Modulation on the Onset of Convection in Walters B Viscoelastic Fluid-Saturated Porous Medium

The linear stability of Walters B viscoelastic fluid-saturated horizontal porous layer is examined theoretically when the walls of the porous layer are subjected to time-periodic temperature modulation. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and the corresponding wave number. The shift in critical Rayleigh number is calculated as a function of modulation frequency, viscoelastic parameter, and Prandtl number. The effect of all three types of modulations is found to be destabilizing as compared to the unmodulated system. This result is in contrast to the system with other types of fluids. Besides, the influence of physical parameters on the control of convective instability of the system is discussed.     

Tollmien-Schlichting Waves in a Hypersonic Boundary Layer Flow in the Neighborhood of a Cooled Surface

A nonlinear time-dependent model of the development of longwave perturbations in a hypersonic boundary layer flow in the neighborhood of a cooled surface is constructed. The pressure in the flow is assumed to be induced the combined variation of the thicknesses of the near-wall and main parts of the boundary layer. Numerical and analytic solutions are obtained in the linear approximation. It is shown that if the main part of the boundary layer is subsonic as a whole, its action reduces the perturbation damping upstream and the perturbation growth downstream, while a supersonic, as a whole, main part of the boundary layer creates the opposite effects. An analysis of the solutions obtained makes it possible to conclude that the asymptotic model proposed can describe the three-dimensional instability of the Tollmien-Schlichting waves.     

Interaction of a Traveling Pressure Perturbation with a Boundary Layer

The interaction between a traveling pressure perturbation and a laminar compressible boundary layer is investigated for a perturbation level higher than that needed to initiate steady-state flow separation. It is found that if the velocity of the pressure perturbation is fairly high the flow may remain unseparated and its direction of motion determines the nature of the perturbation propagation in the boundary layer. It is shown that even in the linear approximation the perturbations are mainly induced by the linear wall layer and not by the critical layer, which always remains nonlinear. It is also found that in the unsteady case shortwave perturbation oscillations are damped with time while the longwave ones grow and that the growth of the perturbations with time amplifies their damping along the streamwise coordinate while damping reduces it.     

Stability of subsonic laminar boundary layer on a flat plate with volume energy supply

Reducing frction drag and delaying the laminar-turbulent transition are topical problems of modern aerodynamics. A series of methods of delaying transition are known: creation of a favorable pressure gradient, boundary layer suction, surface cooling, etc., [1, 2]. Here, the possibility of delaying transition by means of volume heat supply to the boundary layer is considered. For this purpose, a subsonic compressible laminar boundary layer with volume energy supply is subjected to a stability analysis. The nonself-similar flow in the boundary layer is determined by means of a finite-difference marching method. The flow stability characteristics are calculated on the basis of the linear theory in the plane-parallel approximation. It is shown that even on a thermally insulated surface volume energy supply to the flow leads to significant flow stabilization and reduced perturbation growth rates.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 62–67, March–April, 1988.     

Effect of electrohydrodynamic interaction on the stability of a flat plate laminar boundary layer

The stability of a unipolarly charged electrohydrodynamic boundary layer on a flat dielectric plate along which an electric current flows between electrodes located on the plate is investigated within the framework of the linear theory. The solution of the steady-state problem is obtained on the basis of methods developed earlier for conditions typical of aerodynamical experiments and various electric currents and electrode voltages. The effect of the interaction between perturbations of the electric and hydrodynamic flow parameters on the flow stability is estimated within the framework of the locally homogeneous approximation. This effect turns out to be insignificant under the conditions considered. It is shown that steady-state electrohydrodynamic action on the main flow makes it possible to obtain “accelerating” velocity profiles with increased absolute values of the second derivative in the transverse direction. This ensures a significant increase in the critical Reynolds numbers of loss of stability and a narrowing of the growing perturbation wavenumber range.     

The effects of variable specific heat on the stability of hypersonic boundary layer on a flat plate

The gas temperature within hypersonic boundary layer flow is so high that the specific heat of gas is no longer a constant but relates to temperature. How variable specific heat influences on boundary layer flow stability is worth researching. The effect of the variable specific heat on the stability of hypersonic boundary layer flows is studied and compared with the case of constant specific heat based on the linear stability theory. It is found that the variable specific heat indeed has some effects on the neutral curves of both the first-mode and the second-mode waves and on the maximum rate of growth also. Therefore, the relationship between specific heat and temperature should be considered in the study of the stability of the boundary layer.     

Subcritical spatial transition of swept Hiemenz flow

It is known from experimental investigations that the leading-edge boundary layer of a swept wing exhibits transition to turbulence at subcritical Reynolds numbers, i.e. at Reynolds numbers which lie below the critical Reynolds number predicted by linear stability theory. In the present work, we investigate this subcritical transition process by direct numerical simulations of a swept Hiemenz flow in a spatial setting. The laminar base flow is perturbed upstream by a pair of stationary counter-rotating vortex-like disturbances. This perturbation generates high- and low-speed streaks by a non-modal growth mechanism. Further downstream, these streaky structures exhibit a strong instability to secondary perturbations which leads to a breakdown to turbulence.The observed transition mechanism has strong similarities to by-pass transition mechanisms found for two-dimensional boundary layers. It can be shown that transition strongly depends on the amplitude of the primary perturbation as well as on the frequency of the secondary perturbation.     

On Tollmien–Schlichting-like waves in streaky boundary layers

The linear stability of the boundary layer developing on a flat plate in the presence of finite-amplitude, steady and spanwise periodic streamwise streaks is investigated. The streak amplitudes considered here are below the threshold for onset of the inviscid inflectional instability of sinuous perturbations. It is found that, as the amplitude of the streaks is increased, the most unstable viscous waves evolve from two-dimensional Tollmien–Schlichting waves into three-dimensional varicose fundamental modes which compare well with early experimental findings. The analysis of the growth rates of these modes confirms the stabilising effect of the streaks on the viscous instability and that this stabilising effect increases with the streak amplitude. Varicose subharmonic modes are also found to be unstable but they have growth rates which typically are an order of magnitude lower than those of fundamental modes. The perturbation kinetic energy production associated with the spanwise shear of the streaky flow is found to play an essential role in the observed stabilisation. The possible relevance of the streak stabilising role for applications in boundary layer transition delay is discussed.     

含悬浮固粒的旋转射流剪切层稳定性研究

本文在理想不可压旋转圆射流的运动方程中添加了固粒作用项，由此推得了时间增长率的表达式，进而得关于含悬浮固粒放置射流稳定性的修正瑞利稳定性准则，求出了不同固粒质量密度固－气脉动速度比值，固气脉动速度相位差及Ｓｔｏｋｅｓ数情况下旋转射流场的增长率与径向空间波数的关系曲线，在比较这些曲线的基础上，给出了关于固粒属性对旋转射流场稳定性影响的几个重要结论为控制旋转射流场和后续发展提供了依据。     

Regime diagram of the development of long-wave near-wall disturbances in a hypersonic boundary layer

A regime diagram of the development of slow near-wall disturbances induced by an unsteady self-induced pressure perturbation in a hypersonic boundary layer is constructed for a disturbance wavelength greater than the boundary layer thickness. It is shown that the main factors shaping the perturbed flow are the gas enthalpy near the body surface, the intensity of the viscous-inviscid interaction, and the nature (sub- or supersonic) of the main part of the boundary layer. Nonlinear boundary-value problems are formulated for regimes in which the near-wall boundary layer region plays a decisive role. Numerical and analytical solutions are obtained in the linear approximation. It is shown that intensification of the viscous-inviscid interaction or an increase in the role of the supersonic main region of the boundary layer impart generally supersonic properties to the main part of the boundary layer, i.e. the upstream propagation of the disturbances is damped and the disturbance growth downstream becomes more intense. Damping of the viscous-inviscid interaction and an increase in the role of the subsonic main part of the boundary layer have the opposite effect. Surface cooling increases the effect of the main part of the boundary layer on the formation of pressure disturbances and surface heating leads to an increase in the effect of the near-wall boundary layer region. It is also shown that for the regimes considered disturbances propagate in a direction opposite to that of the free stream from the turbulent flow region located downstream of the local disturbance development region.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 59–71. Original Russian Text Copyright © 2004 by Bogolepov and Neiland.     

Linear stability of solutal convection in solidifying mushy layers: permeable mush–melt interface

The linear stability theory is used to investigate analytically the effect of a permeable mush–melt boundary condition on the stability of solutal convection in a mushy layer of homogenous permeability at the near eutectic (solid) limit. The results clearly show that, in contrast to the impermeable mush–melt interface boundary condition, the application of the permeable mush–melt interface boundary condition destabilizes the convection in a mushy layer.     

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.     

Ligament formation in sheared liquid–gas layers

We perform numerical simulations of two-phase liquid–gas sheared layers, with the objective of studying atomization. The Navier–Stokes equations for two-dimensional incompressible flow are solved in a periodic domain. A volume-of-fluid method is used to track the interface. The density ratio is kept around 10. The calculations show good agreement with a fully viscous Orr–Sommerfeld linear theory over several orders of magnitude of interface growth. The nonlinear development shows the growth of finger-like structures, or ligaments, and the detachment of droplets. The effect of the Weber and Reynolds numbers, the boundary layer width and the initial perturbation amplitude are discussed through a number of typical cases. Inversion of the liquid boundary layer is shown to yield more readily ligaments bending upwards and is thus more likely to produce droplets.     

Study of effect of a smooth hump on hypersonic boundary layer instability

Effect of a two-dimensional smooth hump on linear instability of hypersonic boundary layer is studied by using parabolized stability equations. Linear evolution of mode S over a hump is analyzed for Mach 4.5 and 5.92 flat plate and Mach 7.1 sharp cone boundary layers. Mean flow for stability analysis is obtained by solving the parabolized Navier–Stokes equations. Hump with height smaller than local boundary layer thickness is considered. The case of flat plate and sharp cone without the hump are also studied to provide comparable data. For flat plate boundary layers, destabilization and stabilization effect is confirmed for hump located at upstream and downstream of synchronization point, respectively. Results of parametric studies to examine the effect of hump height, location, etc., are also given. For sharp cone boundary layer, stabilization influence of hump is also identified for a specific range of frequency. Stabilization influence of hump on convective instability of mode S is found to be a possible cause of previous experimental observations of delaying transition in hypersonic boundary layers.     

Weakly nonlinear EHD stability of slightly viscous jet

A weakly nonlinear approach is utilized here to study the electrohydrodynamic (EHD) instability of an incompressible viscous liquid jet stressed by an axial electric field. The linear motion equations is solved in the light of nonlinear boundary conditions. The viscosity is assumed to be small. The study takes into account both the shear and radial components of the stresses at the interface. In the linear theory, we discuss the breakup phenomena of liquid jets. Also, it is found that, the electrical shearing stresses have no effect at the linear marginal state, while the linear cutoff wavenumber depends on the electrical shearing stresses. A nonlinear perturbation method is introduced. This method can be described our problem precisely. The nonlinear stability is compared with the linear stability condition in the weak viscosity case. It is found that, the weak viscosity has effect on the nonlinear stability condition, in contrast with the linear analysis, whereas the nonlinear cutoff wavenumber doesn't depend on the weak viscosity in both the linear and nonlinear theory.