Controlling supersonic boundary layer stability by means of distributed mass transfer through a porous wall

The interaction between disturbances in a compressible boundary layer in the presence of distributed mass transfer (injection or suction) through a permeable porous wall is considered in the linear and nonlinear approximations (weakly nonlinear stability theory). The regimes of moderate and high supersonic velocities (Mach numbers M = 2 and 5.35) are studied. The boundary conditions for the disturbances on a permeable wall are derived with account for the gas compressibility in pores and the presence of a suction chamber. Maximum pore dimensions, at which the surface properties have no effect on the disturbance characteristics, which are stabilized upon suction and destabilized upon injection, are determined. When the surface properties are taken into account, intense growth of the first-mode vortex disturbances occurs, which can completely undo the stabilizing effect of the suction. Injection leads to the vortex and acoustic mode destabilization on the linear range and the enhancement of the nonlinear processes on the transitional range.     

Three-wave interactions of disturbances in a hypersonic boundary layer on a porous surface

Interactions of disturbances in a hypersonic boundary layer on a porous surface are considered within the framework of the weakly nonlinear stability theory. Acoustic and vortex waves in resonant three-wave systems are found to interact in the weak redistribution mode, which leads to weak decay of the acoustic component and weak amplification of the vortex component. Three-dimensional vortex waves are demonstrated to interact more intensively than two-dimensional waves. The feature responsible for attenuation of nonlinearity is the presence of a porous coating on the surface, which absorbs acoustic disturbances and amplifies vortex disturbances at high Mach numbers. Vanishing of the pumping wave, which corresponds to a plane acoustic wave on a solid surface, is found to assist in increasing the length of the regions of linear growth of disturbances and the laminar flow regime. In this case, the low-frequency spectrum of vortex modes can be filled owing to nonlinear processes that occur in vortex triplets.     

Three-wave interactions between disturbances in the hypersonic boundary layer on impermeable and porous surfaces

The interaction between disturbances in the hypersonic boundary layer on impermeable and porous surfaces is considered within the framework of weakly-nonlinear stability theory. It is established that on the impermeable surface nonlinear interactions between different waves (acoustic and vortex) occur in the parametric resonance regime. The role of pumping wave is played by a plane acoustic wave. The nonlinear interactions take place over a wide frequency range and can lead to the packet growth of Tollmien-Schlichting waves. On the porous surface the analogous interactions are fairly weak and result in a slight decay of the acoustic mode and a slight amplification of the vortex mode. This leads to the dragging out of the laminar flow regime and the regions of linear disturbance growth. In this situation the low-frequency spectrum of the vortex modes may be filled on account of the nonlinear processes occurring in the three-wave systems between the vortex components.     

Vortex Instability of the Asymptotic Dissipation Profile in a Porous Medium

In this note we consider the thermoconvective stability of the recently-discovered asymptotic dissipation profile (ADP). The ADP is a uniform thickness, parallel-flow boundary layer which is induced by a cold surface in a warm saturated porous medium in the presence of viscous dissipation. We have considered destabilisation in the form of stream-wise vortex disturbances. The critical wavenumber and Rayleigh number for the onset of convection have been determined for all angles of the cooled surface between the horizontal and the vertical for which the ADP exists. The paper closes with a presentation of some strongly nonlinear computations of steady vortices.     

Development of slow disturbances in a hypersonic boundary layer

The mechanisms of development of slow time-dependent disturbances in the wall region of a hypersonic boundary layer are established and a diagram of the disturbed flow patterns is plotted; the corresponding nonlinear boundary value problem is formulated for each of these regimes. It is shown that the main factors that form the disturbed flow are the gas enthalpy near the body surface, the local viscous-inviscid interaction level, and the type, either subsonic or supersonic, of the boundary layer as a whole. Numerical and analytical solutions are obtained in the linear approximation. It is established that enhancement of the local viscous-inviscid interaction or an increased role for the main supersonic region of the boundary layer makes the disturbed flow by and large “supersonic”: the upstream propagation of the disturbances becomes weaker, while their downstream growth is amplified. Contrariwise, local viscous-inviscid interaction attenuation or an increased role for the main subsonic region of the boundary layer has the opposite effect. Surface cooling favors an increased effect of the main region of the boundary layer while heating favors an increased wall region effect. It is also found that in the regimes considered disturbances travel from the turbulent flow region downstream of the disturbed region under consideration counter to the oncoming flow, which may be of considerable significance in constructing the nonlinear stability theory.     

Direct numerical simulation of the transition to turbulence in a supersonic boundary layer on smooth and rough surfaces

Direct numerical simulations of instability development and transition to turbulence in a supersonic boundary layer on a flat plate are performed. The computations are carried out for moderate supersonic (free-stream Mach number M = 2) and hypersonic (M = 6) velocities. The boundary layer development is simulated, which includes the stages of linear growth of disturbances, their nonlinear interaction, stochastization, and turbulent flow formation. A laminar–turbulent transition initiated by distributed roughness of the plate surface at the Mach number M = 2 is also considered.     

Tangential gas blowing and boundary layer stability of a compressible gas

The effect of distributed blowing of a gas mass through a porous surface on the stability characteristics of a supersonic boundary layer is studied at a moderate supersonicMach numberM= 2. Tangential blowing when only the U-component of the mean velocity is not equal to zero on the wall is considered. The effect of the porous surface parameters on vortex perturbations is investigated and a comparison with the variant of the so-called “cutoff” regime is carried out. Different-density gas blowing is simulated by means of variation of the temperature factor (wall heating or cooling), namely, blowing of a heavy gas is simulated via blowing of a cold gas and vice versa.     

The influence of geostrophic force on the stability of an heterogeneous conducting fluid with a radial gravitional force

The linear and nonlinear stability of a heterogeneous incompressible inviscid perfectly conducting fluid between two cylinders is investigated in the presence of a radial gravitational force and geostrophic force. The stability for linear disturbances is investigated using the normal mode method, while the nonlinear stability is investigated by applying the energy method. In the case of linear theory, it is found that a necessary condition for in stability is that the algebraic sum of hydrodynamic, hydromagnetic and rotation Richardson number is less than one quarter somewhere in the fluid. A semi-circle theorem similar to that of Howard is also obtained. In the case of nonlinear disturbances a universal stability estimate namely a stability limit for motions subject to arbitrary nonlinear disturbances is obtained in the form $$E \leqslant E_0 \exp ( - 2M\tau ).$$ The motion is asymptotically stable if $$\delta \leqslant 1 + J_m + J_H $$ somewhere in the fluid. This asymptotic stability limit is improved using the calculus of variation technique. We also find that whenδ=1/4, andJ _R=1, both the linear and nonlinear stability criteria coincide and in that particular case, we have a necessary and sufficient condition for stability.     

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.     

Nonlinear Electrohydrodynamic Stability of Two Superposed Streaming Finite Dielectric Fluids in Porous Medium with Interfacial Surface Charges

A weakly nonlinear stability analysis of wave propagation in two superposed dielectric fluids streaming through porous media in the presence of vertical electric field producing surface charges is investigated in three dimensions. The method of multiple scales is used to obtain a dispersion relation for the linear problem and a nonlinear Klein–Gordon equation with complex coefficients describing the behavior of the perturbed system at the critical point of the neutral curve. In the linear case, we found that the system is always unstable for all physical quantities (including the dimension l), even in the presence of electric fields and porous medium, in the nonlinear case, novel stability conditions are obtained, and the effects of various parameters on the stability of the system are discussed numerically in detail.     

Joint effect of heat and mass transfer on the compressible boundary layer stability

The study continues the cycle of investigations concerned with the modeling of the methods of controlling flow regimes in compressible boundary layers. The effect of distributed heat and mass transfer on the stability parameters of a supersonic boundary layer is considered at amoderate supersonic Mach number M = 2. Emphasis is placed on the modeling of both the normal injection, when only the V component of the mean velocity is nonzero, and injection in other directions, including the tangential injection, when only the U component is nonzero on the wall. The formulation of the problem is similar with that of the gas curtain influence on the small fluctuation development. It is assumed that the effect of the injection of a similar gas with different temperatures is analogous to the injection of a gas with different densities, namely, the cold gas injection mimics the heavy gas injection, and vice versa. For this reason, in this study this modeling is realized by means of varying the temperature factor (wall heating or cooling). The case, in which the so-called “cutoff” regime is realized, that is, the velocity disturbances on a porous surface can be taken to be zero, is also considered.     

Linear and Nonlinear Stability of Double-Diffusive Convection with the Soret Effect

The onset of double-diffusive convection in a horizontal fluid-saturated porous layer is examined by taking the Soret effect into consideration. The linear and nonlinear stability analyses are derived, and the corresponding eigenvalue problems are solved. The nonlinear stability analysis is achieved by using the energy method. In both the cases of linear and nonlinear stability theories, the onset criterion for all possible modes is derived analytically. For numerical computations of the eigenvalue problem, the Chebyshev tau method is employed. It is observed that the effect of stabilization or destabilization caused by the Soret parameter is significant for the Soret parameters which are less than \(Sr = 2\). In the absence of the Soret effect, the linear and nonlinear stability thresholds coincide.     

Bispectral analysis of nonlinear processes in the hypersonic boundary layer on a porous cone surface

The results of the experimental investigation of the nonlinear stage of laminar-turbulent transition in the hypersonic boundary layer on porous and impermeable cone surfaces are presented. The bispectral analysis is applied to show that the porous surface suppresses the subharmonic resonance due to second mode disturbances. It is established that on the porous surface nonlinear processes develop more slowly than on the impermeable surface. This behavior indicates that the subharmonic resonance plays the role of a catalyzer in transferring energy from the mean flow to low-frequency disturbances in transition process, in much the same way as it occurs in a subsonic boundary layer.     

Numerical Investigation of the Second Transition in the Problem of Plane Convective Flow through a Porous Medium

The problem of convective flow through a porous medium in a plane rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. Single-parameter families of steady-state regimes resulting from the existence of cosymmetry of the corresponding differential equations are investigated using the Galerkin method. The onset and development of instability on these families and the characteristics of convective regimes as functions of the seepage Rayleigh number and the rectangle side ratio are studied. It is shown that the number of regimes which lose stability, the instability type, the number of convective rollers developed, and the heat transfer depend significantly on the vessel geometry. Several bifurcations of single-parameter families of steady-state regimes are identified and investigated.     

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.     

Overstability Analysis of Thermo-bioconvection Saturating a Porous Medium in a Suspension of Gyrotactic Microorganisms

A linear stability analysis is performed to analyze bioconvection in a dilute suspension of gyrotactic microorganisms in horizontal shallow fluid layer cooling from below and saturated by a porous medium, in the rigid boundary case. It is established that due to cooling from below thermally stratified layer is stabilized, which opposes the development of bioconvection and the situations for oscillatory convection may take place. The stability criterion is obtained in terms of thermal Rayleigh number, bioconvection Rayleigh number, gyrotactic number, bioconvection Peclet number, measure of cell eccentricity, Prandtl number, and Lewis number. It is observed that the presence of porous medium results in decrease of the magnitude of critical bioconvection Rayleigh number in comparison with its non-existence; hence due to porous effect, the system becomes less stable.     

Joint permeability and roughness effect on the supersonic flat-plate boundary layer stability and transition

The joint effect of the permeability and the roughness of the flat plate surface on the boundary layer stability and laminar-turbulent transition is experimentally and theoretically investigated at the freestream Mach number M_∞ = 2. It is shown that, as a certain roughness value is reached, and with increase in the porous coating thickness (on a certain range), the boundary layer stability against natural disturbances diminishes and laminar-turbulent transition is displaced toward the leading edge of the model.     

Stability and nonlinear wavy regimes in downward film flows on a corrugated surface

The linear and nonlinear stability of downward viscous film flows on a corrugated surface to freesurface perturbations is analyzed theoretically. The study is performed with the use of an integral approach in ranges of parameters where the calculated results and the corresponding solutions of Navier-Stokes equations (downward wavy flow on a smooth wall and waveless flow along a corrugated surface) are in good agreement. It is demonstrated that, for moderate Reynolds numbers, there is a range of corrugation parameters (amplitude and period) where all linear perturbations of the free surface decay. For high Reynolds numbers, the waveless downward flow is unstable. Various nonlinear wavy regimes induced by varying the corrugation amplitude are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 110–120, January–February, 2007.     

Experimental study of the effect of a passive porous coating on disturbances in a hypersonic boundary layer. 1. Effect of the porous coating length

The effect of passive porous coatings of different lengths on the second mode of disturbances in a hypersonic boundary layer is considered. The experiments are performed in a flow with a free-stream Mach number M_∞ = 5.8 and five values of the unit Reynolds number around a sharp cone with an apex half-angle equal to 7°, which is aligned at a zero angle of attack. One half of the model surface along its generatrix is covered by a porous material, and the other part is a solid surface. Pressure fluctuations on the model surface are measured. It is found that application of a passive porous coating can either decrease or increase the amplitude of the second mode. The length of the passive porous coating corresponding to the maximum efficiency of its action on flow disturbances and the coating length that increases the amplitude of the second mode are found.