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.     

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.     

Linear evolution and interaction of disturbances in the boundary layers on impermeable and porous surfaces in the presence of heat transfer

The interaction of disturbances in the compressible boundary layers on both impermeable and porous surfaces is considered in the linear and nonlinear approximations (weakly-nonlinear stability theory) in the presence of surface cooling. The regimes of moderate (Mach number M = 2) and high (M = 5.35) supersonic velocities are considered. It is established that surface cooling leads a considerable change in the linear evolution of the disturbances, namely, the first-mode vortex disturbances are stabilized, whereas the second-mode acoustic disturbances are destabilized, the variation degree being determined by the temperature factor. A porous coating used for controlling flow regimes influences the stability in the opposite fashion. For vortex waves the nonlinear interactions in three-wave systems at M = 2 are considerably attenuated in the presence of cooling. It might be expected that the cooling of the surface can delay the laminar regime for M = 2 and accelerate transition to turbulence for M = 5.35.     

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.     

On secondary instability with respect to three dimensional subharmonic disturbances in boundary layer

The general equations of secondary instability with respect to three-dimensional subharmonic disturbances are derived and applied to Blasius boundary layer in the present paper. The theoretical results of evolution and spatial distribution of subharmonic disturbances are compared with experimental data. The results show the important role of the process of route to transition in low-disturbance environments, and indicate that spatial mode is more rational than temporal mode. Project supported by the National Natural Science Foundation of China Current address: Graduate School, University of Science and Technology of China, Beijing, 100039     

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.     

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.     

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.     

Linear Stability and Convection in a Gravity Modulated Porous Layer Heated from Below: Transition from Synchronous to Subharmonic Solutions

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The linear stability results are presented for both the synchronous and subharmonic solutions and the exact point for the transition from synchronous to subharmonic solutions is computed. It is also demonstrated that increasing the excitation frequency rapidly stabilizes the convection up to the transition point from synchronous to subharmonic convection. Beyond the transition point, the effect of increasing the frequency is to slowly destabilize the convection.     

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.     

Nonlinear evolution of Klebanoff type second mode disturbances in supersonic flat-plate boundary layer简

Studying the evolution of 3D disturbances is of crucial theoretical importance for understanding the transition process. The present study concerns the nonlinear evolution of second mode unstable disturbances in a supersonic boundary layer by the numerical simulation, and discusses the selectivity of 3D disturbances and possibility to transition. The results indicate that a Klebanoff type nonlinear interaction between 2D and 3D disturbances with the same frequency may amplify a band of 3D disturbances centered at a finite spanwise wavenumber. That is, certain 3D disturbances can be selectively and rapidly amplified by the unstable 2D disturbances, and certain small-scale 3D structures will appear.     

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.     

Modeling motion-induced fluid release from partially saturated fibrous media onto surfaces with different hydrophilicity

Modeling the rate of fluid release from moving partially saturated nonwoven sheets in contact with a solid surface is a challenge, as the release rate depends on many parameters, some of which are difficult to quantify. In this paper, we report on a diffusion-controlled boundary treatment which we have developed to simulate fluid release from partially saturated porous materials onto surfaces with different hydrophilicy. The new boundary treatment considers the solid impermeable surface as a fictitious porous layer with a known fluid diffusive coefficient. Motion of the porous sheet on the surface is incorporated in the simulations by periodically resetting the saturation of the fictitious layer equal to zero, with a period obtained from the sheet’s speed of motion. Fluid transport inside the fibrous sheets is calculated by solving Richards’ equation of two-phase flows in porous media. Our numerical simulations are accompanied with experimental data obtained using a custom-made test rig for the release of liquid from partially saturated media at different speeds. It is demonstrated that the novel mathematical formulations presented here can correctly predict the rate of fluid release from moving fibrous sheets onto solid surfaces with different hydrophilicity as a function of time.     

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.     

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.     

Experiments on localized disturbances in a flat plate boundary layer. Part 2. Interaction between localized disturbances and TS-waves

Previous studies on boundary layer transition at moderate levels of free stream turbulence (FST) have shown that the transition process can be promoted by the introduction of Tollmien-Schlichting (TS) waves. In the present work the interaction between localized boundary layer disturbances and controlled TS-waves is studied experimentally. The localized disturbances are generated either from a controlled free stream perturbation, or by means of suction or injection through a slot in the flat plate surface. Both methods result in boundary layer disturbances dominated by elongated streamwise streaks of high and low velocity in the streamwise component. A strong interaction is observed preferably for high frequency TS-waves, which are damped when generated separately, and the interaction starts as a local amplification of a wide band of low-frequency oblique waves. The later stages of the transition process can be identified as a non-linear interaction between the oblique structures, leading to regeneration of new and stronger streamwise streaks.     

Similarity solutions for buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces

In the present paper similarity solutions for the convective flow induced by buoyancy in a saturated porous medium adjacent to horizontal impermeable surfaces are obtained. The analysis incorporates the variation of permeability from the wall and expressions for boundary layer thickness, local and overall surface heat-flux are obtained. Applications of the results to convective flows in a geothermal reservoir are discussed.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

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.     

Onset of Convection in a Porous Rectangular Channel with External Heat Transfer to Upper and Lower Fluid Environments

The conditions for the onset of convection in a horizontal rectangular channel filled with a fluid saturated porous medium are studied. The vertical sidewalls are assumed to be impermeable and adiabatic. The horizontal upper and lower boundary walls are considered as impermeable and subject to external heat transfer, modelled through a third-kind boundary condition on the temperature field. The external fluid environments above and below the channel, kept at different temperatures, provide the heating-from-below mechanism which may lead to the onset of the thermal instability in the porous medium. The linear response of the fluid saturated porous channel, in a basic motionless state, is tested with respect to three-dimensional normal mode disturbances of the temperature field and of the pressure field. The linearised disturbance equations are solved analytically leading to an implicit-form expression of the neutral stability condition, formulated as a functional relationship between the Darcy?CRayleigh number and the continuous longitudinal wave number of the normal modes, for any assigned aspect ratio of the cross-section and for any given Biot number. The analysis of the neutral stability is carried out. The analysis is extended to the case of a channel with a finite length in the longitudinal direction, and with adiabatic and impermeable capped ends.