Wall Effect on the Convective–Absolute Boundary for the Compressible Shear Layer

The linear stability of inviscid compressible shear layers is studied. When the layer develops at the vicinity of a wall, the two parallel flows can have a velocity of the same sign or of opposite signs. This situation is examined in order to obtain first hints on the stability of separated flows in the compressible regime. The shear layer is described by a hyperbolic tangent profile for the velocity component and the Crocco relation for the temperature profile. Gravity effects and the superficial tension are neglected. By examining the temporal growth rate at the saddle point in the wave-number space, the flow is characterized as being either absolutely unstable or convectively unstable. This study principally shows the effect of the wall on the convective–absolute transition in compressible shear flow. Results are presented, showing the amount of the backflow necessary to have this type of transition for a range of primary flow Mach numbers M ₁ up to 3.0. The boundary of the convective–absolute transition is defined as a function of the velocity ratio, the temperature ratio and the Mach number. Unstable solutions are calculated for both streamwise and oblique disturbances in the shear layer. Received 9 May 2001 and accepted 21 August 2001     

Linear stability analysis in compressible, flat-plate boundary-layers

The stability problem of two-dimensional compressible flat-plate boundary layers is handled using the linear stability theory. The stability equations obtained from three-dimensional compressible Navier–Stokes equations are solved simultaneously with two-dimensional mean flow equations, using an efficient shoot-search technique for adiabatic wall condition. In the analysis, a wide range of Mach numbers extending well into the hypersonic range are considered for the mean flow, whereas both two- and three-dimensional disturbances are taken into account for the perturbation flow. All fluid properties, including the Prandtl number, are taken as temperature-dependent. The results of the analysis ascertain the presence of the second mode of instability (Mack mode), in addition to the first mode related to the Tollmien–Schlichting mode present in incompressible flows. The effect of reference temperature on stability characteristics is also studied. The results of the analysis reveal that the stability characteristics remain almost unchanged for the most unstable wave direction for Mach numbers above 4.0. The obtained results are compared with existing numerical and experimental data in the literature, yielding encouraging agreement both qualitatively and quantitatively.      

超声速边界层/混合层组合流动的稳定性分析

利用可压缩线性稳定性理论研究了超声速混合层考虑壁面影响流动时的失稳特性. 基本流场选取了具有不同速度特征的2 股均匀来流,进入存在上下壁面的流道中. 混合层与边界层的距离为1~3 个边界层厚度,其中壁面取为绝热壁. 分析了该流动在超声速情况下的稳定性特征,同时还讨论了不同波角下的三维扰动波的演化特点,并与二维扰动波进行了比较和分析. 研究结果表明,在此流动情况下,边界层流动和混合层流动的稳定性特征同时存在,并互有影响,其流动稳定性特征既有别于单纯的平板边界层,也有别于单纯的平面混合层,呈现出了新的稳定性特征.      

Direct numerical simulation of disturbances generated by periodic suction-blowing in a hypersonic boundary layer

A numerical algorithm and code are developed and applied to direct numerical simulation (DNS) of unsteady two-dimensional flow fields relevant to stability of the hypersonic boundary layer. An implicit second-order finite-volume technique is used for solving the compressible Navier–Stokes equations. Numerical simulation of disturbances generated by a periodic suction-blowing on a flat plate is performed at free-stream Mach number 6. For small forcing amplitudes, the second-mode growth rates predicted by DNS agree well with the growth rates resulted from the linear stability theory (LST) including nonparallel effects. This shows that numerical method allows for simulation of unstable processes despite its dissipative features. Calculations at large forcing amplitudes illustrate nonlinear dynamics of the disturbance flow field. DNS predicts a nonlinear saturation of fundamental harmonic and rapid growth of higher harmonics. These results are consistent with the experimental data of Stetson and Kimmel obtained on a sharp cone at the free-stream Mach number 8.     

超声速边界层/混合层组合流动的稳定性分析简

利用可压缩线性稳定性理论研究了超声速混合层考虑壁面影响流动时的失稳特性. 基本流场选取了具有不同速度特征的2 股均匀来流,进入存在上下壁面的流道中. 混合层与边界层的距离为1~3 个边界层厚度,其中壁面取为绝热壁. 分析了该流动在超声速情况下的稳定性特征,同时还讨论了不同波角下的三维扰动波的演化特点,并与二维扰动波进行了比较和分析. 研究结果表明,在此流动情况下,边界层流动和混合层流动的稳定性特征同时存在,并互有影响,其流动稳定性特征既有别于单纯的平板边界层,也有别于单纯的平面混合层,呈现出了新的稳定性特征.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

Experimental study of vortex structures in a hypersonic shock layer on a flat plate

The characteristics of travelling perturbations of density in a hypersonic shock layer on a flat plate for the Mach number M_∞=21 and unit Reynolds numberRe ₁=6·10⁵ m⁻¹ were experimentally studied by the method of electron-beam fluorescence. The perturbations were generated by interaction of the shock layer behind an oblique gas-dynamic whistle and the leading edge of the plate. The cases of unsteady and quasi-steady interaction were considered. In both cases, vortex disturbances of finite amplitude were generated. The measurements were performed at the fundamental frequency F=0.6·10⁻⁴ and at the harmonic; the streamwise phase velocities, the growth rates of the disturbances, and the angles of wave propagation were obtained. The measurement results are compared with some experimental data for subsonic flows, some particular results of the linear stability theory for compressible flows, and the results obtained on the basis of a simple model of the nonlinear stage of disturbance evolution in a hypersonic boundary layer. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 41–47, November–December, 1999.     

Laminar–turbulent transition in oscillating boundary layer: experimental and numerical analysis using continuous wavelet transform

The laminar–turbulent transition of a forced oscillating boundary layer with a varying pressure gradient is experimentally and numerically investigated for two Strouhal numbers. Time-dependent characterization of the natural instability modes is carried out using continuous wavelet analysis of velocity signals. The periodic evolution of the total growth rates of the most unstable disturbances are measured and compared to the results of the linear stability theory. The “Tollmien–Schlichting” (TS) and “convective” transition modes are identified. It is shown that they correspond to the extrema of opposite signs of the skewness factor of unstable wavelet transform of the most unstable frequencies.     

Verification of parabolized stability equations for its application to compressible boundary layers

Parabolized stability equations (PSE) were used to study the evolution of disturbances in compressible boundary layers.The results were compared with those ob- tained by direct numerical simulations (DNS),to check if the results from PSE method were reliable or not.The results of comparison showed that no matter for subsonic or supersonic boundary layers,results from both the PSE and DNS method agreed with each other reasonably well,and the agreement between temperatures was better than those between velocities.In addition,linear PSE was used to calculate the neutral curve for small amplitude disturbances in a supersonic boundary layer.Compared with those obtained by linear stability theory (LST),the situation was similar to those for incom- pressible boundary layer.     

Analysis of the secondary stability of compressible boundary layer flows over a two-dimensional plate

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｃｅｎｔｌｙ,ｉｎｔｈｅａｎａｌｙｓｉｓｏｆｂｏｕｎｄａｒｙｌａｙｅｒｔｒａｎｓｉｔｉｏｎｆｒｏｍｌａｍｉｎａｒｔｏｔｕｒｂｕｌｅｎｔ,ｔｈｅｉｎｔｅｒａｃｔｉｏｎｏｆｔｗｏ＿ｄｉｍｅｎｓｉｏｎａｌａｎｄｔｈｒｅ＿ｄｉｍｅ...     

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.     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

New parabolized system of equations of stability of a compressible boundary layer

Based on estimates for the critical layer, a system of equations of stability of a compressible boundary layer is obtained. The system is parabolic and free from the known restriction on the step of the marching scheme related to ellipticity, which could not be eliminated within the framework of the previous method. A numerical scheme is described, and calculation results for the boundary layer on a heat-insulated plate are presented. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 63–69, January–February, 2000.     

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.     

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.     

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M _t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M _g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M _t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.      

Stability of a compressible swirling flow in a circular pipe

The stability of an axisymmetric flow of viscous gas in a circular pipe, which models the Burgers vortex in the pipe axis neighborhood, is studied within the linear theory framework. Neutral curves for the most unstable disturbances are calculated. The influence of the characteristic Mach number on the flow stability is investigated. It is shown that for a given model velocity distribution the Mach number affects only the temperature and pressure profiles of the main undisturbed flow. In this case, for the disturbance types considered, as the Mach number increases, the critical Reynolds number corresponding to loss of stability decreases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 35–41, January–February, 1999. The work received financial support from the Russian Foundation for Basic Research (project No. 96-01-00586).     

Asymptotic Behavior of Solutions of the Compressible Navier�CStokes Equation Around the Plane Couette Flow

Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L ² Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L ² norm as solutions of an n − 1 dimensional linear heat equation with a convective term.     

Stability of a hypersonic shock layer on a flat plateStabilité d'une couche de choc hypersonique sur une plaque plane.

Stability of a hypersonic shock layer on a flat plate is examined with allowance for disturbances conditions on the shock wave within the framework of the linear stability theory. The characteristics of the main flow are calculated on the basis of the Full Viscous Shock Layer model. Conditions for velocity, pressure, and temperature perturbations are derived from steady Rankine–Hugoniot relation on the shock wave. These conditions are used as boundary conditions on the shock wave for linear stability equations. The growth rates of disturbances and density fluctuations are compared with experimental data obtained at ITAM by the method of electron-beam fluorescence and with theoretical data of other authors. To cite this article: A.A. Maslov et al., C. R. Mecanique 332 (2004).