Absolute and Convective Instabilities in the Compressible Boundary Layer on a Rotating Disk

A numerical study has been undertaken to investigate the nature of inviscid instability of the three-dimensional compressible boundary layer flow due to a rotating disk. The compressible Rayleigh equation is integrated using a spectral Chebyshev-collocation method together with a fourth-order Runge–Kutta integrator. In the context of spatio-temporal stability analysis, the singularities of the resulting dispersion relation are determined and the ones that satisfy the Briggs–Bers pinching criterion have been selected. In certain finite parameter regions of eigenvalues (wave numbers and wave angles, for instance) it is found that by varying the Mach number, absolute instability occurs in the compressible boundary layer on a rotating disk. The range corresponding to the incompressible flow case given in Lingwood (1995) (ε between 14.615° and 38.114°) is verified. The results of Cole (1995) are also verified. The overall effect of compressibility is to reduce the extent of absolute instability at higher Mach numbers. The effect of heating the wall is to enhance the absolute instability properties, however, cooling the wall is found to decrease greatly the region of absolute instability regime for the range of Mach numbers studied. It is also shown in this study that for non-insulated walls a direct spatial resonance of the eigenmodes is possible and this raises the possibility of large local algebraic growth of perturbations being important in some instances. Received 15 October 1999 and accepted 10 December 1999     

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     

Stability of a compressible boundary layer

The problem of stability in a compressible boundary layer, as opposed to an incompressible layer, involves many parameters and requires consideration of three-dimensional perturbations. The transverse component of the velocity, the thermal regime at the wall, etc., take on great significance. Investigation of all aspects of this problem requires systematic calculations performed by electronic computers. There do exist a few calculations of stability of a compressible boundary layer with respect to three-dimensional disturbances for particular cases. It follows from those studies (see, for example, [1]) that consideration of three-dimensional perturbations and of the transverse component of the basic flow velocity is important. Many aspects of this problem remain uninvestigated. Aside from the sheer cumbersomeness of the problem, there exist purely mathematical difficulties connected with the presence of a small parameter with higher derivatives in the differential equations for the perturbations, which causes losses in accuracy of calculation. In this present study an algorithm will be developed for solution of the problem of stability of a compressible boundary layer relative to three-dimensional disturbances with consideration of the transverse component of the basic velocity. Calculations are performed for a boundary layer on a plane thermally insulating plate, and the effects of the transverse velocity component and the three-dimensionality of the perturbations on stability at various Mach numbers are demonstrated.     

超声速尾涡的稳定性分析

从Ｎ－Ｓ方程出发,通过正则模方法,研究了超声速尾涡的绝对／对流不稳定性性质．计算了流动的稳定性特征随马赫数Ｍ,周向波数ｎ．,轴向自由流速度Ｗ０和旋转度ｑ等流动参数的变化规律,找到了绝对／对流不稳定区域的边界．通过比较发现,马赫数的增加使流动由绝对不稳定向对流不稳定乃至稳定转化．在所计算的参数范围,周向波数的增加加速了这一转化过程,而且,轴向速度的增加,同样使流动向着稳定的方向转化．同时还分析了不同旋拧程度的流动受可压缩影响的不同．这些结果对于了解旋拧流动稳定性的物理机理以及进行流动控制都有着重要意义．     

Linear stability of three-dimensional boundary layers

Stability of compressible three-dimensional boundary layers on a swept wing model is studied within the framework of the linear theory. The analysis based on the approximation of local self-similarity of the mean flow was performed within the Falkner-Skan-Cooke solution extended to compressible flows. The calculated characteristics of stability for a subsonic boundary layer are found to agree well with the measured results. In the case of a supersonic boundary layer, the results calculated for a Mach number M = 2 are also in good agreement with the measured spanwise scales of nonstationary vortices of the secondary flow. The calculated growth rates of disturbances, however, are substantially different from the measured values. This difference can be attributed to a high initial amplitude of disturbances generated in the experiment, which does not allow the linear stability theory to be applied. The evolution of natural disturbances with moderate amplitudes is fairly well predicted by the theory. The effect of compressibility on crossflow instability modes is demonstrated to be insignificant. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 3–14, March–April, 2008.     

Simplified system of equations for calculating the stability of the boundary layer gas flow on a swept wing

A composite system of equations for calculating the stability of a compressible flow in a three-dimensional boundary layer is developed. A comparison with the calculations in accordance with the complete system of equations shows that the system proposed almost accurately predicts the growth rates of both the Tollmien-Schlichting mode and the transverse flow instability mode over a wide range of frequencies and transverse wavenumbers at moderate Mach numbers. At the same time, the computation time reduces by an order.     

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.      

Direct Spatial Resonance in the Compressible Boundary Layer on a Rotating-disk

This paper focuses on a resonance mechanism that can lead to significant disturbance amplification at conditions which are sub-critical to nonlinear instabilities. Particularly, direct spatial resonance instability is investigated, which is present in the basic three-dimensional viscous compressible boundary-layer flow due to a rotating-disk. Within this purpose, the linearized system of stability equations is treated numerically making use of a spectral Chebyshev collocation method. The analysis provides critical resonant Reynolds numbers above which growth occurs. Amplitudes of the response of the degeneracies decaying rapidly due to their high damping rates are shown to exist for small enough Reynolds numbers while the flow is still in the laminar state. If the flow is restricted to the incompressible case, the results of Turkyilmazoglu and Gajjar (in Sadhana Acad P Engs 25:601–617, 2000) are completely reproduced. The influences of compressibility are then explored by means of varying the Mach and Prandtl numbers in the cases of heating/cooling the wall as well as the isothermal wall. In general, compressibility effects are found strongly in favor of stabilizing as the Mach number increases, while a strong destabilization is observed by lowering the critical values of Reynolds numbers in the cases of wall heating and insulation. The modal interaction and coalescence of the eigenmodes calculated here create local algebraic growth by rapid development of relatively large amplitudes which might then provide the onset of nonlinear effects followed by transition.     

Structure and stability of a laminar diffusion flame in a compressible,three-dimensional mixing layer

In this paper we consider the structure and stability of a three-dimensional, finite rate, reacting compressible mixing layer lying between two streams of reactants with different freestream speeds and temperatures. Using both numerical calculations and asymptotic analyses, the structure of the ignition and diffusion flame regimes was studied; in particular the effect of crossflow on the structure. The results of both approaches are in good agreement. In the stability analysis the general case of three-dimensional disturbances was studied. It was shown that certain general results, in particular the circle theorem, could be easily extended to three-dimensional disturbances in a compressible fluid with crossflow. The introduction of a chemical reaction, in the form of a flame sheet, was found to have complex effects on flow stability. These included the appearance of additional neutral modes compared with the nonreacting case and substantial changes in growth rates with heat release, the skew angle of the mean flow, and the direction of propagation of the disturbance wave. These results are discussed in detail.The first author was supported, in part, by AFOSR Contract 91-0180, and in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-19480 while in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001, U.S.A. The second author was supported, in part, by AFOSR Contract 91-0250, and in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-19480 while in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23681-0001, U.S.A.     

Hypersonic viscous flow over large roughness elements

Viscous flow over discrete or distributed surface roughness has great implications for hypersonic flight due to aerothermodynamic considerations related to laminar?Cturbulent transition. Current prediction capability is greatly hampered by the limited knowledge base for such flows. To help fill that gap, numerical computations are used to investigate the intricate flow physics involved. An unstructured mesh, compressible Navier?CStokes code based on the space?Ctime conservation element, solution element (CESE) method is used to perform time-accurate Navier?CStokes calculations for two roughness shapes investigated in wind tunnel experiments at NASA Langley Research Center. It was found through 2D parametric study that at subcritical Reynolds numbers, spontaneous absolute instability accompanying by sustained vortex shedding downstream of the roughness is likely to take place at subsonic free-stream conditions. On the other hand, convective instability may be the dominant mechanism for supersonic boundary layers. Three-dimensional calculations for both a rectangular and a cylindrical roughness element at post-shock Mach numbers of 4.1 and 6.5 also confirm that no self-sustained vortex generation from the top face of the roughness is observed, despite the presence of flow unsteadiness for the smaller post-shock Mach number case.     

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.     

Stability properties of the vertical boundary layers in differentially heated cavities

In the present study, the two-dimensional (2-D) stability properties of the vertical boundary layers in a cavity that is differentially heated over two opposing vertical walls is considered. The study is performed by introducing artificial, controlled perturbations at the base of the vertical boundary layer along the hot cavity wall and by following the evolution of these disturbances. For small initial perturbations, the evolution is governed by linear effects. This method accurately predicts the frequency of the bifurcation, which occurs for (much) larger Rayleigh numbers. Convective instability sets in for Rayleigh numbers much smaller than those at which the absolute instability (i.e., the bifurcation) occurs, and these Rayleigh numbers are in reasonable agreement with those for the boundary layer along a plate. The absolute instability does not result from the first wave which becomes unstable. For small Prandtl numbers (≤ 2), the unstable waves which lead to the absolute instability are shear-driven, and a single frequency is introduced in the flow after the bifurcation. For larger Prandtl numbers, the unstable waves are buoyancy driven and no single-frequency unsteady flow is observed after the bifurcation.     

Liquid jet in a high Mach number air stream

The instability of circular liquid jet immersed in a coflowing high velocity air stream is studied assuming that the flow of the viscous gas and liquid is irrotational. The basic velocity profiles are uniform and different. The instabilities are driven by Kelvin–Helmholtz instability due to a velocity difference and neckdown due to capillary instability. Capillary instabilities dominate for large Weber numbers. Kelvin–Helmholtz instability dominates for small Weber numbers. The wavelength for the most unstable wave decreases strongly with the Mach number and attains a very small minimum when the Mach number is somewhat larger than one. The peak growth rates are attained for axisymmetric disturbances (n = 0) when the viscosity of the liquid is not too large. The peak growth rates for the first asymmetric mode (n = 1) and the associated wavelength are very close to the n = 0 mode; the peak growth rate for n = 1 modes exceeds n = 0 when the viscosity of the liquid jet is large. The effects of viscosity on the irrotational instabilities are very strong. The analysis predicts that breakup fragments of liquids in high speed air streams may be exceedingly small, especially in the transonic range of Mach numbers.     

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.     

Electrohydrodynamic instability of a dielectric compressible liquid sheet streaming into an ambient stationary compressible gas

The effect of compressibility of fluids on the linear electrohydrodynamic instability of a dielectric liquid sheet issued from a nozzle into an ambient dielectric stationary gas in the presence of a horizontal electric field is investigated. It is found that increasing the Mach number from subsonic to transonic causes the maximum growth rate and the dominant wavenumber of the disturbances to increase, and the increase is higher in the presence of the electric field. Liquid compressibility has been found to have a minimal effect on instability. At constant wavenumber and electric field values, the growth rate of disturbances increases as the gas Mach number tends to 1, and then begins to decrease with further increase in the gas Mach number. At small values of wavenumber, antisymmetrical disturbances grow faster than symmetrical ones, while the growth rate of both types of disturbances approach each other at large wavenumbers, which increases by increasing the electric field values. At small Weber numbers, antisymmetrical disturbances exhibit a higher maximum growth rate and a lower dominant wavenumber than symmetrical disturbances. However, the maximum growth rate and dominant wavenumber of the two types of disturbances are almost identical when both Weber number and electric field values become large. An increase in the gas to liquid density ratio enhances the instability, and this effect is enhanced for high electric field values. Surface tension and electric fields always oppose and increase the development of instability, respectively; and they have opposite effects for long wavelengths and high Weber numbers.     

The inviscid compressible Görtler problem in three-dimensional boundary layers

In this paper we investigate numerical solutions for the growth rates of Görtler vortices in a compressible three-dimensional flow in the inviscid limit of a large Görtler number. We look at a range of Mach numbers and find that there are three different types of behaviour for the mode growth-rate, corresponding to whether the flow is incompressible, has a Mach number small enough so that temperature-adjustment-layer modes do not appear in the two-dimensional case, or has a Mach number large enough so that they do. We find that it takes a considerably greater crossflow to destroy the Görtler vortices for moderate Mach numbers than it did in the incompressible case looked at by Bassom and Hall (1991). From this we believe that Görtler vortices may well still be a cause of transition for many practical compressible inviscid three-dimensional flows.Support for the author from SERC is acknowledged.     

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.     

On the inviscid acoustic-mode instability of supersonic shear flows

The same methods used previously to study acoustic-mode instability in supersonic boundary layers are applied to free shear layers, and new calculations are made for boundary layers with cooling and suction. The objective is to obtain additional information about acoustic-mode instability, and to find what features of the instability are common to boundary layers and free shear flows. Acoustic modes exist whenever there is an embedded region of locally supersonic flow relative to the phase speed of the instability wave. Consequently, they can be found in boundary layers, wakes, and jets, but not in mixing layers unless the flow is confined. In this first part of a two-part paper, attention is directed principally to two-dimensional waves. The linear, inviscid stability theory is used to calculate spatial amplification rates at Mach number 3 for the sinuous and varicose modes of a single wake flow and a single jet flow, each made up of the same mixing-layer profile plus a central region of uniform flow. Along with sequences of sinuous and varicose unstable modes clearly identifiable as acoustic modes, both of these flows, unlike the boundary layer, have a lowest sinuous mode that is the most unstable. The unstable modes include both subsonic and radiating disturbances with large amplification rates. The latter phenomenon is also found for highly cooled boundary layers with suction. In these boundary layers, suction is generally stabilizing for nonradiating acoustic disturbances, but destabilizing for radiating disturbances.The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Support from the Aerodynamics Division of the Office of Aeronautics and Exploration Technology is gratefully acknowledged. A preliminary version of this paper was presented at the Fourth Symposium on Numerical and Physical Aspects of Aerodynamic Flows, California State University, Long Beach, CA, 16–19 January 1989.     

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.     

Linear instability in the near wake of a symmetrical junction of two channels

The growth of linear disturbances in the high-Reynolds number laminar wake of a symmetrical junction of two channels is investigated. Disturbances in the near wake respond according to the Rayleigh equation which is solved numerically for a spatial stability problem. The unperturbed flow is studied in the same way as Goldstein's analysis. The results of this analysis allow for the basic profiles at different streamwise positions to be numerically obtained as solutions of the wake boundary-layer equations.