Stability Of Compressible Boundary Layers Research Articles

Effect of flow–thermodynamics interactions on the stability of compressible boundary layers: insights from Helmholtz decomposition

Helmholtz decomposition of velocity perturbations is performed in conjunction with linear stability analysis to examine the effects of flow-thermodynamics interactions on the stability of high-speed boundary layers. A corresponding decomposition of the pressure field is also proposed. The contributions of perturbation solenoidal kinetic, dilatational kinetic and internal energy to the various instability modes are examined as a function of Mach number ( $M$ ). As expected, dilatational and pressure field effects play an insignificant part in the first-mode behaviour at all Mach numbers. The second (Mack) mode, however, is dominantly dilatational in nature, and perturbation internal energy is significant compared to perturbation kinetic energy. The observed behaviour is explicated by examining the key processes of production and pressure-dilatation. Production of the second-mode dilatational kinetic energy is mostly due to the solenoidal-dilatational covariance stress tensor interacting with the mean (background) velocity gradient. This cross production component also inhibits the first mode. The dilatational pressure facilitates energy transfer from the kinetic to the internal field in the near‐wall region, whereas the energy transfer away from the wall is mostly due to the solenoidal pressure work. The dilatational characters of the fast and slow modes are also examined. The fast mode is dominantly dilatational at both $M=4$ and $M=6$ , while the nature of the slow mode is dependent on $M$ . Finally, Helmholtz decomposition of the perturbation momentum vector is performed. Interestingly, both first and second modes are dominated by solenoidal components of momentum.

Prandtl number effects on the hydrodynamic stability of compressible boundary layers: flow–thermodynamics interactions

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ( $M$ ), Prandtl number ( $Pr$ ) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$ ; and $0.5 \leq Pr \leq 1.3$ . For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

Inviscid instability of compressible exponential boundary layer flows

Based on the linearized Euler equations and the normal-mode approach, the Pridmore-Brown equation for acoustics and the stability of parallel shear flows is solved for a boundary layer flow with an exponential velocity profile. The general solution is derived in terms of the confluent Heun function, and in turn, using the boundary conditions of vanishing disturbances at infinity and zero wall-normal velocity, the boundary value problem is converted to an algebraic eigenvalue problem. Solutions to the eigenvalue problem allow a comprehensive picture of the stability of compressible boundary layers. In particular, the complex eigenvalues ω are calculated as a function of the streamwise wavenumber α and the Mach number M. We observe that with growing wavenumbers, the number of eigenvalues increases discretely. Only for M > 1, unstable modes exist, and the boundary between neutrally stable and unstable modes is defined by the transonic line ω = α to be calculated from a degenerated eigenvalue equation. For large wall distances, the eigenfunctions converge exponentially toward zero. The spatial decay rate defines an “acoustic boundary layer thickness” δa, which indicates how far outside modes are still audible. For M > 1 and large α, δa diverges exponentially, i.e., in this parameter range, modes are perceptible even far from the boundary layer. A particularly steep rise of δa is observed when M > 2. Thus, there is a wavenumber range in which ωi and δa are both large and thus generates a particularly strong noise impact. From the eigenfunction for the unstable modes, a strong increase in the amplitude of acoustic waves can be identified in the vicinity of the wall, which indicates an accumulation and saturation of energy and thereafter leads to temporal instability and sound radiation in the free stream.

Dense-gas effects on compressible boundary-layer stability

A study of dense-gas effects on the stability of compressible boundary-layer flows is conducted. From the laminar similarity solution, the temperature variations are small due to the high specific heat of dense gases, leading to velocity profiles close to the incompressible ones. Concurrently, the complex thermodynamic properties of dense gases can lead to unconventional compressibility effects. In the subsonic regime, the Tollmien–Schlichting viscous mode is attenuated by compressibility effects and becomes preferentially skewed in line with the results based on the ideal-gas assumption. However, the absence of a generalized inflection point precludes the sustainability of the first mode by inviscid mechanisms. On the contrary, the viscous mode can be completely stable at supersonic speeds. At very high speeds, we have found instances of radiating supersonic instabilities with substantial amplification rates, i.e. waves that travel supersonically relative to the free-stream velocity. This acoustic mode has qualitatively similar features for various thermodynamic conditions and for different working fluids. This shows that the leading parameters governing the boundary-layer behaviour for the dense gas are the constant-pressure specific heat and, to a minor extent, the density-dependent viscosity. A satisfactory scaling of the mode characteristics is found to be proportional to the height of the layer near the wall that acts as a waveguide where acoustic waves may become trapped. This means that the supersonic mode has the same nature as Mack’s modes, even if its frequency for maximal amplification is greater. Direct numerical simulation accurately reproduces the development of the supersonic mode and emphasizes the radiation of the instability waves.

Boundary-layer stability of supercritical fluids in the vicinity of the Widom line

We investigate the hydrodynamic stability of compressible boundary layers over adiabatic walls with fluids at supercritical pressure in the proximity of the Widom line (also known as the pseudo-critical line). Depending on the free-stream temperature and the Eckert number that determines the viscous heating, the boundary-layer temperature profile can be either sub-, trans- or supercritical with respect to the pseudo-critical temperature, $T_{pc}$. When transitioning from sub- to supercritical temperatures, a seemingly continuous phase change from a compressible liquid to a dense vapour occurs, accompanied by highly non-ideal changes in thermophysical properties. Using linear stability theory (LST) and direct numerical simulations (DNS), several key features are observed. In the sub- and supercritical temperature regimes, the boundary layer is substantially stabilized the closer the free-stream temperature is to $T_{pc}$ and the higher the Eckert number. In the transcritical case, when the temperature profile crosses $T_{pc}$, the flow is significantly destabilized and a co-existence of dual unstable modes (Mode II in addition to Mode I) is found. For high Eckert numbers, the growth rate of Mode II is one order of magnitude larger than Mode I. An inviscid analysis shows that the newly observed Mode II cannot be attributed to Mack’s second mode (trapped acoustic waves), which is characteristic in high-speed boundary-layer flows with ideal gases. Furthermore, the generalized Rayleigh criterion (also applicable for non-ideal gases) unveils that, in contrast to the trans- and supercritical regimes, the subcritical regime does not contain an inviscid instability mechanism.

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.

A minimal composite theory for stability of non-parallel compressible boundary-layer flow

A new “minimal composite” theory that extends the approach of Govindarajan and Narasimha [1] is proposed here for 2D non-parallel compressible boundary-layer stability subject to 3D disturbances. The mean profiles are obtained from Horton’s analysis, which provides a good approximation for a large range of Prandtl numbers at non-zero pressure gradients. In the lowest order, all effects of order lower than O(R-2/3) anywhere in the boundary-layer are included, R being the local boundary-layer Reynolds number; the resulting non-parallel formulation yields a set of four ordinary differential equations, as compared to the five coupled equations of classical parallel flow theory of Mack [2]. The largest effect on stability of flow non-parallelism is found to be due to the wall-normal advection of velocity and temperature disturbance quantities by the mean flow. The present theory shows that the bulk viscosity, invariably included in compressible stability theories, is irrelevant at the lowest order. In comparison with the “full” [O(R-1)] non-parallel theory, the present theory is marginally better than the parallel flow theory.

Computation of Selected Eigenvalues of Generalized Eigenvalue Problems

We examine and develop techniques for obtaining a few selected eigenvalues of the generalized eigenvalue problem Ax = λBx, where A and B are n × n, nonysmmetric, banded complex matrices. One way of obtaining the desired eigenvalues is to use a direct method to compute all the eigenvalues. Direct methods are computationally intensive and destroy the sparsity of the matrices A and B. Iterative methods, on the other hand, maintain the sparsity of the matrices and compute only a few eigenvalues. The iterative algorithms that we consider are the Arnoldi and the Lanczos methods. We use a shift and invert strategy to increase the rate of convergence towards the desired eigenvalues. We compare these two approaches for a model problem, which arises from considering the linear stability of compressible boundary layers and some other test problems. We present a general scheme to compute the eigenvalues lying inside a "box" in the complex plane. We also outline a procedure to seperate the converged eigenvalues from spurious approximations. In addition, this procedure can also improve the approximations to the eigenvalues of interest. Numerical results obtained on a CRAY Y-MP are presented.

Effect of pressure gradient on the stability of compressible boundary layers

An investigation into the influence of pressure gradient (wall shaping) on the stability of compressible boundary layers is presented. The steady, compressible, nonsimilar boundary-layer equations are solved with a potential flow velocity distribution corresponding to a power law edge Mach number distribution. The stability of the mean flow is investigated using the small-disturbance compressible linear stability theory. The results indicate that two-dimensional second-mode (Mack) waves can be stabilised with pressure gradient. However, the effectiveness of the pressure gradient on the natural laminar flow control of two-dimensional second-mode waves decreases at hypersonic speeds. The maximum growth rate varies almost linearly with the pressure gradient level. The effect of pressure gradients on stabilising three-dimensional first-mode waves is much larger than their effect on stabilising two-dimensional first-mode waves.

Effect of heat transfer on the stability of compressible boundary layers

The effect of heat transfer on the spatial stability of compressible boundary layers is investigated using a formulation that accounts for variable-fluid properties of both the mean-flow and stability problems as well as a constant Prandtl number formulation. It is found that heat transfer is more effective in stabilizing or destabilizing flows at low Mach numbers than at high Mach numbers. Cooling always stabilizes first-mode waves and destabilizes second-mode waves. Cooling increases the peak amplifications of second-mode waves and shifts them towards higher frequencies. The least stable second-mode wave remains two-dimensional in the presence of heat transfer. The theoretical results are in good agreement with the experimental data of Lysenko and Maslov [ J. Fluid Mech. 147, 39 (1984)] for the first mode and in reasonable agreement with their experimental data for the second mode. The theoretical results are also compared with the experimental data of Lebiga et al. Archs Mech. 31, 397 (1979)] and Kosinov et al. [ J. Fluid Mech. 219, 621 (1990)].

Analysis of the linear stability of compressible boundary layers using the PSE

The application of linearized parabolic stability equations (PSE) to compressible flow is considered. The effect of mean-flow nonparallelism is found to be weak on 2D waves and strong on 3D waves. Results for a single choice of free-stream parameters that corresponds to the atmospheric conditions at 15,000 m above sea level are presented.

Effect of transverse curvature on the stability of compressible boundary layers

Linear stability theory is used to determine the effect of transverse curvature on the estimated location of the onset of transition for supersonic boundary layers. Results have been obtained for Mach 5 and Mach 1.25 boundary layers formed over cylinders and sharp cones. As transverse curvature increases, the growth rates for the first mode (asymmetric) instability increase, whereas those for the second mode decrease. The overall effect of increasing curvature is to lower the transition Reynolds number at the Mach numbers studied

Remarks on disputed numerical results in compressible boundary-layer stability theory

Past work on the influence of Mach number on the viscous and inviscid instability of flat-plate boundary layers is reviewed, and new spatial calculations are presented. These calculations support the previous view that viscosity is only stabilizing for both two- and three-dimensional first-mode waves above M1=3.0, and for second-mode waves at all Mach numbers. It is concluded that the calculations of Wazzan, Taghavi, and Keltner that show viscous instability at M1=6.0 for first-mode 50° waves, and at M1=3.0 for two-dimensional second-mode waves, are not correct.

Compressible Boundary-Layer Stability Calculations for Sweptback Wings with Suction

The stability of the laminar boundary layers on two transonic wings of infinite span with distributed suction is investigated with the compressible, parallel-flow stability theory. Both wings have supercritical airfoil sections; one has a sweep angle of 23 deg, the other of 35 deg. Zero-frequency disturbances are used to represent cross-flow instability, and disturbances with the wavenumber vector aligned with the local flow direction represent traveling-wave instability. In both cases, the maximum spatial amplification rate is used as a measure of the instability. For the suction, distributions with constant mass flux downstream of the starting point are used. The main objective is to determine how the maximum amplification rate varies with the magnitude and starting point of the suction. It is found for both types of disturbances that the maximum amplification rate varies almost linearly with the suction magnitude up to at least the point where the amplification rate is halved. Different starting locations for the suction in the first 4% of the chord were found to affect cross-flow instability, but to have little influence on traveling-wave instability.

Uniform asymptotic approximations to the solutions of the Dunn-Lin equations

In this paper, we develop a uniform asymptotic theory for the Dunn-Lin equations which govern the stability of compressible boundary layers at moderate Mach numbers. “First approximations” to the solutions are derived when the Prandtl number is equal to unity, in which case the structure of the approximations is especially simple. The rapidly varying parts of the approximations can be expressed in terms of certain generalized Airy functions and the slowly varying parts can be expressed in terms of quantities all but one of which are well-known from the older heuristic theories. The approximations obtained in this way are uniformly valid in a full neighborhood of the turning point. Because of the simplicity of the present theory, it is expected that the techniques developed in this paper can also be applied to other more general stability equations for compressible boundary layers, such as the Lees-Lin equations.

Boundary-Layer Separation on Slender Cones at Angle of Attack

6 McClure, J. D., On Perturbed Boundary Layer Flows, Fluid Dynamic Research Lab. Rept. 62-2, July 1962, MIT, Cambridge, Mass. 7 Lighthill, M. J., Reflection at a Laminar Boundary Layer of a Weak Steady Disturbance to a Supersonic Stream, Neglecting Velocity and Heat Conduction, Quarterly Journal of Mechanics and Applied Mathematics, Vol. 3, No. 3, March 1950, pp. 302-325. 8 Lighthill, M. J., On Boundary Layers and Upstream Influence II Supersonic Flows Without Separation, Proceedings of the Royal Society, Vol. 217, 1953, pp. 478-507, 9 Benjamin, T. B., Shearing Flow Over a Wavy Boundary, Journal of Fluid Mechanics, Vol. 6, No. 2, 1959, pp. 161-205. 10 Lees, L. and Reshotko, E., Stability of the Compressible Laminar Boundary Layer, Journal of Fluid Mechanics, Vol. 12, No. 4, 1962, pp. 555-590. 11 Inger, G. R., Discontinuous Supersonic Flow Past an Ablating Wavy Wall, AIAA Journal, Vol. 7, No. 4, April 1969, pp. 762-764. 12 Inger, G. R., Rotational Inviscid Supersonic Flow Past a Wavy Wall, Bulletin of the American Physical Society, Vol. 13, No. 11, Nov. 1968, p. 1579. 13 Martellucci, H., Rie, H., and J. F. Somtowski, J. F., Evaluation of Several Eddy Viscosity Models through Comparison with Measurements in Hypersonic Flows, AIAA Paper 69-688, San Francisco, Calif., 1969. 14 Brown, W. B., Stability of Compressible Boundary Layers, AIAA Journal, Vol. 5, No. 10, Oct. 1967, pp. 1753-1759. 15 Lew, H. G. and Li, H., The Role of the Turbulent Viscous Sublayer in the Formation of Surface Patterns, Rept. R68SD12, June 1968, General Electric Missle and Space Div., Valley Forge, Pa. 16 Klein, E. J., Liquid Crystals in Aerodynamic Testing, Astronautics and Aeronautics, Vol. 6, No. 7, July 1968, pp. 20-25. 17 Williams, E. P. and Inger, G. R., Investigations of Ablation Surface Cross Hatching, SAMSO TR-70-246, June 1970, McDonnell Douglas Astronautics, Huhtington Beach, Calif.

Stability of compressible boundary layers.

with emphasis on the structure of the jet and separation ahead of the jet, Separated Flow, Part 2, Proceedings of a Specialists Meeting, AGARD Fluid Dynamics Panel (AGARD, Paris, France 1966), pp. 667-700. 16 Needham, D. A. and Stallery, J. L., Boundary layer separation in-hypersonic flow, AIAA Paper 66-455 (1966). 17 Maurer, F., Interaction effects of lateral control jets expanding into a supersonic flow, Deutsche Versuchsanstalt fur Luftund Raumfahrt, Munich, Germany Rept. 65-04 (1965). 18 Wu, J.-M., personal communication, based on R. E. Wilson, An experimental investigation of separated turbulent supersonic flow at M = 3.0, M.S. Thesis, Univ. of Tennessee (June 5, 1966). 19 Reshotko, E. and Tucker, M., Effect of a discontinuity on turbulent boundary-layer-thickness parameters with application to shock-induced separation, NACA TN 3454 (May 1955). 20 Mager, A., On the model of the free shock-separated turbulent boundary layer, J. Aeronaut. Sci. 23, 181-184 (1956). 21 Zukoski, E. E., Review of data concerning turbulent boundary layer separation in front of a forward facing step, Guggenheim Jet Propulsion Center, California Institute of Technology, Pasadena, Calif., Rept. 66-8-15-1 (August 1966).

Transition Reversal and Tollmien—Schlichting Instability

Transition reversal—the sometimes observed decrease in transition Reynolds number with surface cooling—has been thought to be other than a Tollmien—Schlichting instability phenomenon since it could not be reconciled with the results of the Dunn and Lin calculations of the temperatures required for complete stabilization of supersonic flat plate boundary layers. Recent theoretical developments in the stability of compressible boundary layers to Tollmien—Schlichting waves, however, point out the importance of considering in the analysis the hitherto neglected temperature fluctuations and thermal boundary condition. Accordingly, the temperatures required for complete stabilization of flat plate boundary layers to both two-and three-dimensional disturbances have been recomputed. The new required temperatures are all below the Dunn—Lin values. In the usual diagram of surface temperature vs stream Mach number there are two loops of complete stability. The trends suggested by this diagram tend to substantiate the experimentally observed transition reversal phenomenon. The results are compared with available wind tunnel and flight data.

Stability of Compressible Boundary Layers Induced by a Moving Wave

Summary The problem of determining the stability of compressible vis­ cous flows with nonzero surface velocities is formulated and is shown to be identical to that for conventional boundary layers, with only a redefinition of the Mach and Reynolds numbers re­ quired. Specific consideration is given to the wall boundary layer behind a moving shock wave, and the minimum critical Reynolds numbers are obtained for various shock velocities. The entire stability map is determined for the limiting case of a weak wave, which is analogous to the Rayleigh problem. The minimum critical Reynolds number is found to increase monotonically with shock velocity—i.e., with increasing surface cooling and stream Mach number combined. For the ratio of wall to stream velocity of 2.92 with y = 1.4 (shock Mach number of 2.18) the flow is found to be infinitely stable to two-dimension al disturbances. Experimental transition data do not follow the trends predicted by the theory. In fact, the transition Reynolds numbers are orders of magnitude below the computed minimum critical Reynolds numbers. The lack of correlation between theory and experiment is attributed to disturbances which are external to the boundary layer.