Investigations of stability and transition in rotating boundary layers.
Rotating Ekman boundary layer stability and transition for two-dimensional roll vortices superimposed upon basic boundary layer flow
BOUNDARY LAYER TRANSITION↗
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Nonparallel stability of boundary layers
The asymptotic formulations of the nonparallel linear stability of incompressible growing boundary layers are critically reviewed. These formulations can be divided into two approaches. The first approach combines a numerical method with either the method of multiple scales, or the method of averaging, of the Wentzel-Kramers-Brillouin (WKB) approximation; all these methods yield the same result. The second approach combined a multi-structure theory with the method of multiple scales. The first approach yields results that are in excellent agreement with all available experimental data, including the growth rates as well as the neutral stability curve. The derivation of the linear stability of the incompressible growing boundary layers is explained.
Nayfeh, Ali H.↗
Transition in hypersonic boundary layers
Linear stability theory for hypersonic boundary layers is presented. The theory is used to study the effects of real gas (under the assumption of local chemical equilibrium) and small nose bluntness on hypersonic boundary layer stability. It is found that chemical reactions have a stabilizing effect on the first mode instability and a destabilizing effect on the second mode instability in hypersonic boundary layers. There is also a tendency for the second-mode instability to shift to lower frequencies. The effect of small nose bluntness is found to be stabilizing.
Malik, Mujeeb R.↗
On the stability of the decelerating laminar boundary layer
The stability of a decelerating boundary-layer flow is investigated experimentally and numerically. Experimentally, a flat plate having a Blasius boundary layer is decelerated in an 18 m towing tank. The boundary layer becomes unstable to two-dimensional waves, which break down into three-dimensional patterns, hairpin vortices, and finally turbulent bursts when the vortices lift off the wall. The unsteady boundary-layer equations are solved numerically to generate instantaneous velocity profiles for a range of boundary and initial conditions. A quasi-steady approximation is invoked and the stability of local velocity profiles is determined by solving the Orr-Sommerfeld equation using Chebyshev matrix methods. Comparisons are made between the numerical predictions and the experimentally observed instabilities.
Gad-El-hak, M.↗
Stability of free-convection boundary-layer flows
Stability of free convection boundary layer flow investigated by integration of disturbance differential equations
BOUNDARY LAYER FLOW↗
Stability of boundary layers with porous suction strips: Experiment and theory
Low turbulence tunnel experiments on the stability and transition of 2 D boundary layers on flat plates with and without suction are described. A number of general suction cases are discussed. Test results showed that the maximum stabilization occurred when the suction was moved toward the Branch I neutral point. An analytical study of the stability of two dimensional, incompressible boundary layer flows over plates with suction through porous strips was performed. The mean flow was calculated using linearized triple deck, closed form solutions. The stability results of the triple deck theory are shown to be in good agreement with those of the interacting boundary layers. An analytical optimization scheme for the suction configuration was developd. Numerical calculations were performed corresponding to the experimental configurations. In each case, the theory correctly predicts the experimental results.
Reynolds, G. A.↗
Non-parallel stability of compressible boundary layers
Linear and nonlinear stability of compressible growing boundary layers is studied using parabolized stability equations (PSE). Linear PSE calculations are performed for Mach 1.6 and 4.5 plate-plate flow, and the results are compared with the predictions of the multiple-scales approach. In general, the nonparallel effect appears to be less significant for oblique waves near the lower neutral branch but it progressively becomes important at higher Reynolds numbers near the upper branch. In contrast, the nonparallel effect is more pronounced near the lower branch for two-dimensional first-mode waves. The PSE and multiple-scales results agree for the first mode waves, but in the first-second mode transition region, the latter approach tends to break down. Comparison with the first (oblique) and second mode growth rate data from Kendall's (1967) experiment shows good agreement; however, the peak second mode growth rate is over-predicted. Similar conclusions are drawn for the second mode experiment of Stetson et al. (1983) for Mach 8 flow past a sharp cone. We conjecture that the lower experimental growth rate is due to nonlinear saturation and provide supporting calculations.
Chang, Chau-Lyan↗
Linear and nonlinear PSE for compressible boundary layers
Compressible stability of growing boundary layers is studied by numerically solving the partial differential equations under a parabolizing approximation. The resulting parabolized stability equations (PSE) account for nonparallel as well as nonlinear effects. Evolution of disturbances in compressible flat-plate boundary layers are studied for freestream Mach numbers ranging from 0 to 4.5. Results indicate that the effect of boundary-layer growth is important for linear disturbances. Nonlinear calculations are performed for various Mach numbers. Two-dimensional nonlinear results using the PSE approach agree well with those from direct numerical simulations using the full Navier-Stokes equations while the required computational time is less by an order of magnitude. Spatial simulation using PSE were carried out for both the fundamental and subharmonic type breakdown for a Mach 1.6 boundary layer. The promising results obtained show that the PSE method is a powerful tool for studying boundary-layer instabilities and for predicting transition over a wide range of Mach numbers.
Chang, Chau-Lyan↗
Development of a laminar boundary layer behind a suction point
A theoretical investigation is made of the development of a laminar boundary layer behind a suction slot that is assumed to cut off part of the boundary layer without exerting any sink effect. The development, which is approximate, is based on the heat conduction equation. The heat conduction equation enters the analysis through a linearization of the Prandtl-Mises form of the boundary-layer equation.
EQUATIONS - BOUNDARY LAYER↗
Three-dimensional stability of boundary layers
The most recent efforts on the stability and transition of three-dimensional flows are reviewed. These include flows over swept wings, rotating disks, rotating cones, yawed bodies, corners, and attachment lines. The generic similarities of their stability behavior is discussed. It is shown that the breakdown process is very complex, often leading to contradictory results. Particular attention is paid to opposing observations of stationary and traveling wave disturbances.
Saric, William S.↗
Numerical stability of boundary layers with massive blowing.
A basis is provided for examining the stability of so-called 'shooting methods' and to examine the stability of the method successfully applied by Nachtsheim and Green (1970) to boundary-layer flow with large injection and heat transfer. The stability of that method is examined and confirmed.
Liu, T.-M.↗
Effect of roughness on the stability of boundary layers
An analysis is conducted on the effect of imperfections consisting of humps and dips on the stability of incompressible flows over flat plates. The mean flow is calculated using interacting boundary layers. Linear quasiparallel spatial stability is used to calculate the growth rates and mode shapes of two-dimensional disturbances. Then, the amplification factor is computed. A search for the most dangerous frequency is conducted based on an amplification factor of 9 in the shortest distance. Correlations are made with the transition experiment of Walker and Greening using the e sup 9 method.
Nayfeh, Ali H.↗
Boundary-layer linear stability theory
Most fluid flows are turbulent rather than laminar and the reason for this was studied. One of the earliest explanations was that laminar flow is unstable, and the linear instability theory was first developed to explore this possibility. A series of early papers by Rayleigh produced many notable results concerning the instability of inviscid flows, such as the discovery of inflectional instability. Viscosity was commonly thought to act only to stabilize the flow, and flows with convex velocity profiles appeared to be stable. The investigations that led to a viscous theory of boundary layer instability was reported. The earliest application of linear stability theory to transition prediction calculated the amplitude ratio of the most amplified frequency as a function of Reynolds number for a Blasius boundary layer, and found that this quantity had values between five and nine at the observed Ret. The experiment of Schubauer and Skramstad (1947) completely reversed the prevailing option and fully vindicated the Gottingen proponents of the theory. This experiment demonstrated the existence of instability waves in a boundary layer, their connection with transition, and the quantitative description of their behavior by the theory of Tollmien and Schlichting. It is generally accepted that flow parameters such as pressure gradient, suction and heat transfer qualitatively affect transition in the manner predicted by the linear theory, and in particular that a flow predicted to be stable by the theory should remain laminar. The linear theory, in the form of the e9, or N-factor is today in routine use in engineering studies of laminar flow. The stability theory to boundary layers with pressure gradients and suction was applied. The only large body of numerical results for exact boundary layer solutions before the advent of the computer age by calculating the stability characteristics of the Falkner-Skan family of velocity profiles are given. When the digital computer reached a stage of development which permit the direct solution of the primary differential equations, numerical results were obtained from the linear theory during the next 10 years for many different boundary layer flows: three dimensional boundary layers; free convention boundary layers; compressible boundary layers; boundary layers on compliant walls; a recomputation of Falkner-Skan flows; unsteady boundary layers; and heated wall boundary layers.
Leslie M Mack↗
Effects of Nose Bluntness on Stability of Hypersonic Boundary Layers over Blunt Cone
Receptivity and stability of hypersonic boundary layers are numerically investigated for boundary layer flows over a 5-degree straight cone at a free-stream Mach number of 6.0. To compute the shock and the interaction of shock with the instability waves, we solve the Navier-Stokes equations in axisymmetric coordinates. The governing equations are solved using the 5th-order accurate weighted essentially non-oscillatory (WENO) scheme for space discretization and using third-order total-variation-diminishing (TVD) Runge-Kutta scheme for time integration. After the mean flow field is computed, disturbances are introduced at the upstream end of the computational domain. Generation of instability waves from leading edge region and receptivity of boundary layer to slow acoustic waves are investigated. Computations are performed for a cone with nose radii of 0.001, 0.05 and 0.10 inches that give Reynolds numbers based on the nose radii ranging from 650 to 130,000. The linear stability results showed that the bluntness has a strong stabilizing effect on the stability of axisymmetric boundary layers. The transition Reynolds number for a cone with the nose Reynolds number of 65,000 is increased by a factor of 1.82 compared to that for a sharp cone. The receptivity coefficient for a sharp cone is about 4.23 and it is very small, approx.10(exp -3), for large bluntness.
Kara, K.↗
Curvature effects on the stability of laminar boundary layers on swept wings
The stability of the laminar boundary layer on a swept wing is examined. An improved linear stability theory which includes the effects of body and streamline curvature and compressibility is utilized for the calculations. The computed N-factor is correlated with the onset of transition. For this study, only test conditions where transition is due to the growth of highly amplified crossflow instabilities on convex surfaces are examined. The calculations show that the effect of the curvature terms is to dramatically reduce local amplification rates in regions where body and streamline curvature are large. For the cases where transition occurred ahead of the pressure minimum on the upper surface of the wing, the N-factor at transition onset is near 9 when the effects of body and streamline curvature are included in the computations. When the curvature terms are neglected, the average N-factor is about 17. The calculations show that traveling crossflow waves are most amplified.
Collier, F. S., Jr.↗
On the nonlinear stability of a high-speed, axisymmetric boundary layer
The stability of a high-speed, axisymmetric boundary layer is investigated using secondary instability theory and direct numerical simulation. Parametric studies based on the temporal secondary instability theory identify subharmonic secondary instability as a likely path to transition on a cylinder at Mach 4.5. The theoretical predictions are validated by direct numerical simulation at temporally-evolving primary and secondary disturbances in an axisymmetric boundary-layer flow. At small amplitudes of the secondary disturbance, predicted growth rates agree to several significant digits with values obtained from the spectrally-accurate solution of the compressible Navier-Stokes equations. Qualitative agreement persists to large amplitudes of the secondary disturbance. Moderate transverse curvature is shown to significantly affect the growth rate of axisymmetric second mode disturbances, the likely candidates of primary instability. The influence of curvature on secondary instability is largely indirect but most probably significant, through modulation of the primary disturbance amplitude. Subharmonic secondary instability is shown to be predominantly inviscid in nature, and to account for spikes in the Reynolds stress components at or near the critical layer.
Pruett, C. David↗
On the nonlinear stability of a high-speed, axisymmetric boundary layer
The stability of a high-speed, axisymmetric boundary layer is investigated using secondary instability theory and direct numerical simulation. Parametric studies based on the temporal secondary instability theory identify subharmonic secondary instability as a likely path to transition on a cylinder at Mach 4.5. The theoretical predictions are validated by direct numerical simulation at temporally-evolving primary and secondary disturbances in an axisymmetric boundary-layer flow. At small amplitudes of the secondary disturbance, predicted growth rates agree to several significant digits with values obtained from the spectrally-accurate solution of the compressible Navier-Stokes equations. Qualitative agreement persists to large amplitudes of the secondary disturbance. Moderate transverse curvature is shown to significantly affect the growth rate of axisymmetric second mode disturbances, the likely candidates of primary instability. The influence of curvature on secondary instability is largely indirect but most probably significant, through modulation of the primary disturbance amplitude. Subharamonic secondary instability is shown to be predominantly inviscid in nature, and to account for spikes in the Reynolds stress components at or near the critical layer.
Pruett, C. D.↗
Stability of Supersonic Boundary Layers Over Blunt Wedges
Receptivity and stability of supersonic boundary layers over blunt flat plates and wedges are numerically investigated at a free stream Mach number of 3.5 and at a high Reynolds number of 10(exp 6)/inch. Both the steady and unsteady solutions are obtained by solving the full Navier-Stokes equations using the 5th-order accurate weighted essentially non-oscillatory (WENO) scheme for space discretization and using third-order total-variation-diminishing (TVD) Runge-Kutta scheme for time integration. Computations are performed for a flat plate with leading edge thicknesses of 0.0001, 0.001, 0.005 and 0.01 inches that give Reynolds numbers based on the leading edge thickness ranging from 1000 to 10000. Calculations are also performed for a wedge of 10 degrees half angle with different leading edge radii 0.001 and 0.01 inches. The linear stability results showed that the bluntness has a strong stabilizing effect on the stability of two-dimensional boundary layers. The transition Reynolds number for a flat plate with a leading edge thickness of 0.01 inches is about 3.5 times larger than it is for the Blasius boundary layer. It was also revealed that boundary layers on blunt wedges are far more stable than on blunt flat plates.
Balakumar, Ponnampalam↗