The stability of laminar boundary layers at separation

1965 ◽  
Vol 23 (4) ◽  
pp. 737-747 ◽  
Author(s):  
T. H. Hughes ◽  
W. H. Reid
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Adverse Pressure Gradient ◽  
Neutral Curve ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Laminar Boundary Layers ◽  
The Stability ◽  
Limiting Case

The effect of an adverse pressure gradient on the stability of a laminar boundary layer is considered in the limiting case when the skin friction at the wall vanishes, i.e. when U′(0) = 0. Such flows are not absolutely unstable as might have been expected but have a minimum critical Reynolds number of the order of 25. General results are given for the asymptotic behaviour of both the upper and lower branches of the neutral curve and a complete neutral curve is obtained for Pohlhausen's simple fourth-degree polynomial profile at separation.

On the stability of attachment-line boundary layers. Part 2. The effect of leading-edge curvature

1997 ◽  
Vol 333 ◽  
pp. 125-137 ◽  
Author(s):  
RAY-SING LIN ◽  
MUJEEB R. MALIK
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Leading Edge ◽  
Mean Flow ◽  
Eigenvalue Approach ◽  
Curvature Effects ◽  
The Mean ◽  
The Stability ◽  
Attachment Line

The stability of the incompressible attachment-line boundary layer has been studied by Hall, Malik & Poll (1984) and more recently by Lin & Malik (1996). These studies, however, ignored the effect of leading-edge curvature. In this paper, we investigate this effect. The second-order boundary-layer theory is used to account for the curvature effects on the mean flow and then a two-dimensional eigenvalue approach is applied to solve the linear stability equations which fully account for the effects of non-parallelism and leading-edge curvature. The results show that the leading-edge curvature has a stabilizing influence on the attachment-line boundary layer and that the inclusion of curvature in both the mean-flow and stability equations contributes to this stabilizing effect. The effect of curvature can be characterized by the Reynolds number Ra (based on the leading-edge radius). For Ra = 104, the critical Reynolds number R (based on the attachment-line boundary-layer length scale, see §2.2) for the onset of instability is about 637; however, when Ra increases to about 106 the critical Reynolds number approaches the value obtained earlier without curvature effect.

The Effect of Finite Amplitude Disturbance Magnitude on Departures From Laminar Conditions in Impulsively Started and Steady Pipe Entrance Flows

2001 ◽  
Vol 124 (1) ◽  
pp. 235-240 ◽  
Author(s):  
E. A. Moss ◽  
A. H. Abbot
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shape Parameter ◽  
Critical Reynolds Number ◽  
Flow Instability ◽  
Finite Amplitude ◽  
Pipe Flows ◽  
Displacement Thickness ◽  
Layer Flow ◽  
The Stability

The aim of this study was to investigate first departures from laminar conditions in both impulsively started and steady pipe entrance flows. Wall shear stress measurements were conducted of transition in impulsively started pipe flows with large disturbances. These results were reconciled in a framework of displacement thickness Reynolds number and a velocity profile shape parameter, with existing measurements of pipe entrance flow instability, pipe-Poiseuille and boundary layer flow responses to large disturbances, and linear stability predictions. Limiting critical Reynolds number variations for each type of flow were thus inferred, corresponding to the small and gross disturbance limits respectively. Consequently, insights have been provided regarding the effect of disturbance levels on the stability of both steady and unsteady pipe flows.

On the stability of an axisymmetric plume in a uniform stream

1984 ◽  
Vol 142 ◽  
pp. 171-186 ◽  
Author(s):  
D. S. Riley ◽  
M. Tveitereid
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Flow Analysis ◽  
Neutral Curve ◽  
Forced Flow ◽  
Thermal Plume ◽  
Lower Branch ◽  
Stabilizing Effect ◽  
The Stability ◽  
Uniform Stream

The linear stability equations for a round laminar thermal plume in a coflowing vertical stream have been solved numerically. Both symmetric and asymmetric disturbances have been considered for strengths of the forced flow varying between very weak and very strong. The parallel flow analysis confirms that the forced flow has a stabilizing effect. The upper branch of the neutral curve for sinuous disturbances is qualitatively like that of a round momentum jet. However, neither a critical Reynolds number nor a lower branch of the neutral curve was found. Non-parallel effects are discussed.

Separation in oscillating laminar boundary-layer flows

1971 ◽  
Vol 47 (1) ◽  
pp. 21-31 ◽  
Author(s):  
R. A. Despard ◽  
J. A. Miller
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Experimental Investigation ◽  
Reynolds Number ◽  
Reverse Flow ◽  
Oscillation Amplitude ◽  
Hot Wire ◽  
Boundary Layer Flows ◽  
Laminar Boundary Layers ◽  
History Of

The results of an experimental investigation of separation in oscillating laminar boundary layers is reported. Instantaneous velocity profiles obtained with multiple hot-wire anemometer arrays reveal that the onset of wake formation is preceded by the initial vanishing of shear at the wall, or reverse flow, throughout the entire cycle of oscillation. Correlation of the experimental data indicates that the frequency, Reynolds number and dynamic history of the boundary layer are the dominant parameters and oscillation amplitude has a negligible effect on separation-point displacement.

Pressure gradient effects on the large-scale structure of turbulent boundary layers

2013 ◽  
Vol 715 ◽  
pp. 477-498 ◽  
Author(s):  
Zambri Harun ◽  
Jason P. Monty ◽  
Romain Mathis ◽  
Ivan Marusic
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Amplitude Modulation ◽  
Large Scale ◽  
Outer Region ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Small Scale

AbstractResearch into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more prominently in the outer region not only due to increased Reynolds number (Metzger & Klewicki, Phys. Fluids, vol. 13(3), 2001, pp. 692–701; Hutchins & Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw, J. Fluid Mech., vol. 29, 1967, pp. 625–645; Lee & Sung, J. Fluid Mech., vol. 639, 2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.

An experiment on the stability of hypersonic laminar boundary layers

1960 ◽  
Vol 7 (3) ◽  
pp. 385-396 ◽  
Author(s):  
Anthony Demetriades
Keyword(s):  
Boundary Layer ◽  
Flat Surface ◽  
Hypersonic Boundary Layer ◽  
Natural Disturbances ◽  
Hot Wire ◽  
Amplitude Variation ◽  
Laminar Boundary Layers ◽  
Boundary Layer Instability ◽  
The Stability ◽  
Maximum Amplification

An experimental investigation of the hydrodynamic stability of the laminar hypersonic boundary layer was carried out with the aid of a hot-wire anemometer. The case investigated was that of a flat surface at zero angle of attack and no heat transfer.The streamwise amplitude variation of both natural disturbances and of disturbances artifically excited with a siren mechanism was studied. In both cases it was found that such small fluctuations amplify for certain ranges of frequency and Reynolds number Rθ, and damp for others. The demarcation boundaries for the amplification (instability) zone were found to resemble the corresponding limits of boundary-layer instability at lower speeds. A ‘line of maximum amplification’ of disturbances was also found. The amplification rates and hence the degree of selectivity of the hypersonic layer were found, however, to be considerably lower than those at the lower speeds. The disturbances selected by the layer for maximum amplifications have a wavelength which was estimated to be about twenty times the boundary-layer thickness δ.

Estimating amplitude ratios in boundary layer stability theory: a comparison between two approaches

2001 ◽  
Vol 439 ◽  
pp. 403-412 ◽  
Author(s):  
RAMA GOVINDARAJAN ◽  
R. NARASIMHA
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Small Difference ◽  
Detailed Comparison ◽  
The Other ◽  
Boundary Layer Stability ◽  
Mean Flow ◽  
Amplitude Ratios ◽  
The Stability ◽  
Local Reynolds Number

We first demonstrate that, if the contributions of higher-order mean flow are ignored, the parabolized stability equations (Bertolotti et al. 1992) and the ‘full’ non-parallel equation of Govindarajan & Narasimha (1995, hereafter GN95) are both equivalent to order R−1 in the local Reynolds number R to Gaster's (1974) equation for the stability of spatially developing boundary layers. It is therefore of some concern that a detailed comparison between Gaster (1974) and GN95 reveals a small difference in the computed amplitude ratios. Although this difference is not significant in practical terms in Blasius flow, it is traced here to the approximation, in Gaster's method, of neglecting the change in eigenfunction shape due to flow non-parallelism. This approximation is not justified in the critical and the wall layers, where the neglected term is respectively O(R−2/3) and O(R−1) compared to the largest term. The excellent agreement of GN95 with exact numerical simulations, on the other hand, suggests that the effect of change in eigenfunction is accurately taken into account in that paper.

On the Axisymmetric Vortex Flow Over a Flat Surface

1969 ◽  
Vol 36 (3) ◽  
pp. 614-619 ◽  
Author(s):  
E. W. Schwiderski
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Numerical Study ◽  
Lower Boundary ◽  
Tangential Velocity ◽  
Slow Motion ◽  
Boundary Layer Separation ◽  
Local Boundary ◽  
Self Similar

The numerical study of the interaction of a potential vortex with a stationary surface recently published by Kidd and Farris [1] is extended through a transformation of the boundary-value problem to Volterra integral equations. The new calculations verified the results by Kidd and Farris and improved the bounds of the critical Reynolds number Nc, beyond which no self-similar vortex flows exist, to 5.5 < Nc < 5.6 The breakdown of the self-similar motions develops through an instability in the lower boundary layer, which is indicated by two inflection points in the tangential velocity profile. At the critical Reynolds number the lower inflection point reaches the surface and indicates the beginning of boundary-layer separation in the wake-type flow. If the Stokes linearization is applied, one arrives at a new Stokes paradox. However, this “paradox” can be resolved by correcting the free-stream pressure distortion of the Stokes approximation. The new slow-motion approximation is nonlinear and yields an integral which is also free of the Whitehead paradox. The properties of the new exact solution confirm the novel flow features previously detected in almost self-similar motions, which were constructed by adjustable local boundary-layer approximations.

Governing Parameters of Adverse Pressure Gradient Turbulent Boundary Layers

2018 ◽  
Author(s):  
Yvan Maciel ◽  
Tie Wei ◽  
Ayse G. Gungor ◽  
Mark P. Simens
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Outer Region ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Inertial Parameter ◽  
Velocity Scale ◽  
Gradient Parameter

We perform a careful nondimensional analysis of the turbulent boundary layer equations in order to bring out, without assuming any self-similar behaviour, a consistent set of nondimensional parameters characterizing the outer region of turbulent boundary layers with arbitrary pressure gradients. These nondimensional parameters are a pressure gradient parameter, a Reynolds number (different from commonly used ones) and an inertial parameter. They are obtained without assuming a priori the outer length and velocity scales. They represent the ratio of the magnitudes of two types of forces in the outer region, using the Reynolds shear stress gradient (apparent turbulent force) as the reference force: inertia to apparent turbulent forces for the inertial parameter, pressure to apparent turbulent forces for the pressure gradient parameter and apparent turbulent to viscous forces for the Reynolds number. We determine under what conditions they retain their meaning, depending on the outer velocity scale that is considered, with the help of seven boundary layer databases. We find the impressive result that if the Zagarola-Smits velocity is used as the outer velocity scale, the streamwise evolution of the three ratios of forces in the outer region can be accurately followed with these non-dimensional parameters in all these flows — not just the order of magnitude of these ratios. This cannot be achieved with three other outer velocity scales commonly used for pressure gradient turbulent boundary layers. Consequently, the three new nondimensional parameters, when expressed with the Zagarola-Smits velocity, can be used to follow — in a global sense — the streamwise evolution of the stream-wise mean momentum balance in the outer region. This study provides a clear and consistent framework for the analysis of the outer region of adverse-pressure-gradient turbulent boundary layers.

scholarly journals Non-normal origin of modal instabilities in rotating plane shear flows

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190550
Author(s):  
Sharath Jose ◽  
Rama Govindarajan
Keyword(s):  
Reynolds Number ◽  
Shear Flows ◽  
Critical Reynolds Number ◽  
Flow System ◽  
Neutral Line ◽  
Plane Couette Flow ◽  
Plane Shear ◽  
Perturbation Amplitude ◽  
Fundamental Reason ◽  
The Stability

Small variations introduced in shear flows are known to affect stability dramatically. Rotation of the flow system is one example, where the critical Reynolds number for exponential instabilities falls steeply with a small increase in rotation rate. We ask whether there is a fundamental reason for this sensitivity to rotation. We answer in the affirmative, showing that it is the non-normality of the stability operator in the absence of rotation which triggers this sensitivity. We treat the flow in the presence of rotation as a perturbation on the non-rotating case, and show that the rotating case is a special element of the pseudospectrum of the non-rotating case. Thus, while the non-rotating flow is always modally stable to streamwise-independent perturbations, rotating flows with the smallest rotation are unstable at zero streamwise wavenumber, with the spanwise wavenumbers close to that of disturbances with the highest transient growth in the non-rotating case. The instability critical rotation number scales inversely as the square of the Reynolds number, which we demonstrate is the same as the scaling obeyed by the minimum perturbation amplitude in non-rotating shear flow needed for the pseudospectrum to cross the neutral line. Plane Poiseuille flow and plane Couette flow are shown to behave similarly in this context.

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