Linear and nonlinear receptivity of the boundary layer in transonic flows

2015 ◽  
Vol 786 ◽  
pp. 154-189 ◽  
Author(s):  
A. I. Ruban ◽  
T. Bernots ◽  
M. A. Kravtsova
Keyword(s):  
Boundary Layer ◽  
Acoustic Noise ◽  
Neutral Curve ◽  
Wing Surface ◽  
Roughness Height ◽  
Stokes Layer ◽  
Instability Modes ◽  
Direct Numerical Method ◽  
Tollmien Schlichting ◽  
The Stability

In this paper we analyse the process of the generation of Tollmien–Schlichting waves in a laminar boundary layer on an aircraft wing in the transonic flow regime. We assume that the boundary layer is exposed to a weak acoustic noise. As it penetrates the boundary layer, the Stokes layer forms on the wing surface. We further assume that the boundary layer encounters a local roughness on the wing surface in the form of a gap, step or hump. The interaction of the unsteady perturbations in the Stokes layer with steady perturbations produced by the wall roughness is shown to lead to the formation of the Tollmien–Schlichting wave behind the roughness. The ability of the flow in the boundary layer to convert ‘external perturbations’ into instability modes is termed the receptivity of the boundary layer. In this paper we first develop the linear receptivity theory. Assuming the Reynolds number to be large, we use the transonic version of the viscous–inviscid interaction theory that is known to describe the stability of the boundary layer on the lower branch of the neutral curve. The linear receptivity theory holds when the acoustic noise level is weak, and the roughness height is small. In this case we were able to deduce an analytic formula for the amplitude of the generated Tollmien–Schlichting wave. In the second part of the paper we lift the restriction on the roughness height, which allows us to study the flows with local separation regions. A new ‘direct’ numerical method has been developed for this purpose. We performed the calculations for different values of the Kármán–Guderley parameter, and found that the flow separation leads to a significant enhancement of the receptivity process.

Receptivity of the boundary layer to vibrations of the wing surface

2013 ◽  
Vol 723 ◽  
pp. 480-528 ◽  
Author(s):  
A. I. Ruban ◽  
T. Bernots ◽  
D. Pryce
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
Stokes Equations ◽  
Wing Surface ◽  
Main Concern ◽  
Elastic Vibrations ◽  
Stokes Layer ◽  
Characteristic Wavelength ◽  
Tollmien Schlichting ◽  
Pressure Perturbations

AbstractIn this paper we study the generation of Tollmien–Schlichting waves in the boundary layer due to elastic vibrations of the wing surface. The subsonic flow regime is considered with the Mach number outside the boundary layer $M= O(1)$. The flow is investigated based on the asymptotic analysis of the Navier–Stokes equations at large values of the Reynolds number, $\mathit{Re}= {\rho }_{\infty } {V}_{\infty } L/ {\mu }_{\infty } $. Here $L$ denotes the wing section chord; and ${V}_{\infty } $, ${\rho }_{\infty } $ and ${\mu }_{\infty } $ are the free stream velocity, air density and dynamic viscosity, respectively. We assume that in the spectrum of the wing vibrations there is a harmonic that comes in to resonance with the Tollmien–Schlichting wave on the lower branch of the stability curve; this happens when the frequency of the harmonic is a quantity of the order of $({V}_{\infty } / L){\mathit{Re}}^{1/ 4} $. The wavelength, $\ell $, of the elastic vibrations of the wing is assumed to be $\ell \sim L{\mathit{Re}}^{- 1/ 8} $, which has been found to represent a ‘distinguished limit’ in the theory. Still, the results of the analysis are applicable for $\ell \gg L{\mathit{Re}}^{- 1/ 8} $ and $\ell \ll L{\mathit{Re}}^{- 1/ 8} $; the former includes an important case when $\ell = O(L)$. We found that the vibrations of the wing surface produce pressure perturbations in the flow outside the boundary layer, which can be calculated with the help of the ‘piston theory’, which remains valid provided that the Mach number, $M$, is large as compared to ${\mathit{Re}}^{- 1/ 4} $. As the pressure perturbations penetrate into the boundary layer, a Stokes layer forms on the wing surface; its thickness is estimated as a quantity of the order of ${\mathit{Re}}^{- 5/ 8} $. When $\ell = O({\mathit{Re}}^{- 1/ 8} )$ or $\ell \gg {\mathit{Re}}^{- 1/ 8} $, the solution in the Stokes layer appears to be influenced significantly by the compressibility of the flow. The Stokes layer on its own is incapable of producing the Tollmien–Schlichting waves. The reason is that the characteristic wavelength of the perturbation field in the Stokes layer is much larger than that of the Tollmien–Schlichting wave. However, the situation changes when the Stokes layer encounters a wall roughness, which are plentiful in real aerodynamic flows. If the longitudinal extent of the roughness is a quantity of the order of ${\mathit{Re}}^{- 3/ 8} $, then efficient generation of the Tollmien–Schlichting waves becomes possible. In this paper we restrict our attention to the case when the Stokes layer interacts with an isolated roughness. The flow near the roughness is described by the triple-deck theory. The solution of the triple-deck problem can be found in an analytic form. Our main concern is with the flow behaviour downstream of the roughness and, in particular, with the amplitude of the generated Tollmien–Schlichting waves.

Transition of streamwise streaks in zero-pressure-gradient boundary layers

2002 ◽  
Vol 472 ◽  
pp. 229-261 ◽  
Author(s):  
LUCA BRANDT ◽  
DAN S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
High Speed ◽  
Plate Boundary ◽  
Transition Process ◽  
Free Stream Turbulence ◽  
Tollmien Schlichting ◽  
Streamwise Streaks ◽  
The Stability ◽  
Optimal Disturbances

A transition scenario initiated by streamwise low- and high-speed streaks in a flat-plate boundary layer is studied. In many shear flows, the perturbations that show the highest potential for transient energy amplification consist of streamwise-aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. In a previous investigation (Andersson et al. 2001), the stability of these streaks in a zero-pressure-gradient boundary layer was studied by means of Floquet theory and numerical simulations. The sinuous instability mode was found to be the most dangerous disturbance. We present here the first simulation of the breakdown to turbulence originating from the sinuous instability of streamwise streaks. The main structures observed during the transition process consist of elongated quasi-streamwise vortices located on the flanks of the low-speed streak. Vortices of alternating sign are overlapping in the streamwise direction in a staggered pattern. The present scenario is compared with transition initiated by Tollmien–Schlichting waves and their secondary instability and by-pass transition initiated by a pair of oblique waves. The relevance of this scenario to transition induced by free-stream turbulence is also discussed.

scholarly journals On the long-wave instability of natural-convection boundary layers

1997 ◽  
Vol 335 ◽  
pp. 57-73 ◽  
Author(s):  
P. G. DANIELS ◽  
JOHN C. PATTERSON
Keyword(s):  
Boundary Layer ◽  
Travelling Waves ◽  
Vertical Wall ◽  
Phase Speed ◽  
Neutral Curve ◽  
Neutral Stability ◽  
Layer Width ◽  
Dimensional Boundary ◽  
Prandtl Numbers ◽  
The Stability

This paper considers the stability of the one-dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi-steady approximation is used to analyse the asymptotic form of the lower branch of the neutral curve, corresponding to disturbances of wavelength much greater than the boundary-layer width. This leads to predictions of the critical wavenumber for neutral stability and the maximum phase speed of the travelling waves. Results are obtained for a range of Prandtl numbers and are compared with solutions of the full stability equations and with numerical simulations and experimental observations of cavity flows driven by sudden heating of the sidewalls.

Development and use of a vane device for boundary-layer measurements

1959 ◽  
Vol 6 (1) ◽  
pp. 97-112 ◽  
Author(s):  
J. G. Burns ◽  
W. H. J. Childs ◽  
A. A. Nicol ◽  
M. A. S. Ross
Keyword(s):  
Boundary Layer ◽  
Wind Tunnel ◽  
Boundary Layers ◽  
Free Stream ◽  
Neutral Curve ◽  
Approximate Theory ◽  
Electrical System ◽  
Natural Oscillations ◽  
Free Stream Turbulence ◽  
Tollmien Schlichting

A hinged vane and a sensitive electrical system for recording the motion of the vane have been developed for the observation of fluctuating y-components of velocity in boundary layers. An approximate theory of the natural oscillations of such vanes is presented and experimentally verified. Using vanes as resonant detectors, meassurements have been made of oscillations injected into the laminar boundary layer on a flat plate in a wind tunnel with 0·3% free-stream turbulence. Points on the neutral Tollmien-Schlichting curve have thereby been obtained which lie close to the theoretical neutral curve.

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.

scholarly journals Global stability of swept flow around a parabolic body: the neutral curve

2011 ◽  
Vol 678 ◽  
pp. 589-599 ◽  
Author(s):  
CHRISTOPH J. MACK ◽  
PETER J. SCHMID
Keyword(s):  
Boundary Layer ◽  
Leading Edge ◽  
Neutral Curve ◽  
Parameter Study ◽  
Global Approach ◽  
Complex Flow ◽  
Unstable Boundary Layer ◽  
Critical Reynolds Numbers ◽  
Acoustic Instabilities ◽  
The Stability

The onset of transition in the leading-edge region of a swept blunt body depends crucially on the stability characteristics of the flow. Modelling this flow configuration by swept compressible flow around a parabolic body, a global approach is taken to extract pertinent stability information via a DNS-based iterative eigenvalue solver. Global modes combining features from boundary-layer and acoustic instabilities are presented. A parameter study, varying the spanwise disturbance wavenumber and the sweep Reynolds number, showed the existence of unstable boundary-layer and acoustic modes. The corresponding neutral curve displays two overlapping regions of exponential growth and two critical Reynolds numbers, one for boundary-layer instabilities and one for acoustic instabilities. The employed global approach establishes a first neutral curve, delineating stable from unstable parameter configurations, for the complex flow about a swept parabolic body with corresponding implications for swept leading-edge flow.

The upper branch stability of the Blasius boundary layer, including non-parallel flow effects

1981 ◽  
Vol 375 (1760) ◽  
pp. 65-92 ◽  
Keyword(s):  
Boundary Layer ◽  
Parallel Flow ◽  
Neutral Stability ◽  
Relative Order ◽  
Numerical Work ◽  
Blasius Boundary Layer ◽  
Stability Structure ◽  
Tollmien Schlichting ◽  
Flow Effects ◽  
The Stability

The stability of the Blasius boundary layer is studied theoretically, with the aim of fixing the character of the upper branch of the neutral stability curve(s) and its dependence on non-parallel flow effects. Unlike most previous studies this work has a rational basis since, throughout, we consider the linear stability structure for asymptotically large Reynolds numbers ( Re ). The structure is five-zoned and quite complicated, more so than the structure (discussed in Smith (1979 a )) governing the lower branch stability properties, but nevertheless it lends itself to the systematic determination of the neutral frequency and of the influence of non-parallelism. The four leading terms in the asymptotic expansion of the neutral frequency are determined and then the non-parallel flow effects are considered. The latter are shown to be of relative order Re -3/10 in general, much larger than the relative order Re -1/2 suggested by the parallel flow approximations used extensively in the literature. The cause of this discrepancy lies partly in the relatively large wavelength of the Tollmien-Schlichting modes but, more especially, in a ‘transmission feature’, associated with the stability structure and brought about by the major determining role played by the small curvature of the boundary layer profile at the critical layer. This transmission feature enables even quite small effects in the disturbance velocity field to produce a much more profound effect in the neutral stability criteria. The results of this study are not inconsistent overall with previous numerical work but they do tend to suggest that linear non-parallel flow stability theory may well explain most of the related experimental observations, even near the critical Reynolds number.

scholarly journals Convective instability in ammonium chloride solution directionally solidified from below

1994 ◽  
Vol 276 ◽  
pp. 163-187 ◽  
Author(s):  
Falin Chen ◽  
Jay W. Lu ◽  
Tsung L. Yang
Keyword(s):  
Critical Rayleigh Number ◽  
Neutral Curve ◽  
Mushy Layer ◽  
Bulk Fluid ◽  
Left Hand ◽  
Instability Modes ◽  
Layer Mode ◽  
Fluid Concentration ◽  
The Stability ◽  
The Right

The stabilities of salt-finger and plume convection, two major flows characterizing the fluid dynamics of NH4Cl solutions cooling from below, are investigated by theoretical and experimental approaches. A linear stability analysis is implemented to study theoretically the onset of salt-finger convection. Special emphasis is placed on the competition between different instability modes. It is found that in most of the cases considered, the neutral curve consists of two separated monotonic branches with a Hopf bifurcation branch in between; the right-hand monotonic branch corresponding to the boundary-layer-mode convection is more unstable than the left-hand monotonic branch corresponding to the mushy-layer mode. We also conducted a series of experiments covering wide ranges of bulk fluid concentration C∞ and bottom temperature TB to study the stability characteristics of plume convection. From the measurement of both temperature and concentration of the interstitial fluid in the mushy layer, we verify that during the progress of solidification the melt in the mush is in a thermodynamic equilibrium state except at the melt/mush interface where most of the solidification occurs. The critical Rayleigh number of the onset of plume convection is found to be Rccm = 1.1 × 107Π* (see (22)), where Π* is the permeability of the mush. This relation is believed to be valid up to supereutectic NH4Cl solutions.

On the stability of a laminar incompressible boundary layer over a flexible surface

1962 ◽  
Vol 13 (4) ◽  
pp. 609-632 ◽  
Author(s):  
Marten T. Landahl
Keyword(s):  
Boundary Layer ◽  
Phase Speed ◽  
Travelling Wave ◽  
Wave Number ◽  
Neutral Stability ◽  
Class A ◽  
Flexible Surface ◽  
Flexible Wall ◽  
Tollmien Schlichting ◽  
The Stability

The stability of small two-dimensional travelling-wave disturbances in an incompressible laminar boundary layer over a flexible surface is considered. By first determining the wall admittance required to maintain a wave of given wave-number and phase speed, a characteristic equation is deduced which, in the limit of zero wall flexibility, reduces to that occurring in the ordinary stability theory of Tollmien and Schlichting. The equation obtained represents a slight and probably insignificant improvement upon that given recently by Benjamin (1960). Graphical methods are developed to determine the curve of neutral stability, as well as to identify the various modes of instability classified by Benjamin as ‘Class A’, ‘Class B’, and ‘Kelvin-Helmholtz’ instability, respectively. Also, a method is devised whereby the optimum combination of surface effective mass, wave speed, and damping required to stabilize any given unstable Tollmien-Schlichting wave can be determined by a simple geometrical construction in the complex wall-admittance plane.What is believed to be a complete physical explanation for the influence of an infinite flexible wall on boundary-layer stability is presented. In particular, the effect of damping in the wall is discussed at some length. The seemingly paradoxical result that damping destabilizes class A waves (i.e. waves of the Tollmien-Schlichting type) is explained by considering the related problem of flutter of an infinite panel in incompressible potential flow, for which damping has the same qualitative effect. It is shown that the class A waves are associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only make the wave amplitude increase to compensate for the lowered energy level. The Kelvin-Helmholtz type of instability will occur when the effective stiffness of the panel is too low to withstand, for all values of the phase speed, the pressure forces induced on the wavy wall.The numerical examples presented show that the increase in the critical Reynolds number that can be achieved with a wall of moderate flexibility is modest, and that some other explanation for the experimentally observed effects of a flexible wall on the friction drag must be considered.

Design and Tests of Wind-Tunnel Sidewalls for Receptivity Experiments on a Swept Wing

2013 ◽  
Vol 390 ◽  
pp. 96-102 ◽  
Author(s):  
D.G. Romano ◽  
P.H. Alfredsson ◽  
A. Hanifi ◽  
R. Örlü ◽  
N. Tillmark ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Wind Tunnel ◽  
Design Procedure ◽  
Swept Wing ◽  
Experimental Campaign ◽  
Instability Modes ◽  
First Results ◽  
Tollmien Schlichting ◽  
Set Up ◽  
Wing Airfoil

This document explains in its first part the design procedure adopted to design the contoured sidewalls of a swept-wing airfoil section mounted in a wind tunnel in order to satisfy the infinite swept-wing approximation. In the second part, the experimental set-up is described as well as the first results of the experimental campaign. The sidewalls are shown to play their role properly and satisfactorily provide the infinite swept-wing conditions required for subsequent investigations of the most important vortex receptivity mechanisms responsible for excitation of crossflow and Tollmien-Schlichting instability modes in the airfoil boundary layer.

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