scholarly journals Forced convection and sedimentation past a flat plate

1995 ◽  
Vol 294 ◽  
pp. 301-321 ◽  
Author(s):  
Nikolaos A. Pelekasis ◽  
Andreas Acrivos
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Forced Convection ◽  
Concentration Profile ◽  
Similarity Solution ◽  
Particle Concentration ◽  
Free Stream ◽  
Leading Edge ◽  
Sediment Layer ◽  
Free Region

The steady laminar flow of a well-mixed suspension of monodisperse solid spheres, convected steadily past a horizontal flat plate and sedimenting under the action of gravity, is examined. It is shown that, in the limit as Re → ∞ and ∈ → 0, where Re is the bulk Reynolds number and ∈ is the ratio of the particle radius a to the characteristic length scale L, the analysis for determining the particle concentration profile has several aspects in common with that of obtaining the temperature profile in forced-convection heat transfer from a wall to a fluid stream moving at high Reynolds and Prandtl numbers. Specifically, it is found that the particle concentration remains uniform throughout the O(Re−1/2) thick Blasius boundary layer except for two O(∈2/3) thin regions on either side of the plate, where the concentration profile becomes non-uniform owing to the presence of shear-induced particle diffusion which balances the particle flux due to convection and sedimentation. The system of equations within this concentration boundary layer admits a similarity solution near the leading edge of the plate, according to which the particle concentration along the top surface of the plate increases from its value in the free stream by an amount proportional to X5/6, with X measuring the distance along the plate, and decreases in a similar fashion along the underside. But, unlike the case of gravity settling on an inclined plate in the absence of a bulk flow at infinity considered earlier (Nir & Acrivos 1990), here the concentration profile remains continuous everywhere. For values of X beyond the region near the leading edge, the particle concentration profile is obtained through the numerical solution of the relevant equations. It is found that, as predicted from the similarity solution, there exists a value of X at which the particle concentration along the top side of the plate attains its maximum value ϕm and that, beyond this point, a stagnant sediment layer will form that grows steadily in time. This critical value of X is computed as a function of ϕs, the particle volume fraction in the free stream. In contrast, but again in conformity with the similarity solution, for values of X sufficiently far removed from the leading edge along the underside of the plate, a particle-free region is predicted to form adjacent to the plate. This model, with minor modifications, can be used to describe particle migration in other shear flows, as, for example, in the case of crossflow microfiltration.

Receptivity to free-stream vorticity of flow past a flat plate with elliptic leading edge

2010 ◽  
Vol 653 ◽  
pp. 245-271 ◽  
Author(s):  
L.-U. SCHRADER ◽  
L. BRANDT ◽  
C. MAVRIPLIS ◽  
D. S. HENNINGSON
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Free Stream ◽  
Three Dimensional ◽  
Low Frequency ◽  
Leading Edge ◽  
Sound Waves ◽  
Streamwise Vorticity ◽  
Low Frequencies ◽  
Free Stream Turbulence

Receptivity of the two-dimensional boundary layer on a flat plate with elliptic leading edge is studied by numerical simulation. Vortical perturbations in the oncoming free stream are considered, impinging on two leading edges with different aspect ratio to identify the effect of bluntness. The relevance of the three vorticity components of natural free-stream turbulence is illuminated by considering axial, vertical and spanwise vorticity separately at different angular frequencies. The boundary layer is most receptive to zero-frequency axial vorticity, triggering a streaky pattern of alternating positive and negative streamwise disturbance velocity. This is in line with earlier numerical studies on non-modal growth of elongated structures in the Blasius boundary layer. We find that the effect of leading-edge bluntness is insignificant for axial free-stream vortices alone. On the other hand, vertical free-stream vorticity is also able to excite non-modal instability in particular at zero and low frequencies. This mechanism relies on the generation of streamwise vorticity through stretching and tilting of the vertical vortex columns at the leading edge and is significantly stronger when the leading edge is blunt. It can thus be concluded that the non-modal boundary-layer response to a free-stream turbulence field with three-dimensional vorticity is enhanced in the presence of a blunt leading edge. At high frequencies of the disturbances the boundary layer becomes receptive to spanwise free-stream vorticity, triggering Tollmien–Schlichting (T-S) modes and receptivity increases with leading-edge bluntness. The receptivity coefficients to free-stream vortices are found to be about 15% of those to sound waves reported in the literature. For the boundary layers and free-stream perturbations considered, the amplitude of the T-S waves remains small compared with the low-frequency streak amplitudes.

The generation of Tollmien-Schlichting waves by free-stream turbulence

1996 ◽  
Vol 312 ◽  
pp. 341-371 ◽  
Author(s):  
P. W. Duck ◽  
A. I. Ruban ◽  
C. N. Zhikharev
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Free Stream ◽  
Stokes Equations ◽  
Leading Edge ◽  
Wave Generation ◽  
Generation Process ◽  
Viscous Boundary Layer ◽  
Free Stream Turbulence ◽  
Tollmien Schlichting

The phenomenon of Tollmien-Schlichting wave generation in a boundary layer by free-stream turbulence is analysed theoretically by means of asymptotic solution of the Navier-Stokes equations at large Reynolds numbers (Re → ∞). For simplicity the basic flow is taken to be the Blasius boundary layer over a flat plate. Free-stream turbulence is taken to be uniform and thus may be represented by a superposition of vorticity waves. Interaction of these waves with the flat plate is investigated first. It is shown that apart from the conventional viscous boundary layer of thickness O(Re−1/2), a ‘vorticity deformation layer’ of thickness O(Re−1/4) forms along the flat-plate surface. Equations to describe the vorticity deformation process are derived, based on multiscale asymptotic techniques, and solved numerically. As a result it is shown that a strong singularity (in the form of a shock-like distribution in the wall vorticity) forms in the flow at some distance downstream of the leading edge, on the surface of the flat plate. This is likely to provoke abrupt transition in the boundary layer. With decreasing amplitude of free-stream turbulence perturbations, the singular point moves far away from the leading edge of the flat plate, and any roughness on the surface may cause Tollmien-Schlichting wave generation in the boundary layer. The theory describing the generation process is constructed on the basis of the ‘triple-deck’ concept of the boundary-layer interaction with the external inviscid flow. As a result, an explicit formula for the amplitude of Tollmien-Schlichting waves is obtained.

The effect of buoyancy forces on the boundary-layer flow over a semi-infinite vertical flat plate in a uniform free stream

1969 ◽  
Vol 35 (3) ◽  
pp. 439-450 ◽  
Author(s):  
J. H. Merkin
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Numerical Method ◽  
Boundary Layer Flow ◽  
Free Stream ◽  
Leading Edge ◽  
Series Solutions ◽  
Buoyancy Forces ◽  
Layer Flow ◽  
Vertical Flat Plate

The boundary-layer flow over a semi-infinite vertical flat plate, heated to a constant temperature in a uniform free stream, is discussed in the two cases when the buoyancy forces aid and oppose the development of the boundary layer. In the former case, two series solutions are obtained, one of which is valid near the leading edge and the other is valid asymptotically. An accurate numerical method is used to describe the flow in the region where the series are not valid. In the latter case, a series, valid near the leading edge is obtained and it is extended by a numerical method to the point where the boundary layer is shown to separate.

Evolution of the Laminar Boundary Layer Over a Flat Plate Under a Free Stream Turbulence Intensity of 1.3%

2008 ◽  
Author(s):  
E. J. Walsh ◽  
F. Brighenti ◽  
D. M. McEligot
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Laminar Boundary Layer ◽  
Turbulence Intensity ◽  
Free Stream ◽  
Early Stage ◽  
Turbine Blades ◽  
Leading Edge ◽  
Bypass Transition ◽  
Free Stream Turbulence

The evolution of the laminar boundary layer over a flat plate under a free stream turbulence intensity of 1.3% is analysed. The effect of free stream turbulence on the onset of transition is one of the important sources leading to bypass transition. Such disturbances are of great interest in engineering for the prediction of transition on turbine blades. The study concentrates on the early part of the boundary layer, starting from the leading edge, and is characterised by the presence of streamwise elongated regions of high and low streamwise velocity. It is demonstrated that the so called “Klebanoff modes” are not entirely representative of the flow structures, due to the time-averaged representations used in most studies. For the conditions of this investigation it is found that the urms and the peak disturbances remain constant in the early stages of the transition development. This region, in which the streaks strength is constant, is problematic for many theories as it is not known where on a surface to initiate a growth theory calculation, and hence the prediction of transition onset is difficult. The observation that a constant urms region exists within the boundary layer under these conditions may be the source of great difficulty in predicting transition onset under turbulence levels around 1%. This region suggests that the streaks are either continuously generated and damped, or do not grow during the early stage of transition, and highlights the importance of continuous influence of the free stream turbulence along the boundary layer edge. This work concludes that the first is more likely, and furthermore the measurements are shown to agree with recent direct numerical simulations.

Linear stability of a gas boundary layer flowing past a thin liquid film over a flat plate

2001 ◽  
Vol 436 ◽  
pp. 321-352 ◽  
Author(s):  
NIKOLAOS A. PELEKASIS ◽  
JOHN A. TSAMOPOULOS
Keyword(s):  
Boundary Layer ◽  
Liquid Film ◽  
Flat Plate ◽  
Free Stream ◽  
Leading Edge ◽  
Critical Value ◽  
Stream Velocity ◽  
Absolute Instability ◽  
Interfacial Waves ◽  
The Family

The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water film that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, r˙, compared to the free-stream velocity, U∞, and (c) small film thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid film like x3/4; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the film. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the triple-deck regime. Two distinct families of eigenvalues are identified, one corresponding to the well-known Tollmien–Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air–water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We−1) increases, indicating the increasing effect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We−1 ≈ We−1c, beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr−1), indicating the increasing effect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr−1 ≈ Fr−1c, beyond which the family of interfacial waves becomes convectively unstable. As We−1 and Fr−1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.

scholarly journals Boundary-layer receptivity for a parabolic leading edge. Part 2. The small-Strouhal-number limit

1997 ◽  
Vol 353 ◽  
pp. 205-220 ◽  
Author(s):  
P. W. HAMMERTON ◽  
E. J. KERSCHEN
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Acoustic Waves ◽  
Asymptotic Theory ◽  
Free Stream ◽  
Leading Edge ◽  
Numerical Results ◽  
Two Dimensional ◽  
Mean Flow ◽  
The Mean

In Hammerton & Kerschen (1996), the effect of the nose radius of a body on boundary-layer receptivity was analysed for the case of a symmetric mean flow past a two-dimensional body with a parabolic leading edge. A low-Mach-number two-dimensional flow was considered. The radius of curvature of the leading edge, rn, enters the theory through a Strouhal number, S=ωrn/U, where ω is the frequency of the unsteady free-stream disturbance and U is the mean flow speed. Numerical results revealed that the variation of receptivity for small S was very different for free-stream acoustic waves propagating parallel to the mean flow and those free-stream waves propagating at an angle to the mean flow. In this paper the small-S asymptotic theory is presented. For free-stream acoustic waves propagating parallel to the symmetric mean flow, the receptivity is found to vary linearly with S, giving a small increase in the amplitude of the receptivity coefficient for small S compared to the flat-plate value. In contrast, for oblique free-stream acoustic waves, the receptivity varies with S1/2, leading to a sharp decrease in the amplitude of the receptivity coefficient relative to the flat-plate value. Comparison of the asymptotic theory with numerical results obtained in the earlier paper confirms the asymptotic results but reveals that the numerical results diverge from the asymptotic result for unexpectedly small values of S.

Unsteady boundary-layer transition in low-pressure turbines

2011 ◽  
Vol 681 ◽  
pp. 370-410 ◽  
Author(s):  
JOHN D. COULL ◽  
HOWARD P. HODSON
Keyword(s):  
Boundary Layer ◽  
Shear Layer ◽  
Flat Plate ◽  
Free Stream ◽  
Separation Bubble ◽  
Leading Edge ◽  
Transition Process ◽  
Free Stream Velocity ◽  
Stream Velocity ◽  
Free Stream Turbulence

This paper examines the transition process in a boundary layer similar to that present over the suction surfaces of aero-engine low-pressure (LP) turbine blades. This transition process is of significant practical interest since the behaviour of this boundary layer largely determines the overall efficiency of the LP turbine. Modern ‘high-lift’ blade designs typically feature a closed laminar separation bubble on the aft portion of the suction surface. The size of this bubble and hence the inefficiency it generates is controlled by the transition between laminar and turbulent flow in the boundary layer and separated shear layer. The transition process is complicated by the inherent unsteadiness of the multi-stage machine: the wakes shed by one blade row convect through the downstream blade passages, periodically disturbing the boundary layers. As a consequence, the transition to turbulence is multi-modal by nature, being promoted by periodic and turbulent fluctuations in the free stream and the inherent instabilities of the boundary layer. Despite many studies examining the flow behaviour, the detailed physics of the unsteady transition phenomena are not yet fully understood. The boundary-layer transition process has been studied experimentally on a flat plate. The opposing test-section wall was curved to impose a streamwise pressure distribution typical of modern high-lift LP turbines over the flat plate. The presence of an upstream blade row has been simulated by a set of moving bars, which shed wakes across the test section inlet. Further upstream, a grid has been installed to elevate the free-stream turbulence to a level believed to be representative of multi-stage LP turbines. Extensive particle imaging velocimetry (PIV) measurements have been performed on the flat-plate boundary layer to examine the flow behaviour. In the absence of the incoming bar wakes, the grid-generated free-stream turbulence induces relatively weak Klebanoff streaks in the boundary layer which are evident as streamwise streaks of low-velocity fluid. Transition is promoted by the streaks and by the inherent inflectional (Kelvin–Helmholtz (KH)) instability of the separation bubble. In unsteady flow, the incoming bar wakes generate stronger Klebanoff streaks as they pass over the leading edge, which convect downstream at a fraction of the free-stream velocity and spread in the streamwise direction. The region of amplified streaks convects in a similar manner to a classical turbulent spot: the leading and trailing edges travel at around 88% and 50% of the free-stream velocity, respectively. The strongest disturbances travel at around 70% of the free-stream velocity. The wakes induce a second type of disturbance as they pass over the separation bubble, in the form of short-span KH structures. Both the streaks and the KH structures contribute to the early wake-induced transition. The KH structures are similar to those observed in the simulation of separated flow transition with high free-stream turbulence by McAuliffe & Yaras (ASME J. Turbomach., vol. 132, no. 1, 2010, 011004), who observed that these structures originated from localised instabilities of the shear layer induced by Klebanoff streaks. In the current measurements, KH structures are frequently observed directly under the path of the wake. The wake-amplified Klebanoff streaks cannot affect the generation of these structures since they do not arrive at the bubble until later in the wake cycle. Rather, the KH structures arise from an interaction between the flow disturbances in the wake and localised instabilities in the shear layer, which are caused by the weak Klebanoff streaks induced by the grid turbulence. The breakdown of the KH structures to small-scale turbulence occurs a short time after the wake has passed over the bubble, and is largely driven by the arrival of the wake-amplified Klebanoff streaks from the leading edge. During this process, the re-attachment location moves rapidly upstream. The minimum length of the bubble occurs when the strongest wake-amplified Klebanoff streaks arrive from the leading edge; these structures travel at around 70% of the free-stream velocity. The bubble remains shorter than its steady-flow length until the trailing edge of the wake-amplified Klebanoff streaks, travelling at 50% of the free-stream velocity, convect past. After this time, the reattachment location moves aft on the surface as a consequence of a calmed flow region which follows behind the wake-induced turbulence.

Distortion of a flat-plate boundary layer by free-stream vorticity normal to the plate

1992 ◽  
Vol 237 ◽  
pp. 231-260 ◽  
Author(s):  
M. E. Goldstein ◽  
S. J. Leib ◽  
S. J. Cowley
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Free Stream ◽  
Plate Boundary ◽  
Leading Edge ◽  
Uniform Flow ◽  
Viscous Boundary Layer ◽  
Layer Flow ◽  
Viscous Boundary ◽  
Flat Plate Boundary Layer

We consider a nominally uniform flow over a semi-infinite flat plate. Our analysis shows how a small streamwise disturbance in the otherwise uniform flow ahead of the plate is amplified by leading-edge bluntness effects and eventually leads to a small-amplitude but nonlinear spanwise motion far downstream from the leading edge of the plate. This spanwise motion is then imposed on the viscous boundary-layer flow at the surface of the plate – causing an order-one change in its profile shape. This ultimately reduces the wall shear stress to zero – causing the boundary layer to undergo a localized separation, which may be characterized as a kind of bursting phenomenon that could be related to the turbulent bursts observed in some flat-plate boundary-layer experiments.

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