Classical Boundary-Layer Theory

2018 ◽  
Author(s):  
Anatoly I. Ruban
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Boundary Layer Theory ◽  
Laminar Jet ◽  
Compressible Boundary Layers ◽  
Wedge Surface ◽  
Blasius Boundary Layer ◽  
Classical Boundary ◽  
Fast Rotating

Chapter 1 discusses the flows that can be described in the framework of Prandtl’s 1904 classical boundary-layer theory, including the Blasius boundary layer on a flat plate and the Falkner–Skan solutions for the boundary layer on a wedge surface. It presents Schlichting’s solution for the laminar jet and Tollmien’s solution for the viscous wake. These are followed by analysis of Chapman’s shear layer performed with the help of Prandtl’s transposition theorem. It also considers the boundary layer on the surface of a fast rotating cylinder with the purpose of linking the circulation around the cylinder with the speed of its rotation. It concludes discussion of the classical boundary-layer theory with analysis of compressible boundary layers, including the interactive boundary layers in hypersonic flows.

Correlated incompressible and compressible boundary layers

1949 ◽  
Vol 200 (1060) ◽  
pp. 84-100 ◽  
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Boundary Layers ◽  
Absolute Temperature ◽  
Stream Velocity ◽  
Main Stream ◽  
Boundary Layer Equations ◽  
Compressible Boundary Layers ◽  
Incompressible Boundary ◽  
Similar Solutions

The boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the Prandtl number is unity. Various results in the theory of incompressible boundary layers are then taken over into the compressible theory. In particular, the existence of ‘similar’ solutions is proved, and Howarth’s method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity. A comparison with an exact solution is used to show that this method gives a closer approximation than does Pohlhausen’s.

The Influence of Attached Boundary Layers in Determining the Optimum for Blades in Cascade

1965 ◽  
Vol 69 (655) ◽  
pp. 497-498
Author(s):  
W. K. Allan ◽  
B. S. Stratford
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Flow Model ◽  
Flow System ◽  
Boundary Layer Theory ◽  
Blade Passage ◽  
General Flow ◽  
Flow Through

Dr. Stratford (p. 133, February 1965 Journal) is to be supported in his endeavour to apply boundary layer theory to the prediction of optimum loading requirements in flow through blades in cascade. Inevitably some simplification of the general flow system in a blade passage is necessary if undue complexity is to be avoided. In the simplified flow model, however, care must be taken to avoid over-simplification, and the limitations imposed by legitimate approximations must be appreciated.

A resolution of the blow-off singularity for similarity flow on a flat plate

1974 ◽  
Vol 62 (1) ◽  
pp. 145-161 ◽  
Author(s):  
D. R. Kassoy
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Critical Value ◽  
Injection Rate ◽  
Uniform Flow ◽  
Boundary Layer Theory ◽  
Viscous Effects ◽  
Critical Rate ◽  
Flow Past ◽  
Classical Boundary

A study is made of uniform flow past a semi-infinite flat plate with a similarity injection distribution of boundary-layer magnitude. Attention is focused on a solution at exactly the critical injection rate for which classical boundary-layer theory predicts the blow-off singularity. Following a description of the more recent interaction analyses which also fail at the critical rate, a new theory is developed which leads to physically meaningful results. In particular, it is shown that the non-monotonic variation in wall shear with increasing injection rate near the critical value, noted by Klemp & Acrivos (1972), is real. A delicate interplay of weak pressure interactions and viscous effects is shown to be responsible for this surprising phenomenon.

The continuous spectrum of the Orr-Sommerfeld equation: note on a paper of Grosch & Salwen

1991 ◽  
Vol 226 ◽  
pp. 565-571 ◽  
Author(s):  
A. D. D. Craik
Keyword(s):  
Boundary Layer ◽  
Simple Model ◽  
Continuous Spectrum ◽  
Stability Problems ◽  
Spatial Stability ◽  
Laminar Jet ◽  
Physical Understanding ◽  
Blasius Boundary Layer ◽  
Temporal And Spatial ◽  
Sommerfeld Equation

Grosch & Salwen (1978) discuss the continuous-spectrum contribution in both temporal and spatial stability problems that are governed by the linear Orr–Sommerfeld equation. Their computed temporal continuum eigenfunctions for the Blasius boundary layer and for a laminar jet profile show surprising differences. This note provides an improved physical understanding of the results through a simple model, and shows that these differences are more apparent than real.

scholarly journals Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers

2016 ◽  
Vol 797 ◽  
pp. 683-728 ◽  
Author(s):  
Xuesong Wu ◽  
Ming Dong
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Free Stream ◽  
Base Flow ◽  
Bypass Transition ◽  
Leading Order ◽  
Compressible Boundary Layers ◽  
Characteristic Wavelength ◽  
Incompressible Boundary ◽  
Edge Layer

The fundamental difference between continuous modes of the Orr–Sommerfeld/Squire equations and the entrainment of free-stream vortical disturbances (FSVD) into the boundary layer has been investigated in a recent paper (Dong & Wu, J. Fluid Mech., vol. 732, 2013, pp. 616–659). It was shown there that the non-parallel-flow effect plays a leading-order role in the entrainment, and neglecting it at the outset, as is done in the continuous-mode formulation, leads to non-physical features of ‘Fourier entanglement’ and abnormal anisotropy. The analysis, which was for incompressible boundary layers and for FSVD with a characteristic wavelength of the order of the local boundary-layer thickness, is extended in this paper to compressible boundary layers and FSVD with even shorter wavelengths, which are comparable with the width of the so-called edge layer. Non-parallelism remains a leading-order effect in the present scaling, which turns out to be more general in that the equations and solutions in the previous paper are recovered in the appropriate limit. Appropriate asymptotic solutions in the main and edge layers are obtained to characterize the entrainment. It is found that when the Prandtl number $\mathit{Pr}<1$, free-stream vortical disturbances of relatively low frequency generate very strong temperature fluctuations within the edge layer, leading to formation of thermal streaks. A composite solution, uniformly valid across the entire boundary layer, is constructed, and it can be used in receptivity studies and as inlet conditions for direct numerical simulations of bypass transition. For compressible boundary layers, continuous spectra of the disturbance equations linearized about a parallel base flow exhibit entanglement between vortical and entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in the free stream and vice versa, and this amounts to a further non-physical behaviour. High Reynolds number asymptotic analysis yields the relations between the amplitudes of entangled modes.

An Axial Compressor End-Wall Boundary Layer Theory

1971 ◽  
Vol 93 (2) ◽  
pp. 300-314 ◽  
Author(s):  
G. L. Mellor ◽  
G. M. Wood
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Axial Compressor ◽  
Jump Condition ◽  
Present Theory ◽  
Boundary Layer Theory ◽  
Tip Clearance ◽  
Main Stream ◽  
Wall Boundary ◽  
End Wall

The essential ingredient missing in existing prediction methods for the performance of multistage axial compressors is that which would account for the effect of end-wall boundary layers. It is, in fact, believed that end-wall boundary layers play a major role in compressor performance and the absence of an adequate theory represents a handicap to turbomachinery designers that might be likened to the handicap that designers of wings, for example, would face if Prandtl had not introduced the idea of a boundary layer. In this paper a new theory is developed which retains all elements of classical boundary layer theory; for example, we discuss variables such as momentum thickness and wall shear stress. However, the present theory introduces new concepts such as axial and tangential defect force thickness, a rotor exit-stator inlet “jump condition” and the importance of these concepts is demonstrated. Inherent in the derivation is an identification of the role of secondary flow and tip clearance flow. A proper means of matching the boundary layer calculations to conventional main stream calculations is suggested. Independent of empirical parametization it appears that the theory is capable of correctly modeling boundary layer blockage, losses, and end-wall stall. Near stall, the main stream-boundary layer interaction is very strong.

scholarly journals Viscoplastic boundary layers

2017 ◽  
Vol 813 ◽  
pp. 929-954 ◽  
Author(s):  
N. J. Balmforth ◽  
R. V. Craster ◽  
D. R. Hewitt ◽  
S. Hormozi ◽  
A. Maleki
Keyword(s):  
Boundary Layer ◽  
Yield Stress ◽  
Layer Thickness ◽  
Boundary Layers ◽  
Boundary Layer Thickness ◽  
Boundary Layer Theory ◽  
Classical Solutions ◽  
Bingham Number ◽  
Boundary Layer Equations ◽  
Two Dimensional Flow

In the limit of a large yield stress, or equivalently at the initiation of motion, viscoplastic flows can develop narrow boundary layers that provide either surfaces of failure between rigid plugs, the lubrication between a plugged flow and a wall or buffers for regions of predominantly plastic deformation. Oldroyd (Proc. Camb. Phil. Soc., vol. 43, 1947, pp. 383–395) presented the first theoretical discussion of these viscoplastic boundary layers, offering an asymptotic reduction of the governing equations and a discussion of some model flow problems. However, the complicated nonlinear form of Oldroyd’s boundary-layer equations has evidently precluded further discussion of them. In the current paper, we revisit Oldroyd’s viscoplastic boundary-layer analysis and his canonical examples of a jet-like intrusion and flow past a thin plate. We also consider flow down channels with either sudden expansions or wavy walls. In all these examples, we verify that viscoplastic boundary layers form as envisioned by Oldroyd. For each example, we extract the dependence of the boundary-layer thickness and flow profiles on the dimensionless yield-stress parameter (Bingham number). We find that, while Oldroyd’s boundary-layer theory applies to free viscoplastic shear layers, it does not apply when the boundary layer is adjacent to a wall, as has been observed previously for two-dimensional flow around circular obstructions. Instead, the boundary-layer thickness scales in a different fashion with the Bingham number, as suggested by classical solutions for plane-parallel flows, lubrication theory and, for flow around a plate, by Piau (J. Non-Newtonian Fluid Mech., vol. 102, 2002, pp. 193–218); we rationalize this second scaling and provide an alternative boundary-layer theory.

Second-order boundary-layer theory of laminar jet flows

1984 ◽  
Vol 53 (1-2) ◽  
pp. 115-123 ◽  
Author(s):  
K. Mitsotakis ◽  
W. Schneider ◽  
E. Zauner
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Theory ◽  
Second Order ◽  
Jet Flows ◽  
Laminar Jet

A model for subharmonic resonance within wavepackets in unstable boundary layers

2001 ◽  
Vol 432 ◽  
pp. 409-418 ◽  
Author(s):  
A. D. D. CRAIK
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Initial Phase ◽  
Nonlinear Effects ◽  
Subharmonic Resonance ◽  
Simple Explanation ◽  
Resonance Mechanism ◽  
Space And Time ◽  
Wave Resonance ◽  
Blasius Boundary Layer

In recent experiments on the growth of localized disturbances in a Blasius boundary layer, Medeiros & Gaster (1999a, b) observed that the development of nonlinear effects depends markedly on the initial phase of their imposed disturbance. Here, a simple explanation of this phenomenon is proposed. Because the disturbance is localized in space and time, it has a spread of wavenumbers and frequencies: among these are components which can initiate a pair of resonant subharmonic waves with well-determined phase, which are then amplified by the familiar three-wave resonance mechanism. The amplitude attained after some time is strongly phase-dependent, consistent with the experimental observations.

Boundary Layers in Darcy–Brinkman Flow

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Boundary Layer ◽  
Porous Media ◽  
Fluid Flow ◽  
Boundary Layers ◽  
Darcy Number ◽  
Boundary Layer Theory ◽  
Brinkman Equation ◽  
Solid Inclusion ◽  
Saturated Porous Media ◽  
Shear Forces

Abstract Fluid flow in saturated porous media imbedded with a solid inclusion may be described by the Darcy–Brinkman equation. When the Darcy number is small, a boundary-layer theory, similar to Prandtl's theory for viscous flow, is established. The pressure and shear forces are predicted for Darcy–Brinkman flows over a variety of solid inclusions.

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