Simplified Analysis of Boundary-Layer Oscillations

1960 ◽  
Vol 4 (03) ◽  
pp. 37-54
Author(s):  
Robert Betchov
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Nonlinear Effects ◽  
Adverse Pressure Gradient ◽  
Neutral Curves ◽  
Elastic Wall ◽  
Unstable Boundary Layer ◽  
Incompressible Boundary ◽  
Negative Damping ◽  
The Stability

The stability of an incompressible boundary layer is analyzed in terms of three basic processes. These are (a) the oscillations of a boundary layer when friction is disregarded, (b) the effects of friction at the wall, and (c) the effects of friction at the critical layer. These processes are separately discussed and evaluated. Simple models are presented. A general equation leads to the eigenvalues. The neutral curves corresponding to five typical cases are determined—parabolic and Blasius boundary layers, boundary layers with suction and with adverse pressure gradient, two-dimensional Poiseuille flow. The unstable boundary layer is discussed briefly. The nonlinear effects of the oscillation on the velocity profile are evaluated. Finally, the case of a boundary layer along an elastic wall is considered, and it is found that the wall may have a significant effect on the layer. In particular, a wall with negative damping could completely stabilize the boundary layer.

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.

Recent Developments in Calculation Methods for Turbulent Boundary Layers With Pressure Gradients and Heat Transfer

1966 ◽  
Vol 33 (2) ◽  
pp. 429-437 ◽  
Author(s):  
J. C. Rotta
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Temperature Distribution ◽  
Velocity Profile ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients ◽  
Calculation Methods ◽  
Recent Developments ◽  
Incompressible Boundary

A review is given of the recent development in turbulent boundary layers. At first, for the case of incompressible flow, the variation of the shape of velocity profile with the pressure gradient is discussed; also the temperature distribution and heat transfer in incompressible boundary layers are treated. Finally, problems of the turbulent boundary layer in compressible flow are considered.

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.

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 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.

Direct Simulations of Wake-Perturbed Separated Boundary Layers

2011 ◽  
Author(s):  
Ayse G. Gungor ◽  
Mark P. Simens ◽  
Javier Jime´nez
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Separation Bubble ◽  
Turbine Blades ◽  
Plate Boundary ◽  
Maximum Velocity ◽  
Adverse Pressure Gradient ◽  
Suction Side ◽  
Single Experiment ◽  
The Stability

A wake-perturbed flat plate boundary layer with a stream-wise pressure distribution similar to those encountered on the suction side of typical low-pressure turbine blades is investigated by direct numerical simulation. The laminar boundary layer separates due to a strong adverse pressure gradient induced by suction along the upper simulation boundary, transitions and reattaches while still subject to the adverse pressure gradient. Various simulations are performed with different wake passing frequencies, corresponding to the Strouhal number 0.0043 < fθb/ΔU < 0.0496 and wake profiles. The wake profile is changed by varying its maximum velocity defect and its symmetry. Results indicate that the separation and reattachment points, as well as the subsequent boundary layer development, are mainly affected by the frequency, but that the wake shape and intensity have little effect. Moreover, the effect of the different frequencies can be predicted from a single experiment in which the separation bubble is allowed to reform after having been reduced by wake perturbations. The stability characteristics of the mean flows resulting from the forcing at different frequencies are evaluated in terms of local linear stability analysis based on the Orr-Sommerfeld equation.

Investigation of the Dihedral Angle Effect on the Boundary Layer Development Using Special-Shaped Expansion Pipes

2018 ◽  
Author(s):  
Teng Fei ◽  
Lucheng Ji ◽  
Weilin Yi
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Dihedral Angle ◽  
Adverse Pressure Gradient ◽  
Secondary Flows ◽  
Flow Structures ◽  
Wall Surface ◽  
Geometric Structures ◽  
Corner Region ◽  
End Wall

The corners between the blades and end walls are common geometric structures in turbomachinery, where boundary layers on the blade and end wall surface interact with each other. This boundary layer interaction enlarges the region of low momentum fluid which leads to the boundary layers grow thicker at the corner region. Then the corner separation is likely to occur, and even worse by the adverse pressure gradient along the streamwise as well as secondary flows along the pitchwise. The key issue to design the geometric structures of the corner region is to control the dihedral angle between the blade and end wall surface. However, from the current published literature, few researchers have studied the influence of dihedral angle on the flow structures at the corner region in detail. In this paper, a series of expansion pipes with different cross sections which represent different dihedral angles are simulated. Then, some useful conclusions about how the dihedral angle affects the flow structures at the corner region are drawn. Moreover, a new method to predict the boundary layer thickness at the corner region is introduced, and the predicted results are in good agreement with simulation results.

Linear and Nonlinear PSE for Stability Analysis of the Blasius Boundary Layer Using Compact Scheme

2001 ◽  
Vol 123 (3) ◽  
pp. 545-550 ◽  
Author(s):  
V. Esfahanian ◽  
K. Hejranfar ◽  
F. Sabetghadam
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Nonlinear Effects ◽  
Numerical Investigations ◽  
Blasius Boundary Layer ◽  
Incompressible Boundary ◽  
Linear And Nonlinear ◽  
Tollmien Schlichting ◽  
The Stability ◽  
Good Agreement

A highly accurate finite-difference PSE code has been developed to investigate the stability analysis of incompressible boundary layers over a flat plate. The PSE equations are derived in terms of primitive variables and are solved numerically by using compact method. In these formulations, both nonparallel as well as nonlinear effects are accounted for. The validity of present numerical scheme is demonstrated using spatial simulations of two cases; two-dimensional (linear and nonlinear) Tollmien-Schlichting wave propagation and three-dimensional subharmonic instability breakdown. The PSE solutions have been compared with previous numerical investigations and experimental results and show good agreement.

Pressure Gradient Effects on Smooth- and Rough-Surface Turbulent Boundary Layers—Part II: Adverse Pressure Gradient

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Ju Hyun Shin ◽  
Seung Jin Song
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Boundary Layer Thickness ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Roughness Effects ◽  
Mean Velocity ◽  
Velocity Defect

An experimental investigation has been conducted to identify the effects of pressure gradient and surface roughness on turbulent boundary layers. In Part II, smooth- and rough-surface turbulent boundary layers with and without adverse pressure gradient (APG) are presented at a fixed Reynolds number (based on the length of flat plate) of 900,000. Flat-plate boundary layer measurements have been conducted using a single-sensor, hot-wire probe. For smooth surfaces, compared to the zero pressure gradient (ZPG) boundary layer, the APG boundary layer has a higher mean velocity defect throughout the boundary layer and lower friction coefficient. APG decreases the streamwise normal Reynolds stress for y less than 0.4 times the boundary layer thickness and increases it slightly in the outer region. For rough surfaces, APG reduces the roughness effects of increasing the mean velocity defect and normal Reynolds stress for y less than 23 and 28 times the average roughness height, respectively. Consistently, for the same roughness, APG decreases the integrated streamwise turbulent kinetic energy. APG also decreases the roughness effect on the friction coefficient, roughness Reynolds number, and roughness shift. Compared to the ZPG boundary layers, the roughness effects on integral boundary layer parameters—boundary layer thickness and momentum thickness—are weaker under APG. Thus, contrary to the favorable pressure gradient (FPG) in part I, APG reduces the roughness effects on turbulent boundary layers.

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.

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