Reductions of the stationary boundary layer equation with a pressure gradient

The skin friction for a turbulent boundary layer can be measured and calculated in several ways with varying degrees of accuracy. In particular, the methods of the velocity gradient at the wall, the integrated boundary layer equation and the momentum integral equation are evaluated for both smooth and rough surface boundary layers. These methods are compared to the oil film interferometry technique measurements for the case of smooth surface flows. The integrated boundary layer equation is found to be relatively reliable, and the values computed with this technique are used to investigate the effect of increasing external favorable pressure gradient for both smooth and rough surfaces, and increasing roughness parameter for the rough surfaces.

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Experimental results for the transpired turbulent boundary layer in an adverse pressure gradient

Journal of Fluid Mechanics ◽

10.1017/s0022112075001474 ◽

1975 ◽

Vol 69 (2) ◽

pp. 353-375 ◽

Cited By ~ 45

Author(s):

P. S. Andersen ◽

W. M. Kays ◽

R. J. Moffat

Keyword(s):

Boundary Layer ◽

Shear Stress ◽

Turbulent Boundary Layer ◽

Pressure Gradient ◽

Boundary Layer Equation ◽

Adverse Pressure Gradient ◽

Stream Velocity ◽

Pressure Gradients ◽

Mean Velocity ◽

Displacement Thickness

An experimental investigation of the fluid mechanics of the transpired turbulent boundary layer in zero and adverse pressure gradients was carried out on the Stanford Heat and Mass Transfer Apparatus. Profiles of (a) the mean velocity, (b) the intensities of the three components of the turbulent velocity fluctuations and (c) the Reynolds stress were obtained by hot-wire anemometry. The wall shear stress was measured by using an integrated form of the boundary-layer equation to ‘extrapolate’ the measured shear-stress profiles to the wall.The two experimental adverse pressure gradients corresponded to free-stream velocity distributions of the type u∞ ∞ xm, where m = −0·15 and −0·20, x being the streamwise co-ordinate. Equilibrium boundary layers (i.e. flows with velocity defect profile similarity) were obtained when the transpiration velocity v0 was varied such that the blowing parameter B = pv0u∞/τ0 and the Clauser pressure-gradient parameter $\beta\equiv\delta_1\tau_0^{-1}\,dp/dx $ were held constant. (τ0 is the shear stress at the wall and δ1 is the displacement thickness.)Tabular and graphical results are presented.

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Separation Criterion for Turbulent Boundary Layers Via Similarity Analysis

Journal of Fluids Engineering ◽

10.1115/1.1758262 ◽

2004 ◽

Vol 126 (3) ◽

pp. 297-304 ◽

Cited By ~ 45

Author(s):

Luciano Castillo ◽

Xia Wang ◽

William K. George

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Boundary Layers ◽

Boundary Layer Equation ◽

Outer Part ◽

Adverse Pressure Gradient ◽

Gradient Flows ◽

Boundary Layer Equations ◽

Gradient Parameter ◽

Momentum Boundary Layer

By using the RANS boundary layer equations, it will be shown that the outer part of an adverse pressure gradient turbulent boundary layer tends to remain in equilibrium similarity, even near and past separation. Such boundary layers are characterized by a single and constant pressure gradient parameter, Λ, and its value appears to be the same for all adverse pressure gradient flows, including those with eventual separation. Also it appears from the experimental data that the pressure gradient parameter, Λθ, is also approximately constant and given by Λθ=0.21±0.01. Using this and the integral momentum boundary layer equation, it is possible to show that the shape factor at separation also has to within the experimental uncertainty a single value: Hsep≅2.76±0.23. Furthermore, the conditions for equilibrium similarity and the value of Hsep are shown to be in reasonable agreement with a variety of experimental estimates, as well as the predictions from some other investigators.

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Further Solution of the Corner Boundary-Layer Equations

Aeronautical Quarterly ◽

10.1017/s0001925900006612 ◽

1973 ◽

Vol 24 (3) ◽

pp. 219-226 ◽

Cited By ~ 4

Author(s):

M Zamir

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Existence And Uniqueness ◽

Boundary Layer Equation ◽

Critical Value ◽

Numerical Accuracy ◽

Boundary Layer Equations ◽

The Past ◽

Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1967.0151 ◽

1967 ◽

Vol 299 (1459) ◽

pp. 491-507 ◽

Cited By ~ 34

Keyword(s):

Boundary Layer ◽

Velocity Profile ◽

Pressure Gradient ◽

Asymptotic Behaviour ◽

Power Law ◽

Boundary Layer Equation ◽

Rigid Wall ◽

Velocity Profiles ◽

Boundary Layer Theory ◽

Prandtl Boundary Layer

Consider the Prandtl boundary layer equation for the steady two-dimensional laminar flow of an incompressible viscous fluid past a rigid wall. On the basis of an arbitrary velocity profile at some initial position on the wall, the analysis presented shows that, for a power law streaming speed U ( x ) ═ C ( x + d ) m ( m ≽ 0), the velocity profile which develops downstream is asymptotically given by the well known Falkner-Skan similarity solution. Moreover, for a streaming speed satisfying (6), the velocity profile which develops downstream is asymptotically unique, though of course the particular form of the resulting profile depends on the precise nature of the exterior stream. The rate of convergence for this asymptotic behaviour is estimated, as well as corresponding rates for the convergence of the skin friction coefficient. This result verifies the tacit assumption of a number of writers that the downstream velocity profile is essentially independent of the initial profile, and also supplies a theoretical justification for the role of similar solutions in boundary layer theory. We also prove the existence of concave velocity profiles whenever the pressure gradient is favourable. It follows that, for streaming speeds which correspond to a favourable pressure gradient, concave velocity profiles play somewhat the same role as similarity profiles do for a power law streaming speed.

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On the Applicability of the Boundary Layer Independence Principle to Free and Bounded Turbulent Flows in Absence of Pressure Gradient

Proceeding of THMT-12. Proceedings of the Seventh International Symposium On Turbulence, Heat and Mass Transfer Palermo, Italy, 24-27 September, 2012 ◽

10.1615/ichmt.2012.procsevintsympturbheattransfpal.330 ◽

2012 ◽

Author(s):

I. Wygnanski ◽

P. Tewes ◽

N. Zalewski ◽

L. Taubert

Keyword(s):

Boundary Layer ◽

Pressure Gradient ◽

Turbulent Flows ◽

Independence Principle

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The numerical investigation of the effect of spanwise pressure gradient on heat transfer and a streamwise vortex embedded in a turbulent boundary layer

10.1615/ihtc12.1200 ◽

2002 ◽

Author(s):

InSub Lee ◽

Hong Sun Ryou ◽

Hei Cheon Yang ◽

Sang Kyoo Park

Keyword(s):

Heat Transfer ◽

Boundary Layer ◽

Turbulent Boundary Layer ◽

Pressure Gradient ◽

Numerical Investigation ◽

Streamwise Vortex

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Characteristics of a separated flow past a semicircular leading-edge airfoil model under different imposed pressure gradient

Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering ◽

10.1177/09544100211021234 ◽

2021 ◽

pp. 095441002110212

Author(s):

K Anand ◽

KT Ganesh

Keyword(s):

Boundary Layer ◽

Particle Image Velocimetry ◽

Pressure Gradient ◽

Particle Image ◽

Separation Bubble ◽

Separated Flow ◽

Leading Edge ◽

Field Measurements ◽

Bench Mark ◽

Image Velocimetry

The effect of pressure gradient on a separated boundary layer past the leading edge of an airfoil model is studied experimentally using electronically scanned pressure (ESP) and particle image velocimetry (PIV) for a Reynolds number ( Re) of 25,000, based on leading-edge diameter ( D). The features of the boundary layer in the region of separation and its development past the reattachment location are examined for three cases of β (−30°, 0°, and +30°). The bubble parameters such as the onset of separation and transition and the reattachment location are identified from the averaged data obtained from pressure and velocity measurements. Surface pressure measurements obtained from ESP show a surge in wall static pressure for β = −30° (flap deflected up), while it goes down for β = +30° (flap deflected down) compared to the fundamental case, β = 0°. Particle image velocimetry results show that the roll up of the shear layer past the onset of separation is early for β = +30°, owing to higher amplification of background disturbances compared to β = 0° and −30°. Downstream to transition location, the instantaneous field measurements reveal a stretched, disoriented, and at instances bigger vortices for β = +30°, whereas a regular, periodically shed vortices, keeping their identity past the reattachment location, is observed for β = 0° and −30°. Above all, this study presents a new insight on the features of a separation bubble receiving a disturbance from the downstream end of the model, and these results may serve as a bench mark for future studies over an airfoil under similar environment.

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