An Accurate Calculation Method for Two-dimensional Incompressible Laminar Boundary Layers, Including Cases with Regions of Sharp Pressure Gradient

1977 ◽  
Vol 28 (3) ◽  
pp. 149-162 ◽  
Author(s):  
N Curle
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Calculation Method ◽  
Boundary Layer Separation ◽  
Pressure Gradients ◽  
Accurate Calculation ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Type Boundary

SummaryThe paper develops and extends the calculation method of Stratford, for flows in which a Blasius type boundary layer reacts to a sharp unfavourable pressure gradient. Whereas even the more general of Stratford’s two formulae for predicting the position of boundary-layer separation is based primarily upon an interpolation between only three exact solutions of the boundary layer equations, the present proposals are based upon nine solutions covering a much wider range of conditions. Four of the solutions are for extremely sharp pressure gradients of the type studied by Stratford, and five are for more modest gradients. The method predicts the position of separation extremely accurately for each of these cases.The method may also be used to predict the detailed distributions of skin friction, displacement thickness and momentum thickness, and does so both simply and accurately.

Incompressible Laminar Boundary Layers on a Parabola at Angle of Attack: A Study of the Separation Point

1972 ◽  
Vol 39 (1) ◽  
pp. 7-12 ◽  
Author(s):  
M. J. Werle ◽  
R. T. Davis
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Angle Of Attack ◽  
Adverse Pressure Gradient ◽  
Boundary Layer Separation ◽  
Separation Point ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Type Singularity ◽  
Flow Experiences

The laminar boundary-layer equations were solved for incompressible flow past a parabola at angle of attack. Such flow experiences a region of adverse pressure gradient and thus can be employed to study the boundary-layer separation process. The present solutions were obtained numerically using both implicit and Crank-Nicolson-type difference schemes. It was found that in all cases the point of vanishing shear stress (the separation point) displayed a Goldstein-type singularity. Based on this evidence, it is concluded that a singularity is always present at separation independent of the mildness of the pressure gradient at that point.

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

1975 ◽  
Vol 69 (2) ◽  
pp. 353-375 ◽  
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.

scholarly journals Experimental Hydrodynamics of the Accelerated Turbulent Boundary Layer With and Without Mass Injection

1971 ◽  
Vol 93 (4) ◽  
pp. 373-379 ◽  
Author(s):  
H. L. Julien ◽  
W. M. Kays ◽  
R. J. Moffat
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Damping Factor ◽  
Surface Boundary ◽  
Pressure Gradients ◽  
Mean Velocity ◽  
Boundary Layer Equations ◽  
Surface Boundary Conditions ◽  
Gradient Parameter ◽  
Experimental Hydrodynamics

Mean velocity-profile data are reported for blown, unblown, and sucked accelerated turbulent boundary layers. The pressure gradients investigated are those corresponding to constant values of the pressure-gradient parameter K=νU∞2dU∞dx The two values of K considered in detail are 0.57 × 10−6 and 1.45 × 10−6. For each pressure gradient, the surface boundary conditions cover a range of constant blowing and sucking fractions from F = −0.002 to +0.004. Velocity profiles corresponding to these accelerated flows are shown to differ substantially from those characteristic of zero-pressure-gradient flows. For each case of a constant K acceleration, sequential values of the momentum-thickness Reynolds number approach a specific constant, and the velocity distributions near the wall are similar in both wall coordinates and outer coordinates. Results obtained here can be reproduced by a numerical integration of the boundary-layer equations using a modification of the van Driest damping factor, A+, derived from the data presented here. The A+ correlation is presented.

Characteristics of a trailing-edge flow with turbulent boundary-layer separation

1985 ◽  
Vol 157 ◽  
pp. 305-326 ◽  
Author(s):  
B. E. Thompson ◽  
J. H. Whitelaw
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Separation ◽  
Turbulence Energy ◽  
Pressure Gradients ◽  
Relative Importance ◽  
Calculation Methods ◽  
Trailing Edge ◽  
Boundary Layer Equations ◽  
Streamline Curvature ◽  
Recirculating Flow

Experimental techniques, including flying-hot-wire anemometry, have been used to determine the pressure and velocity characteristics of a flow designed to simulate the trailing-edge region of an airfoil at high angle of attack. Emphasis is placed on the region of recirculating flow and on the downstream wake. It is shown that the effect of this recirculation is large even though the details of the flow within it may be unimportant. Normal stresses and cross-stream pressure gradients are important immediately upstream and downstream of the recirculating flow and are associated with strong streamline curvature. The relative importance of the terms in the transport equations for mean momentum and turbulence energy are quantified and the implications for procedures which solve potential-flow and boundary-layer equations and for alternative calculation methods are discussed.

scholarly journals On the Lagrangian description of unsteady boundary-layer separation. Part 1. General theory

1990 ◽  
Vol 210 ◽  
pp. 593-626 ◽  
Author(s):  
Leon L. Van Dommelen ◽  
Stephen J. Cowley
Keyword(s):  
Boundary Layer ◽  
Continuity Equation ◽  
Three Dimensional ◽  
Boundary Layer Separation ◽  
Momentum Equation ◽  
Lagrangian Description ◽  
Direction Parallel ◽  
Boundary Layer Equations ◽  
Symmetry Properties ◽  
Displacement Thickness

Although unsteady, high-Reynolds-number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary-layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady three-dimensional separating flows follow and depend on the symmetry properties of the flow (e.g. line symmetry, axial symmetry). In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi two-dimensional with a displacement thickness in the form of a crescent-shaped ridge. Physically the singularities can be understood in terms of the behaviour of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer.

Laminar boundary layers

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Momentum Integral ◽  
Aerodynamic Lift

This chapter starts by introducing the concept of a boundary layer and the associated boundary-layer approximations. The laminar boundary-layer equations are then derived from the Navier-Stokes equations. The assumption of velocity-profile similarity is shown to reduce the partial differential boundary-layer equations to ordinary differential equations. The results of numerical solutions to these equations are discussed: Blasius’ equation, for zero-pressure gradient, and the Falkner-Skan equation for wedge flows. Von Kármán’s momentum-integral equation is derived and used to obtain useful results for the zero-pressure-gradient boundary layer. Pohlhausen’s quartic-profile method is then discussed, followed by the approximate method of Thwaites. The chapter concludes with a qualitative account of the way in which aerodynamic lift is generated.

A Method for Designing Wind Tunnel Contractions

1951 ◽  
Vol 55 (483) ◽  
pp. 169-180 ◽  
Author(s):  
R. Harrop
Keyword(s):  
Boundary Layer ◽  
Wind Tunnel ◽  
Pressure Gradient ◽  
Cross Section ◽  
Incompressible Flow ◽  
Circular Cross Section ◽  
Flow Theory ◽  
Boundary Layer Separation ◽  
Experimental Results ◽  
Pressure Gradients

SummaryThe contraction of a wind tunnel should be free from adverse pressure gradients, since this might cause boundary layer separation.A wall contour has been designed for a circular cross-section contraction using incompressible flow theory. This gave a favourable pressure gradient at the beginning of the contraction where separation is likely to occur.Appendix I compares the theory with experimental results obtained from a model of a proposed supersonic tunnel of which the contraction is rectangular in cross-section and which has been based on the results obtained in this report.

The equation of similar profiles in boundary layer theory with strong blowing

1966 ◽  
Vol 294 (1437) ◽  
pp. 208-234 ◽  
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Main Part ◽  
Similarity Solutions ◽  
Boundary Layer Theory ◽  
Main Stream ◽  
Outer Solution ◽  
Boundary Layer Equations ◽  
Displacement Thickness ◽  
Complicated Form

This paper contains a study of the similarity solutions of the boundary layer equations for the case of strong blowing through a porous surface. The main part of the boundary layer is thick and almost inviscid in these conditions, but there is a thin viscous region where the boundary layer merges into the main stream. The asymptotic solutions appropriate to these two regions are matched to one another when the blowing velocity is large. The skin friction is found from the inner solution, which is independent of the outer solution, but the displacement thickness involves both solutions and is of more complicated form.

Effects of Area Ratio and Mean Rise Angle on the Aerodynamics of Interturbine Ducts

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Yanfeng Zhang ◽  
Shuzhen Hu ◽  
Ali Mahallati ◽  
Xue-Feng Zhang ◽  
Edward Vlasic
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Static Pressure ◽  
Computational Study ◽  
Pressure Rise ◽  
Adverse Pressure Gradient ◽  
Boundary Layer Separation ◽  
Wake Flow ◽  
Height Ratio ◽  
Inlet Swirl

This work, a continuation of a series of investigations on the aerodynamics of aggressive interturbine ducts (ITD), is aimed at providing detailed understanding of the flow physics and loss mechanisms in four different ITD geometries. A systematic experimental and computational study was carried out by varying duct outlet-to-inlet area ratios (ARs) and mean rise angles while keeping the duct length-to-inlet height ratio, Reynolds number, and inlet swirl constant in all four geometries. The flow structures within the ITDs were found to be dominated by the boundary layer separation and counter-rotating vortices in both the casing and hub regions. The duct mean rise angle determined the severity of adverse pressure gradient in the casing's first bend, whereas the duct AR mainly governed the second bend's static pressure rise. The combination of upstream wake flow and the first bend's adverse pressure gradient caused the boundary layer to separate and intensify the strength of counter-rotating vortices. At high mean rise angle, the separation became stronger at the casing's first bend and moved farther upstream. At high ARs, a two-dimensional separation appeared on the casing and resulted in increased loss. Pressure loss penalties increased significantly with increasing duct mean rise angle and AR.

Predicting Turbulent Boundary Layer Wall Pressure Fluctuations in Adverse Pressure Gradients

2005 ◽  
Author(s):  
Frank J. Aldrich
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Pressure Fluctuations ◽  
Adverse Pressure Gradient ◽  
Wall Pressure ◽  
Prediction Tool ◽  
Pressure Gradients ◽  
Wall Pressure Fluctuations ◽  
The Difference

A physics-based approach is employed and a new prediction tool is developed to predict the wavevector-frequency spectrum of the turbulent boundary layer wall pressure fluctuations for subsonic airfoils under the influence of adverse pressure gradients. The prediction tool uses an explicit relationship developed by D. M. Chase, which is based on a fit to zero pressure gradient data. The tool takes into account the boundary layer edge velocity distribution and geometry of the airfoil, including the blade chord and thickness. Comparison to experimental adverse pressure gradient data shows a need for an update to the modeling constants of the Chase model. To optimize the correlation between the predicted turbulent boundary layer wall pressure spectrum and the experimental data, an optimization code (iSIGHT) is employed. This optimization module is used to minimize the absolute value of the difference (in dB) between the predicted values and those measured across the analysis frequency range. An optimized set of modeling constants is derived that provides reasonable agreement with the measurements.

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