A Simple Integral Method for the Calculation of Thick Axisymmetric Turbulent Boundary Layers

SummaryA simple integral method is described for the calculation of a thick axisymmetric turbulent boundary layer. It is shown that the development of the boundary layer can be predicted with acceptable accuracy by using an approximate form of the momentum-integral equation, an appropriate skin-friction law, and an entrainment equation obtained for axisymmetric boundary layers. The method also involves the explicit use of a velocity profile family in order to interrelate some of the integral parameters. Available experimental results have been used to demonstrate the general accuracy of the method.

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Velocity Profiles of Turbulent Boundary Layers

The Aeronautical Journal ◽

10.1017/s000192400005329x ◽

1969 ◽

Vol 73 (698) ◽

pp. 143-147 ◽

Cited By ~ 4

Author(s):

M. K. Bull

Keyword(s):

Experimental Data ◽

Boundary Layer ◽

Turbulent Boundary Layer ◽

Velocity Profile ◽

Boundary Layers ◽

Experimental Work ◽

Constant Pressure ◽

Velocity Profiles ◽

Turbulent Boundary Layers ◽

Boundary Layer Equations

Although a numerical solution of the turbulent boundary-layer equations has been achieved by Mellor and Gibson for equilibrium layers, there are many occasions on which it is desirable to have closed-form expressions representing the velocity profile. Probably the best known and most widely used representation of both equilibrium and non-equilibrium layers is that of Coles. However, when velocity profiles are examined in detail it becomes apparent that considerable care is necessary in applying Coles's formulation, and it seems to be worthwhile to draw attention to some of the errors and inconsistencies which may arise if care is not exercised. This will be done mainly by the consideration of experimental data. In the work on constant pressure layers, emphasis tends to fall heavily on the author's own data previously reported in ref. 1, because the details of the measurements are readily available; other experimental work is introduced where the required values can be obtained easily from the published papers.

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Effects of sandpaper roughness and stream turbulence on the turbulent boundary layer

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/0954406991522734 ◽

1999 ◽

Vol 213 (5) ◽

pp. 507-515 ◽

Cited By ~ 1

Author(s):

J. C. Gibbings ◽

S. M. Al-Shukri

Keyword(s):

Boundary Layer ◽

Turbulent Boundary Layer ◽

Velocity Profile ◽

Skin Friction ◽

Boundary Layers ◽

Pipe Flow ◽

Shape Factor ◽

Turbulent Boundary Layers ◽

Experimental Measurements ◽

Two Dimensional

This paper reports experimental measurements of two-dimensional turbulent boundary layers over sandpaper surfaces under turbulent streams to complement the Nikuradse experiments on pipe flow. The study included the recovery region downstream of the end of transition. Correlations are given for the thickness, the shape factor, the skin friction and the parameters of the velocity profile of the layer. Six further basic differences from the pipe flow are described to add to the five previously reported.

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Turbulent Flow Near a Rough Wall

Journal of Fluids Engineering ◽

10.1115/1.2910065 ◽

1992 ◽

Vol 114 (4) ◽

pp. 537-542 ◽

Cited By ~ 8

Author(s):

Yang-Moon Koh

Keyword(s):

Boundary Layer ◽

Turbulent Boundary Layer ◽

Velocity Profile ◽

Boundary Layers ◽

Simple Formula ◽

Turbulent Boundary Layers ◽

Mean Velocity ◽

Friction Factors ◽

Special Cases ◽

The Mean

By introducing the equivalent roughness which is defined as the distance from the wall to where the velocity gets a certain value (u/uτ ≈ 8.5) and which can be represented by a simple function of the roughness, a simple formula to represent the mean-velocity distribution across the inner layer of a turbulent boundary layer is suggested. The suggested equation is general enough to be applicable to turbulent boundary layers over surfaces of any roughnesses covering from very smooth to completely rough surfaces. The suggested velocity profile is then used to get expressions for pipe-friction factors and skin friction coefficients. These equations are consistent with existing experimental observations and embrace well-known equations (e.g., Prandtl’s friction law for smooth pipes and Colebrook’s formula etc.) as special cases.

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The Departure from Equilibrium of Turbulent Boundary Layers

Aeronautical Quarterly ◽

10.1017/s0001925900004418 ◽

1968 ◽

Vol 19 (1) ◽

pp. 1-19 ◽

Cited By ~ 8

Author(s):

H. McDonald

Keyword(s):

Boundary Layer ◽

Shear Stress ◽

Kinetic Energy ◽

Stress Distribution ◽

Boundary Layers ◽

Turbulent Boundary Layers ◽

Two Dimensional ◽

Pressure Distributions ◽

Momentum Integral ◽

State Examination

SummaryRecently two authors, Nash and Goldberg, have suggested, intuitively, that the rate at which the shear stress distribution in an incompressible, two-dimensional, turbulent boundary layer would return to its equilibrium value is directly proportional to the extent of the departure from the equilibrium state. Examination of the behaviour of the integral properties of the boundary layer supports this hypothesis. In the present paper a relationship similar to the suggestion of Nash and Goldberg is derived from the local balance of the kinetic energy of the turbulence. Coupling this simple derived relationship to the boundary layer momentum and moment-of-momentum integral equations results in quite accurate predictions of the behaviour of non-equilibrium turbulent boundary layers in arbitrary adverse (given) pressure distributions.

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Prediction of Turbulent Boundary Layer Development in Conical Diffusers

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1966_008_054_02 ◽

1966 ◽

Vol 8 (4) ◽

pp. 426-436 ◽

Cited By ~ 2

Author(s):

A. D. Carmichael ◽

G. N. Pustintsev

Keyword(s):

Boundary Layer ◽

Kinetic Energy ◽

Turbulent Boundary Layer ◽

Boundary Layers ◽

Shape Factor ◽

Turbulent Boundary Layers ◽

Energy Deficit ◽

Momentum Thickness ◽

Boundary Layer Development ◽

Layer Development

Methods of predicting the growth of turbulent boundary layers in conical diffusers using the kinetic-energy deficit equation were developed. Three different forms of auxiliary equations were used. Comparison between the measured and predicted results showed that there was fair agreement although there was a tendency to underestimate the predicted momentum thickness and over-estimate the predicted shape factor.

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

Journal of Applied Mechanics ◽

10.1115/1.3625061 ◽

1966 ◽

Vol 33 (2) ◽

pp. 429-437 ◽

Cited By ~ 4

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.

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An Integral Method for Turbulent Boundary Layers on Rotating Disks

Journal of Fluids Engineering ◽

10.1115/1.2910188 ◽

1993 ◽

Vol 115 (4) ◽

pp. 614-619 ◽

Cited By ~ 2

Author(s):

S. Abrahamson ◽

S. Lonnes

Keyword(s):

Boundary Layer ◽

Boundary Layers ◽

Integral Method ◽

Radial Profile ◽

Tangential Velocity ◽

Turbulent Boundary Layers ◽

Similarity Solutions ◽

Layer Growth ◽

Rotating Disks ◽

Boundary Layer Growth

An integral method for computing turbulent boundary layers on rotating disks has been developed using a power law profile for the tangential velocity and a new model for the radial profile. A similarity solution results from the formulation. Radial transport, boundary layer growth, and drag on the disk were computed for the case of a forced vortex frees tream flow. The results were compared to previous similarity solutions. The method was extended to a Rankine vortex freestream flow. Differential equations for boundary layer parameters were developed and solved for different Reynolds numbers to look at the net entrainment, boundary layer growth, and drag on the disk.

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Fundamental studies on the control of turbulent boundary layers. (The effects of a rapid pressure decrease at the start of the convex surface curvature on turbulent boundary layer characteristics).

TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series B ◽

10.1299/kikaib.55.67 ◽

1989 ◽

Vol 55 (509) ◽

pp. 67-72

Author(s):

Hajime YAMAGUCHI

Keyword(s):

Boundary Layer ◽

Turbulent Boundary Layer ◽

Boundary Layers ◽

Convex Surface ◽

Turbulent Boundary Layers ◽

Surface Curvature ◽

Pressure Decrease ◽

Rapid Pressure

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Equilibrium-Turbulent Boundary Layer Prediction for a Proposed Prandtl Mixing Length Distribution

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1968_010_064_02 ◽

1968 ◽

Vol 10 (5) ◽

pp. 426-433 ◽

Cited By ~ 1

Author(s):

F. C. Lockwood

Keyword(s):

Experimental Data ◽

Boundary Layer ◽

Turbulent Boundary Layer ◽

Boundary Layers ◽

Length Distribution ◽

Momentum Equation ◽

Turbulent Boundary Layers ◽

Mixing Length ◽

Good Agreement

The momentum equation is solved numerically for a suggested ramp variation of the Prandtl mixing length across an equilibrium-turbulent boundary layer. The predictions of several important boundary-layer functions are compared with the equilibrium experimental data. Comparisons are also made with some recent universal recommendations for turbulent boundary layers since the equilibrium experimental data are limited. Good agreement is found between the predictions, the experimental data, and the recommendations.

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Compressibility and Variable Density Effects in Turbulent Boundary layers

Journal of Heat Transfer ◽

10.1115/1.2709971 ◽

2006 ◽

Vol 129 (4) ◽

pp. 441-448 ◽

Cited By ~ 5

Author(s):

Kunlun Liu ◽

Richard H. Pletcher

Keyword(s):

Boundary Layer ◽

Mach Number ◽

Turbulent Boundary Layer ◽

Boundary Layers ◽

Wall Temperature ◽

Stokes Equations ◽

Variable Density ◽

Turbulent Boundary Layers ◽

Reynolds Analogy ◽

Density Effects

Two compressible turbulent boundary layers have been calculated by using direct numerical simulation. One case is a subsonic turbulent boundary layer with constant wall temperature for which the wall temperature is 1.58 times the freestream temperature and the other is a supersonic adiabatic turbulent boundary layer subjected to a supersonic freestream with a Mach number 1.8. The purpose of this study is to test the strong Reynolds analogy (SRA), the Van Driest transformation, and the applicability of Morkovin’s hypothesis. For the first case, the influence of the variable density effects will be addressed. For the second case, the role of the density fluctuations, the turbulent Mach number, and dilatation on the compressibility will be investigated. The results show that the Van Driest transformation and the SRA are satisfied for both of the flows. Use of local properties enable the statistical curves to collapse toward the corresponding incompressible curves. These facts reveal that both the compressibility and variable density effects satisfy the similarity laws. A study about the differences between the compressibility effects and the variable density effects associated with heat transfer is performed. In addition, the difference between the Favre average and Reynolds average is measured, and the SGS terms of the Favre-filtered Navier-Stokes equations are calculated and analyzed.

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