A Simple, Accurate Integral Solution for Accelerating Turbulent Boundary Layers With Transpiration

1995 ◽  
Vol 117 (3) ◽  
pp. 535-538 ◽  
Author(s):  
James Sucec
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
Skin Friction ◽  
Boundary Layers ◽  
Integral Form ◽  
Skin Friction Coefficient ◽  
Momentum Equation ◽  
Turbulent Boundary Layers ◽  
Integral Solution ◽  
Good Agreement

The inner law for transpired turbulent boundary layers is used as the velocity profile in the integral form of the x momentum equation. The resulting ordinary differential equation is solved numerically for the skin friction coefficient, as well as boundary layer thicknesses, as a function of position along the surface. Predicted skin friction coefficients are compared to experimental data and exhibit reasonably good agreement with the data for a variety of different cases. These include blowing and suction, with constant blowing fractions F for both mild and severe acceleration. Results are also presented for more complicated cases where F varies with x along the surface.

Prediction of Transpired Turbulent Boundary Layers With Arbitrary Pressure Gradients

1997 ◽  
Vol 119 (3) ◽  
pp. 526-532 ◽  
Author(s):  
Miodrag Oljaca ◽  
James Sucec
Keyword(s):  
Experimental Data ◽  
Skin Friction ◽  
Boundary Layers ◽  
Integral Method ◽  
Skin Friction Coefficient ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients ◽  
Wide Range ◽  
Better Than ◽  
Good Agreement

An integral method, using Coles combined inner and outer law as the velocity profile, is developed for calculation of turbulent boundary layers with blowing or suction and pressure gradients. The resulting ordinary differential equations are solved numerically for the distribution of skin friction coefficient and integral thickness along the surface. Comparisons of predicted skin friction coefficients with experimental data are made for a wide range of blowing and suction rates and for various pressure gradients, including adverse, zero and a strong favorable gradient. In addition to good agreement with experimental data for constant blowing fractions F, the method is also successfully tested on cases where the blowing fraction is variable with position. Predictions, in general, exhibit satisfactory agreement with the data. The integral method predictions are comparable to, or better than, a number of finite difference procedures in a limited number of cases where comparisons were made.

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

1968 ◽  
Vol 10 (5) ◽  
pp. 426-433 ◽  
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.

Low-Reynolds-Number Turbulent Boundary Layers in Zero and Favorable Pressure Gradients

1983 ◽  
Vol 27 (03) ◽  
pp. 147-157 ◽  
Author(s):  
A. J. Smits ◽  
N. Matheson ◽  
P. N. Joubert
Keyword(s):  
Friction Coefficient ◽  
Skin Friction ◽  
Boundary Layers ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Skin Friction Coefficient ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Low Reynolds

This paper reports the results of an extensive experimental investigation into the mean flow properties of turbulent boundary layers with momentum-thickness Reynolds numbers less than 3000. Zero pressure gradient and favorable pressure gradients were studied. The velocity profiles displayed a logarithmic region even at very low Reynolds numbers (as low as Rθ = 261). The results were independent of the leading-edge shape, and the pin-type turbulent stimulators performed well. It was found that the shape and Clauser parameters were a little higher than the correlation proposed by Coles [10], and the skin friction coefficient was a little lower. The skin friction coefficient behavior could be fitted well by a simple power-law relationship in both zero and favorable pressure gradients.

Turbulent Boundary Layers With Mass Transfer by the Dorodnitsyn Finite Element Method

1987 ◽  
Vol 54 (1) ◽  
pp. 197-202 ◽  
Author(s):  
C. A. J. Fletcher ◽  
R. W. Fleet
Keyword(s):  
Mass Transfer ◽  
Finite Element ◽  
Boundary Layers ◽  
Adverse Pressure Gradient ◽  
Skin Friction Coefficient ◽  
Turbulent Boundary Layers ◽  
Element Formulation ◽  
Surface Mass ◽  
Surface Mass Transfer ◽  
Good Agreement

The Dorodnitsyn finite element formulation is extended to cover incompressible, two-dimensional turbulent boundary layers with surface mass transfer in the normal direction. The method is shown to give accurate and economical answers with only eleven points spanning the boundary layer. Good agreement is obtained when the computational solutions are compared with the experimental results of McQuaid [13] for skin friction coefficient, displacement and momentum thickness and velocity profiles. Zero and adverse pressure gradient and discontinuous injection cases have been considered.

Approach to an asymptotic state for zero pressure gradient turbulent boundary layers

2007 ◽  
Vol 365 (1852) ◽  
pp. 755-770 ◽  
Author(s):  
Hassan M Nagib ◽  
Kapil A Chauhan ◽  
Peter A Monkewitz
Keyword(s):  
Experimental Data ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Boundary Layers ◽  
Classical Theory ◽  
Reynolds Numbers ◽  
Turbulent Boundary Layers ◽  
Careful Examination ◽  
Mean Velocity ◽  
Logarithmic Law

Flat plate turbulent boundary layers under zero pressure gradient at high Reynolds numbers are studied to reveal appropriate scale relations and asymptotic behaviour. Careful examination of the skin-friction coefficient results confirms the necessity for direct and independent measurement of wall shear stress. We find that many of the previously proposed empirical relations accurately describe the local C f behaviour when modified and underpinned by the same experimental data. The variation of the integral parameter, H , shows consistent agreement between the experimental data and the relation from classical theory. In accordance with the classical theory, the ratio of Δ and δ asymptotes to a constant. Then, the usefulness of the ratio of appropriately defined mean and turbulent time-scales to define and diagnose equilibrium flow is established. Next, the description of mean velocity profiles is revisited, and the validity of the logarithmic law is re-established using both the mean velocity profile and its diagnostic function. The wake parameter, Π , is shown to reach an asymptotic value at the highest available experimental Reynolds numbers if correct values of logarithmic-law constants and an appropriate skin-friction estimate are used. The paper closes with a discussion of the Reynolds number trends of the outer velocity defect which are important to establish a consistent similarity theory and appropriate scaling.

Investigation of Flat-Plate Hypersonic, Turbulent Boundary Layers With Heat Transfer

1961 ◽  
Vol 28 (3) ◽  
pp. 323-329 ◽  
Author(s):  
Eva M. Winkler
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Friction Coefficient ◽  
Mach Number ◽  
Flat Plate ◽  
Skin Friction ◽  
Boundary Layers ◽  
Skin Friction Coefficient ◽  
Turbulent Boundary Layers ◽  
Reynolds Analogy

Naturally turbulent boundary layers on a cooled flat plate have been investigated at several distances from the leading edge of the plate at a Mach number of 5.2 for three rates of steady-state heat transfer to the surface. Measurements of Pitot and static pressures and of total and wall temperatures made it possible to compute velocity profiles, static-temperature profiles, and boundary-layer parameters without resorting to assumptions. The data demonstrate that the Reynolds analogy between skin friction and heat transfer is valid for all conditions of the present experiments. With increasing rate of heat transfer to the surface, the skin-friction coefficient was found to decrease, a phenomenon opposite to that predicted by theories and empirical relations. On the basis of the present data and other published results of compressible and incompressible turbulent boundary-layer skin friction, a simple relation was devised which describes closely the variation of the skin-friction coefficient with Mach number, heat-transfer rate, and momentum-thickness Reynolds number.

On the Skin Friction Coefficient for a Fully Rough Flat Plate

1983 ◽  
Vol 105 (3) ◽  
pp. 364-365 ◽  
Author(s):  
A. F. Mills ◽  
Xu Hang
Keyword(s):  
Experimental Data ◽  
Friction Coefficient ◽  
Velocity Profile ◽  
Skin Friction ◽  
Skin Friction Coefficient ◽  
Momentum Equation ◽  
Average Error ◽  
Friction Theory ◽  
Rough Plate ◽  
New Skin

A comparison of the Prandtl-Schlichting formula for skin friction of a fully rough plate with recently obtained experimental data shows an average error of 17.5 percent. It is suggested that the reason for this discrepancy is a failure to account for the wake component of the velocity profile. The integral momentum equation is used to derive a new skin friction theory which when compared to the same data gives an average error of 2.7 percent. A new skin friction formula is proposed which is valid over a wide parameter range.

An Integral Solution for Skin Friction in Turbulent Flow Over Aerodynamically Rough Surfaces With an Arbitrary Pressure Gradient

2014 ◽  
Vol 136 (8) ◽  
Author(s):  
James Sucec
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Integral Form ◽  
Skin Friction Coefficient ◽  
Momentum Equation ◽  
Data Sets ◽  
Momentum Thickness ◽  
Local Skin ◽  
Law Of The Wall

The combined law of the wall and wake, with the inclusion of the “roughness depression function” for the inner law in the “Log” region, is used as the inner coordinates' velocity profile in the integral form of the x momentum equation to solve for the local skin friction coefficient. The “equivalent sand grain roughness” concept is employed in the roughness depression function in the solution. Calculations are started at the beginning of roughness on a surface, as opposed to starting them using the measured experimental values at the first data point, when making comparisons of predictions with data sets. The dependence of the velocity wake strength on both pressure gradient and momentum thickness Reynolds number are taken into account. Comparisons of the prediction with experimental skin friction data, from the literature, have been made for some adverse, zero, and favorable (accelerating flows) pressure gradients. Predictions of the shape factor, roughness Reynolds number, and momentum thickness Reynolds number and comparisons with data are also made for some cases. In addition, some comparisons with the predictions of earlier investigators have also been made.

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