A Simple Theory for the Two-Dimensional Compressible Turbulent Boundary Layer

1972 ◽  
Vol 94 (3) ◽  
pp. 636-642 ◽  
Author(s):  
F. M. White ◽  
G. H. Christoph
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Flat Plate ◽  
Skin Friction ◽  
Plate Theory ◽  
Momentum Equation ◽  
Shape Factors ◽  
Law Of The Wall ◽  
Compressible Turbulent Flow ◽  
New Equation

A new approach is proposed for analyzing the compressible turbulent boundary layer with arbitrary pressure gradient. Utilizing a compressible law-of-the-wall and a Crocco energy approximation, the new theory integrates the momentum equation across the boundary layer in terms of inner variables only. The result is a single first-order ordinary differential equation for skin friction, devoid of integral thicknesses and shape factors. When analyzed for flat plate flow, this new equation has an exact solution apparently superior in accuracy to any other flat plate theory (Table 1). The new equation also agrees well with supersonic skin friction data in both favorable and adverse pressure gradients. The new theory contains an explicit separation criterion and is the simplest and possibly most accurate existing analysis for compressible turbulent flow.

Turbulent boundary-layer interaction with a shock wave at a compression corner

1984 ◽  
Vol 143 ◽  
pp. 23-46 ◽  
Author(s):  
S. Agrawal ◽  
A. F. Messiter
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Turbulent Boundary Layer ◽  
Momentum Equation ◽  
Wall Layer ◽  
Oblique Shock Wave ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Compression Corner ◽  
Boundary Layer Interaction

The local interaction of an oblique shock wave with an unseparated turbulent boundary layer at a shallow two-dimensional compression corner is described by asymptotic expansions for small values of the non-dimensional friction velocity and the flow turning angle. It is assumed that the velocity-defect law and the law of the wall, adapted for compressible flow, provide an asymptotic representation of the mean velocity profile in the undisturbed boundary layer. Analytical solutions for the local mean-velocity and pressure distributions are derived in supersonic, hypersonic and transonic small-disturbance limits, with additional intermediate limits required at distances from the corner that are small in comparison with the boundary-layer thickness. The solutions describe small perturbations in an inviscid rotational flow, and show good agreement with available experimental data in most cases where effects of separation can be neglected. Calculation of the wall shear stress requires solution of the boundary-layer momentum equation in a sublayer which plays the role of a new thinner boundary layer but which is still much thicker than the wall layer. An analytical solution is derived with a mixing-length approximation, and is in qualitative agreement with one set of measured values.

Wall Temperature and Prandtl Number Effects on Turbulent Boundary Layer Thicknesses and Shape Factors for Subsonic Compressible Gas Flow Over a Flat Plate

1969 ◽  
Author(s):  
W. J. Kelnhofer
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Prandtl Number ◽  
Flat Plate ◽  
Wall Temperature ◽  
Gas Flow ◽  
Density Variation ◽  
Shape Factors ◽  
Heating And Cooling ◽  
Subsonic Gas Flow

Based on n-power velocity and temperature profiles a method of computing various turbulent boundary layer thicknesses and shape factors affected by wall temperature and Prandtl number for fully developed subsonic gas flow over a flat plate is presented. Density variation in the boundary layer is given main consideration. Numerical computations include both heating and cooling of gas. Boundary layer thicknesses and shape factors are shown to be significantly affected by wall temperature and to a lesser degree by Prandtl number. An experiment is described which involved air flow up to 30 m/sec over a flat plate maintained at constant wall temperatures up to 250 C. Comparisons between theory and experiment are good.

Reynolds-number scaling of the flat-plate turbulent boundary layer

2000 ◽  
Vol 422 ◽  
pp. 319-346 ◽  
Author(s):  
DAVID B. DE GRAAFF ◽  
JOHN K. EATON
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Turbulent Boundary Layer ◽  
Flat Plate ◽  
Extensive Study ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
The Mean

Despite extensive study, there remain significant questions about the Reynolds-number scaling of the zero-pressure-gradient flat-plate turbulent boundary layer. While the mean flow is generally accepted to follow the law of the wall, there is little consensus about the scaling of the Reynolds normal stresses, except that there are Reynolds-number effects even very close to the wall. Using a low-speed, high-Reynolds-number facility and a high-resolution laser-Doppler anemometer, we have measured Reynolds stresses for a flat-plate turbulent boundary layer from Reθ = 1430 to 31 000. Profiles of u′2, v′2, and u′v′ show reasonably good collapse with Reynolds number: u′2 in a new scaling, and v′2 and u′v′ in classic inner scaling. The log law provides a reasonably accurate universal profile for the mean velocity in the inner region.

scholarly journals Large-eddy simulation of separation and reattachment of a flat plate turbulent boundary layer

2015 ◽  
Vol 785 ◽  
pp. 78-108 ◽  
Author(s):  
W. Cheng ◽  
D. I. Pullin ◽  
R. Samtaney
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Flat Plate ◽  
Skin Friction ◽  
Separation Bubble ◽  
Reynolds Numbers ◽  
Wall Model ◽  
Large Eddy ◽  
Layer Flow

We present large-eddy simulations (LES) of separation and reattachment of a flat-plate turbulent boundary-layer flow. Instead of resolving the near wall region, we develop a two-dimensional virtual wall model which can calculate the time- and space-dependent skin-friction vector field at the wall, at the resolved scale. By combining the virtual-wall model with the stretched-vortex subgrid-scale (SGS) model, we construct a self-consistent framework for the LES of separating and reattaching turbulent wall-bounded flows at large Reynolds numbers. The present LES methodology is applied to two different experimental flows designed to produce separation/reattachment of a flat-plate turbulent boundary layer at medium Reynolds number $Re_{{\it\theta}}$ based on the momentum boundary-layer thickness ${\it\theta}$. Comparison with data from the first case at $Re_{{\it\theta}}=2000$ demonstrates the present capability for accurate calculation of the variation, with the streamwise co-ordinate up to separation, of the skin friction coefficient, $Re_{{\it\theta}}$, the boundary-layer shape factor and a non-dimensional pressure-gradient parameter. Additionally the main large-scale features of the separation bubble, including the mean streamwise velocity profiles, show good agreement with experiment. At the larger $Re_{{\it\theta}}=11\,000$ of the second case, the LES provides good postdiction of the measured skin-friction variation along the whole streamwise extent of the experiment, consisting of a very strong adverse pressure gradient leading to separation within the separation bubble itself, and in the recovering or reattachment region of strongly-favourable pressure gradient. Overall, the present two-dimensional wall model used in LES appears to be capable of capturing the quantitative features of a separation-reattachment turbulent boundary-layer flow at low to moderately large Reynolds numbers.

scholarly journals Skin Friction in the Laminar Boundary Layer in Compressible Flow

1949 ◽  
Vol 1 (2) ◽  
pp. 137-164 ◽  
Author(s):  
A. D. Young
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mach Number ◽  
Flat Plate ◽  
Skin Friction ◽  
Compressible Flow ◽  
Laminar Boundary Layer ◽  
Momentum Equation ◽  
Forward Movement ◽  
Semi Empirical

SummaryFrom an analysis of the work of Crocco and others, semi-empirical formulae are derived for the skin friction on a flat plate at zero incidence with a laminar boundary layer. These formulae arefor the general case of heat transfer, andwhen there is no heat transfer.The problem of heat transfer and the effect of radiation are discussed in the light of these formula;. The second formula is then utilised in the development of an approximate method for solving the momentum equation of the boundary layer on a cylinder without heat transfer. The method indicates that with increase of Mach number there is a marked forward movement of separation from a flat plate in the presence of a constant adverse velocity gradient.

scholarly journals The Importance Of The Law Of The Wall

2015 ◽  
Vol 20 (4) ◽  
pp. 857-869 ◽  
Author(s):  
B.C. Mandal ◽  
H.P. Mazumdar
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Flat Plate ◽  
Law Of The Wall ◽  
Smooth Pipe ◽  
The Law

Abstract This paper reports some results of turbulent boundary layer computation. The calculation is made assuming that law of the wall is valid throughout the boundary layer. Simple relations are proposed for friction for a smooth pipe and a flat plate at zero incidence. The results are compared with recent measurements. Encouraging results are obtained for both the cases of flows.

Wind Tunnel Tests of Two Lucy Ashton Reflex Geosims

1970 ◽  
Vol 14 (04) ◽  
pp. 241-276
Author(s):  
P. N. Joubert ◽  
N. Matheson
Keyword(s):  
Boundary Layer ◽  
Wind Tunnel ◽  
Turbulent Boundary Layer ◽  
Flat Plate ◽  
Skin Friction ◽  
Viscous Drag ◽  
Two Dimensional ◽  
Local Skin ◽  
Wind Tunnel Tests ◽  
Local Skin Friction

A 9-ft and a 4½-ft reflex model of the Lucy Ashton were tested in a wind tunnel. Both pins and wires were used as stimulators to promote a turbulent boundary layer. The effects of the stimulators could be taken into account by considering the virtual origin of the turbulent boundary layer. Slightly different viscous drag curves were found for each model, both with a slope much steeper than previously anticipated. The skin friction was determined using two independent methods. Large increases and deficits in local skin friction coefficients were found at the bow and stern of the models respectively as compared with those for a two-dimensional flat plate.

Wall Temperature and Prandtl Number Effects on Turbulent Boundary Layer Thicknesses and Shape Factors for Subsonic Compressible Gas Flow Over a Flat Plate

1969 ◽  
Vol 91 (4) ◽  
pp. 281-289
Author(s):  
W. J. Kelnhofer
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Prandtl Number ◽  
Flat Plate ◽  
Wall Temperature ◽  
Gas Flow ◽  
Density Variation ◽  
Shape Factors ◽  
Heating And Cooling ◽  
Subsonic Gas Flow

Based on n-power velocity and temperature profiles a method of computing various turbulent boundary layer thicknesses and shape factors affected by wall temperature and Prandtl number for fully developed subsonic gas flow over a flat plate is presented. Density variation in the boundary layer is given main consideration. Numerical computations include both heating and cooling of the gas. Boundary layer thicknesses and shape factors are shown to be significantly affected by wall temperature and to a lesser degree by Prandtl number. An experiment is described which involved air flow up to 30 m/sec over a flat plate maintained at constant wall temperatures up to 250 C. Comparisons between theory and experiment are good.

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