Mean Velocity, Reynolds Shear Stress, and Fluctuations of Velocity and Pressure Due to Log Laws in a Turbulent Boundary Layer and Origin Offset by Prandtl Transposition Theorem

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Noor Afzal ◽  
Abu Seena
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Skin Friction ◽  
Velocity Vector ◽  
Reynolds Shear Stress ◽  
Mean Velocity ◽  
Time Period ◽  
Log Law ◽  
Turbulent Burst

The maxima of Reynolds shear stress and turbulent burst mean period time are crucial points in the intermediate region (termed as mesolayer) for large Reynolds numbers. The three layers (inner, meso, and outer) in a turbulent boundary layer have been analyzed from open equations of turbulent motion, independent of any closure model like eddy viscosity or mixing length, etc. Little above (or below not considered here) the critical point, the matching of mesolayer predicts the log law velocity, peak of Reynolds shear stress domain, and turbulent burst time period. The instantaneous velocity vector after subtraction of mean velocity vector yields the velocity fluctuation vector, also governed by log law. The static pressure fluctuation p′ also predicts log laws in the inner, outer, and mesolayer. The relationship between u′/Ue with u/Ue from structure of turbulent boundary layer is presented in inner, meso, and outer layers. The turbulent bursting time period has been shown to scale with the mesolayer time scale; and Taylor micro time scale; both have been shown to be equivalent in the mesolayer. The shape factor in a turbulent boundary layer shows linear behavior with nondimensional mesolayer length scale. It is shown that the Prandtl transposition (PT) theorem connects the velocity of normal coordinate y with s offset to y + a, then the turbulent velocity profile vector and pressure fluctuation log laws are altered; but skin friction log law, based on outer velocity Ue, remains independent of a the offset of origin. But if skin friction log law is based on bulk average velocity Ub, then skin friction log law depends on a, the offset of origin. These predictions are supported by experimental and direct numerical simulation (DNS) data.

The accelerated fully rough turbulent boundary layer

1977 ◽  
Vol 82 (3) ◽  
pp. 507-528 ◽  
Author(s):  
Hugh W. Coleman ◽  
Robert J. Moffat ◽  
William M. Kays
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Stress Tensor ◽  
Skin Friction ◽  
Shape Factor ◽  
Reynolds Shear Stress ◽  
Smooth Wall ◽  
Mean Velocity

The behaviour of a fully rough turbulent boundary layer subjected to favourable pressure gradients both with and without blowing was investigated experimentally using a porous test surface composed of densely packed spheres of uniform size. Measurements of profiles of mean velocity and the components of the Reynolds-stress tensor are reported for both unblown and blown layers. Skin-friction coefficients were determined from measurements of the Reynolds shear stress and mean velocity.An appropriate acceleration parameterKrfor fully rough layers is defined which is dependent on a characteristic roughness dimension but independent of molecular viscosity. For a constant blowing fractionFgreater than or equal to zero, the fully rough turbulent boundary layer reaches an equilibrium state whenKris held constant. Profiles of the mean velocity and the components of the Reynolds-stress tensor are then similar in the flow direction and the skin-friction coefficient, momentum thickness, boundary-layer shape factor and the Clauser shape factor and pressure-gradient parameter all become constant.Acceleration of a fully rough layer decreases the normalized turbulent kinetic energy and makes the turbulence field much less isotropic in the inner region (forFequal to zero) compared with zero-pressure-gradient fully rough layers. The values of the Reynolds-shear-stress correlation coefficients, however, are unaffected by acceleration or blowing and are identical with values previously reported for smooth-wall and zero-pressure-gradient rough-wall flows. Increasing values of the roughness Reynolds number with acceleration indicate that the fully rough layer does not tend towards the transitionally rough or smooth-wall state when accelerated.

Evolution of a moderately short turbulent boundary layer in a severe pressure gradient

1974 ◽  
Vol 64 (4) ◽  
pp. 763-774 ◽  
Author(s):  
R. G. Deissler
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Reynolds Shear Stress ◽  
Pressure Gradients ◽  
Mean Velocity ◽  
Mass Injection ◽  
The Mean ◽  
Theoretical Results

The early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analysed. For sufficiently large accelerations (or pressure gradients) and for total normal strains which are not excessive, the equation for the Reynolds shear stress simplifies to give a stress that remains approximately constant as it is convected along streamlines. The theoretical results for the evolution of the mean velocity in favourable and adverse pressure gradients agree well with experiment for the cases considered. A calculation which includes mass injection at the wall is also given.

Calculation of a Turbulent Boundary Layer Downstream of a Small Step Change in Surface Roughness

1975 ◽  
Vol 26 (3) ◽  
pp. 202-210 ◽  
Author(s):  
R A Antonia ◽  
D H Wood
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Reynolds Shear Stress ◽  
Step Change ◽  
Rough Wall ◽  
Small Step ◽  
Mean Velocity ◽  
Good Agreement

SummaryMeasurements of mean velocity and Reynolds shear stress have been made in a turbulent boundary layer downstream of a small step change in surface roughness. Upstream of the step the surface is smooth, while downstream it consists of a d-type rough wall made up by a series of two-dimensional elements of square cross section placed transversely across the flow and spaced one element width apart in the direction of the flow. The calculated mean velocity and Reynolds shear stress profiles obtained using the method of Bradshaw, Ferriss and Atwell are in good agreement with the measurements throughout the relaxation region of the layer. Well downstream the calculation method adequately reproduces the self-preserving features of a d-type rough wall.

On turbulent boundary-layer separation

1968 ◽  
Vol 32 (2) ◽  
pp. 293-304 ◽  
Author(s):  
V. A. Sandborn ◽  
C. Y. Liu
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Analytical Study ◽  
Boundary Layer Separation ◽  
Mean Velocity ◽  
Laminar Separation ◽  
Layer Separation ◽  
Turbulent Separation ◽  
Separation Model

An experimental and analytical study of the separation of a turbulent boundary layer is reported. The turbulent boundary-layer separation model proposed by Sandborn & Kline (1961) is demonstrated to predict the experimental results. Two distinct turbulent separation regions, an intermittent and a steady separation, with correspondingly different velocity distributions are confirmed. The true zero wall shear stress turbulent separation point is determined by electronic means. The associated mean velocity profile is shown to belong to the same family of profiles as found for laminar separation. The velocity distribution at the point of reattachment of a turbulent boundary layer behind a step is also shown to belong to the laminar separation family.Prediction of the location of steady turbulent boundary-layer separation is made using the technique employed by Stratford (1959) for intermittent separation.

Turbulent Boundary Layer Subjected to a Sudden Change in Surface Roughness and Temperature

1999 ◽  
Author(s):  
João Henrique D. Guimarães ◽  
Sergio J. F. dos Santos ◽  
Jian Su ◽  
Atila P. Silva Freire
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Turbulent Boundary Layer ◽  
Skin Friction ◽  
Sudden Change ◽  
Stanton Number ◽  
Temperature Profiles ◽  
Internal Layers ◽  
Mean Velocity ◽  
Wall Boundary Conditions

Abstract In present work, the dynamic and thermal behaviour of flows that develop over surfaces that simultaneously present a sudden change in surface roughness and temperature are discussed. In particular, the work is concerned with the physical validation of a newly proposed formulation for the near wall temperature profile. The theory uses the concept of the displacement in origin, together with some asymptotic arguments, to propose a new expression for the logarithmic region of the turbulent boundary layer. The new expressions are, therefore, of universal applicability, being independent of the type of rough surface considered. The present formulation may be used to give wall boundary conditions for two-equation differential models. The theoretical results are validated with experimental data obtained for flows that develop over flat surfaces with sudden changes in surface roughness and in temperature conditions. Measurements of mean velocity and of mean temperature are presented. A reduction of the data provides an estimate of the skin-friction coefficient, the Stanton number, the displacement in origin for both the velocity and the temperature profiles, and the thickness of the internal layers for the velocity and temperature profiles. The skin-friction co-efficient was calculated based on the chart method of Perry and Joubert (J.F.M., 17, 193–211, 1963) and on a balance of the integral momentum equation. The same chart method was used for the evaluation of the Stanton number and the displacement in origin.

Power Law Velocity Profile in the Turbulent Boundary Layer on Transitional Rough Surfaces

2007 ◽  
Vol 129 (8) ◽  
pp. 1083-1100 ◽  
Author(s):  
Noor Afzal
Keyword(s):  
Boundary Layer ◽  
Functional Equation ◽  
Reynolds Number ◽  
Turbulent Boundary Layer ◽  
Velocity Profile ◽  
Skin Friction ◽  
Power Law ◽  
Wall Roughness ◽  
Closure Model ◽  
Log Law

A new approach to scaling of transitional wall roughness in turbulent flow is introduced by a new nondimensional roughness scale ϕ. This scale gives rise to an inner viscous length scale ϕν∕uτ, inner wall transitional variable, roughness friction Reynolds number, and roughness Reynolds number. The velocity distribution, just above the roughness level, turns out to be a universal relationship for all kinds of roughness (transitional, fully smooth, and fully rough surfaces), but depends implicitly on roughness scale. The open turbulent boundary layer equations, without any closure model, have been analyzed in the inner wall and outer wake layers, and matching by the Izakson-Millikan-Kolmogorov hypothesis leads to an open functional equation. An alternate open functional equation is obtained from the ratio of two successive derivatives of the basic functional equation of Izakson and Millikan, which admits two functional solutions: the power law velocity profile and the log law velocity profile. The envelope of the skin friction power law gives the log law, as well as the power law index and prefactor as the functions of roughness friction Reynolds number or skin friction coefficient as appropriate. All the results for power law and log law velocity and skin friction distributions, as well as power law constants are explicitly independent of the transitional wall roughness. The universality of these relations is supported very well by extensive experimental data from transitional rough walls for various different types of roughnesses. On the other hand, there are no universal scalings in traditional variables, and different expressions are needed for various types of roughness, such as inflectional roughness, monotonic roughness, and others. To the lowest order, the outer layer flow is governed by the nonlinear turbulent wake equations that match with the power law theory as well as log law theory, in the overlap region. These outer equations are in equilibrium for constant value of m, the pressure gradient parameter, and under constant eddy viscosity closure model, the analytical and numerical solutions are presented.

The effects of expansion on the turbulence structure of compressible boundary layers

1998 ◽  
Vol 367 ◽  
pp. 67-105 ◽  
Author(s):  
STEPHEN A. ARNETTE ◽  
MO SAMIMY ◽  
GREGORY S. ELLIOTT
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Reynolds Shear Stress ◽  
Energy Levels ◽  
Plate Boundary ◽  
Velocity Profiles ◽  
Turbulence Structure ◽  
Mean Velocity ◽  
Measured Velocity ◽  
Vorticity Transport

A fully developed Mach 3 turbulent boundary layer subjected to four expansion regions (centred and gradual expansions of 7° and 14°) was investigated with laser Doppler velocimetry. Measurements were acquired in the incoming flat-plate boundary layer and to s/δ≃20 downstream of the expansions. While mean velocity profiles exhibit significant progress towards recovery by the most downstream measurements, the turbulence structure remains far from equilibrium. Comparisons of computed (method of characteristics) and measured velocity profiles indicate that the post-expansion flow evolution is largely inviscid for approximately 10δ. Turbulence levels decrease across the expansion, and the reductions increase in severity as the wall is approached. Downstream of the 14° expansions, the reductions are more severe and reverse transition is indicated by sharp reductions in turbulent kinetic energy levels and a change in sign of the Reynolds shear stress. Dimensionless parameters such as anisotropy and shear stress correlation coefficient highlight the complex evolution of the post-expansion boundary layer. An examination of the compressible vorticity transport equation and estimates of the perturbation impulses attributable to streamline curvature, acceleration, and dilatation both confirm dilatation to be the primary stabilizer. However, the dilatation impulse increases only slightly for the 14° expansions, so the dramatic differences downstream of the 7° and 14° expansions indicate nonlinear boundary layer response. Differences attributable to the varied radii of surface curvature are fleeting for the 7° expansions, but persist through the spatial extent of the measurements for the 14° expansions.

High-order moments of Reynolds shear stress fluctuations in a turbulent boundary layer

1973 ◽  
Vol 58 (3) ◽  
pp. 581-593 ◽  
Author(s):  
R. A. Antonia ◽  
J. D. Atkinson
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Turbulent Boundary Layer ◽  
Fourth Order ◽  
High Speed ◽  
Reynolds Shear Stress ◽  
Digital Data ◽  
Normal Velocity ◽  
The Third ◽  
Digital Data Acquisition System

The cumulant-discard approach is used to predict the third- and fourth-order moments and the probability density of turbulent Reynolds shear stress fluctuations uv, the streamwise and normal velocity fluctuations being represented by u and v respectively. Measurements of these quantities in a turbulent boundary layer are presented, with the required statistics of uv obtained by the use of a high-speed digital data-acquisition system. Including correlations between u and u up to the fourth order, the cumulant-discard predictions are in close agreement with the measurements in the inner region of the layer but only qualitatively follow the experimental results in the outer intermittent region. In this latter region, predictions for the third- and fourth-order moments of uv are also obtained by assuming that the properties of both turbulent and irrotational fluctuations are Gaussian and by using some of the available conditional averages of u, v and uv.

Measurements in an Axisymmetric Turbulent Boundary Layer Along a Circular Cylinder

1976 ◽  
Vol 27 (3) ◽  
pp. 217-228 ◽  
Author(s):  
Noor Afzal ◽  
K P Singh
Keyword(s):  
Boundary Layer ◽  
Circular Cylinder ◽  
Turbulent Boundary Layer ◽  
Constant Pressure ◽  
Reynolds Shear Stress ◽  
Longitudinal Velocity ◽  
Outer Region ◽  
Wall Region ◽  
Mean Velocity ◽  
Dimensional Boundary

SummaryIn an axisymmetric turbulent boundary layer along a circular cylinder at constant pressure, measurements have been made of mean velocity profile and turbulence characteristics: longitudinal velocity fluctuations, Reynolds shear stress, transverse correlation and spectrum. It has been found that the qualitative behaviour of an axisymmetric turbulent boundary layer is similar to that of a two-dimensional boundary layer in the wall region, where as in the outer region the effects of transverse curvature are observed.

scholarly journals Toward a K-Profile Parameterization of Langmuir Turbulence in Shallow Coastal Shelves

2015 ◽  
Vol 45 (12) ◽  
pp. 2869-2895 ◽  
Author(s):  
Nityanand Sinha ◽  
Andres E. Tejada-Martínez ◽  
Cigdem Akan ◽  
Chester E. Grosch
Keyword(s):  
Shear Stress ◽  
Shallow Water ◽  
Water Column ◽  
Reynolds Shear Stress ◽  
Vertical Mixing ◽  
Stokes Drift ◽  
Langmuir Turbulence ◽  
Mean Velocity ◽  
Shear Current ◽  
Log Law

AbstractInteraction between the wind-driven shear current and the Stokes drift velocity induced by surface gravity waves gives rise to Langmuir turbulence in the upper ocean. Langmuir turbulence consists of Langmuir circulation (LC) characterized by a wide range of scales. In unstratified shallow water, the largest scales of Langmuir turbulence engulf the entire water column and thus are referred to as full-depth LC. Large-eddy simulations (LESs) of Langmuir turbulence with full-depth LC in a wind-driven shear current have revealed that vertical mixing due to LC erodes the bottom log-law velocity profile, inducing a profile resembling a wake law. Furthermore, in the interior of the water column, two sources of Reynolds shear stress, turbulent (nonlocal) transport and local Stokes drift shear production, can combine to lead to negative mean velocity shear. Meanwhile, near the surface, Stokes drift shear serves to intensify small-scale eddies leading to enhanced vertical mixing and disruption of the surface log law. A K-profile parameterization (KPP) of the Reynolds shear stress comprising local and nonlocal components is introduced, capturing these basic mechanisms by which Langmuir turbulence in unstratified shallow water impacts the vertical mixing of momentum. Single-water-column, Reynolds-averaged Navier–Stokes simulations with the new parameterization are presented, showing good agreement with LES in terms of mean velocity. Results show that coefficients in the KPP may be parameterized based on attributes of the full-depth LC.

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