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.

Alternate Scales for Turbulent Boundary Layer on Transitional Rough Walls: Universal Log Laws

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Noor Afzal
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Surface Roughness ◽  
Reynolds Number ◽  
Turbulent Boundary Layer ◽  
Skin Friction ◽  
Velocity Profiles ◽  
Pressure Gradients ◽  
Wall Roughness ◽  
Extensive Experimental Data

The present work deals with four new alternate transitional surface roughness scales for description of the turbulent boundary layer. The nondimensional roughness scale ϕ is associated with the transitional roughness wall inner variable ζ=Z+∕ϕ, the roughness friction Reynolds number Rϕ=Rτ∕ϕ, and the roughness Reynolds number Reϕ=Re∕ϕ. The two layer theory for turbulent boundary layers in the variables, mentioned above, is presented by method of matched asymptotic expansions for large Reynolds numbers. The matching in the overlap region is carried out by the Izakson–Millikan–Kolmogorov hypothesis, which gives the velocity profiles and skin friction universal log laws, explicitly independent of surface roughness, having the same constants as the smooth wall case. In these alternate variables, just above the wall roughness level, the mean velocity and Reynolds stresses are universal and do not depend on surface roughness. The extensive experimental data provide very good support to our universal relations. There is no universality of scalings in traditional variables and different expressions are needed for inflectional type roughness, monotonic Colebrook–Moody roughness, k-type roughness, d-type roughness, etc. In traditional variables, the velocity profile and skin friction predictions for the inflectional roughness, k-type roughness, and d-type roughness are supported well by the extensive experimental data. The pressure gradient effect from the matching conditions in the overlap region leads to the universal composite laws, which for weaker pressure gradients yields log laws and for strong adverse pressure gradients provides the half-power laws for universal velocity profiles and in traditional variables the additive terms in the two situations depend on the wall roughness.

scholarly journals On the identification of well-behaved turbulent boundary layers

2017 ◽  
Vol 822 ◽  
pp. 109-138 ◽  
Author(s):  
C. Sanmiguel Vila ◽  
R. Vinuesa ◽  
S. Discetti ◽  
A. Ianiro ◽  
P. Schlatter ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Velocity Profile ◽  
Skin Friction ◽  
Boundary Layers ◽  
Shape Factor ◽  
Outer Region ◽  
The Other ◽  
Diagnostic Plot ◽  
Other Hand

This paper introduces a new method based on the diagnostic plot (Alfredsson et al., Phys. Fluids, vol. 23, 2011, 041702) to assess the convergence towards a well-behaved zero-pressure-gradient (ZPG) turbulent boundary layer (TBL). The most popular and well-understood methods to assess the convergence towards a well-behaved state rely on empirical skin-friction curves (requiring accurate skin-friction measurements), shape-factor curves (requiring full velocity profile measurements with an accurate wall position determination) or wake-parameter curves (requiring both of the previous quantities). On the other hand, the proposed diagnostic-plot method only needs measurements of mean and fluctuating velocities in the outer region of the boundary layer at arbitrary wall-normal positions. To test the method, six tripping configurations, including optimal set-ups as well as both under- and overtripped cases, are used to quantify the convergence of ZPG TBLs towards well-behaved conditions in the Reynolds-number range covered by recent high-fidelity direct numerical simulation data up to a Reynolds number based on the momentum thickness and free-stream velocity $Re_{\unicode[STIX]{x1D703}}$ of approximately 4000 (corresponding to 2.5 m from the leading edge) in a wind-tunnel experiment. Additionally, recent high-Reynolds-number data sets have been employed to validate the method. The results show that weak tripping configurations lead to deviations in the mean flow and the velocity fluctuations within the logarithmic region with respect to optimally tripped boundary layers. On the other hand, a strong trip leads to a more energized outer region, manifested in the emergence of an outer peak in the velocity-fluctuation profile and in a more prominent wake region. While established criteria based on skin-friction and shape-factor correlations yield generally equivalent results with the diagnostic-plot method in terms of convergence towards a well-behaved state, the proposed method has the advantage of being a practical surrogate that is a more efficient tool when designing the set-up for TBL experiments, since it diagnoses the state of the boundary layer without the need to perform extensive velocity profile measurements.

Fine-structure turbulence in the wall region of a turbulent boundary layer

1975 ◽  
Vol 67 (1) ◽  
pp. 125-143 ◽  
Author(s):  
H. Ueda ◽  
J. O. Hinze
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Fine Structure ◽  
Turbulent Boundary Layer ◽  
Power Law ◽  
Outer Part ◽  
Viscous Sublayer ◽  
Wall Region ◽  
Hot Wire ◽  
Frequency Range

Measurements have been made concerning the fine structure of the turbulence in the part adjacent to the wall of the wall region of a plane turbulent boundary layer. The objective was to gain further information concerning the larger-scale disturbance mechanism which is mainly responsible for the generation of turbulence. Hot-wire anemomet.ry was used and information on the fine structure was obtained by differentiating and filtering the hot-wire signal.The distributions of the Kolmogorov microscale and of the flatness and skewness factors of the axial fluctuating velocity u and its first and second derivative determined at two Reynolds numbers suggest the existence of Reynolds number similarity. In the region y+ < 15 the flatness and skewness factors of u increase with decreasing y+. At approximately y+ = 15 the flatness factor shows a minimum value, while the skewness factor becomes zero. This location agrees with that where the turbulence intensity u′ has a maximum value. In the outer part of the wall region (y+ > 100) the flatness and skewness factors approach values obtained in shear-free turbulence at the same turbulence Reynolds number.The fine structure of the turbulence is strongly associated with and dominated by the random, larger-scale, intermittent inrush-ejection cycle. In the viscous sublayer both the fine structure, and the large-scale mechanism of the turbulence are influenced mainly by the inrush phase, while further out in the wall region (y+ > 40) they are influenced by both inrush and ejection. As a result, in the viscous sublayer the average burst periods of the high frequency turbulence components and their flatness factors (of ∂u/∂t and of ∂2u/∂t2) attain values twice those in the outer part.The change in the mechanism of the fine structure with distance from the wall is clearly demonstrated by the spectra of non-negative variables, i.e. (∂u/∂t)2 and (∂2u/∂t2)2. The spectra agree with each other and decrease with increasing frequency, following a power law as predicted by Gurvich & Yaglom (1967). The power law applies to almost the whole frequency range, when the highest, viscous, frequency range is excluded. However, the exponent is different for the viscous sublayer and the outer part of the wall region. In the buffer layer the spectra have two distinct power-law regions. In the lower frequency range the exponent is the same as that for the viscous sublayer, while in the higher frequency range it is the same as that for the outer part of the wall region.

Friction velocity and power law velocity profile in smooth and rough shallow open channel flows

2002 ◽  
Vol 29 (2) ◽  
pp. 256-266 ◽  
Author(s):  
R Balachandar ◽  
D Blakely ◽  
J Bugg
Keyword(s):  
Boundary Layer ◽  
Velocity Profile ◽  
Froude Number ◽  
Power Law ◽  
Friction Velocity ◽  
Open Channel ◽  
Rough Surfaces ◽  
Channel Flows ◽  
Open Channel Flows ◽  
Log Law

This paper examines the mean velocity profiles in shallow, turbulent open channel flows. Velocity measurements were carried out in flows over smooth and rough beds using a laser-Doppler anemometer. One set of profiles, composed of 29 velocity distributions, was obtained in flows over a polished smooth aluminum plate. Three sets of profiles were obtained in flows over rough surfaces. The rough surfaces were formed by two sizes of sand grains and a wire mesh. The flow conditions over the rough surface are in the transitional roughness state. The measurements were obtained along the centerline of the flume at three different Froude numbers (Fr ~ 0.3, 0.8, 1.0). The lowest Froude number was selected to obtain data in the range of most other open channel testing programs and to represent a low subcritical Froude number. For each surface, the Reynolds number based on the boundary layer momentum thickness was varied from about 600 to 3000. In view of the recent questions concerning the applicability of the log-law and the debate regarding log-law versus power law, the turbulent inner region of the boundary layer is inspected. The fit of one type of power law for shallow flows over a smooth surface is considered. The appropriateness of extending this law to flows over rough surfaces is also examined. Alternate methods for determining the friction velocity of flows over smooth and rough surfaces are considered and compared with standard methods currently in use.Key words: power law, open channel flow, velocity profile, surface roughness.

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 effects of a favourable pressure gradient and of the Reynolds number on an incompressible axisymmetric turbulent boundary layer. Part 1. The turbulent boundary layer

1998 ◽  
Vol 359 ◽  
pp. 329-356 ◽  
Author(s):  
H. H. FERNHOLZ ◽  
D. WARNACK
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Boundary Layers ◽  
Reynolds Stresses ◽  
The Mean

The effects of a favourable pressure gradient (K[les ]4×10−6) and of the Reynolds number (862[les ]Reδ2[les ]5800) on the mean and fluctuating quantities of four turbulent boundary layers were studied experimentally and are presented in this paper and a companion paper (Part 2). The measurements consist of extensive hot-wire and skin-friction data. The former comprise mean and fluctuating velocities, their correlations and spectra, the latter wall-shear stress measurements obtained by four different techniques which allow testing of calibrations in both laminar-like and turbulent flows for the first time. The measurements provide complete data sets, obtained in an axisymmetric test section, which can serve as test cases as specified by the 1981 Stanford conference.Two different types of accelerated boundary layers were investigated and are described: in this paper (Part 1) the fully turbulent, accelerated boundary layer (sometimes denoted laminarescent) with approximately local equilibrium between the production and dissipation of the turbulent energy and with relaxation to a zero pressure gradient flow (cases 1 and 3); and in Part 2 the strongly accelerated boundary layer with ‘inactive’ turbulence, laminar-like mean flow behaviour (relaminarized), and reversion to the turbulent state (cases 2 and 4). In all four cases the standard logarithmic law fails but there is no single parametric criterion which denotes the beginning or the end of this breakdown. However, it can be demonstrated that the departure of the mean-velocity profile is accompanied by characteristic changes of turbulent quantities, such as the maxima of the Reynolds stresses or the fluctuating value of the skin friction.The boundary layers described here are maintained in the laminarescent state just up to the beginning of relaminarization and then relaxed to the turbulent state in a zero pressure gradient. The relaxation of the turbulence structure occurs much faster than in an adverse pressure gradient. In the accelerating boundary layer absolute values of the Reynolds stresses remain more or less constant in the outer region of the boundary layer in accordance with the results of Blackwelder & Kovasznay (1972), and rise both in the vincinity of the wall in conjunction with the rising wall shear stress and in the centre region of the boundary layer with the increase of production.

scholarly journals The structure of a three-dimensional turbulent boundary layer

1993 ◽  
Vol 250 ◽  
pp. 43-68 ◽  
Author(s):  
A. T. Degani ◽  
F. T. Smith ◽  
J. D. A. Walker
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Turbulent Boundary Layer ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Outer Layer ◽  
Three Dimensional ◽  
Wall Layer ◽  
Stream Velocity ◽  
Overlap Region

The three-dimensional turbulent boundary layer is shown to have a self-consistent two-layer asymptotic structure in the limit of large Reynolds number. In a streamline coordinate system, the streamwise velocity distribution is similar to that in two-dimensional flows, having a defect-function form in the outer layer which is adjusted to zero at the wall through an inner wall layer. An asymptotic expansion accurate to two orders is required for the cross-stream velocity which is shown to exhibit a logarithmic form in the overlap region. The inner wall-layer flow is collateral to leading order but the influence of the pressure gradient, at large but finite Reynolds numbers, is not negligible and can cause substantial skewing of the velocity profile near the wall. Conditions under which the boundary layer achieves self-similarity and the governing set of ordinary differential equations for the outer layer are derived. The calculated solution of these equations is matched asymptotically to an inner wall-layer solution and the composite profiles so formed describe the flow throughout the entire boundary layer. The effects of Reynolds number and cross-stream pressure gradient on the cross-stream velocity profile are discussed and it is shown that the location of the maximum cross-stream velocity is within the overlap region.

Power Law Turbulent Velocity Profile in Transitional Rough Pipes

2005 ◽  
Vol 128 (3) ◽  
pp. 548-558 ◽  
Author(s):  
Noor Afzal ◽  
Abu Seena ◽  
Afzal Bushra
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Power Law ◽  
Roughness Parameter ◽  
Rough Wall ◽  
Wall Roughness ◽  
Velocity Shift ◽  
Normal Wall ◽  
Good Support ◽  
Grain Roughness

Alternate power law velocity profile u+=Aζα in transitional rough pipe fully turbulent flow, has been proposed, in terms of new appropriate inner rough wall variables (ζ=Z+∕ϕ, uϕ=u∕ϕ), and new parameters Rϕ=Rτ∕ϕ termed as the roughness friction Reynolds number, Reϕ=Re∕ϕ termed as the roughness Reynolds number and ϕ termed as roughness scale (along with normal wall coordinate Z=y+ϵr where ϵr is the shift of the origin of boundary layer due to the rough wall, Z+=Zuτ∕ν and u+=u∕uτ). The envelope of the power law shows that the power law constants α and A depend on the parameter Rϕ (i.e., α=α(Rϕ) and A=A(Rϕ)) but explicitly independent of the wall roughness parameter h∕δ (roughness height h in pipe of radius δ). The roughness scale ϕ has been related to the roughness function ΔU+ of Clauser representing the velocity shift caused by wall roughness. The present results of the velocity profile, just slightly above the wall roughness level h, remain valid for all types of wall roughness. The data of Nikuradse for sand-grain roughness, in transitional and fully rough pipes, has been considered, which provides good support to the predictions of an alternate power law velocity profile, based on single parameter Rϕ, the roughness friction Reynolds number.

Sign in / Sign up

Export Citation Format

Share Document