Power law and log law velocity profiles in fully developed turbulent pipe flow: equivalent relations at large Reynolds numbers

The streamwise velocity component in turbulent pipe flow is assessed to determine whether it exhibits asymptotic behaviour that is indicative of high Reynolds numbers. The asymptotic behaviour of both the mean velocity (in the form of the log law) and that of the second moment of the streamwise component of velocity in the outer and overlap regions is consistent with the development of spectral regions which indicate inertial scaling. It is shown that an ‘inertial sublayer’ in physical space may be considered as a spatial analogue of the inertial subrange in the velocity spectrum and such behaviour only appears for Reynolds numbers R + >5×10 3 , approximately, much higher than was generally thought.

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Turbulent pipe flow development at moderate Reynolds numbers

10.2514/6.1995-2247 ◽

1995 ◽

Author(s):

Stefano Bernero ◽

Sastry Munukutla

Keyword(s):

Pipe Flow ◽

Reynolds Numbers ◽

Turbulent Pipe Flow ◽

Flow Development

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Turbulent Pipe Flow at Extreme Reynolds Numbers

Physical Review Letters ◽

10.1103/physrevlett.108.094501 ◽

2012 ◽

Vol 108 (9) ◽

Cited By ~ 178

Author(s):

M. Hultmark ◽

M. Vallikivi ◽

S. C. C. Bailey ◽

A. J. Smits

Keyword(s):

Pipe Flow ◽

Reynolds Numbers ◽

Turbulent Pipe Flow

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Analysis of a Power Law and Log Law for a Turbulent Wall Jet Over a Transitional Rough Surface: Universal Relations

Journal of Fluids Engineering ◽

10.1115/1.4004763 ◽

2011 ◽

Vol 133 (9) ◽

Cited By ~ 1

Author(s):

Noor Afzal ◽

Abu Seena

Keyword(s):

Experimental Data ◽

Rough Surface ◽

Power Law ◽

Power Laws ◽

Velocity Profiles ◽

Wall Jet ◽

Log Law ◽

Jet Velocity ◽

Turbulent Wall ◽

Momentum Integral

The power law and log law velocity profiles and an integral analysis in a turbulent wall jet over a transitional rough surface have been proposed. Based on open mean momentum Reynolds equations, a two layer theory for large Reynolds numbers is presented and the matching in the overlap region is carried out by the Izakson-Millikan-Kolmogorov hypothesis. The velocity profiles and skin friction are shown to be governed by universal log laws as well as by universal power laws, explicitly independent of surface roughness, having the same constants as a fully smooth surface wall jet (or fully rough surface wall jet, as appropriate). The novel scalings for stream-wise variations of the flow over a rough wall jet have been analyzed, and best fit relations for maximum wall jet velocity, boundary layer thickness at maxima of wall jet velocity, the jet half width, the friction factor, and momentum integral are supported by the experimental data. There is no universality of scalings in traditional variables, and different expressions are needed for transitional roughness. The experimental data provides very good support to our universal relations proposed in terms of alternate variables.

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Pipe Flow Measurements of a Transparent Non-Newtonian Slurry

Journal of Fluids Engineering ◽

10.1115/1.3243648 ◽

1989 ◽

Vol 111 (3) ◽

pp. 331-336 ◽

Cited By ~ 38

Author(s):

J. T. Park ◽

R. J. Mannheimer ◽

T. A. Grimley ◽

T. B. Morrow

Keyword(s):

Velocity Profile ◽

Newtonian Fluid ◽

Power Law ◽

Pipe Flow ◽

Large Scale ◽

Longitudinal Velocity ◽

Tangential Velocity ◽

Turbulent Pipe Flow ◽

Laser Doppler Velocimeter ◽

Flow Measurements

An experimental description of the flow structure of non-Newtonian slurries in the laminar, transitional, and full turbulent pipe flow regimes is the primary objective of this research. Experiments were conducted in a large-scale pipe slurry flow facility with an inside pipe diameter of 51 mm. The transparent slurry formulated for these experiments from silica, mineral oil, and Stoddard solvent exhibited a yield-power-law behavior from concentric-cylinder viscometer measurements. The velocity profile for laminar flow from laser Doppler velocimeter (LDV) measurements had a central plug flow region, and it was in agreement with theory. The range of the transition region was narrower than that for a Newtonian fluid. The mean velocity profile for turbulent flow was close to a 1/7 power-law velocity profile. The rms longitudinal velocity profile was also similar to a classical turbulent pipe flow experiment for a Newtonian fluid; however, the rms tangential velocity profile was significantly different.

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Large Eddy Simulation of Fully Developed Turbulent Pipe Flow

Volume 2, Parts A and B ◽

10.1115/ht-fed2004-56330 ◽

2004 ◽

Author(s):

Sowjanya Vijiapurapu ◽

Jie Cui

Keyword(s):

Reynolds Number ◽

Pipe Flow ◽

Reynolds Numbers ◽

Grid Resolution ◽

Turbulent Pipe Flow ◽

Scale Model ◽

Subgrid Scale ◽

Mean Velocity ◽

Fine Mesh ◽

Large Eddy

Fully developed turbulent pipe flow is investigated by large eddy simulations (LES). The three-dimensional, unsteady, incompressible, filtered continuity and Navier-Stokes equations in cylindrical coordinates are discretized by a finite difference method. The spatial derivatives are approximated by second order conservative schemes. This scheme eliminates the numerical generation or dissipation of energy. The pressure Poisson equation is solved by FFT method and time is advanced through a third order Runge-Kutta method. The commonly used subgrid scale (SGS) models — the Smagorinsky model and the dynamic model are implemented and simulations are performed for fully developed turbulent pipe flow at two different Reynolds numbers. The flow features in terms of mean velocity as well as higher order turbulence intensities and correlations are presented and compared to experimental and DNS data available in literature. Extensive comparisons are made for cases using different grid resolution, different streamwise domain dimension, different sub-grid scale model, and, at two different Reynolds number. For two Reynolds numbers (5,000 and 30,000) tested in this study, the fine mesh (64 × 96 × 64, circumferential × radial × longitudinal) produces better results than the coarse mesh (32 × 48 × 32), indicating the significance of the grid resolution, especially near the pipe surface. On the fine mesh for the two Reynolds numbers, the results exhibit a slight Reynolds number effect, indicating the mesh needs to be further refined at higher Reynolds number. Simulations were performed for two domain sizes, namely 6D and 12D, where D is the pipe diameter. When the streamwise grid resolution remains unchanged, the two simulations show negligible difference. This ensures that a 6D domain is adequate to include the largest eddies in a fully developed turbulent pipe flow at the current Reynolds number. When the fine mesh is used, the subgrid scale models (Smagorinsky and Dynamic) provide limited contribution to the total turbulent kinetic energy. Although the current results agree quite well with other published LES simulations, when compared with the Law of the wall, benchmark experiments and DNS results, the simulated mean velocity in the log region is higher than the experimental and DNS data. Overall, it was observed that the numerical methods work satisfactorily well for turbulent pipe flows at low and high Reynolds numbers, and, the method has capability to be used in the simulation of flows with practical interest.

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Laminar-to-turbulent transition of pipe flows through puffs and slugs

Journal of Fluid Mechanics ◽

10.1017/s0022112008003315 ◽

2008 ◽

Vol 614 ◽

pp. 425-446 ◽

Cited By ~ 76

Author(s):

MINA NISHI ◽

BÜLENT ÜNSAL ◽

FRANZ DURST ◽

GAUTAM BISWAS

Keyword(s):

Reynolds Number ◽

Pipe Flow ◽

Axial Direction ◽

Reynolds Numbers ◽

Turbulent Pipe Flow ◽

Test Facility ◽

Pipe Flows ◽

Laminar To Turbulent Transition ◽

Turbulent Transition ◽

Slug Flows

Laminar-to-turbulent transition of pipe flows occurs, for sufficiently high Reynolds numbers, in the form of slugs. These are initiated by disturbances in the entrance region of a pipe flow, and grow in length in the axial direction as they move downstream. Sequences of slugs merge at some distance from the pipe inlet to finally form the state of fully developed turbulent pipe flow. This formation process is generally known, but the randomness in time of naturally occurring slug formation does not permit detailed study of slug flows. For this reason, a special test facility was developed and built for detailed investigation of deterministically generated slugs in pipe flows. It is also employed to generate the puff flows at lower Reynolds numbers. The results reveal a high degree of reproducibility with which the triggering device is able to produce puffs. With increasing Reynolds number, ‘puff splitting’ is observed and the split puffs develop into slugs. Thereafter, the laminar-to-turbulent transition occurs in the same way as found for slug flows. The ring-type obstacle height, h, required to trigger fully developed laminar flows to form first slugs or puffs is determined to show its dependence on the Reynolds number, Re = DU/ν (where D is the pipe diameter, U is the mean velocity in the axial direction and ν is the kinematic viscosity of the fluid). When correctly normalized, h+ turns out to be independent of Reτ (where h+ = hUτ/ν, Reτ = DUτ/ν and $U_{\tau}\,{=}\,\sqrt{\tau_{w}/ \rho}$; τw is the wall shear stress and ρ is the density of the fluid).

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