Analysis of power law and log law velocity profiles in the overlap region of a turbulent wall jet

2005 ◽  
Vol 461 (2058) ◽  
pp. 1889-1910 ◽  
Author(s):  
Noor Afzal
Keyword(s):  
Power Law ◽  
Asymptotic Expansions ◽  
Friction Velocity ◽  
Matched Asymptotic Expansions ◽  
Large Reynolds Number ◽  
Wall Jet ◽  
Law Of The Wall ◽  
Log Law ◽  
Turbulent Wall ◽  
Overlap Region

The two-dimensional turbulent wall jet on a flat surface without free stream is analysed at a large Reynolds number, using the method of matched asymptotic expansions. The open mean equations of the turbulent boundary layer are analysed in the wall and wake layers by the method of matched asymptotic expansions and the results are matched by the Izakson–Millikan–Kolmogorov hypothesis. In the overlap region, the outer wake layer is governed by the velocity defect law (based on U m , the maximum velocity) and the inner layer by the law of the wall. It is shown that the overlap region possesses a non-unique solution, where the power law region simultaneously exists along with the log law region. Analysis of the power law and log law solutions in the overlap region leads to self-consistent relations, where the power law index, α , is of the order of the non-dimensional friction velocity and the power law multiplication constant, C , is of the order of the inverse of the non-dimensional friction velocity. The lowest order wake layer equation has been closed with generalized gradient transport model and a closed form solution is obtained. A comparison of the theory with experimental data is presented.

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

2011 ◽  
Vol 133 (9) ◽  
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.

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.

A Two∼Dimensional Wall Jet with Suction

1972 ◽  
Vol 1 (4) ◽  
pp. 182-188
Author(s):  
T.B. Hedley ◽  
J.F. Keffer
Keyword(s):  
Viscous Sublayer ◽  
Transport Processes ◽  
Wall Jet ◽  
Two Dimensional ◽  
Mean Flow ◽  
Law Of The Wall ◽  
Free Jet ◽  
The Mean ◽  
Turbulent Wall ◽  
Large Eddies

The mean flow field of a two-dimensional turbulent wall jet which encounters a uniform suction is examined. A marked increase in wall shear stress was observed for all suction levels as the jet moved into the suction zone. When the suction level is moderate a viscous sublayer exists next to the surface. The dominance of the flow by the free jet motion however prevents any law-of-the-wall representation for the adjacent turbulent region and a velocity defect model is found to be more satisfactory. One can interpret this lack of an extensive equilibrium layer to mean that the transport processes are controlled by the action of the large eddies over almost the entire wall jet zone, with or without suction.

Effects of an Initial Gap on the Flow in a Turbulent Wall Jet

1966 ◽  
Vol 70 (666) ◽  
pp. 669-673 ◽  
Author(s):  
K. Sridhar ◽  
P. K. C. Tu
Keyword(s):  
Power Law ◽  
Nozzle Exit ◽  
Leading Edge ◽  
Wall Jet ◽  
Gap Size ◽  
Two Dimensional ◽  
Free Jet ◽  
Layer Velocity ◽  
Turbulent Wall ◽  
Gap Effects

SummaryThe flow in a two-dimensional plane wall jet with different initial gaps between the nozzle exit and the leading edge of the wall was probed at various stations along the jet. The jet slot thickness and the velocity were kept constant. It was found that the region close to the leading edge of the wall behaved like a transforming region where the type of flow changed from a free jet to a wall jet. The length of this region, which depended directly on the gap size, was so short for small gaps that the gap effects were found to be negligible. In addition, it was found that the inner layer velocity distribution of a wall jet did not follow the classic one-seventh power law.

Is there a universal log law for turbulent wall-bounded flows?

2007 ◽  
Vol 365 (1852) ◽  
pp. 789-806 ◽  
Author(s):  
William K George
Keyword(s):  
Boundary Layer ◽  
Power Law ◽  
Velocity Data ◽  
Curve Fit ◽  
Logarithmic Curve ◽  
Layer Velocity ◽  
Log Law ◽  
Wall Bounded Flows ◽  
Stress Layer ◽  
Turbulent Wall

The history and theory supporting the idea of a universal log law for turbulent wall-bounded flows are briefly reviewed. The original idea of justifying a log law from a constant Reynolds stress layer argument is found to be deficient. By contrast, it is argued that the logarithmic friction law and velocity profiles derived from matching inner and outer profiles for a pipe or channel flow are well-founded and consistent with the data. But for a boundary layer developing along a flat plate it is not, and in fact it is a power law theory that seems logically consistent. Even so, there is evidence for at least an empirical logarithmic fit to the boundary-friction data, which is indistinguishable from the power law solution. The value of κ ≈0.38 obtained from a logarithmic curve fit to the boundary-layer velocity data, however, does not appear to be the same as for pipe flow for which 0.43 appears to be the best estimate. Thus, the idea of a universal log law for wall-bounded flows is not supported by either the theory or the data.

Complete Velocity Profile and “Optimum” Skin Friction Formulas for the Plane Wall-Jet

1982 ◽  
Vol 104 (1) ◽  
pp. 59-65 ◽  
Author(s):  
G. P. Hammond
Keyword(s):  
Velocity Profile ◽  
Skin Friction ◽  
Initial Conditions ◽  
Jet Flows ◽  
Wall Jet ◽  
Wall Region ◽  
Law Of The Wall ◽  
Single Formula ◽  
Turbulent Wall ◽  
Good Agreement

An analytic expression for the complete velocity profile of a plane, turbulent wall-jet in “stagnant” surroundings is obtained by coupling Spalding’s single formula for the inner layer with a sine function for the “wake component.” This expression is transformed at the velocity maxima to yield an “optimum log-law” for skin friction. An approximate skin friction formula based on the “initial conditions” of the wall-jet is also presented. The formulas are generally in good agreement with experimental data. The complete velocity profile does not exhibit the conventional “law of the wall” behavior and modifications are consequently recommended to the usual treatment of the near-wall region in numerical calculation procedures for wall-jet flows. The use of the “Clauser plot” method of skin friction measurement is similarly shown to be in error when applied to wall-jets.

ACTION AND PERIOD OF HOMOCLINIC AND PERIODIC ORBITS FOR THE UNFOLDING OF A SADDLE-CENTER BIFURCATION

2003 ◽  
Vol 13 (11) ◽  
pp. 3519-3530
Author(s):  
DAVID C. DIMINNIE ◽  
RICHARD HABERMAN
Keyword(s):  
Saddle Point ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Asymptotic Expansions ◽  
Homoclinic Orbit ◽  
Matched Asymptotic Expansions ◽  
Zero Energy ◽  
Global Branch ◽  
Overlap Region

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

On The Role of a Non-Newtonian Fluid in Short Bearing Theory

1985 ◽  
Vol 107 (1) ◽  
pp. 68-74 ◽  
Author(s):  
R. H. Buckholz
Keyword(s):  
Newtonian Fluid ◽  
Power Law ◽  
Asymptotic Expansions ◽  
Reynolds Equation ◽  
Slenderness Ratio ◽  
Fluid Film ◽  
Matched Asymptotic Expansions ◽  
Film Pressure ◽  
Boundary Shape ◽  
Pressure Distributions

The importance of rheological properties of lubricants has arisen from the realization that non-Newtonian fluid effects are manifested over a broad range of lubrication applications. In this paper a theoretical investigation of short journal bearings performance characteristics for non-Newtonian power-law lubricants is given. A modified form of the Reynolds’ equation for hydrodynamic lubrication is studied in the asymptotic limit of small slenderness ratio (i.e., bearing length to diameter, L/D = λ→0). Fluid film pressure distributions in short bearings of arbitrary azimuthal length are studied using matched asymptotic expansions in the slenderness ratio. The merit of the short bearing approach used in solving a modified Reynolds’ equation by the method of matched asymptotic expansions is emphasized. Fluid film pressure distributions are determined without recourse to numerical solutions to a modified Reynolds’ equation. Power-law rheological exponents less than and equal to one are considered; power-law fluids exhibit reduced load capacities relative to the Newtonian fluid. The cavitation boundary shape is determined from Reynolds’ free surface condition; and the boundary shape is shown to be independent of the bearing eccentricity ratio.

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