The Ultimate Asymptote and Mean Flow Structure in Toms’ Phenomenon

1970 ◽  
Vol 37 (2) ◽  
pp. 488-493 ◽  
Author(s):  
P. S. Virk ◽  
H. S. Mickley ◽  
K. A. Smith
Keyword(s):  
Velocity Profile ◽  
Drag Reduction ◽  
Flow Structure ◽  
Viscous Sublayer ◽  
Mixing Length ◽  
Mean Flow ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
Length Constant ◽  
The Mean

The maximum drag reduction in turbulent pipe flow of dilute polymer solutions is ultimately limited by a unique asymptote described by the experimental correlation: f−1/2=19.0log10(NRef1/2)−32.4 The semilogarithmic mean velocity profile corresponding to and inferred from this ultimate asymptote has a mixing-length constant of 0.085 and shares a trisection (at y+ ∼ 12) with the Newtonian viscous sublayer and law of the wall. Experimental mean velocity profiles taken during drag reduction lie in the region bounded by the inferred ultimate profile and the Newtonian law of the wall. At low drag reductions the experimental profiles are well correlated by an “effective slip” model but this fails progressively with increasing drag reduction. Based on the foregoing a three-zone scheme is proposed to model the mean flow structure during drag reduction. In this the mean velocity profile segments are (a) a viscous sublayer, akin to Newtonian, (b) an interactive zone, characteristic of drag reduction, in which the ultimate profile is followed, and (c) a turbulent core in which the Newtonian mixing-length constant applies. The proposed model is consistent with experimental observations and reduces satisfactorily to the Taylor-Prandtl scheme and the ultimate profile, respectively, at the limits of zero and maximum drag reductions.

Simplified theory of the near-wall turbulent layer of Newtonian and drag-reducing fluids

1982 ◽  
Vol 119 ◽  
pp. 423-441 ◽  
Author(s):  
M. A. Goldshtik ◽  
V. V. Zametalin ◽  
V. N. Shtern
Keyword(s):  
Velocity Profile ◽  
Turbulent Flow ◽  
Drag Reduction ◽  
Outer Boundary ◽  
Viscous Sublayer ◽  
Viscous Layer ◽  
Mean Velocity ◽  
Turbulent Layer ◽  
The Mean ◽  
Near Wall

We propose a simplified theory of a viscous layer in near-wall turbulent flow that determines the mean-velocity profile and integral characteristics of velocity fluctuations. The theory is based on the concepts resulting from the experimental data implying a relatively simple almost-ordered structure of fluctuations in close proximity to the wall. On the basis of data on the greatest contribution to transfer processes made by the part of the spectrum associated with the main size of the observed structures, the turbulent fluctuations are simulated by a three-dimensional running wave whose parameters are found from the problem solution. Mathematically the problem reduces to the solution of linearized Navier-Stokes equations. The no-slip condition is satisfied on the wall, whereas on the outer boundary of a viscous layer the conditions of smooth conjunction with the asymptotic shape of velocity and fluctuation-energy profiles resulting from the dimensional analysis are satisfied. The formulation of the problem is completed by the requirement of maximum curvature of the mean-velocity profile on the outer boundary applied from stability considerations.The solution of the problem does not require any quantitative empirical data, although the conditions of conjunction were formulated according to the well-known concepts obtained experimentally. As a result, the near-wall law for the averaged velocity has been calculated theoretically and is in good agreement with experiment, and the characteristic scales for fluctuations have also been determined. The developed theory is applied to turbulent-flow calculations in Maxwell and Oldroyd media. The elastic properties of fluids are shown to lead to near-wall region reconstruction and its associated drag reduction, as is the case in turbulent flows of dilute polymer solutions. This theory accounts for several features typical of the Toms effect, such as the threshold character of the effect and the decrease in the normal fluctuating velocity. The analysis of the near-wall Oldroyd fluid flow permits us to elucidate several new aspects of the drag-reduction effect. It has been established that the Toms effect does not always result in thickening of the viscous sublayer; on the contrary, the most intense drag reduction takes place without thickening in the viscous sublayer.

Observations on turbulent-drag reduction in a dilute suspension of clay in sea-water

1976 ◽  
Vol 75 (1) ◽  
pp. 29-47 ◽  
Author(s):  
Giselher Gust
Keyword(s):  
Drag Reduction ◽  
Clay Mineral ◽  
Flow Structure ◽  
Sea Water ◽  
Streamwise Velocity ◽  
Viscous Sublayer ◽  
Mean Flow ◽  
Turbulent Drag Reduction ◽  
Turbulent Drag ◽  
The Mean

Hot-wire anemometer measurements have been made in a dilute sea-water/claymineral suspension. For fully developed turbulent flows in an open channel with a smooth mud (from the North Sea) bottom, mean streamwise velocity profiles were measured for Reynolds numbers between 5400 and 27 800 (i.e. non-eroding and eroding flow rates) and compared with Newtonian flows under the same experimental conditions. For the clay-mineral suspensions, measurements of the kinematic viscosityv, Kármán's constantkand the mean streamwise velocity$\overline{u}$of the logarithmic layer seemed to verify a Newtonian flow structure. Although the distributions of concentration showed no substantial increase towards the wall, it was found that beneath this Newtonian core there existed a viscous sublayer whose thickness was enhanced by a factor of 2–5. The friction velocityu*determined by the gradient method in the viscous sublayer was reduced by as much as 40 %. The mean flow structure exhibited an additional new layer in the region 10 <y+< 30.The measurements indicate that turbulent-drag reduction occurs for the experimental clay-mineral suspension at non-eroding and also at eroding velocities. Agglomeration of suspended clay-mineral particles is suggested as possible explanation of this phenomenon.

Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction

2014 ◽  
Vol 745 ◽  
pp. 378-397 ◽  
Author(s):  
Sergio Pirozzoli
Keyword(s):  
Velocity Profile ◽  
Turbulent Flows ◽  
Outer Part ◽  
Wall Layer ◽  
Wall Friction ◽  
Mixing Length ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
Turbulent Wall

AbstractWe reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

A Mean Flow Model for Polymer and Fiber Turbulent Drag Reduction

2005 ◽  
Vol 15 (6) ◽  
pp. 370-389 ◽  
Author(s):  
Anshuman Roy ◽  
Ronald G. Larson
Keyword(s):  
Drag Reduction ◽  
Weissenberg Number ◽  
Extensional Viscosity ◽  
Mean Flow ◽  
Dimensionless Numbers ◽  
Zero Shear Viscosity ◽  
Mean Velocity ◽  
Fiber Concentration ◽  
Turbulent Drag ◽  
The Mean

Abstract We present a one-parameter model that fits quantitatively the mean velocity profiles from experiments and numerical simulations of drag-reduced wall-bounded flows of dilute solutions of polymers and non-Brownian fibers in the low and modest drag reduction regime. The model is based on a viscous mechanism of drag reduction, in which either extended polymers or non-Brownian fibers increase the extensional viscosity of the fluid and thereby suppress both small and large turbulent eddies and reduce momentum transfer to the wall, resulting in drag reduction. Our model provides a rheological interpretation of the upward parallel shift S+ in the mean velocity profile upon addition of polymer, observed by Virk. We show that Virk’s correlations for the dependence on polymer molecular weight and concentration of the onset wall shear stress and slope increment on the Prandtl-Karman plot can be translated to two dimensionless numbers, namely an onset Weissenberg number and an asymptotic Trouton ratio of maximum extensional viscosity to zero-shear viscosity. We believe that our model, while simple, captures the essential features of drag reduction that are universal to flexible polymers and fibers, and, unlike the Virk phenomenology, can easily be extended to flows with inhomogeneous polymer or fiber concentration fields.

scholarly journals On the streamwise evolution of turbulent boundary layers in arbitrary pressure gradients

2002 ◽  
Vol 461 ◽  
pp. 61-91 ◽  
Author(s):  
A. E. PERRY ◽  
IVAN MARUSIC ◽  
M. B. JONES
Keyword(s):  
Boundary Layers ◽  
Scaling Laws ◽  
Power Spectra ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Stream Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
The Mean

A new approach to the classic closure problem for turbulent boundary layers is presented. This involves, first, using the well-known mean-flow scaling laws such as the log law of the wall and the law of the wake of Coles (1956) together with the mean continuity and the mean momentum differential and integral equations. The important parameters governing the flow in the general non-equilibrium case are identified and are used for establishing a framework for closure. Initially closure is achieved here empirically and the potential for achieving closure in the future using the wall-wake attached eddy model of Perry & Marusic (1995) is outlined. Comparisons are made with experiments covering adverse-pressure-gradient flows in relaxing and developing states and flows approaching equilibrium sink flow. Mean velocity profiles, total shear stress and Reynolds stress profiles can be computed for different streamwise stations, given an initial upstream mean velocity profile and the streamwise variation of free-stream velocity. The attached eddy model of Perry & Marusic (1995) can then be utilized, with some refinement, to compute the remaining unknown quantities such as Reynolds normal stresses and associated spectra and cross-power spectra in the fully turbulent part of the flow.

A model for inhomogeneous turbulent flow

1970 ◽  
Vol 317 (1530) ◽  
pp. 417-433 ◽  
Keyword(s):  
Boundary Layer ◽  
Eddy Viscosity ◽  
The Self ◽  
Diffusion Equations ◽  
Mean Flow ◽  
Mean Velocity ◽  
Closed Set ◽  
Law Of The Wall ◽  
The Mean ◽  
Model Equations

A set of model equations is given to describe the gross features of a statistically steady or 'slowly varying’ inhomogeneous field of turbulence and the mean velocity distribution. The equations are based on the idea that turbulence can be characterized by ‘densities’ which obey nonlinear diffusion equations. The diffusion equations contain terms to describe the convection by the mean flow, the amplification due to interaction with a mean velocity gradient, the dissipation due to the interaction of the turbulence with itself, and the dif­fusion also due to the self interaction. The equations are similar to a set proposed by Kolmo­gorov (1942). It is assumed that both an ‘energy density’ and a ‘vorticity density’ satisfy diffusion equations, and that the self diffusion is described by an eddy viscosity which is a function of the energy and vorticity densities; the eddy viscosity is also assumed to describe the diffu­sion of mean momentum by the turbulent fluctuations. It is shown that with simple and plausible assumptions about the nature of the interaction terms, the equations form a closed set. The appropriate boundary conditions at a solid wall and a turbulent interface, with and without entrainment, are discussed. It is shown that the dimensionless constants which appear in the equations can all be estimated by general arguments. The equations are then found to predict the von Kármán constant in the law of the wall with reasonable accuracy. An analytical solution is given for Couette flow, and the result of a numerical study of plane Poiseuille flow is described. The equations are also applied to free turbulent flows. It is shown that the model equations completely determine the structure of the similarity solutions, with the rate of spread, for instance, determined by the solution of a nonlinear eigenvalue problem. Numerical solutions have been obtained for the two-dimensional wake and jet. The agreement with experiment is good. The solutions have a sharp interface between turbulent and non-turbulent regions and the mean velocity in the turbulent part varies linearly with distance from the interface. The equations are applied qualitatively to the accelerating boundary layer in flow towards a line sink, and the decelerating boundary layer with zero skin friction. In the latter case, the equations predict that the mean velocity should vary near the wall like the 5/3 power of the distance. It is shown that viscosity can be incorporated formally into the model equations and that a structure can be given to the interface between turbulent and non-turbulent parts of the flow.

Stokes Mechanism of Drag Reduction

2005 ◽  
Vol 73 (3) ◽  
pp. 483-489 ◽  
Author(s):  
Promode R. Bandyopadhyay
Keyword(s):  
Drag Reduction ◽  
Attenuation Factor ◽  
Viscous Sublayer ◽  
Wall Region ◽  
Mean Velocity ◽  
Turbulence Suppression ◽  
Maximum Drag Reduction ◽  
The Mean ◽  
Stokes Second Problem ◽  
Better Than

The mechanism of drag reduction due to spanwise wall oscillation in a turbulent boundary layer is considered. Published measurements and simulation data are analyzed in light of Stokes’ second problem. A kinematic vorticity reorientation hypothesis of drag reduction is first developed. It is shown that spanwise oscillation seeds the near-wall region with oblique and skewed Stokes vorticity waves. They are attached to the wall and gradually align to the freestream direction away from it. The resulting Stokes layer has an attenuated nature compared to its laminar counterpart. The attenuation factor increases in the buffer and viscous sublayer as the wall is approached. The mean velocity profile at the condition of maximum drag reduction is similar to that due to polymer. The final mean state of maximum drag reduction due to turbulence suppression appears to be universal in nature. Finally, it is shown that the proposed kinematic drag reduction hypothesis describes the measurements significantly better than what current direct numerical simulation does.

scholarly journals High-resolution velocity measurement in the inner part of turbulent boundary layers over super-hydrophobic surfaces

2016 ◽  
Vol 801 ◽  
pp. 670-703 ◽  
Author(s):  
Hangjian Ling ◽  
Siddarth Srinivasan ◽  
Kevin Golovin ◽  
Gareth H. McKinley ◽  
Anish Tuteja ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Velocity Profile ◽  
Drag Reduction ◽  
Slip Velocity ◽  
Reynolds Shear Stress ◽  
Turbulent Boundary Layers ◽  
Mean Velocity ◽  
Roughness Elements ◽  
The Mean

Digital holographic microscopy is used for characterizing the profiles of mean velocity, viscous and Reynolds shear stresses, as well as turbulence level in the inner part of turbulent boundary layers over several super-hydrophobic surfaces (SHSs) with varying roughness/texture characteristics. The friction Reynolds numbers vary from 693 to 4496, and the normalized root mean square values of roughness $(k_{rms}^{+})$ vary from 0.43 to 3.28. The wall shear stress is estimated from the sum of the viscous and Reynolds shear stress at the top of roughness elements and the slip velocity is obtained from the mean profile at the same elevation. For flow over SHSs with $k_{rms}^{+}<1$, drag reduction and an upward shift of the mean velocity profile occur, along with a mild increase in turbulence in the inner part of the boundary layer. As the roughness increases above $k_{rms}^{+}\sim 1$, the flow over the SHSs transitions from drag reduction, where the viscous stress dominates, to drag increase where the Reynolds shear stress becomes the primary contributor. For the present maximum value of $k_{rms}^{+}=3.28$, the inner region exhibits the characteristics of a rough wall boundary layer, including elevated wall friction and turbulence as well as a downward shift in the mean velocity profile. Increasing the pressure in the test facility to a level that compresses the air layer on the SHSs and exposes the protruding roughness elements reduces the extent of drag reduction. Aligning the roughness elements in the streamwise direction increases the drag reduction. For SHSs where the roughness effect is not dominant ($k_{rms}^{+}<1$), the present measurements confirm previous theoretical predictions of the relationships between drag reduction and slip velocity, allowing for both spanwise and streamwise slip contributions.

An experimental investigation of pulsating turbulent water flow in a tube

1971 ◽  
Vol 46 (1) ◽  
pp. 43-64 ◽  
Author(s):  
J. H. Gerrard
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Water Flow ◽  
Cylindrical Tube ◽  
Flow Speed ◽  
Mean Flow ◽  
Mean Velocity ◽  
Central Plateau ◽  
Transition Range ◽  
The Mean

Experiments were made on a pulsating water flow at a mean flow Reynolds number of 3770 in a cylindrical tube of diameter 3·81 cm. Pulsations were produced by a piston oscillating in simple harmonic motion with a period of 12 s. Turbulence was made visible by means of a sheet of dye produced by electrolysis from a fine wire stretched across a diameter. The sheet of dye is contorted by the turbulent eddies, and ciné-photography was used to find the velocity of convection which was shown to be the flow speed except in certain circumstances which are discussed. By subtracting the mean flow velocity profile the profile of the component of the motion oscillating at the imposed frequency was determined.The Reynolds number of these experiments lies in the turbulent transition range, so that large effects of laminarization are observed. In the turbulent phase, the velocity profile was found to possess a central plateau as does the laminar oscillating profile. The level and radial extent of this were little different from the laminar ones. Near to the wall, the turbulent oscillating profile is well represented by the mean velocity power law relationship, u/U ∝ (y/a)1/n. In the laminarized phase, the turbulent intensity is considerably reduced at this Reynolds number. The velocity profile for the whole flow (mean plus oscillating) relaxes towards the laminar profile. Laminarization contributes appreciably to the oscillating component.Extrapolation of the results to higher Reynolds numbers and different frequencies of oscillation is suggested.

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