The late start of the mean velocity overlap log law at – a generic feature of turbulent wall layers in ducts

AbstractUnlike the classical scaling relations for the mean-velocity profiles of wall-bounded uniform turbulent flows (the law of the wall, the defect law and the log law), which are predicated solely on dimensional analysis and similarity assumptions, scaling relations for the turbulent-energy spectra have been informed by specific models of wall turbulence, notably the attached-eddy hypothesis. In this paper, we use dimensional analysis and similarity assumptions to derive three scaling relations for the turbulent-energy spectra, namely the spectral analogues of the law of the wall, the defect law and the log law. By design, each spectral analogue applies in the same spatial domain as the attendant scaling relation for the mean-velocity profiles: the spectral analogue of the law of the wall in the inner layer, the spectral analogue of the defect law in the outer layer and the spectral analogue of the log law in the overlap layer. In addition, as we are able to show without invoking any model of wall turbulence, each spectral analogue applies in a specific spectral domain (the spectral analogue of the law of the wall in the high-wavenumber spectral domain, where viscosity is active, the spectral analogue of the defect law in the low-wavenumber spectral domain, where viscosity is negligible, and the spectral analogue of the log law in a transitional intermediate-wavenumber spectral domain, which may become sizable only at ultra-high$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau }$), with the implication that there exist model-independent one-to-one links between the spatial domains and the spectral domains. We test the spectral analogues using experimental and computational data on pipe flow and channel flow.

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The forced turbulent wall jet

Journal of Fluid Mechanics ◽

10.1017/s0022112092002507 ◽

1992 ◽

Vol 242 ◽

pp. 577-609 ◽

Cited By ~ 32

Author(s):

Y. Katz ◽

E. Horev ◽

I. Wygnanski

Keyword(s):

Maximum Velocity ◽

Wall Stress ◽

Coherent Motion ◽

Settling Chamber ◽

Wall Jet ◽

Appreciable Effect ◽

External Excitation ◽

Mean Velocity ◽

The Mean ◽

Turbulent Wall

The effects of external two-dimensional excitation on the plane turbulent wall jet were investigated experimentally and theoretically. Measurements of the streamwise component of velocity were made throughout the flow field for a variety of imposed frequencies and amplitudes. The present data were always compared to the results generated in the absence of external excitation. Two methods of forcing were used: one global, imposed on the entire jet by pressure fluctuations in the settling chamber and one local, imposed on the shear layer by a small flap attached to the outer nozzle lip. The fully developed wall jet was shown to be insensitive to the method of excitation. Furthermore, external excitation has no appreciable effect on the rate of spread of the jet nor on the decay of its maximum velocity. In fact the mean velocity distribution did not appear to be altered by the external excitation in any obvious manner. The flow near the surface, however, (i.e. for 0 < Y+ < 100) was profoundly different from the unforced flow, indicating a reduction in wall stress exceeding at times 30%. The production of turbulent energy near the surface was also reduced, lowering the intensities of the velocity fluctuations. External excitation enhanced the two-dimensionality and the periodicity of the coherent motion. Spectral analysis and flow visualization suggested that the large coherent structures in this flow might be identified with the most-amplified primary instability modes of the mean velocity profile. Detailed stability analysis confirmed this proposition though not at the same level of accuracy as it did in many free shear flows.

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Effects of a Large Eddy Breakup Device on the Fluctuating Wall Pressure Field (Data Bank Contribution)

Journal of Fluids Engineering ◽

10.1115/1.2910151 ◽

1993 ◽

Vol 115 (3) ◽

pp. 389-397 ◽

Cited By ~ 7

Author(s):

W. L. Keith ◽

J. J. Barclay

Keyword(s):

Coherent Structures ◽

Field Data ◽

Pressure Field ◽

Data Bank ◽

Turbulent Boundary Layers ◽

Wall Pressure ◽

Mean Velocity ◽

Large Eddy ◽

The Mean ◽

Log Law

Wall pressure spectra and mean velocity profiles were measured with and without a large eddy breakup device (LEBU) located upstream in high Rθ turbulent boundary layers. Changes of the order of 5 percent occurred in the mean wall shear stress. The wall pressure spectra of the manipulated flow did not show the existence of any highly energized coherent structures, but rather moderate changes occurring over broad frequency ranges. Reductions in the wall pressure autospectra occurred at lower frequencies associated with turbulence activity in the outer layer and the outer portion of the log law layer. Increases occurred at higher frequencies associated with the inner portion of the log law layer. The changes in the wall pressure coherence levels were similar, but generally more complex with greater spatial persistence. The changes in both the autospectra and coherence indicate a conversion of energy from lower to higher convective wavenumbers.

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.101 ◽

2014 ◽

Vol 745 ◽

pp. 378-397 ◽

Cited By ~ 13

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.

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Mean-flow scaling of turbulent pipe flow

Journal of Fluid Mechanics ◽

10.1017/s0022112098002419 ◽

1998 ◽

Vol 373 ◽

pp. 33-79 ◽

Cited By ~ 454

Author(s):

MARK V. ZAGAROLA ◽

ALEXANDER J. SMITS

Keyword(s):

Friction Factor ◽

Friction Velocity ◽

Outer Region ◽

Reynolds Numbers ◽

Velocity Profiles ◽

Mean Velocity ◽

Velocity Scale ◽

The Mean ◽

Log Law ◽

High Reynolds

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×103 to 35×106. Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60<y+<500 or y+<0.15R+, the outer limit depending on whether the Kármán number R+ is greater or less than 9×103; and a log law for 600<y+<0.07R+. The log law is only evident if the Reynolds number is greater than approximately 400×103 (R+>9×103). Von Kármán's constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600<y+<0.07R+, and the additive constant was shown to be 6.15 when the log law is expressed in inner scaling variables.A new theory is developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. A reasonable candidate for the outer velocity scale is the velocity deficit in the pipe, UCL−Ū, which is a true outer velocity scale, in contrast to the friction velocity which is a velocity scale associated with the near-wall region which is ‘impressed’ on the outer region. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity.The friction factor data at high Reynolds numbers were found to be significantly larger (>5%) than those predicted by Prandtl's relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×103 and 35×106, and includes a term to account for the near-wall velocity profile.

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Spectral derivation of the classic laws of wall-bounded turbulent flows

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2017.0354 ◽

2017 ◽

Vol 473 (2204) ◽

pp. 20170354 ◽

Cited By ~ 4

Author(s):

Gustavo Gioia ◽

Pinaki Chakraborty

Keyword(s):

Turbulent Flows ◽

Turbulent Energy ◽

Wall Turbulence ◽

Sufficient Condition ◽

Physical Origin ◽

Mean Velocity ◽

Turbulent Friction ◽

Law Of The Wall ◽

The Mean ◽

Log Law

We show that the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law’—can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also show that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.

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A description of turbulent wall-flow vorticity consistent with mean dynamics

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.565 ◽

2013 ◽

Vol 737 ◽

pp. 176-204 ◽

Cited By ~ 27

Author(s):

J. C. Klewicki

Keyword(s):

Reynolds Number ◽

Wall Region ◽

Dynamical Structure ◽

Mean Velocity ◽

Flow Vorticity ◽

Wall Flows ◽

Outer Domain ◽

Wide Range ◽

The Mean ◽

Turbulent Wall

AbstractA depiction of the mean and fluctuating vorticity structure in turbulent wall flows is presented and described within the context of the self-similar properties admitted by the mean dynamical equation. Data from a relatively wide range of numerical and physical experiments are used to explore and clarify the structure postulated. The mean vorticity indicator for the onset of the four-layer regime of the mean dynamics is revealed. With increasing Reynolds number, the mean vorticity is shown to segregate into two increasingly well-defined domains. Half of the mean vorticity concentrates into a near-wall region of width (relative to the overall flow width) that diminishes proportionally to the inverse square root of Reynolds number. The remainder of the mean vorticity is spread, with diminishing amplitude, over an outer domain that approaches the overall flow width at high Reynolds number. Vorticity stretching and reorientation are surmised to be the characteristic mechanisms accounting for the inner domain behaviour of both the mean and fluctuating vorticity. Vorticity dispersion via advective transport is surmised to be the characteristic mechanism in the outer domain. In this domain, the fluctuating enstrophy approaches that of the instantaneous enstrophy with increasing Reynolds number. This underpins an emerging self-similarity between the mean and r.m.s. vorticity in the domain where the mean velocity profile is logarithmic. The Reynolds number dependence of a number of properties associated with the vorticity field is explored and quantified. The study closes with brief account of the combined vortical and mean dynamical structure of turbulent wall flows.

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Self-similar mean dynamics in turbulent wall flows

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.626 ◽

2013 ◽

Vol 718 ◽

pp. 596-621 ◽

Cited By ~ 43

Author(s):

J. C. Klewicki

Keyword(s):

Boundary Layer ◽

Similarity Solution ◽

Reynolds Numbers ◽

Channel Height ◽

Self Similarity ◽

Mean Velocity ◽

Wall Flows ◽

Self Similar ◽

The Mean ◽

Turbulent Wall

AbstractThis study investigates how and why dynamical self-similarities emerge with increasing Reynolds number within the canonical wall flows beyond the transitional regime. An overarching aim is to advance a mechanistically coherent description of turbulent wall-flow dynamics that is mathematically tractable and grounded in the mean dynamical equations. As revealed by the analysis of Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst.A, vol. 24, 2009, pp. 781–807), the equations that respectively describe the mean dynamics of turbulent channel, pipe and boundary layer flows formally admit invariant forms. These expose an underlying self-similar structure. In all cases, two kinds of dynamical self-similarity are shown to exist on an internal domain that, for all Reynolds numbers, extends from$O(\nu / {u}_{\tau } )$to$O(\delta )$, where$\nu $is the kinematic viscosity,${u}_{\tau } $is the friction velocity and$\delta $is the half-channel height, pipe radius, or boundary layer thickness. The simpler of the two self-similarities is operative on a large outer portion of the relevant domain. This self-similarity leads to an explicit analytical closure of the mean momentum equation. This self-similarity also underlies the emergence of a logarithmic mean velocity profile. A more complicated kind a self-similarity emerges asymptotically over a smaller domain closer to the wall. The simpler self-similarity allows the mean dynamical equation to be written as a closed system of nonlinear ordinary differential equations that, like the similarity solution for the laminar flat-plate boundary layer, can be numerically integrated. The resulting similarity solutions are demonstrated to exhibit nearly exact agreement with direct numerical simulations over the solution domain specified by the theory. At the Reynolds numbers investigated, the outer similarity solution is shown to be operative over a domain that encompasses${\sim }40\hspace{0.167em} \% $of the overall width of the flow. Other properties predicted by the theory are also shown to be well supported by existing data.

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The flow of liquid in a stream from the standard turbine impeller

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19790700 ◽

1979 ◽

Vol 44 (3) ◽

pp. 700-710 ◽

Cited By ~ 10

Author(s):

Ivan Fořt ◽

Hans-Otto Möckel ◽

Jan Drbohlav ◽

Miroslav Hrach

Keyword(s):

Reynolds Number ◽

Kinematic Viscosity ◽

Momentum Flux ◽

Relative Size ◽

Mean Velocity ◽

Axial Profile ◽

The Mean ◽

Hot Film ◽

Turbine Impeller

Profiles of the mean velocity have been analyzed in the stream streaking from the region of rotating standard six-blade disc turbine impeller. The profiles were obtained experimentally using a hot film thermoanemometer probe. The results of the analysis is the determination of the effect of relative size of the impeller and vessel and the kinematic viscosity of the charge on three parameters of the axial profile of the mean velocity in the examined stream. No significant change of the parameter of width of the examined stream and the momentum flux in the stream has been found in the range of parameters d/D ##m <0.25; 0.50> and the Reynolds number for mixing ReM ##m <2.90 . 101; 1 . 105>. However, a significant influence has been found of ReM (at negligible effect of d/D) on the size of the hypothetical source of motion - the radius of the tangential cylindrical jet - a. The proposed phenomenological model of the turbulent stream in region of turbine impeller has been found adequate for values of ReM exceeding 1.0 . 103.

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