Law of bounded dissipation and its consequences in turbulent wall flows

2021 ◽  
Vol 933 ◽  
Author(s):  
Xi Chen ◽  
Katepalli R. Sreenivasan
Keyword(s):  
Dissipation Rate ◽  
Scaling Law ◽  
Random Variable ◽  
Energy Dissipation Rate ◽  
Shear Stresses ◽  
Mean Velocity ◽  
Wall Flows ◽  
Maximum Value ◽  
Turbulent Wall ◽  
Promising Perspective

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number $Re_\tau$ . This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of $Re_\tau$ , owing to the natural constraint that the dissipation rate is bounded. Specifically, $\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$ where $\varPhi$ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript $\infty$ denotes the bounded asymptotic value of $\varPhi$ , and the coefficient $C_\varPhi$ depends on $\varPhi$ but not on $Re_\tau$ . Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form $\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$ , where $\varphi$ represents the streamwise or spanwise velocity fluctuation, and $\alpha _q$ and $\beta _q$ are independent of $Re_\tau$ . Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear $q$ -norm Gaussian’, is proposed to explain the observed linear dependence of $\alpha _q$ on $q$ .

scholarly journals Energy dissipation rate limits for flow through rough channels and tidal flow across topography

2016 ◽  
Vol 808 ◽  
pp. 562-575 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Energy Dissipation ◽  
Dissipation Rate ◽  
Stokes Equations ◽  
Bottom Topography ◽  
Scaling Law ◽  
Tidal Flow ◽  
Energy Dissipation Rate ◽  
Mean Flow ◽  
Smooth Channel ◽  
Flow Through

An upper bound on the energy dissipation rate per unit mass, $\unicode[STIX]{x1D700}$, for pressure-driven flow through a channel with rough walls is derived for the first time. For large Reynolds numbers, $Re$, the bound – $\unicode[STIX]{x1D700}\leqslant cU^{3}/h$ where $U$ is the mean flow through the channel, $h$ the channel height and $c$ a numerical prefactor – is independent of $Re$ (i.e. the viscosity) as in the smooth channel case but the numerical prefactor $c$, which is only a function of the surface heights and surface gradients (i.e. not higher derivatives), is increased. Crucially, this new bound captures the correct scaling law of what is observed in rough pipes and demonstrates that while a smooth pipe is a singular limit of the Navier–Stokes equations (data suggest $\unicode[STIX]{x1D700}\sim 1/(\log Re)^{2}U^{3}/h$ as $Re\rightarrow \infty$), it is a regular limit for current bounding techniques. As an application, the bound is extended to oscillatory flow to estimate the energy dissipation rate for tidal flow across bottom topography in the oceans.

Manning’s formula and Strickler’s scaling explained by a co-spectral budget model

2017 ◽  
Vol 812 ◽  
pp. 1189-1212 ◽  
Author(s):  
S. Bonetti ◽  
G. Manoli ◽  
C. Manes ◽  
A. Porporato ◽  
G. G. Katul
Keyword(s):  
Ad Hoc ◽  
Numerical Solutions ◽  
Energy Dissipation Rate ◽  
Theoretical Explanation ◽  
Inertial Subrange ◽  
Wall Region ◽  
Mean Velocity ◽  
Wide Range ◽  
Turbulent Wall ◽  
Spectral Shapes

Manning’s empirical formula in conjunction with Strickler’s scaling is widely used to predict the bulk velocity$V$from the hydraulic radius$R_{h}$, the roughness size$r$and the slope of the energy grade line$S$in uniform channel and pipe flows at high bulk Reynolds numbers. Despite their importance in science and engineering, both Manning’s and Strickler’s formulations have waited for decades before finding a theoretical explanation. This was provided, for the first time, by Gioia & Bombardelli (Phys. Rev. Lett., vol. 88, 2002, 014501), labelled as GB02, using phenomenological arguments. Perhaps their most remarkable finding was the link between the Strickler and the Kolmogorov scaling exponents, the latter pertaining to velocity fluctuations in the inertial subrange of the turbulence spectrum and presumed to be universal. In this work, the GB02 analysis is first revisited, showing that GB02 employed severalad hocscaling assumptions for the turbulent kinetic energy dissipation rate and, although implicitly, for the mean velocity gradient adjacent to the roughness elements. The similarity constants arising from the GB02 scaling assumptions were presumed to be independent of$r/R_{h}$, which is inconsistent with well-known flow properties in the near-wall region of turbulent wall flows. Because of the dependence of these similarity constants on$r/R_{h}$, this existing theory requires the validity of the Strickler scaling to cancel the dependence of these constants on$r/R_{h}$so as to arrive at the Strickler scaling and Manning’s formula. Here, the GB02 approach is corroborated using a co-spectral budget (CSB) model for the wall shear stress formulated at the cross-over between the roughness sublayer and the log region. Assuming a simplified shape for the spectrum of the vertical velocity$w$, the proposed CSB model (subject to another simplifying assumption that production is balanced by pressure–velocity interaction) allows Manning’s formula to be derived. To substantiate this approach, numerical solutions to the CSB over the entire flow depth using different spectral shapes for$w$are carried out for a wide range of$r/R_{h}$. The results from this analysis support the simplifying hypotheses used to derive Manning’s equation. The derived equation provides a formulation for$n$that agrees with reported values in the literature over seven decades of$r$variations. While none of the investigated spectral shapes allows the recovery of the Strickler scaling, the numerical solutions of the CSB reproduce the Nikuradse data in the fully rough regime, thereby confirming that the Strickler scaling represents only an approximate fit for the friction factor for granular roughness.

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

2013 ◽  
Vol 737 ◽  
pp. 176-204 ◽  
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.

scholarly journals Deriving Six Components of Reynolds Stress Tensor from Single-ADCP Data

Water ◽  
2021 ◽  
Vol 13 (17) ◽  
pp. 2389
Author(s):  
Sergey Bogdanov ◽  
Roman Zdorovennov ◽  
Nikolay Palshin ◽  
Galina Zdorovennova
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Energy Dissipation Rate ◽  
New Method ◽  
Shear Stresses ◽  
Depth Range ◽  
Mean Velocity ◽  
Reynolds Stress Tensor ◽  
Adcp Data ◽  
Acoustic Doppler Current Profilers

Acoustic Doppler current profilers (ADCP) are widely used in geophysical studies for mean velocity profiling and calculation of energy dissipation rate. On the other hand, the estimation of turbulent stresses from ADCP data still remains challenging. With the four-beam version of the device, only two shear stresses are derivable; and even for the five-beam version (Janus+), the calculation of the full Reynolds stress tensor is problematic currently. The known attempts to overcome the problem are based on the “coupled ADCP” experimental setup and include some hard restrictions, not to mention the essential complexity of performing experiments. In this paper, a new method is presented which allows to derive the stresses from single-ADCP data. Its essence is that interbeam correlations are taken into account as producing the missing equations for stresses. This method is applicable only for the depth range, for which the distance between the beams is comparable to the scales, where the turbulence is locally isotropic and homogeneous. The validation of this method was carried out for convectively-mixed layer in a boreal ice-covered lake. The results of computations turned out to be physically sustainable in the sense that realizability conditions were basically fulfilled. The additional verification was carried out by comparing the results, obtained by the new method and “coupled ADCPs” one.

scholarly journals Inertia–Gravity Waves Breaking in the Middle Atmosphere at High Latitudes: Energy Transfer and Dissipation Tensor Anisotropy

2020 ◽  
Vol 77 (9) ◽  
pp. 3193-3210
Author(s):  
Tiago Pestana ◽  
Matthias Thalhammer ◽  
Stefan Hickel
Keyword(s):  
Reynolds Number ◽  
Energy Transfer ◽  
Energy Dissipation ◽  
Kinetic Energy ◽  
Gravity Waves ◽  
Energy Budget ◽  
Dissipation Rate ◽  
Scaling Law ◽  
Energy Dissipation Rate ◽  
Inertia Gravity Waves

Abstract We present direct numerical simulations of inertia–gravity waves breaking in the middle–upper mesosphere. We consider two different altitudes, which correspond to the Reynolds number of 28 647 and 114 591 based on wavelength and buoyancy period. While the former was studied by Remmler et al., it is here repeated at a higher resolution and serves as a baseline for comparison with the high-Reynolds-number case. The simulations are designed based on the study of Fruman et al., and are initialized by superimposing primary and secondary perturbations to the convectively unstable base wave. Transient growth leads to an almost instantaneous wave breaking and secondary bursts of turbulence. We show that this process is characterized by the formation of fine flow structures that are predominantly located in the vicinity of the wave’s least stable point. During the wave breakdown, the energy dissipation rate tends to be an isotropic tensor, whereas it is strongly anisotropic in between the breaking events. We find that the vertical kinetic energy spectra exhibit a clear 5/3 scaling law at instants of intense energy dissipation rate and a cubic power law at calmer periods. The term-by-term energy budget reveals that the pressure term is the most important contributor to the global energy budget, as it couples the vertical and the horizontal kinetic energy. During the breaking events, the local energy transfer is predominantly from the mean to the fluctuating field and the kinetic energy production is in balance with the pseudo kinetic energy dissipation rate.

Self-similar mean dynamics in turbulent wall flows

2013 ◽  
Vol 718 ◽  
pp. 596-621 ◽  
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.

Energy dissipation rate in a baffled vessel with pitched blade turbine impeller

1991 ◽  
Vol 56 (9) ◽  
pp. 1856-1867 ◽  
Author(s):  
Zdzisław Jaworski ◽  
Ivan Fořt
Keyword(s):  
Energy Dissipation ◽  
Cross Sections ◽  
Mechanical Energy ◽  
Vessel Diameter ◽  
Energy Dissipation Rate ◽  
Mean Velocity ◽  
Agitated Vessel ◽  
Radial Profiles ◽  
Pitched Blade Turbine ◽  
Turbine Impeller

Mechanical energy dissipation was investigated in a cylindrical, flat bottomed vessel with four radial baffles and the pitched blade turbine impeller of varied size. This study was based upon the experimental data on the hydrodynamics of the turbulent flow of water in an agitated vessel. They were gained by means of the three-holes Pitot tube technique for three impeller-to-vessel diameter ratio d/D = 1/3, 1/4 and 1/5. The experimental results obtained for two levels below and two levels above the impeller were used in the present study. Radial profiles of the mean velocity components, static and total pressures were presented for one of the levels. Local contribution to the axial transport of the agitated charge and energy was presented. Using the assumption of the axial symmetry of the flow field the volumetric flow rates were determined for the four horizontal cross-sections. Regions of positive and negative values of the total pressure of the liquid were indicated. Energy dissipation rates in various regions of the agitated vessel were estimated in the range from 0.2 to 6.0 of the average value for the whole vessel. Hydraulic impeller efficiency amounting to about 68% was obtained. The mechanical energy transferred by the impellers is dissipated in the following ways: 54% in the space below the impeller, 32% in the impeller region, 14% in the remaining part of the agitated liquid.

Statistical properties of streamline geometry in turbulent wall-flows

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Rina Perven ◽  
Jimmy Philip ◽  
Joseph Klewicki
Keyword(s):  
Statistical Properties ◽  
Wall Flows ◽  
Turbulent Wall
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