Spectral scaling in boundary layers and pipes at very high Reynolds numbers

2015 ◽  
Vol 771 ◽  
pp. 303-326 ◽  
Author(s):  
M. Vallikivi ◽  
B. Ganapathisubramani ◽  
A. J. Smits
Keyword(s):  
Reynolds Number ◽  
Boundary Layers ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Spectral Peak ◽  
Wall Region ◽  
Thin Wall ◽  
Pipe Flows ◽  
Wide Range ◽  
Turbulent Wall

One-dimensional energy spectra in flat plate zero pressure gradient boundary layers and pipe flows are examined over a wide range of Reynolds numbers ($2600\leqslant \mathit{Re}_{{\it\tau}}\leqslant 72\,500$). The spectra show excellent collapse with Kolmogorov scaling at high wavenumbers for both flows at all Reynolds numbers. The peaks associated with the large-scale motions (LSMs) and superstructures (SS) in boundary layers behave as they do in pipe flows, with some minor differences. The location of the outer spectral peak, associated with SS or very large-scale motions (VLSMs) in the turbulent wall region, displays only a weak dependence on Reynolds number, and it occurs at the same wall-normal distance where the variances establish a logarithmic behaviour and where the amplitude modulation coefficient has a zero value. The results suggest that with increasing Reynolds number the energy is largely confined to a thin wall layer that continues to diminish in physical extent. The outer-scaled wavelength of the outer spectral peak appears to decrease with increasing Reynolds number. However, there is still significant energy content in wavelengths associated with the SS and VLSMs. The location of the outer spectral peak appears to mark the start of a plateau that is consistent with a $k_{x}^{-1}$ slope in the spectrum and the logarithmic variation in the variances. This $k_{x}^{-1}$ region seems to occur when there is sufficient scale separation between the locations of the outer spectral peak and the outer edge of the log region. It does not require full similarity between outer and wall-normal scaling on the wavenumber. The extent of $k_{x}^{-1}$ region depends on the wavelength of the outer spectral peak (${\it\lambda}_{OSP}$), which appears to emerge as a new length scale for the log region. Finally, based on the observations from the spectra together with the statistics presented in Vallikivi et al. (J. Fluid Mech., 2015 (submitted)), five distinct wall-normal layers are identified in turbulent wall flows.

scholarly journals Assessment of inner–outer interactions in the urban boundary layer using a predictive model

2019 ◽  
Vol 875 ◽  
pp. 44-70 ◽  
Author(s):  
Karin Blackman ◽  
Laurent Perret ◽  
Romain Mathis
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Predictive Model ◽  
Boundary Layers ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Limited Range ◽  
Rough Wall ◽  
Wall Boundary ◽  
Layer Flow

Urban-type rough-wall boundary layers developing over staggered cube arrays with plan area packing density, $\unicode[STIX]{x1D706}_{p}$, of 6.25 %, 25 % or 44.4 % have been studied at two Reynolds numbers within a wind tunnel using hot-wire anemometry (HWA). A fixed HWA probe is used to capture the outer-layer flow while a second moving probe is used to capture the inner-layer flow at 13 wall-normal positions between $1.25h$ and $4h$ where $h$ is the height of the roughness elements. The synchronized two-point HWA measurements are used to extract the near-canopy large-scale signal using spectral linear stochastic estimation and a predictive model is calibrated in each of the six measurement configurations. Analysis of the predictive model coefficients demonstrates that the canopy geometry has a significant influence on both the superposition and amplitude modulation. The universal signal, the signal that exists in the absence of any large-scale influence, is also modified as a result of local canopy geometry suggesting that although the nonlinear interactions within urban-type rough-wall boundary layers can be modelled using the predictive model as proposed by Mathis et al. (J. Fluid Mech., vol. 681, 2011, pp. 537–566), the model must be however calibrated for each type of canopy flow regime. The Reynolds number does not significantly affect any of the model coefficients, at least over the limited range of Reynolds numbers studied here. Finally, the predictive model is validated using a prediction of the near-canopy signal at a higher Reynolds number and a prediction using reference signals measured in different canopy geometries to run the model. Statistics up to the fourth order and spectra are accurately reproduced demonstrating the capability of the predictive model in an urban-type rough-wall boundary layer.

Hot-wire spatial resolution issues in wall-bounded turbulence

2009 ◽  
Vol 635 ◽  
pp. 103-136 ◽  
Author(s):  
N. HUTCHINS ◽  
T. B. NICKELS ◽  
I. MARUSIC ◽  
M. S. CHONG
Keyword(s):  
Reynolds Number ◽  
Boundary Layers ◽  
Spatial Resolution ◽  
Reynolds Numbers ◽  
Energy Spectra ◽  
Small Scale ◽  
Wire Length ◽  
Hot Wire ◽  
Wide Range ◽  
Near Wall

Careful reassessment of new and pre-existing data shows that recorded scatter in the hot-wire-measured near-wall peak in viscous-scaled streamwise turbulence intensity is due in large part to the simultaneous competing effects of the Reynolds number and viscous-scaled wire length l+. An empirical expression is given to account for these effects. These competing factors can explain much of the disparity in existing literature, in particular explaining how previous studies have incorrectly concluded that the inner-scaled near-wall peak is independent of the Reynolds number. We also investigate the appearance of the so-called outer peak in the broadband streamwise intensity, found by some researchers to occur within the log region of high-Reynolds-number boundary layers. We show that the ‘outer peak’ is consistent with the attenuation of small scales due to large l+. For turbulent boundary layers, in the absence of spatial resolution problems, there is no outer peak up to the Reynolds numbers investigated here (Reτ = 18830). Beyond these Reynolds numbers – and for internal geometries – the existence of such peaks remains open to debate. Fully mapped energy spectra, obtained with a range of l+, are used to demonstrate this phenomenon. We also establish the basis for a ‘maximum flow frequency’, a minimum time scale that the full experimental system must be capable of resolving, in order to ensure that the energetic scales are not attenuated. It is shown that where this criterion is not met (in this instance due to insufficient anemometer/probe response), an outer peak can be reproduced in the streamwise intensity even in the absence of spatial resolution problems. It is also shown that attenuation due to wire length can erode the region of the streamwise energy spectra in which we would normally expect to see kx−1 scaling. In doing so, we are able to rationalize much of the disparity in pre-existing literature over the kx−1 region of self-similarity. Not surprisingly, the attenuated spectra also indicate that Kolmogorov-scaled spectra are subject to substantial errors due to wire spatial resolution issues. These errors persist to wavelengths far beyond those which we might otherwise assume from simple isotropic assumptions of small-scale motions. The effects of hot-wire length-to-diameter ratio (l/d) are also briefly investigated. For the moderate wire Reynolds numbers investigated here, reducing l/d from 200 to 100 has a detrimental effect on measured turbulent fluctuations at a wide range of energetic scales, affecting both the broadband intensity and the energy spectra.

scholarly journals Low-Reynolds-number turbulent boundary layers

1991 ◽  
Vol 230 ◽  
pp. 1-44 ◽  
Author(s):  
Lincoln P. Erm ◽  
Peter N. Joubert
Keyword(s):  
Reynolds Number ◽  
Boundary Layers ◽  
Low Reynolds Number ◽  
Turbulent Boundary Layers ◽  
Stream Velocity ◽  
Wall Region ◽  
Mean Flow ◽  
Low Reynolds ◽  
High Reynolds ◽  
Turbulent Wall

An investigation was undertaken to improve our understanding of low-Reynolds-number turbulent boundary layers flowing over a smooth flat surface in nominally zero pressure gradients. In practice, such flows generally occur in close proximity to a tripping device and, though it was known that the flows are affected by the actual low value of the Reynolds number, it was realized that they may also be affected by the type of tripping device used and variations in free-stream velocity for a given device. Consequently, the experimental programme was devised to investigate systematically the effects of each of these three factors independently. Three different types of device were chosen: a wire, distributed grit and cylindrical pins. Mean-flow, broadband-turbulence and spectral measurements were taken, mostly for values of Rθ varying between about 715 and about 2810. It was found that the mean-flow and broadband-turbulence data showed variations with Rθ, as expected. Spectra were plotted using scaling given by Perry, Henbest & Chong (1986) and were compared with their models which were developed for high-Reynolds-number flows. For the turbulent wall region, spectra showed reasonably good agreement with their model. For the fully turbulent region, spectra did show some appreciable deviations from their model, owing to low-Reynolds-number effects. Mean-flow profiles, broadband-turbulence profiles and spectra were found to be affected very little by the type of device used for Rθ ≈ 1020 and above, indicating an absence of dependence on flow history for this Rθ range. These types of measurements were also compared at both Rθ ≈ 1020 and Rθ ≈ 2175 to see if they were dependent on how Rθ was formed (i.e. the combination of velocity and momentum thickness used to determine Rθ). There were noticeable differences for Rθ ≈ 1020, but these differences were only convincing for the pins, and there was a general overall improvement in agreement for Rθ ≈ 2175.

Reynolds number scaling of inertial particle statistics in turbulent channel flows

2014 ◽  
Vol 758 ◽  
Author(s):  
Matteo Bernardini
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Large Scale ◽  
Spatial Organization ◽  
Reynolds Numbers ◽  
Stokes Number ◽  
Wall Region ◽  
Inertial Particles ◽  
Turbulent Channel Flows ◽  
Strong Segregation

AbstractThe effect of the Reynolds number on the behaviour of inertial particles in wall-bounded turbulent flows is investigated through large-scale direct numerical simulations (DNS) of particle-laden canonical channel flow spanning almost a decade in the friction Reynolds number, from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau } = 150$ to $\mathit{Re}_{\tau } = 1000$. Lagrangian particle tracking is used to study the motion of six different particle sets, described by a Stokes number in the range $\mathit{St} = 1\text {--}1000$. At all Reynolds numbers a strong segregation in the near-wall region is observed for particles characterized by intermediate Stokes number, in the range $\mathit{St} =10\text {--}100$. The wall-normal concentration profiles of such particles collapse in inner scaling, thus suggesting the independence of the turbophoretic drift from the large-scale outer motions. This observation is also supported by the spatial organization of the suspended phase in the inner layer, which is found to be universal with the Reynolds number. The deposition rate coefficient increases with $\mathit{Re}_{\tau }$ for a given $\mathit{St}$. Suitable inner and outer scalings are proposed to collapse the deposition curves across the available ranges of Reynolds and Stokes numbers for the different deposition regimes.

Very large scale motions in the atmospheric surface layer: a field investigation

2016 ◽  
Vol 802 ◽  
pp. 464-489 ◽  
Author(s):  
Guohua Wang ◽  
Xiaojing Zheng
Keyword(s):  
Reynolds Number ◽  
Spectral Analysis ◽  
Surface Layer ◽  
Boundary Layers ◽  
Large Scale ◽  
Atmospheric Surface Layer ◽  
Reynolds Numbers ◽  
Turbulent Boundary Layers ◽  
Western Desert ◽  
Wavenumber Region

A field observation array for the atmospheric surface layer (ASL) was built on a dry flat bed of Qingtu Lake in Minqin (China) as the Qingtu Lake Observation Array (QLOA) site, which is similar to the Surface Layer Turbulence and Environmental Science Test (SLTEST) site in the Utah (USA) Western desert. The present observation array can synchronously perform multi-point measurements of wind velocity and temperature at different vertical and streamwise positions. In other words, three-dimensional turbulent ASL flows can be measured at the QLOA station and Reynolds numbers as high as $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{6})$ can be achieved with steady wind conditions. By careful selection and pretreatment for measured data of more than 1200 h, the QLOA data have been validated to be reliable for high Reynolds number turbulent boundary layer research. Results from correlation and spectral analysis confirm that very large scale motions (VLSMs) exist in the ASL at a Reynolds number up to $Re_{\unicode[STIX]{x1D70F}}\approx 4\times 10^{6}$. Through premultiplied spectral analysis, it is revealed that the spectral energy in the high-wavenumber region decreases with height, similar to turbulent boundary layers at low or moderate Reynolds numbers, while it increases with height in the low-wavenumber region resulting in a log–linear increase of VLSMs energy with height, which is different from turbulent boundary layers at low or moderate Reynolds numbers. The present analyses support the view that the evolution of the VLSMs cannot be fully attributed to a ‘bottom-up’ mechanism alone, and probably other mechanisms, including a ‘top-down’ mechanism, also play a role.

scholarly journals Turbulent plane Couette flow at moderately high Reynolds number

2014 ◽  
Vol 751 ◽  
Author(s):  
V. Avsarkisov ◽  
S. Hoyas ◽  
M. Oberlack ◽  
J. P. García-Galache
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Wall Layer ◽  
Wall Region ◽  
Plane Couette Flow ◽  
Turbulent Plane ◽  
High Reynolds ◽  
Near Wall

AbstractA new set of numerical simulations of turbulent plane Couette flow in a large box of dimension ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}20\pi h,\, 2h,\, 6\pi h$) at Reynolds number $(\mathit{Re}_{\tau }) =125$, 180, 250 and 550 is described and compared with simulations at lower Reynolds numbers, Poiseuille flows and experiments. The simulations present a logarithmic near-wall layer and are used to verify and revise previously known results. It is confirmed that the fluctuation intensities in the streamwise and spanwise directions do not scale well in wall units. The scaling failure occurs both near to and away from the wall. On the contrary, the wall-normal intensity scales in inner units in the near-wall region and in outer units in the core region. The spectral ridge found by Hoyas & Jiménez (Phys. Fluids, vol. 18, 2003, 011702) for the turbulent Poiseuille flow can also be seen in the present flow. Away from the wall, very large-scale motions are found spanning through all the length of the channel. The statistics of these simulations can be downloaded from the webpage of the Chair of Fluid Dynamics.

Turbulence spectra in smooth- and rough-wall pipe flow at extreme Reynolds numbers

2013 ◽  
Vol 731 ◽  
pp. 46-63 ◽  
Author(s):  
B. J. Rosenberg ◽  
M. Hultmark ◽  
M. Vallikivi ◽  
S. C. C. Bailey ◽  
A. J. Smits
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
Large Scale ◽  
Reynolds Numbers ◽  
Turbulent Pipe Flow ◽  
Rough Wall ◽  
Inertial Subrange ◽  
Wall Region ◽  
Turbulence Structure ◽  
High Reynolds

AbstractWell-resolved streamwise velocity spectra are reported for smooth- and rough-wall turbulent pipe flow over a large range of Reynolds numbers. The turbulence structure far from the wall is seen to be unaffected by the roughness, in accordance with Townsend’s Reynolds number similarity hypothesis. Moreover, the energy spectra within the turbulent wall region follow the classical inner and outer scaling behaviour. While an overlap region between the two scalings and the associated${ k}_{x}^{- 1} $law are observed near${R}^{+ } \approx 3000$, the${ k}_{x}^{- 1} $behaviour is obfuscated at higher Reynolds numbers due to the evolving energy content of the large scales (the very-large-scale motions, or VLSMs). We apply a semi-empirical correction (del Álamo & Jiménez,J. Fluid Mech., vol. 640, 2009, pp. 5–26) to the experimental data to estimate how Taylor’s frozen field hypothesis distorts the pseudo-spatial spectra inferred from time-resolved measurements. While the correction tends to suppress the long wavelength peak in the logarithmic layer spectrum, the peak nonetheless appears to be a robust feature of pipe flow at high Reynolds number. The inertial subrange develops around${R}^{+ } \gt 2000$where the characteristic${ k}_{x}^{- 5/ 3} $region is evident, which, for high Reynolds numbers, persists in the wake and logarithmic regions. In the logarithmic region, the streamwise wavelength of the VLSM peak scales with distance from the wall, which is in contrast to boundary layers, where the superstructures have been shown to scale with boundary layer thickness throughout the entire shear layer. Moreover, the similarity in the streamwise wavelength scaling of the large- and very-large-scale motions supports the notion that the two are physically interdependent.

On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers, and slugs of vorticity in channel and pipe flows

1990 ◽  
Vol 428 (1875) ◽  
pp. 255-281 ◽  
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Wall Layer ◽  
Amplitude Dependence ◽  
Thin Wall ◽  
Pipe Flows ◽  
High Reynolds Numbers ◽  
High Reynolds

This theoretical study is motivated by the experimental observations ( a ) on the thickening of a turbulent boundary layer compared with its laminar counterpart, ( b ) on the erupting tongue of fluid that forms the leading edge of a turbulent spot in a boundary layer, ( c ) on the wall-layer and mid-flow scales, and ( d ) on the slugs of vorticity that occur in the middle of turbulent channel and pipe flows. It appears that no previous rational explanation has been put forward for these experimental observations. The present tentative suggestions for ( a ), ( b ) and ( d ) centre on the existence of small-deficit fast-travelling zones of concentrated vorticity governed by the nonlinear Euler equations to leading order at high Reynolds numbers Re but crucially influenced by viscosity nevertheless. In the boundary-layer case these zones travel outside the original boundary layer and hence act to increase the effective boundary-layer thickness. The structure of such zones and their scales, governing equations and amplitude dependence are discussed for assumed planar boundary layers and channel flows and for three-dimensional pipe flows in turn. Allied with this, the theory addresses the closure of the amplitude-dependent neutral curve at high Reynolds numbers, the connection with other Euler-type flows and the possibility of delay in sublayer bursting, as well as aiming to give some guidance on nonlinear aspects of unsteady two- and three-dimensional computations for Euler and related flows. The aspects in ( c ) above, concerning the turbulent scales both of the thin wall layer ( O ( Re -1 In Re ), from a renormalizing and scale-cascade argument) and of the thicker mid-flow zone (containing the Kolmogorov microscale O ( Re -3/4 )) which lies between that layer and the extensive small-deficit outer zone, are also discussed tentatively in terms of their dynamics, leading to apparently good agreement with turbulent-flow experiments and empirical models, for those scales. Other qualitative comparisons are presented.

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 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers

2007 ◽  
Vol 579 ◽  
pp. 1-28 ◽  
Author(s):  
N. HUTCHINS ◽  
IVAN MARUSIC
Keyword(s):  
Reynolds Number ◽  
Boundary Layers ◽  
High Speed ◽  
Large Scale ◽  
Sonic Anemometer ◽  
Streamwise Velocity ◽  
Turbulent Boundary Layers ◽  
Wall Region ◽  
Velocity Fluctuations ◽  
Near Wall

A regime of very long meandering positive and negative streamwise velocity fluctuations, that we term ‘superstructures’, are found to exist in the log and lower wake regions of turbulent boundary layers. Measurements are made with a spanwise rake of 10 hot-wires in two separate facilities (spanning more than a decade of Reτ) and are compared with existing PIV and DNS results. In all cases, we note evidence of a large-scale stripiness in the streamwise velocity fluctuations. The length of these regions can commonly exceed 20δ. Similar length scales have been previously reported for pipes and DNS channel flows. It is suggested that the true length of these features is masked from single-point statistics (such as autocorrelations and spectra) by a spanwise meandering tendency. Support for this conjecture is offered through the study of a synthetic flow composed only of sinusoidally meandering elongated low- and high-speed regions. From detailed maps of one-dimensional spectra, it is found that the contribution to the streamwise turbulence intensities associated with the superstructures appears to be increasingly significant with Reynolds number, and scales with outer length variables (δ). Importantly, the superstructure maintains a presence or footprint in the near-wall region, seeming to modulate or influence the near-wall cycle. This input of low-wavenumber outer-scaled energy into the near-wall region is consistent with the rise in near-wall streamwise intensities, when scaled with inner variables, that has been noted to occur with increasing Reynolds number. In an attempt to investigate these structures at very high Reynolds numbers, we also report on recent large-scale sonic anemometer rake measurements, made in the neutrally stable atmospheric surface layer. Preliminary results indicate that the superstructure is present in the log region of this atmospheric flow at Reτ = 6.6×105, and has a size consistent with outer scaling.

Sign in / Sign up

Export Citation Format

Share Document