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.

scholarly journals Transition to turbulence in the rotating disk boundary layer of a rotor–stator cavity

2018 ◽  
Vol 848 ◽  
pp. 631-647 ◽  
Author(s):  
Eunok Yim ◽  
J.-M. Chomaz ◽  
D. Martinand ◽  
E. Serre
Keyword(s):  
Boundary Layer ◽  
Similarity Solution ◽  
Rotating Disk ◽  
Transition To Turbulence ◽  
Mean Flow ◽  
Self Similarity ◽  
Mean Velocity ◽  
Global Mode ◽  
Spatial Growth ◽  
The Mean

The transition to turbulence in the rotating disk boundary layer is investigated in a closed cylindrical rotor–stator cavity via direct numerical simulation (DNS) and linear stability analysis (LSA). The mean flow in the rotor boundary layer is qualitatively similar to the von Kármán self-similarity solution. The mean velocity profiles, however, slightly depart from theory as the rotor edge is approached. Shear and centrifugal effects lead to a locally more unstable mean flow than the self-similarity solution, which acts as a strong source of perturbations. Fluctuations start rising there, as the Reynolds number is increased, eventually leading to an edge-driven global mode, characterized by spiral arms rotating counter-clockwise with respect to the rotor. At larger Reynolds numbers, fluctuations form a steep front, no longer driven by the edge, and followed downstream by a saturated spiral wave, eventually leading to incipient turbulence. Numerical results show that this front results from the superposition of several elephant front-forming global modes, corresponding to unstable azimuthal wavenumbers $m$, in the range $m\in [32,78]$. The spatial growth along the radial direction of the energy of these fluctuations is quantitatively similar to that observed experimentally. This superposition of elephant modes could thus provide an explanation for the discrepancy observed in the single disk configuration, between the corresponding spatial growth rates values measured by experiments on the one hand, and predicted by LSA and DNS performed in an azimuthal sector, on the other hand.

scholarly journals Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels

2013 ◽  
Vol 734 ◽  
pp. 275-316 ◽  
Author(s):  
Rashad Moarref ◽  
Ati S. Sharma ◽  
Joel A. Tropp ◽  
Beverley J. McKeon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Low Rank ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
The Mean ◽  
High Reynolds

AbstractWe study the Reynolds-number scaling and the geometric self-similarity of a gain-based, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier–Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (${Re}_{\tau } \approx 1{0}^{3} {\unicode{x2013}} 1{0}^{10} $). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.

scholarly journals Turbulent boundary layers and channels at moderate Reynolds numbers

2010 ◽  
Vol 657 ◽  
pp. 335-360 ◽  
Author(s):  
JAVIER JIMÉNEZ ◽  
SERGIO HOYAS ◽  
MARK P. SIMENS ◽  
YOSHINORI MIZUNO
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Turbulent Flows ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Streamwise Velocity ◽  
Rotational Flow ◽  
Mean Velocity ◽  
The Mean ◽  
External Flows

The behaviour of the velocity and pressure fluctuations in the outer layers of wall-bounded turbulent flows is analysed by comparing a new simulation of the zero-pressure-gradient boundary layer with older simulations of channels. The 99 % boundary-layer thickness is used as a reasonable analogue of the channel half-width, but the two flows are found to be too different for the analogy to be complete. In agreement with previous results, it is found that the fluctuations of the transverse velocities and of the pressure are stronger in the boundary layer, and this is traced to the pressure fluctuations induced in the outer intermittent layer by the differences between the potential and rotational flow regions. The same effect is also shown to be responsible for the stronger wake component of the mean velocity profile in external flows, whose increased energy production is the ultimate reason for the stronger fluctuations. Contrary to some previous results by our group, and by others, the streamwise velocity fluctuations are also found to be higher in boundary layers, although the effect is weaker. Within the limitations of the non-parallel nature of the boundary layer, the wall-parallel scales of all the fluctuations are similar in both the flows, suggesting that the scale-selection mechanism resides just below the intermittent region, y/δ = 0.3–0.5. This is also the location of the largest differences in the intensities, although the limited Reynolds number of the boundary-layer simulation (Reθ ≈ 2000) prevents firm conclusions on the scaling of this location. The statistics of the new boundary layer are available from http://torroja.dmt.upm.es/ftp/blayers/.

Near-Wall Measurements in Turbulent Boundary Layers Using Laser Doppler Anemometry

2002 ◽  
Author(s):  
T. Gunnar Johansson ◽  
Luciano Castillo
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Pressure Gradient ◽  
Laser Doppler Anemometry ◽  
Reynolds Numbers ◽  
Mean Velocity ◽  
The Mean ◽  
Wall Shear ◽  
Near Wall

Near wall measurements have been performed in a zero pressure gradient turbulent boundary layer at low to moderate local Reynolds numbers using Laser-Doppler Anemometry in order to investigate how accurately the wall shear stress can be determined. Also, scaling problems are particularly difficult at low Reynolds numbers since they involve simultaneous influences of both inner and outer scales and this is most clearly observed in the near-wall region. In order to fully describe the zero pressure gradient turbulent boundary layer at low to moderate local Reynolds numbers it is necessary to accurately measure a number of quantities. These include the mean velocity and Reynolds stresses, and their spatial derivatives all the way down to the wall (y+∼1). Integral parameters that need to be measured are the wall shear stress and boundary layer thickness, particularly the momentum thickness. Problems with the measurement of field properties get worse close to a wall, and they get worse for increasing local Reynolds number. Three different approaches to measure the wall shear stress were examined. It was found that small measurement errors in the mean velocity close to the wall significantly reduced the accuracy in determining the wall shear stress by measuring the velocity gradient at the wall. The constant stress layer was found to be affected by the advection terms. However, it was found that taking the small pressure gradient into account and improving on the spatial resolution in the outer part of the boundary layer made the momentum integral method reliable.

scholarly journals Simultaneous skin friction and velocity measurements in high Reynolds number pipe and boundary layer flows

2019 ◽  
Vol 871 ◽  
pp. 377-400 ◽  
Author(s):  
R. Baidya ◽  
W. J. Baars ◽  
S. Zimmerman ◽  
M. Samie ◽  
R. J. Hearst ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Large Scale ◽  
Streamwise Velocity ◽  
Momentum Equation ◽  
The Self ◽  
Self Similarity ◽  
Self Similar ◽  
The Mean ◽  
High Reynolds

Streamwise velocity and wall-shear stress are acquired simultaneously with a hot-wire and an array of azimuthal/spanwise-spaced skin friction sensors in large-scale pipe and boundary layer flow facilities at high Reynolds numbers. These allow for a correlation analysis on a per-scale basis between the velocity and reference skin friction signals to reveal which velocity-based turbulent motions are stochastically coherent with turbulent skin friction. In the logarithmic region, the wall-attached structures in both the pipe and boundary layers show evidence of self-similarity, and the range of scales over which the self-similarity is observed decreases with an increasing azimuthal/spanwise offset between the velocity and the reference skin friction signals. The present empirical observations support the existence of a self-similar range of wall-attached turbulence, which in turn are used to extend the model of Baarset al.(J. Fluid Mech., vol. 823, p. R2) to include the azimuthal/spanwise trends. Furthermore, the region where the self-similarity is observed correspond with the wall height where the mean momentum equation formally admits a self-similar invariant form, and simultaneously where the mean and variance profiles of the streamwise velocity exhibit logarithmic dependence. The experimental observations suggest that the self-similar wall-attached structures follow an aspect ratio of$7:1:1$in the streamwise, spanwise and wall-normal directions, respectively.

The intermediate wake of a body of revolution at high Reynolds numbers

2010 ◽  
Vol 659 ◽  
pp. 516-539 ◽  
Author(s):  
JUAN M. JIMÉNEZ ◽  
M. HULTMARK ◽  
A. J. SMITS
Keyword(s):  
Reynolds Number ◽  
Reynolds Numbers ◽  
Stress Distributions ◽  
Body Of Revolution ◽  
Self Similarity ◽  
Mean Velocity ◽  
High Reynolds Numbers ◽  
Order Of Magnitude ◽  
The Mean ◽  
High Reynolds

Results are presented on the flow field downstream of a body of revolution for Reynolds numbers based on a model length ranging from 1.1 × 106 to 67 × 106. The maximum Reynolds number is more than an order of magnitude larger than that obtained in previous laboratory wake studies. Measurements are taken in the intermediate wake at locations 3, 6, 9, 12 and 15 diameters downstream from the stern in the midline plane. The model is based on an idealized submarine shape (DARPA SUBOFF), and it is mounted in a wind tunnel on a support shaped like a semi-infinite sail. The mean velocity distributions on the side opposite the support demonstrate self-similarity at all locations and Reynolds numbers, whereas the mean velocity distribution on the side of the support displays significant effects of the support wake. None of the Reynolds stress distributions of the flow attain self-similarity, and for all except the lowest Reynolds number, the support introduces a significant asymmetry into the wake which results in a decrease in the radial and streamwise turbulence intensities on the support side. The distributions continue to evolve with downstream position and Reynolds number, although a slow approach to the expected asymptotic behaviour is observed with increasing distance downstream.

Behavior of Separation Bubble and Reattached Boundary Layer Around a Circular Leading Edge

1994 ◽  
Author(s):  
B. K. Hazarika ◽  
C. Hirsch
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Separation Bubble ◽  
Low Frequency ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Mean Velocity ◽  
Low Frequency Oscillations ◽  
Lower Reynolds Number ◽  
The Mean

The flow around a circular leading edge airfoil is investigated in an incompressible, low turbulence freestream. Hot-wire measurements are performed through the separation bubble, the reattachment and the recovery region till development of the fully turbulent boundary layer. The results of the experiments in the range of Reynolds numbers 1.7×103 to 11.8×103 are analysed and presented in this paper. A separation bubble is present near the leading edge at all Reynolds numbers. At the lowest Reynolds number investigated, the transition is preceded by strong low frequency oscillations. The correlation given by Mayle for prediction of transition of short separation bubbles is successful at the lower Reynolds number cases. The length of the separation bubble reduces considerably with increasing Reynolds number in the range investigated. The turbulence in the reattached flow persists even when the Reynolds number based on momentum thickness of the reattached boundary layer is small. The recovery length of the reattached layer is relatively short and the mean velocity profile follows logarithmic law within a short distance downstream of the reattachment point and the friction coefficient conforms to Prandtl-Schlichting skin-friction formula for a smooth flat plate at zero incidence.

scholarly journals Scaling of the turbulent/non-turbulent interface in boundary layers

2014 ◽  
Vol 751 ◽  
pp. 298-328 ◽  
Author(s):  
Kapil Chauhan ◽  
Jimmy Philip ◽  
Ivan Marusic
Keyword(s):  
Boundary Layer ◽  
Outer Part ◽  
Tangential Velocity ◽  
Reynolds Numbers ◽  
Momentum Balance ◽  
Normal Velocity ◽  
Mean Velocity ◽  
Order Of Magnitude ◽  
Theoretical Reasoning ◽  
The Mean

AbstractScaling of the interface that demarcates a turbulent boundary layer from the non-turbulent free stream is sought using theoretical reasoning and experimental evidence in a zero-pressure-gradient boundary layer. The data-analysis, utilising particle image velocimetry (PIV) measurements at four different Reynolds numbers ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\delta u_{\tau }/\nu =1200\mbox{--}14\, 500$), indicates the presence of a viscosity dominated interface at all Reynolds numbers. It is found that the mean normal velocity across the interface and the tangential velocity jump scale with the skin-friction velocity$u_{\tau }$and are approximately$u_{\tau }/10$and$u_{\tau }$, respectively. The width of the superlayer is characterised by the local vorticity thickness$\delta _{\omega }$and scales with the viscous length scale$\nu /u_{\tau }$. An order of magnitude analysis of the tangential momentum balance within the superlayer suggests that the turbulent motions also scale with inner velocity and length scales$u_{\tau }$and$\nu /u_{\tau }$, respectively. The influence of the wall on the dynamics in the superlayer is considered via Townsend’s similarity hypothesis, which can be extended to account for the viscous influence at the turbulent/non-turbulent interface. Similar to a turbulent far-wake the turbulent motions in the superlayer are of the same order as the mean velocity deficit, which lends to a physical explanation for the emergence of the wake profile in the outer part of the boundary layer.

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 A comparative study of the velocity and vorticity structure in pipes and boundary layers at friction Reynolds numbers up to

2019 ◽  
Vol 869 ◽  
pp. 182-213 ◽  
Author(s):  
S. Zimmerman ◽  
J. Philip ◽  
J. Monty ◽  
A. Talamelli ◽  
I. Marusic ◽  
...  
Keyword(s):  
Boundary Layer ◽  
Turbulent Flows ◽  
Structural Difference ◽  
Numerical Data ◽  
Reynolds Numbers ◽  
Outer Edge ◽  
Boundary Layer Flows ◽  
Outer Flow ◽  
Wall Flows ◽  
Turbulent Wall

This study presents findings from a first-of-its-kind measurement campaign that includes simultaneous measurements of the full velocity and vorticity vectors in both pipe and boundary layer flows under matched spatial resolution and Reynolds number conditions. Comparison of canonical turbulent flows offers insight into the role(s) played by features that are unique to one or the other. Pipe and zero pressure gradient boundary layer flows are often compared with the goal of elucidating the roles of geometry and a free boundary condition on turbulent wall flows. Prior experimental efforts towards this end have focused primarily on the streamwise component of velocity, while direct numerical simulations are at relatively low Reynolds numbers. In contrast, this study presents experimental measurements of all three components of both velocity and vorticity for friction Reynolds numbers$Re_{\unicode[STIX]{x1D70F}}$ranging from 5000 to 10 000. Differences in the two transverse Reynolds normal stresses are shown to exist throughout the log layer and wake layer at Reynolds numbers that exceed those of existing numerical data sets. The turbulence enstrophy profiles are also shown to exhibit differences spanning from the outer edge of the log layer to the outer flow boundary. Skewness and kurtosis profiles of the velocity and vorticity components imply the existence of a ‘quiescent core’ in pipe flow, as described by Kwonet al. (J. Fluid Mech., vol. 751, 2014, pp. 228–254) for channel flow at lower$Re_{\unicode[STIX]{x1D70F}}$, and characterize the extent of its influence in the pipe. Observed differences between statistical profiles of velocity and vorticity are then discussed in the context of a structural difference between free-stream intermittency in the boundary layer and ‘quiescent core’ intermittency in the pipe that is detectable to wall distances as small as 5 % of the layer thickness.

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