Developments in Measuring Reynolds Stresses

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+ to 0.1–0.2 R+ in a pipe of radius R. Here, we propose a fundamentally new approach based on fractional calculus to model the entire mean velocity profile from the wall to the centerline of the pipe. Specifically, we represent the Reynolds stresses with a non-local fractional derivative of variable-order that decays with the distance from the wall. Surprisingly, we find that this variable fractional order has a universal form for all Reynolds numbers and for three different flow types, i.e., channel flow, Couette flow, and pipe flow. We first use existing databases from direct numerical simulations (DNSs) to lean the variable-order function and subsequently we test it against other DNS data and experimental measurements, including the Princeton superpipe experiments. Taken together, our findings reveal the continuous change in rate of turbulent diffusion from the wall as well as the strong nonlocality of turbulent interactions that intensify away from the wall. Moreover, we propose alternative formulations, including a divergence variable fractional (two-sided) model for turbulent flows. The total shear stress is represented by a two-sided symmetric variable fractional derivative. The numerical results show that this formulation can lead to smooth fractional-order profiles in the whole domain. This new model improves the one-sided model, which is considered in the half domain (wall to centerline) only. We use a finite difference method for solving the inverse problem, but we also introduce the fractional physics-informed neural network (fPINN) for solving the inverse and forward problems much more efficiently. In addition to the aforementioned fully-developed flows, we model turbulent boundary layers and discuss how the streamwise variation affects the universal curve.

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The Use of Subgrid Transport Equations in a Three-Dimensional Model of Atmospheric Turbulence

Journal of Fluids Engineering ◽

10.1115/1.3447047 ◽

1973 ◽

Vol 95 (3) ◽

pp. 429-438 ◽

Cited By ~ 190

Author(s):

J. W. Deardorff

Keyword(s):

Boundary Layer ◽

Three Dimensional ◽

Transport Equations ◽

Subgrid Scale ◽

Reynolds Stresses ◽

Dimensional Model ◽

Three Dimensional Model ◽

Truncation Errors ◽

Subgrid Scales ◽

Grid Points

A three-dimensional numerical model of turbulence in an atmospheric boundary layer has been revised to utilize subgrid transport equations for the subgrid Reynolds stresses and fluxes rather than subgrid eddy coefficients. It was applied to a daytime boundary layer over heated ground in a region of horizontal area 8km square and 2km deep, utilizing 40×40×40 grid points. The constraints involved in selecting four important subgrid closure constants are discussed in some detail, along with maintenance of realizability on the subgrid scale. The results indicate that the subgrid transport equations produce subgrid Reynolds stresses and fluxes which realistically simulate the transfer of larger scale variance to subgrid scales, provided truncation errors due to advective terms are not too large. They also show the superiority of this method over the use of (nonstability dependent) nonlinear eddy coefficients in maintaining the sharpness of the inversion base which lies above the mixed layer.

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Study of the Standard k-ε Model for Tip Leakage Flow in an Axial Compressor Rotor

International Journal of Turbo and Jet Engines ◽

10.1515/tjj-2015-0039 ◽

2016 ◽

Vol 33 (4) ◽

Cited By ~ 7

Author(s):

Yanfei Gao ◽

Yangwei Liu ◽

Luyang Zhong ◽

Jiexuan Hou ◽

Lipeng Lu

Keyword(s):

Flow Field ◽

Large Scale ◽

Turbulent Viscosity ◽

Axial Compressor ◽

Leakage Flow ◽

Reynolds Stresses ◽

Mean Flow ◽

Tip Leakage Flow ◽

Tip Leakage ◽

Compressor Rotor

AbstractThe standard k-ε model (SKE) and the Reynolds stress model (RSM) are employed to predict the tip leakage flow (TLF) in a low-speed large-scale axial compressor rotor. Then, a new research method is adopted to “freeze” the turbulent kinetic energy and dissipation rate of the flow field derived from the RSM, and obtain the turbulent viscosity using the Boussinesq hypothesis. The Reynolds stresses and mean flow field computed on the basis of the frozen viscosity are compared with the results of the SKE and the RSM. The flow field in the tip region based on the frozen viscosity is more similar to the results of the RSM than those of the SKE, although certain differences can be observed. This finding indicates that the non-equilibrium turbulence transport nature plays an important role in predicting the TLF, as well as the turbulence anisotropy.

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Tilting at wave beams: a new perspective on the St. Andrew’s Cross

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.615 ◽

2017 ◽

Vol 830 ◽

pp. 660-680 ◽

Cited By ~ 1

Author(s):

T. Kataoka ◽

S. J. Ghaemsaidi ◽

N. Holzenberger ◽

T. Peacock ◽

T. R. Akylas

Keyword(s):

Wave Field ◽

Finite Length ◽

Line Source ◽

Cutoff Frequency ◽

Three Dimensional ◽

Resonance Phenomenon ◽

Buoyancy Frequency ◽

Reynolds Stresses ◽

Mean Flow ◽

Viscous Effects

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

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Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography

Journal of Physical Oceanography ◽

10.1175/2009jpo4218.1 ◽

2010 ◽

Vol 40 (2) ◽

pp. 257-278 ◽

Cited By ~ 58

Author(s):

Andrew F. Thompson

Keyword(s):

Southern Ocean ◽

Potential Vorticity ◽

Baroclinic Instability ◽

Length Scale ◽

Reynolds Stresses ◽

Mean Flow ◽

Jet Formation ◽

Meridional Transport ◽

Jet Behavior ◽

Jet Variability

Abstract Satellite altimetry and high-resolution ocean models indicate that the Southern Ocean comprises an intricate web of narrow, meandering jets that undergo spontaneous formation, merger, and splitting events, as well as rapid latitude shifts over periods of weeks to months. The role of topography in controlling jet variability is explored using over 100 simulations from a doubly periodic, forced-dissipative, two-layer quasigeostrophic model. The system is forced by a baroclinically unstable, vertically sheared mean flow in a domain that is large enough to accommodate multiple jets. The dependence of (i) meridional jet spacing, (ii) jet variability, and (iii) domain-averaged meridional transport on changes in the length scale and steepness of simple sinusoidal topographical features is analyzed. The Rhines scale, ℓβ = 2πVe/β, where Ve is an eddy velocity scale and β is the barotropic potential vorticity gradient, measures the meridional extent of eddy mixing by a single jet. The ratio ℓβ /ℓT, where ℓT is the topographic length scale, governs jet behavior. Multiple, steady jets with fixed meridional spacing are observed when ℓβ ≫ ℓT or when ℓβ ≈ ℓT. When ℓβ < ℓT, a pattern of perpetual jet formation and jet merger dominates the time evolution of the system. Zonal ridges systematically reduce the domain-averaged meridional transport, while two-dimensional, sinusoidal bumps can increase transport by an order of magnitude or more. For certain parameters, bumpy topography gives rise to periodic oscillations in the jet structure between purely zonal and topographically steered states. In these cases, transport is dominated by bursts of mixing associated with the transition between the two regimes. Topography modifies local potential vorticity (PV) gradients and mean flows; this can generate asymmetric Reynolds stresses about the jet core and can feed back on the conversion of potential energy to kinetic energy through baroclinic instability. Both processes contribute to unsteady jet behavior. It is likely that these processes play a role in the dynamic nature of Southern Ocean jets.

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Publisher’s Note: “Tensorial time scale in turbulent gradient transport of Reynolds stresses” [Phys. Fluids 17, 071701 (2005)]

Physics of Fluids ◽

10.1063/1.2074647 ◽

2005 ◽

Vol 17 (9) ◽

pp. 099901

Author(s):

Choong Won Cho ◽

Myung Kyoon Chung ◽

Kyoungyoun Kim ◽

Hyung Jin Sung

Keyword(s):

Time Scale ◽

Reynolds Stresses

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A Study of Reynolds Stresses, Triple Products and Turbulence States in a Radially Stirred Tank with 3-D Laser Anemometry

Chemical Engineering Research and Design ◽

10.1205/cerd.82.9.1214.44151 ◽

2004 ◽

Vol 82 (9) ◽

pp. 1214-1228 ◽

Cited By ~ 16

Author(s):

C. Galletti ◽

E. Brunazzi ◽

S. Pintus ◽

A. Paglianti ◽

M. Yianneskis

Keyword(s):

Stirred Tank ◽

Reynolds Stresses ◽

Laser Anemometry

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