scholarly journals A Framework for Parameterizing Eddy Potential Vorticity Fluxes

2012 ◽  
Vol 42 (4) ◽  
pp. 539-557 ◽  
Author(s):  
David P. Marshall ◽  
James R. Maddison ◽  
Pavel S. Berloff
Keyword(s):  
Stress Tensor ◽  
Potential Vorticity ◽  
Large Scale ◽  
Eddy Diffusivity ◽  
Linear Instability ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Conservation Of Energy ◽  
The Mean ◽  
Energy And Momentum

Abstract A framework for parameterizing eddy potential vorticity fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy flux. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and nondimensional parameters describing the mean shape and orientation of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are nondimensional and bounded in magnitude by unity. Moreover, these nondimensional geometric parameters have strong connections with classical stability theory. When applied to the Eady problem, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability. Moreover, in the limit in which Reynolds stresses are neglected, the framework reduces to a Gent and McWilliams type of eddy closure where the eddy diffusivity can be interpreted as the form proposed by Visbeck et al. Simulations of three-layer wind-driven gyres are used to diagnose the eddy shape and orientations in fully developed geostrophic turbulence. These fields are found to have large-scale structure that appears related to the structure of the mean flow. The eddy energy sets the magnitude of the eddy stress tensor and hence the eddy potential vorticity fluxes. Possible extensions of the framework to ensure potential vorticity is mixed on average are discussed.

scholarly journals A Geometric Interpretation of Eddy Reynolds Stresses in Barotropic Ocean Jets

2016 ◽  
Vol 46 (8) ◽  
pp. 2285-2307 ◽  
Author(s):  
Talia Tamarin ◽  
James R. Maddison ◽  
Eyal Heifetz ◽  
David P. Marshall
Keyword(s):  
Stress Tensor ◽  
Ray Tracing ◽  
Analytic Solutions ◽  
Geometric Interpretation ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Nonlinear Simulation ◽  
Fully Nonlinear ◽  
Geometric Decomposition ◽  
The Mean

AbstractBarotropic eddy fluxes are analyzed through a geometric decomposition of the eddy stress tensor. Specifically, the geometry of the eddy variance ellipse, a two-dimensional visualization of the stress tensor describing the mean eddy shape and tilt, is used to elucidate eddy propagation and eddy feedback on the mean flow. Linear shear and jet profiles are analyzed and theoretical results are compared against fully nonlinear simulations. For flows with zero planetary vorticity gradient, analytic solutions for the eddy ellipse tilt and anisotropy are obtained that provide a direct relationship between the eddy tilt and the phase difference of a normal-mode solution. This allows a straightforward interpretation of the eddy–mean flow interaction in terms of classical stability theory: the initially unstable jet gives rise to eddies that are tilted “against the shear” and extract energy from the mean flow; once the jet stabilizes, eddies become tilted “with the shear” and return their energy to the mean flow. For a nonzero planetary vorticity gradient, ray-tracing theory is used to predict ellipse geometry and its impact on eddy propagation within a jet. An analytic solution for the eddy tilt is found for a Rossby wave on a constant background shear. The ray-tracing results broadly agree with the eddy tilt diagnosed from a fully nonlinear simulation.

Stable and unstable shear modes of rotating parallel flows in shallow water

1987 ◽  
Vol 184 ◽  
pp. 477-504 ◽  
Author(s):  
Y.-Y. Hayashi ◽  
W. R. Young
Keyword(s):  
Energy Density ◽  
Shallow Water ◽  
Wave Energy ◽  
Potential Vorticity ◽  
The Other ◽  
Shear Mode ◽  
Mean Flow ◽  
Shear Modes ◽  
The Mean ◽  
Energy And Momentum

This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial β-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is ½ h(u2 + v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh′u′). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability.A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously.For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.

scholarly journals Simulations of rib-roughened rough-to-smooth turbulent channel flows

2018 ◽  
Vol 843 ◽  
pp. 419-449 ◽  
Author(s):  
Umair Ismail ◽  
Tamer A. Zaki ◽  
Paul A. Durbin
Keyword(s):  
Large Scale ◽  
Wall Layer ◽  
Shear Stresses ◽  
Thin Wall ◽  
Computational Domain ◽  
Reynolds Stresses ◽  
Smooth Wall ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

High-fidelity simulations of turbulent flow through a channel with a rough wall, followed by a smooth wall, demonstrate a high degree of non-equilibrium within the recovery region. In fact, the recovery of all the flow statistics studied is incomplete by the streamwise exit of the computational domain. Above a thin wall layer, turbulence intensities significantly higher than fully developed, smooth-wall levels persist in the developing region. Within the thin wall layer, the profile shapes for turbulence stresses recover very quickly and wall-normal locations of characteristic peaks are established. However, even in this thin layer, complete recovery of magnitudes of turbulence stresses is exceptionally slow. A similar initially swift but eventually incomplete and slow relaxation behaviour is also shown by the skin friction. Between the turbulence shear and streamwise stresses, the turbulence shear stress shows a comparatively quick rate of recovery above a thin wall layer. Over the developing smooth wall, the balance is not merely between fluxes due to pressure and shear stresses. Strong momentum fluxes, which are directly influenced by the upstream roughness size, contribute significantly to this balance. Approximate curve fits estimate the streamwise distance required by the outer peaks of Reynolds stresses to attain near-fully-developed levels at approximately $20\unicode[STIX]{x1D6FF}{-}25\unicode[STIX]{x1D6FF}$, with $\unicode[STIX]{x1D6FF}$ being the channel half height. An even longer distance, of more than $50\unicode[STIX]{x1D6FF}$, might be needed by the mean velocity to approach near-fully-developed magnitudes. Visualizations and correlations show that large-scale eddies that are created above the roughness persist downstream, and sporadically perturb the elongated streaks. These streaks of alternating high and low momentum appear almost instantly after the roughness is removed. The mean flow does not re-establish an equilibrium log layer within the computational domain, and the velocity deficit created by the roughness continues throughout the domain. On the step change in roughness, near the wall, profiles for turbulence kinetic energy dissipation rate, $\unicode[STIX]{x1D716}$, and energy spectra indicate a sharp reduction in energy at small scales. Despite this, reversion towards equilibrium smooth-wall levels is slow, and ultimately incomplete, due to a rather slow adjustment of the turbulence cascade. The non-dimensional roughness height, $k^{+}$ ranges from 42 to 254 and the friction velocity Reynolds number at the smooth wall, $Re_{\unicode[STIX]{x1D70F}S}$, ranges from 284 to 1160 in the various simulations.

scholarly journals Instability wave models for the near-field fluctuations of turbulent jets

2011 ◽  
Vol 689 ◽  
pp. 97-128 ◽  
Author(s):  
K. Gudmundsson ◽  
Tim Colonius
Keyword(s):  
Flow Field ◽  
Large Scale ◽  
Microphone Array ◽  
Near Field ◽  
Turbulent Jets ◽  
Instability Wave ◽  
Mean Flow ◽  
Velocity Fluctuations ◽  
The Mean ◽  
Scale Structures

AbstractPrevious work has shown that aspects of the evolution of large-scale structures, particularly in forced and transitional mixing layers and jets, can be described by linear and nonlinear stability theories. However, questions persist as to the choice of the basic (steady) flow field to perturb, and the extent to which disturbances in natural (unforced), initially turbulent jets may be modelled with the theory. For unforced jets, identification is made difficult by the lack of a phase reference that would permit a portion of the signal associated with the instability wave to be isolated from other, uncorrelated fluctuations. In this paper, we investigate the extent to which pressure and velocity fluctuations in subsonic, turbulent round jets can be described aslinearperturbations to the mean flow field. The disturbances are expanded about the experimentally measured jet mean flow field, and evolved using linear parabolized stability equations (PSE) that account, in an approximate way, for the weakly non-parallel jet mean flow field. We utilize data from an extensive microphone array that measures pressure fluctuations just outside the jet shear layer to show that, up to an unknown initial disturbance spectrum, the phase, wavelength, and amplitude envelope of convecting wavepackets agree well with PSE solutions at frequencies and azimuthal wavenumbers that can be accurately measured with the array. We next apply the proper orthogonal decomposition to near-field velocity fluctuations measured with particle image velocimetry, and show that the structure of the most energetic modes is also similar to eigenfunctions from the linear theory. Importantly, the amplitudes of the modes inferred from the velocity fluctuations are in reasonable agreement with those identified from the microphone array. The results therefore suggest that, to predict, with reasonable accuracy, the evolution of the largest-scale structures that comprise the most energetic portion of the turbulent spectrum of natural jets, nonlinear effects need only be indirectly accounted for by considering perturbations to the mean turbulent flow field, while neglecting any non-zero frequency disturbance interactions.

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

2016 ◽  
Vol 33 (4) ◽  
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.

scholarly journals Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography

2010 ◽  
Vol 40 (2) ◽  
pp. 257-278 ◽  
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.

Large-scale Structure and Turbulence Transport in the Young Solar Wind – Comparison of Parker Solar Probe Observations with a Global 3D Reynolds-averaged MHD Model

2021 ◽  
Author(s):  
Rohit Chhiber ◽  
Arcadi Usmanov ◽  
William Matthaeus ◽  
Melvyn Goldstein ◽  
Riddhi Bandyopadhyay
Keyword(s):  
Solar Wind ◽  
Large Scale ◽  
Transport Equations ◽  
Turbulence Energy ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Coulomb Collisions ◽  
Flow Equations ◽  
Simulation Results ◽  
Turbulence Transport

<div>Simulation results from a global <span>magnetohydrodynamic</span> model of the solar corona and the solar wind are compared with Parker Solar <span>Probe's</span> (<span>PSP</span>) observations during its first several orbits. The fully three-dimensional model (<span>Usmanov</span> <span>et</span> <span>al</span>., 2018, <span>ApJ</span>, 865, 25) is based on Reynolds-averaged mean-flow equations coupled with turbulence transport equations. The model accounts for effects of electron heat conduction, Coulomb collisions, Reynolds stresses, and heating of protons and electrons via nonlinear turbulent cascade. Turbulence transport equations for turbulence energy, cross <span>helicity</span>, and correlation length are solved concurrently with the mean-flow equations. We specify boundary conditions at the coronal base using solar synoptic <span>magnetograms</span> and calculate plasma, magnetic field, and turbulence parameters along the <span>PSP</span> trajectory. We also accumulate data from all orbits considered, to obtain the trends observed as a function of heliocentric distance. Comparison of simulation results with <span>PSP</span> data show general agreement. Finally, we generate synthetic fluctuations constrained by the local rms turbulence amplitude given by the model, and compare properties of this synthetic turbulence with PSP observations.</div>

The structure of a highly decelerated axisymmetric turbulent boundary layer

2021 ◽  
Vol 929 ◽  
Author(s):  
N. Agastya Balantrapu ◽  
Christopher Hickling ◽  
W. Nathan Alexander ◽  
William Devenport
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Shear Layer ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Large Scale ◽  
Convection Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

Experiments were performed over a body of revolution at a length-based Reynolds number of 1.9 million. While the lateral curvature parameters are moderate ( $\delta /r_s < 2, r_s^+>500$ , where $\delta$ is the boundary layer thickness and r s is the radius of curvature), the pressure gradient is increasingly adverse ( $\beta _{C} \in [5 \text {--} 18]$ where $\beta_{C}$ is Clauser’s pressure gradient parameter), representative of vehicle-relevant conditions. The mean flow in the outer regions of this fully attached boundary layer displays some properties of a free-shear layer, with the mean-velocity and turbulence intensity profiles attaining self-similarity with the ‘embedded shear layer’ scaling (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642). Spectral analysis of the streamwise turbulence revealed that, as the mean flow decelerates, the large-scale motions energize across the boundary layer, growing proportionally with the boundary layer thickness. When scaled with the shear layer parameters, the distribution of the energy in the low-frequency region is approximately self-similar, emphasizing the role of the embedded shear layer in the large-scale motions. The correlation structure of the boundary layer is discussed at length to supply information towards the development of turbulence and aeroacoustic models. One major finding is that the estimation of integral turbulence length scales from single-point measurements, via Taylor's hypothesis, requires significant corrections to the convection velocity in the inner 50 % of the boundary layer. The apparent convection velocity (estimated from the ratio of integral length scale to the time scale), is approximately 40 % greater than the local mean velocity, suggesting the turbulence is convected much faster than previously thought. Closer to the wall even higher corrections are required.

Analysis of Turbulence in the Tip Region of a Waterjet Pump Rotor

2010 ◽  
Author(s):  
Huixuan Wu ◽  
Rinaldo L. Miorini ◽  
Joseph Katz
Keyword(s):  
Energy Flux ◽  
Large Scale ◽  
Multiple Scales ◽  
Tip Clearance ◽  
Mean Flow ◽  
Production Characteristic ◽  
Pump Rotor ◽  
The Mean ◽  
Turbulence Production ◽  
Waterjet Pump

A series of high resolution planar particle image velocimetry measurements performed in a waterjet pump rotor reveal the inner structure of the tip leakage vortex (TLV) which dominates the entire flow field in the tip region. Turbulence generated by interactions among the TLV, the shear layer that develops as the backward leakage flow emerges from the tip clearance as a “wall jet”, the passage flow, and the endwall is highly inhomogeneous and anisotropic. We examine this turbulence in both RANS and LES modelling contexts. Spatially non-uniform distributions of Reynolds stress components are explained in terms of the local mean strain field and associated turbulence production. Characteristic length scales are also inferred from spectral analysis. Spatial filtering of instantaneous data enables the calculation of subgrid scale (SGS) stresses, along with the SGS energy flux (dissipation). The data show that the SGS energy flux differs from the turbulence production rate both in trends and magnitude. The latter is dominated by energy flux from the mean flow to the large scale turbulence, which is resolved in LES, whereas the former is dominated by energy flux from the mean flow to the SGS turbulence. The SGS dissipation rate is also used for calculating the static and dynamic Smagorinsky coefficients, the latter involving filtering at multiple scales; both vary substantially in the tip region, and neither is equal to values obtained in isotropic turbulence.

scholarly journals Self-consistent triple decomposition of the turbulent flow over a backward-facing step under finite amplitude harmonic forcing

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190018
Author(s):  
E. Yim ◽  
P. Meliga ◽  
F. Gallaire
Keyword(s):  
Turbulent Flow ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Backward Facing Step ◽  
Eddy Viscosity Model ◽  
Coherent Interaction ◽  
Self Consistent ◽  
The Mean

We investigate the saturation of harmonically forced disturbances in the turbulent flow over a backward-facing step subjected to a finite amplitude forcing. The analysis relies on a triple decomposition of the unsteady flow into mean, coherent and incoherent components. The coherent–incoherent interaction is lumped into a Reynolds averaged Navier–Stokes (RANS) eddy viscosity model, and the mean–coherent interaction is analysed via a semi-linear resolvent analysis building on the laminar approach by Mantič-Lugo & Gallaire (2016 J. Fluid Mech. 793 , 777–797. ( doi:10.1017/jfm.2016.109 )). This provides a self-consistent modelling of the interaction between all three components, in the sense that the coherent perturbation structures selected by the resolvent analysis are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations, while also accounting for the effect of the incoherent scale. The model does not require any input from numerical or experimental data, and accurately predicts the saturation of the forced coherent disturbances, as established from comparison to time-averages of unsteady RANS simulation data.

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