Barotropic theory for the velocity profile of Jupiter’s turbulent jets: an example for an exact turbulent closure

2018 ◽  
Vol 860 ◽  
pp. 577-607
Author(s):  
E. Woillez ◽  
F. Bouchet
Keyword(s):  
Velocity Profile ◽  
Turbulent Jets ◽  
Equilibrium Structure ◽  
Injection Rate ◽  
Velocity Shear ◽  
Small Scale ◽  
Reynolds Stresses ◽  
Zonal Jets ◽  
Time Scale Separation ◽  
Turbulent Closure

We model the dynamics of Jupiter’s jets by the stochastic barotropic $\unicode[STIX]{x1D6FD}$-plane model. In this simple framework, by analytic computation of the averaged effect of eddies, we obtain three new explicit results about the equilibrium structure of jets. First we obtain a very simple explicit relation between the Reynolds stresses, the energy injection rate and the averaged velocity shear. This predicts the averaged velocity profile far from the jet edges (extrema of zonal velocity). Our approach takes advantage of a time-scale separation between the inertial dynamics on one hand, and the spin-up (or spin-down) time on the other. Second, a specific asymptotic expansion close to the eastward jet extremum explains the formation of a cusp at the scale of energy injection, characterized by a curvature that is independent of the forcing spectrum. Finally, we derive equations that describe the evolution of the westward tip of the jets. The analysis of these equations is consistent with the previously discussed picture of barotropic adjustment, explaining the relation between the westward jet curvature and the $\unicode[STIX]{x1D6FD}$-effect. Our results give a consistent overall theory of the stationary velocity profile of inertial barotropic zonal jets, in the limit of small-scale forcing.

Micro Fluidic Jets for Thermal Management of Electronics

Volume 4 ◽  
2004 ◽  
Author(s):  
Jivtesh Garg ◽  
Mehmet Arik ◽  
Stanton Weaver ◽  
Seyed Saddoughi
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Heated Surface ◽  
Turbulent Jets ◽  
Harmonic Signal ◽  
Synthetic Jet ◽  
Synthetic Jets ◽  
Small Scale ◽  
Air Cooling ◽  
Infrared Thermal Imaging

Micro fluidics devices are conventionally used for boundary layer control in many aerospace applications. Synthetic Jets are intense small scale turbulent jets formed from entrainment and expulsion of the fluid in which they are embedded. The idea of using synthetic jets in confined electronic cooling applications started in late 1990s. These micro fluidic devices offer very efficient, high magnitude direct air-cooling on the heated surface. A proprietary synthetic jet designed in General Electric Company was able to provide a maximum air velocity of 90 m/s from a 1.2 mm hydraulic diameter rectangular orifice. An experimental study for determining the thermal performance of a meso scale synthetic jet was carried out. The synthetic jets are driven by a time harmonic signal. During the experiments, the operating frequency for jets was set between 3 and 4.5 kHz. The resonance frequency for a particular jet was determined through the effect on the exit velocity magnitude. An infrared thermal imaging technique was used to acquire fine scale temperature measurements. A square heater with a surface area of 156 mm2 was used to mimic the hot component and extensive temperature maps were obtained. The parameters varied during the experiments were jet location, driving jet voltage, driving jet frequency and heater power. The output parameters were point wise temperatures (pixel size = 30 μm), and heat transfer enhancement over natural convection. A maximum of approximately 8 times enhancement over natural convection heat transfer was measured. The maximum coefficient of cooling performance obtained was approximately 6.6 due to the low power consumption of the synthetic jets.

Effects of varying injection rate on dynamic slip nucleation along a frictional weakening fault

2021 ◽  
Author(s):  
Federico Ciardo ◽  
Antonio Pio Rinaldi ◽  
Stefan Wiemer
Keyword(s):  
Boundary Element ◽  
Shear Crack ◽  
Fractured Reservoirs ◽  
Injection Rate ◽  
Numerical Results ◽  
Small Scale ◽  
Rate Variation ◽  
Injection Strategy ◽  
Seismicity Rates ◽  
Dynamic Slip

<div> <p><span>Anthropogenic injection of fluid into tight fractured reservoirs is known to alter the stress state of the Earth`s crust,  inducing micro-seismicity and eventually significant earthquakes. The injection scenario, in terms of injection pressure or injection rate, is one of the key controlling parameters for injection-induced seismicity. Although a number of studies have been carried out on understanding the effects of injection strategy on seismicity rates, less is known about its effect on the nucleation of dynamic slip on a pressurized fault, especially for non-stationary injection protocols.</span></p> </div><div> <p><span>In this contribution we study the effects of injection rate variation on the transition between aseismic and seismic slip along a frictional weakenig fault. Notably, we parametrize the injection strategy by assuming an initial linear increase of injection rate in time, up to a value after which it remains constant. We perform a scalying analysis and identify the governing parameters that control the fault response. We solve numerically the coupled hydro-mechanical problem using a fast boundary element solver for localized inelastic deformations [1]. Upon benchmarking the numerical results with the semi-analytical ones of Garagash and Germanovich [2] for the specific case of constant injection rate, we investigate the effect of injection rate variation on critically stressed and marginally pressurized faults. We derive analytical expressions for nucleation time and we confirm them via numerical results. Furthermore, we present a small scale yielding solution for marginallly pressurized faults and investigate the influence of injection scenario on shear crack run-out distances (when occuring).</span></p> </div><div> <p><span> </span></p> </div><div> <p><strong><span>References   </span></strong></p> </div><div> <p><span>[1] Ciardo, F., Lecampion, B., Fayard, F., and Chaillat, S. (2020), A fast boundary element based solver for localized inelastic deformations, </span><em>Int J Numer Methods Eng</em>. 2020; 1–23.</p> </div><div> <p><span>[2] Garagash, D., and L. N. Germanovich (2012), Nucleation and arrest of dynamic slip on a pressurized fault, <em>J. Geophys. Res</em>., 117, </span>B10310<span>.</span></p> </div>

scholarly journals The Dynamics of Baroclinic Zonal Jets*

2015 ◽  
Vol 72 (3) ◽  
pp. 1137-1151 ◽  
Author(s):  
Paul D. Williams ◽  
Christopher W. Kelsall
Keyword(s):  
Root Mean Square ◽  
Power Law ◽  
Zonal Flow ◽  
Small Scale ◽  
Mean Square ◽  
Initial State ◽  
Zonal Jets ◽  
Rotation Periods ◽  
Inverse Energy Cascade ◽  
Geostrophic Turbulence

Abstract Multiple alternating zonal jets are a ubiquitous feature of planetary atmospheres and oceans. However, most studies to date have focused on the special case of barotropic jets. Here, the dynamics of freely evolving baroclinic jets are investigated using a two-layer quasigeostrophic annulus model with sloping topography. In a suite of 15 numerical simulations, the baroclinic Rossby radius and baroclinic Rhines scale are sampled by varying the stratification and root-mean-square eddy velocity, respectively. Small-scale eddies in the initial state evolve through geostrophic turbulence and accelerate zonally as they grow in horizontal scale, first isotropically and then anisotropically. This process leads ultimately to the formation of jets, which take about 2500 rotation periods to equilibrate. The kinetic energy spectrum of the equilibrated baroclinic zonal flow steepens from a −3 power law at small scales to a −5 power law near the jet scale. The conditions most favorable for producing multiple alternating baroclinic jets are large baroclinic Rossby radius (i.e., strong stratification) and small baroclinic Rhines scale (i.e., weak root-mean-square eddy velocity). The baroclinic jet width is diagnosed objectively and found to be 2.2–2.8 times larger than the baroclinic Rhines scale, with a best estimate of 2.5 times larger. This finding suggests that Rossby wave motions must be moving at speeds of approximately 6 times the turbulent eddy velocity in order to be capable of arresting the isotropic inverse energy cascade.

scholarly journals Turbulence, entrainment and low-order description of a transitional variable-density jet

2017 ◽  
Vol 836 ◽  
pp. 1009-1049 ◽  
Author(s):  
B. Viggiano ◽  
T. Dib ◽  
N. Ali ◽  
L. G. Mastin ◽  
R. B. Cal ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Turbulent Jets ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Quadrant Analysis ◽  
Shear Stresses ◽  
Reynolds Stresses ◽  
Buoyant Jet ◽  
Entrainment Ratio ◽  
Turbulent Reynolds Number

Geophysical flows occur over a large range of scales, with Reynolds numbers and Richardson numbers varying over several orders of magnitude. For this study, jets of different densities were ejected vertically into a large ambient region, considering conditions relevant to some geophysical phenomena. Using particle image velocimetry, the velocity fields were measured for three different gases exhausting into air – specifically helium, air and argon. Measurements focused on both the jet core and the entrained ambient. Experiments considered relatively low Reynolds numbers from approximately 1500 to 10 000 with Richardson numbers near 0.001 in magnitude. These included a variety of flow responses, notably a nearly laminar jet, turbulent jets and a transitioning jet in between. Several features were studied, including the jet development, the local entrainment ratio, the turbulent Reynolds stresses and the eddy strength. Compared to a fully turbulent jet, the transitioning jet showed up to 50 % higher local entrainment and more significant turbulent fluctuations. For this condition, the eddies were non-axisymmetric and larger than the exit radius. For turbulent jets, the eddies were initially smaller and axisymmetric while growing with the shear layer. At lower turbulent Reynolds number, the turbulent stresses were more than 50 % higher than at higher turbulent Reynolds number. In either case, the low-density jet developed faster than a comparable non-buoyant jet. Quadrant analysis and proper orthogonal decomposition were also utilized for insight into the entrainment of the jet, as well as to assess the energy distribution with respect to the number of eigenmodes. Reynolds shear stresses were dominant in Q1 and Q3 and exhibited negligible contributions from the remaining two quadrants. Both analysis techniques showed that the development of stresses downstream was dependent on the Reynolds number while the spanwise location of the stresses depended on the Richardson number.

The evolution of a uniformly sheared thermally stratified turbulent flow

1997 ◽  
Vol 334 ◽  
pp. 61-86 ◽  
Author(s):  
PAUL PICCIRILLO ◽  
CHARLES W. VAN ATTA
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Richardson Number ◽  
Initial Conditions ◽  
Velocity Shear ◽  
Spectral Energy ◽  
Small Scale ◽  
Mean Flow ◽  
Mean Velocity ◽  
Small Scales

Experiments were carried out in a new type of stratified flow facility to study the evolution of turbulence in a mean flow possessing both uniform stable stratification and uniform mean shear.The new facility is a thermally stratified wind tunnel consisting of ten independent supply layers, each with its own blower and heaters, and is capable of producing arbitrary temperature and velocity profiles in the test section. In the experiments, four different sized turbulence-generating grids were used to study the effect of different initial conditions. All three components of the velocity were measured, along with the temperature. Root-mean-square quantities and correlations were measured, along with their corresponding power and cross-spectra.As the gradient Richardson number Ri = N2/(dU/dz)2 was increased, the downstream spatial evolution of the turbulent kinetic energy changed from increasing, to stationary, to decreasing. The stationary value of the Richardson number, Ricr, was found to be an increasing function of the dimensionless shear parameter Sq2/∈ (where S = dU/dz is the mean velocity shear, q2 is the turbulent kinetic energy, and ∈ is the viscous dissipation).The turbulence was found to be highly anisotropic, both at the small scales and at the large scales, and anisotropy was found to increase with increasing Ri. The evolution of the velocity power spectra for Ri [les ] Ricr, in which the energy of the large scales increases while the energy in the small scales decreases, suggests that the small-scale anisotropy is caused, or at least amplified, by buoyancy forces which reduce the amount of spectral energy transfer from large to small scales. For the largest values of Ri, countergradient buoyancy flux occurred for the small scales of the turbulence, an effect noted earlier in the numerical results of Holt et al. (1992), Gerz et al. (1989), and Gerz & Schumann (1991).

scholarly journals PIV Measurements in the Atmospheric Boundary Layer within and above a Mature Corn Canopy. Part II: Quadrant-Hole Analysis

2007 ◽  
Vol 64 (8) ◽  
pp. 2825-2838 ◽  
Author(s):  
W. Zhu ◽  
R. van Hout ◽  
J. Katz
Keyword(s):  
Dissipation Rate ◽  
Large Scale ◽  
Reynolds Shear Stress ◽  
Small Scale ◽  
Shear Stresses ◽  
Reynolds Stresses ◽  
Piv Measurements ◽  
Dissipation Rates ◽  
Scale Shear ◽  
Corn Canopy

Quadrant-hole (Q-H) analysis is applied to PIV data acquired just within and above a mature corn canopy. The Reynolds shear stresses, transverse components of vorticity, as well as turbulence production and cascading part of dissipation rates are conditionally sampled in each quadrant, based on stress and vorticity magnitudes. The stresses are representative of large-scale events, while the vorticity is dominated by small-scale shear. Dissipation rates (cascading energy fluxes) are evaluated by fitting −5/3 slope lines to the conditionally sampled and averaged spatial energy spectra, while the Reynolds stresses, vorticity, and production rates are calculated directly from the spatial distributions of two velocity components. The results demonstrate that sweep (quadrant 4) and ejection (quadrant 2) events are the dominant contributors to the Reynolds shear stress, consistent with previous observations. The analysis also shows a strong correlation between magnitudes of dissipation rate and vorticity. The dissipation rates and vorticity magnitudes are higher in quadrants 1 and 4, that is, when the horizontal component of the fluctuating velocity is positive, peaking in quadrant 1. Both are weakly correlated with the Reynolds stresses except for rare quadrant 1 events. However, the more frequently occurring quadrant 4 events are the largest contributors to the dissipation rate. The production rate inherently increases with increasing stress magnitude, but lacks correlation with vorticity. Quadrants 2 and 4 contribute the most to production. However, the contribution of quadrant 1 events to negative production should not be ignored above canopy. The results show a strong disconnection between small-scale- and large-scale-dominated phenomena.

scholarly journals Self-similar properties of decelerating turbulent jets

2017 ◽  
Vol 833 ◽  
Author(s):  
Dong-hyuk Shin ◽  
A. J. Aspden ◽  
Edward S. Richardson
Keyword(s):  
Velocity Profile ◽  
Axial Velocity ◽  
Turbulent Jets ◽  
Entrainment Rate ◽  
Velocity Statistics ◽  
Radial Profiles ◽  
Axial Velocity Profile ◽  
Self Similar ◽  
The Mean ◽  
Analytical Relations

The flow in a decelerating turbulent round jet is investigated using direct numerical simulation. The simulations are initialised with a flow field from a statistically stationary turbulent jet. Upon stopping the inflow, a deceleration wave passes through the jet, behind which the velocity field evolves towards a new statistically unsteady self-similar state. Assumption of unsteady self-similar behaviour leads to analytical relations concerning the evolution of the centreline mean axial velocity and the shapes of the radial profiles of the velocity statistics. Consistency between these predictions and the simulation data supports the use of the assumption of self-similarity. The mean radial velocity is predicted to reverse in direction near to the jet centreline as the deceleration wave passes, contributing to an approximately threefold increase in the normalised mass entrainment rate. The shape of the mean axial velocity profile undergoes a relatively small change across the deceleration transient, and this observation provides direct evidence in support of previous models that have assumed that the mean axial velocity profile, and in some cases also the jet spreading angle, remain approximately constant within unsteady jets.

The trapping and vertical focusing of internal waves in a pycnocline due to the horizontal inhomogeneities of density and currents

1985 ◽  
Vol 158 ◽  
pp. 199-218 ◽  
Author(s):  
S. I. Badulin ◽  
V. I. Shrira ◽  
L. Sh. Tsimring
Keyword(s):  
Wave Packet ◽  
Viscous Dissipation ◽  
Internal Waves ◽  
Equations Of Motion ◽  
Local Maximum ◽  
Velocity Shear ◽  
Vertical Shear ◽  
Small Scale ◽  
The Mean ◽  
Mean Current

This paper studies the propagation of a wave packet in regions where the central packet frequency ω is close to the local maximum of the effective Väisälä frequency Nf(z) = N(z)/[1 − k·U(z)/ω], where k is the central wavevector of the packet and U is the mean current with a vertical velocity shear. The wave approaches the layer ω = Nfm asymptotically, i.e. trapping of the wave takes place. The trapping of guided internal waves is investigated within the framework of the linearized equations of motion of an incompressible stratified fluid in the WKB approximation, with viscosity, spectral bandwidth of the packet, vertical shear of the mean current and non-stationarity of the environment taken into account. As the packet approaches the layer of trapping, the growth of the wavenumber k is restricted only by possible wave-breaking and viscous dissipation. The growth of k is accompanied by the transformation of the vertical structure of internal-wave modes. The wave motion focuses at a certain depth determined by the maximum effective Väisälä frequency Nfm. The trapping of the wave packet results in power growth of the wave amplitude and steepness. At larger times the viscous dissipation becomes a dominating factor of evolution as a result of strong slowing down of the packet motion.The role of trapping in the energy exchange of internal waves, currents and small-scale turbulence is discussed.

Coherent Structures in the Similarity Region of Two-Dimensional Turbulent Jets

1984 ◽  
Vol 106 (2) ◽  
pp. 187-192 ◽  
Author(s):  
J. W. Oler ◽  
V. W. Goldschmidt
Keyword(s):  
Velocity Profile ◽  
Coherent Structures ◽  
Turbulent Jets ◽  
Jet Flows ◽  
Two Dimensional ◽  
Ordered Structure ◽  
Mean Velocity ◽  
The Mean ◽  
Plane Jet ◽  
Average Position

The strongest indication of an ordered structure in the similarity region of plane jet flows is the well documented (but controversial) apparent “flapping” behavior. Previously, the negative correlation between probes placed on opposite sides of the jet centerline has been attributed to the periodic displacement of the mean velocity profile centerline about its average position, i.e., a flapping motion. The present investigation is directed at evaluating the premise of an essentially two-dimensional von Karman vortex street as being responsible for the apparent “flapping” behavior.

scholarly journals Numerical Simulation of Turbulent Jets With Rectangular Cross-Section

1998 ◽  
Vol 120 (2) ◽  
pp. 285-290 ◽  
Author(s):  
R. V. Wilson ◽  
A. O. Demuren
Keyword(s):  
Reynolds Number ◽  
Cross Section ◽  
Stokes Equations ◽  
Rectangular Cross Section ◽  
Turbulent Jets ◽  
Reynolds Numbers ◽  
Scale Model ◽  
Reynolds Stresses ◽  
Spatial Discretization ◽  
Low Reynolds

Three-dimensional turbulent jets with rectangular cross-section are simulated with a finite-difference numerical method. The full Navier-Stokes equations are solved at a low Reynolds number, whereas at a higher Reynolds number filtered forms of the equations are solved along with a sub-grid scale model to approximate effects of the unresolved scales. A 2-N storage, third-order Runge-Kutta scheme is used for temporal discretization and a fourth-order compact scheme is used for spatial discretization. Divergence-free velocity field is obtained by solving a Poisson equation for pressure with the same spatial discretization scheme for consistent accuracy. Computations are performed for different inlet conditions which represent different types of jet forcing within the shear layer. The phenomenon of axis-switching is observed in some cases. At low Reynolds numbers, it is based on self-induction of the vorticity field, whereas at higher Reynolds numbers, the turbulent structure becomes the dominant mechanism in natural jets. Budgets of the mean streamwise velocity show that convection is balanced by gradients of the Reynolds stresses and the pressure.

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