Bifurcation structure of two-dimensional viscous zonal flows on a rotating sphere

2015 ◽  
Vol 774 ◽  
pp. 224-244 ◽  
Author(s):  
Eiichi Sasaki ◽  
Shin-ichi Takehiro ◽  
Michio Yamada
Keyword(s):  
Reynolds Number ◽  
Dimensional Space ◽  
Zonal Flow ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Bifurcation Structure ◽  
Zonal Flows ◽  
Rotating Sphere ◽  
High Reynolds Numbers ◽  
High Reynolds

We study the bifurcation structure of zonal flows on a rotating sphere. The setting of our problem is similar to the Kolmogorov problem on a flat torus, where the vorticity forcing is given by a single eigenfunction of the Laplacian. First we prove the global stability of two-jet zonal flow for arbitrary Reynolds number and the rotation rate of the sphere. Then we study the bifurcation structure of steady solutions arising from three-jet zonal flow. In the non-rotating case, we find that two steady travelling-wave solutions bifurcate from a three-jet zonal flow via Hopf bifurcation. As the Reynolds number increases, steady-travelling solutions arise via pitchfork bifurcation from the steady-travelling solutions. On the other hand, in the rotating case, we find saddle-node bifurcations and closed-loop branches. We carry out time integration to study the properties of unsteady solutions at high Reynolds numbers. In the non-rotating case, the unsteady solution is chaotic and it wanders around the steady-travelling solutions bifurcating from three-jet zonal flow. We show that a linear combination of the steady and steady-travelling solutions gives a good approximation of the zonal-mean zonal flow of the unsteady solution, suggesting that the chaotic solution at high Reynolds numbers exists mostly within a relatively low-dimensional space spanned by the steady and steady-travelling solutions, which become unstable at low Reynolds numbers.

scholarly journals The effect of Reynolds number on turbulent drag reduction by streamwise travelling waves

2014 ◽  
Vol 759 ◽  
pp. 28-55 ◽  
Author(s):  
Edward Hurst ◽  
Qiang Yang ◽  
Yongmann M. Chung
Keyword(s):  
Reynolds Number ◽  
Drag Reduction ◽  
Flow Control ◽  
Travelling Waves ◽  
Reynolds Numbers ◽  
Travelling Wave ◽  
Control Parameters ◽  
Scaling Parameter ◽  
High Reynolds Numbers ◽  
High Reynolds

AbstractThis paper exploits the turbulent flow control method using streamwise travelling waves (Quadrio et al. J. Fluid Mech., vol. 627, 2009, pp. 161–178) to study the effect of Reynolds number on turbulent skin-friction drag reduction. Direct numerical simulations (DNS) of a turbulent channel flow subjected to the streamwise travelling waves of spanwise wall velocity have been performed at Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=200$ to 1600. To the best of the authors’ knowledge, this is the highest Reynolds number attempted with DNS for this type of flow control. The present DNS results confirm that the effectiveness of drag reduction deteriorates, and the maximum drag reduction achieved by travelling waves decreases significantly as the Reynolds number increases. The intensity of both the drag reduction and drag increase is reduced with the Reynolds number. Another important finding is that the value of the optimal control parameters changes, even in wall units, when the Reynolds number is increased. This trend is observed for the wall oscillation, stationary wave, and streamwise travelling wave cases. This implies that, when the control parameters used are close to optimal values found at a lower Reynolds number, the drag reduction deteriorates rapidly with increased Reynolds number. In this study, the effect of Reynolds number for the travelling wave is quantified using a scaling in the form $\mathit{Re}_{{\it\tau}}^{-{\it\alpha}}$. No universal constant is found for the scaling parameter ${\it\alpha}$. Instead, the scaling parameter ${\it\alpha}$ has a wide range of values depending on the flow control conditions. Further Reynolds number scaling issues are discussed. Turbulent statistics are analysed to explain a weaker drag reduction observed at high Reynolds numbers. The changes in the Stokes layer and also the mean and root-mean-squared (r.m.s.) velocity with the Reynolds number are also reported. The Reynolds shear stress analysis suggests an interesting possibility of a finite drag reduction at very high Reynolds numbers.

Temperature Control of Blow-Down, Supersonic Tunnels

1956 ◽  
Vol 60 (541) ◽  
pp. 67-70
Author(s):  
T. A. Thomson
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Reynolds Number ◽  
Mach Number ◽  
Reynolds Numbers ◽  
Stagnation Temperature ◽  
Blow Down ◽  
High Reynolds Numbers ◽  
Experimental Errors ◽  
High Reynolds

The blow-down type of intermittent, supersonic tunnel is attractive because of its simplicity and because relatively high Reynolds numbers can be obtained for a given size of test section. An adverse characteristic, however, is the fall of stagnation temperature during runs, which can affect experiments in several ways. The Reynolds number varies and the absolute velocity is not constant, even if the Mach number and pressure are; heat-transfer cannot be studied under controlled conditions and the experimental errors arising from the effect of heat-transfer on the boundary layer vary in time. These effects can become significant in quantitative experiments if the tunnel is large and the variation of temperature very rapid; the expense required to eliminate them might then be justified.

Relationship of roll and pitch oscillations in a fin flapping at transitional to high Reynolds numbers

2012 ◽  
Vol 702 ◽  
pp. 298-331 ◽  
Author(s):  
Promode R. Bandyopadhyay ◽  
David N. Beal ◽  
J. Dana Hrubes ◽  
Arun Mangalam
Keyword(s):  
Reynolds Number ◽  
Stagnation Point ◽  
Strouhal Number ◽  
High Efficiency ◽  
Oscillation Amplitude ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Stokes Wave ◽  
High Reynolds Numbers ◽  
High Reynolds

AbstractHydrodynamic effects of the relationship between the roll and pitch oscillations in low-aspect-ratio fins, with a laminar section and a rounded leading edge, flapping at transitional to moderately high Reynolds numbers, are considered. The fin is hinged at one end and its roll amplitude is large. Also examined is how this relationship is affected by spanwise twist, which alters the pitch oscillation amplitude and its phase relative to the roll motion. Force, efficiency and surface hot-film-anemometry measurements, and flow visualization are carried out in a tow tank. A fin of an abstracted penguin-wing planform and a NACA 0012 cross-section is used, and the chord Reynolds number varies from 3558 to 150 000 based on total speed. The fin is forced near the natural shedding frequency. Strouhal number and pitch amplitude are directly related when thrust is produced, and efficiency is maximized in narrow combinations of Strouhal number and pitch amplitude when oscillation of the leading-edge stagnation point is minimal. Twist makes the angle of attack uniform along the span and enhances thrust by up to 24 %, while maintaining high efficiency. Only 5 % of the power required to roll is spent to pitch, and yet roll and pitch are directly related. During hovering, dye visualization shows that a diffused leading-edge vortex is produced in rigid fins, which enlarges along the span; however, twist makes the vortex more uniform and the fin in turn requires less power to roll. Low-order phase maps of the measurements of force oscillation versus its derivative are modelled as due to van der Pol oscillators; the higher-order maps show trends in the sub-regimes of the transitional Reynolds number. Fin oscillation imparts a chordwise fluid motion, yielding a Stokes wave in the near-wall vorticity layer. When the roll and pitch oscillations are directly related, the wave is optimized: causing vorticity lift-up as the fin is decelerated at the roll extremity; the potential energy at the stagnation point is converted into kinetic energy; a vortex is produced as the lifted vorticity is wrapped around the leading edge; and free-stream reattachment keeps the vortex trapped. When the twist oscillation is phased along the span, this vortex becomes self-preserving at all amplitudes of twist, indicating the most stable (low-bandwidth) tuned nature.

An Intermittent High-Reynolds-Number Wind Tunnel

1977 ◽  
Vol 28 (4) ◽  
pp. 259-264 ◽  
Author(s):  
J L Stollery ◽  
A V Murthy
Keyword(s):  
Reynolds Number ◽  
Wind Tunnel ◽  
High Reynolds Number ◽  
Reynolds Numbers ◽  
Simple Method ◽  
High Reynolds Numbers ◽  
Reservoir Conditions ◽  
High Reynolds ◽  
Cryogenic Wind Tunnel ◽  
Performance Estimates

SummaryThe paper suggests a simple method of generating intermittent reservoir conditions for an intermittent, cryogenic wind tunnel. Approximate performance estimates are given and it is recommended that further studies be made because this type of tunnel could be valuable in increasing the opportunities for research at high Reynolds numbers.

Investigations of the Flow in Curved Ducts at Large Reynolds Numbers

1948 ◽  
Vol 15 (4) ◽  
pp. 344-348
Author(s):  
J. R. Weske
Keyword(s):  
Reynolds Number ◽  
Pressure Drop ◽  
Boundary Layers ◽  
Reynolds Numbers ◽  
Equation Of Motion ◽  
Radius Ratio ◽  
High Reynolds Numbers ◽  
Net Pressure ◽  
High Reynolds ◽  
Curved Ducts

Abstract It is found that the flow in curved ducts at high Reynolds numbers may be analyzed by methods adapted from the theory of boundary layers. Integration of the equation of motion of the “shedding layer” led to relations for the net pressure drop of curved ducts as a function of radius ratio and of Reynolds number.

Heat Transfer Coefficient Distribution of W and Broken W Turbulators At High Reynolds Numbers

2021 ◽  
pp. 1-29
Author(s):  
Sam Ghazi-Hesami ◽  
Dylan Wise ◽  
Keith Taylor ◽  
Étienne Robert ◽  
Peter Ireland
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Thermal Performance ◽  
Reynolds Numbers ◽  
High Reynolds Numbers ◽  
Smooth Channel ◽  
Wide Range ◽  
High Reynolds

Abstract An experimental and numerical study of the convective heat transfer enhancement provided by two rib families (W and Broken W) is presented, covering Reynolds numbers (Re) between 300,000 to 900,000 in a straight channel with a rectangular cross section (AR=1.29). These high Reynolds numbers were selected for the current study since most data in the available literature typically pertain to investigations at lower Reynolds numbers. The objective of this study is to assess the local heat transfer coefficient (HTC) enhancement (compared with a smooth channel) and the overall thermal performance, taking into account the effect of increased roughness on the friction factor, of a group of W shaped turbulators over a wide range of Reynolds numbers. Furthermore, the effects of increasing the rib spacing on the thermal performance of the Broken W configuration are presented and discussed. The numerical results are compared against heat transfer measurements obtained using the Transient Liquid Crystal (TLC) method. The research shows that for the Broken W turbulators, increasing the Reynolds number is associated with an overall decrease of the thermal performance while the thermal performance of the W configuration is relatively insensitive to Reynolds number. Nevertheless, the Broken W configuration delivers higher thermal performance and heat transfer compared with the W configuration for the range of Re investigated. The Broken W configuration with a pitch spacing of 10 times the rib height was shown to provide the optimal thermal performance in the configurations investigated here.

Turbulent boundary layer statistics at very high Reynolds number

2015 ◽  
Vol 779 ◽  
pp. 371-389 ◽  
Author(s):  
M. Vallikivi ◽  
M. Hultmark ◽  
A. J. Smits
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
High Reynolds Number ◽  
Reynolds Numbers ◽  
Working Fluid ◽  
Mean Velocity ◽  
High Reynolds Numbers ◽  
Layer Flow ◽  
High Reynolds ◽  
Very High

Measurements are presented in zero-pressure-gradient, flat-plate, turbulent boundary layers for Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=2600$ to $\mathit{Re}_{{\it\tau}}=72\,500$ ($\mathit{Re}_{{\it\theta}}=8400{-}235\,000$). The wind tunnel facility uses pressurized air as the working fluid, and in combination with MEMS-based sensors to resolve the small scales of motion allows for a unique investigation of boundary layer flow at very high Reynolds numbers. The data include mean velocities, streamwise turbulence variances, and moments up to 10th order. The results are compared to previously reported high Reynolds number pipe flow data. For $\mathit{Re}_{{\it\tau}}\geqslant 20\,000$, both flows display a logarithmic region in the profiles of the mean velocity and all even moments, suggesting the emergence of a universal behaviour in the statistics at these high Reynolds numbers.

Planar Oscillatory Flow Forces at High Reynolds Numbers

1993 ◽  
Vol 115 (1) ◽  
pp. 31-39 ◽  
Author(s):  
J. R. Chaplin
Keyword(s):  
Surface Roughness ◽  
Reynolds Number ◽  
Circular Cylinder ◽  
Oscillatory Flow ◽  
Reynolds Numbers ◽  
Number Range ◽  
High Reynolds Numbers ◽  
Reynolds Number Range ◽  
Fine Surface ◽  
High Reynolds

Measurements of pressures around a circular cylinder with fine surface roughness in planar oscillatory flow reveal considerable changes in drag and inertia coefficients over the Reynolds number range 2.5 × 105 to 7.5 × 105, and at Keulegan-Carpenter numbers between 5 and 25. In most respects, these results are shown to be compatible with previous measurements in planar oscillatory flow, and with previous measurements in which the same 0.5-m-dia cylinder was tested in waves.

scholarly journals Solitary waves on falling liquid films in the inertia-dominated regime

2018 ◽  
Vol 837 ◽  
pp. 491-519 ◽  
Author(s):  
Fabian Denner ◽  
Alexandros Charogiannis ◽  
Marc Pradas ◽  
Christos N. Markides ◽  
Berend G. M. van Wachem ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Liquid Films ◽  
Flow Reversal ◽  
Experimental Measurements ◽  
Flow Recirculation ◽  
High Reynolds Numbers ◽  
High Reynolds ◽  
Falling Liquid Films

We offer new insights and results on the hydrodynamics of solitary waves on inertia-dominated falling liquid films using a combination of experimental measurements, direct numerical simulations (DNS) and low-dimensional (LD) modelling. The DNS are shown to be in very good agreement with experimental measurements in terms of the main wave characteristics and velocity profiles over the entire range of investigated Reynolds numbers. And, surprisingly, the LD model is found to predict accurately the film height even for inertia-dominated films with high Reynolds numbers. Based on a detailed analysis of the flow field within the liquid film, the hydrodynamic mechanism responsible for a constant, or even reducing, maximum film height when the Reynolds number increases above a critical value is identified, and reasons why no flow reversal is observed underneath the wave trough above a critical Reynolds number are proposed. The saturation of the maximum film height is shown to be linked to a reduced effective inertia acting on the solitary waves as a result of flow recirculation in the main wave hump and in the moving frame of reference. Nevertheless, the velocity profile at the crest of the solitary waves remains parabolic and self-similar even after the onset of flow recirculation. The upper limit of the Reynolds number with respect to flow reversal is primarily the result of steeper solitary waves at high Reynolds numbers, which leads to larger streamwise pressure gradients that counter flow reversal. Our results should be of interest in the optimisation of the heat and mass transport characteristics of falling liquid films and can also serve as a benchmark for future model development.

Flow separation from a stationary meniscus

2009 ◽  
Vol 633 ◽  
pp. 137-145 ◽  
Author(s):  
J. SÉBILLEAU ◽  
L. LIMAT ◽  
J. EGGERS
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Critical Reynolds Number ◽  
Cylindrical Tube ◽  
Reynolds Numbers ◽  
Flow Lines ◽  
Low Reynolds Numbers ◽  
High Reynolds Numbers ◽  
High Reynolds ◽  
Meniscus Region

We consider the steady flow near a free surface at intermediate to high Reynolds numbers, both experimentally and theoretically. In our experiment, an axisymmetric capillary meniscus is suspended from a cylindrical tube, held slightly above a horizontal water surface. A flow of dyed water is released through the tube into the reservoir, and flow lines are thus recorded. At low Reynolds numbers, flow lines follow the free surface, and injected water spreads horizontally inside the container. Increasing the Reynolds number, the injected fluid penetrates to a certain distance into the bath, but ultimately follows the free surface. Above a critical Reynolds number of approximately 60, the flow separates from the free surface in the meniscus region and a jet projects vertically into the bath. We find no indication that the flow reattaches at higher Reynolds numbers, nor are our findings sensitive to surface contamination. We show theoretically and confirm experimentally that the separating streamline forms a right angle with the free surface.

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