Papers
Decay of turbulence generated by a square-fractal-element grid
- R. J. Hearst, P. Lavoie
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- Published online by Cambridge University Press:
- 17 February 2014, pp. 567-584
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A novel square-fractal-element grid was designed in order to increase the downstream measurement range of fractal grid experiments relative to the largest element of the grid. The grid consists of a series of square fractal elements mounted to a background mesh with spacing $L_0 = 100\, {\rm mm}$. Measurements were performed in the region $3.5 \le x/L_0 \le 48.5$, which represents a significant extension to the $x/L_0 < 20$ of previously reported square fractal grid measurements. For the region $x/L_0 \gtrsim 24$ it was found that a power-law decay region following $\langle {q}^2 \rangle \sim (x - x_0)^m$ exists with decay exponents of $m = -1.39$ and $-1.37$ at $\mathit{Re}_{L_0} = 57\, 000$ and $65\, 000$, respectively. This agrees with decay values previously measured for regular grids ($-1 \gtrsim m \gtrsim -1.4$). The turbulence in the near-grid region, $x/L_0 < 20$, is shown to be inhomogeneous and anisotropic, in apparent contrast with previous fractal grid measurements. Nonetheless, power-law fits to the decay of turbulent kinetic energy in this region result in $m = -2.79$, similar to $m \approx -2.5$ recently reported by Valente & Vassilicos (J. Fluid Mech., vol. 687, 2011, pp. 300–340) for space-filling square fractals. It was also found that $C_\epsilon $ is approximately constant for $x/L_0 \ge 25$, while it grows rapidly for $x/L_0 < 20$. These results reconcile previous fractal-generated turbulence measurements with classical grid turbulence measurements.
Wave turbulence in a rotating channel
- Julian F. Scott
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- Published online by Cambridge University Press:
- 13 February 2014, pp. 316-349
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This paper describes wave-turbulence closure and its consequences for rapidly rotating (i.e. small Rossby number) turbulence confined by two infinite, parallel walls perpendicular to the rotation axis. Expressing the flow as a combination of inertial waveguide modes leads to a spectral matrix, whose diagonal elements express the distribution of energy over modes and whose off-diagonal elements represent correlations between modes of different orders. In preparation for wave-turbulence closure, the flow is decomposed into two-dimensional and wave components. The former is found to evolve as if it were a classical, two-dimensional, non-rotating flow, but with wall friction due to Ekman pumping by the boundary layers. Evolution equations for the wave-component elements of the spectral matrix are derived using a wave-turbulence approach. Detailed analysis of these equations shows that, surprisingly, the two-dimensional component has no effect on wave-component energetics. As expected for wave turbulence, energy transfer between wave modes is via resonant triads and takes place at times $O(\varepsilon ^{-2})$ multiples of the rotational period, where $\varepsilon $ is the Rossby number. Despite playing no role in wave-mode energetics, the two-dimensional component produces decay of the off-diagonal elements of the spectral matrix on a time scale that is small compared with $O(\varepsilon ^{-2})$ rotation periods. There are thus three asymptotically distinct stages in the evolution of the turbulence in the limit of small Rossby number: the two-dimensional flow begins to evolve at the usual large-eddy turnover time scale ($O(\varepsilon ^{-1})$ multiples of the rotation period) and continues to develop thereafter. This is followed by decorrelation of different wave orders and finally evolution of the wave energy spectra due to resonant interactions.
Modelling size segregation of granular materials: the roles of segregation, advection and diffusion
- Yi Fan, Conor P. Schlick, Paul B. Umbanhowar, Julio M. Ottino, Richard M. Lueptow
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- Published online by Cambridge University Press:
- 21 February 2014, pp. 252-279
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Predicting segregation of granular materials composed of different-sized particles is a challenging problem. In this paper, we develop and implement a theoretical model that captures the interplay between advection, segregation and diffusion in size bidisperse granular materials. The fluxes associated with these three driving factors depend on the underlying kinematics, whose characteristics play key roles in determining particle segregation configurations. Unlike previous models for segregation, our model uses parameters based on kinematic measures from discrete element method simulations instead of arbitrarily adjustable fitting parameters, and it achieves excellent quantitative agreement with both experimental and simulation results when applied to quasi-two-dimensional bounded heaps. The model yields two dimensionless control parameters, both of which are only functions of control parameters (feed rate, particle sizes, and system size) and kinematic parameters (diffusion coefficient, flowing layer depth, and percolation velocity). The Péclet number, $\mathit{Pe}$, captures the interplay of advection and diffusion, and the second dimensionless parameter, $\Lambda $, describes the interplay between segregation and advection. A parametric study of $\Lambda $ and $\mathit{Pe}$ demonstrates how the particle segregation configuration depends on the interplay of advection, segregation and diffusion. The model can be readily adapted to other flow geometries.
Sidewall effects in Rayleigh–Bénard convection
- Richard J. A. M. Stevens, Detlef Lohse, Roberto Verzicco
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- 17 February 2014, pp. 1-27
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We investigate the influence of the temperature boundary conditions at the sidewall on the heat transport in Rayleigh–Bénard (RB) convection using direct numerical simulations. For relatively low Rayleigh numbers $Ra$ the heat transport is higher when the sidewall is isothermal, kept at a temperature $T_c+\Delta /2$ (where $\Delta $ is the temperature difference between the horizontal plates and $T_c$ the temperature of the cold plate), than when the sidewall is adiabatic. The reason is that in the former case part of the heat current avoids the thermal resistance of the fluid layer by escaping through the sidewall that acts as a short-circuit. For higher $Ra$ the bulk becomes more isothermal and this reduces the heat current through the sidewall. Therefore the heat flux in a cell with an isothermal sidewall converges to the value obtained with an adiabatic sidewall for high enough $Ra$ (${\simeq }10^{10}$). However, when the sidewall temperature deviates from $T_c+\Delta /2$ the heat transport at the bottom and top plates is different from the value obtained using an adiabatic sidewall. In this case the difference does not decrease with increasing $Ra$ thus indicating that the ambient temperature of the experimental apparatus can influence the heat transfer. A similar behaviour is observed when only a very small sidewall region close to the horizontal plates is kept isothermal, while the rest of the sidewall is adiabatic. The reason is that in the region closest to the horizontal plates the temperature difference between the fluid and the sidewall is highest. This suggests that one should be careful with the placement of thermal shields outside the fluid sample to minimize spurious heat currents. When the physical sidewall properties (thickness, thermal conductivity and heat capacity) are considered the problem becomes one of conjugate heat transfer and different behaviours are possible depending on the sidewall properties and the temperature boundary condition on the ‘dry’ side. The problem becomes even more complicated when the sidewall is shielded with additional insulation or temperature-controlled surfaces; some particular examples are illustrated and discussed. It has been observed that the sidewall temperature dynamics not only affects the heat transfer but can also trigger a different mean flow state or change the temperature fluctuations in the flow and this could explain some of the observed differences between similar but not fully identical experiments.
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Interaction of two axisymmetric bodies falling side by side at moderate Reynolds numbers
- Patricia Ern, Nicolas Brosse
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- Published online by Cambridge University Press:
- 11 February 2014, R6
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We consider the interaction of two identical disks freely falling side by side in a fluid at rest for Reynolds numbers ranging from 100 to 300, corresponding to rectilinear and oscillatory paths. For the three aspect ratios of the disks investigated, we observed that the bodies always repel one another when the horizontal distance between their centres of gravity is less than 4.5 diameters. They never come closer for distances spanning between 4.5 and 6 diameters. Beyond the latter distance, the disks appear indifferent to each other. For both rectilinear and periodic paths, the repulsion effect is weak, leading to an overall horizontal drift lower than 3 % of the vertical displacement. We propose a model for the repulsion coefficient Cr, which decreases with the separation distance between the bodies and is inversely proportional to the aspect ratio of the bodies, Cr thus being stronger for the thicker ones. Furthermore, in the case of the oscillatory paths, we show that the effect of the interaction reduces to the repulsion effect, since the characteristics of the oscillatory motion of each disk appear unaffected by the presence of the companion disk and no synchronization is observed between the paths, nor between the wakes, of the two disks.
Papers
On the impact of swirl on the growth of coherent structures
- K. Oberleithner, C. O. Paschereit, I. Wygnanski
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- 07 February 2014, pp. 156-199
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Spatial linear stability analysis is applied to the mean flow of a turbulent swirling jet at swirl intensities below the onset of vortex breakdown. The aim of this work is to predict the dominant coherent flow structure, their driving instabilities and how they are affected by swirl. At the nozzle exit, the swirling jet promotes shear instabilities and, less unstable, centrifugal instabilities. The latter stabilize shortly downstream of the nozzle, contributing very little to the formation of coherent structures. The shear mode remains unstable throughout generating coherent structures that scale with the axial shear-layer thickness. The most amplified mode in the nearfield is a co-winding double-helical mode rotating slowly in counter-direction to the swirl. This gives rise to the formation of slowly rotating and stationary large-scale coherent structures, which explains the asymmetries in the mean flows often encountered in swirling jet experiments. The co-winding single-helical mode at high rotation rate dominates the farfield of the swirling jet in replacement of the co- and counter-winding bending modes dominating the non-swirling jet. Moreover, swirl is found to significantly affect the streamwise phase velocity of the helical modes rendering this flow as highly dispersive and insensitive to intermodal interactions, which explains the absence of vortex pairing observed in previous investigations. The stability analysis is validated through hot-wire measurements of the flow excited at a single helical mode and of the flow perturbed by a time- and space-discrete pulse. The experimental results confirm the predicted mode selection and corresponding streamwise growth rates and phase velocities.
Absolute instabilities in eccentric Taylor–Couette–Poiseuille flow
- Colin Leclercq, Benoît Pier, Julian F. Scott
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- 17 February 2014, pp. 543-566
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The effect of eccentricity on absolute instabilities (AI) in the Taylor–Couette system with pressure-driven axial flow and fixed outer cylinder is investigated. Five modes of instability are considered, characterized by a pseudo-angular order $m$, with here $\vert m\vert \leq 2$. These modes correspond to toroidal ($m=0$) and helical structures ($m\neq 0$) deformed by the eccentricity. Throughout the parameter range, the mode with the largest absolute growth rate is always the Taylor-like vortex flow corresponding to $m=0$. Axial advection, characterized by a Reynolds number ${\mathit{Re}_z}$, carries perturbations downstream, and has a strong stabilizing effect on AI. On the other hand, the effect of the eccentricity $e$ is complex: increasing $e$ generally delays AI, except for a range of moderate eccentricites ${0.3\lesssim e \lesssim 0.6}$, where it favours AI for large enough ${\mathit{Re}_z}$. This striking behaviour is in contrast with temporal instability, always inhibited by eccentricity, and where left-handed helical modes of increasing $\vert m\vert $ dominate for larger ${\mathit{Re}_z}$. The instability mechanism of AI is clearly centrifugal, even for the larger values of ${\mathit{Re}_z}$ considered, as indicated by an energy analysis. For large enough ${\mathit{Re}_z}$, critical modes localize in the wide gap for low $e$, but their energy distribution is shifted towards the diverging section of the annulus for moderate $e$. For highly eccentric geometries, AI are controlled by the minimal annular clearance, and the critical modes are confined to the vicinity of the inner cylinder. Untangling the AI properties of each $m$ requires consideration of multiple pinch points.
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The quasi-static growth of CO2 bubbles
- Oscar R. Enríquez, Chao Sun, Detlef Lohse, Andrea Prosperetti, Devaraj van der Meer
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- 04 March 2014, R1
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We study experimentally the growth of an isolated gas bubble in a slightly supersaturated water–CO2 solution at 6 atm pressure. In contrast to what was found in previous experiments at higher supersaturation, the time evolution of the bubble radius differs noticeably from existing theoretical solutions. We trace the differences back to several combined effects of the concentration boundary layer around the bubble, which we disentangle in this work. In the early phase, the interaction with the surface on which the bubble grows slows down the process. In contrast, in the final phase, before bubble detachment, the growth rate is enhanced by the onset of density-driven convection. We also show that the bubble growth is affected by prior growth and detachment events, though they are up to 15 min apart.
Papers
Solutal-convection regimes in a two-dimensional porous medium
- Anja C. Slim
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- 17 February 2014, pp. 461-491
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We numerically characterize the temporal regimes for solutal convection from almost first contact to high dissolved solute concentration in a two-dimensional ideal porous layer for Rayleigh numbers $\mathcal{R}$ between $100$ and $5\times 10^4$. The lower boundary is impenetrable. The upper boundary is saturated with dissolved solute and either impermeable or partially permeable to fluid flow. In the impermeable case, initially there is pure diffusion of solute away from the upper boundary, followed by the birth and growth of convective fingers. Eventually fingers interact and merge, generating complex downwelling plumes. Once the inter-plume spacing is sufficient, small protoplumes reinitiate on the boundary layer and are swept into the primary plumes. The flow is now in a universal regime characterized by a constant (dimensionless) dissolution flux $F=0.017$ (the rate at which solute dissolves from the upper boundary). The horizontally averaged concentration profile stretches as a simple self-similar wedge beneath a diffusive horizontal boundary layer. Throughout, the plume width broadens proportionally to $\sqrt{t}$, where $t$ is (dimensionless) time. The above behaviour is parameter independent; the Rayleigh number only controls when transition occurs to a final $\mathcal{R}$-dependent shut-down regime. For the constant-flux and shut-down regimes, we rigourously derive upscaled equations connecting the horizontally averaged concentration, vertical advective flux and plume widths. These are partially complete; a universal expression for the plume width remains elusive. We complement these governing equations with phenomenological boundary conditions based on a marginally stable diffusive boundary layer at the top and zero advective flux at the bottom. Making appropriate approximations in each regime, we find good agreement between predictions from this model and simulated results for both solutal and thermal convection. In the partially permeable upper boundary case, fluid from the convecting layer can penetrate an overlying separate-phase-solute bearing layer where it immediately saturates. The regime diagram remains almost the same as for the impermeable case, but the dissolution flux is significantly augmented. Our work is motivated by dissolution of carbon dioxide relevant to geological storage, and we conclude with a simple flux parameterization for inclusion in gravity current models and suggest that the upscaled equations could lay the foundation for accurate inclusion of dissolution in reservoir simulators.
Back Cover (IBC, OBC) and matter
FLM volume 741 Cover and Back matter
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- Published online by Cambridge University Press:
- 03 March 2014, pp. b1-b3
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Front Cover (OFC, IFC) and matter
FLM volume 741 Cover and Front matter
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- Published online by Cambridge University Press:
- 03 March 2014, pp. f1-f4
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