Papers
Ship waves on uniform shear current at finite depth: wave resistance and critical velocity
- Yan Li, Simen Å Ellingsen
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- 24 February 2016, pp. 539-567
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We present a comprehensive theory for linear gravity-driven ship waves in the presence of a shear current with uniform vorticity, including the effects of finite water depth. The wave resistance in the presence of shear current is calculated for the first time, containing in general a non-zero lateral component. While formally apparently a straightforward extension of existing deep water theory, the introduction of finite water depth is physically non-trivial, since the surface waves are now affected by a subtle interplay of the effects of the current and the sea bed. This becomes particularly pronounced when considering the phenomenon of critical velocity, the velocity at which transversely propagating waves become unable to keep up with the moving source. The phenomenon is well known for shallow water, and was recently shown to exist also in deep water in the presence of a shear current (Ellingsen, J. Fluid Mech., vol. 742, 2014, R2). We derive the exact criterion for criticality as a function of an intrinsic shear Froude number $S\sqrt{b/g}$ ($S$ is uniform vorticity, $b$ size of source), the water depth and the angle between the shear current and the ship’s motion. Formulae for both the normal and lateral wave resistance forces are derived, and we analyse their dependence on the source velocity (or Froude number $Fr$) for different amounts of shear and different directions of motion. The effect of the shear current is to increase wave resistance for upstream ship motion and decrease it for downstream motion. Also the value of $Fr$ at which $R$ is maximal is lowered for upstream and increased for downstream directions of ship motion. For oblique angles between ship motion and current there is a lateral wave resistance component which can amount to 10–20 % of the normal wave resistance for side-on shear and $S\sqrt{b/g}$ of order unity. The theory is fully laid out and far-field contributions are carefully separated off by means of Cauchy’s integral theorem, exposing potential pitfalls associated with a slightly different method (Sokhotsky–Plemelj) used in several previous works.
Flux expulsion with dynamics
- Andrew D. Gilbert, Joanne Mason, Steven M. Tobias
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- 24 February 2016, pp. 568-588
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In the process of flux expulsion, a magnetic field is expelled from a region of closed streamlines on a $TR_{m}^{1/3}$ time scale, for magnetic Reynolds number $R_{m}\gg 1$ ($T$ being the turnover time of the flow). This classic result applies in the kinematic regime where the flow field is specified independently of the magnetic field. A weak magnetic ‘core’ is left at the centre of a closed region of streamlines, and this decays exponentially on the $TR_{m}^{1/2}$ time scale. The present paper extends these results to the dynamical regime, where there is competition between the process of flux expulsion and the Lorentz force, which suppresses the differential rotation. This competition is studied using a quasi-linear model in which the flow is constrained to be axisymmetric. The magnetic Prandtl number $R_{m}/R_{e}$ is taken to be small, with $R_{m}$ large, and a range of initial field strengths $b_{0}$ is considered. Two scaling laws are proposed and confirmed numerically. For initial magnetic fields below the threshold $b_{core}=O(UR_{m}^{-1/3})$, flux expulsion operates despite the Lorentz force, cutting through field lines to result in the formation of a central core of magnetic field. Here $U$ is a velocity scale of the flow and magnetic fields are measured in Alfvén units. For larger initial fields the Lorentz force is dominant and the flow creates Alfvén waves that propagate away. The second threshold is $b_{dynam}=O(UR_{m}^{-3/4})$, below which the field follows the kinematic evolution and decays rapidly. Between these two thresholds the magnetic field is strong enough to suppress differential rotation, leaving a magnetically controlled core spinning in solid body motion, which then decays slowly on a time scale of order $TR_{m}$.
The effect of a salinity gradient on the dissolution of a vertical ice face
- Craig D. McConnochie, Ross C. Kerr
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- 24 February 2016, pp. 589-607
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We investigate experimentally the effect of stratification on a vertical ice face dissolving into cold salty water. We measure the interface temperature, ablation velocity and turbulent plume velocity over a range of salinity gradients and compare our measurements with results of similar experiments without a salinity gradient (Kerr & McConnochie, J. Fluid Mech., vol. 765, 2015, pp. 211–228; McConnochie & Kerr, J. Fluid Mech., vol. 787, 2016, pp. 237–253). We observe that stratification acts to reduce the ablation velocity, interface temperature, plume velocity and plume acceleration. We define a stratification parameter, $S=N^{2}Q/{\it\Phi}_{o}$, that describes where stratification will be important, where $N$ is the Brunt–Väisälä frequency, $Q$ is the height-dependent plume volume flux and ${\it\Phi}_{o}$ is the buoyancy flux per unit area without stratification. The relevance of this stratification parameter is supported by our experiments, which deviate from the homogeneous theory at approximately $S=1$. Finally, we calculate values for the stratification parameter at a number of ice shelves and conclude that ocean stratification will have a significant effect on the dissolution of both the Antarctic and Greenland ice sheets.
Using stratification to mitigate end effects in quasi-Keplerian Taylor–Couette flow
- Colin Leclercq, Jamie L. Partridge, Pierre Augier, Stuart B. Dalziel, Rich R. Kerswell
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- 24 February 2016, pp. 608-630
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Efforts to model accretion disks in the laboratory using Taylor–Couette flow apparatus are plagued with problems due to the substantial impact the end plates have on the flow. We explore the possibility of mitigating the influence of these end plates by imposing stable stratification in their vicinity. Numerical computations and experiments confirm the effectiveness of this strategy for restoring the axially homogeneous quasi-Keplerian solution in the unstratified equatorial part of the flow for sufficiently strong stratification and moderate layer thickness. If the rotation ratio is too large, however (e.g. ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{3/2}$, where ${\it\Omega}_{o}/{\it\Omega}_{i}$ is the angular velocity at the outer/inner boundary and $r_{i}/r_{o}$ is the inner/outer radius), the presence of stratification can make the quasi-Keplerian flow susceptible to the stratorotational instability. Otherwise (e.g. for ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{1/2}$), our control strategy is successful in reinstating a linearly stable quasi-Keplerian flow away from the end plates. Experiments probing the nonlinear stability of this flow show only decay of initial finite-amplitude disturbances at a Reynolds number $Re=O(10^{4})$. This observation is consistent with most recent computational (Ostilla-Mónico, et al.J. Fluid Mech., vol. 748, 2014, R3) and experimental results (Edlund & Ji, Phys. Rev. E, vol. 89, 2014, 021004) at high $Re$, and reinforces the growing consensus that turbulence in cold accretion disks must rely on additional physics beyond that of incompressible hydrodynamics.
The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow
- Vivekanand Dabade, Navaneeth K. Marath, Ganesh Subramanian
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- 24 February 2016, pp. 631-703
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It is well known that, under inertialess conditions, the orientation vector of a torque-free neutrally buoyant spheroid in an ambient simple shear flow rotates along so-called Jeffery orbits, a one-parameter family of closed orbits on the unit sphere centred around the direction of the ambient vorticity (Jeffery, Proc. R. Soc. Lond. A, vol. 102, 1922, pp. 161–179). We characterize analytically the irreversible drift in the orientation of such torque-free spheroidal particles of an arbitrary aspect ratio, across Jeffery orbits, that arises due to weak inertial effects. The analysis is valid in the limit $Re,St\ll 1$, where $Re=(\dot{{\it\gamma}}L^{2}{\it\rho}_{f})/{\it\mu}$ and $St=(\dot{{\it\gamma}}L^{2}{\it\rho}_{p})/{\it\mu}$ are the Reynolds and Stokes numbers, which, respectively, measure the importance of fluid inertial forces and particle inertia in relation to viscous forces at the particle scale. Here, $L$ is the semimajor axis of the spheroid, ${\it\rho}_{p}$ and ${\it\rho}_{f}$ are the particle and fluid densities, $\dot{{\it\gamma}}$ is the ambient shear rate, and ${\it\mu}$ is the suspending fluid viscosity. A reciprocal theorem formulation is used to obtain the contributions to the drift due to particle and fluid inertia, the latter in terms of a volume integral over the entire fluid domain. The resulting drifts in orientation at $O(Re)$ and $O(St)$ are evaluated, as a function of the particle aspect ratio, for both prolate and oblate spheroids using a vector spheroidal harmonics formalism. It is found that particle inertia, at $O(St)$, causes a prolate spheroid to drift towards an eventual tumbling motion in the flow–gradient plane. Oblate spheroids, on account of the $O(St)$ drift, move in the opposite direction, approaching a steady spinning motion about the ambient vorticity axis. The period of rotation in the spinning mode must remain unaltered to all orders in $St$. For the tumbling mode, the period remains unaltered at $O(St)$. At $O(St^{2})$, however, particle inertia speeds up the rotation of prolate spheroids. The $O(Re)$ drift due to fluid inertia drives a prolate spheroid towards a tumbling motion in the flow–gradient plane for all initial orientations and for all aspect ratios. Interestingly, for oblate spheroids, there is a bifurcation in the orientation dynamics at a critical aspect ratio of approximately 0.14. Oblate spheroids with aspect ratios greater than this critical value drift in a direction opposite to that for prolate spheroids, and eventually approach a spinning motion about the ambient vorticity axis starting from any initial orientation. For smaller aspect ratios, a pair of non-trivial repelling orbits emerge from the flow–gradient plane, and divide the unit sphere into distinct basins of orientations that asymptote to the tumbling and spinning modes. With further decrease in the aspect ratio, these repellers move away from the flow–gradient plane, eventually coalescing onto an arc of the great circle in which the gradient–vorticity plane intersects the unit sphere, in the limit of a vanishing aspect ratio. Thus, sufficiently thin oblate spheroids, similar to prolate spheroids, drift towards an eventual tumbling motion irrespective of their initial orientation. The drifts at $O(St)$ and at $O(Re)$ are combined to obtain the drift for a neutrally buoyant spheroid. The particle inertia contribution remains much smaller than the fluid inertia contribution for most aspect ratios and density ratios of order unity. As a result, the critical aspect ratio for the bifurcation in the orientation dynamics of neutrally buoyant oblate spheroids changes only slightly from its value based only on fluid inertia. The existence of Jeffery orbits implies a rheological indeterminacy, and the dependence of the suspension shear viscosity on initial conditions. For prolate spheroids and oblate spheroids of aspect ratio greater than 0.14, inclusion of inertia resolves the indeterminacy. Remarkably, the existence of the above bifurcation implies that, for a dilute suspension of oblate spheroids with aspect ratios smaller than 0.14, weak stochastic fluctuations (residual Brownian motion being analysed here as an example) play a crucial role in obtaining a shear viscosity independent of the initial orientation distribution. The inclusion of Brownian motion leads to a new smaller critical aspect ratio of approximately 0.013. For sufficiently large $Re\,Pe_{r}$, the peak in the steady-state orientation distribution shifts rapidly from the spinning- to the tumbling-mode location as the spheroid aspect ratio decreases below this critical value; here, $Pe_{r}=\dot{{\it\gamma}}/D_{r}$, with $D_{r}$ being the Brownian rotary diffusivity, so that $Re\,Pe_{r}$ measures the relative importance of inertial drift and Brownian rotary diffusion. The shear viscosity, plotted as a function of $Re\,Pe_{r}$, exhibits a sharp transition from a shear-thickening to a shear-thinning behaviour, as the oblate spheroid aspect ratio decreases below 0.013. Our results are compared in detail to earlier analytical work for limiting cases involving either nearly spherical particles or slender fibres with weak inertia, and to the results of recent numerical simulations at larger values of $Re$ and $St$.
Effect of wall suction on rotating disk absolute instability
- Joanna Ho, Thomas C. Corke, Eric Matlis
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- 24 February 2016, pp. 704-737
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This research investigates the effect of uniform suction on the absolute instability of Type I cross-flow modes in the boundary layer on a rotating disk. Specifically, it is designed to investigate whether wall suction would transform the absolute instability into a global mode, as first postulated in the numerical simulations of Davies & Carpenter (J. Fluid Mech., vol. 486, 2003, pp. 287–329). The disk is designed so that with a suction parameter of $0.2$, the radial location of the absolute instability critical Reynolds number, $Re_{c_{A}}=650$, occurs on the disk. Wall suction is applied from $Re=317$ to 696.5. The design for wall suction follows that of Gregory & Walker (J. Fluid Mech., 1960, pp. 225–234) where an array of holes through the disk communicate between the measurement side of the disk and the underside of the disk which is inside of an enclosure that is maintained at a slight vacuum. The enclosure pressure is adjustable so that a range of suction or blowing parameters can be investigated. The holes in the measurement surface are covered by a compressed wire porous mesh to aid in uniformizing the suction on the measurement surface of the disk. The mesh is covered by a thin porous high-density polyethylene sheet featuring a $20~{\rm\mu}\text{m}$ pore size which provides a smooth finely porous surface. A companion numerical simulation is performed to investigate the effect that the size and vacuum pressure of the underside enclosure have on the uniformity of the measurement surface suction. Temporal disturbances are introduced using the method of Othman & Corke (J. Fluid Mech., 2006, pp. 63–94). The results document the evolution of disturbance wavepackets in space and time. These show a temporal growth of the wavepackets as the location of the absolute instability is approached which is in strong contrast to the temporal evolution without suction observed by Othman and Corke. The results appear to support the effect of wall suction on the absolute instability postulated by Thomas (PhD thesis, 2007, Cardiff University, UK) and Thomas & Davies (J. Fluid Mech., vol. 663, 2010, pp. 401–433).
Stable equilibrium configurations of an oblate capsule in simple shear flow
- C. Dupont, F. Delahaye, D. Barthès-Biesel, A.-V. Salsac
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- 24 February 2016, pp. 738-757
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The objective of the paper is to determine the stable mechanical equilibrium states of an oblate capsule subjected to a simple shear flow, by positioning its revolution axis initially off the shear plane. We consider an oblate capsule with a strain-hardening membrane and investigate the influence of the initial orientation, capsule aspect ratio $a/b$, viscosity ratio ${\it\lambda}$ between the internal and external fluids and the capillary number $Ca$ which compares the viscous to the elastic forces. A numerical model coupling the finite element and boundary integral methods is used to solve the three-dimensional fluid–structure interaction problem. For any initial orientation, the capsule converges towards the same mechanical equilibrium state, which is only a function of the capillary number and viscosity ratio. For $a/b=0.5$, only four regimes are stable when ${\it\lambda}=1$: tumbling and swinging in the low and medium $Ca$ range ($Ca\lesssim 1$), regimes for which the capsule revolution axis is contained within the shear plane; then wobbling during which the capsule experiences precession around the vorticity axis; and finally rolling along the vorticity axis at high capillary numbers. When ${\it\lambda}$ is increased, the tumbling-to-swinging transition occurs for higher $Ca$; the wobbling regime takes place at lower $Ca$ values and within a narrower $Ca$ range. For ${\it\lambda}\gtrsim 3$, the swinging regime completely disappears, which indicates that the stable equilibrium states are mainly the tumbling and rolling regimes at higher viscosity ratios. We finally show that the $Ca$–${\it\lambda}$ phase diagram is qualitatively similar for higher aspect ratio. Only the $Ca$-range over which wobbling is stable increases with $a/b$, restricting the stability ranges of in- and out-of-plane motions, although this phenomenon is mainly visible for viscosity ratios larger than 1.
Rapids
Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers
- Juan Sánchez, Ferran Garcia, Marta Net
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- 15 February 2016, R1
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A numerical study of the onset of convection in rotating internally heated self-gravitating fluid spheres is presented. The exploration of the stability of the conduction state versus the Taylor and Prandtl numbers supplies a detailed idea of the laws that fulfil the four types of solutions obtained at low Prandtl numbers. The main result found is that axisymmetric (torsional) modes of convection are preferred at high Taylor numbers in the zero-Prandtl-number limit. This instability appears at low Rayleigh numbers and gives rise to an oscillating single vortex of very high frequency.
Moment generating functions and scaling laws in the inertial layer of turbulent wall-bounded flows
- Xiang I. A. Yang, Ivan Marusic, Charles Meneveau
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- 16 February 2016, R2
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Properties of single- and two-point moment generating functions (MGFs) are examined in the inertial region of wall-bounded flows. Empirical evidence for power-law scaling of the single-point MGF $\langle \text{exp}(qu^{+})\rangle$ (where $u^{+}$ is the normalized streamwise velocity fluctuation and $q$ a real parameter) with respect to the wall-normal distance is presented, based on hot-wire data from a $Re_{{\it\tau}}=13\,000$ boundary-layer experiment. The parameter $q$ serves as a ‘dial’ to emphasize different parts of the signal such as high- and low-speed regions, for positive and negative values of $q$, respectively. Power-law scaling $\langle \text{exp}(qu^{+})\rangle \sim (z/{\it\delta})^{-{\it\tau}(q)}$ can be related to the generalized logarithmic laws previously observed in higher-order moments, such as in $\langle u^{+2p}\rangle ^{1/p}$, but provide additional information not available through traditional moments if considering $q$ values away from the origin. For two-point MGFs, the scalings in $\langle \text{exp}[qu^{+}(x)+q^{\prime }u^{+}(x+r)]\rangle$ with respect to $z$ and streamwise displacement $r$ in the logarithmic region are investigated. The special case $q^{\prime }=-q$ is of particular interest, since this choice emphasizes rare events with high and low speeds at a distance $r$. Applying simple scaling arguments motivated by the attached eddy model, a ‘scaling transition’ is predicted to occur for $q=q_{cr}$ such that ${\it\tau}(q_{cr})+{\it\tau}(-q_{cr})=1$, where ${\it\tau}(q)$ is the set of scaling exponents for single-point MGFs. This scaling transition is not visible to traditional central moments, but is indeed observed based on the experimental data, illustrating the capabilities of MGFs to provide new and statistically robust insights into turbulence structure and confirming essential ingredients of the attached eddy model.
Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition
- Xiaozhou He, Eberhard Bodenschatz, Guenter Ahlers
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- 17 February 2016, R3
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We present measurements of the orientation ${\it\theta}_{0}$ and temperature amplitude ${\it\delta}$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh–Bénard convection (RBC) with aspect ratio ${\it\Gamma}\equiv D/L=1.00$ ($D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr\simeq 0.8$. The results for ${\it\theta}_{0}$ revealed a preferred orientation with up-flow in the west, consistent with a broken azimuthal invariance due to the Earth’s Coriolis force (see Brown & Ahlers (Phys. Fluids, vol. 18, 2006, 125108)). They yielded the azimuthal diffusivity $D_{{\it\theta}}$ and a corresponding Reynolds number $Re_{{\it\theta}}$ for Rayleigh numbers over the range $2\times 10^{12}\lesssim Ra\lesssim 1.5\times 10^{14}$. In the classical state ($Ra\lesssim 2\times 10^{13}$) the results were consistent with the measurements by Brown & Ahlers (J. Fluid Mech., vol. 568, 2006, pp. 351–386) for $Ra\lesssim 10^{11}$ and $Pr=4.38$, which gave $Re_{{\it\theta}}\propto Ra^{0.28}$, and with the Prandtl-number dependence $Re_{{\it\theta}}\propto Pr^{-1.2}$ as found previously also for the velocity-fluctuation Reynolds number $Re_{V}$ (He et al., New J. Phys., vol. 17, 2015, 063028). At larger $Ra$ the data for $Re_{{\it\theta}}(Ra)$ revealed a transition to a new state, known as the ‘ultimate’ state, which was first seen in the Nusselt number $Nu(Ra)$ and in $Re_{V}(Ra)$ at $Ra_{1}^{\ast }\simeq 2\times 10^{13}$ and $Ra_{2}^{\ast }\simeq 8\times 10^{13}$. In the ultimate state we found $Re_{{\it\theta}}\propto Ra^{0.40\pm 0.03}$. Recently, Skrbek & Urban (J. Fluid Mech., vol. 785, 2015, pp. 270–282) claimed that non-Oberbeck–Boussinesq effects on the Nusselt and Reynolds numbers of turbulent RBC may have been interpreted erroneously as a transition to a new state. We demonstrate that their reasoning is incorrect and that the transition observed in the Göttingen experiments and discussed in the present paper is indeed to a new state of RBC referred to as ‘ultimate’.
The eruptive regime of mass-transfer-driven Rayleigh–Marangoni convection
- Thomas Köllner, Karin Schwarzenberger, Kerstin Eckert, Thomas Boeck
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- 19 February 2016, R4
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The transfer of an alcohol, 2-propanol, from an aqueous to an organic phase causes convection due to density differences (Rayleigh convection) and interfacial tension gradients (Marangoni convection). The coupling of the two types of convection leads to short-lived flow structures called eruptions, which were reported in several previous experimental studies. To unravel the mechanism underlying these patterns, three-dimensional direct numerical simulations and corresponding validation experiments were carried out and compared with each other. In the simulations, the Navier–Stokes–Boussinesq equations were solved with a plane interface that couples the two layers including solutal Marangoni effects. Our simulations show excellent agreement with the experimentally observed patterns. On this basis, the origin of the eruptions is explained by a two-step process in which Rayleigh convection continuously produces a concentration distribution that triggers an opposing Marangoni flow.
On a suspension of nearly spherical colloidal particles under large-amplitude oscillatory shear flow
- Aditya S. Khair
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- 22 February 2016, R5
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The dynamics of a dilute, monodisperse suspension of nearly spherical particles that undergo Brownian rotations in an oscillatory simple shear flow is quantified, as a paradigm for large-amplitude oscillatory shear (LAOS) rheology of complex fluids. We focus on the ‘strongly nonlinear’ regime of LAOS, defined by ${\it\beta}\gg 1$ and ${\it\beta}/{\it\alpha}\gg 1$, where ${\it\beta}$ is a dimensionless shear rate (or Weissenberg number) and ${\it\alpha}$ is a dimensionless oscillation frequency (or Deborah number). We derive an asymptotic solution for the long-time periodic orientation probability density function of the particles. Our analysis reveals that the orientation dynamics consists of ‘core’ regions of rapid oscillation (on the time scale of the inverse of the shear-rate amplitude), separated by comparatively short ‘turning’ regions of slow evolution when the imposed flow vanishes. Uniformly valid approximations to the shear stress and normal stress differences (NSDs) of the suspension are then constructed: the non-Newtonian contribution to the shear stress, first NSD and second NSD, decays as ${\it\beta}^{-3/2}$, ${\it\beta}^{-1}$ and ${\it\beta}^{-1/2}$, respectively, at large ${\it\beta}$. These stress scalings originate from the orientation dynamics at the turning regions. Therefore, it is the occasions when the flow vanishes that dominate the rheology of this paradigmatic complex fluid under LAOS.
Penetrative internally heated convection in two and three dimensions
- David Goluskin, Erwin P. van der Poel
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- 24 February 2016, R6
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Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows proportionally to the heating rate ($H$) without convection, grows proportionally to $H^{4/5}$ when convection is strong in both two and three dimensions. The ratio of the heating rate to the fluid temperature is likened to the Nusselt number of Rayleigh–Bénard convection. Simulations are reported for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in terms of the heating rate) up to $5\times 10^{10}$.
Analytical formulae for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci
- Darren G. Crowdy
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- 24 February 2016, R7
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This paper reports new analytical formulae for the longitudinal slip lengths for simple shear over a superhydrophobic surface, or bubble mattress, comprising a periodic array of unidirectional circular menisci, or bubbles, protruding into, or out of, the fluid. The accuracy of the formulae is tested against results from full numerical simulations; they are found to give small relative errors even at large no-shear fractions. In the dilute limit the formulae reduce to an earlier result by Crowdy (Phys. Fluids, vol. 22, 2010, 121703). They also extend analytical results of Sbragaglia & Prosperetti (Phys. Fluids, vol. 19, 2007, 043603) beyond the limit of a small protrusion angle.
Turbulent–laminar patterns in shear flows without walls
- Matthew Chantry, Laurette S. Tuckerman, Dwight Barkley
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- 24 February 2016, R8
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Turbulent–laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow – the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotemporal phenomena observed in transition, from spot growth to turbulent bands and uniform turbulence. The model substantially reduces the cost of simulating intermittent turbulent structures while maintaining the essential physics and a direct connection to the Navier–Stokes equations. We demonstrate the generic nature of this process by introducing stress-free equivalent flows for plane Poiseuille and pipe flows that again capture the turbulent–laminar structures seen in transition.
Front Cover (OFC, IFC) and matter
FLM volume 791 Cover and Front matter
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- 07 March 2016, pp. f1-f2
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Back Cover (OBC, IBC) and matter
FLM volume 791 Cover and Back matter
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- 07 March 2016, pp. b1-b3
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