Ian Eames

Mechanics of inhomogeneous turbulence and interfacial layers

The mechanics of inhomogeneous turbulence in and adjacent to interfacial layers bounding turbulent and non-turbulent regions are analysed. Different mechanisms are identified according to the straining by the turbulent eddies in relation to the strength of the mean shear adjacent to, or across, the interfacial layer. How the turbulence is initiated and the topology of the region of turbulence are also significant factors. Specifically the cases of a layer of turbulence bounded on one, or two, sides by a uniform and/or shearing flow, and a circular region of a rotating turbulent vortex are considered and discussed. The entrainment processes at fluctuating interfaces occur both at the outer edges of turbulent shear layers, with and without free-stream turbulence (e.g. jets, wakes and boundary layers), at internal boundaries such as those at the outside of the non-turbulent core of swirling flows (e.g. the ‘eye-wall’ of a hurricane) or at the top of the viscous sublayer and roughness elements in turbulent boundary layers. Conditionally sampled data enables these concepts to be tested. These concepts lead to physically based estimates for critical modelling parameters such as eddy viscosity near interfaces, entrainment rates, maximum velocity and displacement heights.

Segregation patterns in gas-fluidized systems

The formation of segregation patterns in initially homogeneous, fluidized, binary mixtures of particles has been studied. The adjustment of the bed depends on the proportions of fine and coarse particles in the mixture and the gas flow rate relative to the minimum fluidization velocities of the two components. The particles are immobile until the gas flow rate is sufficiently large to fluidize the mixture of particles. When the gas how rate exceeds this critical value, alternating vertical bands of coarse and fine particles form. At a second critical gas velocity this pattern breaks down and the more familiar pattern of a mixed horizontal band on top of a layer of coarse particles forms. A phase diagram, constructed from experimental observations, shows the conditions for which each of these regimes exists. Its structure is explained in terms of the fluidization and consequent mobility of the mixture components. When horizontal bands are present, the thickness of the lower layer of coarse particles decreases with increasing gas flow rate depending on the proportion of fine particles in the bed. This, and its development, can be understood by analogy with the sedimentation of particles through a turbulent fluid. The experiments imply that the efficiency of mixing lay the bubbles in the fluidized bed is very much less than that expected from gas bubbles in a liquid.

Infiltration into inclined fibrous sheets

The flow from line and point sources through an inclined fibrous sheet is studied experimentally and theoretically for wicking from a saturated region and flow from a constant-flux source. Wicking from a saturated line generates a wetted region whose length grows diffusively, linearly or tends to a constant, depending on whether the sheet is horizontal or inclined downwards or upwards. A constant-flux line source generates a wetted region which ultimately grows linearly with time, and is characterized by a capillary fringe whose thickness depends on the relative strength of the source, gravitational and capillary forces. Good quantitative agreement is observed between experiments and similarity solutions.Capillary-driven and constant-flux source flows issuing from a point on a horizontal sheet generate a wetted patch whose radius grows diffusively in time. The flow is characterized by the relative strength of the source and spreading induced by the action of capillary forces, gamma. As gamma increases, the fraction of the wetted region which is saturated increases. Wicking from a saturated point corresponds to gamma = gamma(c), and spreads at a slower rate than from a line source. For gamma < gamma(c), the flow is partially saturated everywhere. Good agreement is observed between measured moisture profiles, rates of spreading, and similarity solutions.Numerical solutions are developed for point sources on inclined sheets. The moisture profile is characterized by a steady region circumscribed by a narrow boundary layer across which the moisture content rapidly changes. An approximate analytical solution describes the increase in the size of the wetted region with time and source strength; these conclusions are confirmed by numerical calculations. Experimental measurements of the downslope length are observed to be slightly in excess of theoretical predictions, though the dependence on time, inclination and flow rate obtained theoretically is confirmed. Experimental measurements of cross-slope width are in agreement with numerical results and solutions for short and long times. The affect of a percolation threshold is observed to ultimately arrest cross-slope transport, placing a limitation on the long-time analysis.

Aerated granular flow over a horizontal rigid surface

The effect of a vertical gas flow on the dynamics of a coulombic granular material moving over a horizontal rigid porous surface has been studied experimentally and theoretically. The presence of a fluidizing gas significantly alters the granular flow dynamics. When the gas velocity, u g , is below the minimum fluidization velocity, u mf , the effect of the gas is to reduce the angle of repose θ from the value measured in the absence of a gas flow. When material is poured from a point source onto a horizontal surface it forms a pile, which adjusts through episodic avalanching to a self-similar conical shape. Under these conditions, the development of the pile is determined by the local force balance on individual particles and its extent may be expressed in terms of the volume of particles added and the angle of repose. A volume of material is poured continuously from a point source onto a surface according to Qt α . Below the minimum uidization velocity, a quasi-static description gives the encroachment distance of the granular pile as l = (2 Q /(2π/3) n −1 tan θ) 1/ n +1 t α/ n +1 where n = 1 for a planar release and n = 2 for an axisymmetric release. A continuum description of fluidized granular flow has been developed by vertically averaging the mass and momentum conservation equations and including the momentum exchange between the gas and granular flow. The bulk movement is driven along the ground by horizontal gradients of particle pressure and is resisted by a viscous drag force due to the particles moving horizontally through a vertical gas flow. Above the minimum fluidization velocity the character of the granular flow is significantly altered and takes on fluid-like properties. The model predicts the shape of the fluidized granular pile and that the encroachment distance grows as l = λ n α ( Q ( u g + u mf ) / e ) 1/ n +2 t α+1/ n +2 , where e is the void fraction in the bed and λ n α is a constant. Below the conditions for fluidization ( u g u mf ), the pile of granular material grows quasi-statically when t > t ∗, where t ∗ ∼ ( e n +1 Q u g + u mf ) / μ 2+ n ( u mf − u g ) 2+ n ) 1/1+ n −α corresponds to the critical time when frictional forces are comparable to gradients of particle pressure and the drag force. Numerical solutions describing the granular flow are presented. Experimental observations of the shape and extent of planar and axisymmetric granular flows when α = 1 compare well with theoretical predictions for various values of particle volume flux Q , time t , and gas flow rate u g . The mathematical description of fluidized granular flows along rigid surfaces indicates a strong analogy with buoyancy-driven flows in a porous medium. This analogy permits extension of our description to include flows down slopes and the effect of internal stratification.

Interfaces and inhomogeneous turbulence

A Euromech colloquium, on interfacial processes and inhomogeneous turbulence, was held in London on 28–30 June 2010. Papers were presented describing and analysing the influence of interfaces that separate turbulent/non-turbulent regions, between regions of contrasting fluid properties, or at the edge of boundaries. This paper describes a summary of the work presented, giving a snapshot of the current progress in this area, along with discussions about future research directions.

Dynamics of monopolar vortices on a topographic beta-plane

The dynamics of a cyclonic monopolar vortex on a topographic beta-plane are studied experimentally and theoretically. Detailed measurements of the vortex structure are conducted using high-resolution quantitative velocity measurements. The initial velocity profiles were described in terms of a radius R vm , maximum azimuthal velocity v θ m , and a dimensionless parameter α which characterizes the steepness of the velocity profile. The initial direction of motion of the monopolar vortex is critically dependent on α and weakly dependent of the initial strength and size of the vortex: isolated vortices (α ∼ 3) move north, whereas non-isolated vortices characterized by α ∼ 1 move northwest. When the azimuthal velocity decays slowly with radial distance (α < 1.4), Rossby wave generation dominates the vortex dynamics and the translational speed of the vortex correlates with the Rossby wave speed. When the azimuthal velocity decays rapidly with radial distance (α > 1.4) the vortex is isolated and the translational speed is much slower than the Rossby wave speed. To interpret the effect of the vortex structure on the direction of motion, a mechanistic model is developed which includes the Rossby force and a lift force arising from circulation around the vortex, but does not include the effect of Rossby waves. The Rossby force results from the integrated effect of the Coriolis force on the vortex and drives the vortex north; the lift force is determined from the circulation around the vortex and drives the vortex west. Comparison with the experimental data reveals two regimes: α < 1.4, where the vortex dynamics are dominated by Rossby waves whereas for α > 1.4 Rossby waves are weak and favourable agreement is found with the mechanistic model.

Vortical Interactions with Interfacial Shear Layers

Physical models based on linearised calculations are developed to provide insight into some of the complex processes which occur adjacent to shearing interfaces. Specifically the case of vortices interacting with strong and weak shearing interfaces are developed. These provide a starting point to interpret previous detailed experimental and theoretical studies.

Analytical Model of Valveless Micropumps

The flow driven by a valveless micropump with a single cylindrical pump chamber and two diffuser/nozzle elements is studied theoretically using a 1-D model. The pump cavity is driven at an angular frequency omega so that its volume oscillates with an amplitude <i>V</i> _m. The presence of diffuser/nozzle elements with pressure-drop coefficients zeta₊, zeta_- (> zeta +) and throat cross-sectional area <i>A</i> ₁ creates a rectified mean flow. In the absence of frictional forces the maximum mean volume flux (with zero pressure head) is <i>Q</i> ₀ where <i>Q</i> ₀/<i>V</i> _momega = (zeta_- - zeta₊)pi/16(zeta_-+ zeta₊), while the maximum pressure that can be overcome is Delta<i>P</i> _max where Delta<i>P</i> _max <i>A</i> ₁ ²/<i>V</i> _m ² omega² = (zeta_- - zeta₊)/16. These analytical results agree with numerical calculations for the coupled system of equations and compare well with the experimental results of Stemme and Stemme.

Intraocular fluid dynamics and retinal shear stress after vitrectomy and gas tamponade.

PURPOSE To evaluate fluid dynamics and fluid shear stress on the retinal wall in a model eye after vitrectomy and gas tamponade in relation to saccadic eye movements and sudden head movements and to correlate the results with gas fill fraction (GF). Methods. Analyses was undertaken using high-resolution computational fluid dynamic software. The fluid volume within the eye was discretized using 6 × 10(5) elements and solved with a volume-of-fluid METHOD The eye was abstracted to a sphere. Vertical and horizontal saccades and sudden rectilinear displacement of the head were examined. GF was varied from 20% to 80% of the eye height filled with gas. RESULTS Maximum shear stress during horizontal and vertical saccades was 1.0 Pa (Pascal) and 2.5 Pa, respectively, and was dependent on GF. Rapid rectilinear acceleration of the head caused a maximum shear stress of 16 Pa, largely independent of GF. Fluid sloshing within the eye decayed within 0.1 second. Stresses were maximum at the contact line and equator of the eye and were parallel to the direction of motion. CONCLUSIONS This study predicts that saccadic eye movements and normal head movements after vitrectomy and gas tamponade generate only small fluid shear stresses on the retina that are below published norms for retinal adhesion strength. Sudden, jerking head movements generate fluid shear forces similar to retinal adhesion strength that localize to the area of gas-fluid interface. Fluid sloshing occurs after movement, but rapidly decays on cessation of movement. These results suggest that restrictive posturing after vitrectomy and gas tamponade may be unnecessary. Patients should avoid sudden head movements.

Fluid displacement by stokes flow past a spherical droplet

The concept of ‘drift’, which has been exploited in many high Reynolds number and inviscid flow problems, is here applied to examine transport by a spherical viscous droplet (of radius