31 results
Assessment time of the Welfare Quality® protocol for dairy cattle
- M de Vries, B Engel, I den Uijl, G van Schaik, T Dijkstra, IJM de Boer, EAM Bokkers
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- Journal:
- Animal Welfare / Volume 22 / Issue 1 / February 2013
- Published online by Cambridge University Press:
- 01 January 2023, pp. 85-93
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The Welfare Quality® (WQ) protocols are increasingly used for assessing welfare of farm animals. These protocols are time consuming (about one day per farm) and, therefore, costly. Our aim was to assess the scope for reduction of on-farm assessment time of the WQ protocol for dairy cattle. Seven trained observers quantified animal-based indicators of the WQ protocol in 181 loose-housed and 13 tied Dutch dairy herds (herd size from 10 to 211 cows). Four assessment methods were used: avoidance distance at the feeding rack (ADF, 44 min); qualitative behaviour assessment (QBA, 25 min); behavioural observations (BO, 150 min); and clinical observations (CO, 132 min). To simulate reduction of on-farm assessment time, a set of WQ indicators belonging to one assessment method was omitted from the protocol. Observed values of omitted indicators were replaced by predictions based on WQ indicators of the remaining three assessment methods, resources checklist, and interview, thus mimicking the performance of the full WQ protocol. Agreement between predicted and observed values of WQ indicators, however, was low for ADF, moderate for QBA, slight to moderate for BO, and poor to moderate for CO. It was concluded that replacing animal-based WQ indicators by predictions based on remaining WQ indicators shows little scope for reduction of on-farm assessment time of the Welfare Quality® protocol for dairy cattle. Other ways to reduce on-farm assessment time of the WQ protocol for dairy cattle, such as the use of additional data or automated monitoring systems, should be investigated.
Comparison of locomotion scoring for dairy cows by experienced and inexperienced raters using live or video observation methods
- A Schlageter-Tello, EAM Bokkers, PWG Groot Koerkamp, T Van Hertem, S Viazzi, CEB Romanini, I Halachmi, C Bahr, D Berckmans, K Lokhorst
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- Journal:
- Animal Welfare / Volume 24 / Issue 1 / February 2015
- Published online by Cambridge University Press:
- 01 January 2023, pp. 69-79
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Lameness is considered a major problem in dairy production. Lameness is commonly detected with locomotion scores assigned to cows under farm conditions, but raters are often trained and assessed for reliability and agreement by using video recordings. The aim of this study was to evaluate intra- and inter-rater reliability and agreement of experienced and inexperienced raters for locomotion scoring performed live and from video, and to calculate the influence of raters and the method of observation (live or video) on the probability of classifying a cow as lame. Using a five-level locomotion score, cows were scored twice live and twice from video by three experienced and two inexperienced raters for three weeks. Every week different cows were scored. Intra- and inter-rater reliability (expressed as weighted kappa, kw) and agreement (expressed as percentage of agreement, PA) for live/live, live/video and video/video comparisons were determined. A logistic regression was performed to estimate the influence of the rater and method of observation on the probability of classifying a cow as lame in live and video observation. Experienced raters had higher values for intra-rater reliability and agreement for video/video than for live/live and live/video comparison. Inexperienced raters, however, did not differ for intra- and inter-rater reliability and agreement for live/live, live/video and video/video comparisons. The logistic regression indicated that raters were responsible for the main effect and the method of observation (live or from video) had a minor effect on the probability for classifying a cow as lame (locomotion score ≥ 3). In conclusion, under the present experimental conditions, experienced raters performed better than unexperienced raters when locomotion scoring was done from video. Since video observation did not show any important influence in the probability of classifying a cow as lame, video observation seems to be an acceptable method for locomotion scoring and lameness assessment in dairy cows.
The effect of an unsteady flow incident on an array of circular cylinders
- C. A. Klettner, I. Eames, J. C. R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 872 / 10 August 2019
- Published online by Cambridge University Press:
- 13 June 2019, pp. 560-593
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In this paper we investigate the effect of an inhomogeneous and unsteady velocity field incident on an array of rigid circular cylinders arranged within a circular perimeter (diameter $D_{G}$) of varying solid fraction $\unicode[STIX]{x1D719}$, where the unsteady flow is generated by placing a cylinder (diameter $D_{G}$) upwind of the array. Unsteady two-dimensional viscous simulations at a moderate Reynolds number ($Re=2100$) and also, as a means of extrapolating to a flow with a very high Reynolds number, inviscid rapid distortion theory (RDT) calculations were carried out. These novel RDT calculations required the circulation around each cylinder to be zero which was enforced using an iterative method. The two main differences which were highlighted was that the RDT calculations indicated that the tangential velocity component is amplified, both, at the front and sides of the array. For the unsteady viscous simulations this result did not occur as the two-dimensional vortices (of similar size to the array) are deflected away from the boundary and do not penetrate into the boundary layer. Secondly, the amplification is greater for the RDT calculations as for the unsteady finite Reynolds number calculations. For the two highest solid fraction arrays, the mean flow field has two recirculation regions in the near wake of the array, with closed streamlines that penetrate into the array which will have important implications for scalar transport. The increased bleed through the array at the lower solid fraction results in this recirculation region being displaced further downstream. The effect of inviscid blocking and viscous drag on the upstream streamwise velocity and strain field is investigated as it directly influences the ability of the large coherent structures to penetrate into the array and the subsequent forces exerted on the cylinders in the array. The average total force on the array was found to increase monotonically with increasing solid fraction. For high solid fraction $\unicode[STIX]{x1D719}$, although the fluctuating forces on the individual cylinders is lower than for low $\unicode[STIX]{x1D719}$, these forces are more correlated due to the proximity of the cylinders. The result is that for mid to high solid fraction arrays the fluctuating force on the array is insensitive to $\unicode[STIX]{x1D719}$. For low $\unicode[STIX]{x1D719}$, where the interaction of the cylinders is weak, the force statistics on the individual cylinders can be accurately estimated from the local slip velocity that occurs if the cylinders were removed.
Flow and passive transport in planar multipolar flows
- M. A. Zouache, I. Eames, C. A. Klettner, P. J. Luthert
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- Journal:
- Journal of Fluid Mechanics / Volume 858 / 10 January 2019
- Published online by Cambridge University Press:
- 02 November 2018, pp. 184-227
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We study the flow and transport of heat or mass, modelled as passive scalars, within a basic geometrical unit of a three-dimensional multipolar flow – a triangular prism – characterised by a side length $L$, a normalised thickness $0.01\leqslant \unicode[STIX]{x1D700}\leqslant 0.1$ and an apex angle $0<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}$, and connected to inlet and outlet pipes of equal normalised radius $0.01\leqslant \unicode[STIX]{x1D6FF}\leqslant 0.1$ perpendicularly to the plane of the flow. The flow and scalar fields are investigated over the range $0.1\leqslant Re_{p}\leqslant 10$ and $0.1\leqslant Pe_{p}\leqslant 1000$, where $Re_{p}$ and $Pe_{p}$ are respectively the Reynolds and Péclet numbers imposed at the inlet pipe when either a Dirichlet ($\text{D}$) or a Neumann ($\text{N}$) scalar boundary condition is imposed at the wall unattached to the inlets and outlets. A scalar no-flux boundary condition is imposed at all the other walls. An axisymmetric model is applied to understand the flow and scalar transport in the inlet and outlet regions, which consist of a turning region close to the pipe centreline and a channel region away from it. A separate two-dimensional model is then developed for the channel region by solving the integral form of the momentum and scalar advection–diffusion equations. Analytical relations between geometrical, flow and scalar transport parameters based on similarity and integral methods are generated and agree closely with numerical solutions. Finally, three-dimensional numerical calculations are undertaken to test the validity of the axisymmetric and depth-averaged analyses. Dominant flow and scalar transport features vary dramatically across the flow domain. In the turning region, the flow is a largely irrotational straining flow when $\unicode[STIX]{x1D700}\geqslant \unicode[STIX]{x1D6FF}$ and a dominantly viscous straining flow when $\unicode[STIX]{x1D700}\ll \unicode[STIX]{x1D6FF}$. The thickness of the scalar boundary layer scales to the local Péclet number to the power $1/3$. The diffusive flux $j_{d}$ and the scalar $C_{s}$ at the wall where ($\text{D}$) or ($\text{N}$) is imposed, respectively, are constant. In the channel region, the flow is parabolic and dominated by a source flow near the inlet and an irrotational straining flow away from it. When $(\text{D})$ is imposed the scalar decreases exponentially with distance from the inlet and the normalised scalar transfer coefficient converges to $\unicode[STIX]{x1D6EC}_{\infty }=2.5694$. When $(\text{N})$ is imposed, $C_{s}$ varies proportionally to surface area. Transport in the straining region downstream of the inlet is diffusion-limited, and $j_{d}$ and $C_{s}$ are functions of the geometrical parameters $L$, $\unicode[STIX]{x1D700}$, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$. In addition to describing the fundamental properties of the flow and passive transport in multipolar configurations, the present work demonstrates how geometrical and flow parameters should be set to control transfers in the different regions of the flow domain.
The effect of a uniform through-surface flow on a cylinder and sphere
- C. A. Klettner, I. Eames, S. Semsarzadeh, A. Nicolle
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- Journal:
- Journal of Fluid Mechanics / Volume 793 / 25 April 2016
- Published online by Cambridge University Press:
- 23 March 2016, pp. 798-839
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The effect of a uniform through-surface flow (velocity $U_{b}$) on a rigid and stationary cylinder and sphere (radius $a$) fixed in a free stream (velocity $U_{\infty }$) is analysed analytically and numerically. The flow is characterised by a dimensionless blow velocity ${\it\Lambda}\,(=U_{b}/U_{\infty })$ and Reynolds number $Re\,(=2aU_{\infty }/{\it\nu}$, where ${\it\nu}$ is the kinematic viscosity). High resolution numerical calculations are compared against theoretical predictions over the range $-3\leqslant {\it\Lambda}\leqslant 3$ and $Re=1,10,100$ for planar flow past a cylinder and axisymmetric flow past a sphere. For $-{\it\Lambda}\gg 1$, the flow is viscously dominated in a thin boundary layer of thickness ${\it\nu}/|U_{b}|$ adjacent to the rigid surface which develops in a time ${\it\nu}/U_{b}^{2}$; the surface vorticity scales as $Re|{\it\Lambda}|U_{\infty }/a$ for a cylinder and sphere. A boundary layer analysis is developed to analyse the unsteady viscous forces. Numerical results show that the surface pressure and vorticity distribution within the boundary layer agrees with a steady state analysis. The flow downstream of the body is irrotational so the wake volume flux, $Q_{w}$, is zero and the drag force is $F_{D}=-{\it\rho}U_{\infty }Q_{b}$, where ${\it\rho}$ is the density of the fluid and $Q_{b}$ is the normal flux through the body surface. The drag coefficient is therefore $-2{\rm\pi}{\it\Lambda}$ or $-8{\it\Lambda}$ for a cylinder or sphere, respectively. A dissipation argument is applied to analyse the drag force; the rate of working of the drag force is balanced by viscous dissipation, flux of stagnation pressure and rate of work by viscous stresses due to sucking. At large $Re|{\it\Lambda}|$, the drag force is largely determined by viscous dissipation for a cylinder, with a weak contribution by the normal viscous stresses, while for a sphere, only $3/4$ of the drag force is determined by viscous dissipation with the remaining $1/4$ due to the flux of stagnation pressure through the sphere surface. When ${\it\Lambda}\gg 1$, the boundary layer thickness initially grows linearly with time as vorticity is blown away from the rigid surface. The vorticity in the boundary layer is weakly dependent on viscous effects and scales as $U_{\infty }/a{\it\Lambda}$ or $U_{\infty }/a{\it\Lambda}^{3/2}$ for a cylinder and sphere, respectively. For large blow velocity, the vorticity is swept into two well-separated shear layers and the maximum vorticity decreases due to diffusion. The drag force is related to the vorticity distribution on the body surface and an approximate expression can be derived by considering the first term of a Fourier expansion in the surface vorticity. It is found that the drag coefficient $C_{D}$ for a cylinder (corrected for flow boundedness) is weakly dependent on ${\it\Lambda}$ while for a sphere, $C_{D}$ decreases with ${\it\Lambda}$.
Low-Reynolds-number flow past a cylinder with uniform blowing or sucking
- C. A. Klettner, I. Eames
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- Journal:
- Journal of Fluid Mechanics / Volume 780 / 10 October 2015
- Published online by Cambridge University Press:
- 09 September 2015, R2
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We analyse the low-Reynolds-number flow generated by a cylinder (of radius $a$) in a stream (of velocity $U_{\infty }$) which has a uniform through-surface blowing component (of velocity $U_{b}$). The flow is characterized in terms of the Reynolds number $Re$ ($=2aU_{\infty }/{\it\nu}$, where ${\it\nu}$ is the kinematic viscosity of the fluid) and the dimensionless blow velocity ${\it\Lambda}$ ($=U_{b}/U_{\infty }$). We seek the leading-order symmetric solution of the vorticity field which satisfies the near- and far-field boundary conditions. The drag coefficient is then determined from the vorticity field. For the no-blow case Lamb’s (Phil. Mag., vol. 21, 1911, pp. 112–121) expression is retrieved for $Re\rightarrow 0$. For the strong-sucking case, the asymptotic limit, $C_{D}\approx -2{\rm\pi}{\it\Lambda}$, is confirmed. The blowing solution is valid for ${\it\Lambda}<4/Re$, after which the flow is unsymmetrical about ${\it\theta}={\rm\pi}/2$. The analytical results are compared with full numerical solutions for the drag coefficient $C_{D}$ and the fraction of drag due to viscous stresses. The predictions show good agreement for $Re=0.1$ and ${\it\Lambda}=-5,0,5$.
Blood flow in the choriocapillaris
- M. A. Zouache, I. Eames, P. J. Luthert
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- Journal:
- Journal of Fluid Mechanics / Volume 774 / 10 July 2015
- Published online by Cambridge University Press:
- 02 June 2015, pp. 37-66
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The choriocapillaris is a capillary bed located in a thin layer adjacent to the outer retina and is part of the oxygen delivery system to the photoreceptors of the eye. The blood flow is approximately planar and is serviced by microvessels, which join the choriocapillaris through inlets perpendicular to its plane. Capillaries are densely organised and separated by avascular septal posts, which direct the blood flow. The capillary bed is composed of a juxtaposition of tessellating vascular units called lobules, which are filled and drained independently from each other. A theoretical analysis of the blood flow in an idealised model of a lobule of the choriocapillaris is developed and studied. Lobules are modelled as tessellating polygonal prisms, where the upper and lower surfaces correspond to planar parallel membranes. The septae are modelled as cylinders randomly distributed between the two membranes. Feeding arterioles and draining venules are modelled as inlets and outlets connecting at the lower surface of the prism perpendicularly to the plane of the lobule. An inlet is placed inside the lobule, while an outlet is placed at each of the vertices. The polygonal prism can be formally subdivided into a set of triangular prisms with one inlet and two outlets, each of them located at one of the vertices. The triangular prisms are taken to be isosceles, and are therefore characterised by a vertex angle ${\it\omega}$ at the inlet and a span $L$. The flow is viscously dominated, and is investigated in the lubrication limit, in which the characteristic thickness of the prism is much smaller than the diameter of the cylinders. As a result of the geometry, a stagnation point is located midway between the outlets. A separation streamline joins the inlet and the stagnation point. The pressure drop ${\rm\Delta}\tilde{p}$ and the average fluid particle residence time $\langle \tilde{T}\rangle$ are analysed as a function of the angle at the inlet ${\it\omega}$ and the septae volume fraction ${\it\Phi}$. When no cylinders are present (${\it\Phi}=0$), an analytical expression for the pressure field is calculated by conformal mapping. Close to the triangle walls, the flow is quasi-parallel and characterised by a shorter fluid particle residence time. In the vicinity of the stagnation point, the velocity decreases and the residence time diverges logarithmically with the distance to the stagnation streamline. The minimum in pressure drop corresponds to a maximum in residence time, and is obtained for ${\it\omega}={\rm\pi}/2$. Asymptotic expressions for the pressure drop and average residence time are formulated in both the limits $\Vert {\it\omega}\Vert \ll 1$ and $\Vert {\rm\pi}-{\it\omega}\Vert \ll 1$. The impact of ${\it\Phi}$ on the flow is characterised by solving the equations for the flow numerically and using the Darwin drift framework. We show that the pressure drop is approximately proportional to $1+2{\it\Phi}$ for relatively small ${\it\Phi}$, and that $\langle \tilde{T}\rangle$ is proportional to $1-{\it\Phi}$ regardless of the void fraction or shape of the septae. In the case ${\it\Phi}=0$, the average residence time equals the volume of the domain divided by the volumetric flux. This analysis provides a new perspective on the blood flow dynamics within the choriocapillaris. Lobules form systems, where perfusion and corpuscle transport are a function of the angle that any two venular openings make with an arteriolar opening, the surface area perfused and the void volume fraction. The blood flow velocity and residence time are significantly heterogeneous, which may be responsible for the high degree of selective localisation observed in the pathogenesis of some inflammatory and degenerative diseases of the eye.
Force acting on a square cylinder fixed in a free-surface channel flow
- Z. X. Qi, I. Eames, E. R. Johnson
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- Journal:
- Journal of Fluid Mechanics / Volume 756 / 10 October 2014
- Published online by Cambridge University Press:
- 04 September 2014, pp. 716-727
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We describe an experimental study of the forces acting on a square cylinder (of width $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}b$) which occupies 10–40 % of a channel (of width $w$), fixed in a free-surface channel flow. The force experienced by the obstacle depends critically on the Froude number upstream of the obstacle, ${\mathit{Fr}}_1$ (depth $h_1$), which sets the downstream Froude number, ${\mathit{Fr}}_2$ (depth $h_2$). When ${\mathit{Fr}}_1<{\mathit{Fr}}_{1c}$, where ${\mathit{Fr}}_{1c}$ is a critical Froude number, the flow is subcritical upstream and downstream of the obstacle. The drag effect tends to decrease or increase the water depth downstream or upstream of the obstacle, respectively. The force is form drag caused by an attached wake and scales as $\overline{F_{D}}\simeq C_D \rho b u_1^2 h_1/2$, where $C_D$ is a drag coefficient and $u_1$ is the upstream flow speed. The empirically determined drag coefficient is strongly influenced by blocking, and its variation follows the trend $C_D=C_{D0}(1+C_{D0}b/2w)^2$, where $C_{D0}=1.9$ corresponds to the drag coefficient of a square cylinder in an unblocked turbulent flow. The r.m.s. lift force is approximately 10–40 % of the mean drag force and is generated by vortex shedding from the obstacle. When ${\mathit{Fr}}_1={\mathit{Fr}}_{1c}\, (<1)$, the flow is choked and adjusts by generating a hydraulic jump downstream of the obstacle. The drag force scales as $\overline{F}_D\simeq C_K \rho b g (h_1^2-h_2^2)/2$, where experimentally we find $C_K\simeq 1$. The r.m.s. lift force is significantly smaller than the mean drag force. A consistent model is developed to explain the transitional behaviour by using a semi-empirical form of the drag force that combines form and hydrostatic components. The mean drag force scales as $\overline{F_{D}}\simeq \lambda \rho b g^{1/3} u_1^{4/3} h_1^{4/3}$, where $\lambda $ is a function of $b/w$ and ${\mathit{Fr}}_1$. For a choked flow, $\lambda =\lambda _c$ is a function of blocking ($b/w$). For small blocking fractions, $\lambda _c= C_{D0}/2$. In the choked flow regime, the largest contribution to the total drag force comes from the form-drag component.
Turbulent acidic jets and plumes injected into an alkaline environment
- H. Ülpre, I. Eames, A. Greig
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- Journal:
- Journal of Fluid Mechanics / Volume 734 / 10 November 2013
- Published online by Cambridge University Press:
- 08 October 2013, pp. 253-274
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The characteristics of an acidic turbulent jet and plume injected into an alkaline environment are examined theoretically and experimentally. Fluid-flow and chemistry models are combined to understand how the concentration of acid in a parcel of fluid changes as it reacts with alkaline fluid entrained from the ambient. The resulting model is tested in an experimental study in which nitric acid jets or plumes are injected into a large tank containing a variety of alkaline substances. A video camera records a pH-sensitive dye in the jet or plume, which changes colour with variations in the pH. The results were time averaged and processed to measure distance from the source to the point of neutralization. The agreement between predictions and observations of neutralization distances is good, confirming that the model captures the salient physics of the problem. Using empirically determined titration curves, a combined fluid flow and chemistry model is applied to discuss the environmental implications of a warm acidic turbulent plume injected into an alkaline river or sea.
Impulsively started planar actuator surfaces in high-Reynolds-number steady flow
- P. B. Johnson, A. Wojcik, K. R. Drake, I. Eames
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- Journal:
- Journal of Fluid Mechanics / Volume 733 / 25 October 2013
- Published online by Cambridge University Press:
- 23 September 2013, pp. 302-324
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The characteristics of unbounded flow past an impulsively started planar energy extracting device, such as a wind or tidal turbine, are studied theoretically, numerically and experimentally. The initial thrust on an impulsively started device, which can be more than double the steady thrust, is an important consideration for design and safe operation. The energy sink is modelled here as an ‘actuator surface’ which imposes a uniform pressure discontinuity in the fluid proportional to the square of the fluid speed normal to the surface, the fluid density, and a dimensionless resistance coefficient. The flow past the actuator is studied theoretically for the case of weak resistance using an unsteady model which recovers steady linear momentum theory in the limit of long time. For the case of strong resistance the flow is studied numerically using the point vortex method. Experimental measurements of thrust on a mesh towed through static water are compared to the numerical results and show good agreement. The thrust on an impulsively started device is estimated, for a typical installation, to fall to within 10 % of the steady value within ∼1 min. The numerical model is also used to simulate the gradual startup of a device, yielding estimates of the time constant necessary in a control system in order to reduce peak thrusts in practice.
Dense gravity currents moving beneath progressive free-surface water waves
- T. O. Robinson, I. Eames, R. Simons
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- Journal:
- Journal of Fluid Mechanics / Volume 725 / 25 June 2013
- Published online by Cambridge University Press:
- 23 May 2013, pp. 588-610
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The characteristics of dense gravity currents in coastal regions, where free-surface gravity waves are dominant, have yet to be studied in the laboratory. This paper provides a first insight into the dynamics of dense saline gravity currents moving beneath regular progressive free-surface water waves. The gravity currents were generated by releasing a finite volume of saline into a large wave tank with an established periodic wave field. After the initial collapse, the gravity currents propagated horizontally with two fronts, one propagating in the wave direction and the other against the wave direction. The fronts of the gravity currents oscillated with an amplitude and phase that correlated with the orbital velocities within a region close to the bed. To leading order, the overall length of the gravity current was found to be weakly affected by the wave action and the dynamics of the current could be approximated by simply considering the buoyancy of the released fluid. Other characteristics such as the position of the gravity current centre and the shape of the two leading profiles were found to be significantly affected by the wave action. The centre was displaced at constant speed dependent on the second-order wave-induced mean Lagrangian velocity. For long waves, the centre was advected downstream in the direction of wave propagation owing to the dominance of Stokes drift. For short waves, the gravity current centre moved upstream against the wave direction, as under these wave conditions Stokes drift is negligible at the bed. An asymmetry in the shape of the upstream and downstream current heads was observed, with the gravity current front moving against the waves being much thicker and the front steeper, similar to the case of a current moving in a stream.
Numerical study of flow through and around a circular array of cylinders
- A. NICOLLE, I. EAMES
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- Journal of Fluid Mechanics / Volume 679 / 25 July 2011
- Published online by Cambridge University Press:
- 27 May 2011, pp. 1-31
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This paper describes a study of the local and global effect of an isolated group of cylinders on an incident uniform flow. Using high resolution two-dimensional computations, we analysed the flow through and around a localised circular array of cylinders, where the ratio of array diameter (DG) to cylinder diameter (D) is 21. The number of cylinders varied from NC = 7 to 133, and they were arranged in a series of concentric rings to allow even distribution within the array with an average void fraction φ = NC(D/DG)2, which varied from 0.016 to 0.30. The characteristic Reynolds number of the array was ReG = 2100. A range of diagnostic tools were applied, including the lift/drag forces on each cylinder (and the whole array), Eulerian and Lagrangian average velocity within the array, and the decay of maximum vorticity with distance downstream. To interpret the flow field, we used vorticity and the dimensionless form of the second invariant of the velocity gradient tensor. A mathematical model, based on representing the bodies as point forces, sources and dipoles, was applied to interpret the results. Three distinct flow regimes were identified. For low void fractions (φ < 0.05), the cylinders have uncoupled individual wake patterns, where the vorticity is rapidly annihilated by wake intermingling downstream and the forces are similar to that of an isolated cylinder. At intermediate void fractions (0.05 < φ < 0.15), a shear layer is generated at the shoulders of the array and the force acting on the cylinders is steady. For high void fractions (φ > 0.15), the array generates a wake in a similar way to a solid body of the same scale. For low void fraction arrays, the mathematical model provides a reasonable assessment of the forces on individual bodies within the array, the Eulerian mean velocity and the upstream velocity field. While it broadly captures the change in the rate of decay of the maximum vorticity magnitude Ωmax downstream, the magnitude is underpredicted.
Inviscid coupling between point symmetric bodies and singular distributions of vorticity
- I. EAMES, M. LANDERYOU, J. B. FLÓR
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- Journal:
- Journal of Fluid Mechanics / Volume 589 / 25 October 2007
- Published online by Cambridge University Press:
- 08 October 2007, pp. 33-56
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We study the inviscid coupled motion of a rigid body (of density ρb, in a fluid of density ρ) and singular distributions of vorticity in the absence of gravity, using for illustration a cylinder moving near a point vortex or dipolar vortex, and the axisymmetric interaction between a vortex ring and sphere.
The coupled motion of a cylinder (radius a) and a point vortex, initially separated by a distance R and with zero total momentum, is governed by the parameter R4/(ρb/ρ+1)a4. When R4/(ρb/ρ+1)a4,≤,1, a (positive) point vortex moves anticlockwise around the cylinder which executes an oscillatory clockwise motion, with a mixture of two frequencies, centred around its initial position. When R4/(ρb/ρ+1)a4≫1, the initial velocity of the cylinder is sufficiently large that the dynamics become uncoupled, with the cylinder moving off to infinity. The final velocity of the cylinder is related to the permanent displacement of the point vortex.
The interaction between a cylinder (initially at rest) and a dipolar vortex starting at infinity depends on the distance of the vortex from the centreline (h), the initial separation of the vortical elements (2d), and ρb/ρ. For a symmetric encounter (h=0) with a dense cylinder, the vortical elements pass around the cylinder and move off to infinity, with the cylinder being displaced a finite distance forward. However, when ρb/ρ<1, the cylinder is accelerated forward to such an extent that the vortex cannot overtake. Instead, the cylinder ‘extracts’ a proportion of the impulse from the dipolar vortex. An asymmetric interaction (h>0) leads to the cylinder moving off in the opposite direction to the dipolar vortex.
To illustrate the difference between two- and three-dimensional flows, we consider the axisymmetric interaction between a vortex ring and a rigid sphere. The velocity perturbation decays so rapidly with distance that the interaction between the sphere and vortex ring is localized, but the underlying processes are similar to two-dimensional flows.
We briefly discuss the general implications of these results for turbulent multiphase flows.
Drift, partial drift and Darwin's proposition
- I. Eames, S. E. Belcher, J. C. R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 275 / 25 September 1994
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- 26 April 2006, pp. 201-223
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A body moves at uniform speed in an unbounded inviscid fluid. Initially, the body is infinitely far upstream of an infinite plane of marked fluid; later, the body moves through and distorts the plane and, finally, the body is infinitely far downstream of the marked plane. Darwin (1953) suggested that the volume between the initial and final positions of the surface of marked fluid (the drift volume) is equal to the volume of fluid associated with the ‘added-mass’ of the body.
We re-examine Darwin's (1953) concept of drift and, as an illustration, we study flow around a sphere. Two lengthscales are introduced: ρmax, the radius of a circular plane of marked particles; and x0, the initial separation of the sphere and plane. Numerical solutions and asymptotic expansions are derived for the horizontal Lagrangian displacement of fluid elements. These calculations show that depending on its initial position, the Lagrangian displacement of a fluid element can be either positive – a Lagrangian drift – or negative – a Lagrangian reflux. By contrast, previous investigators have found only a positive horizontal Lagrangian displacement, because they only considered the case of infinite x0. For finite x0, the volume between the initial and final positions of the plane of marked fluid is defined to be the ‘partial drift volume’, which is calculated using a combination of the numerical solutions and the asymptotic expansions. Our analysis shows that in the limit corresponding to Darwin's study, namely that both x0 and ρmax become infinite, the partial drift volume is not well-defined: the ordering of the limit processes is important. This explains the difficulties Darwin and others noted in trying to prove his proposition as a mathematical theorem and indicates practical, as well as theoretical, criteria that must be satisfied for Darwin's result to hold.
We generalize our results for a sphere by re-considering the general expressions for Lagrangian displacement and partial drift volume. It is shown that there are two contributions to the partial drift volume. The first contribution arises from a reflux of fluid and is related to the momentum of the flow; this part is spread over a large area. It is well-known that evaluating the momentum of an unbounded fluid is problematic since the integrals do not converge; it is this first term which prevented Darwin from proving his proposition as a theorem. The second contribution to the partial drift volume is related to the kinetic energy of the flow caused by the body: this part is Darwin's concept of drift and is localized near the centreline. Expressions for partial drift volume are generalized for flow around arbitrary-shaped two- and three-dimensional bodies. The partial drift volume is shown to depend on the solid angles the body subtends with the initial and final positions of the plane of marked fluid. This result explains why the proof of Darwin's proposition depends on the ratio ρmax/x0.
An example of drift due to a sphere travelling at the centre of a square channel is used to illustrate the differences between drift in bounded and unbounded flows.
Displacement of inviscid fluid by a sphere moving away from a wall
- I. Eames, J. C. R. Hunt, S. E. Belcher
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- Journal:
- Journal of Fluid Mechanics / Volume 324 / 10 October 1996
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- 26 April 2006, pp. 333-353
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We develop a theoretical analysis of the displacement of inviscid fluid particles and material surfaces caused by the unsteady flow around a solid body that is moving away from a wall. The body starts at position hs from the wall, and the material surface is initially parallel to the wall and at distance hL from it. A volume of fluid Df+ is displaced away from the wall and a volume Df- towards the wall. Df+ and Df- are found to be sensitive to the ratio hL/hs. The results of our specific calculations for a sphere can be extended in general to other shapes of bodies.
When the sphere moves perpendicular to the wall the fluid displacement and drift volume Df+ are calculated numerically by computing the flow around the sphere. These numerical results are compared with analytical expressions calculated by approximating the flow around the sphere as a dipole moving away from the wall. The two methods agree well because displacement is an integrated effect of the fluid flow and the largest contribution to displacement is produced when the sphere is more than two radii away from the wall, i.e. when the dipole approximation adequately describes the flow. Analytic expressions for fluid displacement are used to calculate Df+ when the sphere moves at an acute angle α away from the wall.
In general the presence of the wall reduces the volume displaced forward and this effect is still significant when the sphere starts 100 radii from the wall. A sphere travelling perpendicular to the wall, α = 0, displaces forward a volume Df+(0) = 4πa3hL/33/2hS when the marked surface starts downstream, or behind the sphere, and displaces a volume Df+(0) ∼ 2πa3/3 forward when it is marked upstream or in front of the body. A sphere travelling at an acute angle away from the wall displaces a volume Df+(α) ∼ Df+(0) cos α forward when the surface starts downstream of the sphere. When the marked surface is initially upstream of the sphere, there are two separate regions displaced forward and a simple cosine dependence on α is not found.
These results can all be generalized to calculate material surfaces when the sphere moves at variable speed, displacements no longer being expressed in terms of time, but in relation to the distance travelled by the sphere.
Mechanics of inhomogeneous turbulence and interfacial layers
- J. C. R. HUNT, I. EAMES, J. WESTERWEEL
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- Journal:
- Journal of Fluid Mechanics / Volume 554 / 10 May 2006
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- 24 April 2006, pp. 499-519
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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.
Infiltration into inclined fibrous sheets
- M. LANDERYOU, I. EAMES, A. COTTENDEN
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- Journal:
- Journal of Fluid Mechanics / Volume 529 / 25 April 2005
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- 01 April 2005, pp. 173-193
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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 effect of a percolation threshold is observed to ultimately arrest cross-slope transport, placing a limitation on the long-time analysis.
The effect of an ambient flow on the spreading of a viscous gravity current
- I. EAMES, M. A. GILBERTSON, M. LANDERYOU
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- Journal of Fluid Mechanics / Volume 523 / 25 January 2005
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- 21 January 2005, pp. 261-275
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The influence of an external laminar flow on the spreading of a viscous gravity current moving over a horizontal floor is studied theoretically and experimentally. The viscous stress exerted by the ambient flow drives the viscous gravity current streamwise with a velocity proportional to the local height of the current. The one-way coupling between the ambient flow and the spread of the current is examined. Similarity and numerical solutions are developed to describe viscous gravity currents spreading from line and point sources. An experimental study of the spreading of viscous gravity currents issuing from a point source in a channel flow, for both constant-flux and instantaneous releases, confirms the essential character of this description.
Inviscid mean flow through and around groups of bodies
- I. EAMES, J. C. R. HUNT, S. E. BELCHER
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- Journal:
- Journal of Fluid Mechanics / Volume 515 / 25 September 2004
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- 09 September 2004, pp. 371-389
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General estimates are derived for mean velocities through and around groups or arrays of fixed and moving bodies, in unbounded and bounded domains, which lie within a defined perimeter. Robust kinematic flow concepts are introduced, namely the Eulerian spatial mean velocity $\overline{u}_E$ in the fluid volume between the bodies, the Eulerian flow outside the group, ${\bm u}_E^{(0)}$, and the Lagrangian mean velocity of material surfaces or fluid particles as they pass through the group of bodies ($\overline{u}_L^{(S)}$, $\overline{u}_L^{(P)}$). The Eulerian mean velocity is related to the momentum in the fluid domain, and is mainly influenced by fast moving regions of the flow. The Lagrangian mean velocity weights slowly moving regions of flow and is related to how material sheets deform as they are advected through groups of bodies. When the bodies are well-separated, the interstitial Eulerian and Lagrangian mean velocities ($\overline{u}_E^{(I)}$, $\overline{u}_L^{(I)}$), are defined and calculated in terms of the far-field contributions from the velocity or displacement field within the group of bodies.
In unbounded flow past well-separated bodies situated within a rectangular perimeter, the difference between the Eulerian and Lagrangian mean velocity is negligible (as the void fraction of the bodies, $\alpha\,{\rightarrow}\,0$). Within wide and short rectangular arrays, the Eulerian mean velocity is faster than the free-stream velocity $U$ because most of the incident flow passes through the array and $\overline{u}_E\,{=}\,U(1-\alpha)^{-1}$. Within long and thin rectangular arrays (and other cases where the reflux velocity is negligible), the Eulerian mean velocity, $\overline{u}_E\,{=}\,U(1-(1+C_m)\alpha)/(1-\alpha)$, is slower than the free-stream velocity, because most of the incident flow passes around the array. For a spherical or circular arrays of bodies, the particle Lagrangian mean velocity is $\overline{u}_L^{(P)}\,{=}\,U(1+C_m\alpha)^{-1}$ and differs from $\overline{u}_E$. These calculations are extended to examine the mean and interstitial flow through clouds of bodies in bounded channel flows.
The new concepts are applied to calculate the mean flow and pressure between and outside clouds of bodies, the average velocity of bubbly flows as a function of void fraction, and the tendency of clouds of bubbles to be distorted depending on their shape.
Forces on bodies moving unsteadily in rapidly compressed flows
- I. EAMES, J. C. R. HUNT
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- Journal:
- Journal of Fluid Mechanics / Volume 505 / 25 April 2004
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- 21 April 2004, pp. 349-364
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The inviscid compressible flow generated by a rigid body of volume ${\cal V}$ moving unsteadily with a velocity ${\bm U}$ in a rapidly compressed homentropic flow is considered. The fluid is compressed isentropically at a rate ${\bm \nabla}\,{\bm \cdot}\,{\bm v}_0$ uniformly over a scale much larger than the size of the body and the body moves slowly enough that the Mach number $M$ is low. The flow is initially irrotational and remains so during compression. The perturbation to the flow generated by the body moving unsteadily is non-divergent within an evolving region ${\cal D}$ of distance $\int_0^t c_1\,{\rm d}t$ from the body, where $c_1$ is the speed of sound. Within ${\cal D}$, the flow is dominated by a source of strength $({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0){\cal V}$ and a dipolar contribution which is independent of the rate of compression, while outside ${\cal D}$, compressional waves propagate away from the body. When the body is much smaller than the characteristic distance $\|({\bm \nabla}{\bm v}_0)|_{{\bm x}_0}\|/\|({\bm \nabla} {\bm \nabla} {\bm v}_0 )|_{{\bm x}_0}\|$ and the size of the region ${\cal D}$, the separation of length scales enables the force on the body to be calculated analytically from the momentum flux far from the body (but within the region ${\cal D}$). The contribution to the total force arising from fluid compression is $\rho(t) ({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0) {\cal V} ({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha} $, where ${\bm v}_0$ is the velocity field in the absence of the particles and $\boldsymbol{\alpha}$ is the virtual inertia tensor. Thus a body experiences a drag (thrust) force during fluid compression (expansion) because the density of the fluid displaced forward by the body increases (decreases) with time. The analysis indicates that the sum of the compressional and added-mass force is equal to the rate of decrease of fluid impulse ${\bm P} = \rho(t){\cal V}({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha}$. Thus the concept of fluid impulse naturally extends to the class of flows where the fluid density changes with time, but is spatially uniform.
These new results are applied to consider the inviscid dynamics of a rigid sphere and cylinder projected into a uniformly compressed or expanded fluid. When the fluid rapidly expands, a rigid body ultimately moves with a constant velocity because the total force, which is proportional to the density of the fluid, tends rapidly to zero. When the body moves perpendicular to the axis of compression, it slows down and stops when the density of the fluid is comparable to the density of the body. However, a body moving parallel to the axis of compression is accelerated by pressure gradients which are proportional to fluid density and increases in time.