12 results
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.
Vultures in Cambodia: population, threats and conservation
- TOM CLEMENTS, MARTIN GILBERT, HUGO J. RAINEY, RICHARD CUTHBERT, JONATHAN C. EAMES, PECH BUNNAT, SENG TEAK, SONG CHANSOCHEAT, TAN SETHA
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- Journal:
- Bird Conservation International / Volume 23 / Issue 1 / March 2013
- Published online by Cambridge University Press:
- 25 April 2012, pp. 7-24
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Asian vultures have undergone dramatic declines of 90–99% in the Indian Subcontinent, as a consequence of poisoning by veterinary use of the drug diclofenac, and are at a high risk of extinction. Cambodia supports one of the only populations of three species (White-rumped Vulture Gyps bengalensis, Slender-billed Vulture G. tenuirostris and Red-headed Vulture Sarcogyps calvus) outside of South Asia where diclofenac use is not widespread. Conservation of the Cambodian sub-populations is therefore a global priority. This study analyses the results of a long-term research programme into Cambodian vultures that was initiated in 2004. Population sizes of each species are estimated at 50–200+ individuals, ranging across an area of approximately 300 km by 250 km, including adjacent areas in Laos and Vietnam. The principal causes of vulture mortality were poisoning (73%), probably as an accidental consequence of local hunting and fishing practices, and hunting or capture for traditional medicine (15%). This represents a significant loss from such a small population of long-lived, slow breeding, species such as vultures. Cambodian vultures are severely food limited and are primarily dependent on domestic ungulate carcasses, as wild ungulate populations have been severely depleted over the past 20 years. Local people across the vulture range still follow traditional animal husbandry practices, including releasing livestock into the open deciduous dipterocarp forest areas when they are not needed for work, providing the food source. Reducing threats through limiting the use of poisons (which are also harmful for human health) and supplementary food provisioning in the short to medium-term through ‘vulture restaurants’ is critical if Cambodian vultures are to be conserved.
Rapid assessment of influenza vaccine effectiveness: analysis of an internet-based cohort
- K. T. D. EAMES, E. BROOKS-POLLOCK, D. PAOLOTTI, M. PEROSA, C. GIOANNINI, W. J. EDMUNDS
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- Journal:
- Epidemiology & Infection / Volume 140 / Issue 7 / July 2012
- Published online by Cambridge University Press:
- 12 September 2011, pp. 1309-1315
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The effectiveness of influenza vaccination programmes is seldom known during an epidemic. We developed an internet-based system to record influenza-like symptoms and response to infection in a participating cohort. Using self-reports of influenza-like symptoms and of influenza vaccine history and uptake, we estimated vaccine effectiveness (VE) without the need for individuals to seek healthcare. We found that vaccination with the 2010 seasonal influenza vaccine was significantly protective against influenza-like illness (ILI) during the 2010–2011 influenza season (VE 52%, 95% CI 27–68). VE for individuals who received both the 2010 seasonal and 2009 pandemic influenza vaccines was 59% (95% CI 27–77), slightly higher than VE for those vaccinated in 2010 alone (VE 46%, 95% CI 9–68). Vaccinated individuals with ILI reported taking less time off work than unvaccinated individuals with ILI (3·4 days vs. 5·3 days, P<0·001).
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
- Published online by Cambridge University Press:
- 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
- Published online by Cambridge University Press:
- 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 of Fluid Mechanics / Volume 554 / 10 May 2006
- Published online by Cambridge University Press:
- 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.
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
- Published online by Cambridge University Press:
- 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 of Fluid Mechanics / Volume 505 / 25 April 2004
- Published online by Cambridge University Press:
- 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.
The disappearance of laminar and turbulent wakes in complex flows
- J. C. R. HUNT, I. EAMES
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- Journal of Fluid Mechanics / Volume 457 / 25 April 2002
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- 09 April 2002, pp. 111-132
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The singular effects of steady large-scale external strain on the viscous wake generated by a rigid body and the overall flow field are analysed. In an accelerating flow strained at a positive rate, the vorticity field is annihilated owing to positive and negative vorticity either side of the wake centreline diffusing into one another and the volume flux in the wake decreases with downwind distance. Since the wake disappears, the far-field flow changes from monopolar to dipolar. In this case, the force on the body is no longer proportional to the strength of the monopole, but is proportional to the strength of the far field dipole. These results are extended to the case of strained turbulent wakes and this is verified against experimental wind tunnel measurements of Keffer (1965) and Elliott & Townsend (1981) for positive and negative strains. The analysis demonstrates why the total force acting on a body may be estimated by adding the viscous drag and inviscid force due to the irrotational straining field.
Applying the analysis to the wake region of a rigid body or a bubble shows that the wake volume flux decreases even in uniform flows owing to the local straining flow in the near-wake region. While the wake volume flux decreases by a small amount for the flow over streamline and bluff bodies, for the case of a clean bubble the decrease is so large as to render Betz's (1925) drag formula invalid.
To show how these results may be applied to complex flows, the effects of a sequence of positive and negative strains on the wake are considered. The average wake width is much larger than in the absence of a strain field and this leads to diffusion of vorticity between wakes and the cancellation of vorticity. The latter mechanism leads to a net reduction in the volume flux deficit downstream which explains why in calculations of the flow through groups of moving or stationary bodies the wakes of upstream bodies may be ignored even though their drag and lift forces have a significant effect on the overall flow field.
The importance of Vu Quang Nature Reserve, Vietnam, for bird conservation, in the context of the Annamese Lowlands Endemic Bird Area
- J. C. Eames, R. Eve, A. W. Tordoff
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- Journal:
- Bird Conservation International / Volume 11 / Issue 4 / December 2001
- Published online by Cambridge University Press:
- 15 February 2002, pp. 247-285
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Vu Quang Nature Reserve, Vietnam, was brought to the attention of the world scientific community following the discovery of two previously undescribed large mammal species in the early 1990s. In light of the identification of other sites of high biodiversity value in the Annamite mountains of Vietnam and Laos, the relative importance for biodiversity conservation of Vu Quang needs to be reassessed. In this paper we evaluate the importance of the site for bird conservation, in relation to 13 other protected areas in the Annamese Lowlands Endemic Bird Area (EBA) and present species lists for all 14 sites. Whilst Vu Quang supports one of the highest numbers of recorded bird species of all 14 protected areas, a complementarity analysis revealed that Vu Quang does not fall within the critical subset of sites necessary to conserve 95% of the avifaunal diversity of the EBA. The site should not, therefore, be considered a regional bird conservation priority. Furthermore, of the nine restricted-range species known from the Annamese Lowlands EBA, only three are known from Vu Quang, which is not, therefore, a priority site for the conservation of endemic bird species. We also evaluate the conservation status of the avifauna of Vu Quang, and propose potential conservation measures to enhance its importance for bird conservation.
Inviscid flow around bodies moving in weak density gradients without buoyancy effects
- I. EAMES, J. C. R. HUNT
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- Journal:
- Journal of Fluid Mechanics / Volume 353 / 25 December 1997
- Published online by Cambridge University Press:
- 25 December 1997, pp. 331-355
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We examine the inviscid flow generated around a body moving impulsively from rest with a constant velocity U in a constant density gradient, ∇ρ0, which is assumed to be weak in the sense ε=a[mid ]∇ρ0[mid ] /ρ0[Lt ]1, where a is the length scale of the body. In the absence of a density gradient (ε=0), the flow is irrotational and no force acts on the body. When 0<ε[Lt ]1, vorticity is generated by a baroclinic torque and vortex stretching, which introduce a rotational component into the flow. The aim is to calculate both the flow around the body and the force acting on it.
When a two-dimensional body moves perpendicularly to the density gradient U·∇ρ0=0, the density and velocity field are both steady in the body's frame of reference and the vorticity field decays with distance from the body. When a three-dimensional body moves perpendicularly to the density gradient, the vorticity field is regular in the main flow region, [Dscr ]M, but is singular in a thin inner region [Dscr ]I located adjacent to the body and to the downstream-attached streamline, and the flow is characterized by trailing horseshoe vortices. When the body moves parallel to the density gradient U×∇ρ0=0, the density field is unsteady in the body's frame of reference; however to leading order the flow is steady in the region [Dscr ]M moving with the body for Ut/a[Gt ]1. In the thin region [Dscr ]I of thickness O(aε), the density gradient and vorticity are singular. When U×∇ρ0=0 this singularity leads to a downstream ‘jet’ with velocities of O(−(U·∇ρ0) Ua/(ρ0U)) on the downstream attached streamline(s). In the far field the flow is characterized by a sink of strength CM[Vscr ] (U·∇ρ0) /2ρ0, located at the origin, where CM is the added-mass coefficient of the body and [Vscr ] is the body's volume.
The forces acting on a body moving steadily in a weak density gradient are calculated by considering the steady relative velocity field in region [Dscr ]M and evaluating the momentum flux far from the body. When U·∇ρ0=0, a lift force, CL[Vscr ] (U·∇ρ0)×U, pushes the body towards the denser fluid, where the lift coefficient is CL=CM/2 for a three-dimensional body, that is axisymmetric about U, and is CL=(CM+1)/2 for a two-dimensional body. The direction of the lift force is unchanged when U is reversed. A general expression for the forces on bodies moving in a weak shear and perpendicularly to a density gradient is calculated. When U×∇ρ0=0, a drag force −CD[Vscr ] (U·∇ρ0)U retards the body as it moves into denser fluid, where the drag coefficient is CD=CM/2, for both two- and three-dimensional axisymmetric bodies. The direction of the drag force changes sign when U is reversed. There are two contributions to the drag calculation from the far field; the first is from the wake ‘jet’ on the attached streamline(s) caused by the rotational component of the flow and this leads to an accelerating force. The second and larger contribution arises from a downstream density variation, caused by the distortion of the isopycnal surfaces by the primary irrotational flow, and this leads to a drag force.
When cylinders or spheres move with a velocity U at arbitrary orientation to the density gradient, it is shown that they are acted on by a linear combination of lift and drag forces. Calculations of their trajectories show that they initially slow down or accelerate on a length scale of order ρ0/[mid ]∇ρ0[mid ] (independent of [Vscr ] and U) as they move into regions of increasing or decreasing density, but in general they turn and ultimately move parallel to the density gradient in the direction of increasing density gradient.