The disappearance of laminar and turbulent wakes in complex flows

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 disappearance of viscous and laminar wakes in complex flows

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

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

Quantitative LEED I-V and ab initio study of the Si(111)-3x2-Sm surface structure and the missing half order spots in the 3x1 diffraction pattern

We have used Low Energy Electron Diffraction (LEED) I-V analysis and ab initio calculations to quantitatively determine the honeycomb chain model structure for the Si(111)-3x2-Sm surface. This structure and a similar 3x1 recontruction have been observed for many Alkali-Earth and Rare-Earth metals on the Si(111) surface. Our ab initio calculations show that there are two almost degenerate sites for the Sm atom in the unit cell and the LEED I-V analysis reveals that an admixture of the two in a ratio that slightly favours the site with the lower energy is the best match to experiment. We show that the I-V curves are insensitive to the presence of the Sm atom and that this results in a very low intensity for the half order spots which might explain the appearance of a 3x1 LEED pattern produced by all of the structures with a 3x2 unit cell.Comment: 10 pages, 13 figures. Preliminary work presented at the the APS March meeting, Baltimore MD, 2006. To be published in Phys. Rev. B. April/May 200

Displacement of inviscid fluid by a sphere moving away from a wall

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) [similar] 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) [similar] 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

Low-Reynolds-number flow past a cylinder with uniform blowing or sucking

We analyse the low-Reynolds-number flow generated by a cylinder (of radius a) in a stream (of velocity U∞) which has a uniform through-surface blowing component (of velocity Ub). The flow is characterized in terms of the Reynolds number Re (=2aU∞/ν, where ν is the kinematic viscosity of the fluid) and the dimensionless blow velocity Λ (=Ub/U∞). 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→0. For the strong-sucking case, the asymptotic limit, CD≈−2πΛ, is confirmed. The blowing solution is valid for Λ<4/Re, after which the flow is unsymmetrical about θ=π/2. The analytical results are compared with full numerical solutions for the drag coefficient CD and the fraction of drag due to viscous stresses. The predictions show good agreement for Re=0.1 and Λ=−5,0,5

Conformally mapped viscous gravity current

We show how the conformal mapping technique can be applied to analyse specific problems in the context of viscous gravity current theory. We examine the edge of steady thin planar viscous gravity currents in the presence of complex external low Reynolds flows. In addition to the uniform ambient flow we look at the case of viscous gravity currents spreading in positively strained flows and around cylindrical bodies. These external flows exert shear stress on the gravity current, which drives it in the streamwise direction. The idealised conditions are re-created in the laboratory using a Hele–Shaw cell with a point source on the bottom plate where the saline is introduced into the flow. The mapped laboratory results are compared to a known similarity solution and the agreement is good. We conclude by identifying a broad class of viscous gravity current problems where this technique may be applied

Environmental policy constraints for acidic exhaust gas scrubber discharges from ships.

Increasingly stringent environmental legislation on sulphur oxide emissions from the combustion of fossil fuels onboard ships (International Maritime Organization (IMO) Regulation 14) can be met by either refining the fuel to reduce sulphur content or by scrubbing the exhaust gases. Commonly used open loop marine scrubbers discharge warm acidic exhaust gas wash water into the sea, depressing its pH. The focus on this paper is on the physics and chemistry behind the disposal of acidic discharges in seawater. The IMO Marine Environment Protection Committee (MEPC 59/24/Add.1 Annex 9) requires the wash water to reach a pH greater than 6.5 at a distance of 4m from the point of discharge. We examine the engineering constraints, specifically size and number of ports, to identify the challenges of meeting regulatory compliance

Numerical study of flow through and around a circular array of cylinders

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 (D-G) to cylinder diameter (D) is 21. The number of cylinders varied from N-C = 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 phi = N-C(D/DG)(2), which varied from 0.016 to 0.30. The characteristic Reynolds number of the array was Re-G = 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 (phi 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 Omega(max) downstream, the magnitude is underpredicted