Abstract
Buoyancydriven flow, which is flow driven by spatial variations in fluid density^{1}, lies at the heart of a variety of physical processes, including mineral transport in rocks^{2}, the melting of icebergs^{3} and the migration of tectonic plates^{4}. Here we show that buoyancydriven flows can also generate propulsion. Specifically, we find that when a neutrally buoyant wedgeshaped object floats in a densitystratified fluid, the diffusiondriven flow at its sloping boundaries generated by molecular diffusion produces a macroscopic sideways thrust. Computer simulations reveal that thrust results from diffusiondriven flow creating a region of low pressure at the front, relative to the rear of an object. This discovery has implications for transport processes in regions of varying fluid density, such as marine snow aggregation at ocean pycnoclines^{5}, and wherever there is a temperature difference between immersed objects and the surrounding fluid, such as particles in volcanic clouds^{6}.
Main
A fluid system with spatially varying density, resulting from temperature and/or salinity variations, for example, is stablystratified when increasing density is parallel to the direction of gravity. When an object is released in a quiescent, stablystratified fluid, it is expected to settle or rise to the neutral buoyancy level at which the density of the object matches that of the surrounding fluid, and remain stationary thereafter. We carried out a control experiment in which a 19.05mmdiameter sphere of density ρ=1,115 kg m^{−3} was released in a tank of height H=0.40 m, width W=0.20 m and length L=0.40 m, filled with saltstratified water with density gradient dρ/dz=−511±3 kg m^{−4}. As expected, the sphere settled to its neutral buoyancy height, where it remained stationary for 24 hours. A similar experiment was then carried out using a triangular wedge (Fig. 1) of length l=99.9±0.1 mm, base h=17.6±0.10 mm (corresponding to slope angle α=5.0±0.1^{°}) and width w=25.1±0.10 mm. In striking contrast to the stationary sphere, the wedge moved at a constant speed u=10.2±0.1×10^{−3} m h^{−1} (2.83±0.03×10^{−6} m s^{−1}) in the direction of its tip, without any obvious cause (Fig. 1 and its top inset; also see Supplementary Movie S1).
To investigate the cause of this spontaneous propulsion, we visualized the velocity fields in the horizontal and vertical midplanes of the moving wedge using particle image velocimetry (PIV; Fig. 2). Here, z is the vertical coordinate antiparallel to gravity and x and y are the coordinates in the horizontal plane, parallel and perpendicular to the long axis of the wedge, respectively. These experiments reveal that fluid is drawn in towards the wedge tip in the horizontal plane to supply upslope flow in a thin boundary layer above the sloping surface. Furthermore, fluid immediately behind the wedge moves with the same speed as the wedge; this phenomenon, known as blocking, occurs for obstacles in stratified flow when buoyancy forces dominate inertia and viscous forces^{1}. As a whole, these experiments reveal a localized flow field that extends ∼50 mm in front and behind, ∼30 mm above and below and ∼80 mm to either side of the moving wedge.
The upslope flow in Fig. 2 reveals the cause of propulsion to be diffusiondriven flow^{2,7}. This is a buoyancydriven flow that occurs when a stablystratified fluid encounters a sloping boundary, for the following reason. Molecular diffusion requires that surfaces of constant density (isopycnals) approach a sloping boundary at an appropriate angle to ensure continuity of diffusive flux across the boundary. For the triangular wedge that is impermeable to salt, isopycnals must be locally perpendicular to the slope to satisfy the noflux boundary condition ∂ ρ/∂ η=0, where η is the wallnormal coordinate (Fig. 1, bottom inset). Fluid adjacent to the sloping boundary therefore differs in density to fluid at the same level away from the boundary, and experiences a buoyancy force that drives flow up or downslope for underlying and overlying walls, respectively, the kinetic energy of motion being drawn from the internal energy of the fluid through diffusion.
For constant slope, and assuming alongslope flow that is a function only of the wallnormal coordinate (that is, unidirectional flow), the velocity profile for steady diffusiondriven flow is^{2,7} , where u_{0}=2κ γcotα is the characteristic tangential flow velocity, γ^{−1}=(N^{2}sin^{2}α/4ν κ)^{−1/4} is the boundarylayer thickness, is the buoyancy frequency, ρ_{0} is the characteristic fluid density, κ is the molecular diffusivity and ν is the kinematic viscosity. The characteristic scales of u_{0} and γ^{−1} for a salt stratification (κ∼10^{−9} m^{2} s^{−1},ν∼10^{−6} m^{2} s^{−1}) are 1×10^{−5} m s^{−1} and 0.1 mm, respectively, consistent with the PIV visualization (Fig. 2); for a thermal stratification (κ∼10^{−7} m^{2} s^{−1},ν∼10^{−6} m^{2} s^{−1}), these scales increase to 1×10^{−3} m s^{−1} and 0.3 mm, respectively.
Motivated by the observation of a localized flow field around the wedge, our experiments found that for L/l>2,W/w>6 and H/h>5 the propulsion speed u is independent of the size of the experimental tank, and the wedge behaves as if unconfined (see Supplementary Information). In this limit, there are eight variables on which u can depend (ρ,dρ/dz,g,ν,κ,α,l,w), and through the combination of experiment and dimensional analysis we seek to establish what sets u. Buckingham’s pitheorem^{8} requires six independent dimensionless parameters that characterize the system. Although there is freedom in selecting these, five natural choices are: the shape parameters α and w/l, the ratios of viscous and buoyancy forces to inertia forces, Re=u_{0}l/ν and Fr=u_{0}/N l, respectively referred to as the Reynolds and Froude numbers, and the ratio of molecular and viscous diffusivity, Sc=ν/κ, referred to as the Schmidt number. Choosing the dimensionless speed u/u_{0} as the final dimensionless parameter, we investigate the functional relation u/u_{0}=f(α,w/l,Re,Fr,Sc).
In the first set of experiments, for the wedge shown in Fig. 1, it was found that u∝(−dρ/dz)^{0.25±0.02} for stratifications using KCl (Sc=527), NaCl (Sc=810) and MgCl_{2} (Sc=1,592); the results are shown in Fig. 3a. Thus, u/u_{0} is a constant for each stratifying agent, because u_{0} scales similarly with the stratification. These results reveal that u/u_{0}=f(α,w/l,Sc) is independent of Re and Fr. Furthermore, using the wedge shown in Fig. 1, the second set of experiments found that within experimental uncertainty u/u_{0} is independent of Sc for 527<Sc<3,143 (Fig. 3b), and thus u/u_{0}=f(α,w/l) depends only on the wedge shape for the range of Schmidt numbers investigated. For our third set of experiments, several differentshaped wedges were used to investigate the effect of varying the aspect ratio w/l for a constant slope angle (α=5^{°}) and constant NaCl stratification (dρ/dz=−460 kg m^{−4}). It was found that the propulsion speed u increases noticeably with increasing w/l (Fig. 3c) and, consistent with the prediction of dimensional analysis, is independent of wedge size, as demonstrated by the results obtained for wedges of the same aspect ratio but different size. The final set of experiments determined that for a constant aspect ratio (w/l=0.25) and NaCl stratification (dρ/dz=−460 kg m^{−4}), the propulsion speed u drops off rapidly with increasing α (Fig. 3d), because an increasing slope angle both decreases the strength of diffusiondriven flow and increases drag by virtue of increasing the size of the wedge. Significantly, for α<5^{°} there is a diminution in the rate at which u increases with decreasing α, coinciding with the breakdown of diffusiondriven flow^{9}.
To determine the origin of thrust, we carried out a threedimensional (3D) numerical simulation of diffusiondriven flow around a fixed wedge. The simulation used essentially the same configuration as the experiments in Fig. 1, with the one exception being Sc=1 rather than Sc≈1,000, as the former is more tractable numerically. In contrast to the boundarylayer solution, u_{0}(η), which applies only to a semiinfinite domain and produces no thrust, there is an intricate flow field with characteristic flow speeds of 1×10^{−3} m s^{−1} surrounding the wedge (Fig. 4a). This flow field is similar to u_{0}(η) within the boundarylayer scale (γ^{−1}≈2 mm) over most of the top and bottom sloping surfaces, with a slight (∼25%) reduction in peak velocity. The flow field diverges noticeably from the analytical form within two millimetres of the sides, front and rear of the wedge, as well as outside the boundary layer, where fore–aftasymmetric flow structures exist to balance the rearward volume flux along the sloping surfaces^{10,11}; the same qualitative features are present in our PIV data set (Fig. 2a). Asymmetric densityfield perturbations of the order of Δρ≈0.1 kg m^{−3} are induced by the flow field^{10,11} (Fig. 4b), creating a hydrostatic pressure drop of Δp≈10 mPa on the sloping surfaces (Fig. 4c). The thrust on the fixed wedge because of this pressure drop exceeds everywhere the viscous stresses associated with the boundarylayer flow, and there is a net forward force F≈0.4 μN that is calculated from the numerical data using a control volume containing the wedge. A related mechanism has recently been proposed for the propulsion of colloidal particles by virtue of a surface chemical reaction creating a lateral osmoticpressure imbalance^{12}.
To estimate the propulsion speed resulting from the thrust, we carried out a numerical simulation that calculated the drag on a slowly moving wedge, finding that a propulsion speed u≈4.5×10^{−4} m s^{−1} produces a drag of 0.4 μN for the wedge in Fig. 1. This propulsion speed gives u/u_{0}∼0.02, which is consistent with the experimental result obtained for larger values of Sc than the numerical simulation. We note that this mechanism rationalizes why the propulsion speed is only a function of aspect ratio (and not size), because both thrust and viscous drag scale proportionally with the dimensions of the wedge. Furthermore, we recognize that propulsion occurs because of the asymmetric shape of the wedge; a symmetric object, such as a sphere, remains stationary because there is no unbalanced horizontal force produced by diffusiondriven flow.
Our discovery that a propulsive force can arise on any floating object that influences the surrounding fluid density greatly broadens the scope of buoyancydriven flow, and such a fundamental discovery inevitably opens many avenues for future research. For example, we investigated density stratifications resulting from salinity variations, for which Sc>500. Extrapolating these results, one might anticipate significantly faster propulsion speeds for an insulating wedge in a thermal stratification (Sc≈7). Another related scenario that is commonplace in nature is a floating object that is hotter or colder than the surrounding fluid, in which case the resulting surface flows, analogous to valley^{13} and glacier winds^{14}, are much stronger than diffusiondriven flow, also raising the prospect of significantly faster propulsion speeds. Indeed, if sufficiently large thrusts can be produced by surface heating and cooling, this could enable the design of silent underwater vehicles without any moving mechanical components. In another set of (preliminary) experiments, we found that interactions occur between a pair of wedges (see Supplementary Information), suggesting a new mechanism for collective behaviour of floating objects. Finally, an important avenue to explore is the energy balance of these processes. It is particularly interesting that in our experiments the kinetic energy of macroscale motion is actually drawn from the internal energy of the fluid via microscale molecular diffusion. This influences internal and gravitational potential energies of the system, and could connect with present debate regarding ocean energetics^{15}.
Methods
General.
The upper and lower halves of each wedge were machined from highdensity polyethylene (ρ=955 kg m^{−3}) and Ultem 1,000 polyetherimide (ρ=1,280 kg m^{−3}), respectively; this ensured that a wedge’s centre of gravity lay below its centre of buoyancy, producing a stabilizing torque that maintained a horizontal orientation. A stratification was established using the doublebucket method and measured by a calibrated conductivity and temperature probe. At the start of an experiment, a cage of six vertical, 35cmlong stainlesssteel rods was lowered slowly (∼1×10^{−3} m s^{−1}) into the stratification using a computercontrolled traverse. The wedge initially settled within the confine of the rods. The system was left to stand for one hour before the cage was withdrawn slowly, leaving the wedge floating in the quiescent stratification. The top of the experimental tank was sealed to eliminate atmospheric disturbances. Horizontal motion of the wedge was photographed from above, the length of the wedge providing a calibration length scale in each image.
Schmidt number.
Experiments were run for different Schmidt numbers (Sc=ν/κ) by using different salts (KCl,NaCl,MgCl_{2}) dissolved in water, and by using NaCl dissolved in water/glycerol mixtures. The kinematic viscosity, ν, was measured using an Ostwald viscometer. The molecular diffusivity, κ, was measured by establishing a density stratification and introducing a triangular ledge that spanned the entire width of the tank, so as to induce a nominally 2D flow. The ledge had upper and lower slope angles of α=10.0±0.1^{°} with respect to the horizontal plane. The volume flux drawn towards the tip of the ledge in the vertical midplane, Q, was measured using PIV, and κ was calculated using the relation Q=2κcotα.
Visualizations.
A LaVision PIV system was used to obtain flow visualizations in the vertical and horizontal midplane of the wedge. The stratification was seeded with 10μm glass spheres, with a range of densities, the motion of which was tracked to obtain velocity field data. Given the slow flow speeds, to ensure that data were not contaminated by particle settling, the stratification was left to stand for four days after filling, allowing sufficient time for particles to reach their neutral buoyancy heights.
Numerical simulations.
3D numerical simulations of stratified flow past a fixed wedge were computed using an adaptive, blockstructured, twoway nested, secondorder accurate finitevolume method^{16}. This method is designed for the study of flows that are multiscale, incompressible, of variable density and within arbitrarily complex domains. The 3D simulations were run on a Cray XT4 supercomputer. The 3D computational fluid dynamics flow domain was a 0.3×0.3×0.3 m^{3}, with the wedge tip centre located at x=0.17 m,y=0.15 m,z=0.15 m from the lower corner of the cube. The mesh was locally refined using blockstructured refinement, the finest scale of the mesh along the wedge being 5.8×10^{−4} m, yielding an effective resolution of 134 million cells. Insulating, noslip boundary conditions were imposed on the wedge. To determine that the simulations had achieved a steady state, time series of both the force on the wedge and the total kinetic energy of the flow around the wedge were calculated. The system was considered to be in a steady state when both of these quantities varied by less than 1% over a physical timescale of 1,000 s.
A 3D COMSOL simulation was used to calculate the drag on a slowly moving wedge in homogeneous water. The wedge was fixed in the centre of the domain. The front of the domain had constantvelocity inlet conditions and the rear of the domain had outlet boundary conditions. All of the sidewalls had stressfree boundary conditions to remove boundary layer effects. The size of the domain and the mesh resolution were increased until each had a less than 1% impact on the drag. The calculation of the drag was provided by the COMSOL software, by means of boundary integration on the surfaces of the wedge, which was independently checked using our own control volume approach.
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Acknowledgements
We thank R. Stocker, E. Lauga and C. C. Mei for discussions; W. Etheridge, P. Steinmetz and C. Lenahan for experimental assistance; A. Gallant for fabrication of experimental apparatus; O. Fringer for access to computational resources; and G. Buck, J. W. M. Bush, E. Johnson, W. R. Young, C. Wunsch and J. H. Peacock for reading the manuscript. This work was supported by the NSF CBET programme. M.F.B. acknowledges the support of the National Science Foundation MSPRF programme. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under Contract No. DEAC0205CH11231.
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T.P. conceived the original idea, supervised the research and wrote the manuscript. M.R.A. carried out the experiments, ran some numerical simulations and analysed the numerical data. M.F.B. ran the stratified flow simulations.
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Allshouse, M., Barad, M. & Peacock, T. Propulsion generated by diffusiondriven flow. Nature Phys 6, 516–519 (2010). https://doi.org/10.1038/nphys1686
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DOI: https://doi.org/10.1038/nphys1686
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