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Previous studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e.? velocities are δ-function correlated in time). However, many turbulent (geophysical and enginee...
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Previous studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e.? velocities are δ-function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale τ _L. A non-zero τ _L results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale τ _D) times. The longitudinal (along-flow) shear-induced diffusivity D _S is derived, accurate for all τ _L, using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine D _S. The non-dimensionalized D _S depends on time and two parameters: the ratio of Lagrangian to diffusive time scales τ _L/τ _D and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time D _S is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with √τ _L. At moderate τ _L/τ _D = 0. 1 this enhancement is approximately a factor of 3. For classic shear dispersion with τ _L = 0, the diffusive time scale τ _D determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large τ _L, a shear time scale τ _S = (τ _Lτ _D) ~(1/2), anticipated by a simple analysis of the particle's domain-crossing time, determines both the D _S time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time D _S using properties of wall-bounded turbulence.
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We report a series of experiments in which a cylinder, with a vertical axis, is moved back and forth along a long narrow channel containing fresh water at Reynolds numbers Re = 3220-13 102. We examine the mixing of a cloud of dye ...
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We report a series of experiments in which a cylinder, with a vertical axis, is moved back and forth along a long narrow channel containing fresh water at Reynolds numbers Re = 3220-13 102. We examine the mixing of a cloud of dye along the channel by the oscillatory motion of the cylinder. Using light attenuation techniques to measure the time evolution of the concentration of dye along the channel, we find that at early times the concentration profile collapses to a Gaussian profile with dispersivity, D = (2.4 +/- 0.5)fdW, where f is the frequency of the cylinder oscillation, d is the diameter of the cylinder and W is the width of the channel, respectively. For times much longer than L-2/D, with L being the length of the channel, the concentration becomes progressively more uniform over the whole length of the channel, and we show that the long-time non-uniform component decays with time dependence exp(-4 pi(2)Dt/L-2). We consider the implications of these experiments for the dispersal of viral aerosols along poorly ventilated corridors, with implications for infection transmission in hospitals and public buildings.
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We develop a process-based model for the dispersion of a passive scalar in the turbulent flow around the buildings of a city centre. The street network model is based on dividing the airspace of the streets and intersections into ...
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We develop a process-based model for the dispersion of a passive scalar in the turbulent flow around the buildings of a city centre. The street network model is based on dividing the airspace of the streets and intersections into boxes, within which the turbulence renders the air well mixed. Mean flow advection through the network of street and intersection boxes then mediates further lateral dispersion. At the same time turbulent mixing in the vertical detrains scalar from the streets and intersections into the turbulent boundary layer above the buildings. When the geometry is regular, the street network model has an analytical solution that describes the variation in concentration in a near-field downwind of a single source, where the majority of scalar lies below roof level. The power of the analytical solution is that it demonstrates how the concentration is determined by only three parameters. The plume direction parameter describes the branching of scalar at the street intersections and hence determines the direction of the plume centreline, which may be very different from the above-roof wind direction. The transmission parameter determines the distance travelled before the majority of scalar is detrained into the atmospheric boundary layer above roof level and conventional atmospheric turbulence takes over as the dominant mixing process. Finally, a normalised source strength multiplies this pattern of concentration. This analytical solution converges to a Gaussian plume after a large number of intersections have been traversed, providing theoretical justification for previous studies that have developed empirical fits to Gaussian plume models. The analytical solution is shown to compare well with very high-resolution simulations and with wind tunnel experiments, although re-entrainment of scalar previously detrained into the boundary layer above roofs, which is not accounted for in the analytical solution, is shown to become an important process further downwind from the source.
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We describe new experiments to examine the buoyancy-induced turbulent mixing which results from the injection of a small constant volume flux of dense fluid at the top of a long narrow vertical tank with square cross-section, in w...
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We describe new experiments to examine the buoyancy-induced turbulent mixing which results from the injection of a small constant volume flux of dense fluid at the top of a long narrow vertical tank with square cross-section, in which a steady laminar upward flow of less dense fluid is present. To conserve volume of fluid in the tank, fluid leaves the tank through two small openings near the top of the tank. Dense source fluid vigorously mixes with the less dense fluid of the upward flow, such that a dense mixing region of turbulent fluid propagates downwards during the transient mixing phase of the experiment. Eventually, the transport of dense fluid associated with the buoyancy-induced turbulent flow is balanced by the transport of less-dense fluid associated with the steady upward flow, such that the mixing region evolves into a layer of finite extent which stays approximately constant in height during a statistically steady mixing phase of the experiment. With an ideal source of downward constant buoyancy flux Bs at the top of the tank, tank width d, and speed of the upward flow uu, we perform experiments with Froude numbers Fr D uud1=3=B1=3 s ranging between O.0:01/ and O.1/. The steady-state height of the mixing region and the maximum reduced gravity as found near the source of buoyancy flux at the top of the tank increase with decreasing Froude number. For the experiments with intermediate values of the Froude number, we find that the steady-state mixing region is small enough to be contained in the experimental tank, but large enough not to be dominated by developing turbulence near the source of buoyancy flux. For these experiments, we show that the key buoyancy-induced turbulent mixing properties are not significantly affected by the upward flow. We use a dye-attenuation technique to obtain vertical profiles of the time- and horizontally averaged reduced gravity to show a good agreement between the experimental profiles and the solution of a nonlinear turbulent advection-diffusion equation during the steady mixing phase. Furthermore, we discuss the characteristic time scale of the transient mixing phase. We compare our experimental results with the numerical solution of a time-dependent nonlinear turbulent advection-diffusion equation during the transient mixing phase. We also describe three reduced models for the evolution of the reduced gravity distribution in the mixing region, and we demonstrate these models' usefulness by comparison with our experimental results and the numerical solution of the time-dependent nonlinear turbulent advection-diffusion equation.
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We describe new experiments to examine the buoyancy-induced mixing which results from the injection of a small constant volume flux of fluid of density ρ s at the top of a long narrow vertical tank with square cross-section which...
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We describe new experiments to examine the buoyancy-induced mixing which results from the injection of a small constant volume flux of fluid of density ρ s at the top of a long narrow vertical tank with square cross-section which is filled with fluid of density ρ 0 < ρ s. The injected fluid vigorously mixes with the less dense fluid which initially occupies the tank, such that a dense mixed region of turbulent fluid propagates downwards during the initial mixing phase of the experiment. For an ideal source of constant buoyancy flux B s, we show that the height of the mixed region grows as h ～ B _s ~(1/6) d ~(1/3) t ~(1/2) and that the horizontally averaged reduced gravity g ′ = g(ρ-ρ 0)0 at the top of tank increases as g(0) ～ Bs ~(5/6)d- ~(7/3)t ~(1/2), where d is the width of the tank. Once the mixed region reaches the bottom of the tank, the turbulent mixing continues in an intermediate mixing phase, and we demonstrate that the reduced gravity at each height increases approximately linearly with time. This suggests that the buoyancy flux is uniformly distributed over the full height of the tank. The overall density gradient between the top and bottom of the mixed region is hence time-independent for both the mixing phases before and after the mixed region has reached the bottom of the tank. Our results are consistent with previous models developed for the mixing of an unstable density gradient in a confined geometry, based on Prandtls mixing length theory, which suggest that the turbulent diffusion coefficient and the magnitude of the local turbulent flux are given by the nonlinear relations κ _T ~(nl) = λ ~2 d ~2 g ′ z) ~(1/2) and J _(nl) = λ ~2 d ~2(g′ z) ~(3/2), respectively. The O(1) constant λ relates the width of the tank to the characteristic mixing length of the turbulent eddies. Since the mixed region is characterized by a time-independent overall density gradient, we also tested the predictions based on a linear model in which the turbulent diffusion coefficient is approximated by a constant κT _l. We solve the corresponding nonlinear and linear turbulent diffusion equations for both mixing phases, and show a good agreement with experimental profiles measured by a dye attenuation technique, in particular for the solutions based on the nonlinear model.
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Results from large-eddy simulations of short-range dispersion of a passive scalar from a point source release in an urban-like canopy are presented. The computational domain is that of a variable height array of buildings immersed...
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Results from large-eddy simulations of short-range dispersion of a passive scalar from a point source release in an urban-like canopy are presented. The computational domain is that of a variable height array of buildings immersed in a pressure-driven, turbulent flow with a roughness Reynolds number Reτ = 433. A comparative study of several cases shows the changes in plume behaviour for different mean flow directions and source locations. The analysis of the results focuses on utilizing the high-fidelity datasets to examine the three-dimensional flow field and scalar plume structure. The detailed solution of the flow and scalar fields within the canopy allows for a direct assessment of the impact of local features of the building array geometry. The staggered, skewed and aligned arrangements of the buildings with respect to the oncoming flow were shown to affect plume development. Additional post-processing quantified this development through parameters fundamental to reduced-order Gaussian dispersion models. The parameters include measures of concentration decay with distance from the source as well as plume trajectory and spread. The horizontal plume trajectory and width were found to be more sensitive to source location variations, and hence local geometric features, than vertical plume parameters.
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We perform experiments to study the mixing of passive scalar by a buoyancy-induced turbulent flow in a long narrow vertical tank. The turbulent flow is associated with the downward mixing of a small flux of dense aqueous saline so...
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We perform experiments to study the mixing of passive scalar by a buoyancy-induced turbulent flow in a long narrow vertical tank. The turbulent flow is associated with the downward mixing of a small flux of dense aqueous saline solution into a relatively large upward flux of fresh water. In steady state, the mixing region is of finite extent, and the intensity of the buoyancy-driven mixing is described by a spatially varying turbulent diffusion coefficient k_v(z) which decreases linearly with distance z from the top of the tank. We release a pulse of passive scalar into either the fresh water at the base of the tank, or the saline solution at the top of the tank, and we measure the subsequent mixing of the passive scalar by the flow using image analysis. In both cases, the mixing of the passive scalar (the dye) is well-described by an advection–diffusion equation, using the same turbulent diffusion coefficient k_v(z) associated with the buoyancy-driven mixing of the dynamic scalar. Using this advection–diffusion equation with spatially varying turbulent diffusion coefficient k_v(z), we calculate the residence time distribution (RTD) of a unit mass of passive scalar released as a pulse at the bottom of the tank. The variance in this RTD is equivalent to that produced by a uniform eddy diffusion coefficient with value k_e = 0:88(k_v), where (k_v) is the vertically averaged eddy diffusivity. The structure of the RTD is also qualitatively different from that produced by a flow with uniform eddy diffusion coefficient. The RTD using k_v has a larger peak value and smaller values at early times, associated with the reduced diffusivity at the bottom of the tank, and manifested mathematically by a skewness γ_1≈1:60 and an excess kurtosis γ_2≈4:19 compared to the skewness and excess kurtosis of γ_1≈1:46, γ_2≈3:50 of the RTD produced by a constant eddy diffusion coefficient with the same variance.
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A theoretical description of the turbulent mixing within and the draining of a dense fluid layer from a box connected to a uniform density, quiescent environment through openings in the top and the base of the box is presented in ...
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A theoretical description of the turbulent mixing within and the draining of a dense fluid layer from a box connected to a uniform density, quiescent environment through openings in the top and the base of the box is presented in this paper. This is an extension of the draining model developed by Linden et al. (Annu. Rev. Fluid Mech. vol. 31, 1990, pp. 201-238) and includes terms that describe localized mixing within the emptying box at the density interface. Mixing is induced by a turbulent flow of replacement fluid into the box and as a consequence we predict, and observe in complementary experiments, the development of a three-layer stratification. Based on the data collated from previous researchers, three distinct formulations for entrainment fluxes across density interfaces are used to account for this localized mixing. The model was then solved numerically for the three mixing formulations. Analytical solutions were developed for one formulation directly and for a second on assuming that localized mixing is relatively weak though still significant in redistributing buoyancy on the timescale of the draining process. Comparisons between our theoretical predictions and the experimental data, which we have collected on the developing layer depths and their densities show good agreement. The differences in predictions between the three mixing formulations suggest that the normalized flux turbulently entrained across a density interface tends to a constant value for large values of a Froude number FrT, based on conditions of the inflow through the top of the box, and scales as the cube of FrT for small values of FrT. The upper limit on the rate of entrainment into the mixed layer results in a minimum time (tD) to remove the original dense layer. Using our analytical solutions, we bound this time and show that 0.2t_E ≈t_D t_E, i.e. the original dense layer may be depleted up to five times more rapidly than when there is no internal mixing and the box empties in a time t_E.
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This paper extends the analysis of solute dispersion in electrohydrodynamic flows to the case of band broadening in polyelectrolyte-grafted (soft) capillaries by accounting for the effects of ion partitioning, irreversible catalyt...
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This paper extends the analysis of solute dispersion in electrohydrodynamic flows to the case of band broadening in polyelectrolyte-grafted (soft) capillaries by accounting for the effects of ion partitioning, irreversible catalytic reaction and pulsatile flow actuation. In the Debye-Huckel limit, we present the benchmark solutions of electric potential and velocity distribution pertinent to steady and oscillatory mixed electroosmotic-pressure-driven flows in soft capillaries. Afterwards, the mathematical models of band broadening based on the Taylor-Aris theory and generalized dispersion method are presented to investigate the late-time asymptotic state and all-time evolution of hydrodynamic dispersion, respectively. Also, to determine the heterogeneous dispersion behaviour of solute through all spatiotemporal stages and to relax the constraint of small zeta potentials, a full-scale numerical simulation of time-dependent solute transport in soft capillaries is presented by employing the second-order-accurate finite difference method. Then, by inspecting the dispersion of passive tracer particles in Poiseuille flows, we examine the accuracy of two analytical approaches against the simulation results of a custom-built numerical algorithm. Our findings from hydrodynamic dispersion in Poiseuille flows reveal that, compared to rigid capillaries, more time is required to approach the longitudinal normality and transverse uniformity of injected solute in soft capillaries. For the case of dispersion in mixed electrohydrodynamic flows, it is found that the characteristics of the soft interface, including the thickness, permittivity, fixed charge density and friction coefficient of the polymer coating layer, play a significant role in determining the Taylor diffusion coefficient, advection speed and dispersion rate of solutes in soft capillaries.
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The Earth's mantle is chemically heterogeneous and probably includes primordial material that has not been affected by melting and attendant depletion of heat-producing radioactive elements. One consequence is that mantle internal...
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The Earth's mantle is chemically heterogeneous and probably includes primordial material that has not been affected by melting and attendant depletion of heat-producing radioactive elements. One consequence is that mantle internal heat sources are not distributed uniformly. Convection induces mixing, such that the flow pattern, the heat source distribution and the thermal structure are continuously evolving. These phenomena are studied in the laboratory using a novel microwave-based experimental set-up for convection in internally heated systems. We follow the development of convection and mixing in an initially stratified fluid made of two layers with different physical properties and heat source concentrations lying above an adiabatic base. For relevance to the Earth's mantle, the upper layer is thicker and depleted in heat sources compared to the lower one. The thermal structure tends towards that of a homogeneous fluid with a well-defined time constant that scales with is the Rayleigh-Roberts number for the homogenized fluid. We identified two convection regimes. In the dome regime, large domes of lower fluid protrude into the upper layer and remain stable for long time intervals. In the stratified regime, cusp-like upwellings develop at the edges of large basins in the lower layer. Due to mixing, the volume of lower fluid decreases to zero over a finite time. Empirical scaling laws for the duration of mixing and for the peak temperature difference between the two fluids are derived and allow extrapolation to planetary mantles.
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