3 results
Spatially varying mixing of a passive scalar in a buoyancy-driven turbulent flow
- Daan D. J. A. van Sommeren, C. P. Caulfield, Andrew W. Woods
-
- Journal:
- Journal of Fluid Mechanics / Volume 742 / 10 March 2014
- Published online by Cambridge University Press:
- 24 February 2014, pp. 701-719
-
- Article
- Export citation
-
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 $\kappa _v(z)$ 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 $\kappa _v(z)$ associated with the buoyancy-driven mixing of the dynamic scalar. Using this advection–diffusion equation with spatially varying turbulent diffusion coefficient $\kappa _v(z)$, 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 $\kappa _e= 0.88 \langle \kappa _v \rangle $, where $\langle \kappa _v \rangle $ 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 $\kappa _v$ 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 $\gamma _1\approx 1.60$ and an excess kurtosis $\gamma _2\approx 4.19 $ compared to the skewness and excess kurtosis of $\gamma _1\approx 1.46$, $\gamma _2 \approx 3.50$ of the RTD produced by a constant eddy diffusion coefficient with the same variance.
Advection and buoyancy-induced turbulent mixing in a narrow vertical tank
- Daan D. J. A. van Sommeren, C. P. Caulfield, Andrew W. Woods
-
- Journal:
- Journal of Fluid Mechanics / Volume 724 / 10 June 2013
- Published online by Cambridge University Press:
- 29 April 2013, pp. 450-479
-
- Article
- Export citation
-
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 ${B}_{s} $ at the top of the tank, tank width $d$, and speed of the upward flow ${u}_{u} $, we perform experiments with Froude numbers $\mathit{Fr}= {u}_{u} {d}^{1/ 3} / { B}_{s}^{1/ 3} $ 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.
Turbulent buoyant convection from a maintained source of buoyancy in a narrow vertical tank
- Daan D. J. A. van Sommeren, C. P. Caulfield, Andrew W. Woods
-
- Journal:
- Journal of Fluid Mechanics / Volume 701 / 25 June 2012
- Published online by Cambridge University Press:
- 10 May 2012, pp. 278-303
-
- Article
- Export citation
-
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 at the top of a long narrow vertical tank with square cross-section which is filled with fluid of density . 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 , we show that the height of the mixed region grows as and that the horizontally averaged reduced gravity at the top of tank increases as , where 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 Prandtl’s mixing length theory, which suggest that the turbulent diffusion coefficient and the magnitude of the local turbulent flux are given by the nonlinear relations and , respectively. The constant 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 . 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.