2 results
Heat/mass transport in shear flow over a reactive surface with inert defects
- Preyas N. Shah, Tiras Y. Lin, Eric S. G. Shaqfeh
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- Journal:
- Journal of Fluid Mechanics / Volume 811 / 25 January 2017
- Published online by Cambridge University Press:
- 13 December 2016, pp. 372-399
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We study the problem of mass transport to surfaces with heterogeneous reaction rates in the presence of shear flow over these surfaces. The reactions are first order and the heterogeneity is due to the existence of inert regions on the surfaces. Such problems are ubiquitous in the field of heterogeneous catalysis, electrochemistry and even biological mass transport. In these problems, the microscale reaction rate is characterized by a Damköhler number $\unicode[STIX]{x1D705}$, while the Péclet number $P$ is the dimensionless ratio of the bulk shear rate to the inverse diffusion time scale. The area fraction of the reactive region is denoted by $\unicode[STIX]{x1D719}$. The objective is to calculate the yield of reaction, which is directly related to the mass flux to the reactive region, denoted by the dimensionless Sherwood number $S$. Previously, we used boundary element simulations and examined the case of first-order reactive disks embedded in an inert surface (Shah & Shaqfeh J. Fluid Mech., vol. 782, 2015, pp. 260–299). Various correlations for the Sherwood number as a function of $(\unicode[STIX]{x1D705},P,\unicode[STIX]{x1D719})$ were obtained. In particular, we demonstrated that the ‘method of additive resistances’ provides a good approximation for the Sherwood number for a wide range of values of $(\unicode[STIX]{x1D705},P)$ for $0<\unicode[STIX]{x1D719}<0.78$. When $\unicode[STIX]{x1D719}\approx 0.78$, the reactive disks are in a close packed configuration where the inert regions are essentially disconnected from each other. In this work, we develop an understanding of the physics when $\unicode[STIX]{x1D719}>0.78$, by examining the inverse problem of inert disks on a reactive surface. We show that the method of resistances approach to obtain the Sherwood number fails in the limit as $\unicode[STIX]{x1D719}\rightarrow 1$, i.e. in the dilute limit of periodic inert disks, due to the existence of a surface concentration boundary layer around each disk that scales with ($1/\unicode[STIX]{x1D705}$). This boundary layer controls the screening length between inert disks and allows us to introduce a new theory, thus providing new correlations for the Sherwood number that are highly accurate in the limit of $\unicode[STIX]{x1D719}\rightarrow 1$.
Buoyancy-induced turbulent mixing in a narrow tilted tank
- Tiras Y. Lin, C. P. Caulfield, Andrew W. Woods
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- Journal:
- Journal of Fluid Mechanics / Volume 773 / 25 June 2015
- Published online by Cambridge University Press:
- 20 May 2015, pp. 267-297
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We describe a series of experiments in which a constant buoyancy flux $B_{s}$ of salty dyed water of density ${\it\rho}_{s}$ is introduced at the top of a long narrow tank of square cross-section tilted at an angle ${\it\theta}$ to the vertical. The tank is initially filled with fresh clear water of density ${\it\rho}_{0}<{\it\rho}_{s}$, and we investigate the resulting buoyancy-driven turbulent mixing at various tilt angles ${\it\theta}$. Using a light-attenuation image analysis method, we determine the evolution of the reduced gravity $g^{\prime }=g({\it\rho}-{\it\rho}_{0})/{\it\rho}_{0}$ of the mixed fluid in time and space as it propagates towards the bottom of the tank. For all tilt angles tested (${\it\theta}=0^{\circ }$ to ${\it\theta}=45^{\circ }$), we focus exclusively on high-Reynolds-number experiments, where the flow remains turbulent both along the length and across the width of the tank. We find that when ${\it\theta}>0^{\circ }$, the cross-tank component of gravity acts to segregate the dense fluid from the relatively lighter fluid, and a statically stable gradient of $g^{\prime }$ across the width of the tank occurs more frequently than a statically unstable gradient, i.e. $(\partial g^{\prime }/\partial x)<0$ occurs more frequently than $(\partial g^{\prime }/\partial x)>0$. This is in contrast to the case when ${\it\theta}=0^{\circ }$, where instantaneous cross-tank gradients of reduced gravity may be positive or negative, but are equal to zero in an ensemble average. We observe that when ${\it\theta}>0^{\circ }$, the cross-tank gradient of reduced gravity induces a turbulent counterflow where dense fluid flows down the upward-facing surface of the tank and lighter fluid flows in the opposing direction above. We model the evolution of the cross-tank averaged, ensemble averaged reduced gravity $\langle \overline{g^{\prime }}\rangle _{e}$ as a diffusive process using Prandtl’s mixing length theory, building on the model of van Sommeren et al. (J. Fluid Mech., vol. 701, 2012, pp. 278–303) who considered purely vertical tanks. We model the fluctuations (from the cross-tank averaged quantity) of reduced gravity $\langle {\hat{g}}^{\prime }\rangle _{e}$ and counterflow velocity $\langle {\hat{w}}\rangle _{e}$ by characterising the mixing across the width of the tank with a cross-tank turbulent diffusivity ${\it\kappa}_{T,x}$, which we assume is constant in the cross-tank coordinate $x$. We show that the counterflow that exists when ${\it\theta}>0^{\circ }$ acts directly to enhance the effective along-tank turbulent diffusivity ${\it\kappa}_{T,z}$, and from experiments, we find that the mixing length increases approximately linearly with ${\it\theta}$, and that both ${\it\kappa}_{T,x}$ and ${\it\kappa}_{T,z}$ are proportional to $(\partial \langle \overline{g^{\prime }}\rangle _{e}/\partial z)^{1/2}$.