Mixing in axisymmetric gravity currents

Abstract

We present new experiments to measure the rate of entrainment of ambient fluid into a high Reynolds number, axisymmetric, turbulent gravity current. The current is produced by the rapid release of a finite volume of aqueous salt solution from a lock of length $r_{o}$ into a diverging channel, $r>0$, of angle $9.5^{\circ }$, filled with a finite depth, $H$, of fresh water. Using light attenuation we measure the evolving density of the flow, and using dye studies we illustrate the process of mixing between the current and ambient fluid. After an initial adjustment, a circulation develops in the head of the flow: current fluid reaches the nose of the flow, rises up and moves backwards relative to the nose. We find that, owing to the mixing, the volume of the current increases as $V\sim 0.2r_{n}^{7/4}r_{o}^{1/4}H$ while the maximum depth of the head decreases as $h_{n}\sim 0.5H(r_{o}/r_{n})^{1/4}$, where $r_{n}$ is the location of the front of the current. Combining these results, we estimate that the recirculating current fluid mixes with a fraction $E=0.33\pm 0.09$ of the ambient fluid that is directly ahead of the current and displaced upwards by it. Some of the mixed fluid supplies the tail of the flow, while the remainder recirculates into the head, which becomes progressively more dilute. In accord with Huppert & Simpson (J. Fluid Mech., vol. 99, 1980, pp. 785–799), we find that the position of the front increases with time as $r_{n}\approx (1.28\pm 0.05)B^{1/4}t^{1/2}$, where $B$ is the total buoyancy of the flow. We also find that the maximum value of the vertical integral of the buoyancy $(\overline{g^{\prime }}h)_{n}$ decreases with the position of the nose according to the relation $(\overline{g^{\prime }}h)_{n}\approx (0.89\pm 0.12)Br_{n}^{-2}$, consistent with a Froude number $0.86\pm 0.07$. We compare our measurements with a new idealised self-similar solution of the depth-averaged equations that accounts for the mixing at the nose, the vertical shear in the velocity and the lateral stratification of the buoyancy within the current.