Technical Report PHD-2007-19

Title: Simulation of non-homogeneous flow fields subject to rotation and gravity effects
Authors: Tamar Zemach
Supervisors: Marius Ungarish
Abstract: The work is concerned with the investigation of gravity currents and intrusions in various circumstances. Gravity currents are formed by fluid flowing primarily horizontally, under the influence of gravity into another fluid of different density. The gravity current which is released from rest and then propagates horizontally at the neutral buoyancy level in a vertically-stratified ambient fluid is called intrusive gravity current or intrusion. The density of the intrusion or current is constant; the ambient fluid density change is linear over the full-depth of the container. Both two-dimensional and axisymmetric geometries are of interest. A closed one-layer shallow-water inviscid formulation is used to obtain time-dependent solutions of the initial-value problem and an extension of Long's model is used to obtain steady-state solutions. We present some new results as follows:

1). A closed one-layer shallow-water Boussinesq inviscid formulation is presented for the axisymmetric intrusions. In general, the solution of the resulting hyperbolic system is obtained by a finite-differences \mar scheme. However, for the large-time developed motion an analytical similarity solutions is derived. The self-similar result indicates radial expansion with $t^{1/3}$ but the shape is peculiar: the intruding fluid propagates like a ring with a fixed ratio of inner to outer radii; the inner domain (between the axis and the inner radius of the ring) contains clear ambient fluid. It is verified that the initial-value lock-release finite-differences \mar solution indeed approaches the similarity predictions after an initial spread of the outer radius to about 2.5 times the initial radius. To avoid accumulation of numerical errors the problem is reformulated in terms of new variables. It is shown that the numerical solution has a ``tail-ring" shape. The ``tail" decays like $t^{-2}$, and the ``ring" tends to the analytical similarity prediction. The initial geometry of the lock does not influence this result. Comparison with the non-stratified case is also presented. It has been \mar obtained that for the non-stratified case there is a stage of propagation in which the intrusion has a similar ``tail-ring" form; however this stage is only a transient to a self-similar shape which is different from the one \mar that was obtained for the stratified ambient. The shallow-water results are corroborated by numerical solutions of the full axisymmetric Navier-Stokes. It is concluded that the shallow-water model is a versatile and accurate predictive tool, and that the peculiar ring-shape prediction reproduces an interesting physical property of the axisymmetric intrusion. The interaction between the internal gravity waves and the head is less significant than in the two-dimensional geometry. However, a practical limitation on the applicability of the inviscid model is imposed by the prediction that the ratio of viscous to inertia forces increases like $r_N^7$ (the radius of propagation, scaled with the initial value).

2). The behavior of the two-dimensional steady intrusive gravity currents spreading into a stratified ambient fluid is considered. The intrusive gravity current of thickness $h$ and density $\rho_{c}$ which propagates with speed $U$ at the neutral buoyancy level of a long horizontal channel of height $H$ into a stratified ambient fluid whose density increases linearly from $\rho_{o}$ to $\rho_{b}$ is considered. The intrusive and the ambient fluids are assumed to be asymmetric with respect to the horizontal plane passing through the tip of the intrusion. The Boussinesq, high-Reynolds number two-dimensional configuration is discussed. Long's model combined with the flow-force balance over the width of the channel and the pressure balances over a density current are used to obtain the desired results. It is shown that the intrusion velocity decreases with decreasing of system's asymmetry \mar and approaches its minimum for the symmetric configuration. In additional, the comparison between asymmetric and symmetric configurations shows no significant differences between the models.

3). In practical cases, the motion of intrusive gravity currents is affected by additional external effects. We discuss an important extension of the fixed-volume classical intrusions: a flow generated by an external line source with constant volumetric flow rate $Q$ which is assumed to be located at the origin. Inertia-buoyancy controlled axisymmetric intrusive gravity currents with external line source are discussed. The shallow-water theory is presented for this case. The problem is solved numerically and long-time similarity solutions are found. A new approach to find the analytical similarity solution is presented. The results obtained numerically and analytically are compared with the experimental data and show good agreement.

4). An additional external effect of interest is the \mar rotation. The effects of rotation of the system on the boundary gravity currents is discussed. Some experiments are presented and compared with the theoretical results. As expected, the rotation reduces the velocity of propagation and even stops the radial motion after a short time interval (less than one period of revolution).

In general, in the framework of the present work some difficult problems are solved. This research work contributes to the understanding of the intrusive gravity currents behavior. The theory and the methods of solution presented here can be used in the future for \mar investigation of the behavior of intrusions in various circumstances. We hope that the progress achieved in this study will encourage additional research of this problem in the future.

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