arXiv subjects

Gerasimos Politis

Publications and source records attributed to Gerasimos Politis.

2 recordsLinked to original sources

arXiv2022

Lock-exchange problem for Boussinesq fluids revisited: exact shallow-water solution

An exact solution to the lock-exchange problem, which is a two-layer analogue of the classical dam-break problem, is obtained in the shallow-water (SW) approximation for two immiscible fluids with slightly different densities. The problem is solved by the method of characteristics using analytic expressions for the Riemann invariants. The obtained solution, which represents an inviscid approximation to the high-Reynolds-number limit, is in general discontinuous containing up to three hydraulic jumps which are due to either multivaluedness or instability of the continuous SW solution. Hydraulic jumps are resolved by applying the Rankine-Hugoniot conditions for the SW mass and generalized momentum conservation equations. The latter contains a free parameter $α$ which defines the relative contribution of each layer to the interfacial pressure gradient. We consider a solution for $α=0,$ which corresponds to both layers affecting the interfacial pressure gradient with equal weight coefficients. This solution is compared with the solutions resulting from the application of the classical Benjamin's front condition as well as the circulation conservation condition, which correspond to $α=-1$ and $α\rightarrow\infty,$ respectively. The SW solution reproduces all principal features of 2D numerical solution for viscous fluids. The gravity current speed is found to agree well with experimental and numerical results when the front acquires the largest stable height which occurs at $α=\sqrt{5}-2.$ We show that two-layer SW equations for the mass and generalized momentum conservation can describe interfacial waves containing hydraulic jumps in a self-contained way without external closure conditions.

physics.flu-dyn↗

arXiv2021

Fractality of metal pad instability threshold in rectangular cells

We analyse linear stability of interfacial waves in an idealised model of an aluminium reduction cell consisting of two stably stratified liquid layers which carry a vertical electric current in a collinear external magnetic field. If the product of electric current and magnetic field exceeds a certain critical threshold depending on the cell design, the electromagnetic coupling of gravity wave modes can give rise to a self-amplifying rotating interfacial wave which is known as the metal pad instability. Using the eigenvalue perturbation method, we show that, in the inviscid limit, rectangular cells of horizontal aspect ratios $α=\sqrt{m/n}$, where $m$ and $n$ are any two odd numbers, can be destabilised by an infinitesimally weak electromagnetic interaction while cells of other aspect ratios have finite instability thresholds. This fractal distribution of critical aspect ratios, which form an absolutely discontinuous dense set of points interspersed with aspect ratios with non-zero stability thresholds, is confirmed by accurate numerical solution of the linear stability problem. Although the fractality vanishes when viscous friction is taken into account, the instability threshold is smoothed out gradually and its principal structure, which is dominated by the major critical aspect ratios corresponding to moderate values of $m$ and $n$, is well-preserved up to relatively large dimensionless viscous friction coefficients $γ\sim 0.1$. With a small viscous friction, the most stable are cells with $α^{2}\approx2.13$ which have the highest stability threshold corresponding to the electromagnetic interaction parameter $β\approx 4.7$.

physics.flu-dyn↗