32 research outputs found
Fully conservative hydraulic jumps and solibores in two-layer Boussinesq fluids
We consider a special type of hydraulic jumps (internal bores) which, in the
vertically bounded system of two immiscible fluids with slightly different
densities, conserve not only the mass and impulse but also the circulation and
energy. This is possible only at specific combinations of the upstream and
downstream states. Two such combinations are identified with arbitrary upstream
and downstream interface heights. The first has a cross symmetry between the
interface height and shear on both sides of the jump. This symmetry, which is
due to the invariance of the two-layer shallow-water system with swapping the
interface height and shear, ensures the automatic conservation of the impulse
and energy as well as the continuity of characteristic velocities across the
jump. The speed at which such jumps propagate is uniquely defined by the
conservation of the mass and circulation. The other possibility is a marginally
stable shear flow which can have fully conservative jumps with discontinuous
characteristic velocities. Both types of conservative jumps are shown to
represent a long-wave approximation to the so-called solibores which appear as
smooth permanent-shape solutions in a weakly non-hydrostatic model. A new
analytical solution for solibores is obtained and found to agree very well with
the previous DNS results for partial-depth lock release flow. The finding that
certain large-amplitude hydraulic jumps can be fully conservative, while most
are not such even in the inviscid approximation, points toward the wave
dispersion as a primary mechanism behind the lossy nature of internal bores.Comment: 14 pages, 4 figures (to appear in J. Fluid. Mech.
Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct
We analyse numerically the linear stability of a liquid metal flow in a
rectangular duct with perfectly electrically conducting walls subject to a
uniform transverse magnetic field. A non-standard three dimensional vector
stream function/vorticity formulation is used with Chebyshev collocation method
to solve the eigenvalue problem for small-amplitude perturbations. A relatively
weak magnetic field is found to render the flow linearly unstable as two weak
jets appear close to the centre of the duct at the Hartmann number Ha \approx
9.6. In a sufficiently strong magnetic field, the instability following the
jets becomes confined in the layers of characteristic thickness \delta \sim
Ha^{-1/2} located at the walls parallel to the magnetic field. In this case the
instability is determined by \delta, which results in both the critical
Reynolds and wavenumbers numbers scaling as \sim \delta^{-1}. Instability modes
can have one of the four different symmetry combinations along and across the
magnetic field. The most unstable is a pair of modes with an even distribution
of vorticity along the magnetic field. These two modes represent strongly
non-uniform vortices aligned with the magnetic field, which rotate either in
the same or opposite senses across the magnetic field. The former enhance while
the latter weaken one another provided that the magnetic field is not too
strong or the walls parallel to the field are not too far apart. In a strong
magnetic field, when the vortices at the opposite walls are well separated by
the core flow, the critical Reynolds and wavenumbers for both of these
instability modes are the same: Re_c \approx 642Ha^{1/2}+8.9x10^3Ha^{-1/2} and
k_c \approx 0.477Ha^{1/2}. The other pair of modes, which differs from the
previous one by an odd distribution of vorticity along the magnetic field, is
more stable with approximately four times higher critical Reynolds number.Comment: 16 pages, 8 figures, revised version, to appear in J. Fluid Mec
Contactless Electromagnetic Phase-Shift Flowmeter for Liquid Metals
We present a concept and test results of an eddy-current flowmeter for liquid
metals. The flow rate is determined by applying a weak ac magnetic field to a
liquid metal flow and measuring the flow-induced phase disturbance in the
external electromagnetic field. The phase disturbance is found to be more
robust than that of the amplitude used in conventional eddy-current flowmeters.
The basic characteristics of this type of flowmeter are analysed using simple
theoretical models, where the flow is approximated by a solid body motion.
Design of such a flowmeter is presented and its test results reported.Comment: 19 pages, 13 figures, to appear in Meas. Sci. Technol (final version
Lock-exchange problem for Boussinesq fluids revisited: exact shallow-water solution
An exact solution of the lock-exchange problem for a two-layer shallow-water
system of Boussinesq fluids is obtained using the method of characteristics in
combination with analytic expressions for the Riemann invariants of the
underlying system of two hyperbolic differential equations. The multivaluedness
and instability of the simple-wave solution gives rise to a number of hydraulic
jumps which are resolved by imposing the conservation of mass and momentum. The
respective Rankine-Hugoniot jump conditions contain a free parameter
which defines the relative contribution of each layer to the interfacial
pressure gradient in the generalised shallow-water momentum conservation
equation. We consider the solution produced by 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 and
We also consider an alternative formulation of the
problem where the initial quiescent state is substituted by a gravity current
of certain critical depth which depends on and may form due to the
instability of the original gravity current of a larger depth. The resulting
gravity current speed agrees well with experimental and numerical results when
the front is assumed to collapse to the largest stable height which is produced
by Comment: 19 pages, 11 figure
Elementary model of internal electromagnetic pinch-type instability
We analyse numerically a pinch-type instability in a semi-infinite planar
layer of inviscid conducting liquid bounded by solid walls and carrying a
uniform electric current. Our model is as simple as possible but still captures
the salient features of the instability which otherwise may be obscured by the
technical details of more comprehensive numerical models and laboratory
experiments. Firstly, we show the instability in liquid metals, which are
relatively poor conductors, differs significantly from the
astrophysically-relevant Tayler instability. In liquid metals, the instability
develops on the magnetic response time scale, which depends on the conductivity
and is much longer than the Alfv\'en time scale, on which the Tayler
instability develops in well conducting fluids. Secondly, we show that this
instability is an edge effect caused by the curvature of the magnetic field,
and its growth rate is determined by the linear current density and independent
of the system size. Our results suggest that this instability may affect future
liquid metal batteries when their size reaches a few meters.Comment: 14 pages, 5 figures (to appear in J Fluid Mech
Weakly nonlinear stability analysis of MHD channel flow using an efficient numerical approach
We analyze weakly nonlinear stability of a flow of viscous conducting liquid
driven by pressure gradient in the channel between two parallel walls subject
to a transverse magnetic field. Using a non-standard numerical approach, we
compute the linear growth rate correction and the first Landau coefficient,
which in a sufficiently strong magnetic field vary with the Hartmann number as
and
. These
coefficients describe a subcritical transverse velocity perturbation with the
equilibrium amplitude
which exists at Reynolds numbers below the linear stability threshold
We find that the flow
remains subcritically unstable regardless of the magnetic field strength. Our
method for computing Landau coefficients differs from the standard one by the
application of the solvability condition to the discretized rather than
continuous problem. This allows us to bypass both the solution of the adjoint
problem and the subsequent evaluation of the integrals defining the inner
products, which results in a significant simplification of the method.Comment: 16 pages, 10 figures, revised version (to appear in Phys Fluids
Pseudo–magnetorotational instability in a Taylor-Dean flow between electrically connected cylinders
We consider a Taylor-Dean-type flow of an electrically conducting liquid in
an annulus between two infinitely long perfectly conducting cylinders subject
to a generally helical magnetic field. The cylinders are electrically connected
through a remote, perfectly conducting endcap, which allows a radial electric
current to pass through the liquid. The radial current interacting with the
axial component of magnetic field gives rise to the azimuthal electromagnetic
force, which destabilizes the base flow by making its angular momentum decrease
radially outwards. This instability, which we refer to as the
pseudo--magnetorotational instability (MRI), looks like an MRI although its
mechanism is basically centrifugal. In a helical magnetic field, the radial
current interacting with the azimuthal component of the field gives rise to an
axial electromagnetic force, which drives a longitudinal circulation. First,
this circulation advects the Taylor vortices generated by the centrifugal
instability, which results in a traveling wave as in the helical MRI (HMRI).
However, the direction of travel of this wave is opposite to that of the true
HMRI. Second, at sufficiently strong differential rotation, the longitudinal
flow becomes hydrodynamically unstable itself. For electrically connected
cylinders in a helical magnetic field, hydrodynamic instability is possible at
any sufficiently strong differential rotation. In this case, there is no
hydrodynamic stability limit defined in the terms of the critical ratio of
rotation rates of inner and outer cylinders that would allow one to distinguish
a hydrodynamic instability from the HMRI. These effects can critically
interfere with experimental as well as numerical determination of MRI.Comment: 10 pages, 5 figures, minor revision, to appear in Phys. Rev.