arXiv subjects

Jānis Priede

Publications and source records attributed to Jānis Priede.

At least 19 recordsLinked to original sources

arXiv2023

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.

physics.flu-dyn↗

arXiv2023

Realization of Bullard's disc dynamo

We report experimental results from three successful runs of a Bullard-type homopolar disc dynamo. The set-up consisted of a copper disc with a radius of 30 cm and thickness of 3 cm which was placed co-axially beneath a flat, multi-arm spiral coil of the same size and connected to it electrically at the centre and along the circumference by sliding liquid-metal contacts. The magnetic field was measured using Hall probes which were fixed on the top face of the coil. We measured also the radial voltage drop across the coil. When the disc rotation rate reached $Ω\approx 7$ Hz, the magnetic field increased steeply approaching $B_{0}\approx 40$ mT in the central part of the coil. This field was more than two orders of magnitude stronger than the background magnetic field. In the first two runs, the electromagnetic torque braking the disc in the dynamo regime exceeded the breakdown torque of the electric motor driving the disc. As a result, the motor stalled and the dynamo was interrupted. Stalling did not occur in the third run when the driving frequency was set higher and increased faster. We also propose an extended disc dynamo model which qualitatively reproduces the experimental results.

physics.geo-ph↗

arXiv2022

Self-contained two-layer shallow water theory of strong internal bores

We show that interfacial gravity waves comprising strong hydraulic jumps (bores) can be described by a two-layer hydrostatic shallow-water (SW) approximation without invoking additional front conditions. The theory is based on a new SW momentum equation which is derived in locally conservative form containing a free parameter $α.$ This parameter, which defines the relative contribution of each layer to the pressure at the interface, affects only hydraulic jumps but not continuous waves. The Rankine-Hugoniot jump conditions for the momentum and mass conservation equations are found to be mathematically equivalent to the classical front conditions, which were previously thought to be outside the scope of SW approximation. Dimensional arguments suggest that $α$ depends on the density ratio. For nearly equal densities, both layers are expected to affect interfacial pressure with approximately equal weight coefficients, which corresponds to $α\approx0.$ The front propagation velocity for $α=0$ agrees well with experimental and numerical results in a wide range of bore strengths. A remarkably better agreement with high-accuracy numerical results is achieved by $α=\sqrt{5}-2,$ which yields the largest height that a stable gravity current can have.

physics.flu-dyn↗

arXiv2022

Comment on "Linear stability of a rotating channel flow subjected to a static magnetic field" [Phys. Fluids 34, 054116 (2022)]

Recently, Sengupta and Ghosh [Phys. Fluids 34, 054116, (2022)] published a linear stability analysis of a pressure-driven channel flow which is subject to rotation around a spanwise axis and a uniform magnetic field applied in the same direction. Unfortunately, the formulation of the magnetohydrodynamic part of the problem contains an elementary error which makes the obtained results unphysical. The error is due to unfounded omission of the electric potential contribution in the induced electric current which, thus, does not satisfy the charge conservation.

physics.flu-dyn↗

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↗

arXiv2018

Concept of a next-generation electromagnetic phase-shift flowmeter for liquid metals

We present a concept of an electromagnetic phase-shift flowmeter that has a significantly reduced sensitivity to the variation of the electrical conductivity of a liquid metal. A simple theoretical model of the flowmeter is considered where the flow is approximated by a solid finite-thickness conducting layer which moves in the presence of an ac magnetic field. In contrast to the original design [Priede et al., Meas. Sci. Technol. 22 (2011) 055402], where the flow rate is determined by measuring only the phase shift between the voltages induced in two receiving coils, the improved design measures also the phase shift between the sending and the upstream receiving coils. These two phase shifts are referred to as internal and external ones, respectively. We show that the effect of electrical conductivity on the internal phase shift, which is induced by the flow, can be strongly reduced by rescaling it with the external phase shift, which depends mostly on the conductivity of medium. Two different rescalings are found depending on the ac frequency. At low frequencies, when the shielding effect is weak, the effect of conductivity is strongly reduced by rescaling the internal phase shift with the external one squared. At higher frequencies, the same is achieved by rescaling the internal phase shift directly with the external one.

physics.ins-det↗

arXiv2018

Alternative transient eddy-current flowmetering methods for liquid metals

We present a comprehensive numerical analysis of alternative transient eddy-current flowmetering methods for liquid metals. This type of flowmeter operates by tracking eddy-current markers excited by the magnetic field pulses in the flow of a conducting liquid. Using a simple mathematical model, where the fluid flow is replaced by a translating cylinder, a number possible alternative measurement schemes are considered. The velocity of the medium can be measured by tracking zero crossing points and spatial or temporal extrema of the electromotive force (emf) induced by transient eddy currents in the surrounding space. Zero crossing points and spatial extrema of the emf travel synchronously with the medium whereas temporal extrema experience an initial time delay which depends on the conductivity and velocity of the medium. Performance of transient eddy-current flowmetering depends crucially on the symmetry of system. Eddy current asymmetry of a few per cent makes the detection point drift with a velocity corresponding to a magnetic Reynolds number $Rm\sim 0.1.$ With this level of asymmetry transient eddy-current flowmetering can be reliably applicable only to flows with $Rm\gtrsim 0.1.$ A more accurate symmetry adjustment or calibration of flowmeters may be necessary at lower velocities.

physics.ins-det↗

arXiv2017

The effect of finite-conductvity Hartmann walls on the linear stability of Hunt's flow

We analyse numerically the linear stability of the fully developed liquid metal flow in a square duct with insulating side walls and thin electrically conducting horizontal walls with the wall conductance ratio $c=0.01\cdots 1$ subject to a vertical magnetic field with the Hartmann numbers up to $Ha=10^{4}.$ In a sufficiently strong magnetic field, the flow consists of two jets at the side walls walls and a near-stagnant core with the relative velocity $\sim(cHa)^{-1}.$ We find that for $Ha\gtrsim300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $cHa.$ For $c\ll 1,$ the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $Ha\approx30/c.$ In a stronger magnetic field with $cHa\gtrsim 30,$ the destabilizing effect vanishes and the asymptotic results of Priede et al. [J. Fluid Mech. 649, 115, 2010] for the ideal Hunt's flow with perfectly conducting Hartmann walls are recovered.

physics.flu-dyn↗

arXiv2017

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 Alfvén 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.

physics.flu-dyn↗

arXiv2016

Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

This study is concerned with numerical linear stability analysis of liquid metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow which is valid not only for high but also moderate magnetic fields. This solution shows that for low wall conductance ratios $c\ll1,$ an extremely strong magnetic field with the Hartmann number $Ha\sim c^{-4}$ is required to attain the asymptotic flow regime considered in the previous studies. We use a vector stream function/vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1,0.1,0.01$ and Hartmann numbers up to $10^{4}.$ As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the side-wall jets with the characteristic thickness $δ\sim Ha^{-1/2}.$ This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $Re_{c}\sim110Ha^{1/2}$ and $k_{c}\sim0.5Ha^{1/2}.$ The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll1$ is $\bar{Re}_{c}\approx520.$ Although this value is somewhat larger than$\bar{Re}_{c}\approx313$ found by Ting et al. (1991) for the asymptotic side-wall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flows.

physics.flu-dyn↗

arXiv2015

Metamorphosis of helical magnetorotational instability in the presence axial electric current

This paper presents numerical linear stability analysis of a cylindrical Taylor-Couette flow of liquid metal carrying axial electric current in a generally helical external magnetic field. Axially symmetric disturbances are considered in the inductionless approximation corresponding to zero magnetic Prandtl number. Axial symmetry allows us to reveal an entirely new electromagnetic instability. First, we show that the electric current passing through the liquid can extend the range of helical magnetorotational instability (HMRI) indefinitely by transforming it into a purely electromagnetic instability. Two different electromagnetic instability mechanisms are identified. The first is an internal pinch-type instability, which is due to the interaction of the electric current with its own magnetic field. Axisymmetric mode of this instability requires a free-space component of the azimuthal magnetic field. When the azimuthal component of the magnetic field is purely rotational and the axial component is nonzero, a new kind of electromagnetic instability emerges. The latter driven by the interaction of electric current with a weak collinear magnetic field in a quiescent fluid gives rise to a steady meridional circulation coupled with azimuthal rotation.

physics.flu-dyn↗

arXiv2014

Two-dimensional nonlinear travelling waves in magnetohydrodynamic channel flow

The present study is concerned with the 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. Although the magnetic field has a strong stabilizing effect, this flow, similarly to its hydrodynamic counterpart -- plane Poiseuille flow, is known to become turbulent significantly below the threshold predicted by linear stability theory. We investigate the effect of the magnetic field on 2D nonlinear travelling-wave states which are found at substantially subcritical Reynolds numbers starting from $Re_n=2939$ without the magnetic field and from $Re_n\sim6.50\times10^3Ha$ in a sufficiently strong magnetic field defined by the Hartmann number $Ha.$ Although the latter value is by a factor of seven lower than the linear stability threshold $Re_l\sim4.83\times10^4Ha$,it is still more by an order of magnitude higher than the experimentally observed value for the onset of turbulence in the MHD channel flow.

physics.flu-dyn↗

arXiv2013

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 $μ_{1}\sim(0.814-\mathrm{i}19.8)\times10^{-3}\textit{Ha}$ and $μ_{2}\sim(2.73-\mathrm{i}1.50)\times10^{-5}\textit{Ha}^{-4}$. These coefficients describe a subcritical transverse velocity perturbation with the equilibrium amplitude $|A|^{2}=\Re[μ_{1}]/\Re[μ_{2}](\textit{Re}_{c}-\textit{Re})\sim29.8\textit{Ha}^{5}(\textit{Re}_{c}-\textit{Re})$ which exists at Reynolds numbers below the linear stability threshold $\textit{Re}_{c}\sim 4.83\times10^{4}\textit{Ha}.$ 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.

physics.flu-dyn↗

arXiv2013

Fundamental solutions of MHD Stokes flow

A simple analytical solution is obtained for the MHD stokeslet in a homogeneous magnetic field. This solution represents the flow past a small particle and can also be interpreted as the flow sufficiently far away from a body of finite size. Fundamental solutions are found in terms of velocity, pressure and scalar potential distributions for the flows due to either a concentrated force or a current source. The former consists of two basic solutions for the force parallel and transverse to the magnetic field, respectively. All fundamental solutions have the characteristic length scale of the Hartmann boundary layer and two parabolic wakes developing along the magnetic field.

physics.flu-dyn↗

arXiv2013

Feasible homopolar dynamo with sliding liquid-metal contacts

We present a feasible homopolar dynamo design consisting of a flat, multi-arm spiral coil, which is placed above a fast-spinning metal ring and connected to the latter by sliding liquid-metal electrical contacts. Using a simple, analytically solvable axisymmetric model, we determine the optimal design of such a setup. For small contact resistance, the lowest magnetic Reynolds number, Rm~34.6, at which the dynamo can work, is attained at the optimal ratio of the outer and inner radii of the rings Ri/Ro~0.36 and the spiral pitch angle 54.7 degrees. In a setup of two copper rings with the thickness of 3 cm, Ri=10 cm and Ro=30 cm, self-excitation of the magnetic field is expected at a critical rotation frequency around 10 Hz.

physics.geo-ph↗

arXiv2012

Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method

We present a simple technique for avoiding physically spurious eigenmodes that often occur in the solution of hydrodynamic stability problems by the Chebyshev collocation method. The method is demonstrated on the solution of the Orr-Sommerfeld equation for plane Poiseuille flow. Following the standard approach, the original fourth order differential equation is factorised into two second-order equations using a vorticity-type auxiliary variable with unknown boundary values which are then eliminated by a capacitance matrix approach. However the elimination is constrained by the conservation of the structure of matrix eigenvalue problem, it can be done in two basically different ways. A straightforward application of the method results in a couple of physically spurious eigenvalues which are either huge or close to zero depending on the way the vorticity boundary conditions are eliminated. The zero eigenvalues can be shifted to any prescribed value and thus removed by a slight modification of the second approach.

physics.comp-ph↗

arXiv2012

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 δ\sim Ha^{-1/2} located at the walls parallel to the magnetic field. In this case the instability is determined by δ, which results in both the critical Reynolds and wavenumbers numbers scaling as \sim δ^{-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.

physics.flu-dyn↗