In this article the stability loss of the Hartmann flow are investigated by applying the equations for disturbances. The velocity and electric potential quasi-static MHD model is used. The equations allow us to calculate time-dependent disturbance fields using a base flow and an initial disturbance. Two type of initial perturbations are considered: the eigenfunction of the linearized MHD equations and a fluid injection into the flow. These two approaches lead to identical stability results. However, we found a significant difference in the practical implementation of the two approaches. Dealing with the eigenproblem of the linearized MHD system is a laborious task. In terms of calculation costs it is equal to a series of nonlinear perturbation simulations, and if the Hartmann number is increased, the proportion becomes worse. The non-linear stability analysis produced by these two methods shows that the injection technique can also be used in numerical analysis, and that this method is less expensive in terms of calculation costs.
In the article the authors present a numerical method for modelling a laminar-turbulent transition in magnetohydrodynamic flows. The equations in the small magnetic Reynolds numbers approach is considered. Speed, pressure and electrical potential are decomposed to the sum of the state values and the finite amplitude perturbations. A solver based on the Nectar++ framework is described. The authors suggest to use of small-length local disturbances as a transition trigger. They can be imposed by blowing or electrical enforcing. The stability of the Hartmann flow and the flow in the bend are consi…
In this paper, we investigate the magnetohydrodynamic viscous flow with second order slip flow model over a permeable shrinking surface. We have obtained the closed form of exact solution of Navier-Stokes equations by using similarity variable technique. The effects of slip, suction and magnetic parameter have been investigated in detail. The results show that there are two solution branches, namely lower and upper solution branch. The behavior of velocity and shear stress profiles for different values of slip, suction and magnetic parameters has been discussed through graphs.
We derive the flow field disturbance produced by point viscosity variations in a heterogeneous fluid when subject to a background flow while neglecting fluid inertia. The disturbance flow field is found to be identical to that generated by a force-dipole called stresslet. Using a combination of theory and numerical simulations, we show how the hydrodynamics of an active rigid particle is altered due to the presence of point viscosity variations, and how this can be exploited to manipulate and steer them in microfluidic environments.
We report on a numerical study of axisymmetric flow of liquid metal in a circular duct with rectangular cross-section. The flow is forced through the combination of an axial magnetic field and a radial current. Sweeping a wide range of forcing parameters, we identify the different regimes which characterize the flows and explicit the associate scaling laws. Experimental results are interpreted in the light of our numerical simulations.
We examine the response of an inertia-less viscoelastic channel flow to localized perturbations. A simplified model shows that the non-linear interaction between the velocity and elastic stress fields can split an initial pulsed disturbance into two separate pulses if the initial disturbance and the base elastic stress are sufficiently high. In accordance, we demonstrate that a transition to a pulse-splitting regime can be achieved experimentally. These results suggest a possible new direction for studying the elastic instability of viscoelastic channel flows at high elasticity through the gro…
For Gagen-Poiseuille flow, we show that exponential instability (to extremely small, axially symmetric disturbances represented by Galerkin's approximation) is possible only if there exists bi-periodic variability of the disturbances along the pipe axis when the threshold Reynolds number depends on the ratio of two longitudinal periods. Absolute minimum (for) is obtained that corresponds to the observed conditions of transition from the laminar resistance law to the turbulent one and Tollmien-Schlichting waves exciting in the boundary layer.
Laminar flows through pipes driven at steady, pulsatile or oscillatory rates undergo a sub-critical transition to turbulence. We carry out an extensive linear non-modal stability analysis of these flows and show that for sufficiently high pulsation amplitudes the stream-wise vortices of the classic lift-up mechanism are outperformed by helical disturbances exhibiting an Orr-like mechanism. In oscillatory flow, the energy amplification depends solely on the Reynolds number based on the Stokes-layer thickness and for sufficiently high oscillation frequency and Reynolds number, axisymmetric distu…
A general, two-way coupled, point-particle formulation that accounts for the disturbance created by the dispersed particles in obtaining the undisturbed fluid flow field needed for accurate computation of the force closure models is presented. Specifically, equations for the disturbance field created by the presence of particles are first derived based on the inter-phase momentum coupling force in a finite-volume formulation. Solution to the disturbance field is obtained using two approaches: (i) direct computation of the disturbance velocity and pressure using the reaction force due to partic…
We showcase the dynamic mode decomposition (DMD) code developed for applications in two-phase flow analysis. Vertical bubble chain flow in a rectangular vessel filled with liquid gallium is studied without and with applied static horizontal magnetic field (MF) and DMD is applied to the velocity fields computed via volume of fluid simulations. Flow patterns are investigated in the vessel and bubble reference frames. We demonstrate the effect of applied MF and gas flow rate on bubble wake flow and larger scale flow structures within the liquid metal vessel by examining velocity field mode statis…
Comment on "Similarity analysis in magnetohydrodynamics: effects of Hall and ion-slip currents on free convection flow and mass transfer of a gas past a semi-infinite vertical plate," A.A. Megahed, S.R. Komy, A.A. Afify
We study effects of both magnetic field and radiation on unsteady free convective magnetohydrodynamic (MHD) flow under the influence of heat and mass transfer. An implicit finite difference scheme of Crank-Nicolson method is used to analyze the results. The governing boundary layer equations along with the initial and boundary conditions are casted into a dimensionless form which results in a more precise solution. We determine the change in velocity under the influence of radiation and magnetic field with temperature and concentration taken into account. A code is constructed (FORTRAN 6.5) to…
Non-modal transient growth of disturbances in a viscous mixing layer flow is studied for the Reynolds numbers varying from 100 up to 5000 at different streamwise and spanwise wavenumbers. By comparing results of several mathematical approaches, it is concluded that the non-modal optimal disturbances growth results from the discrete part of the spectrum only. It is found that in the linearly unstable configurations the non-modal growth cannot be larger than the exponential one. At the same time, the largest non-modal growth takes place at the wavenumbers for which the mixing layer flow is stabl…
The Okubo [2]-Weiss [3] criterion is recast by using the 2D hydrodynamic Beltrami condition (Shivamoggi et al.[13]) that approximates the slow flow-variation ansatz imposed in its derivation. This turns out to provide an interesting interpretation of the Okubo-Weiss criterion very logically in terms of the topological properties of the underlying vorticity manifold. These developments are then extended to 2D quasi-geostrophic flows (via the potential divorticity framework), magnetohydrodynamic flows and electron magnetohydrodynamic flows (via the generalized magnetic flux framework) and the Ok…
Ongoing development of fuel storage and delivery systems for space probes, interplanetary vehicles, satellites and orbital platforms continues to drive interest in propellant management systems that utilize surface tension to retain, channel and control flow in microgravity environments. Although it has been known for decades that capillary flows offer an ideal method of fuel management, there has been little research devoted to the general stability properties of such flows. In this work, we demonstrate theoretically why capillary flows which channel wetting liquids in slender open triangular…
An integrable localized approximation for interaction of two nearly anti-parallel sheets of the generalized vorticity in 2D ideal electron-magnetohydrodynamic flows
We present the first simultaneous mapping of two-dimensional, time-dependent velocity and pressure fields in a plane Couette flow pervaded by a transverse magnetic field. While electromagnetic forces are strongest in fluids of high electric conductivity such as liquid metals, their opacity excludes optical optical measurement methods. We circumvent this difficulty using a transparent electrolyte (Sulfuric acid), whose weaker conductivity is offset by higher magnetic fields. We describe an experimental rig based on this idea, where the Couette flow is entrained by a tape immersed in sulfuric ac…
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 fr…
Here we build some effective boundary conditions to be used in numerical calculations in order to avoid the thin meshing usually required in problems involving Hartmann layers near a locally plane wall. Wall model are provided for both tangential and normal electric current density and velocity. In particular, a condition on the normal derivative of the tangential velocity is derived. A wide variety of problems is covered as the only restriction is that the magnetic Reynolds number has to be large at the scale of the Hartmann layer. The cases of perfectly conducting or insulating wall are exam…
We investigate the evolution of magnetohydrodynamic perturbations in presence of stochastic noise in rotating shear flows. The particular emphasis is the flows whose angular velocity decreases but specific angular momentum increases with increasing radial coordinate. Such flows, however, are Rayleigh stable, but must be turbulent in order to explain astrophysical observed data and, hence, reveal a mismatch between the linear theory and observations/experiments. The mismatch seems to have been resolved, at least in certain regimes, in presence of weak magnetic field revealing magnetorotational …
We analyze time series stemming from experiments and direct numerical simulations of hydrodynamic and magnetohydrodynamic turbulence. Simulations are done in periodic boxes, but with a volumetric forcing chosen to mimic the geometry of the flow in the experiments, the von Kármán swirling flow between two counter-rotating impellers. Parameters in the simulations are chosen to (within computational limitations) allow comparisons between the experiments and the numerical results. Conducting fluids are considered in all cases. Two different configurations are considered: a case with a weak externa…
Nonmodal amplification of stochastic disturbances in elasticity-dominated channel flows of Oldroyd-B fluids is analyzed in this work. For streamwise-constant flows with high elasticity numbers $μ$ and finite Weissenberg numbers $We$, we show that the linearized dynamics can be decomposed into slow and fast subsystems, and establish analytically that the steady-state variances of velocity and polymer stress fluctuations scale as $O (We^2)$ and $O (We^4)$, respectively. This demonstrates that large velocity variance can be sustained even in weakly inertial stochastically driven channel flows of …
More serious works on 2D2C, 2D3C, 2C2Dcw1C3D, 3D3C, rotating turbulence, thin-layer flows, quasi-static magnetohydrodynamics (QSMHD), and all that are wanted, but we report timely here some studies on locally and globally 2C2Dcw1C3D flows, with the hope to promote smarter and deeper works.
In this paper, the instability of shallow water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow $c-U(y)$ matches the Alfvén wave velocities $\pm B(y)/\sqrt{μρ}$, based on the local magnetic field $B(y)$, the magnetic permeability $μ$ and the mass density of the fluid $ρ$. It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at …
In the theory of hydrodynamic stability, the procedure to decompose an incompressible flow field into its basic motion and disturbances is imprecise and problematic because the disturbances, infinitesimal or finite, are ill-defined quantities in analysis. The linearised equations contravene the first principles of classical mechanics while the disturbance-driven non-linear formulation can hardly be considered as exact science. The notion that unstable and amplified disturbances precipitating the early stages of laminar-turbulent transition is vague, speculative or fundamentally flawed. Similar…
Negative viscosity seems to be an impossible parameter for any thermodynamic system. But for some special boundary conditions the viscosity of a fluid has apparently become negative, like for secondary flow of a fluid or in a plasma flow interacting with a dominant magnetic field. This work studied the effect of negative viscosity for a fluid flow over a cylinder. Four different viscosities are considered, in which the positive viscosities of Air and CO2 has been considered at 300 K temperature and their negative pair of viscosities are considered in this work. The results show a vast differen…
The transport of angular momentum in the outward direction is the fundamental requirement for accretion to proceed in an accretion disc. This objective can be achieved if the accretion flow is turbulent. Instabilities are one of the sources for the turbulence. We study a differentially rotating compressive flow in the presence of non vanishing radial and azimuthal magnetic field and demonstrate the occurrence of a high growth rate instability. This instability operates in a region where magnetic energy density exceeds the rotational energy density.
Two approaches to the problem of transition to turbulence of shear flows are popular in the literature. The first is the linear one of transient growth which focuses on the likely form of the most 'dangerous' (lowest energy) turbulence-triggering disturbances. The second is the nonlinear calculation of the laminar-turbulent boundary which instead focuses on their typical amplitudes. We look to bridge the gap between these two perspectives by considering the fully nonlinear transient growth problem to estimate both the form and amplitude of the most dangerous disturbance. We thereby discover a …
The occurrence of magnetohydrodynamic (MHD) quasiperiodic flows with four fundamental frequencies in differentially rotating spherical geometry is understood in terms of a sequence of bifurcations breaking the azimuthal symmetry of the flow as the applied magnetic field strength is varied. These flows originate from unstable periodic and quasiperiodic states with broken equatorial symmetry but having four-fold azimuthal symmetry. A posterior bifurcation gives rise to two-fold symmetric quasiperiodic states, with three fundamental frequencies, and a further bifurcation to a four-frequency quasi…
We investigate the origins of $k^{-2}$ spectrum in a decaying Taylor-Green magnetohydrodynamic flow with zero large scale magnetic flux that was reported in Lee et al. (2010). A possible candidate for this scaling exponent has been the weak turbulence phenomenology. From our numerical simulations, we observe that current sheets in the magnetic Taylor-Green flow are formed in regions of magnetic discontinuities. Based on this observation and by studying the influence of the current sheets on the energy spectrum, using a filtering technique, we argue that the discontinuities are responsible for …
In this work we present a general strategy for constructing multidimensional Riemann solvers with a single intermediate state, with particular attention paid to detailing the two-dimensional Riemann solver. This is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressur…
We study the transition in dimensionality of a three-dimensional magnetohydrodynamic flow forced only mechanically, when the strength of a magnetic guiding field is gradually increased. We use numerical simulations to consider cases in which the mechanical forcing injects (or not) helicity in the flow. As the guiding field is increased, the strength of the magnetic field fluctuations decrease as a power law of the guiding field intensity. We show that for strong enough guiding fields, the helical magnetohydrodynamic flow can become almost two-dimensional. In this case, the mechanical energy ca…
Various forms of exact laws governing magnetohydrodynamic (MHD) turbulence have been derived either in the incompressibility limit, or for isothermal compressible flows. Here we propose a more general method that allows us to obtain such laws for any turbulent isentropic flow (i.e., constant entropy). We demonstrate that the known MHD exact laws (incompressible and isothermal) and the new (polytropic) one can be obtained as specific cases of the general law when the corresponding closure equation is stated. We also recover all known exact laws of hydrodynamic (HD) turbulence (incompressible, i…
A subcritical transition to turbulence in magnetized Keplerian shear flows is investigated using a statistical approach. Three-dimensional numerical simulations of the shearing box equations with zero net magnetic flux are employed to determine the transition from decaying to sustained turbulence as a function of the magnetic Reynolds number Rm. The results reveal no clear transition to sustained turbulence as the average lifetime of the transients grows as an exponential function of Rm, in accordance with a type-II supertransient law.
We present a generalization of the Bantilan-Ishi-Romatschke (BIR) solution of relativistic hydrodynamics to relativistic magnetohydrodynamics (RMHD). Using the symmetries of the boundary of the Kerr-AdS5 black hole, and certain simplifying assumptions we solve the equations of RMHD on this boundary for a highly conductive fluid. We then transform the resulting solution to the flat spacetime. Furthermore, we show that the force-free condition causes the magnetic field to become singular at particular points and propose a regularization process for removing the singularities. The regularization …
Relaminarization of wall-bounded turbulent flows by means of external static magnetic fields is a long-known phenomenon in the physics of electrically conducting fluids at low magnetic Reynolds numbers. Despite the large literature on the subject, it is not yet completely clear what combination of the Hartmann ($M$) and the Reynolds number has to be used to predict the laminar-turbulent transition in channel or pipe flows fed by upstream turbulent flows free of magnetic perturbations. Relying upon standard phenomenological approaches related to mixing length and structural concepts, we put for…
The magnetohydrodynamic equations system for heavy fluid over an arbitrary surface in shallow water approximation is studied in the present paper. It is shown that simple wave solutions exist only for underlying surfaces that are slopes of constant inclination. All self-similar discontinuous and continuous solutions are found. The exact explicit solutions of initial discontinuity decay problem over a flat plane and a slope are found. It is shown that the initial discontinuity decay solution is represented by one of five possible wave configurations. For each configuration the necessary and suf…
This article deals with rotating magnetohydrodynamic flows of a thin stratified layer of astrophysical plasma in a gravitational field with a free-surface in a vertical external magnetic field. Magnetohydrodynamic equations are obtained in the two-layer shallow water approximation in an external magnetic field when plasma is divided into two layers of different densities. In the beta-plane approximation a system of shallow water equations for rotating stratified plasma in an external magnetic field is obtained. For stationary initial conditions in the form of vertical or horizontal magnetic fi…
We investigate the evolution of magnetohydrodynamic/hydromagnetic perturbations in the presence of stochastic noise in rotating shear flows. The particular emphasis is the flows whose angular velocity decreases but specific angular momentum increases with increasing radial coordinate. Such flows, however, are Rayleigh stable, but must be turbulent in order to explain astrophysical observed data and, hence, reveal a mismatch between the linear theory and observations/experiments. The mismatch seems to have been resolved, atleast in certain regimes, in the presence of weak magnetic field reveali…
We report on a numerical observation of recurrence phenomenon in a three dimensional magneto-hydrodynamic (MHD) plasma for certain classes of initial flow profiles. Our simulations discover such a behaviour for an initial Taylor-Green (TG) type of flow whereas under identical conditions an initial Arnold-Beltrami-Childress (ABC) flow fails to show recurrent behaviour. This difference in the dynamical behaviours is traced to the nature of the Rayleigh quotient in each case. The Rayleigh quotient which is an approximate measure of the number of active degrees of freedom in a dynamical system is …
The emergence of turbulence in shear flows is a well-investigated field. Yet, one of major issues is the apparent contradiction between linear stability analysis quoting a flow to be stable and results from experiments and simulations proving it to be otherwise. There is some success, in particular in astrophysical systems, based on Magneto-Rotational Instability (MRI). However, MRI requires the system to be weakly magnetized, which is not a feature of general magnetohydrodynamic (MHD) flows. Nevertheless, linear perturbations of such flows are nonnormal in nature which argues for an origin of…
