The characteristics of two-phase stratified magnetohydrodynamic (MHD) flow in horizontal rectangular ducts are investigated for a system consisting of a conductive liquid and a non-conductive gas. Numerical and analytical solutions of the governing equations for the velocity and induced magnetic field intensity of fully developed laminar MHD flow are obtained for various combinations of bottom- and side-wall conductivities and for different orientations of an externally applied transverse magnetic field. The relevant set of dimensionless parameters governing the problem is identified. Unlike in single-phase MHD flows, the presence of a non-conductive gas layer breaks the flow symmetry, leading to a significantly different dependence of the flow characteristics on duct aspect ratio, wall-conductivity configuration, and the strength and orientation of the applied magnetic field. Using mercury-air flow as a representative test case, the solutions are employed to quantify the influence of the gas phase on the in-situ liquid holdup, velocity field, pressure gradient, flow lubrication, and pumping-power requirements. It is shown that, regardless of the magnetic Reynolds number, these characteristics are strongly affected by the wall-conductivity configuration and by the orientation of the external magnetic field.
The paper presents 3D numerical results for the laminar liquid metal flow in a rectangular duct compared with experimental results. It is shown that the magnetic interaction parameter $N$ is the main parameter governing the flow provided turbulent pulsations are locally suppressed by magnetic field.
The linear stability of stratified two-phase flows in rectangular ducts is studied numerically. The linear stability analysis takes into account all possible infinitesimal three-dimensional disturbances and is carried out by solution of the associated eigenproblem. The neutral stability boundary and the corresponding critical wave number are obtained for liquid - liquid and air - water systems. Depending on the problem parameters, the instability sets in owing to short, intermediate, of long wave most unstable perturbations. Patterns of the most unstable disturbances are reported and discussed…
We present the analytical solution for the two-dimensional velocity and density fields within an approximation for laminar stratified inclined duct (SID) flows where diffusion dominates over inertia in the along-channel momentum equation but it is negligible in the density transport equation. We refer to this approximation as the hydrostatic/gravitational/viscous in momentum and advective in density (HGV-A) approximation due to the leading balances in the governing equations. The analytical solution is valid for laminar flows in a two-layer configuration in the limit of long ducts. Under such …
We carry out Direct Numerical Simulation (DNS) of flows in closed rectangular ducts with several aspect ratios. The Navier-Stokes equations are discretized through a second-order finite difference scheme, with non-uniform grids in two directions. The duct cross-sectional area is maintained constant as well as the flow rate, which allows to investigate which is the appropriate length scale in the Reynolds number for a good scaling in the laminar and in the fully turbulent regimes. We find that the Reynolds number based on the half length of the short side leads to a critical Reynolds number whi…
Solid particles can be transported as a mobile granular bed, known as bed-load, by pressure-driven flows. A common case in industry is the presence of bed-load in stratified gas-liquid flows in horizontal ducts. In this case, an initially flat granular bed may be unstable, generating ripples and dunes. This three-phase flow, although complex, can be modeled under some simplifying assumptions. This paper presents a model for the estimation of some bed-load characteristics. Based on parameters easily measurable in industry, the model can predict the local bed-load flow rates and the celerity and…
The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously-forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms with (i) recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu & Atoufi et al., arXiv:2301.09773, 2023), (ii)…
This paper reports high resolution simulations using a fifth-order weighted essentially non-oscillatory (WENO) scheme with a third order TVD Runge-Kutta time stepping method to examine the features of detonation front and physics in square ducts. The simulations suggest that two and three-dimensional detonation wave front formations are greatly enhanced by the presence of transverse waves. The motion of transverse waves generates triple points (zones of high pressure and large velocity coupled together), which cause the detonation front to become locally overdriven and thus form "hot spots". T…
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 …
Sidewall effects on the onset of interfacial Holmboe waves in stratified exchange flows at high Schmidt number
Univ Toulouse, Toulouse INP, CNRS, IMFT（图卢兹大学，图卢兹国立理工学院，法国国家科学研究中心，流体力学与传热研究所）；Eindhoven University of Technology（埃因霍温理工大学）；Imperial College London（伦敦帝国学院）
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 consider linear and nonlinear waves in a stratified hydrostatic fluid within a channel of variable area, under the restriction of one-dimensional flow. We derive a modified version of Riemann's invariant that is related to the wave luminosity. This quantity obeys a simple dynamical equation in linear theory, from which the rules of wave reflection are easily discerned; and it is adiabatically conserved in the high-frequency limit. Following a suggestion by Lighthill, we apply the linear adiabatic invariant to predict mildly nonlinear waves. This incurs only moderate error. We find that Ligh…
The stability of the flow under the magnetic force is one of the classical problems in fluid mechanics. In this paper, the flow in a rectangular duct with different Hartmann (Ha) number is simulated. The finite volume method and the SIMPLE algorithm are used to solve a system of equations and the energy gradient theory is then used to study the (associated) stability of magnetohydrodynamics (MHD). The flow stability of MHD flow for different Hartmann (Ha) number, from Ha=1 to 40, at the fixed Reynolds number, Re=190 are investigated. The simulation is validated firstly against the simulation i…
We study the experimental properties of exchange flows in a stratified inclined duct (SID), which are simultaneously turbulent, strongly stratified by a mean vertical density gradient, driven by a mean vertical shear, and continuously forced by gravity. We focus on the core shear layer away from the duct walls, where these flows are excellent experimentally-realisable approximations of canonical hyperbolic-tangent stratified shear layers, whose forcing allows mean and turbulent properties to reach quasi steady states. We analyse state-of-the-art data sets of the time-resolved density and veloc…
The dynamics of dilute micron-sized spherical inertial particles in turbulent duct flows is studied by means of direct numerical simulations of the carrier phase turbulence with one-way coupled Lagrangian particles. The geometries are a square and a rectangular duct with width-to-height aspect ratio $AR$ of 3 operating at $Re_{τ,c}=360$ . The present study is designed to determine the effect of turbulence-driven secondary motion on the particle dynamics. Our results show that a weak cross-flow secondary motion significantly changes the cross-sectional map of the particle concentration, mean ve…
Turbulent flows of viscoplastic fluids at high Reynolds numbers have been investigated recently with direct numerical simulations (DNS) but experimental results have been limited. For this reason, we carry out an experimental study of fully turbulent flows of a yield stress fluid in a rectangular aspect ratio channel with a high-resolution laser doppler velocimetry (LDA) setup. We employ aqueous Carbopol solutions, often considered to be a simple yield stress fluid. We formulate different concentrations to address the effect of the rheology of the fluid on the turbulence statistics at an appro…
Soft hydraulics, which addresses the interaction between an internal flow and a compliant conduit, is a central problem in microfluidics. We analyze Newtonian fluid flow in a rectangular duct with a soft top wall at steady state. The resulting fluid--structure interaction (FSI) is formulated for both vanishing and finite flow inertia. At the leading-order in the small aspect ratio, the lubrication approximation implies that the pressure only varies in the streamwise direction. Meanwhile, the compliant wall's slenderness makes the fluid--solid interface behave like a Winkler foundation, with th…
In this Part 2 we study further experimental properties of two-layer exchange flows in a stratified inclined duct (SID), which are turbulent, strongly-stratified, shear-driven, and continuously-forced. We analyse the same state-of-the-art data sets using the same core shear layer methodology as in Part 1, but we focus here on turbulent energetics and mixing statistics. The detailed analysis of kinetic and scalar energy budgets reveals the specificity and scalings of SID turbulence, while energy spectra provide insight into the current strengths and limitations of our experimental data. The ani…
The paper examines the impact of different modelling choices in second-moment closures by assessing model performance in predicting 3-D duct flows. The test-cases (developing flow in a square duct [Gessner F.B., Emery A.F.: {\em ASME J. Fluids Eng.} {\bf 103} (1981) 445--455], circular-to-rectangular transition-duct [Davis D.O., Gessner F.B.: {\em AIAA J.} {\bf 30} (1992) 367--375], and \tsn{S}-duct with large separation [Wellborn S.R., Reichert B.A., Okiishi T.H.: {\em J. Prop. Power} {\bf 10} (1994) 668--675]) include progressively more complex strains. Comparison of experimental data with s…
We consider the influence of transverse confinement on the instability properties of velocity and density distributions reminiscent of those pertaining to exchange flows in stratified inclined ducts, such as the recent experiment of Lefauve et al. (J. Fluid Mech. 848, 508-544, 2018). Using a normal mode streamwise and temporal expansion for flows in ducts with various aspect ratios $B$ and non-trivial transverse velocity profiles, we calculate two-dimensional (2D) dispersion relations with associated eigenfunctions varying in the 'crosswise' direction, in which the density varies, and the span…
A conservative coupled finite difference-boundary element computational procedure for the simulation of turbulent magnetohydrodynamic flow in a straight rectangular duct at finite magnetic Reynolds number is presented. The flow is assumed to be periodic in the streamwise direction and is driven by a mean pressure gradient. The duct walls are considered to be electrically insulating. The co-evolution of the velocity and magnetic fields as described respectively by the Navier-Stokes and the magnetic induction equations, together with the coupling of the magnetic field between the conducting doma…
This work applies resolvent analysis to incompressible flow through a rectangular duct, in order to identify dominant linear energy-amplification mechanisms present in such flows. In particular, we formulate the resolvent operator from linearizing the Navier--Stokes equations about a two-dimensional base/mean flow. The laminar base flow only has a nonzero streamwise velocity component, while the turbulent case exhibits a secondary mean flow (Prandtl's secondary flow of the second kind). A singular value decomposition of the resolvent operator allows for the identification of structures corresp…
Many practical applications exploit an external local magnetic field -- magnetic obstacle -- as an essential part of their constructions. Recently, it has been demonstrated that the flow of an electrically conducting fluid influenced by an external field can show several kinds of recirculation. The present paper reports a 3D numerical study whose some results are compared with an experiment about such a flow in a rectangular duct.
Inspired by the experiment from Moresco \& Alboussière (2004, J. Fluid Mech.), we study the stability of a liquid metal flow in a rectangular, electrically insulating duct with a steady homogeneous transverse magnetic field. The Lorentz force tends to eliminate velocity variations along the magnetic field, leading to a quasi-two dimensional base flow with Hartmann boundary layers near the walls perpendicular to the magnetic field, and Shercliff layers near the walls parallel to the field. Since the Lorentz force strongly opposes the growth of perturbations with a dependence along the magnetic …
Inertial focusing dynamics of spherical particles in curved microfluidic ducts with a trapezoidal cross-section
School of Mathematics and Statistics, Victoria University of Wellington（维多利亚大学惠灵顿校区数学与统计学院）；School of Mathematical Sciences, University of Adelaide（阿德莱德大学数学科学学院）
We analyse numerically the linear stability of the fully developed flow of a liquid metal in a rectangular duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow t…
