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 field direction too, we represent the flow with Sommeria \& Moreau's (1982, J. Fluid Mech.) model, a two-dimensional shallow water model with linear friction accounting for the effect of the Hartmann layer.
The simplicity of this model makes it possible to study the stability and transient growth of quasi-two dimensional perturbations over an extensive range of parameters up to the limit of high magnetic fields, where the Reynolds number based on the Shercliff layer thickness $Re/H^{1/2}$ becomes the only relevant parameter. Tollmien-Schlichting waves are the most linearly unstable mode, with a further unstable mode $H \gtrsim 42$. The flow is linearly unstable for $Re/H^{1/2}\gtrsim 48350$ and energetically stable for $Re/H^{1/2}\lesssim 65.32$. Between these two bounds, non-modal quasi-two dimensional perturbations undergo significant transient growth (between 2 and 7 times more than in the case of a purely 2D Poiseuille flow, and for much more subcritical values of $Re$). In the limit of a high magnetic field, the maximum gain $G_{max}$ associated to this transient growth varies as $G_{max} \sim (Re/Re_c)^{2/3}$ and occurs at time $t_{Gmax}\sim(Re/Re_c)^{1/3}$ for streamwise wavenumbers of the same order of magnitude as the critical wavenumber for the linear stability.
Decay of honeycomb-generated turbulence in a duct with a static transverse magnetic field is studied via direct numerical simulations. The simulations follow the revealing experimental study of Sukoriansky et al. (1986), in particular the paradoxical observation of high-amplitude velocity fluctuations, which exist in the downstream portion of the flow when the strong transverse magnetic field is imposed in the entire duct including the honeycomb exit, but not in other configurations. It is shown that the fluctuations are caused by the large-scale quasi-two-dimensional structures forming in the…
The downward flow in a vertical duct with one heated and three thermally insulated walls is analyzed numerically using the two-dimensional approximation valid in the asymptotic limit of an imposed strong transverse magnetic field. The work is motivated by the design of liquid metal blankets with poloidal ducts for future nuclear fusion reactors, in which the main component of the magnetic field is perpendicular to the flow direction and very strong heating is applied at the wall facing the reaction chamber. The flow is found to be steady-state or oscillating depending on the strengths of the h…
Linear stability of horizontal, laminar fully developed, quasi-two-dimensional liquid metal duct flow under a transverse magnetic field and heated from below
This manuscript has been accepted for publication in Physical Review Fluids, see https://journals.aps.org/prfluids/accepted/53075Se8O0b1b109b1cc0061b280aaa122f0f92dc. The stability of a pulsatile quasi-two-dimensional duct flow was numerically investigated. Flow was driven, in concert, by a constant pressure gradient and by the synchronous oscillation of the lateral walls. This prototypical setup serves to aid understanding of unsteady magnetohydrodynamic flows in liquid metal coolant ducts subjected to transverse magnetic fields, motivated by the conditions expected in magnetic confinement fu…
This study seeks to elucidate the linear transient growth mechanisms in a uniform duct with square cross-section applicable to flows of electrically conducting fluids under the influence of an external magnetic field. A particular focus is given to the question of whether at high magnetic fields purely two-dimensional mechanisms exist, and whether these can be described by a computationally inexpensive quasi-two-dimensional model. Two Reynolds numbers of $5000$ and $15\,000$ and an extensive range of Hartmann numbers $0 \leq Ha \leq 800$ were investigated. Three broad regimes are identified in…
The cochlea is our fluid-filled organ of hearing with a unique spiral shape. The physiological role of this shape remains unclear. Previous research has paid only little attention to the occurrence of transverse flow in the cochlea, in particular in relation to the cochlea's shape. To better understand its influence on fluid dynamics, this study aims to characterize transverse flow due to harmonically oscillating axial flow in square ducts with curvature and torsion, similar to the shape of human cochleae. Four geometries were investigated to study curvature and torsion effects on axial and tr…
A liquid metal flow in the form of a submerged round jet entering a square duct in the presence of a transverse magnetic field is studied experimentally. A range of high Reynolds and Hartmann numbers is considered. Flow velocity is measured using electric potential difference probes. A detailed study of the flow in the duct's cross-section about seven jet's diameters downstream of the inlet reveals the dynamics, which is unsteady and dominated by high-amplitude fluctuations resulting from instability of the jet. The flow structure and fluctuation properties are largely determined by the value …
Direct numerical simulations and linear stability analysis are carried out to study mixed convection in a horizontal duct with constant-rate heating applied at the bottom and imposed transverse horizontal magnetic field. A two-dimensional approximation corresponding to the asymptotic limit of very strong magnetic field effect is validated and applied, together with full three-dimensional analysis, to investigate the flow's behaviour in the previously unexplored range of control parameters corresponding to typical conditions of a liquid metal blanket of a nuclear fusion reactor (Hartmann number…
We have performed direct numerical simulation of the turbulent flow of a polymer solution in a square duct, with the FENE-P model used to simulate the presence of polymers. First, a simulation at a fixed moderate Reynolds number is performed and its results compared with those of a Newtonian fluid to understand the mechanism of drag reduction and how the secondary motion, typical of the turbulent flow in non-axisymmetric ducts, is affected by polymer additives. Our study shows that the Prandtl's secondary flow is modified by the polymers: the circulation of the streamwise main vortices increas…
In this paper, we present the results of a numerical study of air-water turbulent bubbly flow in a periodic vertical square duct. The study is conducted using a novel numerical technique which leverages Volume of Fluid method for interface capturing and Sharp Surface Force method for accurate representation of the surface tension forces. A three-dimensional geometry construction method is employed during solution of interface equation which gives absolute conservation of mass and sharp interface between gas and liquid phases. The entire algorithm has been implemented on a data parallel mode on…
Linear stability of magnetohydrodynamic flow in a perfectly conducting rectangular duct
本文用数值方法分析了在均匀横向磁场作用下，具有理想导电壁的矩形管道中液态金属流动的线性稳定性。采用非标准的三维矢量流函数/涡量列式和Chebyshev配点法求解小扰动的特征值问题。一个相对较弱的磁场被发现，使流动线性不稳定的两个弱射流出现接近中心的管道在哈特曼数哈\approx 9.6。在足够强的磁场中，射流后的不稳定性被限制在位于平行于磁场的壁处的特征厚度为δ \sim Ha^{-1/2}的层中。在这种情况下，不稳定性由δ确定，这导致临界雷诺数和波数均按\sim δ^{-1}缩放。不稳定模式可以具有沿着和跨越磁场的四种不同对称组合中的一种。最不稳定的是一对模式，其涡量沿磁场均匀分布。这两种模式代表与磁场对齐的强烈非均匀涡流，其在磁场上以相同或相反的方向旋转。只要磁场不太强或平行于磁场的壁相距不太远，前者增强而后者彼此减弱。在强磁场中，当中心流很好地分离了相对壁面上的涡时，这两种不稳定模式的临界雷诺数和波数是相同的：Re_c \approx 642 Ha ^{1/2}+8.9x10^3 Ha ^{-1/2}和k_c \approx 0.477 Ha ^{1/2}。另一对模式与前一对模式的不同之处在于沿磁场的涡旋量呈奇数分布，这对模式在临界雷诺数约为前一对模式的四倍时更加稳定。
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 series of reduced-order numerical simulations on a specific bluff body type (v-gutters) in a subsonic duct flow is done to assess the unsteady wake dynamics. Two of the v-gutter's geometrical parameters are varied: the v-gutter's base angle ($θ$) and the size of a slit ($ξ$) at the leading-edge of the v-gutter. Turbulent flow kinematics and pressure field are analyzed to evaluate the unsteadiness at a freestream Mach number of $M_\infty = 0.25$ and a freestream Reynolds number based on bluff body's transverse length (L) of $Re_L=0.1 \times 10^6$. Five v-gutter angles are considered ($θ$, $^\…
This manuscript has been accepted for publication in Physical Review Fluids, see https://journals.aps.org/prfluids/accepted/d5074S28J6b11905012b7cb06505e8f2149dd5f20. This work investigates the mechanisms that underlie transitions to turbulence in a three-dimensional domain in which the variation of flow quantities in the out-of-plane direction is much weaker than any in-plane variation. This is achieved using a model for the quasi-two-dimensional magnetohydrodynamic flow in a duct with moving lateral walls and an orthogonal magnetic field. In this environment, conventional subcritical routes …
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…
To investigate the mechanism of wave trapping, acoustic embedded trapped modes associated with two-resonant-mode interference in two-dimensional duct-cavity structures are calculated by the feedback-loop closure principle, which allows us to analyse the travelling modes that construct the trapped modes. The exact two coexisting resonant modes that underpin an embedded trapped mode in a cavity open to two semi-infinite ducts are numerically demonstrated. At the interface between the cavity segment and a duct, total reflection can occur for a particular combination of two propagative guided wave…
Institut für Strömungsmechanik und Technische Akustik（流体力学与技术声学研究所）
Acoustic models of resonant duct systems with turbulent flow depend on fitted constants based on expensive experimental test series. We introduce a new model of a resonant cavity, flush mounted in a duct or flat plate, under grazing turbulent flow. Based on previous work by Goody, Howe and Golliard, we present a more universal model where the constants are replaced by physically significant parameters. This enables the user to understand and to trace back how a modification of design parameters (geometry, fluid condition) will affect acoustic properties. The derivation of the model is supporte…
In the framework of the concept of time correlation functions, we develop a self-consistent relaxation theory of the transverse collective particle dynamics in liquids. The theory agrees with well-known results in both the short-wave (free particle dynamics) and the long-wave (hydrodynamic) limits. We obtain a general expression for the spectral density~$C_T(k,ω)$ of transverse particle current realized in the range of wave numbers $k$. In domain of microscopic spatial scales comparable to action scale of effective forces of interparticle interaction, the theory reproduces a transition from a …
