Internal Jumps Research Articles

Magnetic field based finite element method for magneto-static problems with discontinuous electric potential distributions

We introduce two finite element formulations to approximate magneto-static problems with discontinuous electric potential based respectively on the electrical scalar potential and the magnetic field. This work is motivated by our interest in Liquid Metal Batteries (LMBs), a promising technology for storing intermittent renewable sources of energy in large scale energy storage devices. LMBs consist of three liquid layers stably stratified and immiscible, with a light liquid metal on top (negative electrode), a molten salt in the middle (electrolyte) and a heavier liquid metal on bottom (positive electrode). Energy is stored in electrical potential differences that can be modeled as jumps at each electrode-electrolyte interface. This paper focuses on introducing new finite element methods for computing current and potential distributions, which account for internal voltage jumps in liquid metal batteries. Two different formulations that use as primary unknowns the electrical potential and magnetic field, respectively, are presented. We validate them using various manufactured test cases, and discuss their applications for simulating the current distribution during the discharge phase in a liquid metal battery.

Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.

Development of supercritical motion and internal jumps within lock-release radial currents and draining flows

When released from rest in a cylindrical reservoir, relatively dense fluid flows radially and may drain freely over an edge or may propagate along the underlying boundary as a gravity current. Here we show that in both scenarios the unsteady fluid motions develop the same supercritical state, a phenomenon that does not occur in equivalent two-dimensional flows and which strongly influences the ensuing dynamics, even at long times after the release. In particular, we demonstrate the onset and persistence of supercritical outflow when fluid drains over an edge, and the development of internal bores within gravity current flow.

A computational method for rotating, multi-layer spheroids with internal jumps

Abstract We discuss the structure of differentially rotating, multilayer spheroids containing mass-density jumps and rotational discontinuities at the interfaces. The study is based upon a scale-free, numerical method. Polytropic equations of state and cylindrical rotation profiles are assumed. The Bernoulli equation and the Poisson equation for the gravitational potential are solved for each layer separately on a common computational grid. The 2-layer (core-envelope) case is first investigated in detail. We find that the core and the envelope are not, in general, homothetical in shape (cores are more than spherical than the envelope). Besides, the occurence of a mass-density jump all along the interface is prone to a rotational discontinuity (unless the polytropic indices are the same). In particular, for given rotation laws, the mass-density jump is not uniform along the interface. Tests, trends and examples (e.g. false bipolytrope, critical rotation, degenerate configurations) are given. Next, we consider the general case of systems made of ${\cal L}>2$ layers. This includes the full equation set, the virial equation, a comprehensive step-by-step procedure and two examples of tripolytropic systems. The properties observed in the 2-layer case hold for any pairs of adjacent layers. In spite of a different internal structure, two multilayer configurations can share the same mass, same axis ratio, same size and same surface velocity (which is measured through a degeneracy parameter). Applications concern the determination of the interior of planets, exoplanets, stars and compact objects.

Generalized A-GPSFS and modeling of vibrating elements with internal complexities

Mechanical components containing internal complexities in free vibration mode are analyzed through generalized bases made of global piecewise smooth functions. Such bases introduced in 2002, as GPSFs, were made adaptive in 2015 to reduce the computational efforts (A-GPSFs) while, a recent study in 2018 used supplementary analytical conditions along with A-GPSFs to study free vibrations of thin-walled beams and plates. In this paper, generalized bases of A-GPSFs are introduced (i) in a self-contained manner, id est, without needing any supplementary analytical conditions and even (ii) extending the possibility of modeling mechanical components containing internal jumps. It is shown how such generalized A-GPSFs can be used to model continuous functions which have different degrees of smoothness; even discontinuous functions (i.e. containing jumps) can be modeled and always without involving significant approximations or letting rise to any Gibbs phenomenon. Thus, a free vibration analysis of mechanical components including internal complexities of engineering interest becomes part of a simple unified procedure based on a global approximation where unknown eigenvalues and eigenfunctions can be singled out only through generalized bases of A-GPSFs; the results of previous formulations are then obtained as particular cases. The efficiency and ability of the proposed models result from the comparison between calculated eigen-parameters and those of other models presented in the open literature.

Stepwise internal potential jumps caused by multiple-domain polarization flips in metal/ferroelectric/metal/paraelectric/metal stack

Negative capacitance (NC) effects in ferroelectric/paraelectric (FE/PE) stacks have been recently discussed intensively in terms of the steep subthreshold swing (SS) in field-effect transistors (FETs). It is, however, still disputable to stabilize quasi-static-NC effects. In this work, stepwise internal potential jumps in a metal/FE/metal/PE/metal system observed near the coercive voltage of the FE layer are reported through carefully designed DC measurements. The relationship of the internal potential jumps with the steep SS in FETs is also experimentally confirmed by connecting a FE capacitor to a simple metal-oxide-semiconductor FET. On the basis of the experimental results, the observed internal potential jumps are analytically modelled from the viewpoint of bound charge emission associated with each domain flip in a multiple-domain FE layer in a FE/PE stack. This view is different from the original NC concept and should be employed for characterizing FE/PE gate stack FETs.

Modeling discontinuous potential distributions using the finite volume method, and application to liquid metal batteries

The electrical potential in a battery jumps at each electrode-electrolyte interface. We present a model for computing three-dimensional current and potential distributions, which accounts for such internal voltage jumps. Within the framework of the finite volume method we discretize the Laplace and gradient operators such that they account for internal jump boundary conditions. After implementing a simple battery model in OpenFOAM we validate it using an analytical test case, and show its capabilities by simulating the current distribution and discharge curve of a Li‖Bi liquid metal battery.

Validity of judo-specific tests to assess neuromuscular performance of judo athletes

ABSTRACT Uchi-komi Fitness Test (UFT) is a specific judo test that evaluates physiological fitness of judo athletes in similar conditions to judo matches. Neuromuscular parameters obtained by generic and judo-specific tests would aid to get more information about its criterion validity. This study aimed to analyse the relationship between UFT and shoulder external (PTEX) and internal (PTINT) rotation torque, handgrip strength (HGS) and vertical jumps (VJs) performance. The relationship between UFT and Judogi grip strength test (JGST) was also investigated. Eighteen male judo athletes participated in this study. Athletes performed neuromuscular tests (VJ, PTEX, PTINT and HGS) and judo-specific tests (JGST and UFT). Pearson’s correlation was used with the level set at p < 0.05. Significant correlation was found between UFT and all VJ variables (r = 0.50–0.72, p < 0.004), UFT a + b (two first series of UFT) and PTEX (r = 0.49, p = 0.033), UFT and PTINT (r = 0.47, p = 0.044). Also, UFT was correlated to JGST (r = 0.50–0.72, p < 0.044, respectively). We conclude that muscle power of lower limbs, PTEX and PTINT was related to UFT. Strength-endurance in the upper limbs (JGST) was also related to the UFT performance.

A local discontinuous Galerkin scheme for Darcy flow with internal jumps

We present a new version of the local discontinuous Galerkin method which is capable of dealing with jump conditions along a submanifold ΓLG (i.e., Henry’s Law) in instationary Darcy flow. Our analysis accounts for a spatially and temporally varying, non-linear permeability tensor in all estimates which is also allowed to have a jump at ΓLG and gives a convergence order result for the primary and the flux unknowns. In addition to this, different approximation spaces for the primary and the flux unknowns are investigated. The results imply that the most efficient choice is to choose the degree of the approximation space for the flux unknowns one less than that of the primary unknown. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown.

Models of internal jumps and the fronts of gravity currents: unifying two-layer theories and deriving new results

The steady speeds of the front of a gravity current and of an internal jump on a two-layer stratification are often sought in terms of the heights of the relatively dense fluid both up- and downstream from the front or jump, the height of the channel within which they flow, the densities of the two fluids and gravitational acceleration. In this study a unifying framework is presented for calculating the speeds by balancing mass and momentum fluxes across a control volume spanning the front or jump and by ensuring the assumed pressure field is single-valued, which is shown to be equivalent to forming a vorticity balance over the control volume. Previous models have assumed the velocity field is piecewise constant in each layer with a vortex sheet at their interface and invoked explicit or implicit closure assumptions about the dissipative effects to derive the speed. The new formulation yields all of the previously presented expressions and demonstrates that analysing the vorticity balance within the control volume is a useful means of constraining possible closure assumptions, which is arguably more effective than consideration of the flow energetics. However the new approach also reveals that a novel class of models may be developed in which there is shear in the velocity field in the wake downstream of the front or the jump, thus spreading the vorticity over a layer of non-vanishing thickness, rather than concentrating it into a vortex sheet. Mass, momentum and vorticity balances applied over the control volume allow the thickness of the wake and the speed of the front/jump to be evaluated. Results from this vortex-wake model are consistent with published numerical simulations and with data from laboratory experiments, and improve upon predictions from previous formulae. The results may be applied readily to Boussinesq and non-Boussinesq systems and because they arise as simple algebraic expressions, can be straightforwardly incorporated as jump conditions into spatially and temporally varying descriptions of the motion.

Thin-layer models for gravity currents in channels of general cross-section area, a review

We present a brief review of the recent investigations on gravity currents in horizontal channels with non-rectangular cross-section area (such as triangle, $$\bigvee $$ -valley, circle/semi-circle, trapezoid) which occur in nature (e.g., rivers) and constructed environment (tunnels, reservoirs, canals). To be specific, we discuss the propagation of a gravity current (GC) in a horizontal channel along the horizontal coordinate x, with gravity g acting in the $$-z$$ direction, and y the horizontal–lateral coordinate. The bottom and top of the channel are at $$z=0,H$$ . The “standard” problem is concerned with 2D flow in a channel with rectangular (or laterally unbounded) cross-section area (CSA). Recent investigations have successfully extended the standard knowledge to the channels of CSA given by the quite general $$-f_1(z)\le y \le f_2(z)$$ for $$0 \le z \le H$$ . This includes the practical $$\bigvee $$ -valley, triangle, circle/semi-circle and trapezoid; these geometries may be in “up” or “down” setting with respect to gravity, e.g., $$\bigtriangleup $$ and $$\bigtriangledown $$ . The major objective of the extended theory is to predict the height of the interface $$z=h(x,t)$$ and the velocity (averaged over the CSA) u(x, t), where t is time; the prediction includes the speed and position of the nose $$u_N(t), x_N(t)$$ . We show that the motion is governed by a set of simplified equations, called “model,” that provides versatile and insightful solutions and trends. The emphasis in on a high-Reynolds-number current whose motion is dominated by buoyancy–inertia balance; in particular a GC released from a lock, which also contains general effects such as front and internal jumps (shocks), and reflected bore. We discuss two-layer, one-layer, and box models; Boussinesq and non-Boussinesq systems; compositional and particle-driven cases; and the effect of stratification of the ambient fluid. The models are self-contained, and admit realistic initial and boundary conditions. The governing equations are amenable to analytical solutions in some special circumstances. Some salient features of the buoyancy-viscous regime, and the estimate for the length at which transition to this regime takes place, are also presented. Some experimental support to the theory, and open questions for further investigations, are also mentioned. The major conclusions are (1) The CSA geometry has significant influence on the motion of the GC; and (2) The new theory is a useful, very significant, extension of the standard two-dimensional GC problem. The standard current is just a particular case, $$f_{1,2} =$$ constants, among many other covered by the new theory .

Measuring Multivalent Binding Interactions by Isothermal Titration Calorimetry.

Multivalent glycoconjugate-protein interactions are central to many important biological processes. Isothermal titration calorimetry (ITC) can potentially reveal the molecular and thermodynamic basis of such interactions. However, calorimetric investigation of multivalency is challenging. Binding of multivalent glycoconjugates to proteins (lectins) often leads to a stoichiometry-dependent precipitation process due to noncovalent cross-linking between the reactants. Precipitation during ITC titration severely affects the quality of the baseline as well as the signals. Hence, the resulting thermodynamic data are not dependable. We have made some modifications to address this problem and successfully studied multivalent glycoconjugate binding to lectins. We have also modified the Hill plot equation to analyze high quality ITC raw data obtained from multivalent binding. As described in this chapter, ITC-driven thermodynamic parameters and Hill plot analysis of ITC raw data can provide valuable information about the molecular mechanism of multivalent lectin-glycoconjugate interactions. The methods described herein revealed (i) the importance of functional valence of multivalent glycoconjugates, (ii) that favorable entropic effects contribute to the enhanced affinities associated with multivalent binding, (iii) that with the progression of lectin binding, the microscopic affinities of the glycan epitopes of a multivalent glycoconjugate decrease (negative cooperativity), (iv) that lectin binding to multivalent glycoconjugates, especially to mucins, involves internal diffusion jumps, (bind and jump) and (v) that scaffolds of glycoconjugates influence their entropy of binding.

Two-layer shallow-water dam-break solutions for gravity currents in non-rectangular cross-area channels

AbstractWe consider the propagation of a high-Reynolds-number gravity current in a horizontal channel along the horizontal coordinate $x$. The bottom and top of the channel are at $z= 0, H$, and the cross-section is given by the quite general $- {f}_{1} (z)\leq y\leq {f}_{2} (z)$ for $0\leq z\leq H$. We develop a two-layer shallow-water (SW) formulation, and we implement it for the solution of the dam-break problem. The dependent variables are the position of the interface, $h(x, t)$, and the velocity (averaged over the area of the current), $u(x, t)$. The non-rectangular cross-section geometry enters the formulation via $f(h)$ and integrals of $f(z)$ and $zf(z)$, where $f(z)= {f}_{1} (z)+ {f}_{2} (z)$ is the width of the channel. For a given geometry $f(z)$, the free input parameters of the lock-release problem are: (i) the height ratio $H/ {h}_{0} $ of ambient to lock; and (ii) the density ratio of ‘light’ to ‘heavy’ fluids, $R$. We show that the dam-break problem is amenable to solution by the method of characteristics, but various complications (internal jumps, critical restrictions) appear when the return flow in the ambient is significant; these features are not captured by a one-layer shallow-water model. The role of the non-standard geometry in the detection and calculation of the internal and reflected jumps, and on the dam-break flow field, is elucidated. The methodology is illustrated for Boussinesq flow in typical geometries: power-law ($f(z)= b{z}^{\alpha } $, where $b, \alpha $ are positive constants), trapezoidal and circle (full or sector); we solved for full-depth ($H= 1$) and part-depth ($H\gt 1$) lock-release configurations. The approach seems to work well: in all the tested cases, the solution was obtained with simple mathematical and numerical tools (a Runge–Kutta integrator and a secant-method solver); the insights are compatible with our previous understanding of the problem; and the standard (rectangular) case is recovered in the appropriate limit (i.e. when the side-boundaries of the trapezium cross-section are vertical or far apart). A comparison of the speed of propagation with available experiments is also presented, and the agreement is satisfactory. The present solution is a significant generalization of the classical two-layer gravity–current dam-break problem. The classical formulation for a rectangular (or laterally unbounded) channel is now just a particular case, $f(z)= $ const., in the wide domain of cross-sections covered by the new model.

Internal hydraulic jumps in a long narrow lake

High‐resolution three‐dimensional numerical modeling and field observations were used to describe the nonlinear response of Cayuga Lake to surface wind forcing. The degeneration of the basin‐scale internal wave field was characterized according to the composite Froude number (G2), Wedderburn number (WN), and Lake number (LN), which are measures of hydraulic control and bulk and integral wind disturbance force to the baroclinic restoring force, respectively. The typical Cayuga Lake response was a nonlinear surge when ∼ 1 &lt; WN (or LN) &lt; ∼ 2–12 and a surge with emergent nonlinear internal waves when WN or LN &lt; ∼ 2, in agreement with published laboratory studies. An observed shock front was simulated to be an internal hydraulic jump, occurring at midbasin during strong winds when WN &lt; 0.8. To our knowledge, this is the first simulation of a midbasin seiche‐induced hydraulic jump (supported by field data) due to supercritical conditions (G2 &gt; 1) in a lake. The occurrence of the hydraulic jump was correctly predicted using scaling parameters. It was also shown that topographically induced internal hydraulic jumps form when the nonlinear surges interact with a sill‐contraction topographic feature. In contrast to published literature, the observed high‐frequency nonlinear internal waves were preferentially associated with internal jumps, as opposed to steepened internal surges. Computed vertical diffusivities showed mixing was enhanced by two orders of magnitude within both the surges and hydraulic jumps as they propagated through the basin and interacted with topography. Our results can be generalized to other lakes and fjords with similar long‐narrow geometry and topographically separate side basins.

Internal wave shocks in continuously stratified flows with velocity shear

The eigenvalue problem for momentum-conserving internal jumps in a stratified shear flow is developed, and solutions for several classes of flows are considered. It is shown that the presence of velocity shear significantly enhances the parameter range within which energetically-permissible shock solutions exist, and that the maximum allowable amplitude is increased when the ambient state possesses a baroclinic shear. It is also shown that the presence of steep regions in the stratification allow a class of larger-amplitude solutions compared to the case with smooth density gradients.

Upstream Internal Jumps in Stratified Sill Flow: Observations of Formation, Evolution, and Release

Abstract The time-dependent response of upstream undular bores and internal hydraulic jumps from initial formation to eventual release is documented. Two events, characterized by qualitatively different responses, are discussed. In the first case, an undular bore develops upstream of the sill crest. This disturbance remains upstream through the ebb tidal flow but is transformed to a hydraulic jump as its amplitude increases. Toward the end of ebb tide, it is released and subsequently disperses into a group of solitary-like waves. During the second event, an upstream jump also develops at an early stage of the tide. However, it is subsequently swept downstream by the tidal flow such that the upstream region then appears featureless. Approaching slack tide, as an exchange flow becomes established, a large bore or gravity current is emitted. The different responses seen in these two events are interpreted in terms of the Froude number associated with the near-surface stratification.

Internal Hydraulic Jumps and Overturning Generated by Tidal Flow over a Tall Steep Ridge

Abstract Recent observations from the Hawaiian Ridge indicate episodes of overturning and strong dissipation coupled with the tidal cycle near the top of the ridge. Simulations with realistic topography and stratification suggest that this overturning has its origins in transient internal hydraulic jumps that occur below the shelf break at maximum ebb tide, and then propagate up the slope as internal bores when the flow reverses. A series of numerical simulations explores the parameter space of topographic slope, barotropic velocity, stratification, and forcing frequency to identify the parameter regime in which these internal jumps are possible. Theoretical analysis predicts that the tidally driven jumps may occur when the vertical tidal excursion is large, which is shown to imply steep topographic slopes, such that dh/dxN/ω &amp;gt; 1. The vertical length scale of the jumps is predicted to depend on the flow speed such that the jump Froude number is of order unity. The numerical results agree with the theoretical predictions, with finite-amplitude internal hydraulic jumps and overturning forming during strong offslope tidal flow over steep slopes. These results suggest that internal hydraulic jumps may be an important mechanism for local tidally generated mixing at tall steep topography.

Internal hydraulic jumps and mixing in two-layer flows

Internal hydraulic jumps in two-layer flows are studied, with particular emphasis on their role in entrainment and mixing. For highly entraining internal jumps, a new closure is proposed for the jump conditions. The closure is based on two main assumptions: (i) most of the energy dissipated at the jump goes into turbulence, and (ii) the amount of turbulent energy that a stably stratified flow may contain without immediately mixing further is bounded by a measure of the stratification. As a consequence of this closure, surprising bounds emerge, for example on the amount of entrainment that may take place at the location of the jump. These bounds are probably almost achieved by highly entraining internal jumps, such as those likely to develop in dense oceanic over flows. The values obtained here are in good agreement with the existing observations of the spatial development of oceanic downslope currents, which play a crucial role in the formation of abyssal and intermediate waters in the global ocean.

DEVELOPMENT OF CIP-m-FEM IN A FLOW INCLUDING BOTH RAPID AND STREAMING FLOW

Internal waves in a bay have an effect on vertical mixing and mass transfer. Toestimate sources and roles of the internal waves, we have to carry out computation with high resolution by using a three-dimensional non-hydrostatic model. As it is hard to carry out it now, a multi-layer model is considered to be one of the most suitable model. However, internal hydrolic jumps occur frequently on an interface between layers because of small density difference and great velocity (Internal Froude number>1.0). In previous studies, a flow field including both rapid and streaming flow has been solved by CIP-m scheme in which a finite difference method is used. But it is better to use a finite difference method (FEM) because the shape of a bay are complicated and FEM can be applied to any shape. In this study our purpose is to develop CIP-m FEM. The result of an experiment was consistent with that of computation.

Tidal circulation and buoyancy effects in the St. Lawrence Estuary

The action of tides on density‐driven circulation, internal gravity waves, and mixing was investigated in the St. Lawrence Estuary between Rimouski and Québec City. Time‐varying fields of water level, currents and density were computed under typical summer conditions using a three‐dimensional hydrostatic coastal ocean model that incorporates a second order turbulence closure submodel. These results are compared with current meter records and other observations. The model and the observations reveal buoyancy effects produced by tidal forcing. The semi‐diurnal tide raises the isopycnals over the sills at the head of the Laurentian Trough and English Bank, producing internal tides radiating seaward. Relatively dense intermediate waters rise from below 75‐m depth to the near surface over the sills, setting up gravity currents on the inner slopes. Internal hydraulic controls develop over the outer sills; during flood, surface flow separation occurs at the entrances of the Saguenay Fjord and the upper estuary west of Ilet Rouge Bank. Early during ebb flow (restratification), the surface layer deepens to encompass the tops of the sills. As the ebb current intensifies, the model predicts the formation of seaward internal jumps over the outer sills, which were confirmed from acoustic reflection observations. As the internal Froude number increases further, flow separation migrates up to sill height. As a result of these transitions, internal bores emanate from the head region one to two hours before low water. We find that the mixing of oceanic and surface waters near the sills is driven by the vertical shear produced during ebb in the channel south of Ilet Rouge, the shear produced in the bottom gravity flood currents, and, to a lesser extent, the processes over the sills.