Choked flows in open capillary channel

A forced liquid flow through an open capillary channel with free liquid surfaces is investigated. Since the free surfaces can only withstand a certain difference between the liquid pressure and the ambient pressure the flow rate in the channel is limited. The maximum flow rate is achieved when the surfaces collapse. A theoretical approach is presented which shows that the limitation of flow rate occurs due to choking. This theory confirms the results of an experiment performed on board the sounding rocket TEXUS‐37 to measure the maximum stationary volume flux of a forced flow through an open capillary channel.     

Critical Velocities and Flow Rate Limitations in Open Capillary Channel Flows (CCF)

In the present study a forced liquid flow through an open capillary channel is investigated experimentally and numerically under reduced gravity conditions (microgravity). An open capillary channel is a structure that establishes a liquid flow path at low Bond numbers, when the capillary pressure caused by the surface tension force dominates in comparison to the hydrostatic pressure induced by gravitational or residual accelerations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resonance phenomena for water waves in channels of arbitrary cross‐section

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ω at the free surface. It is shown that there is a countably infinite set of frequencies {ω₀, ω₁, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t^1/2 as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary‐value problems in non‐smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Solutions for Initial-Boundary-Value Problems Representing Gravity Currents Arising from Variable Inflow

In this article, we report on theoretical and numerical studies of models for suddenly initiated variable inflow gravity currents in rectangular geometry. These gravity currents enter a lighter, deep ambient fluid at rest at a time‐dependent rate from behind a partially opened lock gate and their subsequent dynamics is modeled in the buoyancy‐inertia regime using ½‐layer shallow water theory. The resistance to flow that is exerted by the ambient fluid on the gravity current is accounted for by a front condition which involves a non‐dimensional parameter that can be chosen in accordance with experimental observations. Flow filament theory is used to arrive at expressions for the variable inflow velocity under the assumptions of an inviscid and incompressible fluid moving through an opening of fixed area which is suddenly opened under a lock gate at one end of a large rectangular tank. The fluid in the lock is subjected to a (possibly) time varying pressure applied uniformly over its surface and the finite movement of the free surface is accounted for. Finding this time‐dependent inflow velocity, which will then serve as a boundary condition for the solution of the shallow‐water equations, involves solving forced non‐linear ordinary differential equations and the form of this velocity equation and its attendant solutions will, in general, rule out finding self‐similar solutions for the shallow‐water equations. The existence of self‐similar solutions requires that the gravity currents have volumes proportional to t ^α , where α≥ 0 and t is the time elapsed from initiation of the flow. This condition requires a point source of fluid with very special properties for which both the area of the gap and the inflow velocity must vary in a related and prescribed time‐dependent manner in order to preserve self‐similarity. These specialized self‐similar solutions are employed here as a check on our numerical approach. In the more natural cases that are treated here in which fluids flow through an opening of fixed dimensions in a container an extra dimensional parameter is introduced thereby ruling out self‐similarity of the solutions for the shallow‐water equations so that the previous analytical approaches to the variable inflow problem, involving the use of phase‐plane analysis, will be inapplicable. The models developed and analyzed here are expected to provide a first step in the study of situations in which a storage container is suddenly ruptured allowing a heavy fluid to debouch at a variable rate through a fixed opening over level terrain. They also can be adapted to the study of other situations where variable inflow gravity currents arise such as, for example, flows of fresh water from spring run‐off into lakes and fjords, flows from volcanoes and magma chambers, discharges from locks and flash floods.     

Irrotational solitary waves with surface-tension

Purely capillary solitary waves cannot be obtained under the systematic shallow water theory developed by Friedrichs. In fact, if we neglect the gravity and take into account the surface-tension only at the free surface and proceed on the lines of Keller, who obtained cnoidal and solitary waves using the Friedrichs’ shallow water systematic theory, we get nothing other than uniform flow. In this article, on the lines of Friedrichs and Hyers, we find the solitary wave motion when surface-tension is also taken into account along with the gravity andgh/U² < 1, whereg is the acceleration due to gravity,h and U are the depth and the horizontal velocity of the liquid at infinity. The solution is sought in the form of infinite series in ascending powers of a suitably defined parameter after giving a stretching in the horizontal direction.     

Effect of weak gravitation on the plane Poiseuille flow of a highly rarefied gas

Plane Poiseuille flow of a highly rarefied gas that flows horizontally in the presence of weak gravitation is studied based on the Boltzmann equation for a hard sphere molecular gas and the diffuse reflection boundary condition. The behavior of the solution in the regime of large mean free path and small strength of gravity is studied numerically based on the one-dimensional Boltzmann equation derived by means of the asymptotic analysis for a slow variation in the flow direction. It is clarified that the effect of weak gravity on the flow is not negligible when the gas is so rarefied that the mean free path is comparable to the maximum range that the molecules travel along the parabolic path within the channel. When the mean free path is much larger than this range, the effect of gravity that makes the molecules fall plays the dominant role in determining the distribution function, and thus the over-concentration in the distribution function as well as the flow velocity does not increase further even if the mean free path is increased. The upper bound of the flow velocity and the mass flow rate of the gas are obtained as a function of the gravitational acceleration.     

Annular liquid jets and other axisymmetric free-surface flows at high Reynolds numbers

A perturbation method based on a long wavelength approximation is used to obtain the leading order equations governing the fluid dynamics of laminar, annular, round and compound liquid jets and liquid films on convex and concave cylindrical surfaces. An approximate, integral balance method is also used to determine the inviscid core and the thickness of the boundary layers of annular liquid jets near the nozzle exit. The steady state equations are transformed into parabolic ones by means of the von Mises transformation and solved in an adaptive, staggered grid to determine the axial velocity distribution and the location of the free surfaces. It is shown that, for free surface flows subject to inertia, gravity and surface tension, there is a contraction near the nozzle which increases as the Reynolds and Froude numbers are decreased, and is nearly independent of the Weber number for Weber numbers larger than about one hundred. It is also shown that this contraction depends on the flow considered, and is larger for films on convex surfaces. It is also shown that, for round jets, the acceleration of the jet's free surface is larger than that of the jet's centerline, although, sufficiently far from the nozzle exit, the axial velocity is uniform across the jet.     

Self-Similarity and the Evaluation of Long-Time Nonhydrostatic Effects in Compositionally Driven Gravity Flows in Deep Surroundings

In the study of compositionally driven gravity currents involving one or more homogeneous fluid layers, it has been customary to adopt the hydrostatic assumption for the pressure field in each layer which, in turn, leads to a depth‐independent horizontal velocity field in each of these layers and significant simplifications to the governing equations. Under this hydrostatic paradigm, each layer will then have its motion governed by the well‐known reduced dimension shallow‐water equations. For the so‐called ‐layer or reduced gravity shallow‐water equations, similarity solutions for fixed volume gravity currents released in rectangular geometry have been found. Very few attempts have been made to evaluate contributions arising from the possible loss of hydrostatic balance in the context of the problems treated using the classic shallow‐water approach. Where such attempts have been pursued, they have usually been carried out in a time‐independent context or using layer‐averaged equations and very small amplitude disturbances. The vast majority of these studies into nonhydrostatic effects do not include any relevant numerical work to assess these effects. In this paper, we develop an approach for evaluating nonhydrostatic contributions to the flow field for bottom gravity currents in deep surroundings and rectangular geometry. Our approach makes no assumptions on the amplitudes of the disturbances and does not depend on layer‐averaging in the governing equations. We seek asymptotic expansions of the solutions to the Euler equations for a shallow fluid by using the small parameter δ², where δ is the aspect ratio of the flow regime. At leading order the equations enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects which we evaluate. Our method for evaluation of these first‐order contributions employs the self‐similar nature of the solution to the leading‐order equations in the new first‐order equations without any vertical averaging procedures being employed.     

A circulating gravity wave in a cylindrical tank

A circular cylindrical vertical tank is partially filled with a liquid (Water) and a gas (air) above it. The top lid of the container rotates around the cylinder axis and induces a flow in the gas and the liquid. Above a critical rotational speed a large amplitude circulating gravity wave forms. The two phase flow problem will be reduced to a free surface single phase flow problem and the critical parameters, when the axis-symmetric flow becomes unstable with respect to the circulating gravity wave will be determined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free-surface instabilities during electromagnetic shaping of liquid metals

Electromagnetic shaping of free surfaces of liquid metals is a well-known EPM technology used in a couple of metallurgic processes like cold crucible melting, semi-levitation, and electromagnetic slit sealing, among others. However, the stability of such free surfaces is the most important problem and stability control is crucial for success. Within this context we investigate experimentally the stability behavior of liquid metal free surfaces submitted to a high-frequency magnetic field. In this case, the induced Lorentz forces act as an electromagnetic pressure directly on the free surface of the liquid met al. We consider three experimental model configurations: (i) Sessile liquid metal drop (ii) liquid metal ring, and (iii) liquid metal disc. In each model experiment, upon increasing the feeding current beyond a certain threshold value, I_C, we observe that the initial surface contour becomes unstable resulting in (i) drop oscillations (ii) electromagnetic pinching and (iii) static disc deformations. In each configuration the threshold value depends in a similar manner on the frequency of the applied magnetic field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces

In this paper, we prove the decay estimates of the velocity and H¹ scattering for the two‐dimensional linearized Euler equations around a class of monotone shear flow in a finite channel. Our result is consistent with the decay rate predicted by Case in 1960. © 2016 Wiley Periodicals, Inc.     

Error analysis of the L2 least‐squares finite element method for incompressible inviscid rotational flows

In this article we analyze the L² least‐squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity‐vorticity‐pressure formulation. The least‐squares functional is defined in terms of the sum of the squared L² norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first‐order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H¹ norm for velocity and pressure and a suboptimal rate of convergence in the L² norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Numerical simulation of 1‐D incompressible laminar micro‐flow convection behaviour with temperature dependent fluid properties

This paper investigates the qualitative behaviour of single‐phase laminar convection for microchannels and conventionallysized channels formed between two parallel plates, captured by a numerical simulation on water flow. The convection parameters are obtained by separate numerical calculations on a series of parallel plates at constant temperatures. The pairs of parallel plates are maintained at progressively greater temperatures, to simulate the condition of increasing fluid temperature in a channel. The governing one‐dimensional (1‐D) momentum and energy equations are formulated to incorporate the dependence on temperature of both fluid viscosity (μ) and thermal conductivity (k). The qualitative behaviour of Nusselt number (Nu) decreasing with increasing Reynolds number (Re), exhibited by reported experimental data in literature, is simulated. Results show that it is practically dif_cult to observe this behaviour in the conventionally‐sized channels, but the effect easily surfaces in microchannels for practical lengths of flow and allowable high heat flux (q^″_W). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Capillary driven Oscillations of a Free Liquid Interface under Non‐Isothermal Conditions

In this study the reorientation behavior of a free liquid interface in a partly filled right circular cylinder upon a step reduction in gravity is investigated experimentally. The experiments focus on the investigation of non‐isothermal boundary conditions on the liquid reorientation. The situation is similar to a spacecraft which enters a ballistic flight after the end of thrust. Heat flux between the liquid propellant and the tank wall occurs and influences the behavior of the liquid reorientation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive finite element solution of the porous medium equation in pressure formulation

The pressure formulation of the porous medium equation has been commonly used in theoretical studies due to its much better regularities than the original formulation. The goal here is to study its use in the adaptive moving mesh finite element solution. The free boundary is traced explicitly through Darcy's law. The method is shown numerically second‐order in space and first‐order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second‐order in the maximum norm. However, numerical results also show that the convergence order in the original variable stays between first‐order and second‐order in L¹ norm or between 0.5th‐order and first‐order in L² norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired.     

Advection of methane in the hydrate zone: model,analysis and examples

A two‐phase two‐component model is formulated for the advective–diffusive transport of methane in liquid phase through sediment with the accompanying formation and dissolution of methane hydrate. This free‐boundary problem has a unique generalized solution in L¹; the proof combines analysis of the stationary semilinear elliptic Dirichlet problem with the nonlinear semigroup theory in Banach space for an m‐accretive multi‐valued operator. Additional estimates of maximum principle type are obtained, and these permit appropriate maximal extensions of the phase‐change relations. An example with pure advection indicates the limitations of these estimates and of the model developed here. We also consider and analyze the coupled pressure equation that determines the advective flux in the transport model. Copyright © 2015 John Wiley & Sons, Ltd.     

An analysis of laminar boundary layers on liquid curtains

A simplified analysis of the laminar boundary layer along an isothermal liquid curtain falling under gravity is presented. The analysis uses a von Kármán-Pohlhausen integral method and includes the effects of gravity, pressure differences, surface tension and nozzle exit geometry on the convergence length of liquid curtains which have applications as chemical reactors and as protection systems in laser fusion reactors. It is shown that the effects of the surrounding gases on the curtain shape and convergence length are small, and that good approximations to the liquid curtain shape can be obtained by using inviscid flow analyses.     

Singularities on Free Surfaces of Fluid Flows

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120° angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120° angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120°. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130° angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.     

Modeling of mold filling of Al gravity casting and validation with X-ray in-situ observation

In the mold filling simulation, element parameters including volume filled ratio, surface dimensionless distance, and surface filled ratio, were proposed to describe the shape and location of free surfaces in DFDM (Direct Finite Difference Method) elements. A model of the filling process was established, specially taking into account the mass, momentum and heat transfer in the vicinity of free surfaces. It was applied to an experimental AC4C (Al–7Si–0.4Mg) gravity casting. With a special X-ray apparatus, in-situ observation and record of actual mold filling process of the casting were carried out. The simulation results were validated and analyzed by comparing with the observation. The liquid flowing in the casting runner and ingate as well as the evolution of free surfaces were analyzed and discussed.     

Analysis of Two Linearized Problems Modeling Viscous Two‐Layer Flows

Two problems that appear in the linearization of certain free boundary value problems of the hydrodynamics of two viscous fluids are studied in the strip‐like domain Π = {x = (x₁, x₂) ∈ ℝ² : x₁ ∈ ℝ¹, (0 < x₂ < h_*) ∨ (h_* < x₂ < 1)}. The first problem arises in the linearization of a two‐layer flow down a geometrically perturbed inclined plane. The second one appears after the linearization of a two‐layer flow in a geometrically perturbed inclined channel with one moving (smooth) wall. For this purpose the unknown flow domain was mapped onto the double strip Π. The arising linear elliptic problems contain additional unknown functions in the boundary conditions. The paper is devoted to the investigation of these boundary problems by studying the asymptotics of the eigenvalues of corresponding operator pencils. It can be proved that the boundary value problems are uniquely solvable in weighted Sobolev spaces with exponential weight. The study of the full (nonlinear) free boundary value problems will be the topic of a forthcoming paper.