A continuum method for modeling surface tension
01 Jun 1992-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 100, Iss: 2, pp 335-354
TL;DR: In this paper, a force density proportional to the surface curvature of constant color is defined at each point in the transition region; this force-density is normalized in such a way that the conventional description of surface tension on an interface is recovered when the ratio of local transition-reion thickness to local curvature radius approaches zero.
About: This article is published in Journal of Computational Physics.The article was published on 1992-06-01. It has received 7863 citations till now. The article focuses on the topics: Capillary surface & Capillary length.
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TL;DR: In this article , the authors derived basic principles for liquid jet breakup using upscaled nozzles to increase the liquid mass flow rate, and applied these results as reference data, highly resolved numerical simulations have been performed to gain a deeper understanding of the effect of mass flow scaling.
Abstract: Mass flow scaling of gas-assisted coaxial atomizers from laboratory to industrial scale is of major interest for a wide field of applications. However, there is only scarce knowledge and research concerning the effect of atomizer scale-up on liquid breakup and spray characteristics. The main objective of this study is therefore to derive basic principles for liquid jet breakup using upscaled nozzles to increase the liquid mass flow rate [Formula: see text]. For that purpose, atomizers with the same geometrical setup but increased sizes have been designed and experimentally investigated for [Formula: see text], 50, 100, and 500 kg/h, while the aerodynamic Weber number Weaero and gas-to-liquid ratio GLR have been kept constant. The primary jet breakup was recorded via high-speed imaging, and the liquid core length LC and the frequency of the Kelvin–Helmholtz instability fK were extracted. Applying these results as reference data, highly resolved numerical simulations have been performed to gain a deeper understanding of the effect of mass flow scaling. In the case of keeping Weaero and GLR constant, it has been shown by both experiments and simulations that the breakup morphology, given by a pulsating liquid jet with the disintegration of fiber-type liquid fragments, remains almost unchanged with the degree of upscaling n. However, the normalized breakup length [Formula: see text] has been found to be considerably increased with increasing n. The reason has been shown to be the decreased gas flow velocity vgas at the nozzle exit with n, which leads to a decreased gas-to-liquid momentum flux ratio j and an attenuated momentum exchange between the phases. Accordingly, the calculated turbulence kinetic energy of the gas flow and the specific kinetic energy in the liquid phase decrease with n. This corresponds to a decreased fKHI with n or [Formula: see text], respectively, which has been confirmed by both experiments and simulations. The same behavior has been shown for two liquids with different viscosities and at different Weaero. The obtained results allow a first-order estimate of the liquid breakup characteristics, where the influence of nozzle upscaling can be incorporated into j and Reliq in terms of n.
1 citations
01 Jan 2011
TL;DR: In this article, a numerical solution method for multi-phase flow analysis based on local volume and time averaged conservation equations is presented, where the entropy concept analytical reduction to a pressure-velocity coupling is found.
Abstract: This chapter presents a numerical solution method for multi-phase flow analysis based on local volume and time averaged conservation equations. The emphasis of this development was to create a computer code architecture that absorb all the constitutive physics and functionality from the past 25years development of the three fluid multi-component IVA-entropy concept for multiphase flows into a boundary fitted orthogonal coordinate framework. Collocated discretization for the momentum equations is used followed by weighted averaging for the staggered grids resulting in analytical expressions for the normal velocities. Using the entropy concept analytical reduction to a pressure-velocity coupling is found. The performance of the method is demonstrated by comparison of two cases for which experimental results and numerical solution with the previous method are available. The agreement demonstrates the success of this development.
1 citations
TL;DR: Wang et al. as mentioned in this paper found that wall properties have great influence on gas-liquid flow in micro channels not only on flow regime but also flow characteristics, and analyzed the combined effect of curve channel and wall properties on flow resistance.
1 citations
01 May 2009
TL;DR: In this article, a new fluid solver that combines the advantages of both a Lagrangian scheme and an Eulerian scheme was developed for dam breaking, wave breaking in shallow water, impact pressure acting on a vertical wall.
Abstract: We developed a new fluid solver that combines the advantages of both a Lagrangian scheme and an Eulerian scheme. The massless Lagrangian marker particles are put into the Eulerian grid and advected to capture accurately the free surface. The applicability of the present method was demonstrated for dam breaking, wave breaking in shallow water, impact pressure acting on a vertical wall. The efficiency and the accuracy were also investigated. The numerical results showed good agreement with numerical and experimental results performed by other researchers.
1 citations
TL;DR: In this article, a micro-pin-fin based membrane separator was proposed, where an enclosed membrane with micron scale holes was symmetrically populated in a rectangular duct.
1 citations
References
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TL;DR: In this paper, the concept of a fractional volume of fluid (VOF) has been used to approximate free boundaries in finite-difference numerical simulations, which is shown to be more flexible and efficient than other methods for treating complicated free boundary configurations.
11,567 citations
Book•
01 Jan 1967TL;DR: The dynamique des : fluides Reference Record created on 2005-11-18 is updated on 2016-08-08 and shows improvements in the quality of the data over the past decade.
Abstract: Preface Conventions and notation 1. The physical properties of fluids 2. Kinematics of the flow field 3. Equations governing the motion of a fluid 4. Flow of a uniform incompressible viscous fluid 5. Flow at large Reynolds number: effects of viscosity 6. Irrotational flow theory and its applications 7. Flow of effectively inviscid liquid with vorticity Appendices.
11,187 citations
Book•
01 Jan 1978
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
10,258 citations
TL;DR: In this paper, the Navier-Stokes equation is derived for an inviscid fluid, and a finite difference method is proposed to solve the Euler's equations for a fluid flow in 3D space.
Abstract: This brief paper derives Euler’s equations for an inviscid fluid, summarizes the Cauchy momentum equation, derives the Navier-Stokes equation from that, and then talks about finite difference method approaches to solutions. Typical texts for this material are apparently Acheson, Elementary Fluid Dynamics and Landau and Lifschitz, Fluid Mechanics. 1. Basic Definitions We describe a fluid flow in three-dimensional space R as a vector field representing the velocity at all locations in the fluid. Concretely, then, a fluid flow is a function ~v : R× R → R that assigns to each point (t, ~x) in spacetime a velocity ~v(t, ~x) in space. In the special situation where ~v does not depend on t we say that the flow is steady. A trajectory or particle path is a curve ~x : R→ R such that for all t ∈ R, d dt ~x(t) = ~v(t, ~x(t)). Fix a t0 ∈ R; a streamline at time t0 is a curve ~x : R→ R such that for all t ∈ R, d dt ~x(t) = ~v(t0, ~x(t)). In the special case of steady flow the streamlines are constant across times t0 and any trajectory is a streamline. In non-steady flows, particle paths need not be streamlines. Consider the 2-dimensional example ~v = [− sin t cos t]>. At t0 = 0 the velocities all point up and the streamlines are vertical straight lines. At t0 = π/2 the velocities all point left and the streamlines are horizontal straight lines. Any trajectory is of the form ~x = [cos t + C1 sin t + C2] >; this traces out a radius-1 circle centered at [C1 C2] >. Indeed, all radius-1 circles in the plane arise as trajectories. These circles cross each other at many (in fact, all) points. If you find it counterintuitive that distinct trajectories can pass through a single point, remember that they do so at different times. 2. Acceleration Let f : R × R → R be some scalar field (such as temperature). Then ∂f/∂t is the rate of change of f at some fixed point in space. If we precompose f with a 1 Fluid Dynamics Math 211, Fall 2014, Carleton College trajectory ~x, then the chain rule gives us the rate of change of f with respect to time along that curve: D Dt f := d dt f(t, x(t), y(t), z(t)) = ∂f ∂t + ∂f ∂x dx dt + ∂f ∂y dy dt + ∂f ∂z dz dt = ( ∂ ∂t + dx dt ∂ ∂x + dy dt ∂ ∂y + dz dt ∂ ∂z ) f = ( ∂ ∂t + ~v · ∇ ) f. Intuitively, if ~x describes the trajectory of a small sensor for the quantity f (such as a thermometer), then Df/Dt gives the rate of change of the output of the sensor with respect to time. The ∂f/∂t term arises because f varies with time. The ~v ·∇f term arises because f is being measured at varying points in space. If we apply this idea to each component function of ~v, then we obtain an acceleration (or force per unit mass) vector field ~a(t, x) := D~v Dt = ∂~v ∂t + (~v · ∇)~v. That is, for any spacetime point (t, ~x), the vector ~a(t, ~x) is the acceleration of the particle whose trajectory happens to pass through ~x at time t. Let’s check that it agrees with our usual notion of acceleration. Suppose that a curve ~x describes the trajectory of a particle. The acceleration should be d dt d dt~x. By the definition of trajectory, d dt d dt ~x = d dt ~v(t, ~x(t)). The right-hand side is precisely D~v/Dt. Returning to our 2-dimensional example ~v = [− sin t cos t]>, we have ~a = [− cos t − sin t]>. Notice that ~v · ~a = 0. This is the well-known fact that in constant-speed circular motion the centripetal acceleration is perpendicular to the velocity. (In fact, the acceleration of any constant-speed trajectory is perpendicular to its velocity.) 3. Ideal Fluids An ideal fluid is one of constant density ρ, such that for any surface within the fluid the only stresses on the surface are normal. That is, there exists a scalar field p : R × R → R, called the pressure, such that for any surface element ∆S with outward-pointing unit normal vector ~n, the force exerted by the fluid inside ∆S on the fluid outside ∆S is p~n ∆S. The constant density condition implies that the fluid is incompressible, meaning ∇ · ~v = 0, as follows. For any region of space R, the rate of flow of mass out of the region is ∫∫ ∂R ρ~v · ~n dS = ∫∫∫
9,804 citations
"A continuum method for modeling sur..." refers background in this paper
...Fortunately, there is an alternative and computationally much simpler expression for K [6], which one can also derive by considering the net surface force per unit area, F,, , on any given element of the surface S [9]....
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...Examples can be found in the studies of capillarity [2, 31, low-gravity fluid flow [4, 51, hydrodynamic stability [6], surfactant behavior [7, 81, cavitation [9], and droplet dynamics [2] in clouds [lo] and in fuel sprays used in internal combustion engines [ 111....
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TL;DR: In this paper, a new technique is described for the numerical investigation of the time-dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier-Stokes equations are written in finite-difference form, and the solution is accomplished by finite-time step advancement.
Abstract: A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement The primary dependent variables are the pressure and the velocity components Also used is a set of marker particles which move with the fluid The technique is called the marker and cell method Some examples of the application of this method are presented All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired
5,841 citations