Numerical Studies of Droplet Deformation and Breakup


 Beverly Melton
 2 years ago
 Views:
Transcription
1 ILASS Americas 14th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2001 Numerical Studies of Droplet Deformation and Breakup B. T. Helenbrook Department of Mechanical and Aeronautical Engineering Clarkson University, Potsdam, NY Abstract Numerical results are presented on the deformation and stability of liquid droplets in a uniform gaseous flow. The shape response of the droplets is categorized over a wide range of conditions, and a new parameter that predicts this response is identified. The stability limits of the droplets are also investigated. It is shown that the critical Weber number of breakup can be easily predicted by the onset of an exponentially growing oscillatory instability. Predicted critical Weber numbers of breakup are in good agreement with experiments. Introduction Both statistical [1] and Lagrangian [2] spray simulations require information on the behavior of individual liquid droplets in a gaseous flow. This information typically includes a drag law, an evaporation law, a droplet merging model, and a droplet breakup model. In this paper, we perform detailed numerical simulations of nonevaporating, isolated, liquid droplets to better understand droplet deformation and breakup properties. The droplet behavior is examined in a very fundamental flow, a uniform stream. There is a vast amount of experimental data for this flow obtained predominantly from droptower and shocktube experiments [3]. By doing numerical simulations however, we can obtain precise information for an extremely wide range of conditions. This enables us to provide new insights into the experiments and examine conditions that are difficult to study experimentally. A wellknown difficulty in studying droplet behavior is the droplet s extreme sensitivity to surface contaminants [4]. In this work, we study uncontaminated droplets. Most experimental data is to some degree affected by surface contaminants so differences between our results and the experimental data are expected. However, it is important to study uncontaminated liquid droplets because in most spray systems it is difficult or impossible to determine the degree/nature of the contamination. Numerical results for uncontaminated droplets can bound this limit of the droplet behavior. In our future work, we will perform parametric studies of contaminated droplets. There have been several previous numerical studies of droplet behavior [5, 6]. The most relevant is that of Dandy and Leal. They studied axisymmetric falling droplets and categorized the drop shapes, drag response, and flow fields at various conditions. This work extends those efforts including more than 3000 simulations over the entire range of physically relevant conditions. We focus mainly on the drop deformation response and the breakup limits. The ability to perform this study is due to a new numerical algorithm we have developed for twophase flows which is both accurate and efficient and, in addition, can simulate drops at highly deformed conditions such as those that occur near the breakup limits. This algorithm is described in [7]. Formulation The physical problem is that of an axisymmetric liquid drop being driven through a quiescent gas by a body force such as gravity. We study the deformation and stability of the drop at its terminal velocity, U. This problem is relevant not only to systems with gravity, but also to spray systems in which the injected droplet s deformation rate is fast relative to its velocity decay rate. This will be discussed further after we introduce the nondimensional parameters governing the problem. Assuming that the droplet velocities are small relative to the speed of sound, we approximate both the gas and the liquid as incompressible. We also neglect temperature gradients and evaporation. This is appropriate for noncombusting systems and for combusting systems in which droplet deformation
2 and breakup occurs in a zone with nearly constant temperature while evaporation occurs downstream in the vicinity of the flame. We can approximate the above conditions as two fluids with constant densities, ρ l and ρ g, and viscosities, µ l and µ g, separated by an interface with constant surface tension, σ. The subscripts denote either liquid or gas. Both fluids must satisfy the axisymmetric form of the incompressible Navier Stokes equations. At the interface, we enforce the conditions that the interfacial mass flux is zero and that the jump in stress across the interface is balanced by surface tension. For the mathematical form of the governing equations and the interface conditions, see [7]. Physical Parameters If we nondimensionalize the problem, there are four independent parameters. We choose the liquid to gas density ratio ρ l /ρ g, the liquid to gas dynamic viscosity ratio µ l /µ g, the Weber number W e = ρ g Ud 2 /σ, and the Ohnesogre number, Oh = µ l / ρ l σd to describe the problem where d is the volume equivalent diameter of the droplet. The body force on the droplet does not appear in any of the independent parameters because we have assumed the drops are at their terminal velocity. At the terminal velocity, the body force and the drop velocity are not independent. For a given pair of fluids such as hexane/air, the liquid to gas density ratio varies mainly with the gas density which is function of the ambient pressure and temperature. For combustion systems, the pressures of interest are between atmospheric and the critical pressure of the mixture, and the ambient temperature range is between ambient and approximately 2500 K, the approximate adiabatic flame temperature. To investigate the effect of density ratio over this range of conditions, we study density ratios of 5, 50, and 500. The liquid to gas viscosity ratio is primarily a function of the temperature. If we assume that the gas temperature varies between ambient and 2500 K while the liquid temperature is fixed near the boiling temperature, the viscosity ratio varies between 5 and 15. We study the values 5, 10, and 15. It is somewhat inconsistent with our formulation to assume that the gas temperature is much greater than the liquid temperature because we have neglected temperature gradients, but by examining this range we can bound the effect of viscosity ratio. Results are obtained by fixing the Ohnesogre number and increasing the Weber number from small values, usually 0.1 or less, up to the critical Weber number of breakup. The method of determining the critical Weber number of breakup will be discussed in the results. For fixed fluid properties, the Ohnesogre number only varies with drop diameter, thus increasing the Weber number with Oh constant corresponds to an experiment in which drops of a fixed size are driven through a flow at increasing velocities. For a given set of conditions, the simulations determine the steady drop shape, the flow field in and around the drop, and the required body force to hold the drop at the specified conditions. In some cases, a more traditional unsteady simulation is performed where the body force is fixed and we let the drop evolve to its terminal state. The maximum gasphase Reynolds number, ρ g Ud/µ g, we study is 200. Beyond 200, the flow probably becomes nonaxisymmetric based on results for flow over a sphere [8]. This also limits the smallest Ohnesogre number case (largest drop size) for which we can study drop breakup; for Oh < ρ l /ρ g µ l /µ g 1/200, the Reynolds number exceeds 200 before the Weber number reaches unity. In the results, the Ohnesgore number is increased from this minimum value by factors of over a range of 2 to 3 orders of magnitude. Time Scale Analysis To justify our statement that the behavior of falling droplets is predictive of injected drops, we analyze the time scales in the problem. We begin by transforming the governing equations to a coordinate system that moves with the center of mass of the droplet. This transformation results in one additional term in the momentum equation for the axial velocity equation [6], ρ k U c / t, where U c is the center of mass velocity of the drop. If the time scale associated with this term is large relative to the time scale of the droplet response, we can neglect it. Physically, this means that the drop will respond in a quasisteady manner to the instantaneous relative velocity between the drop and the gas even though this velocity is changing with time. To determine the velocity decay rate, we assume that the drag on the drop is similar to that on a solid sphere. In this case, we can approximate the drag using the Stokes law, C D = 24/Re where C D is the coefficient of drag. There are more accurate curvefits for drag on a sphere, but for our purposes, Stokes law is sufficient. Based on this drag, we arrive at an exponential velocity decay rate for the droplet of 18(ρ g /ρ l )ν g /d 2 where ν g is the kinematic
3 viscosity of the gas. To determine the deformation time scale of a liquid droplet, we examine an isolated liquid droplet with no gas effects. The primary mode of oscillation of an isolated liquid droplet can be modeled by a damped spring mass system. The characteristic rates are then λ = 16ν l d 2 ± (16νl d 2 ) 2 64σ ρ l d 3 (1) The constant multiplying the σ term is chosen such that for the inviscid case the oscillation frequency is the same as that predicted by linear analysis [9] for the primary mode of oscillation. We determined the constant multiplying the ν l term by performing numerical calculations of the decay rate of the primary oscillation mode of a viscous drop. This model faithfully represents smallamplitude, primarymode droplet oscillation for all values of ρ l, d, σ, and µ l. The model shows that when Oh = 1/2, the drops are critically damped. For Oh less than 1/2 the decay rate of drop oscillations is 16ν l /d 2. If we divide this by the velocity decay rate of a drop in a gas we arrive at 16/18 µ l /µ g. Thus, in the small Ohnesogre number limit, the drops will respond in a quasisteady manner to the flow if the liquid to gas viscosity ratio is large. For the conditions we are studying this is a reasonable leading order approximation. When the Ohnesogre number is much greater than one, we can expand equation (1) to determine the decay rate of a perturbation as 2σ/(dµ l ). Dividing by the velocity decay rate, we get the ratio Oh 2 /9 µ l /µ g. Thus, when the Ohnesogre number is of order µ l /µ g, the quasisteady approximation breaks down. In this limit, our simulations will only be relevant to falling liquid droplets and not to injected droplets. Results The solutions are calculated on a trapezoidal domain given by the r, z points (0,10), (0,15), (10,7.5), and (10,12.5) with the drop positioned at r, z = (0, 0). At the lower boundary of the domain an inflow condition is enforced with a nondimensional velocity of unity. At the right and upper boundaries, a nostress condition is enforced. We have performed drag calculations for flow over a sphere with the boundaries at various distances from the sphere in [7]. With the boundaries given, the change in drag due to increasing the distance of the boundaries by 25% is less than a percent. Deformation Response We begin the analysis by categorizing the drop deformation response. This will help us to understand the breakup behavior. Figure 1 shows the three most prevalent drop shapes: prolate, oblate, and dimpled. The gas flow direction is from the bottom to the top of the figure. The first two shapes are obviously mutually exclusive while the third can occur in both prolate and oblate drops. Figure 1: Prolate, oblate and dimpled drop shapes. The transition between prolate and oblate can be characterized by the ratio of magnitude of the dynamic pressures in the liquid to the magnitude of the dynamic pressures in the gas. The dynamic pressure in the gas is of order ρ g U 2 with high pressures occurring at the leading and trailing edge of the drop and low pressures occurring at the equator. This tends to cause oblate shapes. Inside the drop, internal circulation also causes high pressures at the leading and trailing edges. This opposes the effects of the gas and tends to cause prolate drop shapes. To estimate the magnitude of the dynamic pressures in the liquid, we must estimate the magnitude of the internal circulation velocity. Stokes solution for flow around a spherical liquid drop predicts that the internal circulation velocity will be of magnitude U/(2 + 2µ l /µ g ). Comparing this to our numerical results, we find that it is fairly predictive even for cases that have large Reynolds numbers. Thus, the liquid to gas dynamic pressure ratio is approximately P l/g = ρ l ρ g ( µg µ l ) 2 (2) where we have assumed the viscosity ratio is large
4 relative to one. Figure 2 shows the length to width aspect ratio of all calculations performed versus P l/g. Each vertical line of data points consists of all the Oh and W e conditions simulated for a single pair of parameters, ρ l /ρ g and µ l /µ g. There are 9 vertical lines in all corresponding to the 3 x 3 matrix of density ratios and viscosity ratios studied. From this figure we see that P l/g is a reasonable predictor of the prolate to oblate transition; cases with P l/g less than one are predominantly oblate while those with P l/g greater than one are prolate. This is true independent of the Weber number and the Ohnesogre number although the magnitude of the deviation from spherical is dependent on these parameters. Length / Width Ratio inary calculations with a contaminant model and shown that the internal circulation pattern is radically different. In some cases, this causes the drop shape to change from prolate when uncontaminated to oblate when contaminated. Thus, the P l/g transition may be only observable in contaminantfree experiments. The dimpled shape is defined by a concave region at the rear of the drop. This phenomenon is primarily due to viscous effects and occurs when the Capillary number, µ g U/σ = W e/re is O(1). To confirm this, on Figure 3 we plot the Capillary number vs. Ohnesogre number of all the calculated points that have a dimpled shape with a dark square and the remaining cases with a gray triangle. A line of constant Weber number is also shown on the plot, this is given by Ca = W e Oh µ g /µ l ρl /ρ g = W e Pl/g Oh. For Oh < P 1/2 l/g, with increasing flow velocity the Weber number will exceed unity before the Capillary number. Thus, the drops will tend to breakup before the Capillary number exceeds unity. This explains why there are no calculations of stable droplets in the upperleft quadrant of the figure. For Oh > P 1/2 l/g we see dimpling when the Capillary number becomes O(1) ρ l / ρ g (µ g / µ l ) 2 Figure 2: Length to width ratio versus the parameter P l/g With increasing Weber number, the prolate drops continue to increase in aspect ratio. In some cases, we find stable axisymmetric solutions with aspect ratios of two or greater. This leads us to question the validity of the axisymmetric assumption. Simple experiments of a falling solid ellipsoid have shown that the ellipsoid tends to align itself with its largest crosssectional area normal to the flow. Thus, we suspect that the prolate cases are not axisymmetrically stable but rather will tumble as they move through the gas. For this reason, we have not tried to use the axisymmetric simulations to determine the breakup limits of prolate drops. According to these results, there should be qualitatively different behaviors for P l/g greater or less than one, however this has not been observed in experiments. This is most likely due to the effect of contaminants. We have performed some prelim Capillary Number We = c Oh Figure 3: Capillary number of drops with dimpled shape. There are two distinct trends in the appearance of dimpling: one showing Ca 1 over a range of Oh and the other showing a linear increase in Ca number from 0.1 to 1.0. This a secondary effect due to P l/g. The cases with P l/g less than one tend to be oblate which flattens the back of the drop and makes it more likely to dimple. These are the cases that give the linear increase in Ca from 0.1
5 to 1.0. The cases with P l/g greater than one tend to be prolate which makes it harder for the end to become dimpled due to the increased opposing curvature. In this case, the Ca must be approximately one independent of Oh. Stability Limits Having classified the deformation behavior, we now examine the stability limits of the droplets. We do not study any prolate cases, P l/g > 1, because these probably do not remain axisymmetric. The stability limits are determined by increasing the Weber number of the drop until the drop becomes unstable. Figure 4 shows two examples of this for ρ l /ρ g = 50, µ l /µ g = 15, and Oh = 0.063, The axes of the plot are the length to width ratio of the drop and the Weber number. For these conditions, P l/g = 0.2 and the drops become oblate as the Weber number is increased. For the case of Oh = 0.063, at W e 14.0 there is transition in stability and the drop begins to oscillate. We have simulated the growth of these oscillations up to a point at which the minimum droplet length is less than 0.2. Some improvements in the numerical algorithm are needed to be able to continue the simulation. However, up to this point the amplitude grows exponentially which leads us to believe that this transition in stability is an accurate predictor of the breakup point of the droplet. Length/Width Ratio Oh = 0.25 Oh = Weber Number Figure 4: Deformation response characterized by the length to width ratio versus Weber number for ρ l /ρ g = 50, µ l /µ g = 15 and Oh = 0.063, The triangle denotes onset of instability The second case, Oh = 0.25, provides further confirmation of this statement. In this case, as the Weber number is increased the solution reaches a turning point. At the turning point solutions are obtained by specifying a body force and then determining the terminal Weber number. As the body force is increased, the terminal velocity initially increases as expected, but beyond the turning point it decreases with increased body force which indicates that the increase in drag with body force is greater than the linear increase in force on the drop. This unexpected phenomenon indicates that there are no stable steady solutions for a single drop beyond the turning point. Thus, we can take the value of the Weber number at the turning point, 12.5, as the critical Weber number of breakup. Further increases in the body force result in a transition to unstable oscillations as observed for Oh = The value of the Weber number at this point is within 2% of the turning point value so either is a reasonable prediction of the breakup limit. Figure 5 shows the critical Weber number of drop stability versus Ohnesogre number. Four cases are shown, ρ l /ρ g, µ l /µ g = (50,15), (5,15), (5,10) and (5,5). There is some scatter in the data because we have only determined the stability boundary to approximately 10% accuracy. The data shows several distinct trends. First, for Oh < 1, the critical Weber number is between 10 and 14 and there is very little sensitivity to the density ratio and viscosity ratio. This is in very good agreement with both shocktube studies and droptower studies [3] of drop breakup. For Oh > µ g /µ l ρl /ρ g, the Capillary number exceeds unity before the Weber number. Thus, for large Oh the Capillary number, Ca = W e Oh µ g /µ l ρl /ρ g is the correct parameter. This explains the sensitivity to the density ratio and viscosity ratio in this limit. If we examine the results in terms of the Capillary number, for Oh > 3 they begin to collapse on Ca 2.0 which confirms that the Capillary number is the relevant parameter in this limit. The breakup modes for Ohnesogre number greater or less than unity are different. For small Oh, the drop flattens then transitions to unstable oscillations. For moderate Oh, the drop develops a broad dimple at the rear of the drop and transitions to unstable oscillations. For large Oh the dimple at the rear of the drop becomes very sharp and difficult to resolve numerically. Because of this, we are not able to determine whether there is a transition to oscillation or not. This limits the maximum Oh for which we can determine the stability limits. If we compare the large Oh trends to those observed in shocktube experiments, we see that they are very different. The shock tube experiments pre
6 References Weber Number ρ l /ρ g, µ l /µ g 50, 15 5, 15 5, 10 5, Ohnesogre Number Figure 5: Stability limits of liquid droplets. dict that the critical Weber number increases at large Oh while our numerical experiments predict that the critical Weber number decreases. As discussed previously, this is because for large Oh the velocity decay rate of the drop is fast relative to the deformation rate. In this limit, the velocity of the drop will decay before the drop has a chance to breakup, and the quasisteady response is not relevant. For falling droplets however, the large Oh limits are correct. [1] M. R. Archambault and C. F. Edwards. In Proceedings of the Eighth International Conference on Liquid Atomization and Spray Systems (ICLASS), pages , Pasadena, CA, July [2] J. C. Oefelein. In Proceedings of ILASS America s 12th Annual Meeting, May [3] L. P. Hsiang and G. M. Faeth. Int. J. Multiphase Flow, 18(5): , [4] R. Clift, J. R. Grace, and M. E. Weber. Bubbles, Drops, and Particles. Academic Press, [5] D. S. Dandy and L. G. Leal. J. Fluid Mech., 208: , [6] R. J. Haywood, M. Renksizbulut, and G. D. Raithby. Numer. Heat Transfer, Part A, 26: , [7] B. T. Helenbrook. accepted to Comp. Meth. App. Mech. Eng., [8] T. A. Johnson and V. C. Patel. J. Fluid Mech., 378:19 70, [9] H. Lamb. Hydrodynamics. Dover Publications, New York, Conclusions We have categorized the behavior of uncontaminated liquid droplets over a wide range of parameters. This has allowed us to obtain new insight into drop behavior including the identification of a nondimensional number, P l/g, that characterizes the deformation and breakup modes of an uncontaminated liquid droplet. We have also seen unexpected new physical phenomena such as the nonmonotonic response of the droplet terminal velocity to body force. Most importantly, we have established that a numerical approach can accurately predict the stability limits of liquid droplets and used this method to provide new insight into dropstability in the large and small Ohnesogre number limits. This approach will eventually allow us to answer many difficult questions about drop breakup such as the effect of unsteady flows, contaminants, droplet timehistory etc...
Lecture 9 Laminar Diffusion Flame Configurations
Lecture 9 Laminar Diffusion Flame Configurations 9.1 Different Flame Geometries and Single Droplet Burning Solutions for the velocities and the mixture fraction fields for some typical laminar flame configurations.
More informationEnhancement of Heat Transfer by an Electric Field for a Drop Translating at Intermediate Reynolds Number
Rajkumar Subramanian M. A. Jog 1 email: milind.jog@uc.edu Department of Mechanical, Industrial, and Nuclear Engineering, University of Cincinnati, Cincinnati, OH 452210072 Enhancement of Heat Transfer
More informationDetailed Numerical Simulation of Liquid Jet in Cross Flow Atomization: Impact of Nozzle Geometry and Boundary Condition
ILASSAmericas 25th Annual Conference on Liquid Atomization and Spray Systems, Pittsburgh, PA, May 23 Detailed Numerical Simulation of Liquid Jet in Cross Flow Atomization: Impact of Nozzle Geometry and
More information12.1 Viscous potential flow (VPF)
1 Energy equation for irrotational theories of gasliquid flow:: viscous potential flow (VPF), viscous potential flow with pressure correction (VCVPF), dissipation method (DM) 1.1 Viscous potential flow
More informationReduced Order Drag Modeling of Liquid Drops
ILASSAmericas 24th Annual Conference on Liquid Atomization and Spray Systems, San Antonio, TX, May 2012 Reduced Order Drag Modeling of Liquid Drops B. T. Helenbrook Department of Mechanical and Aeronautical
More informationReduced Order Modeling of Steady and Unsteady Flow over a Sphere
ILASSAmericas 22nd Annual Conference on Liquid Atomization and Spray Systems, Cincinnati, OH, May 21 Reduced Order Modeling of Steady and Unsteady Flow over a Sphere B. T. Helenbrook and D. R. Witman
More informationPredicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices
ILASS Americas 2th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 27 Predicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices M. Rohani,
More informationThermocapillary Migration of a Drop
Thermocapillary Migration of a Drop An Exact Solution with Newtonian Interfacial Rheology and Stretching/Shrinkage of Interfacial Area Elements for Small Marangoni Numbers R. BALASUBRAMANIAM a AND R. SHANKAR
More informationCCC Annual Report. UIUC, August 19, Argon Bubble Behavior in EMBr Field. Kai Jin. Department of Mechanical Science & Engineering
CCC Annual Report UIUC, August 19, 2015 Argon Bubble Behavior in EMBr Field Kai Jin Department of Mechanical Science & Engineering University of Illinois at UrbanaChampaign Introduction Argon bubbles
More informationLiquid Jet Breakup at Low Weber Number: A Survey
International Journal of Engineering Research and Technology. ISSN 09743154 Volume 6, Number 6 (2013), pp. 727732 International Research Publication House http://www.irphouse.com Liquid Jet Breakup at
More informationSimulating Interfacial Tension of a Falling. Drop in a Moving Mesh Framework
Simulating Interfacial Tension of a Falling Drop in a Moving Mesh Framework Anja R. Paschedag a,, Blair Perot b a TU Berlin, Institute of Chemical Engineering, 10623 Berlin, Germany b University of Massachusetts,
More informationFlow Field and Oscillation Frequency of a Rotating Liquid Droplet
Flow Field and Oscillation Frequency of a Rotating Liquid Droplet TADASHI WATANABE Center for Computational Science and esystems Japan Atomic Energy Agency (JAEA) Tokaimura, Nakagun, Ibarakiken, 3191195
More informationFigure 11.1: A fluid jet extruded where we define the dimensionless groups
11. Fluid Jets 11.1 The shape of a falling fluid jet Consider a circular orifice of a radius a ejecting a flux Q of fluid density ρ and kinematic viscosity ν (see Fig. 11.1). The resulting jet accelerates
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More information2 D.D. Joseph To make things simple, consider flow in two dimensions over a body obeying the equations ρ ρ v = 0;
Accepted for publication in J. Fluid Mech. 1 Viscous Potential Flow By D.D. Joseph Department of Aerospace Engineering and Mechanics, University of Minnesota, MN 55455 USA Email: joseph@aem.umn.edu (Received
More informationDeformation and Secondary Atomization of Droplets in Technical TwoPhase Flows
Institute for Applied Sustainable Science, Engineering & Technology Roland Schmehl Flow Problem Analysis in Oil & Gas Industry Conference Rotterdam, January 2 Deformation and Secondary Atomization of Droplets
More informationFluid Flow, Heat Transfer and Boiling in MicroChannels
L.P. Yarin A. Mosyak G. Hetsroni Fluid Flow, Heat Transfer and Boiling in MicroChannels 4Q Springer 1 Introduction 1 1.1 General Overview 1 1.2 Scope and Contents of Part 1 2 1.3 Scope and Contents of
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationBeyond the Point Particle: LESStyle Filtering of FiniteSized Particles
ILASS Americas th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 7 Beyond the Point Particle: LESStyle Filtering of FiniteSized Particles Brooks Moses and Chris Edwards Department
More informationContents. I Introduction 1. Preface. xiii
Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................
More informationModule 3: "Thin Film Hydrodynamics" Lecture 12: "" The Lecture Contains: Linear Stability Analysis. Some well known instabilities. Objectives_template
The Lecture Contains: Linear Stability Analysis Some well known instabilities file:///e /courses/colloid_interface_science/lecture12/12_1.htm[6/16/2012 1:39:16 PM] Linear Stability Analysis This analysis
More informationChapter 10: Boiling and Condensation 1. Based on lecture by Yoav Peles, Mech. Aero. Nuc. Eng., RPI.
Chapter 10: Boiling and Condensation 1 1 Based on lecture by Yoav Peles, Mech. Aero. Nuc. Eng., RPI. Objectives When you finish studying this chapter, you should be able to: Differentiate between evaporation
More information6. Basic basic equations I ( )
6. Basic basic equations I (4.24.4) Steady and uniform flows, streamline, streamtube One, two, and threedimensional flow Laminar and turbulent flow Reynolds number System and control volume Continuity
More informationLIQUID FILM THICKNESS OF OSCILLATING FLOW IN A MICRO TUBE
Proceedings of the ASME/JSME 2011 8th Thermal Engineering Joint Conference AJTEC2011 March 1317, 2011, Honolulu, Hawaii, USA AJTEC201144190 LIQUID FILM THICKNESS OF OSCILLATING FLOW IN A MICRO TUBE Youngbae
More informationThe Shape of a Rain Drop as determined from the NavierStokes equation John Caleb Speirs Classical Mechanics PHGN 505 December 12th, 2011
The Shape of a Rain Drop as determined from the NavierStokes equation John Caleb Speirs Classical Mechanics PHGN 505 December 12th, 2011 Derivation of NavierStokes Equation 1 The total stress tensor
More informationChapter 10. Solids and Fluids
Chapter 10 Solids and Fluids Surface Tension Net force on molecule A is zero Pulled equally in all directions Net force on B is not zero No molecules above to act on it Pulled toward the center of the
More informationGravitational effects on the deformation of a droplet adhering to a horizontal solid surface in shear flow
PHYSICS OF FLUIDS 19, 122105 2007 Gravitational effects on the deformation of a droplet adhering to a horizontal solid surface in shear flow P. Dimitrakopoulos Department of Chemical and Biomolecular Engineering,
More informationDynamics of Transient Liquid Injection:
Dynamics of Transient Liquid Injection: KH instability, vorticity dynamics, RT instability, capillary action, and cavitation William A. Sirignano University of California, Irvine  Round liquid columns
More informationDetailed Outline, M E 320 Fluid Flow, Spring Semester 2015
Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of OneDimensional Gas Flow The dissipative processes  viscosity (internal friction) and heat conduction  are connected with existence of molecular
More information2.3 The Turbulent Flat Plate Boundary Layer
Canonical Turbulent Flows 19 2.3 The Turbulent Flat Plate Boundary Layer The turbulent flat plate boundary layer (BL) is a particular case of the general class of flows known as boundary layer flows. The
More informationx j r i V i,j+1/2 r Ci,j Ui+1/2,j U i1/2,j Vi,j1/2
Merging of drops to form bamboo waves Yuriko Y. Renardy and Jie Li Department of Mathematics and ICAM Virginia Polytechnic Institute and State University Blacksburg, VA , U.S.A. May, Abstract Topological
More informationDetailed Outline, M E 521: Foundations of Fluid Mechanics I
Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Secondorder tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationAC & DC Magnetic Levitation and SemiLevitation Modelling
International Scientific Colloquium Modelling for Electromagnetic Processing Hannover, March 2426, 2003 AC & DC Magnetic Levitation and SemiLevitation Modelling V. Bojarevics, K. Pericleous Abstract
More informationBoundary Conditions in Fluid Mechanics
Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial
More informationNonlinear oscillations and rotations of a liquid droplet
Nonlinear oscillations and rotations of a liquid droplet Tadashi Watanabe watanabe.tadashi66@jaea.go.jp Abstract Nonlinear oscillations and rotations of a liquid droplet are simulated numerically by solving
More informationMass and momentum transport from a sphere in steady and oscillatory flows
PHYSICS OF FLUIDS VOLUME 14, NUMBER 11 NOVEMBER 2002 Mass and momentum transport from a sphere in steady and oscillatory flows H. M. Blackburn a) CSIRO Manufacturing and Infrastructure Technology, P.O.
More informationApplication of the immersed boundary method to simulate flows inside and outside the nozzles
Application of the immersed boundary method to simulate flows inside and outside the nozzles E. Noël, A. Berlemont, J. Cousin 1, T. Ménard UMR 6614  CORIA, Université et INSA de Rouen, France emeline.noel@coria.fr,
More informationPaper ID ICLASS EXPERIMENTS ON BREAKUP OF WATERINDIESEL COMPOUND JETS
ICLASS2006 Aug.27Sept.1, 2006, Kyoto, Japan Paper ID ICLASS06047 EXPERIMENTS ON BREAKUP OF WATERINDIESEL COMPOUND JETS ShengLin Chiu 1, RongHorng Chen 2, JenYung Pu 1 and TaHui Lin 1,* 1 Department
More informationThe effect of momentum flux ratio and turbulence model on the numerical prediction of atomization characteristics of air assisted liquid jets
ILASS Americas, 26 th Annual Conference on Liquid Atomization and Spray Systems, Portland, OR, May 204 The effect of momentum flux ratio and turbulence model on the numerical prediction of atomization
More informationOblique Drop Impact on Deep and Shallow Liquid
1 2 3 4 5 6 7 8 9 11 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Commun. Comput. Phys. doi: 10.4208/cicp.XXX.XXX Oblique Drop Impact on Deep and Shallow Liquid B. Ray 1, G. Biswas 1,2, and A. Sharma 3
More informationMass flow determination in flashing openings
Int. Jnl. of Multiphysics Volume 3 Number 4 009 40 Mass flow determination in flashing openings Geanette Polanco Universidad Simón Bolívar Arne Holdø Narvik University College George Munday Coventry University
More informationCorrection of Lamb s dissipation calculation for the effects of viscosity on capillarygravity waves
PHYSICS OF FLUIDS 19, 082105 2007 Correction of Lamb s dissipation calculation for the effects of viscosity on capillarygravity waves J. C. Padrino and D. D. Joseph Aerospace Engineering and Mechanics
More information Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)
2.20  Marine Hydrodynamics, Spring 2005 Lecture 4 2.20  Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities
More informationCENG 501 Examination Problem: Estimation of Viscosity with a Falling  Cylinder Viscometer
CENG 501 Examination Problem: Estimation of Viscosity with a Falling  Cylinder Viscometer You are assigned to design a fallingcylinder viscometer to measure the viscosity of Newtonian liquids. A schematic
More informationPrototype Instabilities
Prototype Instabilities David Randall Introduction Broadly speaking, a growing atmospheric disturbance can draw its kinetic energy from two possible sources: the kinetic and available potential energies
More informationEffect of density ratio on the secondary breakup: A numerical study
ICLASS 2018, 14 th Triennial International Conference on Liquid Atomization and Spray Systems, Chicago, IL, USA, July 2226, 2018 Effect of density ratio on the secondary breakup: A numerical study Suhas.
More informationFluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition
Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow
More informationAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS Hierarchy of Mathematical Models 1 / 29 AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS 2 / 29
More informationModeling of dispersed phase by Lagrangian approach in Fluent
Lappeenranta University of Technology From the SelectedWorks of Kari Myöhänen 2008 Modeling of dispersed phase by Lagrangian approach in Fluent Kari Myöhänen Available at: https://works.bepress.com/kari_myohanen/5/
More informationDetailed 3D modelling of mass transfer processes in two phase flows with dynamic interfaces
Detailed 3D modelling of mass transfer processes in two phase flows with dynamic interfaces D. Darmana, N.G. Deen, J.A.M. Kuipers Fundamentals of Chemical Reaction Engineering, Faculty of Science and Technology,
More informationViscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids
PHYSICS OF FLUIDS 17, 052105 2005 Viscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids. Wang and D. D. oseph a Department of Aerospace Engineering
More informationUNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of boundary layer Thickness and classification Displacement and momentum thickness Development of laminar and turbulent flows in circular pipes
More informationBasic Fluid Mechanics
Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible
More information, where the function is equal to:
Paper ID ILASS08000 ILASS0894 ILASS 2008 Sep. 810, 2008, Como Lake, Italy BINARY COLLISION BETWEEN UNEQUAL SIZED DROPLETS. A NUMERICAL INVESTIGATION. N. Nikolopoulos 1, A. Theodorakakos 2 and G. Bergeles
More informationNUMERICAL SIMULATION OF UNSTEADY CAVITATING FLOWS
NUMERICAL SIMULATION OF UNSTEADY CAVITATING FLOWS Charles C.S. Song and Qiao Qin St. Anthony Falls Laboratory, University of Minnesota Mississippi River at 3 rd Ave. SE, Minneapolis, MN 55414, USA ABSTRACT
More informationNumerical simulation of wave breaking in turbulent twophase Couette flow
Center for Turbulence Research Annual Research Briefs 2012 171 Numerical simulation of wave breaking in turbulent twophase Couette flow By D. Kim, A. Mani AND P. Moin 1. Motivation and objectives When
More informationInvestigation of an implicit solver for the simulation of bubble oscillations using Basilisk
Investigation of an implicit solver for the simulation of bubble oscillations using Basilisk D. Fuster, and S. Popinet Sorbonne Universités, UPMC Univ Paris 6, CNRS, UMR 79 Institut Jean Le Rond d Alembert,
More informationRaindrop shape determined by computing steady axisymmetric solutions for NavierStokes equations
P.26 Raindrop shape determined by computing steady axisymmetric solutions for NavierStokes equations James Q. Feng and Kenneth V. Beard *Boston Scientific Corporation, 3 Scimed Place, Maple Grove, Minnesota
More informationDirect Numerical Simulation of Single Bubble Rising in Viscous Stagnant Liquid
Direct Numerical Simulation of Single Bubble Rising in Viscous Stagnant Liquid Nima. Samkhaniani, Azar. Ajami, Mohammad Hassan. Kayhani, Ali. Sarreshteh Dari Abstract In this paper, direct numerical simulation
More informationModeling Complex Flows! Direct Numerical Simulations! Computational Fluid Dynamics!
http://www.nd.edu/~gtryggva/cfdcourse/! Modeling Complex Flows! Grétar Tryggvason! Spring 2011! Direct Numerical Simulations! In direct numerical simulations the full unsteady NavierStokes equations
More informationSimulation of Liquid Jet Breakup Process by ThreeDimensional Incompressible SPH Method
Seventh International Conference on Computational Fluid Dynamics (ICCFD7), Big Island, Hawaii, July 913, 212 ICCFD7291 Simulation of Liquid Jet Breakup Process by ThreeDimensional Incompressible SPH
More informationarxiv: v1 [physics.fludyn] 8 Mar 2018
Secondary breakup of drops at moderate Weber numbers: Effect of Density ratio and Reynolds number Suhas S Jain a,1, Neha Tyagi a, R. Surya Prakash a, R. V. Ravikrishna a, Gaurav Tomar a, a Department of
More informationSelfExcited Vibration in Hydraulic Ball Check Valve
SelfExcited Vibration in Hydraulic Ball Check Valve L. Grinis, V. Haslavsky, U. Tzadka Abstract This paper describes an experimental, theoretical model and numerical study of concentrated vortex flow
More informationKelvin Helmholtz Instability
Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid
More informationOn the displacement of threedimensional fluid droplets adhering to a plane wall in viscous pressuredriven flows
J. Fluid Mech. (2001), vol. 435, pp. 327 350. Printed in the United Kingdom c 2001 Cambridge University Press 327 On the displacement of threedimensional fluid droplets adhering to a plane wall in viscous
More informationDevelopment and analysis of a LagrangeRemap sharp interface solver for stable and accurate atomization computations
ICLASS 2012, 12 th Triennial International Conference on Liquid Atomization and Spray Systems, Heidelberg, Germany, September 26, 2012 Development and analysis of a LagrangeRemap sharp interface solver
More informationForce analysis of underwater object with supercavitation evolution
Indian Journal of GeoMarine Sciences Vol. 42(8), December 2013, pp. 957963 Force analysis of underwater object with supercavitation evolution B C Khoo 1,2,3* & J G Zheng 1,3 1 Department of Mechanical
More informationMultiphysics CFD simulation of threephase flow with MPS method
APCOM & ISCM 1114 th December, 2013, Singapore Abstract Multiphysics CFD simulation of threephase flow with MPS method *Ryouhei Takahashi¹, Makoto Yamamoto 2 and Hiroshi Kitada 1 1 CMS Corporation,
More informationdynamics of f luids in porous media
dynamics of f luids in porous media Jacob Bear Department of Civil Engineering Technion Israel Institute of Technology, Haifa DOVER PUBLICATIONS, INC. New York Contents Preface xvii CHAPTER 1 Introduction
More information3. FORMS OF GOVERNING EQUATIONS IN CFD
3. FORMS OF GOVERNING EQUATIONS IN CFD 3.1. Governing and model equations in CFD Fluid flows are governed by the NavierStokes equations (NS), which simpler, inviscid, form is the Euler equations. For
More informationBefore we consider two canonical turbulent flows we need a general description of turbulence.
Chapter 2 Canonical Turbulent Flows Before we consider two canonical turbulent flows we need a general description of turbulence. 2.1 A Brief Introduction to Turbulence One way of looking at turbulent
More informationFluid Mechanics. Spring 2009
Instructor: Dr. YangCheng Shih Department of Energy and Refrigerating AirConditioning Engineering National Taipei University of Technology Spring 2009 Chapter 1 Introduction 11 General Remarks 12 Scope
More informationME 144: Heat Transfer Introduction to Convection. J. M. Meyers
ME 144: Heat Transfer Introduction to Convection Introductory Remarks Convection heat transfer differs from diffusion heat transfer in that a bulk fluid motion is present which augments the overall heat
More informationChapter 1: Basic Concepts
What is a fluid? A fluid is a substance in the gaseous or liquid form Distinction between solid and fluid? Solid: can resist an applied shear by deforming. Stress is proportional to strain Fluid: deforms
More informationINTERNAL GRAVITY WAVES
INTERNAL GRAVITY WAVES B. R. Sutherland Departments of Physics and of Earth&Atmospheric Sciences University of Alberta Contents Preface List of Tables vii xi 1 Stratified Fluids and Waves 1 1.1 Introduction
More informationNO SPLASH ON THE MOON
UNIVERSITY OF LJUBLJANA Faculty of Mathematics and Physics Department of Physics NO SPLASH ON THE MOON Mentor: prof. Dr. RUDOLF PODGORNIK Ljubljana, February 2007 ABSTRACT The basic description of a droplet
More information3 BUBBLE OR DROPLET TRANSLATION
3 BUBBLE OR DROPLET TRANSLATION 3.1 INTRODUCTION In the last chapter it was assumed that the particles were rigid and therefore were not deformed, fissioned or otherwise modified by the flow. However,
More informationLinear analysis of threedimensional instability of nonnewtonian liquid jets
J. Fluid Mech. (2006), vol. 559, pp. 451 459. c 2006 Cambridge University Press doi:10.1017/s0022112006000413 Printed in the United Kingdom 451 Linear analysis of threedimensional instability of nonnewtonian
More informationEFFECTS OF INTERFACE CONTAMINATION ON MASS TRANSFER INTO A SPHERICAL BUBBLE
Journal Abdellah of Chemical Saboni, Technology Silvia Alexandrova, Metallurgy, Maria 50, Karsheva 5, 015, 589596 EFFECTS OF INTERFACE CONTAMINATION ON MASS TRANSFER INTO A SPHERICAL BUBBLE Abdellah Saboni
More informationChapter 5 Control Volume Approach and Continuity Equation
Chapter 5 Control Volume Approach and Continuity Equation Lagrangian and Eulerian Approach To evaluate the pressure and velocities at arbitrary locations in a flow field. The flow into a sudden contraction,
More informationheat transfer process where a liquid undergoes a phase change into a vapor (gas)
TwoPhase: Overview TwoPhase twophase heat transfer describes phenomena where a change of phase (liquid/gas) occurs during and/or due to the heat transfer process twophase heat transfer generally considers
More informationTopics in Other Lectures Droplet Groups and Array Instability of Injected Liquid Liquid FuelFilms
Lecture Topics Transient Droplet Vaporization Convective Vaporization Liquid Circulation Transcritical Thermodynamics Droplet Drag and Motion Spray Computations Turbulence Effects Topics in Other Lectures
More informationIHTC DRAFT MEASUREMENT OF LIQUID FILM THICKNESS IN MICRO TUBE ANNULAR FLOW
DRAFT Proceedings of the 14 th International Heat Transfer Conference IHTC14 August 813, 2010, Washington D.C., USA IHTC1423176 MEASUREMENT OF LIQUID FILM THICKNESS IN MICRO TUBE ANNULAR FLOW Hiroshi
More informationLecture notes Breakup of cylindrical jets Singularities and selfsimilar solutions
Lecture notes Breakup of cylindrical jets Singularities and selfsimilar solutions by Stephane Popinet and Arnaud Antkowiak June 8, 2011 Table of contents 1 Equations of motion for axisymmetric jets...........................
More informationTurbulence Modeling I!
Outline! Turbulence Modeling I! Grétar Tryggvason! Spring 2010! Why turbulence modeling! Reynolds Averaged Numerical Simulations! Zero and One equation models! Two equations models! Model predictions!
More informationFundamentals of Fluid Mechanics
Sixth Edition Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department
More informationNUMERICAL INVESTIGATION OF THERMOCAPILLARY INDUCED MOTION OF A LIQUID SLUG IN A CAPILLARY TUBE
Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS March 2630, 2017, Jeju Island, Korea ACTSP00786 NUMERICAL INVESTIGATION OF THERMOCAPILLARY INDUCED MOTION OF A LIQUID SLUG IN A
More informationBoundaryLayer Theory
Hermann Schlichting Klaus Gersten BoundaryLayer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22
More informationMixing and Combustion in Dense Mixtures by William A. Sirignano and Derek DunnRankin
Mixing and Combustion in Dense Mixtures by William A. Sirignano and Derek DunnRankin At very high pressures and densities, what is different and what is similar about the processes of Injection and Atomization,
More informationSIMULATION OF THREEDIMENSIONAL INCOMPRESSIBLE CAVITY FLOWS
ICAS 2000 CONGRESS SIMULATION OF THREEDIMENSIONAL INCOMPRESSIBLE CAVITY FLOWS H Yao, R K Cooper, and S Raghunathan School of Aeronautical Engineering The Queen s University of Belfast, Belfast BT7 1NN,
More informationWall Effects in Convective Heat Transfer from a Sphere to Power Law Fluids in Tubes
Excerpt from the Proceedings of the COMSOL Conference 9 Boston Wall Effects in Convective Heat Transfer from a Sphere to Power Law Fluids in Tubes Daoyun Song *1, Rakesh K. Gupta 1 and Rajendra P. Chhabra
More informationNonlinear shape evolution of immiscible twophase interface
Nonlinear shape evolution of immiscible twophase interface Francesco Capuano 1,2,*, Gennaro Coppola 1, Luigi de Luca 1 1 Dipartimento di Ingegneria Industriale (DII), Università di Napoli Federico II,
More informationShell Balances in Fluid Mechanics
Shell Balances in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University When fluid flow occurs in a single direction everywhere in a system, shell
More informationFluid Mechanics. Chapter 9 Surface Resistance. Dr. Amer Khalil Ababneh
Fluid Mechanics Chapter 9 Surface Resistance Dr. Amer Khalil Ababneh Wind tunnel used for testing flow over models. Introduction Resistances exerted by surfaces are a result of viscous stresses which create
More informationGFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability
GFD 2012 Lecture 1: Dynamics of Coherent Structures and their Impact on Transport and Predictability Jeffrey B. Weiss; notes by Duncan Hewitt and Pedram Hassanzadeh June 18, 2012 1 Introduction 1.1 What
More informationDrop Impact on a Wet Surface: Computational Investigation of Gravity and Drop Shape
Drop Impact on a Wet Surface: Computational Investigation of Gravity and Drop Shape MURAT DINC and DONALD D. GRAY Department of Civil and Environmental Engineering West Virginia University P.O. Box 6103,
More informationReduction of parasitic currents in the DNS VOF code FS3D
M. Boger a J. Schlottke b C.D. Munz a B. Weigand b Reduction of parasitic currents in the DNS VOF code FS3D Stuttgart, March 2010 a Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring
More informationCFD SIMULATIONS OF FLOW, HEAT AND MASS TRANSFER IN THINFILM EVAPORATOR
Distillation Absorption 2010 A.B. de Haan, H. Kooijman and A. Górak (Editors) All rights reserved by authors as per DA2010 copyright notice CFD SIMULATIONS OF FLOW, HEAT AND MASS TRANSFER IN THINFILM
More informationPrinciples of Convection
Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid
More information