Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00004262/00001
## Material Information- Title:
- A level set approach for two-dimensional, two-fluid flows with surface tension and viscosity
- Creator:
- Wilson, John
- Publication Date:
- 1999
- Language:
- English
- Physical Description:
- viii, 92 leaves : illustrations ; 28 cm
## Subjects- Subjects / Keywords:
- Two-phase flow -- Computer simulation ( lcsh )
Fluid dynamics -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 91-92).
- General Note:
- Department of Mechanical Engineering
- Statement of Responsibility:
- by John Wilson.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- |Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 44105082 ( OCLC )
ocm44105082 - Classification:
- LD1190.E55 1999m .W35 ( lcc )
## Auraria Membership |

Full Text |

A LEVEL SET APPROACH FOR TWO-DIMENSIONAL, TWO-FLUID
FLOWS WITH SURFACE TENSION AND VISCOSITY by John Wilson B.S.M.E., University of Colorado at Denver, 1997 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering 1999 This thesis for the Master of Science degree by John Wilson has been approved by Ken Ortega Date Wilson, John (M.S., Mechanical Engineering) A Level Set Approach for Two-dimensional, Two-Fluid Flows with Surface Tension and Viscosity Thesis directed by Assistant Professor Sam W. Welch ABSTRACT A level set approach is applied to two-fluid flows with surface tension and viscosity. The level set method defines a smooth function, distance from the interface. The zero level set of (f> locates the interface. The interface is defined with a set thickness that must be maintained throughout the evolution of the flow. Flow properties are transitioned from the interior to the exterior across the interface thickness. The curvature calculation used in determining the surface tension is formulated using the level set function. The level set function is advected with the flow field velocity. The flow field velocities and pressures are computed using a semi-implicit version of the Mac method. The equations of motion are discretized on a staggered grid. The fluids are modeled with constant densities and viscosities. Simulations are conducted to ascertain the level set methods capability of modeling flow situations in which there are analytical or experimental results with which to compare. An oscillating bubble is simulated to verify the correctness of the surface tension model. To further examine the surface tension accuracy the Raleigh-Taylor instability is simulated. Buoyancy driven flows were considered in flow regimes where there are experimental and computational results available with which to compare. In addition simulations are conducted using the in level set method to determine its ability to accurately model two-fluid flows where merging and breaking of the interface occurs. This abstract accurately represents the content of the candidates thesis. I recommend its publication . Signed IV ACKNOWLEDGMENT I would like to thank Sam Welch for his guidance in everything that is computational fluids. I would like to thank my fiance, Antoinette, for everything that isnt computational fluids. I thank her for all of her support through the times that I wasnt so pleasant to be around. I would like to thank her for hanging my bubbles on the refrigerator as a sign of support and encouragement. I would also like to thank my family that supported me through all of the lean years. CONTENTS Figures.....................................................................viii Chapter 1. Introduction.............................................................1 2. Basic Equations..........................................................3 2.1 Control Volume Containing Interface......................................3 2.2 Level Set Formulation....................................................5 2.3 Free Boundary Condition at the Interface.................................6 2.4 Navier Stokes Equations..................................................9 3. Numerical Method........................................................12 3.1 Equations of Motion.....................................................12 3.2 Mac Method..............................................................13 3.2.1 Pressure Iteration......................................................17 3.3 Spatial Discretization..................................................17 3.3.1 Convection Terms........................................................18 3.3.2 Viscous Stress Terms....................................................20 3.3.3 Surface Tension Terms...................................................22 3.4 Fluid Properties........................................................24 3.5 Maintaining Level Set as a Distance Function............................24 3.6 Convergence of Level Set Method.........................................27 4. Numerical Simulations and Results.......................................28 4.1 Oscillating Bubble......................................................28 4.1.1 Derivation of Analytical Frequency......................................29 4.1.2 Numerical Results.......................................................33 4.2 Raleigh-Taylor Instability..............................................36 4.2.1 Derivation of Critical Wave Number......................................37 4.2.2 Numerical Results......................................................42 4.3 Buoyancy Driven Bubbles.................................................52 4.3.1 Necessity of Reinitialization Procedure.................................54 4.3.2 Spherical Bubble........................................................59 4.3.3 Ellipsoidal Bubble......................................................61 4.3.4 Dimpled Ellipsoidal-Cap Bubble.........................................63 4.3.5 Skirted Bubble.........................................................65 4.4 Merging and Breaking Interfaces.........................................68 vi 4.4.1 Dropping Bubble....................................................68 4.4.2 Two Bubbles........................................................76 4.4.3 Unstable Interface.................................................82 5. Conclusions and Recommendations....................................89 Bibliography............................................................91 vii FIGURES Figure 2.1 Control volume containing interface......................................4 2.2 Unit normals on interface................................................4 3.1 Momentum cells..........................................................14 3.2.2.1 Viscous stress on x-momentum cell.......................................21 3.2.3.1 Curvature stencil.......................................................23 4.1.1.1 Oscillating bubble, initial position....................................35 4.1.1.2 Oscillating bubble, half period.........................................35 4.1.1.3 Oscillating bubble, full period.........................................36 4.2.1.1 Perturbed interface.....................................................37 4.2.2.1 Enlarged interfacial disturbance........................................43 4.2.2.2 Velocity field, k = 50, t = 0.125.......................................43 4.2.2.3 Velocity field, k = 50, t = 0.25........................................44 4.2.2.4 Velocity field, k = 50, t = 0.375.......................................44 4.2.2.5 Pressure field, k = 50, t = 0.375.......................................45 4.2.2.6 Velocity field, k = 95, t = 0.25........................................46 4.2.2.7 Velocity field, k = 95, t = 0.375.......................................46 4.2.2.8 Pressure field, k = 95, t = 0.375.......................................47 4.2.2.9 Velocity field, k = 105, t = 0.25......................................48 4.2.2.10 Velocity field, k = 105, t = 0.375.....................................48 4.2.2.11 Velocity field, k = 105, t = 0.5.......................................49 4.2.2.12 Velocity field, k = 150, t = 0.25......................................50 4.2.2.13 Velocity field, k = 105, t = 0.375.....................................50 4.2.2.14 Velocity field, k = 150, t = 0.5.......................................51 4.3.1 Graphical correlation for buoyancy driven flows.........................53 4.3.1.1 No reinitialization, E = 1.0, M = 104..................................56 4.3.1.2 Reinitialization, E = 1.0, M = 10"4.....................................57 4.3.1.3 Reynolds number, E = 1.0, M = 10"4.....................................58 4.3.1.4 Interface thickness, no reinitialization, E = 1.0, M = 10"4.............58 4.3.1.5 Interface thickness, reinitialization, E = 1.0, M = 10-4................58 4.3.2.1 Spherical bubble shape and velocity field...............................59 4.3.2.2 Spherical bubble Reynolds number.......................................60 4.3.2.3 Spherical bubble pressure field.........................................60 4.3.3.1 Ellipsoidal bubble shape and velocity field.....................61 4.3.3.2 Ellipsoidal bubble Reynolds number.....................................62 viii 4.33.3 Ellipsoidal bubble pressure field........................................62 4.3.4.1 Dimpled ellipsoidal-cap bubble shape and velocity field..................63 43.4.2 Dimpled ellipsoidal-cap bubble pressure field............................64 43.5.1 Skirted bubble shape and velocity field..................................65 43.5.2 Skirted bubble pressure field............................................66 4.3.53 Skirted bubble shape and velocity field after break......................67 4.4.1.1 Dropping bubble velocity field, t = 0.1..................................70 4.4.1.2 Dropping bubble velocity field, t = 0.15.................................71 4.4.1.3 Dropping bubble velocity field, t = 0.17.................................72 4.4.1.4 Dropping bubble velocity field, t = 0.175................................73 4.4.1.5 Dropping bubble velocity field, t = 0.18.................................74 4.4.1.6 Dropping bubble velocity field, t = 0.2..................................75 4.4.2.1 Two bubbles , velocity field, t = 0.02.................................77 4.4.2.2 Two bubbles , velocity field, t = 0.05.................................78 4.4.23 Two bubbles , velocity field, t = 0.1..................................79 4.4.2.4 Two bubbles , velocity field, t = 0.15.................................80 4.4.2.5 Two bubbles , velocity field, t = 0.165................................81 4.43.1 Perturbed interface......................................................82 4.43.2 Unstable interface, velocity field, t = 0.1..............................83 4.4.33 Unstable interface, velocity field, t = 0.2..............................84 4.43.4 Unstable interface, velocity field, t = 03...............................85 4.43.5 Unstable interface, velocity field, t = 0.35.............................86 4.43.6 Unstable interface, velocity field, t = 0.37.............................87 IX 1. Introduction There are many numerical methods used for computing solutions for incompressible two-fluid flow problems. One such method is the level set method. In this work we investigate the level set method's ability to accurately model incompressible two-fluid flows where surface tension plays an integral part in the overall physics. There are two general numerical approaches for solving incompressible fluid flow problems with a moving boundary. One approach is based on front tracking methods where the moving boundary is explicitly tracked. One popular front tracking method is the Volume-of-Fluid (VOF) method. In this method a fluid fraction variable,/ is defined and used to identify the position of the interface [1]. The fluid fraction variable is assigned values of 1 and 0 in the two phases. The location of the interface is identified by fractional values of/ The VOF method reconstructs the interface for each time step based on volume considerations. The reconstruction step requires difficult geometrical calculations. The fluid fraction variable is then tracked as it is advected with the local flow velocity. The second method is based on front capturing. The front capturing method considered here is the level set method introduced by Osher and Sethian [2], The level set method involves defining a smooth function, (j>, as the signed normal distance from the front. The fluid interface is the zero level set of $. The level set function is advected with the local flow velocity. The advection of the level set has the quality that it is uncoupled from the equations of motion for the two fluids and thus doesnt introduce any greater complexity to the solution of the velocity and pressure fields. 1 The level set method involves replacing the singular interface with an interface of a finite thickness. Maintaining the prescribed interface thickness as the interface evolves is paramount to obtaining accurate solutions. This is enforced by preserving function include capturing the interface location as the zero level set of the ability to transition smoothly from interior fluid properties to exterior fluid properties accurately calculating the curvature of the interface based on the level set function The implementation of the level set formulation presented here closely follows the work of Sussman, Smereka, and Osher [3], The solution of the equations of motion in [3] were obtained using the projection method. The solution of the equations of motion in the work presented here is obtained using a semi-implicit version of the Mac method. In Chapter 2 the equations that describe the motion of the interface are derived based on two immiscible fluids with constant fluid properties. In Chapter 3 these equations are discretized and the computational solution technique is outlined. In Chapter 4 simulations are presented that probe the level set method's ability to accurately model surface tension forces and deforming interfaces. The simulations considered are an oscillating bubble, Rayleigh-Taylor instability, buoyancy driven flows, and flows with merging or breaking interfaces. In Chapter 5 recommendations and areas of current improvements to the level set method are discussed. 2 2. Basic Equations In this chapter we summarize the basic principles to fluid flows involving two immiscible fluids. In addition, only flows where temperature remains constant are considered, thus density, viscosity, and surface tension are assumed to be constant. Considering these assumptions the only principles that need to be taken into account are the conservation of mass and linear momentum. These equations will provide the theory to solve for the velocity and pressure field for any time in question. Included with these principles is the level set, which is discussed in section 2.2. 2.1 Control Volume Containing Interface The conservation of mass and linear momentum will be applied to an arbitrary control volume shown in figure 2.1.1. The subscripts / and g denote the liquid and gas regions respectively. The liquid region has a volume of Q, bounded by the surface dd, + 3Q, where 3Q, is the interface separating the two fluids. The same nomenclature is used for the gas region. C/ represents the perimeter of 3Q;. Q without a subscript is used to denote the union of the gas and liquid regions. Figure 2.1.2 shows the unit normal vectors directed out of their respective region. Na represents the unit vector directed outward from Q in the tangent plane of 3Q, 3 I o figure 2.1.2 Unit normals on interface 4 2.2 Level Set Formulation To locate the interface position a smooth level set function, f), is introduced such that dQx = {x| (f>{x,t) = 0} (2.2-1) Equation (2.2-1) states that if /) is the level set of the interface the zero level set of region bounded by the interface and as negative inside the region. The evolution of the level set in time is given by a + u V (f> = 0 (2.2-2) Defining 4>(Jc, /) as a distance function provides a convenient means to determine the location of the interface and as a consequence the fluid properties. The continuous level set function provides a mechanism of transitioning from the interior fluid properties to those of the exterior. p{ p{)= Pg + (pi-Pg)H{0) (2.2-4) 5 Where H{ 0 if 1 if (f> > 0 1 [2 if (2.2-5) An additional benefit of defining <}>(Â£, t) as the signed normal distance from the interface is that it provides a simple means of calculating the curvature of the interface which is discussed in Chapter 3. 2.3 Free Boundary Condition at the Interface By applying the conservation of linear momentum a boundary condition at the interface that relates surface tension to the jump in normal stress can be derived. Below is the conservation of linear momentum for the entire control volume presented in figure 2.1.1 neglecting the contribution from the interfacial momentum. 4 \PludV + ^ {pgudV = jPigdV + lpggdV+ (2.3-1) a a, a og n, ng \ T,dS + \ fgdS + | Tads da, ent c, g is the gravitational acceleration and represents the only body force considered. For the gas and liquid regions: F = T n (2.3-2) 6 where T is the stress tensor. T = -p\ + 2 pD (2.3-3) D is the rate of strain tensor, I is the identity tensor, and p is the pressure. For the interface: fa = Ta K (2-3-4) where T0 is the surface stress tensor given by the following expression when surface viscous stresses are neglected. T. = fla (2.3-5) Where y is the surface tension and I0 is the surface projection tensor. The surface projection tensor is like the identity tensor in three-dimensions. Conservation of linear momentum for the liquid region may be expressed as: 4 \ p,udV = J p,gdV + \f,dS (2.3-6) n, n, | Conservation of linear momentum for the gas region may be expressed as: 7 (2.3-7) 4 J = \pgdV + /2g nt xi'+an, Substituting equations (2.3-2) ,(2.3-4), (2.3-6), and (2.3-7) into equation (2.3-1) gives |T, n,dS + jTg = jTa (2.3-8) an, an, c, Applying the surface divergence theorem [4] equation (2.3-8) becomes j|T(-T,J-,dS=J Applying the surface divergence operator [4] to the surface stress tensor assuming the surface tension to be constant results in the following boundary condition at the interface. PH = 2yKmnl (2.3-10) Where |T| = T, -T2 and represents the jump in T across the interface. Km is the mean curvature of the interface. Equation (2.3-10) shows that there is a discontinuity in the normal stress proportional to the mean curvature of the interface. The mean curvature is given by [5] K=^(ici + k2), (2.3-11) 8 where k, and k2 are the greatest and least curvatures. For a curve in two dimensions the least curvature is zero. This results in K = 2Km (2.3-12) For two-dimensional problems the boundary condition at the interface becomes |T| fi, = yml (2.3-13) 2.4 Navier Stokes Equations Restating the total conservation of linear momentum with the surface divergence theorem and equation (2.3-2) results in the following equation. ^\p(^dV=iMgdV+ (2.4-1) a n n JTrn,dS+ JTg-ngdS+ \yKn,dS an, ans an, Applying the Reynold's transport theorem and Greens theorem to equation (2.4-1) results in \p(
n v 7 Dt n n an,
3.5 Maintaining Level Set as a Distance Function
by |