CFD Studies on Fluid Structure Interaction andMultiphase Flow Problems

MES 801- PhD Seminar

by

MEHTA SAGAR NARESHKUMAR

144100006

Under the Guidance of:

Prof. Amitabh Bhattacharya

Prof. Atul Sharma

Department of Mechanical Engineering

Indian Institute of Technology Bombay

DEPARTMENT OF MECHANICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY, BOMBAY

May 2015

Abstract

Gasliquidsolid three-phase interactions are becoming increasingly utilized in heat transfer

engineering designs for its uniqueness in generating energetics on deformable solid structures.

Due to increasing demands from manufacturing applications, researchers and engineers are

constantly exploring ways of augmented heat transfer techniques to maximize or minimize heat

transfer rates in various industrial processes. One such compound enhancement technique is

using exible micro ns for better ow mixing via formation of secondary ow structures and

utilizing boiling phenomena for improved heat transfer rates due to phase change. However,

the uid dynamics and heat transfer mechanisms during such complex three-phase interactions

are yet to be thoroughly studied for optimized design of heat transfer devices. Hence, the

development of accurate algorithms and solvers for simulating the rich dynamics involved in

the interplay of boiling phenomenon with elasto-capillary eects remains to be a challenging

task.

The present work involves literature survey on the work done in modeling three-phase in-

teractions with phase change phenomena. Comparing to the well-developed uidsolid and

gasliquid interactions, fewer eorts were devoted to model gasliquidsolid interactions simul-

taneously. The most promising recent works done independently for uid-structure interactions

are based on non-body conforming class of methods known as Immersed Boundary methods.

Also for simulation of multiphase ows, despite of many dierent well-established approaches,

level-set method promises great eciency for modeling interfacial forces and phase change

activity due to its inherent accurate interface tracking capability. It has been successfully im-

plemented for the cases of lm as well as nucleate boiling along with conjugate heat transfer

model for surface with rigid ns. Despite these accomplishments, eorts are still needed for

more robust, and extensible algorithms in modeling exible solid-liquid-vapor interaction in

ows involving boiling process.

A numerical framework to model and simulate such multiphysics problems is proposed and

partly implemented in this work. Here, the immersed boundary (IB) method and the level-set

(LS) method that model uidsolid and gasliquid interactions, respectively, is proposed to be

coupled. A simplied IB algorithm is used for modeling uidsolid interactions that accounts

for the dynamics of exible slender micro ns assumed to be one-dimensional. Few test cases are

performed to show the robustness and capability of the numerical framework for solid dynamics.

Also, the future course of action towards development of proposed comprehensive algorithm is

discussed in brief.

i

Contents

Abstract i

Table of Contents ii

List of Figures iv

1 Overview 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Heat Transfer Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Passive Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Active Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Problem Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Nucleate Boiling Heat Transfer Mechanism . . . . . . . . . . . . . . . . . 5

1.4.2 Fluid Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Structure of Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Literature Review 9

2.1 Fluid-structure interaction algorithms . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Arbitrary Lagrangian-Eulerian Method . . . . . . . . . . . . . . . . . . . 9

2.1.2 Immersed Boundary Method . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2.1 The classic IB method . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2.2 Direct forcing IB method . . . . . . . . . . . . . . . . . . . . . 11

2.1.2.3 Penalization method . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2.4 Cut-cell methods . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2.5 Immersed interface methods . . . . . . . . . . . . . . . . . . . . 12

2.1.2.6 Hybrid Cartesian-Immersed boundary method . . . . . . . . . . 13

2.1.2.7 The curvilinear immersed boundary method . . . . . . . . . . . 13

2.2 Algorithms for Multi-uid ow with boiling . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Film Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Nucleate Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Numerical Methodology 18

3.1 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Level Set Method for Nucleate Boiling . . . . . . . . . . . . . . . . . . . 18

3.1.2 Modication for an Immersed Flexible Micro-n . . . . . . . . . . . . . . 21

3.1.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2.2 Numerical Discretization . . . . . . . . . . . . . . . . . . . . . . 22

3.1.2.3 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Tests & Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 Simulation of exible lament in a uniform conservative force eld by the

immersed boundary method . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Closure 28

4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Scope of Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

iii

List of Figures

1.1 Schematic of nucleate boiling in presence of exible thin laments . . . . . . . . 4

1.2 Boiling curve and boiling regimes for water . . . . . . . . . . . . . . . . . . . . . 5

1.3 Heat transfer mechanisms during (a) bubble growth and (b) bubble departure . 6

1.4 Macro and micro regions for nucleate boiling . . . . . . . . . . . . . . . . . . . . 6

2.1 Computational domain for (a) ALE method (b) classic IB method . . . . . . . . 10

2.2 Schematics for (a) Cut-cell method (b) Hybrid CURVIB method . . . . . . . . . 13

2.3 Simulation results for a test case of lm boiling[37] . . . . . . . . . . . . . . . . 15

2.4 Simulation results for a test case of nucleate boiling[55] . . . . . . . . . . . . . . 17

3.1 Schematic of the computational domain[59] . . . . . . . . . . . . . . . . . . . . . 19

3.2 Schematic for discretization of heat ux from a liquid micro-layer[51] . . . . . . 20

3.3 Computational stencil for Lagrangian coordinate system representing solid lament 23

3.4 Schematic for exible lament suspended under gravitational force eld . . . . . 26

3.5 Comparison of results for (a[58],b) case 1 and (c[58],d) case 2 where position

of the laments are plotted at time intervals of 0.02 for a total period of 0.8.

(e) compares the solution obtained to approximate analytical solution for the

problem using perturbation method. . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 1

Overview

1.1 Introduction

As the role of technology has grown, so too has the importance of heat transfer engineering. For

example, in the industrial sector heat transfer concerns are critical to the design of practically

every process. The same is true of such vitally important areas as energy production, conversion,

and the expanding eld of environmental controls. In the generation of electrical power, whether

by nuclear ssion or combustion of fossil fuels, innumerable problems remain to be solved.

Similarly, further miniaturization of advanced computers is limited by the capability of removing

the heat generated in the microprocessors. As technology advances, engineers are constantly

confronted by the need to maximize or minimize heat transfer rates while at the same time

maintaining system integrity.

1.2 Heat Transfer Enhancement

The study of improved heat transfer performance is referred to as heat transfer enhancement,

augmentation, or intensication. The performance of conventional heat exchangers can be

substantially improved by a number of enhancement techniques. On the other hand, certain

systems, particularly those in space vehicles, may require enhancement for successful opera-

tion. A great deal of research eort has been devoted to developing apparatus and performing

experiments to dene the conditions under which an enhancement technique will improve heat

(and mass) transfer.

Enhancement techniques can be classied as passive methods, which require no direct appli-

cation of external power, or as active schemes, which require external power. The eectiveness

of both types depends strongly on the mode of heat transfer, which might range from single-

phase free convection to dispersed-ow lm boiling.

1

1.2.1 Passive Techniques

• Treated surfaces involve ne-scale alternation of the surface nish or coating (continuous

or discontinuous). They are used for boiling and condensing; the roughness height is

below that which aects single-phase heat transfer.

• Rough surfaces are produced in many congurations, ranging from random sand-grain

type roughness to discrete protuberances. The conguration is generally chosen to pro-

mote turbulence rather than to increase the heat transfer surface area. The application

of rough surfaces is directed primarily toward single-phase ow.

• Extended surfaces are routinely employed in many heat exchangers. The development of

non conventional extended surfaces, such as integral inner n tubing, and the improvement

of heat transfer coecients on extended surfaces by shaping or interrupting the surfaces

are of particular interest.

• Displaced enhancement devices are inserted into the ow channel so as to indirectly

improve energy transport at the heated surface. They are used with forced ow.

• Swirl-ow devices include a number of geometric arrangements or tube inserts for forced

ow that create rotating and/or secondary ow: inlet vortex generators, twisted-tape

inserts, and axial-core inserts with a screw-type winding.

• Coiled tubes lead to more compact heat exchangers. The secondary ow leads to higher

single-phase coecients and improvements in most regions of boiling.

• Surface-tension devices consist of grooved surfaces to direct the ow of liquid in boiling

and condensing.

• Additives for liquids include solid particles and gas bubbles in single-phase ows and

liquid trace additives for boiling systems.

• Additives for gases are liquid droplets or solid particles, either dilute phase (gas-solid

suspensions) or dense phase (uidized beds).

1.2.2 Active Techniques

• Mechanical aids stir the uid by mechanical means or by rotating the surface. Surface

"scraping," widely used for batch processing of viscous liquids in the chemical process

industry, is applied to the ow of such diverse uids as high-viscosity plastics and air.

Equipment with rotating heat exchanger ducts is found in commercial practice.

• Surface vibration, at either low or high frequency, has been used primarily to improve

single-phase heat transfer.

2

• Fluid vibration is the most practical type of vibration enhancement, given the mass of

most heat exchangers. The vibrations range from pulsations of about 1 Hz to ultrasound.

Single phase uids are of primary concern.

• Electrostatic elds (dc or ac) are applied in many dierent ways to dielectric uids.

Generally speaking, electrostatic elds can be directed to cause greater bulk mixing of uid

in the vicinity of the heat transfer surface, which enhances heat transfer. An electrical eld

and a magnetic eld may be combined to provide a forced convection or electromagnetic

pumping.

• Injection involves supplying gas to a owing liquid through a porous heat transfer surface

or injecting similar uid upstream of the heat transfer section. Surface degassing of liquids

can produce enhancement similar to gas injection. Only single-phase ow is of interest.

• Suction involves either vapor removal through a porous heated surface in nucleate or lm

boiling, or uid withdrawal through a porous heated surface in single-phase ow.

Two or more of these techniques may be utilized simultaneously to produce an enhancement

larger than that produced by only one technique. This simultaneous use is termed compound

enhancement. It should be emphasized that one reason for studying enhanced heat transfer is

to assess the eect of an inherent condition on heat transfer.

1.3 Motivation and Objective

Due to demands from manufacturing of high-power electronic devices, surfaces have been mod-

ied physically and chemically to increase the heat transfer rate. The study of pool boiling on

micro and nano-structured surfaces show promising indications that the heat transfer rate is

signicantly improved. Therefore, a detailed investigation of the nucleate boiling processes on

enhanced surfaces is of utmost importance in response to the demand from the electronic indus-

try. Another major motivation for this work lies in the observation that the eect of exibility

of active laments tethered to a surface on micro-uidic mixing via formation of secondary ow

structures. However, an understanding of the uid dynamics, heat transfer and phase change

during nucleate boiling on controlled surfaces fabricated with exible and externally actuated

micro-pillars, is an open question.

Given the rich dynamics involved in the interplay of boiling phenomenon with elasto-

capillary eects, the main objective of this seminar is to survey some of the important lit-

erature to explore the computational methods pertaining to interaction of uid with exible

micro structure and nucleate boiling phenomena.

3

Figure 1.1: Schematic of nucleate boiling in presence of exible thin laments

1.4 Problem Denition

The objective of the seminar is to survey literature exploring various computational algorithms

that can be utilized for numerical simulation of nucleate boiling on passive and active exible

micro-structured surfaces. Figure 1.1 shows the schematic of the interaction between a growing

vapor bubble and the surrounding exible pillars. On the heated surface, there is an articial

cavity from which a vapor bubble can be nucleated. The external force eld and the bubble

cause the deformation of the pillars, which in turn induce secondary ows of the surrounding

liquid.

Boiling heat transfer occurs when the temperature of a solid surface is suciently higher

than the saturation temperature of the liquid with which it comes in contact. The solid liquid

heat transfer is accompanied by the transformation of some of the heated liquid into vapor

and by the formation of vapor bubbles, jets, and lms. The vapor and surrounding packets

of heated liquid are carried away by buoyancy (natural convection or pool boiling) or by a

combination of buoyancy and the forced ow of liquid that may be sweeping the solid heater

(mixed convection, or ow boiling).

Initially take under consideration a setup, in which the heater surface (Tw) is immersed

in a pool of initially stagnant liquid (Tl). The fundamental physics problem is to determine

the relationship between the surface heat ux (qw) and the temperature dierence (Tw − Tsat),where Tsat is the saturation temperature of the liquid. When the liquid pool is at the saturation

temperature, the vapor generated at the heater surface reaches the free surface of the pool.

Figure 1.2 shows the main features of the boiling curve, the relationship between qw and the

excess temperature (Tw−Tsat). This particular curve corresponds to the pool boiling of water atatmosphere pressure; however, its non-monotonic relationship is a characteristic of the curves

describing pool boiling of other liquids as well.

4

Figure 1.2: Boiling curve and boiling regimes for water

1.4.1 Nucleate Boiling Heat Transfer Mechanism

Once a bubble nucleates, it grows through evaporation of liquid at the liquid/vapor interface.

Numerous mechanisms are available through which energy is transferred from the wall as illus-

trated in Figure 1.3. A quickly growing, hemispherical bubble can trap a thin layer of liquid

between the growing bubble and the super-heated wall (the micro-layer1), and evaporation of

this liquid contributes to bubble growth (qml). The energy to evaporate this liquid comes from

the energy stored in the super-heated wall. Bubble can also grow through evaporation of the

super-heated liquid layer surrounding the bubble cap (qsl). Another mechanism for bubble

growth is through evaporation at the three-phase contact line (qcl) once a dry patch forms on

the surface due to partial dry out of the micro-layer. The growing bubble can also perturb the

liquid adjacent to the bubble and disrupt the background natural convection boundary layer

(qnc), resulting in energy transfer by micro-convection (qmc). Inertial controlled growth occurs

when the heat transfer to the interface is very fast and the bubble growth is limited by the

rate at which momentum can be transferred to the surrounding liquid. Thermally controlled

growth occurs when the bubble growth is limited by the rate at which heat can be conducted

to the liquid/vapor interface.

In carrying out the analysis ows are taken to be laminar. The uid properties including

density, viscosity and thermal conductivity are assumed to be constant in each phase. The

conservation equations of mass, momentum and energy in the micro-layer (shown in Fig. 1.4)

are derived by using a lubrication theory[1] as:

1Consider a growth of bubble in a saturated uid medium. Here, there is a presence of some velocity of

the liquid vapor interface due to the phase change and bubble growth process. Now, consider a plate passing

through the center of this growing bubble. Since the plate is stationary, and if the vapor is totally non-wetting,

the velocity of the uid near the plate should be zero, a no-slip condition. Hence, some uid is overtaken by

the vapor phase during the bubble growth and the liquid ows in, beneath the bubble, causing a rise to thin

liquid lm termed as Micro-Layer.

5

(a) (b)

Figure 1.3: Heat transfer mechanisms during (a) bubble growth and (b) bubble departure

Figure 1.4: Macro and micro regions for nucleate boiling

6

∂

∂x

δˆ

0

ρluldy = − q

hlv(1.1)

∂pl∂x

= µl∂2ul∂y2

(1.2)

Using a modied ClausiusClapeyron equation, the evaporation heat ux is written as

q = hev[Tint − Tv + (pl − pv)Tv/ρlhlv] (1.3)

where

hev = (2/πRvTv)0.5ρvh

2lv/Tv;Tv = Tsat(pv) (1.4)

The pressures in the vapor and liquid phases are related as

pl = pv − σκ−A

δ3+

q2

ρvh2lv(1.5)

where σ is taken to be a function of temperature and A is the dispersion constant relating

disjoining pressure to the lm thickness. The combination of the mass, momentum, and energy

equations for the micro-layer yields

δ′′′′

= f(δ, δ′δ′′δ′′′) (1.6)

where ′ denotes ∂/∂x. The boundary conditions for the above equation are as follows: at

x = X0,

δ = δ0; δ′= δ

′′′= 0 (1.7)

at x = X1,

δ = δmax; δ′= tanϕ; δ

′′= 0 (1.8)

where δ0 is of the order of molecular size and ϕis an apparent contact angle.

Considering that the heat ux through the vapor phase is much smaller than that through

the liquid phase, we assume that the vapor phase is maintained at the saturation temperature.

The equations governing the conservation of mass, momentum and energy for each phase are

written as

∇ · uf = 0 (1.9)

ρf

(∂uf∂t

+ uf · ∇uf)

= −(∇p)f + ρf [1− βT (Tf − Tsat)]g +∇ · µf (∇u+∇uT )f + f (1.10)

7

ρfcf

(∂Tf∂t

+ uf · ∇Tf)

= ∇ · kf (∇T )f (1.11)

where the subscript f denotes the liquid/vapor phase. The conservation equations for each

phase are coupled through the matching (or jump) conditions at the liquid-vapor interface:

ul − uv = νlvmn (1.12)

n · [µl(∇u+∇uT )l − µv(∇u+∇uT )v]× n = 0 (1.13)

−pl + pv + n · [µl(∇u+∇uT )l − µv(∇u+∇uT )v] · n = σκ− νlvm2 (1.14)

where νlv = ρ−1v − ρ−1l ,n is the normal to the interface and κ is the interface curvature. The

interface temperature is specied as a Dirichlet boundary condition, Tf = Tsat. The mass ux

m is evaluated from the energy balance at the interface

m = n · kl∇Tl/hlv (1.15)

1.4.2 Fluid Structure Interaction

The governing equation for the structure reads as follows:

ρs∂2X

∂t2= ∇ · σs + Fs (1.16)

where Fs is the external force due to uid on the structure and the stress tenser σs is

given by the constitutive equation of the structure. The uid and solid parts are coupled with

each other through coupling boundary conditions at the uidstructure interface. The coupling

conditions usually adopted are:

Continuity of displacement: d(XΓ ) = ds(XΓ )

Continuity of velocity: u(XΓ ) = Us(XΓ )

Equilibrium of stresses: σ(XΓ ) = σs(XΓ )

1.5 Structure of Report

The report consists of mainly two parts: literature review and development of a numerical

algorithm to simulate nucleate pool boiling in presence of exible micro ns. Chapter 2 surveys

some important works done in this context and compares various computational methods for

uid-structure interaction and nucleate boiling. Chapter 3 gives an overview of the solution

algorithm for the problem as discussed in section 1.4 and also discusses detailed methodology

for its uid structure sub-module. Also a few validation tests are carried out for its performance

analysis. The report concludes with summary of the work and future scope in Chapter 4.

8

Chapter 2

Literature Review

2.1 Fluid-structure interaction algorithms

Fluidstructure interaction (FSI) is of great relevance to many engineering applications, such

as uttering and bueting of bridges, wind-exited vibration of tall buildings, vibration of wind

turbine blades, wind-plants interactions, aero-elastic response of airplanes and many biological

ows, such as blood ows in arteries and articial heart valves, ying and swimming.

2.1.1 Arbitrary Lagrangian-Eulerian Method

The conventional approach for simulating FSI problem is the Arbitrary LagrangianEulerian

(ALE) method, which adopts body-conforming grids to follow the movement of the uidstructure

interface as shown in Fig. 2.1(a). Generating these grids proceeds in two sequential steps. First,

a surface grid covering the boundaries is generated. This is then used as a boundary condition

to generate a grid in the volume occupied by the uid. If a nite-dierence method is employed

on a structured grid, then the dierential form of the governing equations is derived from a

curvilinear coordinate system aligned with the grid lines. Because the grid conforms to the

surface of the body, the transformed equations can then be discretized in the computational

domain with relative ease. If a nite-volume technique is employed, then the integral form of

the governing equations is discretized and the geometrical information regarding the grid is

incorporated directly into the discretization. If an unstructured grid is employed, then either

a nite-volume or a nite-element methodology can be used. Both approaches incorporate

the local cell geometry into the discretization and do not resort to grid transformations. Al-

though ALE method has been widely used in many FSI problems, they are cumbersome if not

impossible to apply to FSI problems with large deformations[2].

2.1.2 Immersed Boundary Method

Immersed boundary (IB) methods provide an alternative approach for simulating FSI prob-

lems involving complex geometries and arbitrarily large deformations. In IB methods, the

9

(a) (b)

Figure 2.1: Computational domain for (a) ALE method (b) classic IB method

NavierStokes equations are solved on a xed background grid, which can be Cartesian, curvi-

linear or unstructured. In this approach, the IB would still be represented through some means

such as a surface grid, but the Cartesian volume grid does not conform to this surface grid.

Thus, the solid boundary would cut through this Cartesian volume grid (Fig. 2.1(b)). Because

the grid does not conform to the solid boundary, incorporating the boundary conditions at

uid-structure interfaces would require modifying the equations in the vicinity of the boundary

by introducing ctitious body forces. However, the governing equations would then be dis-

cretized using a nite-dierence, nite-volume, or a nite-element technique without resorting

to coordinate transformation or complex discretization operators. Compared to ALE methods,

IB methods have two advantages: (1) fast grid generation; and (2) use of ecient ow solvers

for stationary grid. Based on the representation of the uidstructure interfaces, immersed

boundary methods can be classied as diused interface and sharp interface methods. In dif-

fused interface IB methods, the immersed boundaries are smeared by distributing the singular

forces to the surrounding background grid nodes using discrete delta functions or mask func-

tions for penalization methods. A class of methods that eliminates this, generally undesirable

numerical feature, is the so called sharp-interface IB methods.

2.1.2.1 The classic IB method

The classical IB method was rstly introduced by Peskin[3] in 1970s to simulate blood ow

in the human heart. In the classic IB method, the forces at the immersed boundaries are

calculated from appropriate constitutive laws, which depends on the conguration of immersed

boundaries. The forces on the background grid are calculated by distributing the forces from

the immersed boundaries using discrete delta functions. The immersed boundaries follow the

motion of the surrounding uid (no slip boundary conditions). Special attention needs to be

paid on the temporal integration scheme for advancing the uid and solid parts. The simplest

approach is to calculate the forces explicitly using the conguration of the interface at the

last time step before solving the uid part. The other approach is using an approximated

conguration (e.g. explicitly compute the conguration at current time step by using the

velocity of the interface at last time step) for the force calculation. Whereas both schemes

10

can be easily implemented, they are often numerically unstable especially for interfaces with

large stiness. The unconditionally stable approach is to calculate the forces implicitly using

the conguration at current time step. In this approach, sub-iterations are required for solving

the uid and solid parts from last time step to current time step[4], which are usually very

time-consuming. The development of IB methods in conjunction with adaptive grid can be

found in Roma et al.[5] and Grith et al.[6]. A version of the IB method accounting for the

mass of the immersed boundary was developed Zhu and Peskin[7].

2.1.2.2 Direct forcing IB method

The classic IB method proposed by Peskin is ideally suited for FSI problems with elastic

boundaries for which the singular force can be calculated from a constitutive law (e.g. Hook's

law). However, this method is dicult to apply to rigid body problems because of numerical

instabilities associated with the sti systems that are inherent to rigid bodies. To remedy this

shortcoming of the classic IB method, a direct forcing method[8] was proposed for problems

with rigid boundaries, which was originally applied to a class of sharp-interface methods. To

incorporate direct forcing in the classic, diused interface, IB method, the quantities on the

background and immersed boundary meshes can be transferred by employing the discrete Dirac

delta functions of the classic formulation. Two dierent approaches have been proposed for

doing so. The rst approach, which is dubbed as explicit direct forcing IB method, was proposed

by Uhlmann[9]. In this method, iterations were used to improve the satisfaction of the velocity

boundary conditions to mimic the eects of the boundary. In the second approach, named

as implicit direct forcing IB method[10], the forces at the immersed boundaries and velocity

interpolation are coupled. However, this method requires computational resources to solve

the linear equations, which could be signicant for problems with large number of immersed

boundary nodes or multiple immersed bodies. A direct forcing IB method based on the discrete

stream function with local mesh renement was developed by Wang and Zhang[11].

2.1.2.3 Penalization method

The penalization method was rstly introduced by Arquis and Caltagirone[12]. In this method,

the solid obstacles are modeled as porous media with vanishing porosity. The ctitious body

forces are then usually calculated by a penalization parameter to be specied in the simulation.

However, the parameter cannot be too large in order to avoid a system with a very large

stiness. In order to ensure numerical stability[13] and avoid numerical oscillations[14], the

ctitious body forces have to be distributed smoothly over the immersed boundaries as in the

classic IB method and direct forcing IB method. A Heaviside function was used by Kevlahan

et al.[15]. Application of the penalization method to compressible ows can also be found in

[16]. The level set method was used for capturing the uidstructure interfaces in [17].

11

2.1.2.4 Cut-cell methods

In cut-cell methods, the grid cells cut by the interfaces are reshaped according to the local geom-

etry of the interface so that a boundary-conforming, albeit locally unstructured grid emerges.

The cut-cells for a relative simple immersed boundary are shown in Fig. 2.2(a). The uxes

across the faces of cut-cells are reconstructed from the surrounding regular cells and immersed

boundaries. The cut-cell method was rstly introduced by Clarke for inviscid ows[18] and has

been applied both on collocated[19] and staggered grids[20]. Most applications of the cut-cell

method were focused on 2D problems[21]. This is because of the inherent diculty in applying

cut-cell methods to 3D problems because of the many possibilities of the geometrical shape

of cut-cells, which may arise when dealing with the arbitrarily complex body shape typically

encountered in real-life applications. As alluded to above, a major challenge in cut-cell for-

mulations is the management of the topology of cut-cells, which could be cumbersome even

in 2D problems with stationary boundaries. To mitigate this diculty and enable simulations

of moving boundary problems, the cut-cell approach has been used in conjunction with the

level-set method[22] to facilitate the tracking of the immersed boundaries[21]. The advantages

of cut-cell methods are their inherent conservation property and also the higher accuracy near

the interfaces. A major drawback, however, is the diculty to extend such methods to 3D

problems with complex geometries as mentioned above. Another drawback is that very small

time steps have to be used because of cut-cells of very small size may arise in the vicinity of

the interface.

2.1.2.5 Immersed interface methods

The immersed interface method was rst proposed by Leveque and Li for elliptical equations[23]

to improve the accuracy of the classic IB method near the immersed boundaries. Later, it was

extended to Stokes equations[24] and to NavierStokes[25]. The immersed interface method

shares the same idea with the classic IB method in that the eects of the immersed interfaces

on the surrounding uids are represented by singular forces. Instead of spreading the singular

forces to the background grid nodes as in the classic IB method, the immersed interface method

introduces jump conditions to the nite dierence scheme to account for the eects of non-

smooth solutions which could be caused by singular forces at the immersed boundaries or

the discontinuous substance properties across the immersed interfaces. The principal jump

conditions for the velocity, the pressure, and their normal derivatives across interfaces have

been derived by Lai and Li[26]. Later, all the necessary spatial and temporal jump conditions

for incompressible viscous ows were systematically derived by Xu and Wang[27] and applied to

2D[28] and 3D problems[29]. The jump conditions are the functions of the singular forces. For

elastic bodies, the singular forces can be computed from some constitutive laws based on the

conguration of the immersed boundaries. For rigid bodies, Xu and Wang[28] used a feedback

approach to calculate the singular forces.

12

(a) (b)

Figure 2.2: Schematics for (a) Cut-cell method (b) Hybrid CURVIB method

2.1.2.6 Hybrid Cartesian-Immersed boundary method

The hybrid Cartesian-immersed boundary method was proposed by Mohod-Yusof in conjunc-

tion with spectral methods and was later extended to nite dierence methods by Fadlun et

al.[8]. The key idea of this class of methods is the direct forcing approach. Namely, the forces

are calculated at the immersed boundaries by satisfying velocity boundary conditions and

then distributed to the surrounding background grid nodes via an appropriate interpolation

approach. Compared with the diused-interface IB methods, the hybrid Cartesian-immersed

boundary methods retain the sharp representation of the immersed boundaries and, given suf-

cient numerical resolution, they can accurately predict the forces acting on the uidstructure

interfaces. Non-physical force oscillations are generated when hybrid Cartesian IB methods are

applied to moving boundary problems. These oscillations are due to the fact that grid nodes

near a moving immersed boundary can change in time from uid to IB to solid nodes and

vice-versa. Lee et al.[30] identied two sources for these non-physical force oscillations: one

is from the spatial discontinuity of pressure when a solid point becomes a uid point because

of moving boundary; the other is from the temporal discontinuity of the velocity when a uid

point becomes a solid point. Seo and Mittal[31] developed a combined hybrid Cartesian im-

mersed boundary and cut-cell method, in which the cut-cell method is applied to the Poisson

equation and velocity correction step. Application of Cartesian-immersed boundary methods

to ows with heat transfer can be found in [32].

2.1.2.7 The curvilinear immersed boundary method

The immersed boundary method based on curvilinear background grids, denoted as the CURVIB

method, was proposed by Ge and Sotiropoulos[33] and is a generalization of the hybrid Cartesian-

immersed boundary method proposed by Gilmanov and Sotiropoulos[34]. A background curvi-

linear grid enhances the versatility and eciency of an IB formulation since it allows the dis-

cretization of geometrically simple boundaries within the computational domain with a body-

13

tted grid while retaining the power of the IB formulation in handling arbitrarily complex

boundaries embedded within this background mesh (see Fig. 2.2(b)). The CURVIB method

has been widely used in biological ows[35] and environmental ows[36].

2.2 Algorithms for Multi-uid ow with boiling

Boiling is one of the most ecient modes of heat transfer and as such is routinely used in

applications such as power generation, propulsion, electronics cooling, chemical processes, etc.

Over the past eight decades, signicant progress has been made in improving our understanding

of the boiling process. In spite of the progress made, we are still unable to accurately predict,

from basic principles, the boiling curve.

2.2.1 Film Boiling

The rst numerical simulation of lm boiling was performed by Son and Dhir[37], wherein

saturated lm boiling on a horizontal surface was studied. The simulations were performed in

axisymmetric two-dimensional curvilinear coordinates. A moving mesh was used to capture the

liquid-vapor interface and the wall temperature was maintained constant. The surface tension

was modeled as a volumetric source term[38] in the momentum equation. The conservation

equations were solved in both the liquid and vapor phases to determine the temperature and

ow elds.

Soon thereafter, Son and Dhir[39] incorporated the level set method into their previous

numerical model. This allowed the simulation to capture the breaking and merging of the

interface eectively. Film boiling of saturated water on a horizontal surface at near critical

pressures was also investigated in this study.

Juric and Tryggvason[40] used the so-called phase-eld formulation to investigate lm boil-

ing on a horizontal surface. A two-dimensional Cartesian coordinate system was used for these

simulations. The liquid-vapor interface was captured using a front-tracking method. A con-

stant wall heat ux boundary condition was imposed on the bottom wall. Numerical results of

the wall heat ux and wall temperature distribution were found to be in good agreement with

experimental data. Banerjee and Dhir[41] performed a three-dimensional Taylor instability

analysis during subcooled lm boiling on a horizontal disk.

Welch and Wilson[42] adopted a volume-of-uid (VOF) method to simulate lm boiling on

a horizontal surface for constant wall superheat conditions. Welch and Rachidi[43] modied

the model of Welch and Wilson[42] to include a solid wall. Constant heat ux was applied at

the lower solid boundary.

Other notable numerical simulations of lm boiling on horizontal surfaces were those by

Esmaeeli and Tryggvason[44], and Agarwal et al.[45]. More recently, Son and Dhir[46] simulated

saturated lm boiling on a horizontal cylinder by using an immersed solid boundary to represent

the cylindrical solid heater in a pool of liquid. A constant wall superheat condition was used

in these simulations.

14

Figure 2.3: Simulation results for a test case of lm boiling[37]

2.2.2 Nucleate Boiling

One of the earliest attempts to model bubble growth and departure from a heated wall was by

Lee and Nydahl[47]. In this study the bubble growth rate was calculated by solving the two

dimensional axisymmetric NavierStokes and energy equations numerically to determine the

associated ow and temperature elds. Due to the fact that a hemispherical bubble and wedge

shaped micro-layer were assumed, the change in bubble shape during growth was not accounted

for. The wall temperature was assumed to be constant. Cooper and Lloyd's[48] formulation

was used for the micro-layer thickness. Mei et al.[49] studied the bubble growth and departure

time using numerical simulations. They assumed that a wedge shaped micro-layer existed

underneath the bubble and that the heat transfer to the bubble was only through the micro-

layer. This assumption is not totally correct for both subcooled and saturated boiling. The

study did not consider the hydrodynamics of the liquid motion induced by the growing bubble

and introduced empiricism through the shape of the growing bubble. However, in their work

the temperature distribution in the heater was solved for numerically.

Welch[50] studied bubble growth using a nite volume method and an interface tracking

method. Conduction in the solid wall was accounted for, but the micro-layer was not modeled

explicitly. The rst complete numerical simulation of bubble growth was performed by Son et

al.[51]. In their study; in addition to the solution of the conservation equations, the liquid-

vapor interface was captured using the level-set (LS) method. This level-set method had been

previously applied to adiabatic incompressible twophase ow by Sussman et al.[52] and to

lm boiling near critical pressures by Son and Dhir[39]. In the model of Son et al.[51], the

15

computational domain was divided into two regions, namely, the micro region and the macro

region.

The micro region is the ultra thin liquid lm that forms between the solid surface and the

evolving liquid-vapor interface. On the inner edge, the micro-layer has a thickness of the order

of a few nanometers (few molecules of liquid adsorbed on the surface and do not evaporate).

The thickness of the non-evaporating (adsorbed) lm depends on the vapor pressure, substrate

temperature, and the disjoining pressure[53]. As such, the solid surface further radially in-

ward is considered to be dry (i.e., nonevaporating micro-layer). On the other hand, at the

outer edge, the micro-layer has a thickness of the order of several microns. Heat is conducted

across this lm and is utilized for evaporation. Lubrication theory similar to that developed

by Wayner[53], Stephan and Hammer[54], and Lay and Dhir[1] was used to solve for the ra-

dial variation of micro-layer thickness. For numerically analyzing the macro region, the level

set formulation modied by Son et al.[51] to accommodate the eect of phase change is used.

When bubbles merge in the lateral direction numerical simulations were carried out in three

dimensions. However, for the micro-layer contribution a two-dimensional model is still em-

ployed under the assumption of axisymmetry around a bubble. In analyzing the micro region,

continuum assumption was considered to hold until the lm became a few molecules thick. In

the micro-layer formulation, capillary pressure gradient is related to change in the curvature

and/or change in temperature of the interface. It also includes recoil pressure which results

from the momentum dierence of vapor leaving the interface and liquid approaching the inter-

face. Inertia terms are neglected in the momentum equation, and convection terms are ignored

in the energy equation. Quasistatic analysis is carried out, and a two-dimensional model for

the micro-layer is used even in three-dimensional (3D) situations under the assumption that no

crossow occurs in the azimuthal direction.

For the macro region, the uid is assumed to be incompressible. Additionally, the ows are

assumed to be laminar, and all properties are evaluated at the mean temperature. The vapor

is assumed to remain at saturation temperature corresponding to the pressure in the bubble.

As such, the energy equation is not solved inside the vapor bubble and heat transfer from

solid to vapor is ignored. A nite dierence scheme is used discretize the governing equations.

All diusion terms are solved implicitly, while the convection terms are solved explicitly. The

projection method is used to solve for pressure. In order to increase the rate of convergence of

the Poisson equation for pressure, the multigrid method is used. A second-order ENO scheme

is adopted for the advection terms when solving for the level-set function. In the original model

developed by Son et al.[51], the heater wall was maintained at a constant temperature, and

symmetry conditions were imposed on the domain boundaries. The eciency of the model was

tested with several standard problems.

Very few numerical simulations have been carried out to predict nucleate boiling heat ux as

a function of wall superheat. Son and Dhir[55] have investigated multiple bubble merger during

saturated nucleate boiling. In these two-dimensional (2D) simulations, the wall superheat was

specied. Both the active nucleation site density[56] and the bubble waiting time[57] were

16

Figure 2.4: Simulation results for a test case of nucleate boiling[55]

specied as a function of wall superheat.

2.3 Summary

After a thorough study of present day advancements in the eld of numerical simulations

involving nucleate boiling as well as uid-structure interaction, one may conclude that the

solution algorithm can be most eciently developed by combination of Level Set method and

Immersed Boundary method for their promising features in simulating nucleate boiling and

uid-structure interaction respectively. The algorithm of such a method is developed in the

next chapter.

17

Chapter 3

Numerical Methodology

In order to solve the problem as stated in section 1.4, a solution methodology has been carved

out from the combination of various previous works done independently in their respective

elds of uid-structure interaction and nucleate boiling simulation. The method proposed is

based mainly on the works of Zhu and Peskin[7] and Huang et al.[58] for Immersed Boundary

formulation and Lee et al.[59] and Gada and Sharma[60] for Nucleate boiling over a micro

cavity on the surface.

3.1 Computational Method

3.1.1 Level Set Method for Nucleate Boiling

Figure 3.1 gives the general conguration for the numerical analysis. The liquidvapor interface

is tracked by the LS function φ, which is dened as a signed distance from the interface. The

positive sign is chosen for the liquid phase and the negative sign for the vapor phase. The

normal n to the interface, the interface curvature κ, and the mass ux m are dened as:

n = ∇φ/|∇φ| (3.1)

κ = ∇ · n (3.2)

m = ρf (U− uf ) · n (3.3)

where U is the interface velocity. The mass ux m dened at the liquid-vapor interface

is extrapolated into the entire domain (or a narrow band near the interface) for its ecient

implementation. Based on the level-set approach, the conservation equations can be rewritten

for the liquidvapor region as

∇ · u = νlvmn · ∇αφ (3.4)

18

Figure 3.1: Schematic of the computational domain[59]

ρ

(∂u

∂t+ uf · ∇uf

)= −[∇p+(σκ− νlvm2)∇αφ] + ρ[1−βT (Tf −Tsat)]g+∇· µ((∇u+∇uT )

− (νlvmn · ∇αφ + (νlvmn · ∇αφ)T )) + f + fs (3.5)

(ρcp)l

(∂T

∂t+ ul · ∇T

)= ∇ · k∇T if φ > 0 (3.6)

T = Tsat(1 + νlvσκ/hlv) if φ < 0 (3.7)

where

αφ =

1 if φ > 0

0 if φ < 0

ul = u+ νlvmn(1− αφ)

uv = u− νlvmnαφ

ρ = ρv(1− Fφ) + ρlFφ

µ−1 = µ−1v (1− Fφ) + µ−1l Fφ

k−1 = k−1l Fφ

19

Figure 3.2: Schematic for discretization of heat ux from a liquid micro-layer[51]

Here, αφ is the discontinuous step function rather than the smoothed step function varying

over several grid spacings, and uf (ul or uv) is the velocity for each phase, which is extrapolated

into the entire domain by using the velocity jump condition. Also, fs represents the force induced

due to presence of solid IB in the uid which is given later in this section. The eective (or

interpolated) properties, ρ, µ, and k, are evaluated from a fraction function Fφ, which is dened

as

Fφ =

1 ifαφ(φA) = αφ(φB) = 1

0 ifαφ(φA) = αφ(φB) = 0

max(φA,φB)max(φA,φB)−min(φA,φB)

otherwise

where the subscripts A and B denote the grid points adjacent to the location where Fφ is

evaluated.

In this method, we use a simplied model for the micro-layer derived in [1]. It is noted from

the results for the micro-layer obtained by Son et al.[55] that δ has a nearly constant slope

except near the nonevaporating region, where the slope decreases to zero rapidly. Thus, we

assume over the whole micro-layer that

dδ

dr= tanϕ

where ϕ is an apparent contact angle for the case plotted in Fig. 3.2. The heat ux from

the control surface including the liquidvaporsolid contact location is thus discretized as

q = klTs − Tint

∆l

(3.8)

where

∆l = ∆r tanϕ/ ln

(δl + kl/hevkl/hev

)(3.9)

20

In the LS formulation, the interface is described as φ = 0. The zero level set of φ is advanced

as

∂φ

∂t+U · ∇φ = 0 (3.10)

where U can be written as U = uf + mn/ρf . The LS function is reinitialized to a distance

function from the interface by obtaining a steady-state solution of the equation

∂φ

∂τ= S(φ)(1− |∇φ|) (3.11)

where

S(φ) =

0 if |φ| < h/2

φ√φ2+h2

otherwise

Here h is a grid spacing and the formulation of sign function S implies that a near-zero level

set rather than φ = 0 is used as the immobile boundary condition during the reinitialization

procedure.

3.1.2 Modication for an Immersed Flexible Micro-n

3.1.2.1 Governing Equations

If the n is very slender i.e. Wf < h Hf (refer Fig. 3.1), then the exible n can be modeled

as a thin one-dimensional lament which is xed at one end to the surface and other end is free.

The stresses that are hereby coming into play are mainly those due to tension and bending in

the structure. Thus, the comprehensive governing equation for the n structure, as derived

from the principle of virtual work taking into account these stresses, in a Lagrangian form[58]

is given as

ρs∂2X

∂t2=

∂

∂s

(T∂X

∂s

)− ∂2

∂s2

(γ∂2X

∂s2

)+ F− Fh (3.12)

where s is the arc length, T is the tension force along the lament axis, γ is the bending

rigidity, and Fh is the Lagrangian forcing exerted on the lament by the surrounding uid ow.

Here, ρs denotes the density dierence between the solid lament and the surrounding uid.

In the present model, the tension force T is determined by the constraint of inextensibility

and is a function of s and t, while the bending rigidity γ is assumed to be constant. The

inextensibility condition can be expressed as

∂X

∂s· ∂X∂s

= 1

By using (∂X/∂s) · (∂/∂s) operator on Eqn. 3.12, the Poisson equation for T is derived as

21

∂X

∂s· ∂

2

∂s2

(T∂X

∂s

)= ρs

[1

2

∂2

∂t2

(∂X

∂s· ∂X∂s

)− ∂2X

∂t∂s· ∂

2X

∂t∂s

]− ∂X

∂s· ∂∂s

(Fb + F− Fh) (3.13)

where

Fb = −∂2

∂s2

(γ∂2X

∂s2

)denotes the bending force. The rst term on the right hand side of Eqn. 3.13 is zero

theoretically. However, this term is not dropped to correct numerical errors of the inextensibility

constraint. At the free end (s = Hf ) , we have

T = 0,∂2X

∂s2= (0, 0),

∂3X

∂s3= (0, 0) (3.14)

At the xed end (s = 0), two types of boundary conditions are considered. One is the

simply supported condition,

X = X0,∂2X

∂s2= (0, 0) (3.15)

The other is the clamped or build-in supported condition,

X = X0,∂X

∂s= X

′

0 (3.16)

The interaction force between the uid and the IB can be calculated by the feedback law[28]

F = α

tˆ

0

(Uib −Us)dt′ + β(Uib −Us) (3.17)

where α and β are large negative free constants, Uib is the uid velocity obtained by inter-

polation at the IB, and Us = dX/dt. To ensure the continuity of velocity at IB, transformation

between Eulerian and Lagrangian variables is realized by a Dirac delta function[7]. The inter-

polation of velocity is expressed as

Uib(s, t) =

ˆ

Ω

u(x, t)δh(X(s, t)− x)dx (3.18)

Spreading of the Lagrangian forcing to the nearby grid points is expressed as

f(x, t) = ρs

ˆ

Γ

F(s, t)δh(x−X(s, t))ds (3.19)

3.1.2.2 Numerical Discretization

For the solution of FSI problems specically involving thin exible soft laments, a Lagrangian

framework for grid is shown in Fig. 3.3. Here, nite dierence method is employed to dene

22

Figure 3.3: Computational stencil for Lagrangian coordinate system representing solid lament

all variables except T on the grid points or vertices after dividing the one-dimensional lament

is into N segments. However, T is dened in a staggered grid fashion at inter-nodal points

as indicated by non-integer indices in Fig. 3.3. For the spatial rst-, second- and third- order

derivatives with respect to arc length s, central dierence approximations are employed, and

a backward dierence scheme is used for temporal marching as detailed for each term in this

section. The spatial derivatives are discretized using central dierence approximation.

The tension force (as in Eqn. 3.12) is thus given as[∂

∂s

(T∂X

∂s

)]i

=Ti+1/2 (Xi+1 −Xi)− Ti−1/2 (Xi −Xi−1)

∆s2(3.20)

and the bending force term is expressed as

(Fb)i =

[− ∂2

∂s2

(γ∂2X

∂s2

)]i

= −γ (Xi+2 − 2Xi+1 +Xi)− 2 (Xi+1 − 2Xi +Xi−1) + (Xi − 2Xi−1 +Xi−2)

∆s4(3.21)

In the governing equation of motion of solid, as T is a function of dX/ds and t, the system

becomes nonlinear and a very sti system of coupled equations evolve. To resolve this issue,

analogous to treatment of pressure term in semi explicit projection method for Navier Stokes

equations, we treat tension force implicitly while all other forces being calculated explicitly for

time marching. The time-marching scheme of Eqn. 3.12 then can be summarized as

ρsXn+1i − 2Xn

i +Xn−1i

∆t2=

[∂

∂s

(T∂X

∂s

)]n+1

i

+ (Fb + F− Fh)ni i=0:N (3.22)

where ∆t denotes the time increment and the boundary conditions (Eqn. 3.14-3.15) should

be accounted for at i = 0 and i = N . However, as the tension force term is treated implicitly,

while the bending force term is treated explicitly, numerical instability is invoked when increas-

ing γ. Thus the maximum time step decreases as γ or N increases which needs to be kept in

mind. Also we need some constitutive law in order to ensure the constraints of the system and

accurately estimate T and X. This constraint is provided by using inextensibility condition for

the solution of T n+1 from Eqn. 3.13. The discretized form for this equation can be written as

23

(Xni+1 −Xn

i

∆s2

)·([

∂

∂s

(T n+1∂X

n

∂s

)]i+1

−[∂

∂s

(T n+1∂X

n

∂s

)]i

)= ρs

∆s2 − 2 [(Xi+1 −Xi) · (Xi+1 −Xi)]n + [(Xi+1 −Xi) · (Xi+1 −Xi)]

n−1

2∆t2∆s2

− ρs[(Ui+1 −Ui) · (Ui+1 −Ui)]

n

∆s2−(Xni+1 −Xn

i

∆s2

)·[(Fb + F− Fh)

ni+1 − (Fb + F− Fh)

ni

]i=0:N-1

where Uj = (Xj−Xj−1)/∆t. The inextensibility constraint is implemented in the rst term

on the right hand side by taking

[(Xi+1 −Xi) · (Xi+1 −Xi)]n+1

∆s2= 1

Such an approach leads to iterative calculation of T n+1 until the numerical errors introduced

in a given time step for inextensibility condition are penalized. once we obtain T n+1, then Xn+1

calculated from Eqn. 3.12 is taken as the new time step value and the time-marching continues.

Other specic feature of this problem is the interaction force due to surrounding uid. This

is calculated as proposed by Huang[58] using a smoothed delta function near IB as

(Fh)ni = α

n∑j=1

((Uib)

ji −Uj

i

)∆t

+ β ((Uib)

ni −Un

i ) (3.23)

where

(Uib)i =∑∀x

uδh(Xi − x)h2 (3.24)

In the above equation h denotes the mesh size and, in the present simulations, a mesh of

uniform size is distributed around the IB in the x- and y-directions, i.e. h = ∆x = ∆y. A

four-point delta function with base of 2h size introduced by Peskin[7] as shown below is used

in the present algorithm.

δh(x) =1

h2φh

(xh

)φh

(yh

)where

φh(r) =

18

(3− 2|r|+

√1 + 4|r| − 4r2

)if 0 ≤ |r| < 1

18

(5− 2|r| −

√−7 + 12|r| − 4r2

)if 1 ≤ |r| < 2

0 if 2 ≤ |r|

On the other hand, the presence of the solid aects the uid ow which is treated as a delta

function based Lagrangian force near IB spread to the Eulerian grid as

24

(fs)nP =

N∑i=1

(Fh)ni δh(xP −Xi)∆s (3.25)

where subscript P represents a representative control volume of Eulerian grid.

3.1.2.3 Solution Algorithm

The solution algorithm for implementing IB method to simulate exible laments in a ow (at

present single phase ow) is summarized as follows:

1. For each time step, interpolate the uid velocity at the IB by Eqn. 3.24 for the solid

lament.

2. Calculate the Lagrangian interaction force by Eqn. 3.23 for each grid point.

3. Spread the Lagrangian interaction force to the Eulerian ow eld grid by using Eqn. 3.25

on the staggered grid positions.

4. Solve the Navier-Stokes equations treating interaction force as a body force to obtain

the updated uid velocity eld and pressure eld. using semi-explicit nite-volume ow

solver.

5. Calculate the tension force by Eqn. 3.13 iteratively as explained in previous subsection.

6. Obtain the lament position at the new time step using Eqn. 3.12.

7. Go to step (1) for next time step.

3.2 Tests & Results

3.2.1 Simulation of exible lament in a uniform conservative force

eld by the immersed boundary method

The motion of a hanging lament without ambient uid under a gravitational force, which

is analogous to a rope pendulum is simulated, the schematic of which is shown in Fig. 3.4. The

lament is initially held stationary at an angle from the vertical, i.e. the initial conditions are

given by

X(s, 0) = X0 + (L− s)(cos k, sin k), ∂X(s, 0)/∂t = (0, 0)

where k is a constant, L is the length of lament, and X0 = (0, 0). At t = 0, it is released

and starts swinging due to the gravity force. In these simulations, we use L = 1, Fr = 10.0,

and k = 0.1π, and compare systems with two dierent bending rigidities: without the bending

25

Figure 3.4: Schematic for exible lament suspended under gravitational force eld

force (γ = 0), and with the bending force included (γ = 0.01). The results for number of

grid-points in Lagrangian system N = 100 based on grid-independence criteria and time-step

∆t = 0.0001 based on CFL criteria of numerical stability are chosen for validation.

As shown in Fig. 3.5(a-b), the lament is totally exible in the absence of the bending force

(γ = 0), and the free end rolls up obviously at the left side, a feature known as a `kick'[58]. When

the bending force is included (γ = 0.01), by contrast, the lament remains straighter during

the pendulum motion and no kick is observed (Fig. 3.5(c-d)). The results from simulation

are also compared with an analytical solution derived using perturbation method using same

parameters. As shown in Fig. 3.5(e), the free end position of the lament obtained from the

analytical solution coincides with the numerical result with the inextensibility condition.

It is also found that the computation is always stable for γ = 0 using the present method.

However, numerical instability is invoked when increasing γ. This is because the tension force

term is treated implicitly, while the bending force term is treated explicitly. It is also observed

that the maximum time step decreases as γ or N increases. However, since we only deal with

soft and slender laments, i.e. with small γ, the numerical instability limitation due to the

bending force term is not signicant.

26

(e)

Figure 3.5: Comparison of results for (a[58],b) case 1 and (c[58],d) case 2 where position of thelaments are plotted at time intervals of 0.02 for a total period of 0.8. (e) compares the solutionobtained to approximate analytical solution for the problem using perturbation method.

27

Chapter 4

Closure

4.1 Conclusion

The literature has been thoroughly explored and examined for its suitability to the numerical

simulation of nucleate boiling on active and passive micro-nned surface. Many recent devel-

opments in this eld have been able to accurately simulate the problems of Fluid-structure

interaction for single-phase ows and Multiphase boiling ow problems independently. How-

ever, very few works have been found on coupling of uid-structure with multiphase ow[61]

and hence provides a great scope for development of comprehensive multiphysics algorithms

in this research area. Hereby, a novel solution algorithm using Immersed Boundary-Level Set

method has been proposed based primarily on the contributions of Huang et al.[58] and Lee

et al.[59]. The sub-module for the Fluid Structure Interaction has been developed and tested

for no ow condition and is being augmented to comply multiphase ow interactions. The

preliminary results of present algorithm development are found to be fairly accurate as well as

computationally ecient.

4.2 Scope of Future Work

An all-inclusive in-house code is to be developed based on the proposed algorithm for simulation

and study of nucleate boiling on micro-nned surfaces. Furthermore, the application can be

extended to other three-phase problems with liquid-gas phase change such as drops on soft

surfaces, piezoelectric actuators [4,5] and dynamic responses of three-phase interactions used

in games and graphics. The future tasks can be organized as:

• Testing of code in single-phase ow. Though the task seems simple but from the stability

and stiness issues experienced in present simulations demands an in-depth study of

time-step criteria for numerical stability and optimization techniques for computational

eciency.

• This task needs to be further extended to two-phase ows without phase change involved.

Level-Set method is to be used for multiphase simulation and additional interfacial forces

28

such as surface tension and contact angle modeling becomes challenging.

• The phase change algorithm needs to be incorporated later and tested for lm boiling as

well as nucleate boiling. The modeling of micro layer heat transfer in nucleate boiling is

an additional subroutine.

• Each step or subroutine needs to be thoroughly validated.

The code needs to be tested for various real-life problems and validated against experimental

data. Thus it may help to study intrinsic physics involved in the ow and heat transfer of such

problems. An extension to three dimensional problems using parallel programming is also an

open end to work.

29

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