Numerical Representation of Free Surfaces in Multi-Phase Flow

Numerical Representation of Free Surfaces in Multi-Phase Flow

CARL-MAGNUS SVENSSON

Master’s Degree Project Stockholm, Sweden 2005

TRITA-NA-E05094

Numerisk analys och datalogi Department of Numerical Analysis KTH and Computer Science 100 44 Stockholm Royal Institute of Technology SE-100 44 Stockholm, Sweden

CARL-MAGNUS SVENSSON

TRITA-NA-E05094

Master’s Thesis in Numerical Analysis (20 credits) at the School of Engineering Physics,

Royal Institute of Technology year 2005 Supervisor at Nada was Gunilla Kreiss

Examiner was Jesper Oppelstrup

Numerical Representation

of Free Surfaces in Multi-Phase Flow

“When I came here I was confused about this subject. Having listenedto your lecture I’m still confused, but on a higher level.”E. Fermi, 1901-1954.

Abstract

This thesis deals with models for surface tension in incompressible viscous flow. Theincompressible Navier-Stokes equations are solved in Matlab using the projection method.The surface tension is implemented using a further development of the Ghost Fluid Method(GFM) as done in [?]. This will keep the surface sharp instead of using a smeared out δ-function that is the case in many other implementations. This method is first implementedin Poisson’s equation by the use of jump conditions. With the right jump conditions in thepressure equation we will get a geometry of the surface that will minimize the energy of thebubble. The bubble will be represented by the zero level set of a signed distance functiondefined over the whole computational domain. This function is then advected using ahyperbolic partial differential equation. We look at a few different Reynolds numbers tocheck that the surface tension effects are properly implemented.

Numerisk representation av fria ytor i flerfasflöde

Sammanfattning

Det har examensarbetet syftar till att simulera ytspanning i inkompressibel viskosstromning. De inkompressibla Navier-Stokesekvationerna loses i Matlab med hjalpav projektionsmetoden. Ytspanningen implementeras genom en vidareutveckling avalgoritmen Ghost Fluid Method (GFM) som beskrivs i [?]. Att anvanda denna me-tod istallet for att anvanda en utsmetad δ-funktion, som ar fallet i manga andrametoder, leder till att ytspanningen kommer att vara koncentrerad till en skarp yta.Metoden implementeras forst i Poissons ekvation med hjalp av hoppvillkor. Medratt valda hoppvillkor i tryckekvationen kommer vi att fa en form pa trycket somkommer att strava efter att minimera bubblans energi. Bubblan representeras avnollnivan hos en teckenfunktion som ar definierad over hela berakningsomradet.Denna funktion flyttas sedan med ett hastighetsfalt med en partiell differentia-lekvation av hyperbolisk typ. Slutligen tittar vi pa denna losning vid nagra olikaReynoldstal for att bekrafta att ytspanningseffekterna ar riktigt implementerade.

Acknowledgments

I want to thank my supervisor Gunilla Kreiss who constructed this thesis for meand also provided guidance. I would also like to thank Elin Olsson for feed-backand help with references.

5

Contents

1 Presentation of the problem 71.1 The incompressible Navier-Stokes equations . . . . . . . . . . . . . . 71.2 The projection method . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Implementation of a surface in Poisson’s equation 112.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Jump conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Implementation in one dimension . . . . . . . . . . . . . . . . . . . . 122.4 Implementation in two dimensions . . . . . . . . . . . . . . . . . . . 152.5 Test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Propagation of the free interface 233.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Discretization and implementation . . . . . . . . . . . . . . . . . . . 243.3 Test runs with a given velocity field . . . . . . . . . . . . . . . . . . 25

4 Introducing a bubble into the Navier-Stokes equations 294.1 Discretization on staggered grid . . . . . . . . . . . . . . . . . . . . . 294.2 Introducing the surface tension . . . . . . . . . . . . . . . . . . . . . 31

5 Results 355.1 Oscillations around a stable circle . . . . . . . . . . . . . . . . . . . . 355.2 Area conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6 Conclusions 39

Bibliography 41

6

Chapter 1

Presentation of the problem

In this master thesis a method for simulating the effect of surface tension in in-compressible Navier-Stokes flow is tested. Other methods often use a regularizedδ-function to implement the surface tension as a volume force, this is for exampledone in [?]. This, however leads to that the surface is smeared out instead of gettinga distinctive interface. This thesis instead uses a more implicit approach to describethe surface tension. By using the methods for jump conditions presented in [?] weare going to implement a pressure jump that corresponds to the surface tension.The background to this is the Ghost Fluid Method (GFM) that is mentioned in [?]and [?]. The interface will be represented using a zero level set of the signed dis-tance function. This interface with jump conditions will finally be introduced intoa Matlab-code that solves the incompressible Navier-Stokes equations. The surfacetension makes an elliptical bubble become circular over time. This effect comesfrom the fact that the bubble strives to minimize it’s surface area. The first partof this thesis will consider how different jump conditions can be implemented intoPoisson’s equation. From there on we look at how the interface can be moved andreshaped on the domain by solving a hyperbolic partial differential equation. Thenwe will look at the incompressible Navier-Stokes equations in two dimensions andhow they can be solved by the projection method. Finally we introduce the inter-face into these equations and see if the effects of surface tension takes place. Thedomain, which we will call Ω, consists of two disjoint pieces, Ω+ and Ω−. Ω+ will beconsidered as the interior of the bubble. The boundary between the the domains,i.e. the surface, will be denoted Γ.

1.1 The incompressible Navier-Stokes equations

The equations of motion for a fluid are the Navier-Stokes equations. The incom-pressible Navier-Stokes equations can be written as

ut + (u · ∇)u +∇p =∇2uRe

+ F + σκδ(Γ)N (1.1)

∇ · u = 0.

7

8 CHAPTER 1. PRESENTATION OF THE PROBLEM

Here u = (U, V ) is the velocity of the fluid in two dimensions, p denotes the scaledpressure, F is external forces such as gravity. The domain, which we will call Ω,consists of two disjoint pieces, Ω+ and Ω−. Ω+ will be considered as the interiorof the bubble. The boundary between the the domains, i.e. the surface, will bedenoted Γ. The term σκδ(Γ)N represents the surface tension forces. This forcedepends on κ, that is the curvature of the bubble interface, and the surface tensioncoefficient σ, that is a constant that is determined by the fluids. The force is locatedat the surface of the bubble, Γ, and N is the normal to Γ, see figure 1.1. This forceacting on a sharp interface will lead to the pressure discontinuity that we want tolook at in this thesis. Finally Re is the Reynolds number,

Re =ρ∞u∞L

µ∞, (1.2)

where ρ∞ is the density in free-stream, u∞ is the free-stream velocity, µ∞ is theviscosity in free-stream and L is the characteristic length. A higher Reynolds num-ber will lead to that we have a more volatile fluid. In equation (1.1) the terms(u ·∇)u are called convection terms and the terms ∇2u

Re are known as viscous terms.Incompressible refers to that the density is constant over time. When the equationis solved numerically we need to limit our computational domain with a boundarythat will be denoted ∂Ω. In this thesis we will use the free slip boundary condition,i.e.

u · n = 0. (1.3)

Here n is the normal at the boundary.

1.2 The projection method

The method that will be used to solve the incompressible Navier-Stokes equationsis called the projection method. The method is described in [?]. In the statementof the equations we can use the incompressibility condition, ∇ · u = 0, to get

∇ · (ut + (u · ∇)u +∇p− ∇2uRe

− F− σκδ(Γ)N) = 0 (1.4)

From this we define u∗ as

u∗ − un

∆t+ (un · ∇)un =

∇2un

Re+ Fn, (1.5)

where n denotes the time level. Then we have to consider the pressure term bydefining

un+1 = u∗ −∆t(∇pn+1 + σκδ(Γ)N). (1.6)

By taking the divergence of equation (1.6) we get

∆pn+1 = ∆t(∇ · u∗ + σκδ(Γ)N)) (1.7)

1.2. THE PROJECTION METHOD 9

Figure 1.1. The division of the computational domain into Ω+ and Ω−.

because of the incompressibility condition. Now u can be updated by equation (1.6)where pn+1 is obtained by solving equation (1.7) and u∗ is defined as in (1.5). Theultimate aim of this thesis is to represent the term σκδ(Γ)N in the equations (1.6)and (1.7) without using any smeared out representation of the δ-function. This willbe done by first introducing a jump condition, [p] = ∆tσκ, into equation (1.7). Tocompensate for the discontinuity in the pressure we also need this term in equation(1.6) to get a smooth function. In equation (1.6) we represent the term by takingspecial care when taking the gradient of the pressure.

Chapter 2

Implementation of a surface inPoisson’s equation

2.1 The equation

The equation into which we want to introduce the jump condition is equation (1.7).This is really a special case of Poisson’s equations. To understand the mathematicalbackground we therefore look at Poisson’s equation. What we have is a computa-tional domain, Ω ⊂ Rn(n = 2), that has a boundary ∂Ω. Then the Poisson’sequation with variable coefficients is defined by

∇(β(x)∇u(x)) = f(x), xεΩu(x) = g(x), xε∂Ω, (2.1)

where x = (x, y)T . The boundary condition, that here is of Dirichlet type, can ofcourse be changed into a Neumann condition by stating that u(x)n = g(x), xε∂Ω.This equation can be solved numerically in a number of ways as for example is donein [?].

2.2 Jump conditions

To represent the free interface we want to place the lower dimensional interface, Γ,that divides Ω into two disjoint pieces, Ω− and Ω+. The coefficient function β(x) isassumed to be continuous in both Ω− and Ω+ but may be discontinuous across Γ.Further let’s assume that β(x) is positive and bounded below by ε > 0. We startby defining two extensions, a(x) and b(x), of the jump conditions over the whole ofΩ. On Γ we want to specify different jump conditions which will be denoted

[u]Γ = a(xΓ), xΓεΓ[βun]Γ = b(xΓ), xΓεΓ. (2.2)

11

12 CHAPTER 2. IMPLEMENTATION OF A SURFACE IN POISSON’S EQUATION

Note that un = ∇u ·N, where N is the normal to Γ.In [?] Γ is defined as the zero level of a signed distance function φ(x). Ω+ is then

defined as the part of Ω where φ(x) ≥ 0 and Ω− is of course the part of the domainwhere φ(x) < 0, see figure 1.1. By considering our two domains we can define twodifferent solutions that coexist on the entire domain Ω. These solutions are u+(x)and u−(x) and they are defined by

u+(x) = u−(x) + a(x) (2.3)β(x)u+

n (x) = β(x)u−n (x) + b(x). (2.4)

This definition is valid throughout Ω but the place where it interesting is on thesurface Γ. On Γ where φ(x) = 0 the jumps are defined as

[u]Γ = u+(x)− u−(x)[βun]Γ = βu+

n (x)− βu+n (x). (2.5)

Note that solutions u+(x) and u−(x) coexist on the whole of Ω and not only at theinterface Γ.

2.3 Implementation in one dimension

To be able to see what is really happening it is best to first look at the one dimen-sional case. Consider the one dimensional Laplace equation

uxx = 0, 0 ≤ x ≤ 1[u]Γ = a(xΓ), xΓεΓ (2.6)

[βun]Γ = b(xΓ), xΓεΓ

with Dirichlet conditions at x = 0 and x = 1. To be able to discretize the equationwe must define

ui,j = u(i ·∆x, j ·∆y). (2.7)

Here is ui,j defined over the whole domain Ω. In other cases we will use the notationu+

i,j or u−i,j to clarify if we want to consider Ω+ or Ω+. For discretization of thisequation let’s use the standard second order scheme[(

ui+1 − ui

∆x

)−(

ui − ui−1

∆x

)]/∆x = 0. (2.8)

Each unknown ui will lead to one row in a linear systems of equations on the formAx = f . The matrix A will be a symmetric tridiagonal matrix and the vectorf contains the boundary conditions, source terms, and, as we will see later, alsothe jump conditions. As a first example let’s assume that φ(x) = x − c0 where0 < c0 < 1. From that follows that Ω− is defined as 0 ≤ x < c0 and Ω+ is

2.3. IMPLEMENTATION IN ONE DIMENSION 13

Figure 2.1. The division of the domain Ω by the interface Γ.

c0 ≤ x ≤ 1. The interface Γ is then just the point x = c0. It’s totally arbitrarywhether you want Γ to be a part of Ω− or Ω+, the important thing is that you makea choice and then stick to it. In this paper Γ will be considered to be a part of Ω+.

In most of the grid points xi equation (??) can be used without problem becauseall the points xi−1, xi and xi+1 is either located in Ω+ or Ω−. But if we’re in Ω−

and are moving right toward Ω+ there will be a grid point k that is the first pointin Ω+, see figure ??. When the term uk is involved in the system of equations wecan’t use use the standard form of equation (??). Instead we must use equation (??)to be able to compare a term that lies in Ω+ with one that lies in Ω−.

By considering the equations (2.2) and (??) we can deduce that

u+(x) = a(x) + u−(x) (2.9)

and

u−(x) = −a(x) + u+(x). (2.10)

This is true on the entire domain Ω but it is only used when the stencil crosses Γ.To get the right jump condition one must interpolate to get a value of a(x) at Γ as

aΓ =ak−1|φk|+ ak|φk−1|

|φk−1|+ |φk|. (2.11)

Please note that if Γ lies exactly upon xk then we will get that φk = 0 and that willof course give

aΓ = ak. (2.12)

Using these equations the discretizations at Γ can be rewritten as[((u+

k − aΓ)− u−k−1

∆x

)−

(u−k−1 − u−k−2

∆x

)]/∆x = 0 (2.13)

and [(u+

k+1 − u+k

∆x

)−

(u+

k − (u−k−1 + aΓ)∆x

)]/∆x = 0. (2.14)


To consider the derivative jump condition first rewrite equation (??) as

(ux)i+1/2 − (ux)i−1/2

∆x= 0. (2.15)

At Γ this will become

(ux)+k+1/2 − (ux)−k−1/2

∆x= 0. (2.16)

By again using equations (2.2) and (??) leads to

u+x (x) = [ux]Γ + u−x (x) = b(x) + u−x (x) (2.17)

and

u−x (x) = −[ux]Γ + u+x (x) = −b(x) + u+

x (x). (2.18)

Note that in equation (??) that β = 1. In the same way as the jump condition wasinterpolated the derivate jump condition is interpolated as

bΓ =bk−1|φk|+ bk|φk−1|

|φk−1|+ |φk|. (2.19)

If both a(x) 6= 0 and b(x) 6= 0 equations (??) and (??) can be written as[(uk − uk−1

∆x

)−(

uk−1 − uk−2

∆x

)]/∆x =

aΓ

(∆x)2(2.20)

and [(uk+1 − uk

∆x

)−(

uk − uk−1

∆x

)]/∆x = − aΓ

(∆x)2+

bΓ

∆x. (2.21)

This will however only be a good approximation if Γ is located exactly on the gridpoint xk. If Γ lies between xk−1 and xk we don’t want to make to use the jumpcondition [βun]Γ in the point xk. Instead the derivate jump condition must beweighted with the distance to the point xk. This is done by defining a function θ as

θ =|φk−1|

|φk−1|+ |φk|. (2.22)

The function θ will take a value between 0 and 1 where θ = 1 means that thedistance from xk−1 to Γ is ∆x, i.e. Γ is located on the grid point xk. This meansthat the derivative jump condition will affect equation (??) as well as (??). Theequations (??) and (??) will then be written as[(

uk − uk−1

∆x

)−(

uk−1 − uk−2

∆x

)]/∆x =

aΓ

(∆x)2+

bΓ(1− θ)∆x

(2.23)

2.4. IMPLEMENTATION IN TWO DIMENSIONS 15

and [(uk+1 − uk

∆x

)−(

uk − uk−1

∆x

)]/∆x = − aΓ

(∆x)2+

bΓθ

∆x. (2.24)

With these equations on the rows k − 1 and k in the system of equations the jumpconditions can be treated.

As we use a second order stencil to discretize equation (2.6) we can expect secondorder accuracy if a(x) 6= 0 and b(x) = 0. If we however have b(x) 6= 0 the secondderivative approximation

(ux)i+1/2 − (ux)i−1/2

∆x= 0 (2.25)

will be first order, here denoted O(1), when approximating the nonzero secondderivative. When this approximation crosses Γ we will however get O(1/∆x). Thisis quite intuitive as the true derivative really is unlimited. The formulation thatwe use will however restore O(1) and therefore we can expect this order for thealgorithm.

2.4 Implementation in two dimensions

To generalize this method we can’t assume that β = 1 and that f(x) = 0 in equation(2.1). In the one dimensional example Ω+ and Ω− are two convex domains, and thatis of course not always the case even in one dimension. While in one dimensionalspace it is fairly easy to make these corrections to the system of equations simplyby considering the problem analytically. In two dimensions however the problembecomes complicated. An algorithm is described in [?] but since this algorithmcontains some errors it will be useful to restate the algorithm in a whole. The figure?? shows what the signed distance function φ(x) can look like in two dimensions.In this case is φ(x, y) = −

√(x− 1)2 + (y − 0.5)2 + 0.2. This function will give the

distance to a circle with radius 0.2 which is centered in (x, y) = (1, 0.5)In figure ??the zero level set of this φ(x, y) is shown. First equation (2.1) is discretized as[

βi+1/2,j

(ui+1,j − ui,j

∆x

)− βi−1/2,j

(ui,j − ui−1,j

∆x

)]/∆x +

+[βi,j+1/2

(ui,j+1 − ui,j

∆y

)− βi,j−1/2

(ui,j − ui,j−1

∆y

)]/∆y (2.26)

= fi,j + F xi,j + F y

i,j .

The function β(x) is discretized using cell averages. A graphical interpretation ofthis stencil is presented in figure ??. In this discretization there are two terms notdeduced from the general Poisson’s equation, namely F x

i,j and F yi,j . These terms

come instead from jump conditions. As was shown in the one dimensional case, ifthe numerical stencil crosses Γ extra terms must be added on the right hand side


Figure 2.2. An example of φ(x) in two dimensions. The function is φ(x, y) =−

p(x− 1)2 + (y − 0.5)2 + 0.2.

Figure 2.3. The zero level set contour of the φ(x) that is shown in figure ??.

of the system of equations. These terms, F xi,j and F y

i,j , are constructed by lookingup, down, left and right in each grid point xi,j . By looking in these four directionswe get four terms FL, FR, FB and FT . Then the right hand terms are constructedas F x

i,j = FL + FR and F yi,j = FB + FT .

Consider the left arm of the stencil. If both φi−1,j < 0 and φi,j < 0 or ifφi−1,j ≥ 0 and φi,j ≥ 0 then FL = 0. This means that both grid points xi−1,j andxi,j lie in the same sub domain, either Ω+ or Ω−. If the signs of φ should differ


Figure 2.4. The standard five point discretization around the point (xi,yj).

between the two grid points the following parameters are calculated:

θ =|φi−1,j |

|φi−1,j |+ |φi,j |, (2.27)

aΓ =ai,j |φi−1,j |+ ai−1,j |φi,j |

|φi,j |+ |φi−1,j |(2.28)

and

bΓ =bi,jn

1i,j |φi−1,j |+ bi−1,jn

1i−1,j |φi,j |

|φi,j |+ |φi−1,j |. (2.29)

In the calculation of bΓ, N = (n1, n2) = ∇φ|∇φ| is used. Please note that these

quantities are independent of whether xi,j is located in Ω+ or Ω−. Hereafter comesa few quantities that differ depending on if the left arm of the stencil goes into Ω+

or out of Ω+. Consider the case φi,j < 0 and φi−1,j ≥ 0, this means that the gridpoint xi,j lies in Ω− and xi−1,j in Ω+. In equation (??) β(x) was discretized usingcell averages i.e. βi−1/2,j = (β(xi−1,j , yj) + β(xi,j , yj))/2. This is sufficiently goodas long as β(x) is a smooth function. As earlier mentioned it is assumed that β(x)is smooth in both Ω+ and Ω− but that it can be discontinuous across Γ. This ishandled by defining φ+ = max(φi−1,j , φi,j) and φ− = min(φi−1,j , φi,j). Then β+ ischosen as the value of β(x) in the point where φ+ is defined. Corresponding β−

will then be defined with help from φ−. This is then used to calculate

βi−1/2,j =β+β−

β+(1− θ) + β−θ. (2.30)

This approximation of βi−1/2,j replaces the earlier one in the system of equationsand is also used to calculate

FL =βi−1/2,jaΓ

(∆x)2−

βi−1/2,jbΓθ

β+∆x. (2.31)


If instead φi,j ≥ 0 and φi−1,j < 0 we determine β+ and β− from the same criteriaas before and then calculate

βi−1/2,j =β+β−

β+θ + β−(1− θ)(2.32)

and

FL = −βi−1/2,jaΓ

(∆x)2−

βi−1/2,jbΓθ

β+∆x. (2.33)

By considering how these terms were deduced in the one dimensional case it’s quiteintuitive why the signs differ depending on if the arm goes into or out of Ω+.

The same procedure as earlier is then used to consider the right arm. If φi+1,j < 0and φi,j < 0 or if φi+1,j ≥ 0 and φi,j ≥ 0 then FR = 0. Otherwise

θ =|φi+1,j |

|φi+1,j |+ |φi,j |, (2.34)

aΓ =ai,j |φi+1,j |+ ai+1,j |φi,j |

|φi,j |+ |φi+1,j |(2.35)

and

bΓ =bi,jn

1i,j |φi+1,j |+ bi+1,jn

1i+1,j |φi,j |

|φi,j |+ |φi+1,j |. (2.36)

Then β+ and β− is determined by the same criteria as earlier and if φi+1,j ≥ 0 andφi,j < 0 we get

βi+1/2,j =β+β−

β+(1− θ) + β−θ(2.37)

and

FR =βi+1/2,jaΓ

(∆x)2+

βi+1/2,jbΓθ

β+∆x. (2.38)

If instead φi+1,j < 0 and φi,j ≥ 0

βi+1/2,j =β+β−

β+θ + β−(1− θ)(2.39)

and

FR = −βi+1/2,jaΓ

(∆x)2−

βi+1/2,jbΓθ

β+∆x. (2.40)

These terms is then added to F xi,j = FL + FR.


This is then repeated in the y-direction. Considering the downward arm givesFB = 0 if φi,j−1 < 0 and φi,j < 0 or if φi,j−1 ≥ 0 and φi,j ≥ 0. If the arm crosses Γwe get

θ =|φi,j−1|

|φi,j−1|+ |φi,j |, (2.41)

aΓ =ai,j |φi,j−1|+ ai,j−1|φi,j |

|φi,j |+ |φi,j−1|(2.42)

and

bΓ =bi,jn

2i,j |φi,j−1|+ bi,j−1n

2i,j−1|φi,j |

|φi,j |+ |φi,j−1|. (2.43)

If φi,j−1 ≥ 0 and φi,j < 0 then

βi,j−1/2 =β+β−

β+(1− θ) + β−θ(2.44)

and

FB =βi,j−1/2aΓ

(∆y)2−

βi,j−1/2bΓθ

β+∆y. (2.45)

If instead φi,j−1 < 0 and φi,j ≥ 0 then

βi,j−1/2 =β+β−

β+θ + β−(1− θ)(2.46)

and

FB = −βi,j−1/2aΓ

(∆y)2+

βi,j−1/2bΓθ

β+∆y. (2.47)

Finally by looking up we set FT = 0 if φi,j+1 < 0 and φi,j < 0 or if φi,j+1 ≥ 0and φi,j ≥ 0. Otherwise

θ =|φi,j+1|

|φi,j+1|+ |φi,j |, (2.48)

aΓ =ai,j |φi,j+1|+ ai,j+1|φi,j |

|φi,j |+ |φi,j+1|(2.49)

and

bΓ =bi,jn

2i,j |φi,j+1|+ bi,j+1n

2i,j+1|φi,j |

|φi,j |+ |φi,j+1|. (2.50)


If φi,j+1 ≥ 0 and φi,j < 0 then

βi,j+1/2 =β+β−

β+(1− θ) + β−θ(2.51)

and

FT =βi,j−1/2aΓ

(∆y)2+

βi,j−1/2bΓθ

β+∆y. (2.52)

Else if φi,j+1 < 0 and φi,j ≥ 0 then

βi,j+1/2 =β+β−

β+θ + β−(1− θ)(2.53)

and

FT = −βi,j−1/2aΓ

(∆y)2−

βi,j−1/2bΓθ

β+∆y. (2.54)

Then we have F yi,j = FB + FT .

2.5 Test results

To check that acceptable results are obtained and to check order of convergencesome test cases were run. In both test cases an evenly spaced grid is chosen, i.e.∆x = ∆y on the two dimensional domain −1 ≤ x ≤ 1,−1 ≤ y ≤ 1.

The first case is the equation ∆u = 0 with jump conditions [u] = ex cos(y) and[un] = 2ex(y sin(y)−x cos(y)). The interface Γ is defined by the circle x2+y2 = 0.25.The exact solution of this equation is u(x, y) = 0 on the exterior and u(x, y) =ex cos(y) on the interior of the circle. Figure 2.5 shows the solution and table 2.1displays the results of the convergence test. This experiment indicates that at leastlinear order of convergence is acquired. If ∆x is cut in half and the error is also cutin half we will get order=1.

∆x L∞-error of u Order L2-error of u Order0.1 0.0152 - 0.0459 -0.05 0.0081 0.91 0.0185 1.310.025 0.0044 0.88 0.0073 1.340.0125 0.0023 0.94 0.0031 1.24

Table 2.1. Order of convergence for the solution in figure ??.

The second test treats the same equation on the same domain but with thejump conditions [u] = x2 − y2 and [un] = 4(y2 − x2). Γ is defined as the circlex2 + y2 = 0.25 exactly as in the first case. The exact solution is u(x, y) = 0 on theexterior and u(x, y) = x2 − y2 on the interior. Figure ?? shows the solution and intable 2.2 the order of convergence test is presented. Also in this case at least linearconvergence is reached. This is consistent with the theory.

2.5. TEST RESULTS 21

∆x L∞-error of u Order L2-error of u Order0.1 0.0072 - 0.0277 -0.05 0.0034 1.08 0.0112 1.270.025 0.0015 1.18 0.0042 1.420.0125 0.0008 0.91 0.0017 1.30

Table 2.2. Order of convergence for the solution in figure ??.

Figure 2.5. Solution of ∆u = 0 with [u] = ex cos(y), [un] = 2ex(y sin(y)− x cos(y)).


Figure 2.6. Solution of ∆u = 0 with [u] = x2 − y2, [un] = 4(y2 − x2).

Chapter 3

Propagation of the free interface

3.1 Equations

The jump conditions will now be used to simulate a bubble moving in viscous fluid.The surface of the bubble will be represented by our interface Γ. This means thatΓ must be able to move and change shape. In [?] a method to advance this frontis presented. This method uses the function φ(x) that was described in chapter 1to represent the bubble. As before we are using the zero level set, φ(x) = 0, as theinterface and that means the interface never has to be explicitly calculated. Thisis good because in [?] it is mentioned that it is hard to implement a function thattracks the interface. Instead the whole function φ(x) is advected. The velocitiesthat advect the interface will be given by the solution of Navier-Stokes equations.However, first a constant velocity field will be used to advance Γ. This also meansthat φ(x) will from here on be dependent on time and will here after be denotedφ(x, t) .

The equation that will advance φ(x, t) is the hyperbolic partial differential equa-tion

φt + (u(x) · ∇)φ = 0 (3.1)φ(x, t = 0) = φ0.

In this equation is u(x) the velocity field. We will use a divergence free field,∇ · u(x) = 0.

One problem that one faces is that equation (??) doesn’t conserve the property ofφ(x, t) being a distance function. In [?] this is fixed by performing a reinitializationstep after the time step. This is done by solving

φτ = S(φ0)(1−√

φ2x + φ2

y) (3.2)

φ(x, τ = 0) = φ0

until steady state is reached. S(φ0) a sign function that assumes the values -1, 1or 0 dependent on the sign of φ(x, t = t2). To avoid numerical difficulties the sign

23

24 CHAPTER 3. PROPAGATION OF THE FREE INTERFACE

function is defined as

S(φ0) =φ0√

φ20 + ε2

, (3.3)

where ε is a small positive number. Please note that this iteration is done within avirtual time τ . This means that equation (??) is updated from a time t1 to a newtime t2 = t1 +∆t. After that τ is set to τ = 0 and hence we get the initial conditionin (??) by setting φ(x, τ = 0) = φ0 = φ(x, t = t2). The time τ is then advanceduntil steady state is reached.

3.2 Discretization and implementation

Both equation (??) and equation (??) are of hyperbolic type and will be solved withupwind schemes. For more details on solving hyperbolic problems see [?]. Theseequations will now be implemented and solved in a two dimensions. If we assumethat we have a divergence free velocity field u(x, y) = (U(x, y), V (x, y)) equation(??) can be rewritten as

φt + (Uφ)x + (V φ)y = 0 (3.4)φ(x, y, t = 0) = φ0.

If φ(x, t) is a smooth function we can approximate φi+1/2,j +φi−1/2,j ≈ φi,j+1/2+φi,j−1/2. If u is divergence free as discussed earlier we can also approximate thatUi+1/2,j − Ui−1/2,j ≈ −(Vi,j+1/2 − Vi,j−1/2). We also assume that the grid is evenlyspaced in both directions, i.e. ∆x = ∆y = h. The time derivative is then approx-imated by standard forward difference and the complete scheme then becomes

φn+1i,j = φn

i,j −∆t((Uni+1/2,j + Un

i−1/2,j)(φni+1/2,j − φn

i−1/2,j)/(2h)+(V n

i,j+1/2 + V ni,j−1/2)(φ

ni,j+1/2 − φn

i,j−1/2)/(2h)). (3.5)

The subscript n denotes that we look at u at the time n ·∆t. The value of φni+1/2,j is

determined by the sign of the velocity field. If the x-component of the velocity fieldu(x, y) is positive in the point xi,j we will use φn

i+1/2,j = φni+1,j and if the component

is negative then φni+1/2,j = φn

i,j . In the same way the other values of φi−1/2,j andφi,j±1/2is determined to get an upwind scheme. The boundary conditions used islinear extrapolation. As an example the boundary condition at the “left” boundarywill become

φ1,j = 2φ2,j − φ3,j . (3.6)

A drawback of this method is that it only uses one sided differences to approximatethe derivatives which means that it only is a first order method.

3.3. TEST RUNS WITH A GIVEN VELOCITY FIELD 25

As in [?] we use the following algorithm is used to solve equation (??). Firstdefine

a = (φi,j − φi−1,j)/∆x,

b = (φi+1,j − φi,j)/∆x,

c = (φi,j − φi,j−1)/∆y,

d = (φi,j+1 − φi,j)/∆y.

(3.7)

This is then used to calculate

G(φ)i,j =

√

max((a+)2, (b−)2, ) + max((c+)2, (d−)2, )− 1, if φ0i,j > 0√

max((a−)2, (b+)2, ) + max((c−)2, (d+)2, )− 1, if φ0i,j < 0

0, otherwise.

Where a+ = max(a, 0) and a− = min(a, 0).The constants b, c and d are then treatedin the same way. The time derivative is also here approximated with the standardforward difference

φτ =φm+1 − φm

∆τ, (3.8)

where m denotes that we looks at the time level m · ∆τ . Will all these quantitiestogether equation (??) can now be updated with the scheme

φm+1i,j = φm

i,j −∆τS(φ0i,j)G(φm

i,j). (3.9)

3.3 Test runs with a given velocity field

To see how this algorithm works a few test runs have been performed. The equationthat is solved is

∆p = 0, 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 (3.10)p(x, 0) = p(x, 2) = p(0, y) = p(2, y) = 0

[p] = 1, [pn] = 0φt + (u(x) · ∇)φ = 0 (3.11)

φ(x, y, 0) = −√

(y − 1.3)2 + (y − 0.7)2 + 0.2

This will give Γ as a circle with radii r = 0.2 that is centered in (x = 0.7, y = 1.3).In figure ?? you can see the constant velocity field, (U = −y + 1, V = x − 1), thatis used to advance the front. In the figure ?? the solution of equation (??) is shownat t = 0 and t = π. We can see that the area that is enclosed by Γ has diminishedconsiderably. We can however see a clear improvement of the result as ∆x and ∆ydecrease. In table ?? the enclosed area is displayed for t = 0 and t = π. Note

26 CHAPTER 3. PROPAGATION OF THE FREE INTERFACE

that the way we measure the error has a numerical error built into it. The area iscalculated using Matlab’s contour-command. This command will produce a resultwith an error that will diminish as the grid is refined. The area of the circle att = 0 should be 0.1257 and in table 3.1 we can see that this error is negligible incomparison with the errors presented in table ??. As we can see in table 3.2 we geta very big error. This is something that must be improved if this method is goingto be successfully used in practical problems.

∆x = ∆y Area at t = 0 Error0.0455 0.1243 0.00140.0312 0.1250 0.00070.0238 0.1253 0.0004

Table 3.1. The numerical error of the contour-command at t = 0.

Figure 3.1. The constant velocity field that is used is (U = −y + 1, V = x− 1).

3.3. TEST RUNS WITH A GIVEN VELOCITY FIELD 27

Figure 3.2. The solution of equation (??) at t = 0 and t = π when ∆x = ∆y = 0.0238.

∆x = ∆y = Area at t = 0 Area at t = π % left0.0455 0.1243 0.0496 39.9 %0.0312 0.1250 0.0738 59.0 %0.0238 0.1253 0.0851 68.7 %

Table 3.2. The amount of the bubble that is left at t = π.

Chapter 4

Introducing a bubble into theNavier-Stokes equations

In chapter 1 the incompressible Navier-Stokes equations were presented. Here thediscretization of these equations is given and then the algorithm that implementsthe surface tension is presented.

4.1 Discretization on staggered grid

To discretize the incompressible Navier-Stokes equation we will use a staggered grid.This means that the velocities U and V and p will be defined in separate grid points.In figure (??) an example of staggered grid is shown. Beside u and p we must alsodefine our signed distance function φ(x, t) on the staggered grid. This means thatpi,j and φi,j is defined in the point xi,j while Ui,j is defined in the point xi+1/2,j andVi,j is defined in xi,j+1/2. The distance function will be defined in the same pointsas the pressure. To decide the value of u∗ by equation (1.5) we must discretize theconvection terms on the staggered grid. The convective terms can in two dimensionsbe written as

(u · ∇)u = (U∂

∂x+ V

∂

∂y)u =

=

(U ∂

∂xU + V ∂∂yU

U ∂∂xV + V ∂

∂yV

). (4.1)

The viscous terms will in two dimensions be

∇2uRe

=1

Re

(Uxx + Uyy

Vxx + Vyy

). (4.2)

To handle the convective terms we first have to do linear interpolation of the velo-cities as

VU =vi,j + vi−1,j + vi,j−1 + vi−1,j−1

4(4.3)

29

30CHAPTER 4. INTRODUCING A BUBBLE INTO THE NAVIER-STOKES

EQUATIONS

Figure 4.1. Example of staggered grid. The p grid points are defined in the samepoints as φ.

and

UV =ui,j + ui−1,j + ui,j−1 + vi−1,j−1

4. (4.4)

These expressions will be used to implement equation (??) on the staggered grid.VU denotes that we have interpolated to determine the value of V in a point xi+1/2,j

where we really have U defined and vice versa for UV . To approximate the deriv-atives we use the standard central difference stencil. The convective terms thenbecome

12(Ui,j

Ui+1,j − Ui−1,j

∆x+ VU

Ui,j+1 − Ui,j−1

∆y) (4.5)

and

12(UV

Vi+1,j − Vi−1,j

∆x+ Vi,j

Vi,j+1 − Vi,j−1

∆y). (4.6)

The viscous terms are not as hard to handle. Equation (??) will simply be discretizedwith the five point stencil that was discussed in chapter 2 and will look like

1Re

(Ui+1,j − 2Ui,j + Ui−1,j

(∆x)2+

Ui,j+1 − 2Ui,j + Ui,j−1

(∆y)2

)(4.7)

and

1Re

(Vi+1,j − 2Vi,j + Vi−1,j

(∆x)2+

Vi,j+1 − 2Vi,j + Vi,j−1

(∆y)2

). (4.8)

4.2. INTRODUCING THE SURFACE TENSION 31

These expressions are then used in equation (1.5) to compute a value of u∗. Byapproximating ∇ · u∗ with central differences we get all that is needed to solveequation (1.7) and then the velocity field can be updated by equation (1.6). Asearlier mentioned this velocity field is used to advance φ(x, t). However as seen infigure ?? the velocity field is not defined in the same points as φ(x, t). This is solvedby linear interpolation between the adjacent velocity points. As an example thevelocities in the grid point xi,j , Uφi,j

and Vφi,j, where φi,j is defined are approximated

as

Uφi,j= (Ui,j + Ui−1,j)/2 (4.9)

Vφi,j= (Vi,j + Vi,j−1)/2

To determine ∆t the expression

∆t =1

max(|u|)/∆x + max(|v|)/∆y + 4/(Re · ((∆x)2 + (∆y)2))(4.10)

is used.

4.2 Introducing the surface tension

Now it is time to introduce the bubble with it’s surface tension into the incompress-ible Navier-Stokes equations. A closer look at equation (1.7) reveals that it reallyis Poisson’s equation, equation (2.1). If we let ∆h denote the standard five pointapproximation of the Laplace-operator. This means we can rewrite equation (1.7)as

∆hpn+1 = fi,j + F xi,j + F y

i,j . (4.11)

The term fi,j is a discrete approximation of ∆t∇ ·u∗ where ∇ ·u∗ is approximatedby standard central differences. The terms F x

i,j and F yi,j are calculated as in chapter

2 and together they represent the term σκδ(Γ)N. When we have continuous densityand viscosity we get from [?] that the jump condition can be formulated as

[p] = σκ. (4.12)

From basic calculus we find that the curvature of a function φ(x) is defined as

κ = −∇ · ∇φ

|∇φ|= −∇ ·N. (4.13)

In [?] a expression for κ in three dimensions is presented. This can easily be rewritteni two dimensions as

κ = −(φ2

xφyy − 2φxφyφxy + φ2yφxx)

(φ2x + φ2

y)1.5. (4.14)


EQUATIONS

All derivatives are then approximated using standard central differences. This jumpcondition will lead to that if Γ is circular the pressure jump will be constant alongΓ. If we further assume that the interior of Γ is Ω+ and that we have no drivingterms, i.e F = 0 in equation (1.1), then p will be constant in Ω+. If instead Γ isdefined by the equation√

(x− 1)2/4 + (y − 0.5) = 0.2, (4.15)

then the pressure will look like in figure ??. As is well known in fluid and thermodynamics things flow from higher pressure to lower. This will lead to a velocity fieldas shown in figure 4.3 and that will give that Γ converges towards a stable circle.This represents the first effect of surface tension namely that for energy purposessurface area that conatins a given volume should be minimized.

Figure 4.2. The upper figure shows the ellipsep

(x− 1)2/4 + (y − 0.5) = 0.2. Thelower displays the pressure when [p] = σκ. Here is σ = 5.

In figure 4.4 we see the bubble contour and pressure at t = 0.3. As expectedthe bubble is now circular and the pressure is constant in both Ω+ and Ω−. Thismeans that we have a steady state. Finally we need to look a bit closer at equation(1.6) and then specially the term ∆t∇pn+1. This is the term in our equations thatinduces a flow from higher pressure to lower. We approximate ∇p by

∇p ≈(

(pi+1,j − pi,j)/∆x(pi,j+1 − pi,j)/∆y

)T

. (4.16)

This will lead to problem when this stencil stretches across Γ in either direction.When this happens we will get a flow from Ω+ to Ω− if as in our case we have a

4.2. INTRODUCING THE SURFACE TENSION 33

Figure 4.3. The velocity field from the conditions shown in figure ??.

Figure 4.4. This figure shows the ellipse in figure ?? at t = 0.3. The bubble is nowcircular.

higher pressure inside the bubble. This is because we have introduced the pressurejump [p]Γ = ∆tσκ. This is countered by the term σκδ(Γ)N in equation (1.6). Inother methods, such as the one in [?], the δ-function is smeared out and the force isapplied. Here we know that we have a pressure discontinuity at Γ and we furtherknow exactly how big the pressure jump is. We can now compensate for the pressurejump when equation (??) crosses Γ. Note that we are not introducing any explicitforce on Γ but we are simply ignoring that we have a discontinuity in the pressure.To explain how this is used let us look at the x-direction. First assume that pi+1,j


EQUATIONS

lies in Ω+ and that pi,j lies in Ω−. This will give that

pi+1,j − pi,j = p+i+1,j − p−i,j = p+

i+1,j − (p+i,j + [p]Γ). (4.17)

This means that if we got this case we use

∇xpΓ ≈ (pi+1,j − pi,j − [p]i,j)/(∆x) (4.18)

to approximate the term. If instead pi+1,j lies in Ω− and pi,j lies in Ω+ we can withsimilar reasoning get

∇xpΓ ≈ (pi+1,j − pi,j + [p]i+1,j)/(∆x). (4.19)

In the y-direction this algorithm extends to

∇ypΓ ≈ (pi,j+1 − pi,j − [p]i,j)/(∆y) (4.20)

if pi,j+1 lies in Ω+ and pi,j lies in Ω−. The last case is if pi,j+1 lies in Ω− and pi,j

lies in Ω+ then

∇ypΓ ≈ (pi,j+1 − pi,j + [p]i,j+1)/(∆y). (4.21)

By using these difference at Γ we can take the gradient of the pressure field andbypassing the problem that it is discontinuous.

Chapter 5

Results

5.1 Oscillations around a stable circle

To see how effective this method is we look at the elliptical bubble that is shown infigure ??. This ellipse is mathematically defined as√

(x− 1)2/4 + (y − 1)2 = 0.2. (5.1)

This ellipse should converge to a circle after some time. What this process lookslike is highly dependent on how viscous the fluid is. A low viscosity, i.e. highReynolds number, will give oscillations of the bubble around the stable circle. Alower Reynolds number will dampen these oscillations. To illustrate this figure ??shows snapshots of Γ in a series of times with Re = 20. On the other hand, figure?? shows Γ when Re = 5. We can clearly see that the oscillations are much smallerin the latter case. We also observed that the frequency of oscillation of the bubbleis much smaller when the Reynolds number is lower.

5.2 Area conservation

A more measurable thing that we can look at is that the area should remain constantover time. It could also be interesting to look at the area loss for different Reynoldsnumbers. As seen in the earlier case a higher Reynolds number will lead to a greatervelocity. This could lead to a bigger numerical error. In table ?? the remaining areaat t = 0.3 with Re = 5 is displayed. In table ?? we see the area conservation whenRe = 10 is used. As the velocities get higher with higher Reynolds number we wantto see if that effects the accuracy of the method. The changes are quite small butfor the finer grids we can see that the errors are a bit bigger when the Reynoldsnumber is higher.

35

36 CHAPTER 5. RESULTS

Figure 5.1. Γ at a number of times with Re = 20. The grid used is ∆x = ∆y = 0.0455.

Figure 5.2. Γ at a number of times with Re = 5. The grid used is ∆x = ∆y = 0.0455.

5.2. AREA CONSERVATION 37

Figure 5.3. The initial and end state of the bubble of which we look at the area.

h Area at t = 0 Area at t = 0.3 percent left order0.0588 0.2508 0.1639 65.4% -0.0455 0.2510 0.2147 85.5% 1.850.0370 0.2512 0.2266 90.2% 1.200.0312 0.2512 0.2327 92.6% 1.12

Table 5.1. Area conservation of the bubble in the two phases seen in figure ?? withRe = 5.

h Area at t = 0 Area at t = 0.3 percent left order0.0588 0.2508 0.1791 65.4% -0.0455 0.2510 0.2120 85.5% 1.850.0370 0.2512 0.2252 90.2% 1.200.0312 0.2512 0.2312 92.6% 1.12

Table 5.2. Area conservation of the bubble in the two phases seen in figure ?? withRe = 10.

Chapter 6

Conclusions

We’ve looked at a special way to simulate a bubble and it’s surface tension in twodimensional incompressible Navier-Stokes flow. To do this we have representedthe surface of the bubble as the zero level set of a signed distance function, φ(x).This zero level set is called Γ and this interface is advanced by the velocity fieldgenerated by the solution of Navier-Stokes equations. The numerical method thatis used to advance and reinitialize Γ is presented in [?]. To solve the incompressibleNavier-Stokes equations the projection method is used that is implemented in aMatlab-code that is based on [?]. In the area enclosed by Γ the method describedin [?] is used to introduce a pressure jump. By using the jump condition describedin [?] we will get a higher pressure jump where the curvature of Γ is high. Thefundamentals of thermo and fluid dynamics will then lead to that an ellipticalbubble will become circular with time. Special care is taken when the gradient ofthe pressure is calculated across Γ. One advantage of this method is that it’s easy toexpand to extend to higher dimensions. Also the pressure jump is strictly confined toΓ and not smeared out as the case is in some of the alternative formulations. Furtherdevelopments of this method is to develop a method with non-smooth viscosity anddensity. It would also be interesting to look at a closer comparison between thismethod and other methods such as the regularized δ-formulation technique. Animportant advantage would also be to implement a higher order method for thesolution of equation (??).

39

Bibliography

[1] X.-D. Liu, R. Fedkiw and M. Kang, A Boundary Condition Capturing Methodfor Poisson’s Equation on Irregular Domains,Journal of Computational Physics160, 151-178, 2000.

[2] A. Iserles, A First Course in the Numerical Analysis of Differential Equations,Cambridge University Press, 1998.

[3] M. Sussman, P. Smereka and S. Osher. (1994)A Level Set Approach for Com-puting Solutions to Incompressible Two-Phase Flow,Journal of ComputationalPhysics 114, 146-159.

[4] M. Kang, R.P. Fedkiw, X.D. Liu.(2000)Journal of Scientific Computing, vol.15, No. 3, 323-360.

[5] R.J. Leveque.Finite Volume Methods for Hyperbolic Problems, Cambridge Uni-versity Press, 2002.

[6] F. Harlow and E. Welch. Numerical calculation of time-dependent viscous in-compressible flow of fluids with free surfaces. Phys. Fluids, 8:2182, 1965.

[7] C. Hirsch. Numerical Computation of Internal and External Flows. Vol. 1:Fundamentals of Numerical Discretization. John Wiley & Sons Ltd. 1988

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Numerical Representation of Free Surfaces in Multi-Phase Flow