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Journal ArticleDOI

Numerical modeling of two-phase flows using the two-fluid two-pressure approach

01 May 2004-Mathematical Models and Methods in Applied Sciences (World Scientific Publishing Company)-Vol. 14, Iss: 05, pp 663-700
TL;DR: In this paper, two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme, to compute two-phase flows using the two-fluid approach.
Abstract: The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows.

Summary (3 min read)

1 Introduction

  • For instance the transport of Liquefied Natural Gas (LNG) in LNG carriers.
  • The numerical modeling of two-phase flow involves dealing with a multitude of jumps of fluid properties across the interface separating the fluids.
  • Numerically such an underresolved interface boundary layer effectively results in a velocity field which has a discontinuity in the tangential direction.
  • Concluding remarks are made in Section 5.

2.1 Primary equations

  • The authors consider incompressible flow for which the mass conservation equations result in a volume constraint on the evolution of the interface d dt |ωl|+.
  • The influence of surface tension is of interest in their application and the authors therefore include it via Laplace’s law −JpK = σκ. (4) Here κ denotes the interface mean curvature and JϕK denotes jump of some flow variable ϕ JϕK := ϕg−ϕl.
  • Note that this model does not impose any smoothness on the tangential velocity component uτ = τ ·u, where τ denotes the interface tangent.

2.2 The pressure Poisson problem

  • Addition of the mass conservation equations, when divided by their respective densities, yields ∫ ∂ω uη dS = ∫ ∂ωl\I ulη dS+ ∫ ∂ωg\I ugη dS = 0, (7) thus showing that the mixture velocity field is divergence free.
  • The latter condition (9) is necessary for having a well-posed coupled Poisson problem.

3.2 Momentum equations

  • In the interior of each of the phases the momentum equations are discretized using the symmetry-preserving finite-volume discretization by Verstappen and Veldman [8].
  • Near the interface the authors choose to discretize the momentum equations in strong form, thereby sacrificing exact momentum conservation at the interface but alleviating difficulties faced with having arbitrarily small cells |cπ|(t) and non-smooth in time face areas | f π|(t).
  • At the interface the authors use a first-order upwind convection scheme per phase, which relies on constant extrapolation of velocities.

3.3 Pressure Poisson problem

  • The solution of the Poisson problem plays a central role in the numerical model, since this is where the two phases are implicitly coupled.
  • The authors will now precisely define the divergence D̄ and the gradient G.

3.3.1 Divergence operator

  • At the interface their velocity field is discontinuous, and therefore the divergence operator needs to be modified.
  • A finite difference approach such as the Ghost Fluid Method (GFM) [4] will result in an incompatible discretization of the Poisson problem ∃u ∈ F̂h0 s.t. ∑ c∈C D̄(u)c 6= 0, (12) where F̂h0 is the set of velocity fields which vanish at the boundary.
  • A consequence of this incompatibility is that the resulting linear system of equations, resulting from the pressure Poisson equation, has no solution.
  • A finite-volume approach however naturally preserves the flux cancellation property which a divergence operator should satisfy, and therefore the authors propose to use the cut-cell method [6] for discretization of the divergence operator.
  • This divergence operator satisfies the discrete equivalent to Gauss’s theorem exactly, and therefore the term in (12) vanishes exactly for all u ∈ F̂h0 .

3.3.2 The gradient operator

  • Near the interface the gradient needs modification to sharply capture the imposed jumps.
  • Hence the authors know the liquid pressure plc on one side of the face and the gas pressure p g c′ on the other side of the face.
  • From the discussion in Section 2.2 the authors know that they should not impose a jump on the full pressure gradient, but rather only on the component normal to the interface.
  • The discretization of the gradient operator for an interface normal face is as follows.
  • For faces f ∈ FIη for which this ratio exceeds 1 the authors interpret the face as an interface tangential face instead.

4 Validation

  • Here the authors consider the validation of the proposed discretization.
  • The authors first assess the accuracy of the discretized Poisson problem and then consider the more exciting dam-break problem.

4.1 Poisson problem

  • The authors compare their proposed method to the Immersed Interface Method (IIM) [3] which sharply imposes jump conditions directly on the Laplacian.
  • The 1d-GFM [4], implemented as described in Section 3.3.2, is also included in the comparison.
  • Whenever the methods 1d-GFM or MdGFM are referred to in the context of a Laplace operator, the composition with the CCM divergence operator (13) is implied.

4.1.1 Mesh refinement

  • As expected, the 1d-GFM is firstorder accurate, whereas the Md-GFM is second-order accurate, and of comparable accuracy to the IIM.
  • The main advantage of using the Md-GFM is that the Laplace operator itself follows from the composition of a divergence operator and a gradient operator which is required in the context of solving incompressible two-phase problems.

4.1.2 Varying the density ratio

  • To asses the dependence of the errors on the density ratio the authors fix the mesh-width h = 2/80, and vary the density ratio.
  • The resulting gradient errors are shown in Table 2.
  • Hence the proposed method can be used to accurately simulate near the one-phase limit ρg→ 0.

4.2 A dam-break problem

  • The proposed discretization has been implemented in their in-house free-surface NavierStokes solver ComFLOW.
  • Local and adaptive mesh refinement is used, as detailed in Van der Plas [7].
  • The authors validate their model using a smooth version of the classic dam-break problem.

4.2.1 Problem description

  • The domain is a rectangle of size 20×12m with an elliptic bathymetry of half lengths 18 and 2.8m whose center lies in the left-hand side bottom corner.
  • Slip boundary conditions are imposed on the velocity field.
  • The liquid density is given by ρl = 103kg/m3, the gas density varies and will always be indicated.
  • Both fluids are initially at rest and separated by the following interface profile y(x) = 7.6+3.6tanh(0.36(x−12.5)) , (25) which will result in a flip-through impact (FTI) [1] in which the wave trough and crest reach the wall at the same time instance, resulting a violent impact.
  • The authors consider several levels of mesh refinement, where they refine the mesh near the interface using blocks of size 16× 16.

4.2.2 Comparison to standard two-phase method

  • Here the authors demonstrate the efficacy of the proposed model when compared to the standard two-phase model2.
  • The authors also show the initial interface profile and at a later time t = 1.47 .
  • Note that the standard two-phase model has a thin region at the interface in which the velocity transitions from gas to liquid.
  • The proposed model captures this transition in a discontinuity, which allows the breaking wave to develop properly, as seen by the interface profile at t = 1.47.

4.2.3 Mesh refinement

  • Results by the CADYF code3[1] are included.
  • Except for the highest level of refinement, l = 4, the authors observe convergence towards the reference solution.
  • For the highest refinement level, the authors observe a small fragmentation of the interface at the location where the velocity discontinuity is largest.
  • The reference solution clearly does not exhibit this behavior, but this could be attributed to numerical damping.

4.2.4 Varying the density ratio

  • Furthermore the authors consider the dependence of the solution on the gas density.
  • For the 2The numerical scheme was identical except for the condition JuτK = 0 and the use of first-order upwind throughout the entire domain.
  • The authors find fairly good agreement in terms of the interface profile.
  • Moreover, for the larger gas densities, oscillations in the tangential velocity jump can be observed just before impact.
  • This suggests that free surface instabilities are about to develop.

5 Conclusion

  • The authors presented a discretization approach for capturing contact discontinuities in two-phase flow.
  • The discretization of the pressure Poisson problem plays a central role since it implicitly couples the phases using a gradient jump condition which corresponds to imposing smoothness of the velocity field only in the interface normal direction.
  • A novel combination of their proposed Md-GFM and the CCM was used to achieve this.
  • Using the dam-break problem the authors then demonstrated that this approach is able to capture contact discontinuities sharply and accurately, even at high density ratios (demonstrated up to 10−6) close to the one-phase limit.
  • The model will be extended with gas compressibility and the effects of viscosity will be included.

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University of Groningen
Numerical modeling of contact discontinuities in two-phase flow
Remmerswaal, Ronald; Veldman, Arthur
Published in:
Computational Methods in Marine Engineering MARINE2019
DOI:
10.1142/S0218202504003404
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Final author's version (accepted by publisher, after peer review)
Publication date:
2019
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Remmerswaal, R., & Veldman, A. (2019). Numerical modeling of contact discontinuities in two-phase flow.
In
Computational Methods in Marine Engineering MARINE2019
https://doi.org/10.1142/S0218202504003404
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Download date: 09-08-2022

VIII International Conference on Computational Methods in Marine Engineering
MARINE 2019
R. Bensow and J. Ringsberg (Eds)
NUMERICAL MODELING OF CONTACT DISCONTINUITIES IN
TWO-PHASE FLOW
RONALD A. REMMERSWAAL
AND ARTHUR E. P. VELDMAN
,
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence
University of Groningen, The Netherlands
r.a.remmerswaal@rug.nl
a.e.p.veldman@rug.nl
Key words: two-phase flow, contact discontinuity, Ghost Fluid Method, Cut Cell Method
Abstract. For convection dominated two-phase flow, velocity components tangential to the
interface can become discontinuous when interface boundary layers are numerically underre-
solved. When sharp interface tracking methods are used it is essential that such discontinuities
are captured in an equally sharp way.
In this paper we propose to model the velocity component tangential to the interface as dis-
continuous using an appropriate interface jump condition on the normal component of the pres-
sure gradient. We achieve this numerically using a novel combination of a Multi-dimensional
Ghost Fluid Method for the gradient and the Cut Cell Method for the divergence operator. The
resulting model is able to accurately and sharply capture discontinuities at large density ratios.
The model is applied to an inviscid dam-break problem. Here we observe that our proposed
model accurately captures the shear layer at the interface with the tangential velocity disconti-
nuity.
In future work we will apply this discretization approach to the modeling of viscous two-
phase sloshing problems with LNG and its compressible vapour, with a particular interest in
studying the development of free surface instabilities.
1 Introduction
Sloshing of fluids in a container is a complex physical phenomena which is present in many
engineering problems. For instance the transport of Liquefied Natural Gas (LNG) in LNG
carriers. In particular the role of free surface instabilities in measured impact pressures during
breaking wave impacts, which may occur during sloshing, is not well understood [2]. Numerical
modeling can facilitate this understanding.
The numerical modeling of two-phase flow involves dealing with a multitude of jumps (dis-
continuities) of fluid properties across the interface separating the fluids. Additional challenges
arise when shear layers develop at the fluid-fluid interface, resulting in an interface boundary
layer. When such an interface boundary layer is underresolved this can result in unphysical

Ronald A. Remmerswaal
and Arthur E. P. Veldman
interaction between the two fluids at the interface. Numerically such an underresolved inter-
face boundary layer effectively results in a velocity field which has a (contact) discontinuity
in the tangential direction. We therefore propose to model the underresolved velocity field as
being discontinuous in the direction tangential to the interface. As a starting point we consider
inviscid two-phase flow modeled by the Euler equations.
In this paper we explore the numerical modeling, in a Finite Volume setting, of contact
discontinuities for incompressible and inviscid two-phase flow. To this end we describe the
governing equations in Section 2, followed by our proposed discretization at the interface in
Section 3. In Section 4 we demonstrate the accuracy of our discretization when applied to a
Poisson problem as well as a time-dependent Euler problem in which we model a dam-break.
Concluding remarks are made in Section 5.
For simplicity in notation we consider a two-dimensional setting.
2 Mathematical model
Here we briefly describe the underlying mathematical model we use, which are the incom-
pressible Euler equations for each of the two phases π = l,g (liquid and gas).
2.1 Primary equations
The primary equations describe the conservation of mass and momentum in each of the
phases in an arbitrary control volume ω = ω
l
ω
g
d
dt
Z
ω
π
ρ
π
dV +
Z
∂ω
π
\I
ρ
π
u
π
η
dS = 0 (1)
d
dt
Z
ω
π
ρ
π
u
π
dV +
Z
∂ω
π
\I
ρ
π
u
π
u
π
η
dS =
Z
∂ω
π
(p
π
ρ
π
g · x)η dS, (2)
where η denotes the face normal, u
π
η
the face normal velocity component, p
π
the pressure, g
the gravitational acceleration and ρ
π
the density per phase. We consider incompressible flow
for which the mass conservation equations result in a volume constraint on the evolution of the
interface
d
dt
|ω
l
| +
Z
∂ω
l
\I
u
l
η
dS = 0, (3)
where |ω
π
| denotes the volume of ω
π
.
The influence of surface tension is of interest in our application and we therefore include it
via Laplace’s law
JpK= σκ. (4)
Here κ denotes the interface mean curvature and JϕKdenotes jump of some flow variable ϕ
JϕK
:
= ϕ
g
ϕ
l
. (5)
We assume immiscible fluids without phase change, and therefore
Ju
η
K= η · (u
g
u
l
) = 0. (6)
2

Ronald A. Remmerswaal
and Arthur E. P. Veldman
Together with appropriate boundary conditions on u
π
and a contact angle boundary condition
on κ this results in a closed system of equations. Note that this model does not impose any
smoothness on the tangential velocity component u
τ
= τ · u, where τ denotes the interface tan-
gent.
2.2 The pressure Poisson problem
Addition of the mass conservation equations, when divided by their respective densities,
yields
Z
∂ω
u
η
dS =
Z
∂ω
l
\I
u
l
η
dS +
Z
∂ω
g
\I
u
g
η
dS = 0, (7)
thus showing that the mixture velocity field is divergence free. Taking the time derivative of
the divergence constraint, substituting the momentum equation and using Ju
η
K= 0, yields an
equation for the pressure
Z
∂ω
1
ρ
η
p dS =
Z
∂ω
η · (u · )u dS. (8)
We supplement the aforementioned equation with Laplace’s law (4), an homogeneous Neumann
boundary condition on the pressure and the following jump condition on the normal derivative
of the pressure gradient
s
1
ρ
η
p
{
=
s
η ·
Du
Dt
{
, (9)
which follows directly from the strong form of the Euler equations. The latter condition (9) is
necessary for having a well-posed coupled Poisson problem.
3 Numerical model
We consider a staggered variable arrangement (Arakawa C grid) on a rectilinear grid. The
grid cells are denoted by the set C , with faces F (c) for c C . The set of all faces is denoted
by F . A subset of the faces are cut by the interface I(t) , we denote this time-dependent
set by F
I
. Every interface face is split into its liquid and gaseous part f = f
g
f
l
. This leads to
the definition of
ˆ
F
π
containing all the (possibly cut) faces which are entirely contained in the
π-phase. Moreover let
ˆ
F =
ˆ
F
l
ˆ
F
g
, see Figure 1.
The space of functions defined on C is denoted by C
h
, with e.g. p C
h
: c 7→ p
c
p(x
c
),
where x
c
is the center of cell c. Similarly we have the function space F
h
, where the approxima-
tions are located at x
f
, the center of the face f . We denote by η
f
the normal of the face f .The
function α : C × F {1,1} encodes the orientation of the face normals such that α
c, f
η
f
points out of cell c.
3.1 Interface advection
The interface is represented using the volume fraction field
¯
χ = |c
l
|/|c| C
h
as per the
Volume-of-Fluid method. Advection of the interface is performed using the Lagrangian-Eulerian
Advection Scheme (LEAS) [9].
3

Ronald A. Remmerswaal
and Arthur E. P. Veldman
3.2 Momentum equations
In the interior of each of the phases the momentum equations are discretized using the
symmetry-preserving finite-volume discretization by Verstappen and Veldman [8]. Near the
interface we choose to discretize the momentum equations in strong form, thereby sacrificing
exact momentum conservation at the interface but alleviating difficulties faced with having ar-
bitrarily small cells |c
π
|(t) and non-smooth in time face areas | f
π
|(t). At the interface we use
a first-order upwind convection scheme per phase, which relies on constant extrapolation of
velocities.
The time integration is performed under a CFL constraint of 0.5 using a second-order accu-
rate explicit method, followed by a pressure correction step
u
f
u
(n)
f
t
= R
3
2
u
(n)
f
1
2
u
(n1)
f
f
, u
(n+1)
f
= u
f
t
ρ
(Gp)
f
, f
ˆ
F . (10)
Here R denotes the convection and gravity terms, and G : C
h
ˆ
F
h
is the gradient operator.
3.3 Pressure Poisson problem
The solution of the Poisson problem plays a central role in the numerical model, since this
is where the two phases are implicitly coupled. The gradient of the resulting pressure is used
to make the mixture velocity field divergence free and it is therefore important that the Laplace
operator can be decomposed in a divergence
¯
D :
ˆ
F
h
C
h
and a gradient G.
Given
¯
D and G we may write the Poisson problem as (for notational convenience we let
t = 1)
¯
D
1
ρ
Gp
c
=
¯
D(u
)
c
, c C (11)
where the gradient operator G contains the value jump due to surface tension as well as the
jump in the normal derivative. We will now precisely define the divergence
¯
D and the gradient
G.
3.3.1 Divergence operator
At the interface our velocity field is discontinuous, and therefore the divergence operator
needs to be modified. A finite difference approach such as the Ghost Fluid Method (GFM) [4]
will result in an incompatible discretization of the Poisson problem
u
ˆ
F
h
0
s.t.
cC
¯
D(u)
c
6= 0, (12)
where
ˆ
F
h
0
is the set of velocity fields which vanish at the boundary. A consequence of this in-
compatibility is that the resulting linear system of equations, resulting from the pressure Poisson
equation, has no solution.
4

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Cites background or methods from "Numerical modeling of two-phase flo..."

  • ...[20], and at last the Godunov scheme of Schwend-...

    [...]

  • ...[20] (see also Guillemaud [24]), the approximation of the convective terms of the system is based on the Rusanov scheme [36] and the so-called VFRoe-ncv scheme [8], these strategies being adapted to the nonconservative framework....

    [...]

  • ...With this in mind, we follow [20] and first observe that the eigenvalues of the Jacobian matrix F0(U) + B(U) of (1) are always real and given by uI, uk, uk ± ck, k = 1, 2, where ck denotes the sound speed of the phase k....

    [...]

  • ...On the ground of the above developments, the definition of the numerical solution to system (26) given in [20] is based on a natural discretization of a1 1 a1 ðtÞ 1⁄4 a1 1 a1 nþ1=2 exp 1 sp R t 0 p1 p2 p1þp2 ðsÞds ; ðak....

    [...]

  • ...The first simulation (Test 1a) is taken from [20] and corresponds to the choice v = 0....

    [...]

Journal ArticleDOI
TL;DR: In this article, a relaxation strategy is proposed to deal with both the nonlinearities associated with the pressure laws and the nonconservative terms that are inherently present in the set of convective equations and that couple the two phases.
Abstract: This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows. We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative terms that are inherently present in the set of convective equations and that couple the two phases. In particular, the proposed approximate Riemann solver is given by explicit formulas, preserves the natural phase space, and exactly captures the coupling waves between the two phases. Numerical evidences are given to corroborate the validity of our approach.

69 citations


Cites background or methods from "Numerical modeling of two-phase flo..."

  • ...3) (u2 remains continuous even if α1 presents a discontinuity, see also [20])....

    [...]

  • ...Rusanov denotes a non conservative version of the Rusanov scheme which is explained in detail in [20]....

    [...]

  • ...4) which corresponds to a particular choice of the closures proposed for instance in [20,25]....

    [...]

  • ...[20] for a comprehensive study of general closure laws concerning uI and pI ....

    [...]

  • ...[20] (see also Guillemaud [25]), the approximation of the convective terms of the system is based on the Rusanov scheme (Rusanov [34]) and the so-called VFRoe-ncv scheme (Buffard et al....

    [...]

References
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TL;DR: In this article, it is shown that these features can be obtained by constructing a matrix with a certain property U, i.e., property U is a property of the solution of the Riemann problem.

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TL;DR: In this paper, the basics of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley, are presented in a way accessible to a wider audience than just mathematicians.
Abstract: The purpose of this book is to make easily available the basics of the theory of hyperbolic conservation laws and the theory of systems of reaction-diffusion equations, including the generalized Morse theory as developed by Charles Conley It presents the modern ideas in these fields in a way that is accessible to a wider audience than just mathematicians The book is divided into four main parts: linear theory, reaction-diffusion equations, shock-wave theory, and the Conley index For the second edition, typographical errors and other mistakes have been corrected and a new chapter on recent results has been added The new chapter contains discussion of the stability of travelling waves, symmetry-breaking bifurcations, compensated compactness, viscous profiles for shock waves, and general notions for constructing travelling-wave solutions for systems of non-linear equations

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Abstract: The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.

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Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "Numerical modeling of contact discontinuities in two-phase flow" ?

In this paper, a discretization of the pressure Poisson problem is proposed to capture contact discontinuities in two-phase flow. 

Eventually the authors want to use this model to study the effects of free surface instabilities in sloshing of LNG and its vapor.