A Coupled Runge Kutta Discontinuous Galerkin-Direct Ghost Fluid (RKDG-DGF) Method to Near-field Early-time Underwater Explosion (UNDEX) Simulations Jinwon Park Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Dr. Alan J. Brown, Chair Dr. Owen F. Hughes Dr. Rakesh K. Kapania Dr. Leigh S. McCue-Weil Dr. Danesh K. Tafti August 28, 2008 Blacksburg, Virginia Keywords: Underwater Explosion (UNDEX), Runge Kutta Discontinuous Galerkin (RKDG), Ghost Fluid Method (GFM), Bubble, Cavitation, Level Set Method (LSM), Multi-fluid Copyright 2008, Jinwon Park
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A Coupled Runge Kutta Discontinuous Galerkin-Direct GhostFluid (RKDG-DGF) Method to Near-field Early-time Underwater
Explosion (UNDEX) Simulations
Jinwon Park
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Equation (2.22) and the expressiondX · dX = dX · I · dX give
dX ·(
(FT · F) − I − 2E)
· dX = 0 (2.23)
whereI is the identity matrix with ones on main diagonal and zeros elsewhere [66, 70, 143]. The
Green-Lagrange strainE is then rearranged as
E =1
2(FT · F − I) =
1
2
(
∂ui
∂Xj+
∂uj
∂Xi+
∂uk
∂Xi
∂uk
∂Xj
)
(2.24)
where
FT · F = FkiFkj =
(
∂ui
∂Xj+
∂uj
∂Xi+
∂uk
∂Xi
∂uk
∂Xj+ δij
)
There are other strains to consider including the following:
X the Euler-Almansi straine
e =1
2(I − (FFT )−1) =
1
2
(
∂ui
∂xj+
∂uj
∂xi−
∂uk
∂xi
∂uk
∂xj
)
(2.25)
is similar to the Green-Lagrange strain except the derivatives of the displacements with respect to
the spatial coordinatesx [66].
30
X the Cauchy infinitesimal strainǫ for small deformation
ǫ =1
2
(
∂ui
∂xj
+∂uj
∂xi
)
(2.26)
is a reduced version of the Euler-Almansi straine. Under the assumption of small deformation
[66], the last term in Equation (2.25) disappears.
X the velocity strainD or the rate-of-deformation
Dij =1
2
(
∂vi
∂xj
+∂vj
∂xi
)
(2.27)
is often recommended in fluid mechanics [115].
Stresses
There are three popular stresses; the Cauchy stressσ, the nominal stressP and the second Piola-
Kirchhoff (or PK2 for brevity) stressS [70, 143]. The Cauchy stress is the ratio of the force applied
and the area in the current configuration as
σ =F
A(2.28)
Since the Cauchy stress follows the symmetry asσ = σT [70, 100, 143], the conservation of
angular momentum can be decoupled from the system of equations [70, 71, 143] and the Cauchy
stress is most commonly used in continuum mechanics [66, 70, 115, 143].
The nominal stress is defined as
P =F
A0(2.29)
whereA0 is the reference area which can be chosen as one in the initialconfiguration. The nominal
stress, or engineering stress is especially useful for measuring the stress in experiments where the
current areaA is not easy to obtain accurately [70, 143].
31
The PK2 stress is the ratio of the force and the area in the reference configuration as
S = F−1
(
F
A0
)
(2.30)
whereF is the deformation gradient (Equation (2.19)). Since the PK2 stress is constant in pure
rotations (or called rigid rotations), the PK2 stress is often used in describing the effects of large
deformation due to rotations [143].
The different stresses are inter-related by transformations in Table2.1with the deformation gradi-
entF and the Jacobian J.
Cauchy stress (σ) Nominal stress (P) PK2 stress (S)
σ - J−1F · P J−1F · S · FT
P JF−1 · σ - S · FT
S JF−1 · σ · F−T P · F−T -
Table 2.1. Transformation between stresses [143]
This section outlined the basics of continuum mechanics that include the fundamental definitions,
kinematic concepts and stress-strain relations which willfrequently be mentioned in this work.
The material time derivative in the Lagrangian descriptionis in the form of ordinary differential
equations (ODE) while those in the Eulerian and the ALE descriptions are in the form of partial
differential equations (PDE) with convection terms. The ALE convection velocitym was defined
to distinguish the ALE description from the others. Note that the convection effect is due to the
difference between material motion and mesh motion.
With these concepts, Section2.2 provides a brief derivation of the fundamental equations which
govern physical phenomena in near-field UNDEX. These equations play a major part in the nu-
merical methodologies, discussed in subsequent chapters.
32
2.2 Governing equations
The fundamental equations which govern the motion of continuums are identical both in a fluid
and a solid, and even applicable to any particular medium. This is a very remarkable property.
The fundamental equations are based on following universallaws of conservation: conservation of
mass, conservation of momentum and conservation of energy.The resulting equations are called
the continuity equation, the momentum equations and the energy equation, respectively.
This section provides a brief derivation of the governing equations using the basic concepts given
in Section2.1. The numerical discretizations of these equations are discussed in the subsequent
chapters. The Reynolds transport theorem, the product rule, and the Gauss’ theorem are useful in
manipulating the mathematical expressions throughout this section [70, 71, 143]. See AppendixA.
The equations are grouped into the Lagrangian expression and the advection expression depending
on the presence of the convection term.
2.2.1 Conservation of mass: continuity equation
The principal of mass conservation requires that the mass within an infinitesimal domain remains
constant; the mass within the domain cannot be created nor bedestructed [27], even if it is rear-
ranged in the domain. The mass within the domain can be transported only by convection. Hence,
there is no diffusion [19].
Lagrangian expression
Let us define massm as [143]
m =
∫
Ω
ρ(X, t)dΩ (2.31)
whereρ(X, t) is density in the domainΩ. For the conservation of mass, the material time derivative
of massm must always vanish as [27, 96, 143]
33
Dm
Dt=
D
Dt
∫
Ω
ρdΩ = 0 (2.32)
which is the integral expression of the continuity equationin the Lagrangian description. As an
alternative, the algebraic form of the continuity equationis often used in the Lagrangian description
as
ρJ = ρ0 (2.33)
where J andρ0 are the determinant of the deformation gradient, det(F) and initial density, respec-
tively [70, 143].
Advection expressions
By applying the Reynolds transport theorem to Equation (2.32), the continuity equation is obtained
as∫
Ω
(
Dρ
Dt+ ρ ·v
)
dΩ = 0 (2.34)
and in the differential form
Dρ
Dt+ ρ ·v = 0 (2.35)
By replacing DDt
in Equation (2.35) by Equation (2.18), we obtain
∂ρ
∂t+ v · ρ + ρ ·v = 0 in the Eulerian description (2.36)
∂ρ
∂t+ m · ρ + ρ ·v = 0 in the ALE description (2.37)
wherev is material velocity andm is the ALE convection velocity which implies there is convec-
tion effect in both descriptions. Equations (2.36-2.37) are in the quasi-linear or non-conservative
form. Applying the product rule to the above equations provides the continuity equations in con-
servative form as
34
∂ρ
∂t+ · (ρv) = 0 in the Eulerian description (2.38)
∂ρ
∂t+ · (ρm) = 0 in the ALE description (2.39)
where the first term is the rate of change of density and the second term is the rate of mass flux
passing out of domain surface [27, 132]. In the steady-state,∂∂t
= 0, Equation (2.38) is simplified
to
·(ρv) = 0 (2.40)
and for incompressible flows
·v = 0 (2.41)
where the densityρ is constant in time.
2.2.2 Conservation of momentum: momentum equation
The principle of momentum conservation requires that the time rate of change of momentum of a
body be equal to the net force exerted on a body [27, 96, 143]
D
Dt(mv) = F (2.42)
wherem, v andF are mass, velocities and net force. Momentum is defined as a vector quantity,mv
and forceF is the resultant of all the forces acting on a body. For constant massm, the momentum
equation is the general expression of Newton’s second law (F = ma). Net forceF consists of body
forceb and surface tractiont.
Lagrangian expression
The momentum equation within the domainΩ is expressed as
D
Dt
∫
Ω
ρvdΩ =
∫
Ω
ρbdΩ +
∫
Γ
tdΓ (2.43)
35
whereρv is linear momentum per unit mass,ρb is body force exerted on a body per unit mass,
andt is surface traction per unit area [143].
The surface integral in Equation (2.43) can be transformed into the volume integral by using the
Cauchy’s relation and the Gauss’ theorem
∫
Γ
tdΓ =
∫
Γ
n · σdΓ =
∫
Ω
· σdΩ (2.44)
whereσ is the Cauchy stress tensor [143]. By replacing the second RHS term of Equation (2.43)
by Equation (2.44), we obtain
D
Dt
∫
Ω
ρvdΩ =
∫
Ω
(ρb + · σ) dΩ (2.45)
where · σ representsσij,i in tensor notation. By applying the Reynolds transport theorem and
the product rule to the LHS of Equation (2.45), it is rewritten as
D
Dt
∫
Ω
ρvdΩ =
∫
Ω
(
D
Dtρv + (ρv) ·v
)
dΩ
=
∫
Ω
(
ρDvDt
+ v(
Dρ
Dt+ ρ ·v
))
dΩ
=
∫
Ω
ρDvDt
dΩ (2.46)
The term (DρDt
+ρ·v) was deleted by the definition of the continuity equation (2.35). Substituting
Equation (2.46) into Equation (2.45), and moving all terms to the left-hand side gives∫
Ω
(
ρDvDt
− ρb − · σ
)
dΩ (2.47)
and in differential form
ρDvDt
= ρb + · σ (2.48)
36
Advection expressions
By applying material time derivatives to Equation (2.48), the momentum equation in the Eulerian
description is obtained as
ρ
(
∂v∂t
+ v · v)
= ρb + · σ (2.49)
and
ρ
(
∂v∂t
+ m · v)
= ρb + · σ (2.50)
in the ALE description. By applying the product rule to the above equations, we obtain the con-
servative form of the momentum equations in terms of linear momentumρv as
∂(ρv)
∂t+ · (ρvv) = ρb + · σ (2.51)
in the Eulerian description and
∂(ρv)
∂t+ · (ρvm) = ρb + · σ (2.52)
in the ALE description.
2.2.3 Conservation of energy: energy equation
The principle of energy conservation requires that the rateof change of total energy within a
domain be equal to the sum of the work done by external forces,and heat energy supplied by heat
sources and heat flux [27, 96, 143]. The conservation of energy is also known as the first law of
thermodynamics:
wtotal = wext + wheat (2.53)
wherewtotal is the rate of change of total energy,wext is the work done by external forces andwheat
is the heat energy added by heat sources and heat flux [27, 96, 143].
37
Lagrangian expression
The rate of change of total energy within the domainΩ is expressed as
wtotal =D
Dt
∫
Ω
(
ρeint +1
2ρv · v
)
dΩ (2.54)
whereρeint is internal energy per unit volume and12ρv · v is kinetic energy per unit volume. The
work done by external forcewext is the sum of the rate of the work done by the body forceb and
the surface tractiont as
wext =D
Dt
∫
Ω
v · ρbdΩ +D
Dt
∫
Γ
v · tdΓ (2.55)
The heat energywheat is the sum of the rate of the work done by heat sources and heat fluxq as
wheat =D
Dt
∫
Ω
ρsdΩ −D
Dt
∫
Γ
n · qdΓ (2.56)
where the sign of heat flux term is negative due to physical reason [143]. For the sake of simplicity,
suppose that no heat sources exist in a control volume,s = 0 and no heat flux flows in and out
through the surface of a control volume,q = 0. These assumptions mean that the system is
thermally isolated [71]. Based on these assumptions, we can decouple the contribution of the
heat energy from the energy equation. Substituting Equations (2.54-2.55) into Equation (2.53), we
obtain the integral form of a simplified energy equation as
D
Dt
∫
Ω
(
ρeint +1
2ρv · v
)
dΩ =
∫
Ω
v · ρbdΩ +
∫
Γ
v · tdΓ (2.57)
By applying the Cauchy’s law and the Gauss’s theorem to the surface integral of Equation (2.57),
the surface integral can be converted into the volume integral as [71]∫
Γ
v · tdΓ =
∫
Γ
v · (σ · n)dΓ =
∫
Ω
· (σ · v)dΩ (2.58)
where · (σ · v) represents(viσij)j in tensor notation. For detail derivations, References [70, 71,
38
143] are recommended. Rearranging all terms to the left-hand side gives
∫
Ω
[
D
Dt
(
ρeint +1
2ρv · v
)
− v · ρb − · (σ · v)
]
dΩ = 0 (2.59)
and the differential form of the energy equation becomes
ρDe
Dt= v · ρb + · (σ · v) (2.60)
wheree is the total energy per unit mass.
Advection expressions
By applying the material time derivatives to Equation (2.60), the energy equations is obtained as
ρ
(
∂e
∂t+ v · e
)
= v · ρb + · (σ · v) (2.61)
in the Eulerian description and
ρ
(
∂e
∂t+ m · e
)
= v · ρb + · (σ · v) (2.62)
in the ALE description. By applying the product rule and the continuity equation, we obtain the
conservative form of the energy equations in terms of conservative variableρe (i.e. total energy
per unit volume)
∂(ρe)
∂t+ · (ρev) = v · ρb + · (σ · v) (2.63)
in the Eulerian description and
∂(ρe)
∂t+ · (ρem) = v · ρb + · (σ · v) (2.64)
in the ALE description.
The universal conservation laws are summarized in Tables2.2and2.3. On the basis of the principle
of continuum mechanics, these equations are applicable to any particular medium. According
to the mesh motion we wish to use, we can choose the Lagrangiangoverning equations or the
advection governing equations.
39
Table 2.2. Summary of universal conservation laws in the non-conservative form
Table 2.3. Summary of universal conservation laws in the conservative form
For the structure, the Lagrangian form is most commonly selected in the literature [70, 91, 100,
143]. If the structure undergoes severe deformation, the ALE governing equations can be chosen
as an alternative to relax severe mesh-distortion associated with large displacements [71]. It is
usually assumed that the densityρ is constant and the entire process is adiabatic, thus allowing
the continuity equation and the energy equation to be eliminated from the system of equations
[91, 100, 143].
Table 2.4. Lagrangian governing equation for the structure
40
For the fluid, the conservative form of the governing equations is used. In Chapter 1, we assumed
that the fluid flow induced by near-field UNDEX is compressible[54]. The non-conservative form
does not provide physically meaningful results in shock-involved flow simulations. It cannot treat
properly the conservation of variables, and also capture the discontinuity of flow variables which
is important in compressible flow simulations [71, 143]. Accordingly, the conservative form of
governing equations has received widespread use in compressible flow applications [7, 18, 71,
143]. For the fluid computation, the following set of ALE governing equations is considered.
Table 2.5. ALE governing equations for the fluid
2.2.4 Additional equation
As shown in Table2.5, there are four unknowns with three equations when we solve 1-D fluid
governing equations: density, velocity, energy and stress. For the structure, there are two unknowns
with the single equation given in Table2.4: velocity and stress. Since a unique solution requires
an equal number of equations and unknowns, one more equationis necessary. The constitutive
relation is the additional equation used to complete the setof governing equations.
2.2.4.1 Constitutive relations
The material’s behaviors are distinguished by their way of resistance against deformation [18]. The
structural behaviors are often defined by
σij = Cijklǫkl (2.65)
41
whereCijkl is the fourth-order tensor of material constants andǫij is the linear strain. Some strains
and stresses were previously described in Subsection2.1.3. Equation (2.65) is a strain-stress re-
lationship for linear elastic materials, also referred to as Hooke’s law [70, 143]. The Kirchhoff
material model which is an extension of the Hooke’s law to large deformations [70, 143], is simply
achieved by replacing the stressσij by the PK2 stressSij , and the linear-strainǫij by the Green-
Lagrange strainEij [70, 143] as
Sij = CijklEkl (2.66)
By the transformation of stresses in Table (2.1), we can convert the PK2 stress to the Cauchy stress
commonly used
σ = J−1F · S · FT (2.67)
where F and J are the deformation gradient and its determinant [143]. For nonlinear plastic mate-
rial models, more complex constitutive relations are required, but these models are not considered
in this work. For the details of such models, References [66, 70, 100, 143] are recommended.
The constitutive relation for the fluid is usually
σij = −pδij + 2µ
(
ǫij − δijǫkk
3
)
(2.68)
whereǫ is the velocity strain andµ is the dynamic (shear) viscosity [115]. Since the displacement
of fluid particles is much larger than those of solid particles, the velocity strain is preferred to
model the motion of fluid particles [71, 115]. For inviscid flows, i.e.µ = 0, Equation (2.68) is
reduced to
σij = −pδij (2.69)
Equation (2.69) is a function of pressurep only. An expression for pressure in Equations (2.68,
2.69) is still required. It is called the equation of state which is a relationship between flow vari-
ables, e.g.p = p(ρ, e, . . .).
42
2.2.4.2 Equations of state
Thermodynamic concepts play an important role in the understanding of compressible flow phe-
nomena that include microscopic properties such as density, pressure and internal energy. This
section provides a brief discussion of thermodynamic concepts and some equations of state for
UNDEX fluid flow simulations.
Thermodynamic relations
For a perfect gas which does not involve intermolecular forces [27, 82], a relation between state
variables, first synthesized by Boyle in the 1600s [82], is defined as
p = ρRT or pv = RT (2.70)
wherep, ρ = 1/v, R andT are pressure, density per unit mass, gas constant and temperature,
respectively. A useful state variable, enthalpy is defined as
h = eint + pv (2.71)
which is a function of internal energyeint, pressurep and specific volumev.
For a calorically perfect gas with constant specific heatscv andcp [82], internal energy per unit
masseint and enthalpyh are given as
eint = cvT (2.72)
h = cpT
By manipulating Equations (2.70-2.72), we obtain the Carnot’s law [53] and other expressions as
cp − cv = R, cp =γR
γ − 1and cv =
R
γ − 1(2.73)
whereγ = cp/cv is the ratio of specific heats. Equations (2.71-2.73) yield a relationship between
pressure, density per unit mass and internal energy per unitvolume as
43
ρeint =p
(γ − 1)(2.74)
which is referred to as the equation of state for perfect gas or the ideal-gas law [27, 53, 82].
Any number of other equations of state can be obtained in a similar way, either theoretically or
experimentally.
Entropy, which indicates how much of the internal energy is available for useful work [27, 82], is
defined as
s = cv log
(
p
ργ
)
+ const. (2.75)
For the isentropic process, which is a reversible and adiabatic process [27, 53], pressure is ex-
pressed as
p = sργ (2.76)
In isentropic flow, the entropy is constant over all the flows excepts at shocks [27, 53, 82]. This
assumption permits a simple treatment of compressible flow complexities which has often been
used in compressible flow simulations [27, 29, 30, 31, 40, 135].
The speed of sound is another useful state variable in thermodynamics. For the relationp =
p(ρ, eint), the speed of sounda is determined by
a =
√
p
ρ2peint + pρ (2.77)
wherepeint andpρ represent partial derivatives of pressurep with respect toeint andρ [27, 135].
For example, the speed of sounda for a perfect gas obeying Equation (2.74) is
a =
√
p
ρ2(γ − 1)ρ + (γ − 1)eint =
√
γp
ρ(2.78)
Since no interaction of particles between microscopic and macroscopic levels exists in incompress-
ible flows [27], incompressible flow simulations do not require the equation of state.
44
Equations of state for explosive gas
The explosion process is largely divided into the detonation phase through a high explosive mass at
rapid constant velocity, and the expansion phase into a surrounding media, e.g. water in UNDEX
events [54, 120]. Currently, this work focuses on the simulation of the expansion phase. Since
the detonation phase can be canceled instantaneously because of its infinite detonation velocity
(e.g. 6930m/s for TNT [17]), the original explosive mass is immediately transformedinto a high
pressure homogeneous gas bubble, and resulting mediums including explosive gas and water are
considered to be compressible and inviscid [1, 54]. For the details of detonation simulations,
References [1, 17, 32] are recommended.
The initial state of UNDEX simulations is usually defined as the moment when a high-pressure
gas bubble is formed after the completion of the detonation process [5, 7, 10, 40, 57, 83, 88, 92,
161]. The initial gas bubble has the same volume and internal energy as the original explosive
in water. See Figure2.6. Both the gas bubble and shock wave initially expand outward[27, 45,
120, 132]. A strong shock wave propagating into the water suddenly raises the flow velocity from
zero, giving rise to the decaying of pressure caused by the rapid decrease of density [120, 131].
As time continues, the shock wave propagates outward fasterthan the interface. The shock wave
propagates at a speed determined by the sum of flow velocity and speed of sound (v + c) in the
water, whereas the interface travels only by flow velocity (v). Across the gas-water interface, two
different equations of state must be applied to describe thestate of each fluid.
Figure 2.6. An illustration of the initial condition and flowfields at timet = t
45
For explosive gas, the JWL EOS and the ideal-gas law have beenextensively employed in the
literature as summarized in Table2.6.
JWL EOS Ideal gas law
A. B. Wardlaw, Jr[7, 5], R. P.Fedkiw[135, 136], J. E.Chisum[83, 162], J. J. Dike[88], HongLuo[57], G. R. Liu[54], K. Webster[99],H. J. Schittke[60], A. Alia[1]
R. R. Nourgaliev[138],A. Pishevar[11], Keh-MingShyue[102], X.Y. Hu[161], C. H.Cooke[30], F. H. Harlow[46], T. G.Liu[147, 148], W. F. Xie[157], J. P.Cocchi[92], M. A. Jamnia[110], R.Saurel[127], Peiran. Ding[116], B.Koren[15], J. Qiu[78], C. Wang[24]
Table 2.6. Use of JWL EOS and ideal gas law for explosive gas
X Jones-Wilkins-Lee (JWL) EOS
p = A
(
1 −ωη
R1
)
e−R1η + B
(
1 −ωη
R2
)
e−R2η + ωρeint (2.79)
where coefficientsA, B, R1, R2 andω are experimentally determined,ρ is density andeint is
internal energy per unit mass,η = ρρ0
is the ratio of current density and initial density [7, 54]. For
typical trinitrotoluene (TNT) and pentaerythritol tetranitrate (PETN), the material properties and
coefficients are listed in Table2.7. These coefficients were determined from the experimental data.
TNT [83] PETN [58]
Initial density,ρ 1630kg/m3 1770kg/m3
Fitting coefficient,A 3.712 × 1011Pa 6.1327 × 1011 Pa
Fitting coefficient,B 0.0321 × 1011 Pa 0.15069× 1011 Pa
The schematic of this slope limiter is shown in Figure3.11.
Figure 3.11. The illustration of a general minmod slope limiter; ux means the slope of solutions and () represents the sign of theslope[84]
74
The limiter compares the slope and its neighbors, then sets the slope to the smaller value in case
(a) while in case (b) or (c) the slope sets equal to zero [84]. Once the slope Uj,x is limited, the
quadratic value Uj,xx is set equal to zero to avoid nonphysical errors. For stable results with no
spurious pressure oscillations, this generalized slope limiter is always applied before computing
the right-hand side of the equations at each level of a singletime step. Below is the pseudo-code
of the slope limiter for a scalar equation.
Algorithm 4 Pseudo-code of the modified minmod slope limiter for the scalar equations
DO i=2,nx-1 ! for internal elements only
! 1. Define minmod argumentsUS=U(i,1) ! slope of element iU1=U(i+1,0)-U(i,0) ! the difference of constant values betweeni + 1 andiU2=U(i,0)-U(i-1,0) ! the difference of constant values betweeni andi − 1
! 2. Apply the minmod limiterUS1=MINMOD(US,U1,U2)IF(US1 .NE. US) THEN ! Check whether the slope is limited or not
U(i,1)=US1; U(i,2)=0ENDIF
ENDDO
FUNCTION MINMOD(US,U1,U2)
IF(US > 0 .AND. U1 > 0 .AND. U2 > 0) THEN ! All signs are positiveMINMOD=MIN(US,U1,U2)
ELSEIF(US< 0 .AND. U1 < 0 .AND. U2 < 0) THEN ! All signs are negativeMINMOD=MAX(US,U1,U2)
ELSEMINMOD=0
ENDIF
ENDFUNCTION
We performed a numerical experiment to assess the performance of this slope limiter by solving
the Fedkiw’s strong shock tube problem given in Subsection3.2.4. Figure3.12 represents the
potential of the slope limiter for suppressing the oscillations near discontinuities. The RKDG
method without the limiting had several kinks near discontinuities which may spoil the numerical
results in many cases.
75
Figure 3.12. The performance of the generalized slope limiter for the Fedkiw’s problem
The slope limiting was applied to local characteristic variables rather than system variables which
are interrelated. The limiting procedure is summarized in AppendixC.
This chapter provided a description of the spatial discretization, the temporal discretization and the
generalized slope limiter in the framework of the RKDG method for compressible inviscid fluid
flows. Section3.1briefly introduced the RKDG method. Section3.2provided a detail of the spatial
discretization based on the DG weak formulation. The DG weakform provides the discontinuous
formulation which is effective for treating shock-involved compressible flows [12, 18]. By relaxing
the inter-element continuity in the discontinuous formulation, the communication between an ele-
ment and its neighbors is conducted only by numerical flux functions. Subsection3.2.4discussed
the first-order FORCE flux function which is better than otherflux functions in accuracy, efficiency
and CPU time [77]. Section3.3provided a description of the temporal discretization based on the
third-step Runge-Kutta time integrator. Section3.4 presented the generalized slope limiter based
on the slope and two neighboring values. Several pseudo-codes for the scalar equation were pro-
vided to explain the implementation of the RKDG method. Thiscompletes the discussion of a
single-fluid flow solution method based on the RKDG method.
This single-fluid method cannot be directly applied to the simulation of two-fluid flows in which
explosive gas and water are presented together since there is no consideration of the discontinuous
adoption of the equations of state across the interface separating fluids. The next chapter discusses
a two-fluid flow solution method incorporated in the RKDG framework.
76
Chapter 4
Solution methods
The previous chapter described the RKDG method consisting of the RK scheme in time, the DG
scheme in space, and the slope limiter in suppressing spurious oscillations. A set of the governing
equations in the ALE description was provided. The numerical approach in Chapter 3 cannot
directly treat two-fluid flows with material discontinuities and handle a deformable fluid mesh
required in near-field UNDEX simulations.
This chapter introduces the solution methods for two-fluid flows in which explosive gas and water
coexists, cavitation effects, and an ALE-based deformablefluid mesh. The solution methods are
integrated in the framework of the RKDG method discussed previously. This provides an ALE-
RKDG compressible two-fluid approach which is a unique coupled method usable for numerically
simulating complex two-fluid flows and fluid-structure interactions in near-field UNDEX events.
Section4.1 provides a description of interface methods and two-fluid methods. A reliable two-
fluid method is suggested for numerically treating explosive gas-water flows. Section4.2presents
a fluid-structure interaction (FSI) algorithm based on a grid moving strategy. Cavitation is dis-
cussed with FSI which may effect the state of the surroundingfluid and the pressure loading on
the structure. The cavitation mechanism consists of cavitation cutoff, its subsequent collapse and
reloading the target. Through several examples in Chapter 5, the effectiveness and robustness of
these approaches are assessed.
77
4.1 Two-fluid method
Multi-fluid/multi-phase compressible flows occur in many industrial situations. There are different
physical and thermodynamic properties across the interface separating fluids. R. Saurel defined a
multi-fluid flow as homogeneous phases separated by well-defined interfaces, and a multi-phase
flow with a number of the interfaces such as a bubbly flow [128]. See Figure4.1.
Figure 4.1. Comparison of a multi-fluid and a multi-phase flow[128]
In Chapter 1, we observed that near-field UNDEX fluid flows consist of homogeneous explosive
gas and water separated by a well-defined interface. Therefore, we restrict our concern to the
two-fluid flows in which immiscible explosive gas and water coexist.
Once the explosion is initiated, the high pressure gas bubble produces a shock wave followed by
the material interface, and a rarefaction wave. At the initial state, both the shock wave and the
material interface propagate radially outward, while the rarefaction wave travels toward the origin
[45, 132]. This distinguishable wave structure provides a good way to identify the location of
the material interface throughout the computation and therefore we can readily distinguish one
fluid from the other across the material interface. Different equations of state (EOS) are then
applied to each fluid. The discontinuous EOS may create spurious pressure oscillations near the
material interface, perhaps resulting from nonphysicallydiffused system variables [31, 92, 135].
This section describes a two-fluid method coupled with an interface method suitable for near-
field UNDEX flow simulations. The first subsection provides a description of interface methods
commonly used.
78
4.1.1 Interface methods
Interface methods can be largely divided into two main classes; Lagrangian approaches and Eule-
rian approaches. Lagrangian approaches can be further subdivided into fitting methods and track-
ing methods. Figure4.2represents the description of interface methods in two-dimensions.
Figure 4.2. Description of interface methods in two-dimensions [76]
The Interface-fitting method (a) employs a set of grids aligned along the interface which is trans-
ported in Lagrangian fashion [10, 57, 79]. This method provides an explicit, accurate interface
location, but for large interface motion the interface-aligned grids become extremely distorted
[57, 145]. In the Interface-tracking method (b), the flow computation is carried out on a spatially-
fixed Eulerian mesh with a set of mass-less markers representing the interface [79]. This method
may enhance the mesh quality in large interface motion, but the motion of markers may still be-
come complicated when the interface motion is extreme [140]. In Lagrangian approaches, the
interface is transported by
dxdt
= v (4.1)
where x = [x, y, z] and v = [u, v, w] are the position and the velocity vectors at every point
[79, 140]. The use of Lagrangian approaches is often limited to the case where interface motion
is not extreme [10, 57, 79, 140]. In case (c), the interface is embedded on a spatially-fixedmesh
and transported by flow velocities in Eulerian fashion [79, 132, 140]. This method may prevent
topological problems as in the Lagrangian approaches when the interface motion is extreme. A
tracer function,φ(x, t), labels every point in the domain as either one fluid or the other, and is
79
transported by
dφ
dt+ v · φ = 0 (4.2)
where v·φ representsuφx + vφy + wφz [79, 132, 140]. Equation (4.2) implies that the interface
defined initially is transported by flow velocities v.
Eulerian methods include theγ-based method, the mass fraction or volume of fluid (VOF) method
and the level set (LS) method [2, 15, 75, 132, 140]. Recently, the LS method has been used in many
disciplines such as graphics, geometry optimization and computational fluid dynamics [79, 140].
The method handles the time-varying shape by solving Equation (4.2). The tracer functionφ(x, t)
becomes the signed distance functionφ = ±d, or the so-called LS function [79, 140]. This method
naturally provides topological information such as curvatures and normal vectors, and is easily
applicable to multi-dimensions with no extra cost associated with parametrizing curves or surfaces
[79, 140]. These characteristics make the LS method a good mathematical tool for modeling the
time-varying nature of complex two-fluid flows. Table4.1 lists the use of interface methods in the
This work uses the LS method to identify the interface location in explosive gas-water flows.
The following sub-subsections detail the mathematical expressions of the interface, the evolution
and reinitialization of the LS function to ensure that it is areal distance function throughout the
computation.
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4.1.1.1 Interface presentation
S. Osher states that the interface can be expressed mathematically as ann − 1 dimensional com-
ponent inn spatial dimension [140]. Therefore, the interface is expressed as points on a line in
one-dimension, a curve on a surface in two-dimensions and a closed surface on a volume in three-
dimensions. Figure4.3gives respective graphical representations of the interface in one-dimension
and two-dimensions.
Figure 4.3. Graphical representations of the interface [140]; external region is denoted byΩ+, internal regionΩ− and interface∂Ω
The LS functionφ(x, t) is the signed distance function±d(x), being negative in one fluid and
positive in the other. The LS function implies thatφ = 0.0 is for points on∂Ω, φ = −d(x) for
points∈ Ω− andφ = d(x) for points∈ Ω+ [140]. A distance functiond labels every point in a
domain with the value of the shortest normal distance to the interface∂Ω [140]
d(x) = min(|x − xi|) where xi ∈ ∂Ω (4.3)
For example, the initial LS functionφ0 is set as
φ0 =
|x| − 1 in 1D√
x2 + y2 − 1 for unit -circle in 2D√
x2 + y2 + z2 − 1 for unit-sphere in 3D
(4.4)
then the LS functionφ is transported by Equation (4.2).
81
Consider a gas bubble in domain[−1, 1] × [−1, 1] as a centered unit-diameter circle. Figure4.4
represents the contours of the initial LS functionφ0 (a), and the LS functionφ (b) transported by a
uniform outward velocity v.
Figure 4.4. Contours of the LS functionφ with a centered circle
For multi-objects, the LS functionφ is expressed as the intersection of two sub-LS functionsφ1
andφ2 [140]
φ = min(φ1, φ2) (4.5)
To illustrate, consider two circles initially embedded in domain[−1, 1]×[−1, 1]. Figure4.5demon-
strates the potential of the LS method showing it is capable of treating the merging of interfaces.
Figure 4.5. Contours of the LS functionφ with two circles
82
This simple initialization and evolution of the LS functionφ makes the LS method a good math-
ematical tool for modeling the time-varying nature of a gas bubble in explosive gas-water flows.
Below is the sample MATLAB code to initialize a unit-diameter circle centered in domain[−1, 1]×
[−1, 1].
Algorithm 5 MATLAB code of the initialization of 2-D LS functionφ for a centered unit-circle[164]
! 1. Define the number of nodes N, the size of elements h and the size of matricesN=40; h=2/N; x=zeros(N,N); y=zeros(N,N); phi0=zeros(N,N); phx=phi0; phy=phi0;
! 2. Set the mesh grid and initial LS functionφ0 by Equation (4.4)[x,y]=meshgrid(-1:h:1,-1:h:1); phi0=SQRT(x.^2+y.^2)-0.5;
! 3. Compute normal vector of internal grids n= φ0
|φ0|by central differential scheme
for i=2,N-1for j=2,N-1phx(i,j)=(phi0(i+1,j)-phi0(i-1,j))/(2*h); phy(i,j)=(phi0(i,j+1)-phi0(i,j-1))/(2*h)
! 4. Plot the contoursfigure(1); surfc(x,y,phi0); hold on; contour(x,y,phi0,zeros(size(phi0)),’k’); axis(’square’);figure(2); quiver(x,y,nx,ny); hold on; contour(x,y,phi0,zeros(size(phi0)),’k’); axis(’square’);
4.1.1.2 Interface evolution
Because of the presence of flow velocity v in the LS Equation (4.6), the system of equations and the
LS equation must be solved simultaneously. Since the LS function φ is a scalar function, Equation
(4.6) can be decoupled from the system of equations. The interface (i.e.,φ = 0) is transported by
dφ
dt+ v · φ = 0 (4.6)
where v is the flow velocity given by the system of governing equations. Note that both the fluid
flow and the LS function do not necessarily have to use the samenumerical approach. For the spa-
tial discretization of Equation (4.6), upwind differencing, Hamilton-Jacobi ENO, Hamilton-Jacobi
83
(HJ) WENO and other schemes have been used [76, 140, 164]. In this work, a FV approximation
is employed to evolve the LS function. A DG approximation forthe LS method is not addressed
here since it would require a separate method development.
The standard FV discretization to Equation (4.6) gives
φn+1i,j = φn
i,j −
[
ui,j
∆x(φi+ 1
2,j − φi− 1
2,j) +
vi,j
∆y(φi,j− 1
2− φi,j− 1
2)
]
(4.7)
whereφi,j is piecewise constant LS values,φi± 12,j andφi,j± 1
2are cell-face LS values. A third-order
flux limiter by Koren [76, 132] is employed to evaluate cell-face LS values as
If ui,j ≥ 0
φi+ 12,j = φi,j + 1
2Φ(ri+ 1
2,j)(φi,j − φi−1,j)
φi− 12,j = φi−1,j + 1
2Φ(ri− 1
2,j)(φi−1,j − φi−2,j)
then
else
φi+ 12,j = φi,j −
12Φ(ri+ 1
2,j)(φi+1,j − φi+2,j)
φi− 12,j = φi,j −
12Φ(ri− 1
2,j)(φi,j − φi+1,j)
(4.8)
where
ri+ 12,j =
φi+1,j − φi,j
φi,j − φi−1,j
and Φ(r) = max
(
min
(
2r, min(1
3+
2
3r, 2)
))
(4.9)
Other limiters in [132] can also be used in evaluating the cell-face values. For they-direction, the
same scheme (4.8) with velocityvi,j is applied to the cell-face valuesφi,j± 12.
The FV spatial scheme is then coupled with third-step Runge-Kutta (RK) temporal scheme which
was discussed in Chapter 3.
4.1.1.3 Interface reinitialization
The LS functionφ in Equation (4.6) is numerically transported by real non-uniform flow velocity
v. Non-uniform velocity generates noisy features that are not appropriate in the LS computation
because the numerical results with distorted LS function become unreasonable at subsequent time
steps [79, 140]. The frequent reinitialization of the distorted LS functionφ is recommended in the
84
literature to correct this problem [15, 79, 122, 135, 140, 147]. The commonly used method is the
reinitialization equation developed by Sussman, Smereka and Osher [79, 140]
∂φ
∂τ+ S(φ0)(| φ| − 1) = 0 (4.10)
whereS(φ0) = 2H(φ0) − 1, τ andH(φ0) are a sign function, fictitious time within the reinitial-
ization equation and a Heaviside function, respectively. The key to this correction is to repeatedly
solve Equation (4.10) until steady state is reached as
∂φ
∂τ= 0 (4.11)
such that
| φ| = 1 (4.12)
At steady state, the distorted LS function can be converted back to real signed distance function
[79, 140]. The absolute gradient of the LS function in Equation (4.10) is determined by a second-
In order to assess the performance of the RKDG-original GFM,a 1D explosive gas-water flow
simulation [78] is performed. The initial conditions are shown in Figure4.12. The domain[0, 1]
m is discretized with 200 elements and the 1D RKDG-original GFM code is run to a final time of
1E-3 second.
Figure 4.12. Initial conditions of an explosive gas-water flow example
In this case, the coupling failed. The erroneous density profile versus the analytic solution at very
early time is provided in Figure4.13. A strong shock wave was generated by the large pressure
jump across the interface. Since a small density error in stiff water generated spurious pressure
oscillations, the density treatment in water is probably a source of numerical instabilities and the
sudden breakdown of the computation.
92
Figure 4.13. Density profile with a overshoot at the interface
At the first time step, density showed a large overshoot at theinterface as observed in Figure4.13
ultimately leading to the sudden end of the computation at a subsequent time step. The density
treatment by the original GFM was not applicable to the explosive gas-water flows we wish to
solve.
T.G. Liu pointed out that the incorrect interface conditions set by the original GFM may cause
nonphysical pressure oscillations [147]. He presented a modified GFM (MGFM), which is sim-
ilar to PC methods using local shock values for correct interface conditions [147]. The MGFM
solve local Riemann problems to set ghost values across the material interface. Compared to the
original GFM which failed at the case shown in Figure4.12, the MGFM has wider application for
various compressible fluid flows [147], but also has several disadvantages: its extension to multi-
dimensions does not seem trivial because of geometric complexities as much as in PC methods,
and it may not be applicable directly to explosive gas-waterflows modeled by a general EOS such
as the JWL EOS. Thus, a GFM based on the original GFM is better suited for explosive gas-water
flow simulations, which may still possess appealing features of the original GFM and does not re-
quire local Riemann solutions. This need required extensive fine-tuning of a variable extrapolation
to obtain reasonable outputs in explosive gas-water flow simulations.
93
4.1.2.2 Direct ghost fluid method
For the water, we directly extrapolate density from the realwater to the ghost water (i.e., in real
gas region) rather than using the indirect variable extrapolation (4.17). For the gas, we directly
extrapolate density from the real gas to the ghost gas (i.e.,in real water region) with the isobaric
fix technique minimizing the “overheating error” [135, 140]. To distinguish it from the indirect
variable extrapolation mentioned above, the current treatment is called the Direct Ghost Fluid
Method (DGFM) which is a variant GFM suitable for explosive gas-water flow simulations. The
DGFM helps decrease the density jump across the interface, thus minimizing spurious pressure
oscillations near the interface. The procedure is described in Figure4.14.
Figure 4.14. The description of the DGFM for explosive gas-water flows
94
Since this approach does not need the entropy calculation asin Equation (4.17) including thermo-
dynamic coefficients, there are no difficulties when a general equation of state such as the JWL
EOS is employed in the computation.
The RKDG-DGF method using the direct extrapolation performs well in the previous 1D explosive
gas-water flow test case. Figure4.15shows that the gas-water interface is fairly well captured at
the correct location, and RKDG-DGF results agree well with the analytic solution. Some density
diffusion occurs in the region near the interface, but no spurious pressure oscillations are observed
in the domain. These results demonstrate the potential of this approach for explosive gas-water
flows common in near-field UNDEX.
Figure 4.15. RKDG-DGF results for a 1-D explosive gas-waterflow
For multi-dimensions, we extend the variables into the ghost region along the normal to the inter-
face as in the one-dimensional manner. To extend discontinuous variables across the interface, we
solve an advection equation only with having to do a few iterations
I,τ ± N · I = 0 (4.18)
95
whereI is a discontinuous variable such as density and tangential velocity andτ is an artificial
time scale [135, 140]. The normal vectors at each point,N = [nx, ny, nz] are provided by the LS
method. To apply the isobaric fix technique, we populate a thin bandǫ = 1.0 ∼ 1.5∆x of real
elements bordering the interface. The positive sign+ is used to extend the variablesI in the real
regionφ < 0 to the ghost regionφ > 0, and the negative sign− is used to extend the variablesI
in the real regionφ > 0 to the ghost regionφ < 0 [135]. Velocities (u, v) in the ghost region are
recomputed by
Vn = unx + vny , Vt = −uny + vnx
u = Vnnx − Vtny and v = Vnny + Vtnx (4.19)
whereVn andVt are the normal velocity directly copied from the real fluid, and the tangential
velocity extrapolated from the other fluid.
Figure 4.16. Description of a two-dimensional DGFM procedure
The DGFM using the direct variable extrapolation performedwell in the 1-D explosive gas-water
flow simulation which failed using the original GFM [135, 140]. Compared to the MGFM, which
uses the local analytic solutions at the interface [147], the DGFM is simpler to extend to multi-
dimensions without solving multi-fluid Riemann problems.
96
Algorithm 6 Pseudo code of the DGFM using the direct variable extrapolation
1. Define two sets of fluidU for water andV for gas
2. Evolve the LS functionφ with flow velocitiesφt + uφx = 0 : Gas denoted by negativeφ and water by positiveφ
For water flow3.1 Set the ghost water located to the left of the interface with densityρR at the right of the interfaceDO i=1, the number of nodes
IF(φi< 0.0 ) THENρi = ρR : copy the densityρR to the density at node imi = ρiui : reassemble momentum flux using modified density at node iEi = ρiei : reassemble energy flux using modified density at node i
END IFEND DO
3.2 Solve the system of equations for waterUt + Fx(U) = 0
For gas flow4.1 Set the ghost gas located to the right of the interface with densityρL nearφ = −1.5∆xDO i=1, the number of nodes
IF(φi > −1.5∆x ) THENρi = ρL : copy the density at node L to the density at node imi = ρiui : reassemble momentum flux using modified density at node iEi = ρiei : reassemble energy flux using modified density at node i
END IFEND DO
4.2 Solve the system of equations for gasVt + Fx(V) = 0
We are ultimately concerned about the effects of underwaterweapons on naval ships. In close-
proximity UNDEX, the hull structure is usually subjected toan extremely high intensity transient
fluid loading perhaps resulting in structural deformation.The structural deformation may cause
a modification of the fluid flow near the structure by a feedbackloop. Therefore, without cou-
pling between the fluid and the structure, we cannot conduct an accurate UNDEX simulation that
97
includes cavitation formation and closure, shock-bubble interaction, and bubble-structure interac-
tion. The following section discusses the fluid-structure interaction with a coupling algorithm, and
a cavitation mechanism including its effect and numerical approach
4.2 Fluid-structure interaction
The fluid-structure interaction (FSI) is commonly considered a feedback loop shown in Figure
4.17[42, 71, 72, 160]. The interface pressures at timen − 1 acting on the structure are provided
by the fluid computation at timen − 1. The structure computation at timen provides interface
displacements to accommodate the fluid mesh against the deformation of the structural interface.
Figure 4.17. A feedback of FSI simulations[71]
In practical FSI simulations, boundary nodes behave in Lagrangian fashion while internal nodes
move independently of flow motion. Accordingly, flow motion and mesh motion are not equal so
the flow variables must be convected to an updated fluid mesh. For the details of this convection ef-
fect, refer to Chapter 2. Since the Eulerian mesh is spatially fixed and the Lagrangian mesh suffers
from excessive mesh distortion, we use the fluid governing equations in the ALE description.
This section presents the solution method of a Lagrangian governing equation for the structure, its
coupling with the fluid , an ALE-based moving mesh algorithm and cavitation associated with FSI.
98
4.2.1 Solution method for the structure
Consider the Lagrangian momentum equation for the structure given in Chapter 2:
ρDvDt
= · σ (4.20)
where the Cauchy stressσ is determined by a constitutive relation. By applying the classical
Galerkin method to the weak form of Equation (4.20), we obtain the matrix form of the momentum
equation
Mdvdt
+ fint = fext (4.21)
where M, fint and fext are mass matrix, internal force vector and external force vector [143]:
M =
∫
Ω
ρNT NdΩ, fint =
∫
Ω
dNT
dxσ and fext =
∫
∂Ω
NT td∂Ω
Now, we temporally discretize Equation (4.21) due to the presence of a time-dependent term. Since
oscillatory structure solutions can contaminate the fluid flow near the structure, a stable solution
method is required. This work uses a two-step Newmarkβ time scheme commonly used in solid
mechanics [91, 100, 143]. Figure4.18represents the flow chart of the two-step Newmarkβ time
scheme used when solving Equation (4.21). Parametersγ andβ play a major role in controlling
the stability of structural solutions. Asγ increases, more artificial viscosity is added [143].
Figure 4.18. Flow chart of the two-step Newmarkβ time scheme [143]
99
To see the effect of artificial viscosity introduced by the Newmarkβ time scheme, a simple axial-
bar example loaded by a distributed external load at the leftend was tested. In Figure4.19, the
largerγ, at a fixedβ, the more artificial viscosity.
Figure 4.19. A simple axial-bar example using the two-step Newmarkβ time schemewith different ratiosγ/β
4.2.2 Interface conditions
The fluid-structure interactions require kinematic and dynamic conditions along the normal to the
interface [71]. No particles may cross the interface and stresses must be continuous across the
interface [71]. To accomplish these requirements, all interface nodes must remain permanently
aligned during the computation by prescribing that the gridvelocities of fluid nodes vg are equal
to the material velocities v of structural nodes on the interface [71].
100
Using permanently aligned interface nodes, we implement the kinematic interface condition. For
inviscid flow, the displacements along the normal to the interface must be continuous as
nf · uf = ns · us (4.22)
wheren andu are normal vector and displacements, and subscriptsf ands represent the fluid and
the structure. This procedure insures that the fluid and structure never detach or overlap during the
computation [71].
The dynamic condition requires that stresses along the normal to the interface be continuous [71].
For inviscid flow, it is achieved by
nfp = nsts (4.23)
wherep andts are fluid pressure and surface traction acting on the structure.
4.2.3 Mesh Coupling algorithm
The two types of FSI mesh coupling schemes, a matching and a non-matching schemes, are shown
in Figure4.20. In a matching scheme, both the fluid and structure meshes arediscretized in a same
mesh density along the interface. A non-matching scheme allows for different mesh densities.
Figure 4.20. FSI mesh coupling algorithms
101
The matching scheme has permanently aligned nodes along theinterface so that the imposition
of interface conditions (4.22-4.23) is straightforward. Since the size of the time step,∆t, may be
determined by the structural mesh density, the computationbecomes expensive.
If a valid data exchange between the fluid and the structure with different mesh densities can be
carried out, computational efficiency can be improved with alarger time step,∆t, compared to the
matching scheme. This method is called the non-matching scheme. The communication is carried
out via a set of virtual interface nodes. Although this scheme requires careful dimensional grid
matching along the interface, it can be considered an alternative coupling method for the matching
method which requires intensive computing cost. This work uses the non-matching scheme with
a coarser structure mesh to increase the size of time step,∆t, which is usually determined by the
structure mesh density. For the details of the grid matchingbetween the fluid and structure meshes,
see AppendixE. Figure4.21depicts an example of the different mesh densities in both methods.
Figure 4.21. An example of mesh couplings: red (fluid) and blue (structure)
Next, we consider a 2-D nodal configuration for near-field UNDEX simulations shown in Figure
4.22with a high pressure circular gas bubble surrounded by water. The high intensity transient
fluid loading (i.e. shock) impinges against the structure, causing the structure to deform. To ac-
commodate this resulting interface deformation, the fluid mesh is adjusted during the computation.
This adjustment is accomplished by allowing the motion of interface nodes in Lagrangian fashion.
102
Figure 4.22. A 2D configuration of FSI in near-field UNDEX
After updating the fluid interface nodes, the remapping of fluid internal nodes is needed to maintain
the quality of the fluid mesh. This is accomplished using an averaging mesh-smoothing scheme
shown in Figure4.23. For other mesh-smoothing schemes, References [109, 143] are recom-
mended.
Figure 4.23. Description of an averaging mesh- smoothing scheme in two-dimensions [20]
Using the updated nodal positionsxn+1, the grid velocity at timen + 1/2 is calculated by
vn+1/2g =
xn+1 − xn
∆t(4.24)
The ALE convection velocitym is defined as
103
mn+1 = vn+1 − vn+1/2g (4.25)
which represents the difference between flow velocity and grid velocity.
The flow computation using any ALE scheme must satisfy the Geometric Conservation Law (GCL)
which states that the numerical scheme preserve the state ofa uniform flow independently of the
grid deformation [69]. M. Lesoinne [107] provided the discussion of a GCL in aeroelastic problems
and its impact on fluid-structure interactions. He stated that the uniform flow can be recovered if
the flow computation is evaluated at the mid-point configuration, and the grid velocity is assumed
to be constant over the time step [107]. The GCL is satisfied by evaluating the flux and integral
terms using the mid-point configurationxn+1/2 and the grid velocityvn+1/2g .
4.2.4 Cavitation in FSI
FSI with cavitation effects is depicted in Figure4.24.
Figure 4.24. The traditional view of cavitation mechanism [6]
The transient intensive pressure loading gives the structure a downward acceleration which contin-
ues until the structure begins to move faster than the surrounding fluid [14, 131]. At this point, the
fluid next to the structure is exposed to tension. Since watercannot sustain tension, a local cavita-
tion region develops near the interface [8, 131]. After reaching its maximum downward velocity,
the structure begins to slow and the local cavitation regioncloses [8, 14, 131]. This closure causes
104
the structure to be reloaded [8, 14, 131]. Due to the presence of cavitating water, it is necessary to
include the cavitating flows to simulate real fluid-structure interactions.
J. M. Brett, et al. [90] performed small-scale experiments showing that cavitation reloading and
bubble effects in near-field UNDEX are as important as the primary shock impact. The duration
and magnitude of the cavitation effects were not negligiblein close-in explosions. Some numerical
studies were also conducted by A.B. Wardlaw [5, 6] and W.F. Xie [157, 158]. To assess cavitation
formation, collapse and reloading, A.B. Wardlaw used a pressure-cutoff model which requires that
cavitation occurs when the water pressure is lower than a pressure limit [6]. The low limit is usually
0.05 bars [6]. He found that the low pressure limit has little impact on the solution as long as it is
positive and much less than 1 bar [6]. When the water pressure drops below the low pressure limit,
the pressure is set to a given low limit as
p = MAX(pcalculated, plimit) (4.26)
wherepcalculated andplmit are the calculated pressure and the given low limit, respectively. The
comparison with experimental data in [6] showed that this simple model performs well in FSI sim-
ulations with cavitation effects. T.G. Liu [149] recently proposed an isentropic one-fluid model
that assumes the cavitating flow is a homogeneous mixture comprising isentropic vapor and liquid
components. This approach calculates the sound speed and pressure of a mixture, separately. W.F.
Xie conducted some numerical experiments using this model that analyze the bulk cavitation near
the free water surface and an internal explosion in a water-filled cylinder [157, 158]. This model
also performed well, but is more complicated than the cutoffmodel. In [149], the comparison
between these two common models shows that the choice of models is not significant to the solu-
tion. For ease of implementation, we use the pressure cutoffmodel, Equation (4.26). In Chapter
5, we examine some cases which include FSI with cavitation effects. For other cavitation models,
Reference [149] is recommended.
The flow-chart of FSI simulations is summarized in Figure4.25. The interface information is
exchanged via a set of virtual interface nodes.
105
Figure 4.25. The flow-chart of FSI simulations using the non-matching scheme
Section4.1 discussed a two-fluid model (i.e. explosive gas-water) based on the LS interface
method. Since the LS method is able to capture any time-varying shape, it has been applied to
a wide range of engineering problems [140]. However, the mixed elements in which fractions of
explosive gas and water coexist make the use of LS method alone inadequate for two-fluid flow
simulations. Physical and thermodynamic properties are discontinuous across the interface so the
classical methods may suffer from nonphysical behaviors. The DGFM using direct variable ex-
trapolation to minimize or remove the spurious pressure oscillations near the interface is used.
This method performed well in the 1D explosive gas-water flowsimulation in which the original
GFM failed. Compared to existing methodologies, this method was simpler to implement and to
extend to multi-dimensions. Section4.2 discussed the procedures for fluid-structure interaction
simulations. Without coupling between the fluid and the structure, we cannot carry out a valid
UNDEX simulation. The structural solution method with the ALE-based moving mesh algorithm
106
was addressed. The fluid mesh which is usually fixed in space must be adjusted to the deformation
of the structural interface.
Chapters 1 to 4 concluded that near-field UNDEX simulations require the treatment of two-fluid
compressible flows and a deformable fluid mesh for fluid-structure interaction simulation. This
required the ALE-based RKDG compressible two-fluid method:the RKDG method, the DGFM
and the ALE methodology. In Chapter 5, we examine 1D and 2D cases to assess the performance
of the suggested solution methods.
107
Chapter 5
Assessment
This chapter describes assessment of RKDG-DGF approach outlined in the previous chapters.
Section 5.1 examines several 1D Cartesian cases (Cases 5.1.1~5.1.4) and 1D symmetric cases
(Cases 5.1.5~5.1.7). RKDG-DGF results are compared with analytic solutions and other reference
results. For the 1D Cartesian cases, the analytic solution is obtained using a Riemann problem so-
lution FORTRAN code provided in [45]. For symmetric cases, an analytic solution is unavailable
[27, 45] so reference results from previous work are used. Section 5.2 examines the extension of
the 1D RKDG-DGF approach to multi-dimensions. 1D symmetricRKDG-DGF results on a fine
mesh are used to assess RKDG-DGF results in multi-dimensions. Section 5.3 extends the RKDG-
DGF approach to practical UNDEX applications that include bubble behavior (Cases 5.3.1 and
5.3.2), shock-bubble interaction and cavitation with fluidstructure interaction (FSI) (Cases 5.3.3
and 5.3.4). Experimental data and reference results are used to assess the RKDG-DGF approach
in these cases. Since the gas bubble caused by UNDEX has spherical symmetry, the 1D spherical
symmetric RKDG-DGF approach can be used to simulate bubble behavior (e.g., peak pressure,
bubble radius-time history and maximum bubble radius). Cases 5.3.3 and 5.3.4 consider an ex-
ternal explosion near a rigid wall, and an internal explosion within a water-filled aluminum tube.
The pressure cutoff model discussed in Section 4.2 is used topredict the formation and collapse of
108
cavitation which is an important phenomenon in these cases.The differences between a rigid-wall
case and a deformable-wall case are discussed to describe the shock-bubble interaction and wall
deformation effects with FSI. The cavitation mechanisms with FSI are described. An ALE-based
deformable fluid mesh is used and explored in the fluid-structure coupling. These cases show the
potential of our approach for near-field UNDEX applicationsand identify the deficiencies we need
to correct.
In all test cases, fluid computations use the RKDG-DGF methodwith CFL=0.2. Our 1D and 2D
RKDG-DGF codes use FORTRAN 90. For the details of the boundary conditions used, see Ap-
pendixF. Unless stated otherwise, we use a uniform mesh of 200 elements along each direction,
a rigid wall boundary condition, and the c-g-s unit system (i.e., density (g/cm3), velocity (cm/s),
pressure (dyne/cm2), length (cm) and time (second)), respectively. The interface is initially lo-
cated at the center of the domain. Variablesρ andp are chosen to show the numerical results.
5.1 One-dimensional assessment
Several 1-D cartesian and 1D symmetric cases are consideredin this section.
5.1.1 Cartesian 1-D cases
This subsection explores the applicability of RKDG-DGF approach for simple high pressure gas-
water flows which are qualitatively similar to explosion-produced gas and water flows in near-field
UNDEX. Assessment of RKDG-DGF approach is carried out by comparing with the Riemann
analytic problem solution [27, 45, 92]. In gas dynamics, the ideal equations of state are often
employed to easily obtain the analytic solution [45, 92]. For consistency, this approach is followed
in Cases 5.1.1~5.1.3. Unless stated otherwise, the Tait equation of state (EOS) withγ = 7.15,
B = 3.31E9 dyne/cm2 andρ0 = 1.0 g/cm3, and the ideal gas law withγ = 1.4 are used to model
the water and gas. Other EOS model are also used in other test cases.
109
Case 5.1.1: This case considers a high pressure gas-water shock tube problem previously modeled
by J. Qiu [78]. The initial conditions are given in Figure5.1. J. Qiu solved this problem using a 1D
two-fluid RKDG method which treats the moving interfaces conservatively [78]. J. Qiu’s method
obtains the solution of the cells near the interface by solving multi-medium Riemann problems
defined at the interface [78]. The solution procedure in J. Qiu’s method [78] is similar to that of
Godunov-type Prediction Correction method which requireslocal Riemann solutions to correct
affected cell values in the region near the interface (See Section 4.2). The 1D RKDG-DGF code is
run to a final time of 0.16E-3 seconds. RKDG-DGF results versus the Riemann problem analytic
solution and J.Qiu’s results are compared in Figure5.2.
Figure 5.1. Initial conditions for Case 5.1.1
Figure5.2 shows that at 0.16E-3 seconds, the shock front and material interface predicted by the
RKDG-DGF method, which are initially located at the center of the domain, are observed at about
86cm and 55cm. Since the shock travels at a speed by the sum of the speed of sound and flow
velocity, the shock front propagates outward faster than the material interface [45]. The material
interface is transported by flow velocity only. The RKDG-DGFmethod for this case resolves
110
Figure 5.2. RKDG-DGF results for Case 5.1.1 at 0.16E-3 seconds
the shock front with only 3 elements. The head and tail of rarefaction wave are also very well
resolved. Small density oscillations are observed in the region near the interface, but no pressure
oscillations occur. Compared to J. Qiu’s results taken from[78], the RKDG-DGF method produces
a less-diffusive and oscillatory density profile.
Case 5.1.2: This case considers a strong shock tube problem previouslyinvestigated by H. Luo
[57]. H. Luo assessed the application of various numerical fluxes on an ALE moving grid. [57]. He
states that the ALE approach is attractive when the interface is not subjected to large deformation
[57]. Since large interface deformation occurs in real UNDEX problems and may lead to significant
distortion of the mesh, this is a major concern which is avoided in the RKDG-DGF method. To
compare the performance of the RKDG-DGF method and H. Luo’s ALE-based method, the same
initial conditions shown in Figure5.3are used. The 1D RKDG-DGF code is run to a final time of
1.55921E-04 seconds. The domain [0, 100] cm is discretized with 100 elements as in [57]. RKDG-
DGF results versus the Riemann problem analytic solution and H. Luo’s results are compared in
Figure5.4.
111
Figure 5.3. Initial conditions for Case 5.1.2
Figure 5.4. RKDG-DGF results for Case 5.1.2 at 1.55921E-04 seconds
112
Compared to the analytic solution, H. Luo’s results are verydiffusive. To improve the resolution of
results, the number of elements must be increased. The RKDG-DGF method using the same num-
ber of elements produces less-diffusive profiles for the density and the pressure compared to those
of H. Luo’s ALE-based interface method without the requirement for interface mesh adjustment.
Case 5.1.3: This case considers a TNT-produced equivalent gas and water flow problem previously
modeled by J. Qiu [78]. J. Qiu considered this problem to assess the application of a 1D RKDG
two-fluid method for an explosion flow simulation. To comparethe performance of the two dif-
ferent RKDG approaches (i.e., J. Qiu’s two-fluid RKDG methodand RKDG-DGF method), the
same initial conditions are considered as shown in Figure5.5. Both initial density and pressure in
the gaseous medium are equivalent to those of TNT-produced gas from [5]. The 1D RKDG-DGF
code is run to a final time of 0.1E-3 seconds. RKDG-DGF resultsversus the analytic solution and
J.Qiu’s results are compared in Figure5.6.
Figure 5.5. Initial conditions for Case 5.1.3
113
Figure 5.6. RKDG-DGF results for Case 5.1.3 at 0.1E-3 seconds
With both methods, the shock front and material interface are sharply captured at the correct loca-
tions and magnitudes. Only 2~3 elements are used to resolve the shock front. As in Case 5.1.1, our
RKDG-DGF approach provides a less-diffusive and oscillatory density profile in the region near
the interface without the requirement of multi-medium Riemann solutions at the interface.
Case 5.1.4: This case considers a TNT-produced gas and water flow problem modeled by A.B.
Wardlaw using an ALE-based interface method [5]. The ALE-based interface method treats the
interface in a Lagrangian manner, and adjusts fluid nodes to the interface deformation [5]. As
discussed in Case 5.1.2, unlike in the RKDG-DGF method, special concern for mesh adjustment is
required. In this case, RKDG-DGF results are compared only to the results from [5]. An analytic
solution is unavailable. The JWL EOS is used to model the explosive gas. Material properties and
coefficients for the JWL EOS are summarized in Table2.7. The initial conditions are shown in
Figure5.7. The 1D RKDG-DGF code is run to a final time of 0.5E-3 seconds. RKDG-DGF results
versus the reference results from [5] are compared in Figure5.8.
114
Figure 5.7. Initial conditions for Case 5.1.4
Figure 5.8. RKDG-DGF results for Case 5.1.4 at 0.5E-3 seconds
115
Both methods provide sharp density and stable pressure profiles. The shock front in the RKDG-
DGF results is sharply captured with only 2 elements. The results show that the RKDG-DGF
method is applicable for numerically simulating explosivegas flows modeled using the JWL EOS.
Unlike A.B. Wardlaw’s ALE approach, our approach does not require special concern for mesh
adjustment.
Cases 5.1.1~5.1.4 were selected to assess the application of the RKDG-DGF method for simple
high pressure gas-water flows which are very similar to TNT-produced gas-water flows in near-
field UNDEX applications. The high pressure gaseous medium produces a strong shock travelling
into the water and simultaneously a rarefaction wave movingtoward the origin. Compared to
the analytic solutions and reference results from previousworks, the RKDG-DGF method sharply
captures both the shock front and interface at the correct locations and magnitudes. The head
and tail of rarefaction are also very well resolved. Some density diffusion occurs in the region
near the interface, but it is less than with other methods andno spurious pressure oscillations
are visible in the domain. Compared to the original GFM and the MGFM methods which add
difficulty and complexity in multi-fluid using the JWL EOS andmulti-dimension implementation,
the RKDG-DGF approach did not increase difficulty and complexity even in the implementation
and computation of the JWL EOS. Since the RKDG-DGF method in the fluid is based on an
Eulerian description, no special concern for mesh adjustment is required as in the ALE-based
methods for interface tracking.
To assess the applicable range of the RKDG-DGF method, initial pressures in the gaseous medium
were increased by 8E+8Pa to 7.81E+9Pa. The RKDG-DGF approach produced no severe error
through Cases 5.1.1~5.1.4 when compared to the analytic solution. Since the initial pressure pro-
duced by typical TNT explosives is around 7.81E+9Pa [5, 17, 32], the RKDG-DGF method is very
applicable to real near-field UNDEX simulations without magnitude limitations.
116
5.1.2 Symmetric 1-D cases
1D symmetric solutions can be used to assess multi-dimensional solution in method development
[45, 102], and to analyze symmetric UNDEX fluid flows [5, 7, 10, 92]. A reliable 1-D symmetric
solution on a very fine mesh can provide very accurate numerical results. Section 5.1.1 presented
a series of 1D assessment for the RKDG-DGF method. This Cartesian approach can be extend to
a 1D symmetric approach as follows. The governing equationsare rewritten as
Ut + F (U)r = S(U) (5.1)
where
U =
ρ
ρu
ρe
, F =
ρu
ρu2 + p
u(ρe + p)
and S = −α
r
ρu
ρu2
u(ρe + p)
wherer is the stream-wise distance andu is the stream-wise velocity. The vectorS(U) with α
is the geometric source vector reflecting the symmetric effect [30, 45, 73, 92]. For cylindrical
symmetry,α is 1.0. For spherical symmetry,α is 2.0. Whenα is zero, Equation (5.1) becomes
equal to the 1-D cartesian conservation laws.
Equation (5.1) can be solved either by a fully coupled method or a splittingmethod [45, 132]. For
ease of implementation, this work uses the coupled method. For splitting methods, References [45,
132] are recommended. Cases 5.1.5 to 5.1.7 explore this symmetrical 1D RKDG-DGF approach
for cylindrical and spherical symmetry.
Case 5.1.5: This case considers a single-gas cylindrical explosion problem previously modeled
by E.F. Toro using a FVM [45]. The non-dimensionalized initial conditions are shown inFigure
5.9. E.F. Toro’s 2D results were assessed by comparing with 1D cylindrical results on a very fine
mesh [45]. To reflect cylindrically symmetry, we setα = 1 in Equation (5.1), and run the 1D
cylindrical RKDG-DGF code to a final time of 0.25 seconds. In Figure5.10, RKDG-DGF results
are compared with E.F. Toro’s 1D cylindrical results taken from [45]. The results show good
agreement.
117
Figure 5.9. Initial conditions for Case 5.1.5
Figure 5.10. RKDG-DGF results for Case 5.1.5 at 0.25 seconds
118
Case 5.1.6: This case considers a spherical UNDEX flow which has been used extensively in
method developments [30, 73, 92, 102, 146]. It was first numerically studied by J. Flores using the
Random Choice Method (RCM) [73]. C.H. Cooke employed the subcell resolution method with
the interface tracking scheme for this problem [30]. The subcell resolution method produced very
good numerical results compared with those reported in [73]. This case requiresα = 2 to reflect
spherical symmetric flow. The initial conditions are shown in Figure5.11. The domain[0, 30.48]
cm is discretized with 200 elements and the 1D spherical RKDG-DGF code is run to a final time
of 0.455E-3 seconds. The interface is initially located 10.16 cm from the origin. In Figure5.12,
RKDG-DGF results are compared to the results of the subcell resolution method in [30]. As in
[30, 73, 92, 102, 146], density, pressure and radial distance on the plots are non-dimensionalized
by ρ′ = ρ/1.007, p′ = p/3.331231E9 andr′ = r/30.48 [30]. The Tait EOS withγ = 7.0 and
B = 3.311301E9 dyne/cm2, and the ideal-gas law withγ = 1.4 are used to model the water and
explosive gas, respectively.
Figure 5.11. Initial conditions for Case 5.1.6
119
Figure 5.12. RKDG-DGF results for Case 5.1.6 at 0.455E-3 seconds
Once the explosion is initiated, a spherical outward shock wave travels into the water and a spheri-
cal inward rarefaction wave propagates toward the origin, simultaneously. Both the location of the
interface and shock front compare very well to those reported in [30]. The interface is sharply cap-
tured, and no spurious pressure oscillations occur. Eulerian methods used in [30, 73, 92, 102, 146]
require local Riemann solutions to correct affected variables across the material interface, but the
RKDG-DGF method does not require this complexity.
Case 5.1.7: This case considers a spherical UNDEX shock problem taken from A.B. Wardlaw’s
test cases [5]. As in Case 5.1.4, A.B. Wardlaw used the ALE-based interface method. The JWL
EOS is used to model the TNT-produced explosive gas. Similarly, Case 5.3.1 considers this prob-
lem to approximate the characteristics of a spherical UNDEXbubble pulse. However, Case 5.3.1
requires a larger mesh and longer time duration for bubble pulses. The initial conditions are shown
in Figure5.13. For initial conditions, we assume that a TNT-produced gas bubble with radius
r0 = 16 cm is surrounded by water at rest. We setα = 2 in Equation (5.1) for spherically symme-
try. The domain[0, 250] cm is discretized with 200 non-uniform elements, and the 1D spherical
RKDG-DGF code is run to 0.5E-3 seconds and 1.0E-3 seconds, respectively.
120
Figure 5.13. Initial conditions for Case 5.1.7
RKDG-DGF results are compared to those from [5]. Figure5.14shows that both methods, which
one method is A.B. Wardlaw’s 1D ALE interface method and the other method is our 1D RKDG-
DGF method, predict very similar pressure profiles at two times. In [5], reference density profiles
are unavailable. At 0.5E-3 seconds, the locations of the material interface and shock front are about
41 cm and 120 cm. A secondary outward shock is created, and themagnitude of peak pressure
is about 1.9Edyne/cm2. At 1.0E-3 seconds, the locations of the material interfaceand shock front
are about 52 cm and 250 cm. A reflected inward shock and a transmitted shock near the interface
are observed, and the magnitude of peak pressure is about 0.75Edyne/cm2. In Figure5.14, the
locations of material interfaces and shock fronts, and the magnitudes of peak pressures compare
very well to those from [5].
The shock fronts propagate outward faster than the materialinterfaces during time interval. The
shock front travels about 130 cm, but the material interfaces travels about only 11 cm. It is mainly
due to the difference of the characteristic speeds as discussed in Case 5.1.1. The magnitudes of
peak pressure are also decreased.
121
(a) Density distributions
(b) Pressure distributions
Figure 5.14. RKDG-DGF results for Case 5.1.7
122
Note in Figure (5.14) (a) that the fluid flow appears incompressible or weakly compressible at 1E-3
seconds. Spatial variation of the density is negligible. Based on this finding in mid-time UNDEX
events, incompressible approaches are often used in the simulations of a gas bubble [121, 9, 41, 23].
This result provides some insight as to why a compressible approach may suffer from a loss of
efficiency and accuracy in mid-to-late time multi-dimensional UNDEX simulations and may not
be necessary. Since the Mach numberM = uc
for incompressible/weakly compressible flows
ranges from 0.0 to 0.3 [142], the usual CFL condition (e.g.∆t = ∆xmax(u+c)
wherec is the sound
speed) strictly requires an extremely small time step size to avoid stability problems at mid-to-late
time simulations. Separate simulations for early-time andmid-to-late time are potentially more
effective in both cost and accuracy. This work focuses on early-time UNDEX simulations. For
low-Mach number flows, References [33, 69, 142] are recommended.
The ALE-based interface method is often used to prevent mixed elements caused by diffused den-
sity, which usually occurs in Eulerian methods. However, its extension to multi-dimensional appli-
cations where the interface motion is large, may not be effective [10, 57]. Early Eulerian methods
and J. Qiu’s two-fluid method [78] require multi-medium Riemann solutions to correct affectval-
ues in the region near the interface. The RKDG-DGF method does not have geometric limitations,
and does not require multi-medium Riemann solutions to treat explosive gas-water interfaces.
Section5.1provides reasonable test cases for 1D Cartesian and 1D symmetric RKDG-DGF meth-
ods. RKDG-DGF results are compared with the Riemann problemanalytic solution for 1D Carte-
sian cases, and the results from previous works for 1D symmetric cases. In these cases, the
shock front and material interface which are important hyperbolic characteristics in compressible
flow simulations, are sharply captured at correct locationsand magnitudes. No spurious pressure
oscillation and excessive density diffusion are observed.Now, to assess the multi-dimensional
applicability of our RKDG-DGF approach, Section 5.2 extends the 1D RKDG-DGF method to
multi-dimensions. Cases 5.2.1 (a 2D cylindrical case) and 5.2.2 (a 3D axi-symmetric case) are
considered.
123
5.2 Multi-dimensional assessment
An alternative way to assess the RKDG-DGF results in multi-dimensions is required because an
analytic solution is unavailable. To accomplish this, our RKDG-DGF results in multi-dimensions
are compared with our equivalent 1D symmetric RKDG-DGF results. The 1D domains are dis-
cretized with 1000 uniform elements to insure well-defined results.
Case 5.2.1: This case considers a 2D cylindrical explosive gas-water flow. Since the analytic
solution is unavailable, 2D RKDG-DGF results are compared with 1D cylindrical results on a fine
mesh. The initial conditions are shown in Figure5.15. The 2D domain[−100, 100]× [−100, 100]
cm is discretized with200 × 200 uniform elements. An initial gas bubble with radiusr0 = 50 cm
is located at the origin. The 2D RKDG-DGF code is run to a final time of 0.1E-3 seconds.
Figure 5.15. Initial conditions for Case 5.2.1 at 0.1E-3 seconds
2D RKDG-DGF results versus 1-D cylindrical results (α = 1.0 in Equation (5.1)) are compared in
Figure5.16. The half profiles of 2D results taken along the horizontal center are compared with the
1D results. The contour plots of density and pressure are shown in Figures5.17and5.18. Results
are very consistent.
124
Figure 5.16. RKDG-DGF results for Case 5.2.1 at 0.1E-3 seconds
Figure 5.17. Density contours for Case 5.2.1 at 0.1E-3 seconds
125
Figure 5.18. Pressure contours for Case 5.2.1 at 0.1E-3 seconds
Case 5.2.2: This case considers a 3D axi-symmetric flow that exhibits symmetry about an axis of
rotation. The fluid flow is described in cylindrical coordinates (r, θ andz). We assume that the
flow has symmetry around the axial coordinatez and the radial coordinater is defined as the dis-
tance from the axisz. Since the flow variables are a function of radial and axial coordinates(r, z)
only, the solutions are independent of the circumferentialcoordinateθ. Under these conditions,
the governing equations can be reduced from 3D to 2D with geometric sourceSsym(U) as
Ut + F (U)r + G(U)z = Ssym(U) (5.2)
where
U =
ρ
ρu
ρv
ρe
, F =
ρu
ρu2 + p
ρuv
u(ρe + p)
, G =
ρv
ρuv
ρv2 + p
v(ρe + p)
and Ssym = −1
r
ρu
ρu2
ρuv
u(ρe + p)
126
For the details of axi-symmetric equations, References [45, 123, 132] are recommended. Equation
(5.2) is often applied to axi-symmetric UNDEX flow applications (See Case 5.3.4) [158]. The
initial conditions are shown in Figure5.19. The initial gas bubble with radiusr0 = 100 cm is
located at the origin of the domain. The domain[0, 200] × [−200, 200] cm is discretized with
100×200 uniform elements. A symmetric boundary condition is applied along the axis z. The 2D
RKDG-DGF code is run to a final time of 0.3E-3 seconds.
Figure 5.19. Initial conditions for Case 5.2.2
Figure5.20compares the 3D axi-symmetric RKDG-DGF results and 1D reference results (i.e.,
α = 2.0 in Equation (5.1)). Figures5.21 and 5.22 represent the contour plots of density and
pressure at 0.3E-3 seconds. The shock front is sharply captured with only 3 elements and the
second outgoing shock wave reflected from the origin is very well resolved. Results are again very
consistent.
127
Figure 5.20. RKDG-DGF results for Case 5.2.2 at 0.3E-3 seconds
Figure 5.21. Density contours for Case 5.2.2 at 0.3E-3 seconds
128
Figure 5.22. Pressure contours for Case 5.2.2 at 0.3E-3 seconds
Cases 5.2.1 and 5.2.2 demonstrate UNDEX fluid-only results in multi-dimensions. Case 5.2.1
shows results at the time before the inward rarefaction wavereaches the origin. Case 5.2.2 shows
results at the time when the low-pressure region near the origin is created immediately after re-
flecting the inward rarefaction wave. The locations of both the interface and shock front are very
similar to those of the 1-D reference results. No spurious pressure oscillations and contour distor-
tions occur.
Section5.2 explored two test cases (i.e. a 2D cylindrical flow and a 3D axi-symmetric flow) to
assess a multi-dimensional RKDG-DGF method for modeling near-field UNDEX events in the
fluid only. In Cases 5.2.1 and 5.2.2, the multi-dimensional extension of the RKDG-DGF method
was examined by comparison of multi-dimensional results to1D symmetric RKDG-DGF results.
The RKDG-DGF results in multi-dimensions compared very well with the 1D reference results.
No contour distortion occurred in the domain. These resultsshow the potential of the RKDG-
DGF method for multi-dimensional UNDEX applications. The RKDG-DGF method in multi-
dimensions is not susceptible to error caused by distortionof the mesh as in ALE interface methods,
and the JWL EOS which provides better early-time/near-fieldaccuracy in the explosive gas model
as discussed in [54], does not require additional effort in implementation or computation.
129
5.3 Applications
Sections 5.1 and 5.2 examined simple fluid-only cases where the analytic solution or reference
results are available. These demonstrated the potential ofthe RKDG-DGF method for real UNDEX
simulations. This section extends the RKDG-DGF method to practical UNDEX applications which
include bubble behavior (Cases 5.3.1 and 5.3.2), shock-bubble interaction and cavitation in FSI
(Cases 5.3.3 and 5.3.4).
5.3.1 1D Spherical Bubble Pulses
Cases 5.3.1 and 5.3.2 consider 1D spherical bubble pulses which are important phenomenon in
near-field UNDEX. Assuming spherical symmetry, the fluid flowcan be modeled using the 1D
spherical RKDG-DGF method based on Equation (5.1) with α = 2.0. Experimental data or ref-
erence results from previous work are used to assess the RKDG-DGF results. In these cases, the
JWL and stiffened-gas EOSs are used to model the TNT-produced gas and water. The inner and
outer boundaries are treated using a rigid wall condition and a outflow condition, respectively.
The outflow condition helps decrease wave distributions at the boundary, even though it may be
troublesome in oblique shock impinging cases [98, 132].
Typical UNDEX gas bubble dynamics is described in Figure5.23. Following the explosion, the
pressure rises instantaneously to its peak value and then decreases exponentially. The high pressure
inside the gas bubble causes the bubble to expand, and pushesout the surrounding water. As the
gas pressure drops to the hydrostatic level, this outward expansion slows down until the expansion
stops. The high-pressure surrounding water pushes in the bubble causing it to be contracted inward.
This process is known as bubble collapse. If the bubble reaches a minimum radius, the bubble
begins to be re-compressed, producing another pulse. Several other cycles may follow at reduced
shock strength. The pressure-time history shows that the magnitudes of pressure are inversely
proportional to the bubble volumes.
130
Figure 5.23. UNDEX gas bubble dynamics [116]
Case 5.3.1: This case considers a 1D spherical bubble collapse previously modeled by A. B. Ward-
law [5], and introduced in Case 5.1.7. This problem has often been used in UNDEX method de-
velopments [57, 89, 103, 142]. A.B. Wardlaw conducted this problem using a 1D ALE interface
method code with highly resolved meshes [5]. It models the UNDEX of a 28kg TNT explosive
surrounded by water at a depth of 178m. In our model, the TNT charge is replaced by a gas bubble
with equivalent initial volume and internal energy for the original High Explosive (HE) material.
The bubble radiusr is initialized usingr0 =(
3V0
4π
)1/3with the approximate initial volumeV0 for the
HE material [17]. The initial conditions are shown in Figure5.24. The 1D domain[0, 10000] cm
is discretized with 2000 non-uniform elements. The center of a gas bubble withr0 = 16 cm is
located at the origin. The 1D spherical RKDG-DGF code is run to a final time of 0.15 seconds.
Figure5.25shows the bubble radius-time history, and the magnitudes ofpeak pressures taken at
121cm from the origin.
131
Figure 5.24. Initial conditions for Case 5.3.1
Figure 5.25. RKDG-DGF results for Case 5.3.1
132
The bubble radius is estimated using the LS functionφ(r) (see Subsection 4.1.2). The maximum
bubble radius is approximately 220 cm at about 0.07 seconds,and the minimum bubble radius is
approximately 30 cm at about 0.13 seconds. In Table5.1, these results are compared with those
from A.B. Wardlaw [5]. The peak pressure at 5E-4 seconds is approximately1.9 E9 dyne/cm2 as
observed in Case 5.1.7. There exists a small difference in bubble radius. This is mainly due to the
fact that A.B. Wardlaw identified the interface location by aLagrangian fashion, but we use the
Case 5.3.2: This case considers an experiment conducted by Swift and Decius in 1950 [43]. Swift
and Decius presented measurements on the oscillations and maximum radii of gas bubbles from
underwater explosions [43]. The experiment provides a useful graph of the bubble radius-time
history for a 300g charge detonated at a depth of 91.46m. J. M.Brett investigated this problem
using a nonlinear explicit FE-DYNA2D code [89]. The 1D domain[0, 9146] cm is discretized
with 2000 non-uniform elements. The initial conditions areshown in Figure5.26. Compared to
Case 5.3.1, we use a smaller-radius gas bubble withr0 = 3.529 cm which corresponds to a 300g
TNT charge. The 1D spherical RKDG-DGF code is run to a final time of 0.045 seconds. The
RKDG-DGF results are compared with the experimental data inFigure5.27and Table5.2.
Both the RKDG-DGF and DYNA2D approaches perform well for thefirst bubble pulse, but devi-
ate from the experimental data during the second pulse. DYNA2D predicts a very different second
pulse. This deviation may result from the fact that the real bubble behavior includes other mech-
anisms such as turbulence, buoyancy producing, asymmetricflows and heat loss from hot gases
[83, 89, 131, 99].
133
Figure 5.26. Initial conditions for Case 5.3.2
Figure 5.27. RKDG-DGF results for Case 5.3.2
134
Experiment data Numerical results Differences (%)
Max. bubble radius (cm) 48.1 48.0 +1.0
Bubble period (sec.) 0.298 0.291 -1.7
Table 5.2. Comparisons with the reference results
The RKDG-DGF approach is particularly useful for estimating the pressure loading associated
with the primary shock wave and the first bubble pulse. However, as discussed in Section5.1,
this compressible approach may lose efficiency and accuracyin simulating mid-to-late time multi-
dimensional UNDEX flows where the fluid flow becomes incompressible/weakly compressible.
The 1D spherical results can effectively be used to a fine initial condition for mid-to-late time
UNDEX simulations [103].
Subsection5.3.1modeled two 1D spherical gas bubble collapses which one is a numerical bench-
mark and the other is an experiment. RKDG-DGF results were compared to those of the results
from References [5, 89] and experimental data [43]. Bubble radius-time histories compared very
well. Tables5.1 and5.2 list maximum bubble radii, bubble periods and peak pressures. Cases
5.3.1 and 5.3.2 show the applicability of the 1D spherical RKDG-DGF method for modeling the
first bubble pulse, and the deficiency surrounding the inaccuracy and inefficiency of a compress-
ible approach to the second pulse. Subsection 5.3.2 extendsthe RKDG-DGF method to cavitating
flows in multi-dimensions.
5.3.2 Cavitating flows in multi-dimensions
An underwater explosion occurring near a structure generates a low pressure zone in the water
next to the structure, that has substantial influence on the response of a structure. Since water
cannot sustain tension, the water subjected to a sufficiently low pressure may be cavitated [6,
111]. Prediction of the formation and collapse of cavitation isan important component in UNDEX
135
simulations. Reference [6] provides a detail description of cavitation mechanisms inUNDEX
applications, and References [149, 156] overview various cavitation prediction methods applicable
to UNDEX. This work uses the pressure cutoff cavitation method described in Section 4.2. The
low pressure limit is taken as 0.05 bar.
To explore the applicability of our RKDG-DGF approach coupled with the pressure cutoff cavi-
tation method, the RKDG-DGF results from Cases 5.3.3 (a 2D cylindrical case) and 5.3.4 (a 3D
axi-symmetric case) are compared with those from previous works.
Case 5.3.3: This case considers a 2D cylindrical external explosion near a rigid wall as previously
modeled by W.F. Xie [156]. W.F. Xie obtained the pressure-time history in this cavitating flow
using the Modified Ghost Fluid Method (MGFM) with a modified Schmidt cavitation method (see
References [149, 156]). The modified Schmidt method models the cavitation as a homogeneous
mixture of vapor and liquid [156]. The initial conditions and the fluid mesh are shown in Figure
5.28. The domain[−600, 600] × [300,−600] cm is discretized with100 × 100 non-uniform ele-
ments. The initial gas bubble with radiusr0 = 100 cm is located at the origin. Our 2D RKDG-DGF
code is run to a final time of 0.01 seconds. The ideal-gas law with γ = 2.0 and the Tait EOS with
γ = 7.15 andB = 3.31E9 dyne/cm2 are used to model the explosive gas and water, respectively.
Figure 5.28. Initial conditions and the fluid mesh for Case 5.3.3
136
The rigid wall is located at the top boundary, and other boundaries are treated using a sponge-layer
non-reflecting boundary condition (NRBC) to minimize nonphysical wave reflections at bound-
aries. For the details of the sponge-layer NRBC, see Appendix F and References [80, 98]. Using
the pressure cut-off cavitation method, the pressure-timehistory at the center of the upper rigid
wall is shown in Figure5.29. Pressure contours at 0.0036 seconds and 0.0051 seconds when the
formation and collapse of cavitation are observed, are provided in Figures5.30and5.31.
RKDG-DGF results are compared with the results from [156]. Figure5.29shows that the RKDG-
DGF method provides similar peak pressure, cavitation cutoff and collapse times, and reloading
in the pressure-time history. Figures5.30and5.31show the cavitation mechanism in FSI with a
rigid wall. At 0.0036 seconds, the cavitation forms in the region below the upper wall as shown in
Figure5.30. At 5.1E-3 seconds, the cavitation collapses from the high-pressure surrounding water,
causing the upper rigid wall to be reloaded. The collapsed cavitations are observed at both sides of
the collapse region. The cavitation closes within a few milliseconds.
Figure 5.29. RKDG-DGF results on the rigid wall for Case 5.3.3
137
Figure 5.30. Pressure contour of Case 5.3.3 at 0.0036 seconds
Figure 5.31. Pressure contour of Case 5.3.3 at 0.0051 seconds
138
There is a small pressure deviation after the cavitation collapse at 5.1E-3 seconds. This may be
due to the difference of methods. The sponge-layer NRBC performs well in preventing significant
interference from reflected waves at boundaries. No contourdistortions and nonphysical waves
occur. For the treatment of interface cells, the MGFM used byW.F. Xie [156] requires local shock
values to be calculated at the gas-water interface, but the RKDG-DGF method does not require
these complexities in the implementation and computation.Compared to the modified Schmidt
method, the pressure cut-off method is simpler and faster. The comparison between both methods
shows that the choice of cavitation methods has no significant impact on the solution of cavitating
flows in FSI. Although the approaches used different cavitation models, the formation and collapse
of cavitation is very similar.
Case 5.3.4: This case considers a 3D axi-symmetric internal explosionmodeled by H. Sandusky
[58], G. Chambers [47], T.G. Liu [149], W.F. Xie [156] and A.B. Wardlaw [6] in separate work.
The experiment by H. Sandusky [58] provides the plastic deformation history of a water-filled
aluminum tube subjected to a 3g PETN explosion. The 2.8g PETNcharge plus 0.2g detonator
were located at the center of a water-filled aluminum tube [58]. The tube bottom was sealed by a
thin plastic sheet and the tube top left open [6, 47, 58]. The pressure-time history is measured at
the inner center of the tube wall. For the details of the experiment, see References [6, 47, 58].
Using experimental data for this case, H. Sandusky [58] assessed the applicability of a coupled
Eulerian-Lagrangian DYSMAS code developed by the Naval Surface Warface Center (NSWC)
Indian Head Division. G. Chambers [47] and A.B. Wardlaw [6] used DYSMAS and GEMINI-
DYNA_N codes (i.e., GEMINI for the fluid and the Navy version of DYNA (DYNA_N) for the
structure) for the same problem. Both used the pressure-cutoff cavitation method withPlimit =
0.05 bar to predict the formation and collapse of cavitation withFSI. More recently, W.F. Xie
and T.G. Liu [149, 156] used the MGFM coupled with the modified Schmidt cavitation method.
In [149, 156], the charge was replaced by an ideal gas bubble. A.B. Wardlaw using DYSMAS
provides a good description of the cavitation mechanisms with UNDEX FSI [6]. The differences
139
between a rigid-wall case and a deformable-wall case are discussed to assess the shock-bubble
interaction and wall deformation effects with FSI [6]. Wardlaw [6] provides useful results for
exploring the RKDG-DGF results.
Based on the experimental arrangement for our model, the initial conditions are shown in Figure
5.32. Since the fluid flow has axi-symmetric flow along the axis z, the fluid computation using
Equation (5.2) treats only the right half of the domain[0, 4.415]× [−8.9, 8.9] cm discretized with
40 × 160 uniform elements. The JWL EOS with PETN parameters in Table2.7, and the Tait EOS
are used to model the explosive gas and water, respectively.
Figure 5.32. Experimental arrangement and initial conditions for Case 5.3.4
The internal explosion case with a rigid-wall tube is considered, first. Along the axisz, a symmetry
condition is applied. Both the top and bottom are treated using a outflow condition to minimize the
interference from reflection waves at boundaries. The 3D axi-symmetric RKDG-DGF code is run
to a final time of 1.5E-4 seconds. The pressure-time history at the inner center of the outer wall is
shown in Figure5.33.
140
The peak pressures at the shock arrival and reloading moments (i.e. at 1.5E-5 seconds and 1.2E-4
seconds) are predicted to be 6.6E9 dyne/cm2 and 3.2E9 dyne/cm2 which are very close to the refer-
ence results taken from [6]. A pressure jump at around 1.4E-4 seconds occurs. This is likely due to
a different boundary condition from Wardlaw [6] used for both top and bottom where we assumed
a simple outflow condition. The condition used in [6] is unknown. Otherwise, the pressure-time
histories show excellent agreement.
Figure 5.33. RKDG-DGF results for Case 5.3.4 with the rigid wall
The pressure contours at various times are shown in Figure5.34. Before the cavitation cutoff at 4E-
5 seconds, interactions between the primary shock, expansion wave and gas bubble are dominant.
After the initial shock reflects off from the rigid wall, the reflected wave interacts with the gas
bubble at 3E-5 seconds. The interaction between the reflected wave and gas bubble creates an
expansion wave traveling back toward the wall. The expansion wave reduces the surface pressure
on the wall. This low pressure next to the wall causes the surrounding water to be cavitated.
141
Figure 5.34. Pressure contours for Case 5.3.4 with the rigidwall; the red line denotes the material interface
142
As in Case 5.3.3, the local cavitation originates from the center of the rigid wall around 4E-5
seconds. The high-pressure surrounding water rushes into the low-pressure zone, and the low-
pressure zone collapses at 1.2E-4 seconds. The cavitation collapse causes the rigid wall to be
reloaded [6, 8, 131]. These results show that the cavitation mechanism in a rigid wall case is
mainly influenced by the shock-bubble interaction.
Next the rigid tube wall is replaced by a deformable wall (i.e., Al5086 cylinder). The wall behav-
ior is described by the Kirchhoff material model (See Subsection 2.2.4.1). Typical properties of
Al5086 from [154] are used in the calculation. The “three-to-one” non-matching mesh technique
described in Section 4.2 and Appendix E is used to match the fluid and structural meshes along the
interface. The initial FSI meshes are shown in Figure5.35. The structural computation is based on
the method described in Section 4.2. Both the top and bottom structural nodes are fixed in r and z-
directions during the computation. The tube wall is treatedusing a moving wall condition, and the
domain[0, 4.415] × [−8.9, 8.9] cm is discretized with60 × 240 uniform elements. Unless stated
otherwise, the same fluid computation used in the rigid-wallcase is applied to this FSI application.
Figure 5.35. The initial FSI meshes for Case 5.3.4
143
Figure 5.36. RKDG-DGF results for Case 5.3.4 with the elastic wall
In Figure5.36, the pressure-time history from the RKDG-DGF calculation is similar to experi-
mental data [58] and results from A.B. Wardlaw’s work [6]. The predicted peak pressure 6.05E9
dyne/cm2 is somewhat lower than the experimental value 6.25E9 dyne/cm2. In the experimental
results, the first cavitation occurs at 3E-5 seconds with reloading and a second cavitation at 6E-5
seconds. References [6, 58] conclude that at about 9 E-5 seconds, final cavitation collapse and
reloading occur. The RKDG-DGF results show pressure reloadings at about 4.7E-5 seconds and
8.5E-5 seconds which are earlier than those of the experiment. Earlier time appearances are likely
due to the elastic model used in our study. Pressure contoursat various times are provided in
Figure5.37. The first cavitation occurs at about 2.8E-5 seconds in the water next to the tube wall
which is earlier than in the rigid wall case. This is mainly due to the wall deformation from the
initial shock impact. The deformed wall generates an extra volume required to be filled by the
surrounding water [6].
144
Figure 5.37. Pressure contours for Case 5.3.4 with the elastic wall; the red line denotes the material interface
145
This volume increase makes the initial shock strength lowerthan that of the rigid wall case after the
peak pressure (Compare Figures5.33and5.36) and a low pressure cavitation region on the wall
is created at about 2.8E-5 seconds. At around 3E-5 seconds, another cavitation occurs midway
between the tube and bubble. This is due to the interaction between the reflected wave from the wall
and the gas bubble. This interaction generates an expansionwave traveling back toward the tube
wall. These observations lead to the conclusion that the first cavitation is related to the lowered wall
pressure and second cavitation is associated with the shock-bubble interaction. The first cavitation
may not occur in the rigid wall case, since it is mainly related to the wall deformation. Two
cavitations merge at about 4E-5 seconds and collapses later[6]. Unlike the DYSMAS calculation,
a small cavitation and collapse at about 4.7E-5~5.8E-5 seconds is reflected in the RKDG-DGF
calculation. The tube wall is re-loaded at about 8.5E-5 seconds.
The RKDG-DGF method using the elastic material model provided a reasonable description of
the significant cavitation mechanisms (one cavitation related to the wall deformation and the other
cavitation due to the shock-bubble interaction). Unlike the rigid wall case, the presence of the
deformable wall in the FSI simulation is very important.
To connect the fluid and structure meshes in FSI simulations,the non-matching mesh technique
described in Ch. 4 is used. To keep the quality of fluid mesh during the calculation, the fluid
nodes are continuously adjusted by the ALE mesh-smoothing scheme. A single FSI framework
based on the ALE scheme is assessed by this case. Compared to the previous NSWC work using
a Coupled Eulerian Lagrangian (CEL) scheme, the ALE-based FSI framework is more attractive
for UNDEX FSI simulation; There is no error caused by mesh-overlapping, no complex mesh
information exchange required, and no geometric complexities.
Sections5.1 and5.2 provided several 1D and 2D test cases to assess the RKDG-DGF method in
the fluid by comparing with analytic solutions and referenceresults. Both the shock front and
material interface were sharply captured at correct locations and magnitudes. No spurious oscil-
lation, excessive diffusion or contour distortion occurred. Compared to existing methods such as
146
the original GFM, the ALE interface method and the MGFM, the RKDG-DGF approach has fewer
difficulties in the implementation and computation: The RKDG-DGF method has no geometric
complexities for multi-dimensional applications where ALE interface method may suffer from er-
rors caused by distortion of the mesh in large interface motion. The RKDG-DGF method also has
no limitations on the use of the JWL EOS and shock strengths. Section5.3extended RKDG-DGF
approach for practical UNDEX applications that include bubble behavior, shock-bubble interac-
tion and cavitation with FSI. Cases 5.3.1 and 5.3.2 investigated the behavior of a spherical gas
bubble produced by TNT explosions. The RKDG-DGF results including maximum bubble radius,
bubble pulse and peak pressures showed good agreement with reference results. Compared to a 3D
flow simulation, this 1D symmetric approach provided a simple and fast way to obtain the charac-
teristics of UNDEX gas bubbles. This 1D symmetric approach is also useful for providing initial
profiles in mid-to-late time UNDEX simulations as in [103]. Case 5.3.3 considered the 2D external
UNDEX application near a rigid wall. The pressure cut-off model was used to detect the formation
and collapse of cavitation on the wall. The performance of the sponge-layer NRBC was shown
in the pressure contours. No nonphysical wave reflections occurred during the computation. Case
5.3.4 considered internal explosions in a water-filled tube. Due to axi-symmetric flow, Equation
(5.2) with the pressure cutoff cavitation model was applied to the half of the tube. The cavitation
mechanisms were explored.
These assessments support the conclusion that the RKDG-DGFmethod has wider applications for
near-field UNDEX applications and is easier to extend to multi-dimensions. Compared to other
multi-fluid methods, the JWL EOS does not increase difficulties and complexities in the imple-
mentation and computation of our RKDG-DGF approach. Compared to the early CFD approaches
which require the solution of local Riemann problems at the interface, the RKDG-DGF approach
does not require these additional computations. Compared to the ALE-based interface method,
error associated with distortion of the mesh is managed evenwhen the interface motion is large in
multi-dimensions.
147
Chapter 6
Conclusions
A coupled solution approach was presented for numerically simulating a near-field early-time UN-
DEX. The approach consists of the Runge Kutta DiscontinuousGalerkin (RKDG) method to dis-
cretize the Euler fluid equations, the Direct Ghost Fluid (DGF) method to treat explosive gas-water
flows and the ALE deformable fluid mesh to adjust grid points tothe structural interface deforma-
tion. The combination of RKDG and DGF (RKDG-DGF) methods forexplosive gas-water flows
is the main contribution of this work. Compared to existing two-fluid methods, the RKDG-DGF
method has wider application for various near-field UNDEX simulations, and is easier to extend
to multi-dimensions.
Several test cases (e.g., 1D Cartesian, 1D symmetric, 2D cylindrical and 3D axi-symmetric cases)
were examined and assessed by comparing the RKDG-DGF results with analytic solutions, ex-
perimental data, results from previous work and results from equivalent 1D symmetric simulation.
These comparisons showed excellent agreement. Both the shock front and material interface were
sharply captured at the correct locations and magnitudes. The comparison between RKDG-DGF
results and results from previous work showed that the RKDG-DGF method produces less diffu-
sive and oscillatory fluid results, allows easier extensionto multi-dimension and simply models
explosive gas governed by the JWL EOS.
148
To assess the applicability of the RKDG-DGF method for near-field UNDEX, practical UNDEX
applications that include bubble pulses, shock-bubble interactions and cavitations with FSI, were
also studied. In spherically symmetric UNDEX flows, the motion of a gas bubble can effectively
be simulated by the 1D spherically symmetric RKDG-DGF approach. An investigation in the
simulation of a small-radius gas bubble showed that the approach predicts very well for the first
bubble pulsation and reasonably well for subsequent pulses. A 2D underwater explosion near a
rigid wall was studied to obtain the pressure-time history in a cavitating fluid flow. The pressure-
time history obtained from the 2D RKDG-DGF method coupled with the pressure-cutoff cavitation
model and the sponge-layer NRBC, agreed well with results from W.F. Xie’s previous work [156].
Although the two approaches used different solution methods, the comparison of peak pressure,
cavitation formation and collapse times showed excellent agreement. It was found that the choice
of a cavitation model does not have significant impact on the pressure-time history. Compared
to W.F. Xie’s one-fluid homogeneous cavitation model, insertion of the pressure cut-off model
was simpler and easier. The sponge layer allowed the shock wave to propagate outward without
any disturbance at boundaries. Compared to other NRBC approaches which require a modifica-
tion of governing equations, the implementation of the sponge layer NRBC was straightforward.
A 3D spherical internal explosion within a water-filled tube, which has received much attention
from many researchers, was studied. Cavitation mechanismswere discussed. The rigid wall case
showed excellent agreement with results from DYSMAS. The cavitation was mainly dominated by
the shock-bubble interaction. When using a deformable walltube, the pressure-time history from
the RKDG-DGF approach was similar to experimental data and results from DYSMAS. A series
of cavitation effects was observed: cavitation formation,cavitation collapse and reloading. This
case provided significant insight into the cavitation mechanisms distinguishable between the rigid
wall and the deformable wall. The ALE deformable fluid mesh was used to adjust fluid nodes to
the structural interface deformation. Compared to previous NSWC work using a CEL DYSMAS
code, the ALE-based deformable fluid mesh has less geometriccomplexity in the communication
between the structure and the fluid.
149
To extend the application of RKDG-DGF method to wider near-field UNDEX applications, the
following future works are recommended:
X Add plastic/nonlinear material models
– In current code
– Linking with a commercial structural code
X Extend applicability to other UNDEX mechanisms
– Contact explosion
– Bubble jetting
150
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Appendix A
Mathematical theorems and rule
Gauss theorem: For a differentiable functionf(x),∫
Ω
∂f(x)
∂xi
dΩ =
∫
Γ
nif(x)dΓ (A.1)
Reynolds transport theorem: The material time derivative in the Lagrangian description is
D
Dt
∫
Ω
f(x, t)dΩ =
∫
Ω
(
Df(x, t)
Dt+ f
∂vi
∂xi
)
dΩ (A.2)
Substituting the material time derivatives (2.16) into Equation (A.2) gives
D
Dt
∫
Ω
f(x, t)dΩ =
∫
Ω
(
∂f(x, t)
∂t+
∂ (vif)
∂xi
)
dΩ (A.3)
Applying Equation (A.1) to Equation (A.3) gives
D
Dt
∫
Ω
f(x, t)dΩ =
∫
Ω
∂f(x, t)
∂tdΩ +
∫
Γ
fvinidΓ (A.4)
Product rule: For differentiable functionf(x) andg(x),
d(f g)
dx= f
dg
dx+ g
df
dx(A.5)
162
Appendix B
FE-FCT algorithm
This work first chose the Finite Element Method-Flux Corrected Transport (FEM-FCT) scheme for
UNDEX flow simulations. Compared to other shock capturing schemes [132, 151], the FEM-FCT
method provides a simple and easy limiting procedure in the implementation and computation
[125]. The key is the combination of a high-order accurate schemewhich is accurate but oscil-
latory, and a low-order accurate scheme which is diffusive but monotonic. In an implicit way, a
high-order accurate scheme is applied to smooth flow region and a low-order accurate scheme
is applied to the region near the discontinuity. For the details of FCT procedure, References
[48, 68, 71, 124, 125] are recommended. The schematic of FCT procedure is shown inFigure
B.1.
Figure B.1. The schematic of FCT procedure
163
A high-order accurate scheme advances the solutions from timetn to timetn+1 as
un+1 = un + ∆uH (B.1)
where∆uH = un+1H − un is the solution increment obtained from a high-order accurate scheme.
diffusion in the region near the discontinuity by allowing avariable jump.
165
Appendix C
The slope limiting procedure
For the system of 1D Euler equations, the limiting procedureis as follows [12, 132]
1. Find eigenvectors of the jacobian matrix A= ∂F/∂U
(a) convert conservative form equation to quasi-linear form with a jacobian matrix A1
U,t + F,x = 0 ⇒ U,t + AU,x = 0 where Ai,j =∂fi(U)
∂uj(C.1)
(b) Compute matrices R and R−1 of eigenvectors2 as
R−1 ∂F∂U
R = R−1AR = Λ so A = RΛR−1 (C.2)
where
Λ = diag[λ1, λ2, λ3] = diag[u − a, u, u + a]
2. Transform system variablesU to local variablesW by multiplying variables by R−1
W = R−1U (C.3)
where inter-related system variables are converted into decoupled local variables.
1Since the system is hyperbolic, the jacobian matrixA can be diagonalized and has distinct real eigenvalues.2Variablesu anda are flow velocity and the speed of sound in a medium.
166
3. Apply the minmod slope limiter to local variablesW.
Wj,k = minmod(Wj,k, Wj+1 − Wj, Wj − Wj−1) (C.4)
4. Transform local variables to original system variables by multiplying local variables by R