An Efficient hp-Adaptive Strategy for a Level-Set Ghost-Fluid Method

Abstract

We present an hp-adaptive discretization for a sharp interface model with a level-set ghost-fluid method to simulate compressible multiphase flows. The scheme applies an efficient p-adaptive discontinuous Galerkin (DG) operator in regions of smooth flow. Shocks and the phase interface are captured by a Finite Volume (FV) scheme on a h-refined element-local sub-grid. The resulting hp-adaptive scheme thus combines both the high order accuracy of the DG method and the robustness of the FV scheme by using p-adaptation in smooth areas and h-refinement at discontinuities, respectively. For the level-set based interface tracking, a similar hybrid DG/FV operator is employed. Both p-refinement and FV shock and interface capturing are performed at runtime and controlled by an indicator, which is based on the modal decay of the solution polynomials. In parallel simulations, the hp-adaptive discretization together with the costly interface tracking algorithm cause a significant imbalance in the processor workloads. To ensure parallel efficiency, we propose a dynamic load balancing scheme that determines the workload distribution by element-local wall time measurements and redistributes elements along a space filling curve. The parallelization strategy is supported by strong scaling tests using up to 8192 cores. The framework is applied to established benchmarks problems for inviscid, compressible multiphase flows. The results demonstrate that the hybrid adaptive discretization can efficiently and accurately handle complex multiphase flow problems involving pronounced interface deformations and merging interface contours.

Full text

https://doi.org/10.1007/s10915-023-02363-7
REVIEW
An Efficient hp-Adaptive Strategy for a Level-Set Ghost-Fluid
Method
Pascal Mossier1·Daniel Appel1·Andrea D. Beck1·Claus-Dieter Munz1
© The Author(s) 2023
Abstract
We present an hp-adaptive discretization for a sharp interface model with a level-set ghost-
fluid method to simulate compressible multiphase flows. The scheme applies an efficient
p-adaptive discontinuous Galerkin (DG) operator in regions of smooth flow. Shocks and the
phase interface are captured by a Finite Volume (FV) scheme on a h-refined element-local
sub-grid. The resulting hp-adaptive scheme thus combines both the high order accuracy of
the DG method and the robustness of the FV scheme by using p-adaptation in smooth areas
and h-refinement at discontinuities, respectively. For the level-set based interface tracking, a
similar hybrid DG/FV operator is employed. Both p-refinement and FV shock and interface
capturing are performed at runtime and controlled by an indicator, which is based on the
modal decay of the solution polynomials. In parallel simulations, the hp-adaptive discretiza-
tion together with the costly interface tracking algorithm cause a significant imbalance in
the processor workloads. To ensure parallel efficiency, we propose a dynamic load balancing
scheme that determines the workload distribution by element-local wall time measurements
and redistributes elements along a space filling curve. The parallelization strategy is supported
by strong scaling tests using up to 8192 cores. The framework is applied to established bench-
marks problems for inviscid, compressible multiphase flows. The results demonstrate that
the hybrid adaptive discretization can efficiently and accurately handle complex multiphase
flow problems involving pronounced interface deformations and merging interface contours.
Keywords Discontinuous Galerkin ·High order schemes ·p-Adaptivity ·Shock capturing ·
Finite Volumen ·Level-set ·Multiphase ·Ghost-fluid ·Dynamic load balancing
1 Introduction
Compressible multiphase flows are encountered in many technical applications like fuel
injection systems of rocket or jet engines. Due to their multi-scale nature, the accurate and
efficient simulation of such flows poses a challenging task and is still an active field of
BPascal Mossier
pascal.mossier@iag.uni-stuttgart.de
1Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, Pfaffenwaldring 21,
70569 Stuttgart, Germany
0123456789().: V,-vol 123
Journal of Scientific Computing (2023) 97:50
Received: 13 December 2022 / Revised: 5 July 2023 / Accepted: 9 September 2023 /
Published online: 11 October 2023
Content courtesy of Springer Nature, terms of use apply. Rights reserved.

research. For the modeling and computation of multiphase flows, two main approaches can
be distinguished in literature: the sharp interface method (SIM) and the diffuse interface
method (DIM). In the DIM, the phase interface is modelled as a smooth transition layer
between two fluids, by introducing a mixture of the phases. Consequently, the DIM requires
a high resolution of the numerical approximation near the smoothed interface to preserve the
physical properties. Well-known implementations of this ansatz comprise the Navier–Stokes-
Korteweg equations [1] and the Baer–Nunziato equations [3]. The SIM, by contrast, models
the interface as a discontinuity separating the bulk phases. This approach is advantageous
when the width of the physical interface is infinitesimal with respect to the macroscopic
flow scale. The SIM consists of two major building blocks: An algorithm to track the phase
interface and a physically consistent coupling of the bulk phases.
In this work, we use a SIM with a level-set approach [58] to track the phase interface. A
common strategy to couple the bulk phases in SIM is the ghost-fluid method of Fedkiw et al.
[22]. It relies on the definition of ghost states at the interface to obtain boundary conditions
for the bulk phases. Merkle and Rohde [43] pioneered a strategy to obtain the required
ghost states with the solution of a two-phase Riemann problem across the phase interface.
Variations of this idea like the modified ghost-fluid method of Xu and Liu [66] have been
recently reported and show promising results for complex, multi-dimensional compressible
droplet dynamics. In the present work, we build on the approach by Fechter et al. [18,20]
that employs a ghost-fluid method in combination with two-phase Riemann solvers at the
phase interface.
The simulation of compressible multiphase flows with the SIM poses demanding require-
ments on a numerical scheme. In smooth regions inside the bulk phases, a high order scheme
is beneficial for its accuracy and efficiency. At the phase interface, a robust, non-oscillatory
scheme is required to cope with the strong discontinuity caused by the sharp interface. Fur-
thermore, for compressible flows, the discretization needs to handle potential shocks in the
bulk phase. There exist a multitude of numerical schemes in literature to cope with the above
requirements. From the class of Finite Volume schemes (FV), the Weighted Essentially
Non-Oscillatory (WENO) [14,62] and the Central Weighted Essentially Non-Oscillatory
(CWENO) [16] schemes have been successfully applied to compressible multiphase flows
e.g.byHuetal.[31] and Tsoutsanis et al. [63]. An alternative is the Discontinuous Galerkin
(DG) Method that has gained popularity due to its efficiency and accuracy [9,27]. To avoid
spurious oscillations related to the Gibb’s instabilities at discontinuous solution features,
different stabilization and shock capturing techniques have been developed. One approach
is to smooth out the oscillations by a local switch-on of artificial viscosity [48,51,68]. A
different method is to combine the DG method with a WENO approach to avoid instabilities
at shocks [13,15].
In this work, we follow the idea of Huerta et al. [32] and Persson et al. [52] to introduce
piecewise constant ansatz functions and apply them on a sub-cell grid inside the original DG
element. The approach presented in this paper is based on the work of Sonntag and Munz
[55] who introduced an a priori FV sub-cell limiting scheme. If a shock or strong gradient
is observed inside a DG element, the element is subdivided into sub-cells and discretized
by a second-order FV scheme. In a recent publication [46], we extended this hybrid DG/FV
approach by introducing a p-adaptive DG operator and allowing for a FV sub-cell resolution
independent of the element-local DG degree N. The resulting scheme combines the high
order accuracy of the DG scheme with the robustness of the FV method by applying p-
adaptivity in smooth regions and a h-refined sub-cell grid to improve shock localization. In
this paper, we introduce this adaptive hybrid scheme [46] to our sharp interface framework
[20,35,47].
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Our sharp interface framework solves a level-set transport equation to compute the position
and geometry of the phase interfaces. To provide accurate geometric properties, high order
accuracy is required at the phase interface. However, in case of merging phases, the level-set
field may exhibit discontinuities and kinks. Therefore we employ a hybrid p-adaptive DG
scheme combined with FV sub-cell limiting in analogy to the proposed fluid discretization.
The sub-cell limiting and the p-refinement of both the fluid and level-set discretization are
controlled at runtime by means of an a priori indicator based on the modal decay of the
solution polynomials [41,46].
The adaptive discretization and the interface tracking algorithm cause an inhomogeneous
distribution of the element-local computational costs, which implies a workload imbalance
across the processor units in parallel simulations. Therefore, a dynamic load balancing (DLB)
is essential to ensure parallel efficiency of the code. Appel et al. [2] introduced a DLB scheme
based on repartitioning along a space filling curve to treat the imbalance due to the interface
tracking. In the present study, we extend this algorithm to cover imbalances caused by the
adaptive discretization and propose an efficient communication strategy to speedup the data
redistribution during rebalancing.
In summary, the goal of the present work is to introduce an hp-adaptive hybrid DG/FV
method into a sharp interface framework. Hereby, the adaptive discretization is applied to
both the interface tracking and the fluid flow. To maintain favorable parallel scaling, the
key challenge of an equal load distribution during run time is addressed with an efficient
communication and dynamic load balancing scheme.
The paper is organized as follows: In Sect. 2we introduce the governing equations for the
bulk flow and the interface tracking algorithm. Section3provides the numerical framework
to solve the conservation equations and the level-set interface tracking with an adaptive
hybrid DG/FV scheme. The parallelization strategy with the focus on an efficient dynamic
load balancing approach is addressed in Sect. 4. Finally, we apply the code framework to
well-known shock-droplet and shock-bubble interactions in Sect. 5. We assess the efficiency
and accuracy of our scheme by comparing with numerical and experimental results from
literature. We close with a conclusion and a summary of our findings.
2 Governing Equations
In this work, we investigate compressible multiphase flows with two pure, immiscible phases
separated by a sharp interface of zero thickness. Therefore, we consider a computational
domain , consisting of a liquid part land a vapor part v, separated by the phase interface
on a finite time interval [0,T]. The bulk fluid behavior is governed by the compressible
Euler equations
∂Q
∂t+∇
x·F(Q)=0in ×[0,T](1)
with the vector of conservative variables Q=[ρ,ρu,E]and the convective flux vector
F=[ρu,ρu⊗u+pI,u(E+p)]given in terms of the density ρ, the velocity vector
u=(u1,u2,u3), the pressure pand the total energy E. The total energy Eis defined as
the sum of the internal energy and the kinetic energy
E=ρe=ρ +1
2ρu·u.(2)
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The equation system is closed with an equation of state (EOS), that provides expressions for
the pressure pand the specific internal energy :
p=p(ρ,),=(ρ, p).(3)
Both algebraic and multi-parameter EOS are implemented in our code framework and can be
efficiently evaluated by the tabulation method introduced by Föll et al. [23]. Here, we model
the liquid phase by a stiffened gas EOS and apply the perfect gas law to model the gaseous
phase. Following Le Métayer et al. [45], the stiffened gas EOS is given as
p=(γ −1)ρ −γp∞(4)
with the ratio of specific heats γand the reference pressure p∞.Forp∞=0, the stiffened
gas equation reduces to the perfect gas law.
A necessary building block for the SIM is the tracking algorithm for the phase interface
. Following Sussman et al. [58], we obtain the position of the phase interface as the root of
a level-set function that is advected by a velocity field s. The level-set transport equation
is defined as
∂
∂t+s·∇
x=0.(5)
An essential feature of the level-set function is the signed distance property. It is necessary
for the accurate derivation of geometric properties such as the unit normal vector nls and the
interface curvature κls. Since the level-set transport equation (5) does not preserve the signed
distance property, a reinitialization procedure is required to regularize the level-set field.
Following Sussman et al. [58], this is accomplished by solving a Hamilton–Jacobi equation
in pseudo time τ:
∂
∂τ +sign ()(
|∇x|−1)=0(6)
Fast marching [54] or fast sweeping [61] methods could be used as an alternative here.
Accordingto[7], geometric properties of the phase boundary can be expressed in terms of
derivatives of the level-set field, with the normal vector nls given as
nls := ∇x
|∇x|(7)
and the curvature κls computed by
κls := 2
x1x2x2+2x1x2x1x2+2
x2x1x1
|∇x|3
+2
x1x3x3+2x1x3x1x3+2
x3x1x1
|∇x|3
+2
x2x3x3+2x2x3x2x3+2
x3x2x2
|∇x|3(8)
As a final building block a velocity field shas to be provided for the level-set transport equation
(5). The velocity sat the phase boundary results from the two-phase Riemann problem [19,
21,43]. To obtain a velocity field in the volume, the velocities computed by the Riemann
solver are first copied to neighboring cells. These pointwise data are then extrapolated into the
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Fig. 1 Spatial discretization of a
droplet in a vapor environment.
Elements in the bulk region
containing a smooth solution are
discretized by a DG method with
a local degree NDG . Elements
containing the phase boundary
are subdivided into NFV
sub-cells per direction to improve
the approximation of the phase
boundary by the surrogate
surface
volume by solving in pseudo time a Hamilton–Jacobi equation for each velocity component
si[50]
∂si
∂τ +sign()nls ·∇
xsi=0,with i=1,...,d(9)
To save computational cost, the level-set transport equation (5), the reinitialization (6)and
the velocity extrapolation (9) are only evaluated in a narrow band around the phase boundary.
3 Numerical Methods
In this section, we first provide the numerical methods used to approximate the governing
equations of the bulk flow and the level-set interface tracking algorithm. We then assemble
the building blocks to obtain a hp-adaptive level-set ghost-fluid framework. All equations
are discretized on a computational domain ⊂R3that is subdivided into K∈Nnon
overlapping hexahedral elements Esuch that =K
e=1Eand K
i=1E
i=∅.
We apply a hybrid discretization based on a p-adaptive Discontinuous Galerkin spectral
element method (DGSEM) with a FV sub-cell shock capturing scheme for both the Euler
equations (1) and the level-set transport equation (5). The hybrid discretization is illustrated
in Fig. 1where a single droplet and the underlying mesh are visualized. DG elements E
containing the physical phase boundary are subdivided into NFV FV sub-cells eper
direction. At the discontinuous phase interface, a switch to the robust FV sub-cells operator
avoids spurious oscillations caused by the jump in the solution. However, there is a further
motivation for the FV sub-cell approach:
The physical phase boundary is approximated on a discrete level by a surrogate phase
interface, obtained as the projection of to the grid. A high spatial resolution is thus vital
to obtain an accurate approximation of the physical interface . Therfore, the FV sub-cell
approach is essential to achieve a precise interface localization. For a detailed derivation of
the DGSEM and FV sub-cell methods, the reader is referred to [38,39,55]. The hp-adaptivity
is covered in more depth in [46], while the foundations of the multiphase framework can be
found in [20,35,47].
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3.1 Spatial Discretization
3.1.1 DGSEM with p-Adaptation
In this section, we apply a p-adaptive formulation of the DGSEM in combination with a FV
sub-cell scheme to both the hyperbolic conservation law of the bulk flow (1) and the level-set
transport equation (5), which is a hyperbolic equation with a non-conservative product.
As a first step, we introduce the mapping from the physical space x=(x1,x2,x3)to
reference space ξ=(ξ1,ξ
2,ξ
3)to transform the physical elements Eto a unit reference
element E=[−1,1]3. We thus obtain the conservation law and the level-set transport
equation in reference space as
Jgeo ∂Q
∂t+∇
ξ·F(Q)=0(10)
Jgeo ∂
∂t+s·∇
ξL=0 (11)
with the Jacobi determinant of the mapping Jgeo, the contravariant flux Fand the contravariant
level-set field Lrespectively. Projection onto a space of test functions ψ∈Pleads to the
weak form of Eqs. (10)and(11)
EJgeo ∂Q
∂tψd+∂E
(F·nξ)ψdS
ξ−E
(F·∇
ξ)ψd=0(12)
EJgeo ∂
∂tψd+∂E
(s·∇
ξL)ψdS
ξ+E
(s·∇
ξL)ψd=0 (13)
The term F·nξdenotes the contravariant flux across a surface element nξand is approximated
by a numerical flux function (F·nξ)∗, provided either by a HLLC [60]orHLLE[17]
approximate Riemann solver in the present study. To evaluate the surface integral of the level-
set transport equation, a path-conservative jump term (D·nξ)∗:= s·∇
ξLis introduced.
Following [6,13,35], it can be approximated by a Rusanov-type Riemann solver as
(D·nξ)∗≈1
4(s++s−)−2smax+−−(14)
with the maximum signal speed smax := max |sL·nξ|,|sR·nξ|. The transport velocities
s±are determined from sLand sRin an upwinding procedure to allow for topological changes
like merging interface contours. A detailed explanation of this procedure is given in [35,67].
The solution vectors Qand as well as the contravariant flux Fand contravariant level-set
Lare approximated as piecewise polynomials
Q(ξ,t)≈
N

i,j,k=0
ˆ
Qijk(t)ζijk(ξ), (ξ,t)≈
N

i,j,k=0
ˆ
ijk(t)ζijk(ξ)(15)
F(ξ,t)≈
N

i,j,k=0
Fijk(ˆ
Qijk)ζijk(ξ), L(ξ,t)≈
N

i,j,k=0
ˆ
Lijk(t)ζijk(ξ)(16)
in the space spanned by tensor products of one-dimensional Lagrange polynomials
ζijk(ξ)=i(ξ1) j(ξ 2)k(ξ 3). (17)
According to the Galerkin ansatz, we use the tensor product Lagrange polynomials for both
the ansatz functions ζand the test functions ψ.
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Fig. 2 Flux calculation for two adjacent elements with degree N=3 (left) and M=4 (right). The node
distribution is indicated by dots in the volume and by squares on the surface. The numerical flux is computed
on the solution representation of the higher polynomial degree
A semi-discrete scheme is eventually obtained by introducing a numeric quadrature for the
integrals in Eqs. (12)and(13). A central idea of the DGSEM is to use the same node distribu-
tion, in our case Legendre–Gauss nodes, for both the quadrature and the interpolation. This
collocation approach results in a tensor product structure of the DG operator and reduces the
number of required operations per degree of freedom (DOF). A multi-dimensional operator
can thus be constructed as a succession of multiple one-dimensional operations.
The extension to a p-adaptive DGSEM discretization is straightforward, by allowing for a
variable element-local degree N∈[Nmin ,Nmax]. The bounds Nmin and Nmax for the allowed
polynomial degree need to be set prior to each computation. In the general case, the degrees
of adjacent elements differ, M= Nwith M>N, such that the flux computation needs to
be adjusted. As depicted in Fig. 2, the surface solution Q−
Nof the element with the lower
degree Nis interpolated to the higher degree Q−
M. With surface solutions Q−
Mand Q+
Mof
common degree M, the numerical flux F∗
Mcan be computed. For the element of degree N,
the flux is subsequently projected to the lower degree F∗
N.
We follow an analogous procedure for the level-set transport equation, using the path-
conservative jump term (D·nξ)∗instead of the numerical flux. When the local degree of
an element changes during runtime, the solution is either interpolated to a higher degree or
projected to a lower degree. For a more detailed derivation of the p-adaptive scheme, we refer
to [46].
3.1.2 h-Refined FV Scheme for DGSEM
In the presence of strong gradients or discontinuities within an element, high order DGSEM
schemes produce oscillatory solutions (Gibb’s phenomenon). For the Euler equations, shock
waves and phase boundaries are typical discontinuous solution features that necessitate a
stabilization technique. The level-set field can develop sharp gradients for large curvatures,
merging interface contours and the cut-off at the edge of the narrow band. To stabilize the
solution, we combine the DGSEM with a robust FV scheme on an h-refined sub-cell grid. The
FV scheme is formally obtained by introducing piecewise constant test and basis functions
and is thus equivalent to a first order DGSEM. With piecewise constant basis functions, the
derivative in the volume integral vanishes, simplifying the weak formulations (12)and(13)
to
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eJgeo ∂Q
∂tψd+∂e
(F·nξ)ψdS
ξ=0 (18)
eJgeo ∂
∂tψd+∂e
(s·∇
ξL)ψdS
ξ=0 (19)
The reduced approximation order is compensated by refining an affected DG element Eto
a sub-cell grid of NFV equidistant sub-cells eper direction. This improves the localization
of shocks and increases the resolution of the surrogate phase boundary significantly. In the
previous implementation of our framework [35], a DG operator with constant degree Nand
a sub-cell resolution NFV =N+1 were used. With this choice, both DG and FV elements
use the same number of DOFs, which allows a common data structure and thereby eases
the implementation of the sub-cell approach. Following Dumbser et al. [15], we choose an
increased sub-cell resolution of NFV =2Nmax +1. This particular choice is motivated by the
time step restriction of the DGSEM compared to the FV scheme. In case of an equidistant
sub-cell grid, NFV =2Nmax +1 is the maximum sub-cell resolution that does not impose a
more restrictive time step compared to a DG element of degree Nmax. Thus, the increased sub-
cell resolution of the phase boundary can be achieved without compromising the time step.
The accuracy of the FV sub-cell scheme is further improved through a TVD second-order
reconstruction scheme, proposed by [56].
Switching between the DG and FV operators requires transformations between piecewise
polynomial and piecewise constant solution representations. The transformation DG FV
can be performed in a conservative manner by integrating the DG polynomial over each sub-
cell e. Conversely, a polynomial representation FV DG can be recovered from FV
sub-cell data with a discrete projection, equivalent to a least square approach with integral
conservation as a constraint. This transformation was proposed by Dumbser et al. [13]for
the case NFV >N+1, which causes an over-determined system. Switching between a DG
and FV solution is performed as a matrix vector operation with the transformation matrices
VFVDG and VFVDG, for which holds
VFVDG ◦VDGFV =I.(20)
Adjacent FV sub-cells are coupled via a numerical flux F∗. Following [56], we use the FV
sub-cell scheme as an a priori limiter to stabilize the bulk fluid flow. Thus, every element is
discretized either by the DG or the FV operator, according to an indicator evaluated prior to
each time step. In case of a mixed interface between a DG and a FV element, as depicted
in Fig. 3, a common solution representation needs to be provided for the flux computation.
To this end, the DG solution is transformed to a FV sub-cell representation at the element
side. With a consistent FV solution representation from both sides, the numerical flux can be
evaluated as usual and is subsequently transformed back to a DG discretization for the DG
element.
For the level-set transport equation, the FV sub-cell scheme is applied as an a posteriori
limiter. We compute the time update for a pure DG representation as well as a pure FV
representation. The update candidate for each element is then chosen according to an a priori
indicator. This procedure is motivated by the occurrence of spurious oscillations if adjacent
DG and FV elements are coupled via the path-conservative jump term [13]. It should be
noted that evaluating both the DG and the FV operator for each element has a minor effect on
the overall computation time since the level-set transport equation is only solved in a narrow
band around the phase interface.
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Fig. 3 Flux computation for a mixed interface with a DG element of degree N=3 (left) and a FV sub-
cell element with NFV =4 sub-cells (right). DG volume nodes and FV sub-cell centers are indicated by
dots. Surface nodes are indicated by squares. The numerical flux is computed on a piecewise constant FV
representation F∗
FV and projected afterwards to a DG solution F∗
DG to provide the flux for the DG element
3.2 Time Discretization
The Euler equations and the level-set transport are both integrated in time with an explicit
fourth-order low-storage Runge–Kutta scheme [37]. The time integration is subject to time
step restrictions, derived from the CFL condition by Courant, Friedrichs and Lewy [10]. For
the DGSEM and FV discretizations of the Euler equations and the level-set transport, time
step restrictions can be formulated as follows
tDG =CFL ·αRK(N)·xDG
(2N+1)|λmax|(21)
tFV =CFL ·αRK(0)·xFV
|λmax|.(22)
They depend on the maximum signal velocity |λmax|, the grid spacing xDG or xFV and
the empirical scaling factor αRK(N)>0 for the Runge–Kutta scheme. The factor αRK (N)
accounts for the size of the stability region of a particular Runge–Kutta method for a spatial
discretization operator and is determined through numerical experiments in [5]. It decreases
monotonously for an increasing order N, thus αRK (N+1)<α
RK(N)holds. The DG time
step tDG is further scaled by the factor (2N+1)to account for the element-local degree
N. A comparison between the time step restrictions (21)and(22) reveals that for xFV >
xDG/(2N+1), the FV method will not impose a smaller time step compared to a DG
element of degree N. This motivates the choice of a sub-cell resolution of NFV =2Nmax +1,
with Nmax denoting the highest DG degree within the computation.
In case of the Euler equations the maximum signal velocity is given by the maximum
absolute eigenvalue λmax =max(|c−u|,|u|,|c+u|)with the speed of sound c.Forthelevel-
set transport, the maximum signal velocity corresponds to the transport velocity λmax =s.
In two-phase flows, an additional time step restriction is imposed by capillary waves of the
phase interface. According to Denner et al. [11], the dynamic capillary time step constraint
depends on the signal velocity
λmax =2πσ
ρxFV +|s|(23)
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with the surface tension σ. Instead of the sum of the liquid and the vapor densities, we only
use the density of the current degree of freedom to simplify the implementation. Since the
sum of liquid and vapor density is always larger than either the liquid or the vapor density,
this is a conservative assumption and thus does not compromise numerical stability. The
capillary time step criterion is only relevant at the phase boundary that is discretized with the
FV sub-cell scheme. Therefore, the signal velocity is formulated for a FV sub-cell element
with the grid spacing xFV.
The maximum signal velocity of the level-set transport sis of the same order of magnitude
as the convective velocity of the bulk flow u. Thus, the signal speed of the Euler equations
c+uis usually larger and therefore the dominant time step restriction in practical applications.
3.3 Indicator for FV Sub-Cell Limiting and p-Refinement
p-adaptivity and FV sub-cell limiting are controlled on an element-local level through a
combination of different indicators. First, we infer an estimate for the error and the smoothness
of the solution from the decay rate of the modal polynomial solution representation [41,46].
To this end, we interpolate the nodal polynomial solution representation ˆ
unod of the DGSEM
scheme to a modal Legendre basis ˆ
umod using a Vandermonde matrix VLeg
ˆ
umod =VLeg ˆ
unod.(24)
The relative contribution ωmof the mth mode in ξ1direction to the solution can be expressed
as
wm=⎡
⎢
⎣N
j,k=0ˆ
u2
mod,ijki=m
N
i,j,k=0ˆ
u2
mod,ijk ⎤
⎥
⎦.(25)
The contributions ωmare subsequently fitted to the exponential function wm=ae−σmfor
every direction and the indicator is obtained as
Imodal =min(|σ1|,|σ2|,|σ3|)(26)
The final indicator value Imodal is then compared against user-defined thresholds to decide
whether p-refinement or FV sub-cell limiting is required. For every degree N, lower thresholds
Tlow(N)and uppers threshold Tup(N)are defined for both the DG p-refinement and FV sub-
cell limiting:
Tlow,FV(N)=Tmin,FV +NTmax,FV −Tmin,FV
Nmax −Nmin (27)
Tup,FV(N)=Tmin,FV +(N+1)Tmax,FV −Tmin,FV
Nmax −Nmin (28)
Tlow,DG(N)=Tmin,DG +NTmax,DG −Tmin,DG
Nmax −Nmin (29)
Tup,DG(N)=Tmin,DG +(N+1)Tmax,DG −Tmin,DG
Nmax −Nmin .(30)
The thresholds depend on four free parameters Tmin,FV,Tmax,FV ,Tmin,DG and Tmax,DG that
are tuned empirically for the bulk flow and the level-set transport in the test problems of
Sect. 5. For a more in-depth discussion of the modal indicator and the threshold definitions,
the reader is referred to [46].
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To stabilize both the bulk flow and the level-set transport in the presence of phase bound-
aries, additional criteria for the FV sub-cell limiting are defined. Using the position of the
level-set zero-line, elements containing a phase boundary are detected and flagged for FV
sub-cell limiting in the Euler equations by the indicator Iinterface. In case the level-set func-
tion changes its sign multiple times within a DG element, we assume approaching level-set
contours and flag affected elements with the indicator Itopo for FV sub-cell limiting [35,67].
In conclusion, p-refinement is controlled for the bulk flow and the level-set by the indicator
Imodal. FV sub-cell limiting depends on the combination of Imodal and Iinterface in the bulk,
and on Imodal and Itopo for the level-set.
3.4 Discretization of the Reinitialization and Velocity Extrapolation
The scalar Hamilton–Jacobi equations of the reinitialization (6) and the velocity extrapolation
(9) are discretized with a fifth-order WENO scheme [33] on the FV sub-cell grid. We apply
a local Lax–Friedrichs flux for the velocity extrapolation and a Godunov-type flux for the
reinitialization [18]. Both Hamilton–Jacobi equations are integrated in pseudo time with an
explicit Runge–Kutta scheme [64]. In case of the velocity extrapolation, scalar equations
are solved for every space dimension. To obtain initial data for the velocity extrapolation,
the velocities provided by the two-phase Riemann solver at the surrogate phase boundary
are copied to adjacent sub-cells. If a sub-cell is affected by multiple two-phase Riemann
problems, an averaging is performed. From these pointwise data, the velocity extrapolation
routine constructs a velocity field in the narrow band region. For a detailed discussion of
reinitialization and velocity extrapolation, we refer the reader to [18,50,58].
3.5 Calculation of Level-Set Normals and Curvature
The evaluation of the two-phase Riemann problem requires the normal vectors and the cur-
vature of the level-set field at the phase boundary. The normals (7) and the curvature (8)
are expressed in terms of first and second derivatives of the level-set field. These derivatives
are approximated by applying a central second-order finite difference scheme to the level-
set field in FV representation. The second derivatives are obtained by reapplying the same
operator to the first derivatives.
3.6 Ghost-Fluid Method and Two-Phase Riemann Solver
The ghost-fluid method of Fedkiw et al. [22] is a common tool in sharp interface models
to couple the bulk phases at the phase boundary. It is based on the idea to define ghost
states next to the phase interface that serve as boundary conditions for the bulk phases. With
these boundary conditions, the bulk phases can be treated with standard methods for single
phase flows. In the present framework, ghost states QG
land QG
vare obtained as the inner
states of a two-phase Riemann problem [43], evaluated at the surrogate phase boundary.
Subsequently, ghost fluxes across the phase boundary are computed with the ghost states
as FG
l=FG
l(Ql,QG
l)and FG
v=FG
v(QG
v,Qv)respectively. The two-phase Riemann
problem and the ghost-fluid method are visualized with an exemplary sketch in Fig. 4.Asa
consequence of the non-unique flux across the phase boundary, the ghost-fluid method is not
conservative. An additional result of the two-phase Riemann solver is the wave speed sof
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Fig. 4 Illustration of an exemplary two-phase Riemann problem at the phase interface (left) with the intermedi-
ate states QG
land QG
v. The intermediate states of the Riemann problem serve as ghost states to define boundary
conditions and to compute the ghost fluxes FG
l(Ql,QG
l)and FG
v(QG
v,Qv)across the phase boundary (right)
the material interface, which serves as point wise data for the extrapolation of the level-set
velocity field, see Sect. 3.4.
If the sign of the level-set changes within a sub-cell, i.e. the phase interface has moved
from one to the next sub-cell, the respective sub-cell changes its phase. In this case, the state
vector Qof the sub-cell is populated by the ghost state QG=QG(x=0,t)given as the
inner state of the two-phase Riemann problem. For a detailed discussion of the ghost-fluid
method and the two-phase Riemann solver, the reader is referred to [18,21,34].
3.7 Assembly of the Level-Set Ghost-Fluid Framework
Finally, we can assemble the introduced building blocks to obtain an adaptive level-set ghost-
fluid framework. Figure5illustrates the interaction of the different operators for one time
step [tn,tn+1]. Within one time step, the following steps are performed:
1. Based on the sign of the level-set scalar, the domain is divided into land v.
2. At the phase boundary, DG elements are switched to the FV sub-cell representation to
provide a stable discretization of the sharp interface and to improve the approximation by
the surrogate phase boundary.
3. The level-set field is switched to the FV representation for the subsequent WENO and FV
operators.
4. Before evaluating geometric properties of the level-set, a WENO-based reinitialization of
the level-set field is performed to restore the signed distance property.
5. Level-set normals and curvature are calculated by a second-order FV method.
6. With the geometric properties of the level-set, a two-phase Riemann problem is solved at
the surrogate phase boundary. It provides ghost states to couple the bulk phases as well
as the local velocity of the phase interface.
7. The pointwise information of the interface velocity at the surrogate phase boundary is
copied to adjacent sub-cells and extrapolated with a WENO routine to obtain a velocity
field for the level-set transport.
8. Based on the indicators of Sect. 3.3, the DG/FV distributions for the bulk flow and level-set
transport are determined. Furthermore, the polynomial degree for DG elements is chosen
based on the indicator Imodal.
9. Finally, the adaptive hybrid DG/FV operators are advanced for the bulk flow and the
level-set.
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Fig. 5 Assembly of the level-set ghost-fluid framework from the main building blocks and chronological
interaction within one time step. Operators applied in every Runge–Kutta stage are colored light gray. Steps
that are only performed once per time step are highlighted in dark gray. The novelty is the hp-refinement step
prior to the application of the spatial operators for the bulk flow and level-set transport (Color figure online)
4 Parallelization Concept
4.1 MPI Communication
In general, the hp-adaptive scheme, employed for the bulk phase (see Sect. 3.1), is formu-
lated for unstructured, curved grids, which has been demonstrated in [46]. While the WENO
scheme discretizing the Hamilton–Jacobi equations (Sect. 3.4) restricts the presented mul-
tiphase framework to Cartesian grids, it internally still operates on an unstructured grid
representation. The grid representation and the associated parallelization concept inherit
from the work by [30] and are optimized towards massively parallelized computations, as
shown in [4,39].
The parallelization concept exploits that the DG operator can be split into two building
blocks, cf. Eqs. (12)and(13): the volume integral, which depends exclusively on the element’s
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inner nodes, and the surface integral, which requires information of the neighbored elements.
In parallel simulations, this information exchange implies a communication operation if the
two elements sharing one side reside on different partitions. The induced idle time can
conveniently be hidden by computing the element-local volume integral while exchanging
the side data for the surface integral through non-blocking communication operations. This
latency hiding approach increases parallel performance as long as the network communication
is faster than the memory buffer operations. The communication routines are provided by the
widelyusedMPIstandard[44], which is commonly availableo n HPCsystems in architecture-
optimized implementations. Moreover, the latency hiding approach is applied to the surface
integral itself. To this end, the data structure separates volume and surface data, with the
solution urepresented at the corresponding interpolation nodes each. Every element face,
except for those at the domain boundaries, forms a pair (side) with the face of the adjacent
element. As the numerical flux evaluated from this data pair is unique, it needs to be computed
only once. We therefore label the element faces as major or minor and evaluate the numerical
flux solely on the major. If two elements share a common side but reside on different partitions,
the surface data of the minor need to be sent to the major prior to the flux computation, and the
resulting flux values are sent back afterwards. Following the aforementioned latency hiding
approach, this communication pattern suggests to first calculate the flux on those MPI-sides
and use the idle time to calculate the flux on the processor-local inner sides.
In the present hp-adaptive framework,the recurring send and receive operations are accom-
plished efficiently by collecting the surface data per polynomial degree Nin a dedicated array
urecv, as depicted in Fig. 6. Once the communication has finished, urecv is unpacked by copy-
ing the received data back to the DG side array uminor. These additional buffer operations
account for the fact that the DG side arrays are only sparsely filled in terms of one specific
degree N. The same strategy is applied to the computed fluxes. That is, transforming uminor
to the common polynomial degree as well as projecting the resulting flux back to N(cf.
Sect.3.1) are both performed locally on the major side. Thus, only surface data of the actual
degree Nare exchanged, which minimizes the communication amount and further enhances
the scalability of the hp-adaptive scheme.
4.2 Domain Decomposition
As indicated, the parallelization strategy of the present framework relies on splitting the
computational domain into separate subdomains (partitions) of one or more grid cells, which
are each assigned to one processor unit. For the given, unstructured mesh representation,
Hindenlang [30] suggested the use of space-filling curves (SFCs) for the domain decompo-
sition. Compared to graph partitioning approaches, they were reported to provide a similar
surface-to-volume ratio, which correlates with the communication amount per data opera-
tion, while greatly simplifying data structures (cf. Sect.4.1), parallel I/O, and dynamic load
balancing (Sect.4.3). Specifically, we employ the Hilbert SFC [29], which returned highly
compact subdomains and thus proved suited for massively parallel simulations.
4.3 Dynamic Load Balancing Scheme
One of the major challenges to the efficient utilization of current and future HPC systems is
load balance, i.e. the even distribution of the application workload across the processor units
[12,40]. Temporal changes in the workload at runtime, in particular, necessitate a dynamic
load balancing (DLB) approach.
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Fig. 6 Flowchart of a major/minor side communication for the hp-adaptive data structure
In the presented hp-adaptive multiphase framework, two main factors contribute to a work-
load imbalance: First, the element-local choice between a p-adaptive DGSEM discretization
and the FV sub-cell scheme, and second, the narrow band approach of the interface track-
ing algorithm [2], which confines the costly solution of the level-set equations to a small
band of 2–3 elements adjacent to the surrogate phase boundary. Since both the element-local
discretization and the position of the phase interface may change over time, the processor
workload needs to be balanced dynamically.
In this work, we extend and improve the DLB scheme of [2] to fit our adaptive dis-
cretization. It relies on a domain decomposition through SFCs, which conveniently reduces
the partitioning problem to one dimension. The partitioning problem seeks to distribute K
elements with the weights wi,i=1,...,Kamong Pavailable processors, such that the
maximum processor load, also referred to as bottleneck B, is minimized:
B=max
1≤k≤P{Lk},with Lk=
sk

i=sk−1+1
wi.(31)
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Here skdenotes the index of the last element associated with the processor k,andLkthe total
processor workload. The separator indices s0=0≤s1≤··· ≤ sk···≤ sP=Ksubdivide
the domain of Kelements into Ppartitions of Nk=(sk−sk−1)elements. The task of the
partitioning algorithm is thus to find the sequence of separator indices (s0,...,sP)with the
lowest feasible bottleneck Bopt .ThevalueofBopt is unknown and bounded below by the
ideal bottleneck
B∗=1
P
N

i=1
wi(32)
which assumes identical loads among all partitions, i.e. Bopt ≥B∗.
The partitioning problem, commonly known as chains-on-chains partitioning problem
[53], has been addressed by a variety of algorithms. Algorithms that always return an opti-
mal partitioning are called exact, opposed to heuristics, which, however, typically execute
faster and can be parallelized. An extensive overview of both, exact methods and heuris-
tics is given in [53]. In our work, we adopt the bisection algorithm “ExactBS+PI” recently
proposed by [40]. Bisection algorithms in general use binary search to find a good bot-
tleneck. They repeatedly call a probing function, that checks a given bottleneck value for
feasibility, by adjusting the provided bottleneck iteratively through bisection. ExactBS+PI
specifically employs an improved probing function and a more accurate initial search interval
for the bisection. It thus features quick execution times while remaining exact. For details,
the reader is referred to the original work, where ExactBS+PI is embedded in a hierarchical
method to design a highly scalable, near-exact partitioning algorithm.
A crucial building block of our load balancing scheme is the determination of the weights
wi. Following [2], we perform element-local wall time measurements. The code instrumenta-
tion exploits that the narrow band approach decomposes the set Dwhich contains all elements
in into three intersecting subsets: The bulk elements Dbulk ≡D, where only the Euler
equations (1) are solved, the elements inside the narrow band DNB ⊂Dbulk where the inter-
face tracking is required additionally, and the elements directly at the interface D⊂DNB,
where the two-phase Riemann problem is evaluated. Evidently, the computational cost of an
individual grid element rises in the listed order of these subsets.
In [2], the measured time was distributed evenly among the elements within each subset,
which directly defined the element weight wi. This averaging ansatz was plausible since all
elements used the same discretization and could therefore be expected to contribute equally
to the cost of the considered subset.
To account for the variable element cost in the bulk due to the hp-adaptivity, the cost share
of the individual discretization schemes needs to be determined. To this end, a calibration run
at the beginning of the simulation calls the bulk operator once for every degree Nand once
for a pure FV sub-cell discretization in order to measure the average wall times per element
τNmin ,τNmin+1,...,τNmax ,τNFV .
At runtime, the measured time τbulk of subset Dbulk is distributed among the elements
according to the relative cost of the different discretizations. This relative cost is evaluated
by first calculating the expected wall time τcal of each subdomain based on the calibrated
values τi∈{τNmin ,...,τNFV }
τcal =
sk

i=sk−1+1
τi.(33)
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Fig. 7 Overview of the dynamic load balancing scheme
This expected wall time, together with the measured wall time τbulk, defines a dimensionless
factor χ,
χ=τbulk
τcal ,(34)
which allows to infer the weight wiof each bulk element from scaling the calibrated weight
wi=τi·χ. (35)
For now, the imbalance due to p-refinement in the discretization of the level-set advection
(5) is neglected since here both the FV and DG operator are evaluated for every element.
Thus, the variable degree Nhas only a minor effect on the computation time per element. In
the subsets DNB and D, the average costs of the additional interface-related operations are
added to wi.
With the element-local weights known, the current workload imbalance
I=B
B∗−1∈[0,∞)(36)
can be evaluated. It assumes zero for identical workload on all processors, I(B=B∗)=0,
and triggers a repartitionig upon exceeding a user-defined threshold. To redistribute the
elements between the processors, we first store the current solution of the bulk flow and the
level-set field. Then, the new partitioning is computed globally, allowing each processor to
determine which elements it needs to send and receive by comparing its old partition to the
new one. During the communication, each processor has to recompute the mesh metrics for
the new elements, thus hiding most of the communication time. Once the communication is
completed, the computation resumes. This repartitioning procedure is summarized in Fig. 7.
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Fig. 8 Setup for the strong scaling test sliced at z=0. The local polynomial degree Nand the elements
discretized by FV sub-cells are indicated with different colors (Color figure online)
4.4 Parallel Performance Analysis
In this section, we investigate the influence of the hp-adaptive discretization on the strong
scaling behavior.We chose a setup similar to the one used by Appel et al. [2], where the parallel
performance of the previous, non-adaptive code version was analyzed. All computations are
run on the HPE Apollo System Hawk at the High Performance Computing Center Stuttgart
(HLRS). For the scaling tests, we used up to 64 compute nodes, each with two 64-core
AMD EPYC 7742 CPUs and 256GB RAM. The test case consists of a resting droplet in
a three-dimensional domain , discretized by 30 ×30 ×30 elements. To study the effects
of the adaptive scheme on the performance, we enforced a checkerboard-like distribution of
the local polynomial degree. For the bulk, the degree varies between N∈[2,4]and for the
level-set, the degree changes between N∈[2,8]. FV elements deploy NFV =9 sub-cells
per spatial direction and the standard smoothness indicator presented in Sect. 3.3 is used.
Figure8visualizes the discretization for a slice through the z=0plane.
It should be noted that the checkerboard-like discretization is the most expensive scenario
for an adaptive scheme, since all element interfaces are of mixed type. To isolate the effects
of the repartitioning and the novel hp-adaptive data structure on the parallel performance,
we perform a single load balancing step after 5 time steps, followed by further 44 time steps.
During these 44 time steps, we measure the performance index PID, which expresses the wall
time required by a single processor to advance one DOF for one stage of the Runge–Kutta
time integration scheme
PID =#total wall time ·#cores P
#DOFs ·#time steps ·#RK-stages .(37)
The second metric to analyze the parallel performance is the parallel efficiency ηwhich
relates the current PID to the PID obtained with one node (P=128)
η=PIDP=128
PID (38)
We compute the setup with a range of 1,2,4,...,64 compute nodes corresponding to P=
128,256,...,8192 cores and supply the results in Fig. 9.
Up to 32 nodes, a parallel efficiency of about 50% is achieved. At 64 nodes, a sharp decrease
of parallel efficiency can be observed. It can be attributed to the limit of the load balancing
capabilities when the cost of the most expensive elements approaches the target weight of a
single processor unit, B∗. There are two main reasons for the decreased parallel efficiency:
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Fig. 9 Strong scaling behavior for the benchmark setup of Fig. 8
First, for processors containing very few elements, the latency hiding of the communication
becomes less efficient and, secondly, the partitioning of loads among processors becomes
less accurate, i.e. Bopt B∗.
It should be noted, that the overhead due to the load balancing is not included in the
measured PID. For the present setup, one repartitioning and data redistribution together have
a maximum cost equivalent to 17.2 time steps at 16 nodes and a minimum cost of 1.4 time
steps at 64 nodes. The overhead due to the time measurements for the weight computation
can be neglected.
5 Numerical Results
To assess the performance of the presented adaptive hybrid DG/FV sharp interface framework,
we simulate a wide range of benchmark problems. The considered test cases are compressible,
inviscid two-phase flows with the gas modeled as an ideal gas and the liquid modeled by the
stiffened gas equation of state. The following list provides an overview of the test problems
and motivates their choice:
–The 1D Gas–liquid Riemann problem allows the validation of the scheme against an
exact solution for a two-phase shock tube.
–The 2D Air-helium shock-bubble interaction demonstrates the schemes ability to cope
with severe interface deformations in the limit of an infinite Weber number.
–The 2D Bubble collapse in water is commonly applied to test the robustness of a two-
phase scheme in the presence of strong shocks during the bubble collapse.
–The 2D Shock-droplet interaction is chosen to validate the framework against the
results of Xiang and Wang [24]. It is further used as a benchmark case to study the
scaling properties of the developed method.
–The 3D Shock-droplet interaction is simulated to demonstrate the applicability of our
method to three-dimensional problems.
For all setups, a hybrid discretization is used with a p-adaptive DG scheme with N∈[2,4]
and a FV resolution of NFV =9 sub-cells per direction in the bulk flow. If not stated otherwise,
the level-set transport is discretized with DG elements of N∈[2,8]and a FV resolution of
NFV =9. The computations were again run on the Hawk system and used up to 32 nodes.
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Table 1 Initial conditions of the
gas–liquid shock tube problem in
non-dimensionalized form,
following Cocchi et al. [8]
ρu1pγp∞
Compressed air 1.241 0.0 2.753 1.4 0.0
Water 0.991 0.0 3.059 ·10−45.5 1.505
Fig. 10 Density, velocity and pressure distribution of the gas–liquid Riemann problem at the final time t=0.2.
The exact solution is compared against a numerical solution computed with a grid of 40 elements. Smooth
regions are discretized by a DG scheme (blue) with N∈[2,4], whereas a FV scheme (red) with NFV =9
sub-cells per element is applied at discontinuities (Color figure online)
5.1 1D Gas–Liquid Riemann Problem
In a first step, the novel scheme is applied to the well-known one-dimensional gas–liquid
shock tube problem of Cocchi et al. [8]. It considers two initially constant states inside
a computational domain =[−1,1]. The left side of the shock tube, x<0, contains
highly compressed air and the right side liquid at ambient pressure. The initial conditions are
provided in non-dimensionalized form in Table 1.
The computational domain is discretized by 40 elements. The simulation runs until
the final time t=0.2. In Fig. 10, the numerical results of our approach are compared
against the reference solution, which relies on the exact Riemann solver devised by Cocchi
et al. [8] for this setup. As expected, the FV sub-cell scheme is only applied in regions of
sharp gradients, that is at the right-moving shock in the liquid phase, the material interface
between the phases and the left-moving rarefaction wave in the gas. Since the other regions
contain nearly constant states, p-refinement is not necessary in those smooth regions and
the DG discretization always uses the lowest possible degree N=2. Despite the coarse
discretization, the numerical result is in good agreement with the exact solution and only a
minor undershoot can be observed at the material interface.
5.2 2D Air-Helium Shock-Bubble Interaction
As a first two-dimensional test case, we consider the interaction of a Ma =6 shock wave
in air with a planar, two-dimensional helium bubble [28,31]. The setup results in severe
deformations of the phase interface during the breakup of the helium bubble and is thus a
challenging benchmark for a sharp interface framework. At time t=0 s, the computational
domain =[0.0,0.356]m×[−0.0445,0.0445]m contains a shock at x1=0.1m and a
helium bubble of radius r=0.025 m is located at x1=0.15 m. The setup is sketched in
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Fig. 11 Initial setup for the
two-dimensional air-helium
shock-bubble interaction
Table 2 Initial conditions of the
two-dimensional air-helium
shock-bubble interaction
ρ(kg m−3)u1(ms
−1)p(MPa)γ
Air (pre-shock) 1.20 0.0 0.101 1.4
Air (post-shock) 6.34 1669.0 4.239 1.4
Helium 0.166 0.0 0.101 1.66
Fig. 11 and the initial conditions for the three constant regions vsbefore the shock, vafter
the shock and vbinside the helium bubble are provided in Table 2.
For the discretization of the physical domain , a resolution of 512×64 elements is chosen.
Since the phase interface is always discretized by FV sub-cells, this results in an effective
resolution of 647 DOFs per bubble diameter. Due to the symmetry of the setup, we only
compute half of the domain and impose symmetry boundary conditions at x2=0.0m.For
the remaining boundaries, we impose inflow boundary conditions on the left, non-reflecting
boundary conditions on the right and wall boundary conditions on the top side. In the bulk
flow, an HLLE approximate Riemann solver is used. The setup is computed until the final
time t=1·10−4s. The current workload imbalance Iis evaluated every 20 time steps and
triggers a repartitioning if I>1.0. The simulation was computed on 4 nodes of the cluster
Hawk and consumed a total of 743 CPU hours for 18,861 time steps. During the computation,
112 DLB steps where performed and required a share of 3.87% of the total wall time. In
Fig. 12, the density fields at the non-dimensional times t∗=(tu1)/(2r)={0.6,1.2,1.8,2.4}
are visualized. Our results are in good agreement with those by Han et al. [28]. An initial
deformation followed by a roll-up of the bubble can be observed. Complex shock patterns
develop and vortical structures evolve at entropy shear layers. Finally, the separation of helium
droplets from the initial bubble and the enclosure of air filaments can be seen.
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Fig. 12 Density plots of the Ma =6 air-helium shock-bubble interaction at the time instances t∗=0.6,
t∗=1.2, t∗=1.8andt∗=2.4
The local polynomial degree Nof DG elements and areas, where the FV sub-cell limiting
is applied, are visualized at t∗=2.4inFig.13 for both the bulk flow and the level-set
field. In the bulk flow, shocks and the phase interface are detected correctly and discretized
by the FV scheme on a refined sub-cell grid. An increased polynomial degree is applied
around vortical structures. To improve the accuracy of the geometry computation, elements
containing the level-set root are discretized with the highest possible degree for the level-set
transport equation. FV sub-cells are used mainly at the level-set cut off and at sharp kinks
of the level-set contour or at merging phase boundaries. All in all, the p-refinement and the
FV sub-cell limiting were adaptively controlled as expected, resulting in an accurate and
stable representation of shocks and the phase interface as well as a high resolution of vortical
structures. The proposed sharp interface framework thereby demonstrates its ability to cope
with complex flow patterns and pronounced interface deformations.
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Fig. 13 Discretization of the bulk flow (left) and the level-set field (right) for the Ma =6 air-helium shock-
bubble interaction at t∗=2.4. The element-local degree Nof DG elements is indicated by color, whereas
FV sub-cell elements are marked in gray. Detail plots highlight the refined resolution of the FV sub-cell grid
compared to the underlying DG mesh (Color figure online)
5.3 2D Bubble Collapse in Water
As a second two-dimensional test case, we study the interaction of a planar Ma =1.72 shock
wave in water with a gas bubble. During the shock-inducedcollapse of the gas bubble, a shock
wave and a high speed liquid jet are emitted. This mechanism is relevant for hydraulic pumps
and turbo machinery, fuel injectors, naval propulsion systems and medical applications like
lithotripsy. Since the setup involves strong pressure gradients, high speed fluid dynamics
and a complex deformation of the phase interface, it has been frequently used to assess the
efficiency of numerical schemes [26,28,49,63].
We consider the computational domain =[0.0,0.024]m×[−0.012,0.012]m initial-
ized at time t=0.0 s with a planar shock in the liquid at x1=0.0066 m and a gas bubble of
radius of r=0.003 m at x1=0.012 m. The setup is sketched in Fig. 14, with the pre and
post shock states as well as the state of the bubble provided in Table 3.
Exploiting the symmetry of the setup, we consider only half of the domain by imposing
symmetry boundary conditions along the x2=0.0 m axis. Inflow boundary conditions are
applied on the left side and the remaining sides are treated as non-reflecting boundaries.
The computational domain is discretized by 360 ×360 elements, resulting in an effective
resolution of 810 FV sub-cells per bubble diameter. For the bulk flow, we employ an HLLC
approximate Riemann solver. The computation is run until a final time t=4.5μs, using the
same load balancing parameters as above. On 2 nodes of the Hawk cluster, 105 CPU hours
were spent on 3514 time steps, required for this test case. Of the wall time, a fraction of
5.7% was spend for 75 DLB steps. Figure15 visualizes the non-dimensionalized density and
pressure distributions ρ∗=ρ/ρ0and p∗=p/p0, with reference density ρ0and pressure
p0set to the pre-shock state of the water.
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Fig. 14 Initial setup of the
two-dimensional shock-induced
bubble collapse
Table 3 Initial conditions of the two-dimensional shock-induced bubble collapse
ρ(kg m−3)u1(ms
−1)p(MPa)γ p∞(MPa)
Water (pre shock) 1000.00.01.0 4.4 6000.0
Water (post shock) 1323.65 681.058 19,000.0 4.4 6000.0
Air 1.00.01.01.4 0.0
The simulation reproduces essential flow phenomena like the reflected rarefaction wave at
t=1.9µs, the kidney shaped deformation of the droplet at t=3.7µs, the high speed water
jet piercing the bubble at t=3.8µs and the shock wave resulting form the bubble collapse at
t=4.0µs. Additionally, very delicate flow features like secondary jets, that further divide the
remnants of the initial bubble, can be observed at t=4.0µsandt=4.2µs. Our numerical
results agree very well with the simulations of Nourgaliev et al. [49]. To provide a qualitative
comparison with literature, we evaluate the pressure field along the x2=0 line in Fig. 16.
Until the final simulation time of t=4.5µs, a maximum pressure peak of 5.1 GPa can be
observed which is in good agreement with the results reported by Tsoutsanis et al. [63].
The adaptive discretization is analyzed for the time instance t=3.8µsinFig.17.In
the bulk flow, shocks and the phase interface are discretized by the FV sub-cell scheme,
while acoustic waves as well as the direct vicinity of shocks are captured with an increased
polynomial degree. Since the gas bubble is displaced by the significantly heavier water
at high speed, the level-set transport velocity is in the order of magnitude of the incident
shock wave. Therefore, the level-set field is only p-refined up to polynomial degree N=4
to avoid a dominant time step constraint through to the level-set transport equation. The
maximum degree is again applied at the level-set root to improve the accuracy of the geometry
computation. The FV sub-cell scheme of the level-set field does not only capture the cut-
off region at the narrow band edge, but also the tips of the separated bubbles due to their
curvature.
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Fig. 15 Non-dimensionalized density and pressure fields of the shock-induced bubble collapse at various time
instances
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Fig. 16 Pressure distribution along the x2=0 line for various time instances
Fig. 17 Discretization of the bulk flow (left) and the level-set field (right) for the shock-induced bubble collapse
at t=3.8µs. The element-local degree Nis indicated by color, whereas FV sub-cell elements are marked
in gray. Detail plots highlight the refined resolution of the FV sub-cell grid compared to the underlying DG
mesh (Color figure online)
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5.4 2D Shock-Droplet Interaction
In this section, we study the interaction of a planar shock wave with a water column, using
the setups investigated by Xiang and Wang [24] and Tsoutsanis et al. [63]. We consider two
configurations: a regular liquid column and a liquid column with an enclosed cavity. Both
configurations are computed with a Weber number of We =1000 to analyze the breakup in
the shear-induced entrainment (SIE) regime [59].
5.4.1 Water Column
First, we consider the interaction of a Ma =2.4 shock wave in gas with a pure liquid column.
The domain =[0.0,0.06]m×[−0.02,0.02]m contains a planar shock at x1=0.0074 m
and a water column of radius r=0.0048 m at x1=0.015 m at the initial time t=0.0s.The
setup is illustrated in Fig. 18, with the initial conditions provided by Table 4. The surface
tension is set to σ=11.90 N/m to obtain We =1000.
The computational domain is discretized by 360 ×240 elements, which corresponds
to a FV sub-cell resolution of 518 DOFs per droplet diameter. We again impose symmetry
boundary conditions along the x1=0 axis, inflow boundaries to the left side and non-
reflecting boundaries at the right and top. For the flux computation in the bulk, an approximate
HLLC Riemann solver is chosen. The setup is advanced until a final computation time
of t=2.0·10−4s. The workload distribution is evaluated every 50 time steps, with the
imbalance threshold again set to I>1. The computational cost on 4 nodes of the Hawk
system amounts to 694.8 CPU hours for 19,510 time steps required for the described setup.
In total, 160 DLB steps were performed, taking up 6.0% of the overall wall time. Figure19
supplies the numerical results at the non-dimensionalized time instances t∗=(tu1)/(2r)=
{0.8,1.62,5.5,11.8}. Schlieren plots visualize the vortical structures and shocks, while the
pressure field is given in non-dimensionalized form p∗= p/p0, with p0being the pre shock
pressure in the gas at t=0.0.
Fig. 18 Initial setup of the 2D
shock-droplet interaction without
cavity
Table 4 Initial conditions of the 2D shock-droplet interaction without cavity
ρ(kg m−3)u1(ms
−1)p(MPa)γ p∞(GPa)
Air (pre-shock) 1.20 0.01.01 1.40.0
Air (post-shock) 3.85 567.36.12 1.40.0
Water 1000 0.01.03479 6.12 0.343
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Fig. 19 Schlieren images and pressure fields of the 2D Ma =2.4 shock-droplet interaction for a Weber
number of We =1000 (SIE regime)
In particular, the wave pattern at time t∗=0.8 has been thoroughly studied in literature,
e.g. by Meng and Colonius [42] and Xiang and Wang [24], and the reported numerical
results are supported by experimental investigations of Sembian et al. [57]. Our computation
reproduces all relevant flow features: the reflected incident shock wave, the weak transmitted
shock wave, the reflected rarefaction inside the water column and the Mach stem and slip
surfaces.
To support our results quantitatively, we compare the pressure distribution along the x2=0
axis at time t∗=0.8 with the results of Xiang and Wang [24]inFig.20. A near perfect
agreement can be observed.
During the later stages of the simulation, the water column is flattened and complex
recirculation zones and vortex patterns develop in the wake. With We =1000, the droplet
breakup occurs in the SIE regime and the formation of filaments, that are stripped from
the main liquid body can be observed. The final column shape at t∗=11.8 agrees well
with the findings of Xiang and Wang [24] and Tsoutsanis et al. [63]. Computations with the
previous code framework, published by Jöns et al. [35], applied a constant number of DOFs
per element. To demonstrate the efficiency of the novel adaptive scheme, we compare the
adaptive computation with N∈[2,4]and NFV =9 against non-adaptive computations with
N=4andNFV =5. Table 5lists the average number of DOFs and the computation time
for all setups. Setup 1 uses the same mesh as the adaptive computation, which results in a
lower effective number of DOFs per droplet diameter due to the coarser FV sub-cell grid.
To achieve the same resolution at the phase interface as the adaptive scheme, setup 2 uses a
finer mesh. In Fig. 21, the temperature fields and element-local discretizations are compared
for the three configurations.
Shocks and the phase interface are detected well for all setups and discretized by FV
sub-cells. Setups 2 and 3 both achieve the same FV resolution at shocks and the phase
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Fig. 20 Pressure distribution along the symmetry line x2=0 of the shock-droplet interaction without cavity
and We =1000 at t∗=0.8. The results are compared against the data from Xiang and Wang [24]
Table 5 Efficiency comparison of the non-adaptive discretization (setups 1 and 2) with the proposed adaptive
scheme (setup 3)
DOFs per droplet diam Discretization Average DOFs Wall time (CPU h)
Setup 1 288 N=4, NFV =51.08 ·106288
Setup 2 518 N=4, NFV =53.50 ·1061561
Setup 3 518 N∈[2,4],NFV =90.60 ·106727
Fig. 21 Temperature field (top) and bulk discretization (bottom) of the 2D shock-droplet interaction with
We =1000 at t∗=11.8, comparing a static scheme with N=4and NFV =5 (left, middle) to the proposed
adaptive scheme with N∈[2,4]and NFV =9 (right). The element-local degree Nis indicated by color,
whereas FV sub-cell elements are marked in gray (Color figure online)
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Fig. 22 Evolution of the load imbalance (left), the average number of DOFs per element and the amount of
FV sub-cell elements (right) during the simulation time
interface. Therefore, shocks appear sharper and more intricate phase interface geometries
can be captured compared to setup 1. In setup 3, p-adaptivity allows to apply a reduced
degree of N=2 in most of the domain, while intricate vortical structures in the wake are
discretized with the highest possible degree. Since setup 2 has a finer DG mesh, slightly more
detailed vortical structures in the recirculation zone can be observed. Overall however, setups
2 and 3 produce comparable results. This is significant, since the adaptive scheme required
on average 5.8 times less DOFs than setup 2. Compared to setup 1, the adaptive scheme still
requires 1.8 times less DOFs, even though a higher resolution is achieved at shocks and the
phase boundary. Due to the increased number of DOFs in the expensive narrow band, setup
3 requires 727 CPU hours and is thus more expensive than setup 1 with 240 CPU hours.
However, compared with setup 2, requiring 1561 CPU hours, the adaptive scheme achieves
a similar result in less than half the wall time.
To study the evolution of the adaptive discretization over time, the average number of
DOFs per element and the relative amount of FV sub-cell is plotted in Fig. 22 for setup 3.
Starting from a pure FV sub-cell discretization to avoid oscillatory initial states, the amount
of required FV sub-cells increases gradually from a minimum of 1.7% to a maximum of
7.1% at the end of the computation. This increase is mainly attributed to the shock patterns
forming in the wake of the droplet and the increased surface of the deformed water column.
Likewise, a steady increase in the average number of DOFs per element from 10.3 to 16.1
can be observed. This can be explained by the increased number of FV sub-cells and by
the p-refinement of structures in the wake of the column. In order to assess the scalability
of the framework in practical applications, we finally investigate its parallel performance
for the We =1000 setup. To this end, we study the strong scaling behavior in a range of
P=32,...,1024 cores on the cluster Hawk. Figure 23 evaluates the parallel efficiency η
and the parallel speedup using the total runtime, opposed to the PID-based results for the
generic setup in Fig. 9.
Up to 512 cores, a parallel efficiency beyond 50% is achieved. A significant drop in the
parallel efficiency occurs for 1024 cores, which is caused by the fact that the target weight
falls below the cost of the most expensive element, B∗<w
max, as discussed in Sect. 4.4.
The imperfect strong scaling behavior can in general be attributed to two main factors: the
overhead of the dynamic load balancing procedure and a remaining load imbalance due to
the fast change in element-local loads. To support this explanation, the imbalance Iover time
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Fig. 23 Parallel efficiency (left) and parallel speedup (right) for the 2D shock-droplet interaction with We =
1000
Fig. 24 Workload distribution and domain decomposition (top) for the bulk discretization (bottom) of the state
giveninFig.21 using P=512 processor units
was evaluated for the computation with P=512 cores. Starting from an initial imbalance
of more than I=15, the dynamic load balancing scheme reduced the imbalance to I=0.7
on average. Thus, the load of the slowest processor still exceeded the target load by a factor
of 1.7.
Furthermore, a partitioning with P=512 or more cores entails that an increasing number
of partitions in the narrow band region host only a singular element due to their high com-
putational cost. As the partitioning operates on the element level, no further decomposition
is possible for these partitions, implying the natural limit of parallelization. Few of these
one-element partitions can be seen in the Fig. 24, which shows the workload distribution
{wi}for the bulk discretization of the state given in Fig. 21. The domain decomposition also
illustrates the general working principle of the dynamic load balancing scheme that makes
small partitions cluster in regions of higher computational cost while cheaper elements of
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N=2 form larger partitions. As computations in practice deploy fewer cores to evade
the depicted granularity limit, an overall acceptable parallel efficiency is achieved with the
proposed hp-adaptive level-set ghost-fluid framework.
5.4.2 Water Column with Cavity
As seen in the previous setup, a local pressure minimum can be observed inside the water
column due to the reflected rarefaction wave. While phase transitions are not included in
the present model, in reality the low pressure could lead to cavitation, i.e. the formation of
a vapor bubble inside the water column. Therefore Xiang and Wang [24] proposed a setup
to study the interaction between a planar incident shock wave and a water column with a
gas-filled cavity. Hence, we extend the previous scenario by introducing a circular cavity of
radius r=0.0036 m inside the water column. All other parameters are adopted from above,
resulting in the setup sketched in Fig. 25. The adapted initial conditions are summarized in
Table 6.
Again, a Weber number of We =1000 is considered to simulate a droplet breakup in the
SIE regime. The computational setup is identical to Sect. 5.4, and consumed 865 CPU hours,
on 4 compute nodes on Hawk. During 19,437 time steps, the domain was repartitioned 155
times by the DLB scheme, requiring 4.9% of the total wall time. Schlieren images and non-
dimensional pressure fields are depicted in Fig. 26 for the time instances t∗=(tu1)/(2r)=
{0.8,1.62,5.9,6.5,9.0,11.8}.Att∗=1.62, the reflected incident shock wave, the Mach step
and the slip line develop similar to the simulation without a cavity. However, the transmitted
shock wave is reflected and retransmitted at the surface of the water ring, creating a more
complex pattern of reflected and transmitted waves inside the water ring and the gas cavity.
Due to the curvature of the bubble, the transmitted waves inside the bubble appear as circular
shapes. Later on, the water column is flattened and a high speed water jet causes the bubble
to collapse, similar to the phenomena observed in Sect. 5.3. Our numerical results agree well
Fig. 25 Initial setup of the 2D
shock-droplet interaction with
cavity
Table 6 Initial conditions of the 2D shock-droplet interaction with a cavity
ρ(kg m−3)u1(ms
−1)p(MPa)γ p∞(GPa)
Air (pre-shock) 1.20 0.01.01 1.40.0
Air (post-shock) 3.85 567.36.12 1.40.0
Air (cavity) 1000 0.01.04306 6.12 0.343
Water 1000 0.01.03479 6.12 0.343
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Fig. 26 Schlieren images and pressure fields of the 2D Ma =2.4 shock-droplet interaction with a gas-filled
cavity for a Weber number of We =1000 (SIE regime)
Fig. 27 Evolution of the phase boundary for the 2D Ma =2.4 shock-droplet interaction with a gas-filled
cavity for a Weber number of We =1000. The separation of the initial gas bubble into four bubbles by liquid
jets and the formation of filaments can be observed
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Fig. 28 Mass conservation error
Em(t)over time for the shock
water column interaction test
case with an air-filled cavity
Fig. 29 Discretization of the bulk flow (left) and the level-set field (right) for the 2D shock-droplet interaction
with cavity for We =1000 at t∗=5.9. The element-local degree Nis indicated by color, whereas FV sub-cell
elements are marked in gray. Detail plots highlight the refined resolution of the FV sub-cell grid compared to
the underlying DG mesh (Color figure online)
with the simulations presented by Xiang and Wang [24] and Tsoutsanis et al. [63]. During
the final phase of the simulation, the gas bubble is further divided by secondary jets until
four separate bubbles are formed. At t∗=9.0, thin filaments have developed at the surface
of the water column, which have detached from the main liquid body at t∗=11.8. The
evolution of the phase boundary in time is visualized in detail in Fig. 27. A well-known
issue of the level-set interface tracking method is the appearance of conservation errors. On
the one hand, this is due to the non-conservative level-set transport equation (5). On the
other hand, the level-set reinitialization procedure is notorious for affecting the interface
position and thus causes further conservation errors [25]. Finally, the ghost-fluid method is
non-conservative, as discussed in [34]. Ultimately, the impact of these conservation errors
depends on the resolution at the phase interface. Therefore, they appear most prominent if
interfacial structures are under-resolved.
To investigate the conservation properties of the novel hp-adaptive framework, we ana-
lyzed the mass conservation error Em(t)=(m(t)−m(0))/m(0)for the challenging test case
at hand. Figure28 shows the mass conservation error over the simulation time in percent.
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Table 7 Initial conditions of the three-dimensional shock-droplet interaction
ρ(kg m−3)u1(ms
−1)p(MPa)γ p∞(GPa)
Air (pre-shock) 1.20 0.01.01 1.40.0
Air (post-shock) 3.85 567.36.12 1.40.0
Water 1000 0.01.257917 6.12 0.343
The conservation error remains below 2% for most parts of the simulation. In the later stages
of the droplet breakup, a conservation error of approximately 3% is encountered. Given the
extreme interface deformation of the droplet and the complex topological changes during the
collapse of the air cavity, the authors consider this conservation error acceptable. Finally, we
look into the element-local discretization of the bulk flow and the level-set transport.
Figure29 visualizes the FV sub-cells and the element-local polynomial degree of the DG
elements at time t∗=5.9. Shocks and the phase boundary are successfully captured by FV
sub-cells and the wake is resolved by a higher polynomial degree to improve the resolution of
vortices or weak acoustics. The level-set field is discretized by the highest possible degree at
the level-set zero position and oscillatory solutions at filaments or merging phase boundaries
are avoided by locally applying the robust and accurate FV sub-cell discretization.
5.5 3D Shock-Droplet Interaction
As a final test case, we extend the two-dimensional shock-water-column setups from the
previous sections to a three-dimensional shock-droplet interaction to evaluate the perfor-
mance of our scheme in three-dimensions. We consider a Ma =2.4 shock wave and a
Weber number of We =100. A similar setup with a lower Mach number of Ma =1.47
was used by Winter et al. [65] to investigate the RTP and SIE breakup regime in three
space dimensions. In analogy to the two-dimensional setup, the computational domain
=[0.0,0.06]m×[−0.02,0.02]m×[−0.02,0.02]m contains a shock at x1=0.0074 m
and a droplet of radius r=0.0048 m centered at x1=0.015 m . The surface tension is chosen
as σ=119.0N/m and the initial conditions are listed in Table 7. To achieve an effective res-
olution of 138 DOFs per droplet diameter, the domain is discretized by 96×48×48 elements.
Due to the symmetry of the setup, it is sufficient to compute a quarter of the domain and
impose symmetry boundary conditions at the x2=0andx3=0 planes. The left boundary
is defined as an inflow plane, while all remaining boundaries are non-reflecting. For the flux
computation, an approximate HLLC Riemann solver is applied. The setup is advanced in
time until the final time t=2.0·10−4s.
The workload distribution is evaluated every 200 time steps and triggers a repartitioning
if I>1.0. We deployed 32 compute nodes of Hawk and spent 7221 CPU hours on the
simulation. During 7231 time steps, DLB was applied 36 times and required a share of
7.0% of the total wall time. Figure30 depicts the evolution of the droplet geometry and the
non-dimensionalized pressure field p∗=p/p0at the time instances t∗=(tu1)/(2r)=
{0.8,2.4,4.7,7.0,9.4,11.8}.Att∗=2.4, the development of interfacial waves can be
observed. Kaiser and Winter [36] suggested, that these waves are triggered by the interaction
of pressure fluctuations with the phase interface. From t∗=0.8 until t∗=9.4 the interface
waves grow in size and merge into larger liquid sheets. At the final simulation time t∗=11.8,
the stripping of the water sheets from the main liquid body can be observed. Figure31
visualizes the velocity magnitude and the element-local bulk discretization at time t∗=7.0.
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Fig. 30 Pressure field of the 3D Ma =2.4 shock-droplet interaction with a Weber number of 100 at different
time instances
Fig. 31 Three-dimensional Ma =2.4 shock-droplet interaction at t∗=7.0. The velocity magnitude in the
x2=0 plane is visualized at the bottom and the element-local bulk discretization in the x2=0 plane is
projected to the background. To visualize the recirculation zone in the wake of the droplet, isocontours of the
velocity magnitude are included
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Fig. 32 Mass conservation error
Em(t)of the 3D shock-droplet
interaction
Due to the significantly lower discretization than in the two-dimensional setups, FV sub-cells
are used to discretize the recirculation zone in the wake of the droplet.
This is necessary to avoid oscillations caused by the aliasing error of a DG discretization in
under-resolved regions. Finally, the mass conservation error Emof the three-dimensional test
case is evaluated, analogous to Sect. 5.4.2. Figure32 depicts the conservation error over time.
Despite the coarser resolution compared to the two-dimensional setups, the conservation error
remains below 2% throughout the simulation. Given the significant deviation from the initial
droplet shape and coarse resolution, the mass loss is in an acceptable range when compared
to 3D sharp interface simulations reported in literature, see e.g. [66].
6 Conclusion
We have introduced an hp-adaptive discretization into a level-set ghost-fluid framework
for compressible multiphase flows by combining a p-adaptive DGSEM scheme with a FV
sub-cell method. The hybrid discretization is applied to the continuum model of the bulk
fluids, as well as the level-set transport equation. It relies on an indicator that evaluates the
modal decay of the polynomial solution representation and additionally takes into account
the position and topology of the phase interface. The element-local discretization can thus
automatically be adapted at runtime to provide a high-order accuracy in smooth regions
while offering stable and accurate results in the presence of shocks and severely deformed
phase boundaries. The adaptive discretization, together with applying the interface tracking
algorithm only in a narrow band around the phase interface, cause pronounced variations in
the element-local computational costs throughout the domain. In parallel simulations, these
translate to significant imbalances in the processor workloads, which necessitate a dynamic
load balancing scheme to ensure parallel scalability. The proposed scheme determines the
current workload distribution through element-local wall time measurements and repartitions
the elements along a space-filling curve. Strong scaling tests show acceptable results, with a
parallel efficiency of 50% on up to 4096 cores. We apply our level-set ghost-fluid framework to
a wide range of inviscid, compressible multiphase flows and obtain results in good agreement
with numerical and experimental findings from literature. When compared to non-adaptive
computations, the presented adaptive scheme produced comparable results with far fewer
DOFs and a significantly reduced wall time. The results confirm that the proposed hybrid
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discretization is well suited for the multi-scale nature of multiphase flows. While the FV
sub-cell grid enables a high resolution and a good localization at the phase interface, the
p-adaptive DG scheme produces accurate results on a relatively coarse mesh. In the future,
we plan to include viscous effects to capture the impact of viscosity on the droplet breakup
regimes. Furthermore, we seek to include phase transfer to model evaporation and cavitation.
Funding Open Access funding enabled and organized by Projekt DEAL. We gratefully acknowledge the
support by the German Research Foundation (DFG) for the research reported in this publication through the
project GRK 2160/1 “Droplet Interaction Technologies” under the project number 270852890 and through
Germany’s Excellence Strategy EXC 2075 under the project number 390740016. Further, the authors thank
the DFG for supporting the presented work through the framework of the research unit FOR 2895 (Grant BE
6100/3-1). All simulations were performed on the national supercomputer HPE Apollo Systems HAWK at the
High Performance Computing Center Stuttgart (HLRS) under the Grant number hpcmphas/44084.
Data Availibility All data generated or analyzed during this study are included in this published article.
Code Availability The open source code FLEXI, on which all extensions are based, is available at https://
www.flexi-project.org under the GNU GPL v3.0 license.
Declarations
Conflict of interest The corresponding author states on behalf of all authors, that there is no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,
and indicate if changes were made. The images or other third party material in this article are included in the
article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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