Phase-resolving direct numerical simulations of particle transport in liquids—From microfluidics to sediment

Abstract

The article describes direct numerical simulations using an Euler–Lagrange approach with an immersed-boundary method to resolve the geometry and trajectory of particles moving in a flow. The presentation focuses on own work of the authors and discusses elements of physical and numerical modeling in some detail, together with three areas of application: microfluidic transport of spherical and nonspherical particles in curved ducts, flows with bubbles at different void fraction ranging from single bubbles to dense particle clusters, some also subjected to electro-magnetic forces, and bedload sediment transport with spherical and nonspherical particles. These applications with their specific requirements for numerical modeling illustrate the versatility of the approach and provide condensed information about main findings.

Full text

Received: 8 March 2022 Revised: 31 May 2022 Accepted: 31 May 2022
DOI: 10.1002/gamm.202200016
ORIGINAL PAPER
Phase-resolving direct numerical simulations of particle
transport in liquids—From microfluidics to sediment
Jochen Fröhlich Thomas E. Hafemann Ramandeep Jain
Institute of Fluid Mechanics, TU Dresden,
Dresden, Germany
Correspondence
Jochen Fröhlich, Institute of Fluid
Mechanics, TU Dresden, Dresden,
Germany.
Email: jochen.froehlich@tu-dresden.de
Funding information
DFG, Grant/Award Numbers: 634058,
625609, 570058; ESF, Grant/Award
Number: 100231947; State of Saxony,
Grant/Award Number: L-201535
Abstract
The article describes direct numerical simulations using an Euler–Lagrange
approach with an immersed-boundary method to resolve the geometry and tra-
jectory of particles moving in a flow. The presentation focuses on own work
of the authors and discusses elements of physical and numerical modeling in
some detail, together with three areas of application: microfluidic transport
of spherical and nonspherical particles in curved ducts, flows with bubbles at
different void fraction ranging from single bubbles to dense particle clusters,
some also subjected to electro-magnetic forces, and bedload sediment transport
with spherical and nonspherical particles. These applications with their specific
requirements for numerical modeling illustrate the versatility of the approach
and provide condensed information about main findings.
KEYWORDS
bubbles, collision modeling, direct numerical simulation, immersed-boundary method, Lorentz
forces, microfluidics, multiphase flow, nonspherical particles
1INTRODUCTION
An instant where the simulation of multiphase flows touched almost everybody’s life in Europe was the outbreak of the
volcano Eyafjallajökull in April 2010. Decisions on shutting down almost the entire air traffic were taken solely based
on simulations of the transport of the ash particle concentration in the atmosphere [25,136]. Beyond particles in the
atmosphere, there are many more instances where multiphase flows and their simulation are decisive.
Multiphase flows (MPFs) are characterized by mobile regions inside of which material properties vary only slightly
or even not at all, so-called phases, separated by sharp phase boundaries across which properties exhibit marked jumps.
Most often phases are related to different states of aggregation, as in the case of solid particles inside a gas or liquid flow,
for example, but emulsions, composed of two liquid phases, also fall into this category. The subject is extremely vast due
to the large number of possible combinations and the resulting number of material parameters, together with boundary
conditions on composition and flow rates. Single-phase flows already exhibit many different regimes. Each, in principle,
can be enhanced by adding one or more additional phases, bringing about further dimensions in parameter space. Due to
the relevance of MPF for process engineering (pneumatic and hydraulic conveying, bubble columns, boiling, etc.) a large
volume of literature on the topic exists [17,66,87,146]. Also numerous environmental phenomena (rain, sediment trans-
port, aerosols, etc.) and biological phenomena (blood flow, plankton, etc.) feature multiphase flows. In recent decades the
simulationofmultiphaseflowshasbeenadvancedbyenormousleaps,madepossiblenotonlybyincreasedcomputational
power but also by substantially improved physical models and better numerical algorithms.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the
original work is properly cited.
© 2022 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.
GAMM - Mitteilungen. 2022;45:e202200016. wileyonlinelibrary.com/journal/gamm 1of26
https://doi.org/10.1002/gamm.202200016

2of26 FRÖHLICH  .
The present contribution will only treat flows where one phase is dispersed in another, so-called continuous phase,
and where properties within each phase can be assumed constant. Furthermore, the elements of the dispersed phase
considered are rigid and the continuous phase a liquid. Still, this defines a broad range of topics, as illustrated below.
Beyond true solid particles, small bubbles in liquids containing a sufficient amount of surfactants remain spherical and
feature a no-slip condition at their surface, so that they can be treated as spherical particles of constant shape as well,
just having a density much smaller than the continuous phase. The ash particles mentioned above were represented as a
continuous concentration field with different size distributions [136]. The present article, in contrast, treats simulations
where the disperse phase is geometrically resolved.
For single-phase flows the term direct numerical simulation (DNS) has a well-defined meaning, designating a simu-
lation where the unsteady turbulent motion of a fluid is entirely resolved. Since a smallest scale exists due to viscosity,
the Kolmogorov scale lK, this can be done, at the corresponding computing cost [90]. Discretization in space and time is
generally made so fine that numerical errors are negligible, when speaking of a single-phase DNS.
With disperse MPF, the term DNS needs careful revisiting, because a new length scale is introduced, the size of the
elements of the disperse phase in terms of a diameter dp. These elements are often summarized under the term “parti-
cle” for simplicity, regardless whether they are true particles, drops, bubbles, and so forth. For particles larger than the
Kolmogorov scale, dp>lK, DNS of a MPF in any case implies that the particles are resolved, that is, their geometry as
well as the details of the surrounding flow. For dp<lK, resolving all fluid motion including those generated by the par-
ticles moving relative to the fluid requires a grid substantially finer than dp, hence, finer than lK. This, and the previous
case, is termed particle-resolving DNS (PR-DNS) in some of the literature. For various reasons, the present authors since
many years use the term phase-resolving DNS, which is more general but in the context discussedhere can be considered
equivalent, so that the same abbreviation PR-DNS is used.
Collisions between particles introduce yet another length, which is the instantaneous distance between the particle
surfaces. It becomes arbitrarily small and, hence, cannot be resolved over the entire collision process. Furthermore, col-
lisions involve a very small time scale related to the elastic deformation of the collision partners. Generally, both issues
are accounted for by models, as discussed in Section 2.6. Since these models do not concern turbulent motion, the term
DNS (PR-DNS, PM-DNS) is still employed.
Depending on the research question, a flow with dp<lKcan also be simulated on a grid with step size larger than dp,
but still smaller than lK. This is termed particle-modeling DNS (PM-DNS), since the interaction between the continuous
phase and the particles is modeled by empirical correlations, like a drag coefficient, for example, still without any turbu-
lence model [71]. With this approach particles are treated as point-particles, that is, without their volume or shape being
geometrically represented, the motion of which is computed as a function of the forces acting on them.
In both cases, PR-DNS and PM-DNS, DNS can also be replaced with large Eddy simulation (LES), with the
corresponding subgrid-scale modeling for the smallest turbulent motion.
Finally,theparticlescanberepresentedasacontinuum with a transportequationfortheirconcentration,withnumer-
ous closure terms being modeled [21,71,97,107,123]. Here, turbulence treatment can vary from DNS over LES to statistical
modeling (RANS). An illustration is given in Figure 1(right). Methods of this type are termed Euler–Euler methods, as
FIGURE 1 Illustration of different modeling approaches for multiphase flows. Left: Velocity field around a rising spherical bubble
with no-slip boundary condition in a particle-resolving DNS (courtesy R. May). Middle: Illustration of what has to be modeled in a
particle-modeling DNS (sketch of physics, not data from simulation). Right: Void fraction represented as a continuum in an Euler–Euler LES
of a bubble plume by Ma et al. [84]. All pictures only show part of the computational domain.

FRÖHLICH  . 3of26
opposed to Euler–Lagrange approaches where the motion of individual particles, with or without geometrical resolution,
is computed in a Lagrangian manner.
During the last two decades the work on algorithms for PR-DNS of multiphase flows has exploded, enhancing and
further developing the classical volume-of-fluid (VOF), level set (LS), cut-cell, chimera, front-tracking, fictitious-domain,
and other approaches, including combinations of individual elements of these, and using finite-volume, finite-element,
Lattice–Boltzmann or higher-order methods as the basic framework. To date, there is no single method outperforming
the others, but depending on the physical situation and the numerical implementation on today’s parallel computers one
or the other method has advantages or disadvantages, which is a matter of investigation in the community.
The present article is devoted to PR-DNS for moving bodies and its application to a broad set of phenomena. It is not
meant as a general review on the topic, but rather assembles some of the work done in the group of the first author during
the last 15 years, with a focus on the most recent developments, including unpublished results obtained in collaboration
with the second and the third author. The goal is to provide an overview and highlight central issues with reference to
the corresponding publications for details.
2NUMERICAL METHOD AND SUBMODELS
2.1 Scales and modeling tasks
In this section both, physical as well as numerical modeling for PR-DNS will be discussed by means of Figure 2featuring
the situation of sediment transport. In other situations, some of the phenomena encountered here might not be present
or not relevant, or additional ones present. This does not alter the principle issue.
Particles in a mobile bed, at rest or mobilized, interact with the surrounding flow on the length scale of the particle
diameter dp. Scales much smaller than dpare responsible for friction between surfaces in contact, and time scales much
shorter than dp∕Ufor collision, elastic or damped, with Uan overall fluid reference velocity. Smaller scales are also
characterizing lubrication forces generated by fluid leaving or entering the gap between colliding particles just before
and after surfaces touch. Here, the particle distance is relevant, instead of the diameter. On spatial scales larger than the
particles, gradients in porosity of the bed can generate particular phenomena. The formation of ordered particle clusters
much larger than dpconstituting large-scale roughness for the outer flow is often observed and an important research
topic. Even larger scales are present when addressing submergence in a river, for example, followed by the width and the
stream-wise stretch, and so on.
Employing a uniform three-dimensional isotropic grid, present computer power allows to cover about three orders of
magnitude between the smallest resolved scales, that is, the step size of the grid, and the largest scales given by the size of
the domain. It is, hence, to be decided which window in scale shall be resolved by a simulation. Effects on smaller scales
have to be modeled by closure expressions and effects on larger scales by appropriately defined boundary conditions.
The work presented here aims at resolving the particle scale and to extend the window in scale as far as possible by
increasing the domain size. By definition, this amounts to resolving all mechanisms of 4-way coupling. Concerning still
larger scales in space and time periodic boundary conditions are suitable in many cases to remove such an impact in
FIGURE 2 Hierarchy of length scales and related phenomena encountered with the transport of heavy particles in a flow. The bottom
row illustrates the interplay between resolved a modeled features with the present PR-DNS approach.

4of26 FRÖHLICH  .
fundamental studies. These conditions are well established and common practice, so that they will not be discussed
further. Euler–Lagrange methods for PR-DNS, hence, require physical and numerical modeling on several scales, which
will now be addressed one by one, as employed by the authors.
2.2 Eulerian description of liquid
The flow of the continuous phase is modeled by the Navier–Stokes equations for a Newtonian fluid of constant density
𝜌fand kinematic viscosity 𝜈,reading
∇⋅u=0,(1)
𝜕u
𝜕t+u⋅(∇u)=−1
𝜌f∇p+𝜈𝜵2u+fIBM,(2)
where urepresents the velocity vector, ttime, ppressure, and fIBM enforces the no-slip condition as described below.
The time advancement uses a predictor-corrector scheme, accomplished by a combined fractional-step and three-step,
second order low-storage Runge–Kutta method [77]. In each stage of the scheme, the convective terms are advanced using
a first-order-explicit scheme, and the viscous terms by a Crank–Nicholson scheme. The pressure term in each substep is
corrected solving a Poisson equation. Spatial discretization employs a homogeneous, isotropic, Cartesian grid minimizing
the computational effort per grid point due to simplicity of stencils and efficiency of memory access.
2.3 Lagrangian representation of particle motion
Calculating the particle trajectory is based on the Newtonian equations of motion (EOM) describing the change of
translational and angular momentum
mpdup
dt =∫Γp
𝝈⋅ndS +Vp𝜌p−𝜌fg+Fc,(3)
Ipd𝝎p
𝜕t=∫Γp
r×(𝝈⋅n)dS +Tc,(4)
where mpis the particle mass, upthe particle velocity, 𝝈the hydrodynamic stress tensor, and nthe normal vector at the
surface Γpof the particle. The buoyancy term contains the particle volume Vp, the difference of the particle density 𝜌pand
the fluid density 𝜌f, and the gravitational acceleration gof magnitude g. The force Fcaccounts for collisions as discussed
below. The angular momentum equation features the moment of inertia of the particle Ip, the angular velocity 𝜔p,the
vector between the center of mass of the particle and a point on its surface r, and the angular collision torque Tc.
Incase of nonspherical particles the implementation makes use of quaternions [72] which is state of the art. Time step-
ping of the Lagrangian solver is accomplished with the same Runge–Kutta scheme as for the fluid solver and is performed
in the same loop. Substepping is used by some groups [10,64,74] and was tried in other projects of the authors. Here,
for efficiency the same size of the time step is employed for fluid and particles, which is possible due to an appropriate
collision model.
2.4 Particle-fluid coupling
The coupling between Eulerian and Lagrangian frames is accomplished by an immersed-boundary method (IBM) based
on a judiciously defined force fIBM in (2). Reviews on this approach have been compiled by Mittal and Iaccarino [88] and
more recently by Griffith and Patankar [36]. The method is particularly suited to efficiently represent moving objects in
a flow, as opposed to stationary boundaries, a goal which underlies all subsequent steps of modeling and discretization
discussed below. The way the coupling between the phases is achieved plays an important role for the stability of the

FRÖHLICH  . 5of26
method, bearing much similarity with fluid-structure coupling, although the communities are somewhat disjoint. The
present approach amounts to a segregated solver strategy, as opposed to a monolithic one, and is based on the method
of Uhlmann [133] as a starting point. Beyond the coupling in time a central issue of such a method is that the boundary
conditions on Γphave to be converted to values fIBM on the Eulerian grid. To this end, the particle surface Γpis covered
with a grid of marker points, L, distributed over the surface of the particle, and a so-called direct forcing method [89] is
applied. Assuming an explicit discretization in time the provisional velocity

u=un−1+Δtrhs (5)
is determined on the Eulerian grid E,withrhsdenoting all terms in (2) except the first and the last. It is then interpolated
from the Eulerian grid points x∈Eto the Lagrangian points X∈Lyielding 
Uon the Lagrangian grid.1Supposing the
velocity Ud
Γon the boundary Γpbeing known, the force is determined via direct forcing at the Lagrangian points using
FIBM =Ud−
U
Δt.(6)
Subsequently, thisvalue is distributed back to the Eulerian grid points in aspreading step. For moving bodies, (6) reduces
the tendency to create numerical oscillations [133] compared to direct forcing on the Eulerian grid as proposed by Fad-
lun et al. [27] Both interpolation and spreading are accomplished using a regularized delta-function 𝛿h(𝝃)with specific
properties in terms of local support and discrete moment conditions when sampled on an equidistant grid [39,105]

U(X)=
x∈E

u(x)𝛿h(x−X)h3,∀X∈L(7)
fIBM(x)= 
X∈L
FIBM 𝛿h(x−X)h3,∀x∈E,(8)
where his the stepsize of the Eulerian grid.
An important consequence of the IBM is that the interior of the particle is filled with fluid obeying the Navier–Stokes
equations. As a consequence, the hydrodynamic forces on the particle covering the domain Ωpcan be written as
∫Γp
𝝈⋅ndS =d
dt ∫Ωp
𝜌fudV −∫Ωp
𝜌ffIBM dV,(9)
∫Γp
r×(𝝈⋅n)dS =d
dt ∫Ωp
𝜌fr×udV −∫Ωp
𝜌fr×fIBM dV.(10)
Uhlmann [134] showed that in the continuous setting the first term on the r.h.s. of (9) reduces to 𝜌fVpdup∕dt for a spherical
particle with a no-slip condition on the interface, so that the stress term can be evaluated from the integral of the IBM
force. The same does not hold for the angular momentum but after tests was employed as an approximation.
The second term on the r.h.s. of (9), (10) can be evaluated as the sum of the forces distributed over the Lagrange points.
This results in the following explicit fluid–solid coupling
Vp(𝜌p−𝜌f)dup
dt =−𝜌f
X∈L
FIBM(Xl)ΔV+Vp𝜌p−𝜌fg+Fc,(11)
Ip
𝜌p(𝜌p−𝜌f)d𝝎p
𝜕t=−𝜌f
X∈L
(Xl−xp)×FIBM(Xl)ΔV+Tc(12)
with ΔVthe volume associated to the Lagrangian points, taken equal to h3.
1Throughout, all quantities pertaining to these markers are written in capital letters, while quantities related to the Eulerian grid are denoted with
small letters.

6of26 FRÖHLICH  .
An IBM for moving particles has the advantage of simplicity and efficiency, compared to moving mesh methods and
remeshing. Both fluid and particle meshes are constant in time, while the particle moves within the Eulerian domain. The
fluid density is used throughout the domain, which is beneficial for the convergence of the pressure solver and different
from front tracking algorithms employing variable density [135]. It also facilitates the use of direct solvers which can be
beneficial in some situations. This treatment of density is different from front tracking algorithms employing a different
density in the different phases [135]. Interpolation and spreading stencils do not depend on direction, and the explicit
determination of the force simplifies the coupling. Employing a smoothed delta function for the transfer of velocity and
forces reduces grid locking. Grid locking results from the fact that the interpolation operation on a given grid is not
translation invariant, as the stencil of contributing points changes discontinuously at certain positions [96].Asaresult,
the computational forces on a moving body do not, or only to a very small extent, exhibit numerical oscillations [133].
On the other hand, convergence of the solution in space generally is first order only [37,127]. Due to the spreading of
the forces the smoothing of the particle interface results in an overestimation of the particle size, as also observed for
other smooth fictitious domain methods [48].Breugem[11] proposed to shift the Lagrangian markers slightly inward to
improve the determination of integral quantities like drag. This was also discussed in the framework of fictitious domain
methods (see Yu and Shao [150] and references therein) but needs calibration and was only rarely applied by the present
authors. We refrain here from a comparative discussion with other methods employing a constant Eulerian fluid grid,
likeVOF,levelset,fronttracking,cutcell,andsoforth,asthisisavasttopicandbeyondthepresentscope.
Another advantage of an IBM is that the forces FIBM,fIBM are readily available for coupling the continuous and the
dispersephase,asseenin(2),(11),and(12).Theaboveschemeamountstoa weak coupling, well suited for heavyparticles,
but not for light particles. Kempe and Fröhlich [59] noticed that using the preliminary velocity 
ucan be improved upon
and proposed an efficient time scheme with better stability and better imposition of the boundary condition by repeated
forcing without repeated solution of the Poisson equation. Together with explicit evaluation of the volume integrals in
(9), (10) to eliminate problems for 𝜌p≈𝜌f, the range down to 𝜋𝜌=𝜌p∕𝜌f=0.3 became accessible.
A further step was accomplished by Schwarz et al. [119] stabilizing the particle EOMs by introducing an additional
term mimicking the added mass effect. While the added mass effect is, of course, resolved with the fluid discretization
and the coupling, it is not present in (3) and (4). Introducing the effect in these equations by a suitable approximation is
efficient and enhances the stability of the coupling by orders of magnitude. As a result, very light particles with mp≈0
representing bubbles could be treated with an IBM for the first time.
Bubbles in purified liquids exhibiting a slip-boundary at their surface require a different forcing approach. This was
developed by Kempe et al. [61], with particular account of the surface curvature, so that this case can be covered as well.
Tschisgale et al. [130] made a substantial step forward in the coupling between continuous and disperse phase by
devising a semi-implicit method. Inserting (9)intotheequationofmotion(3) and observing that the integral over fIBM in
this equation amounts to an integration over FIBM allows to introduce Udvia the direct forcing scheme (6). This quantity is
Ud=up+𝝎p×r,(13)
which makes upappear on the r.h.s. of (3) appear as well. The same can be done for the balance of angular momentum
(10). This strategy creates an implicit contribution in the EOMs providing high stability. The IBM is then formulated in
terms of a coupling force within a small shell Lof thickness haround the particle surface as shown in Figure 3.After
discretization in time the EOMs for a spherical particle without collisions read [130]
un
p=1
mp+mLmpun−1
p+𝜌f∫L
udV +∫Ωp
𝜌f
udVn
n−1
+ΔtV
p(𝜌p−𝜌f)g,(14)
𝝎n
p=1
Ip+ILIp𝝎n−1
p+𝜌f∫Lr×
udV +∫Ωp𝜌fr×
udVn
n−1,(15)
where [𝜙]n
n−1=𝜙n−𝜙n−1and the velocity 
uin these brackets approximates u. In a subsequent publication, this was
generalized to particles of arbitrary shape [131] which is not trivial due to several technical complications. An overview
over the schemes discussed here in terms of their stability is presented in Table 1. The enhanced stability of the last
method makes it robust and easy to use in general situations. The additional cost compared to (11) and (12) is small in
typical simulations.

FRÖHLICH  . 7of26
FIGURE 3 Illustration of the semi-implicit IBM with the horizontal velocity component. The liquid in the cells covering the particle
surface contributes to the implicit part in the EOMs for the bubbles. The gray region receives fIBM through spreading with the three-point
variant of 𝛿nby Roma et al. [105]. The dashed line is the stencil width of the delta function associated to the arbitrarily selected Langrangian
point Xl. Picture from [130]
TABLE 1 Improvement in stability with the immersed
boundary methods discussed here
Scheme Stability in terms of 𝜋p=𝜌p∕𝜌f
Uhlmann [133] ≥1.2
Kempe and Fröhlich [59] ≥0.3
Schwarz and Fröhlich [118] ≥0.001
Tschisgale et al. [130] ≥0
2.5 Particle shape
Almost all fundamental studies employing PR-DNS with particles of fixed shape so far are being conducted with spherical
particles and generate extremely valuable information. Particle shape, however, is another dimension in parameter space
whichis relevant in certainapplications,since all four coupling mechanisms arealtered in such a caseyieldingneweffects.
Examples are solid particles representing biological cells, seeds, fibers, or sediment, to name but these. Sedimentation of
single particles, for example, heavily depends on shape, as reviewed by Ern et al. [26]. An illustrative study of a rotating
seed was conducted by Arranz et al. [4] using an IBM. Bubbles at moderate Reynolds number can also be represented by
a fixed ellipsoidal shape. The same holds for drops and flexible biological cells experiencing sufficient shear. Bubbles with
large Eötvös number can exhibit complicated unsteady shapes which introduces a substantial further complexity which
is not addressed here.
For the IBM discussed in the present text, consideration of nonspherical shape requires several enhancements. The
direct forcing approach with interpolation and spreading can be retained in case of the method of Uhlmann [133],while
the method of Tschisgale et al. [130] required a certain amount of technical work [131]. The distribution of forcing points
on the particle surface, L, becomes an issue to be addressed, though. For spherical particles equidistribution is usually
assumed in the forcing procedure. It is implemented either to very good approximation with dedicated algorithms, as
the one of Leopardi [78], by iteratively minimizing the potential of repulsive forces between these points [133],orother
measures. For nonspherical particles applying these approaches is not straightforward. Also, the geometry may require
smaller distances between forcing points in regions of higher curvature to adequately represent the shape, so that in
general the assumption of constant distance is violated. For a uniform Eulerian grid the uneven distribution of Lcan be
accounted for by an individual weight ΔVfor each forcing point, obtained from a quadrature of the surface grid [118].A
further step was done by Akiki and Balachandar [2] to account for nonuniformity in Eas well. Arbitrary meshes were
addressed by Pinelli et al. [98] but lead to complex local operations which in case of moving objects become expensive.

8of26 FRÖHLICH  .
A second issue to be resolved is the EOMs for the particles which become more complicated, as the scalar moment of
inertiais replaced by the inertiatensor.Thisis commonly addressed by solvingthe angular EOM in abody-fixedcoordinate
system [100]. Third, and one of the larger steps, collision algorithms have to be altered when addressing nonspherical
particles, as discussed in the following subsection.
An issue influencing all of the above parts of the method is the representation of the particle shape. The issue is
common to discrete-element methods, which usually focus on the particle interaction and neglect the coupling to a
surrounding fluid, but exhibits particular requirements for PR-DNS. A closed triangulated surface is a very general rep-
resentation of a particle, also employed in certain problems with fluid–structure coupling [124], but often not needed in
fundamental studies of multiphase flows. A completely different approach is lumping spheres together to create so-called
compoundparticles, which allows for software reuse of routines for spherical particles [31]. Here, realism of fluid–particle
and particle–particle interaction is reduced, so that the approach is reserved for very applied studies [67].
The present authors favor an analytical representation of the particles and employ spheres, ellipsoids,and representa-
tions by spherical harmonics. The latter allow very accurate representation of curvature and regularization of the surface.
Indeed, ellipsoids which in contrast to spheroids have three independent axes constitute a large and versatile family of
shapes well suited for fundamental studies. The advantage of an analytical description is the accuracy of all quantities
related to the shape. In fact, it constitutes a second representation of the shape complementing the one generated by the
forcing points, so that depending on the task one or the other can be employed.
An impression of how the shape of solid particles impacts the behavior in a situation with merely no collisions, that
is, where only the fluid–solid coupling is dominant, can be obtained by comparing the results of a swarm of ellipsoids in
Santarelli et al. [112] with the results for spherical particles in a very similar configuration [108].
Particles of variable shape representing rising bubbles have been addressed in two ways by the present authors. A very
robust approach is to choose a spheroidal shape and determine the aspect ratio as a function of the rise velocity according
to an empirical correlation [120]. This is applied in Figure 7A,B. An expansion in spherical harmonics with the expansion
coefficients computed from the balance of surface load and surface tension was proposed in the same publication and
applied to wobbling ascending bubbles with good success. Figure 7C,D shows a result with this approach.
2.6 Collisions
An important aspect of densely laden flows is the adequate and efficient representation of collisions between particles.
This can be decomposed into several steps: identification of collision pairs, computing of the minimal distances between
surfaces, calculation of repulsive forces, and subgrid models to estimate the lubrication force.
The detection of potential collision pairs has to be done by an efficient search algorithm.
For this task, hierarchical algorithms are typically employed, such as a k-d tree [8,63]. Another method, the sweep
and prune approach, was discussed and employed by Yang et al. [148].
Once such a collision pair is detected, the minimal distance between points on the surfaces has to be computed. For
spheres, this is easily obtained from the radii and the position of the particle centers. For other shapes, even if relatively
simple and analytically described, this can become complex and requires specific algorithms. A comparative study of such
methods for the case of ellipsoids was recently conducted by Girault et al. [34].
When particle surfaces directly touch this can be due to sustained contact or due to a short-term collision. During the
latter, particles deform elastically and rebound. With typical material properties the time scale of this process is by two
orders of magnitude shorter than the characteristic fluid time scale [60]. Hence, fluid forces do not change during this
process and the collision itself behaves like a collision without the surrounding liquid, so that this process is termed dry
collision and usually computed disregarding any change in fluid loads.
For this task, two classes of approaches are known, termed soft-sphere and hard-sphere models. The former is compu-
tational simple and amounts to defining a linear or nonlinear spring-dashpot system [18]. The selection of the constants,
however, is delicate and the model requires extremely small time steps for realistic values. Two strategies can be followed
as a remedy. One is to employ the model with a smaller time step for the particle EOMs alone with, for example, 100–240
time steps within one fluid time step [64]. If for simplicity and efficiency the time step for the particles is chosen to be the
same as for the fluid one can reduce the spring constant. However, this is generally done ad hoc and may lead to loss of
realism and massive overlap of the numerical particles. Kempe and Fröhlich [60] developed the adaptive collision model
(ACM) where the dry collision is stretched over a number of fluid time steps. This can be done without iterations over

FRÖHLICH  . 9of26
the model constants [102] and also covers tangential collision forces. With small technical adjustments this model car-
ries over to sustained contact [10,62]. Similar models are used by other groups working on MPF [3,16,92], and analogous
ones have been proposed in the DEM framework [12,95].
A different approach to collision modeling is the so called hard-sphere approach [49]. Based on momentum conserva-
tion, this method uses the restitution coefficient between two bodies to determine the normal velocity after the collision.
When several bodies collide simultaneously, a collision network (LCP) must be solved before the velocity of the particles
can be estimated [38]. Tschisgale et al. [132] implemented such an iterative method, with later extension to sustained
contact [51].
Prior to direct contact of particle surfaces, the gap between them becomes too small for the lubrication flow between
the surfaces to be resolved on the Eulerian grid of the fluid. The resulting dissipative effect, therefore, has to be modeled
by a lubrication model. A distance typically chosen for employing such a model is d<dlub =2h. Since lubrication forces
diverge for surface distances approaching 0 some regularization is required to avoid discretization effects [10].Jainetal.
[51] recently devised a robust and efficient model avoiding any such problem by setting the lubrication force to a constant
for distances smaller than dlub and introducing a prefactor depending on the Stokes number built with the approach
velocity. Extensive validations for spherical and nonspherical particles were conducted and the model successfully used
in the large scale simulations discussed in Sections 6.3 and 6.4.
Another, more numerical type of model is substepping, which is to employ a separate, substantially reduced time step
for the particle motion with the fluid field unchanged during these substeps. While this can be run without lubrication
model and a simple collision model only [64] it also constitutes an approximation relaxing the fluid particle coupling and
generally mollifying the elastic collision.
In Section 4, applications with bubbles will be discussed, as these can be approximated as very light particles, that
is, without deformation or with very little deformation. If such bubbles collide vigorously or hit a wall, an appropriate
collision model must be devised. The approach, then, is to account for the physics of the deformation in this model, even
if the shape is treated constant throughout. This amounts to allowing some overlap, but with sound physical justification.
A very simple model is based on assuming a plane lamella between the two colliding bubbles, which lends itself to a
static balance of forces, as detailed in the supplementary material of Heitkam et al. [41] A model also accounting for
dynamic effects and justified by detailed physical arguments was proposed subsequently by Heitkam et al. [45], together
with validation by own experiments. The approach combines the advantages of assuming a constant bubble shape with
enhanced realism of collision modeling.
3PARTICLES IN MICRO-FLUIDICS
3.1 Motivation and physical characteristics
Micro-fluidic phenomena play an important role in a wide range of disciplines, among others in the study and analysis
of living cells responsible for particular metabolic and energetic processes. The analysis of living cells, such as early dis-
ease identification, requires the separation, sorting and enrichment of specific cells [155].Anexampleofthisprocessis
the identification of cancer tumor cells at very low concentrations in blood samples at early stages of the disease [93,145].
A similar process is used for the analysis and decontamination of algae reactors from invading species or bacteria [113].
Most of these cells are neutrally buoyant in the solutions where they are dissolved, so that sedimentation methods, sep-
aration methods like centrifuging, for example, are not effective. Micro-fluidics present an opportunity of using small
scale processes to separate and sort these particles. Contrary to membrane or filtering systems, which require stops for
cleaning or changes of the filter, micro-fluidics allows continuous separation of samples without the danger of the foul-
ing of the system [47]. Separation can be obtained using passive systems, making use of hydrodynamic forces, without
the extra complication of electric, acoustic forces or moving parts [151]. Furthermore, this is done without any labeling,
so that cells do not need to be dyed, dried or affected otherwise, which is prioritized in applications [35]. Overviews over
different cell sorting methods are available in the literature [15,125].
The basic principle behind the separation of particles in micro-channels is inertial migration, first observed in the
experiments of Segré and Silberberg [122]. These authors observed that particles transported in a pipe undergo a migra-
tion to a certain position in the cross-section located at roughly 0.6R,withRthe radius of the pipe, where they remained
throughout the entire downstream flow. Only much later, the physical mechanisms responsible for this migration were

10 of 26 FRÖHLICH  .
fully understood [28]. It is based on the balance of two forces, a lift force induced by the shear-rate driving the par-
ticle toward the wall and an opposing wall-induced force, which rapidly decays with the distance from the wall. In
micro-channels with rectangular cross section an additional migration is observed which is generated by a viscous lift
force resulting from particle rotation at low Reynolds number [106]. The component of the particle rotation vector normal
to the wall and the particle relative velocity generate a force toward the center of the wall. It moves the particles parallel
to the wall until they reach the center [154]. This results in the four commonly seen focusing positions in rectangular
channels.
By separating the bulk of the flow from the flow near the walls, where the particles are focused, the particle concentra-
tion in a sample can be increased. This, however, requires that the confinement ratio, dp∕H, the ratio between the particle
diameter and the smallest dimension of the channel, must remain higher than 0.07, otherwise the curvature of the flow
profile around the particle can be neglected [9]. The constraint, in fact, allows the separation of different sized particles,
where some particles focus near the walls while others are dispersed in the flow.
In an attempt to reduce the number of focusing positions in a channel, curved channels were introduced making use
of the Dean effect [9]. In a curved channel, the redirection of the flow generates centrifugal forces resulting in secondary
flow composed of two counter-rotating vortices, commonly known as the Dean flow [20]. The additional drag on the
particles due to Dean vortices is superimposed on the inertial migration, resulting in a single stable focusing position in
the channel, so that particles are easier to separate from the rest of the flow [9].
Although the basic principle of the separation process making use of the inertial migration and the Dean flow is
known, it varies strongly with the configuration used. Considering the wide range of applications the separation pro-
cess not only deals with different particle sizes, but also different shapes, flexibilities and concentrations, diluted in
Newtonian or non-Newtonian fluids. All these configurations affect the migration process and the associated separation
efficiency [5,15].
The study of the migration in different configurations still plays an important role in the design of such devices. Due
to the strong influence of inertia on migration Stokesian dynamics do not apply here[75]. Early studies with perturbation
methods can provide insight only for very low particle Reynolds numbers [28]. Empirical models are also very limited
and only apply for specific setups, sometimes even requiring DNS for their calibration [5,80]. Therefore, DNS is the state
of the art for the study of the migration process in micro-channels [1,58,76].
The setups used in experiments for the separation of particles [1,24,85,145] feature both straight and curved chan-
nels, with square or rectangular cross-sections. Width Wand height Hvary between 50 and 900 μm with channel lengths
between 20 and 100mm. These values are defined considering the range of cell sizes, with diameters dpbetween 2 and
40μm and the necessary confinement ratio dp∕H>0.07. Curved channels make use of spiraling geometries, with radii of
curvature,R, between 1 and 6 mm. The percentualvolume fraction, 𝜙,rangesfromverydilutedsolutionswhere𝜙<0.01%
to diluted solutions with 𝜙<2%. In cases involving blood samples, which contain a large number of flexible red blood
cells, hematocritic fractions up to 25% are used [145]. Most cases feature laminar flow, with the bulk Reynolds num-
ber, Reb=UbH∕𝜈between Reb=10 and Reb=300. The particle Reynolds number, Rep=urdp∕𝜈, based on the relative
velocity, ur, is usually not available from experiments. Alternatively, a modified particle Reynolds number is used, Rebp =
UbH∕𝜈(dp∕H)2,based on the bulk velocity and the size of the particles [116], typicallyassuming valuesbetweenRebp =0.5
and Rebp =3. For the flow in curved channels the Dean number is often defined as De =DhUb∕𝜈Dh∕2R=RebDh∕2R,
with values between De =0andDe =35, where Dhis the hydraulic diameter of the duct.
3.2 Straight channels
The simulation of the inertial migration in straight channels nowadays is standard for the validation and error assessment
of the method used [6]. Working with dimensionless quantities is common practice. Here, the height of the channel, H,
is chosen as the characteristic length, also in the bulk Reynolds number. The aspect ratio is W∕H,whereWis the width
of the cross section. The resulting geometry presents normalized height and width, which for the a square cross-section is
H∕H=W∕H=1. Since the length of the micro-channels is several orders larger than Dhperiodic boundary conditions are
imposed in streamwise direction, allowing the simulation of a shorter domain with length Lvarying between L∕H=2
and L∕H=4. This ratio depends on the case, in particular the particle size, to avoid the influence of the particle wake on
itself. The particles simulated have sizes between dp∕H=0.1anddp∕H=0.2. Furthermore, they are neutrally buoyant
𝜌p∕𝜌f=1, so that no gravity effects are present. The bulk Reynolds numbers used vary between Reb=10 and Reb=200,
so that the flow is laminar.

FRÖHLICH  . 11 of 26
FIGURE 4 Particles in straight channel with 0.03 vol%: Fluid velocity at cross-section of focused particle [40]
Thenumerical method described above was validatedfor the present type of applications with experimental results for
the particle migration [1,24,76] considering the final focusing position in the channel, resulting in deviations lower than
1% of the height of the channel [40]. An interesting research question concerns the perturbation induced by the particles
to the flow in different cross-sections of the channel. The perturbation velocity, in this case, is defined as 
u=u−u0,
where u0is the velocity in the channel without particles. In a straight channel, with a fully developed flow, the secondary
velocity us=(0,v,w)Talso corresponds to the perturbation velocity perpendicular to the streamwise direction, 
vand 
w.
Results for the secondary velocity are presented in Figure 4. Of physical interest is also the relative velocity between
particle and fluid, that is, the particle Reynolds number. If particles are geometrically resolved this is not immediate, as
the fluid velocity at the position of the particle is not directly available (see discussion in [108]). In the present case the
disturbance of the flow by the particles is small so that the flow without particles was used to determine the fluid velocity.
For the straight duct it was obtained from the analytical solution, for the curved duct it was obtained from a separate
single-phase flow simulation. For the cases simulated, the particle Reynolds number Repranged between 0.14 and 4.25.
3.3 Curved channels
A DNS of the particle migration in curved channels is of higher complexity and, therefore, not as common as DNS of
straight channels [6]. This results from the fact, that IBM based simulations typically use regular Cartesian grids [76].
Embedding a curved geometry requires a large domain which increases the computational cost by orders of magnitude,
although a significant amount of the domain is outside the channel [57]. To overcome this problem Hafemann et al.
[40] suggested an approximate model, which uses a coordinate transformation to toroidal coordinates. This generates a
representation of the channel through a regular rectangular grid in computational space instead of physical space. To
furthersimplify the complexity a constant radiusisassumed, with R≫Hand R≫W,sothatR∕(R+𝜁)≈1,where𝜁isthe
radial coordinate, with 𝜁=0 at the inner channel wall. Since Reb>10, and R>10, most of the newly introduced viscous
terms can be neglected. The remaining additional terms correspond to the Coriolis force and centrifugal acceleration
terms. With R≫H, the deformation of the particle shape by the transformation to computational space is also negligible,
so that the same superficial grid can be used as in straight channels.
The modeling of curved channels is also done in dimensionless form here, and uses the same settings as for the
straight channels above, with a rectangular cross-section and an aspect ratio W∕H=2. Additionally, the curvature radius
isnormalized by the characteristic length, resulting in values between R∕H=40 and R∕H=80. Combined with the range
of simulated Reynolds numbers this accounts for Dean numbers between De =2.3andDe =25.8. The whole method
was validated in Hafemann et al. [40] yielding good agreement between the experimental and the calculated focusing
position. Regarding the observed focusing behavior, it was shown that, along with the single focusing position near the
inner wall, in some cases symmetric focusing positions or even oscillating particles were observed due to the large Dean
numbers, as also found in early experiments [9]. A study on the influence of the particle concentration in the channel for
concentrations 𝜙∈{0.2%,1.5%,5%,10%} showed that particles preferentially focus around the so-called null-Dean-drag
planes, and not just at the inner side of the channel. These are the planes characterized by small secondary velocity, such
that the Dean flow does not generate any entrainment by means of a drag force on the particle. They are located around

12 of 26 FRÖHLICH  .
FIGURE 5 Focusing position and migration path over secondary flow in curved channel. (A) Spherical particle, (B) oblate particle, (C)
prolate particle. Secondary flow obtained from simulation without particles
𝜂≈0.25 and 𝜂≈0.75 in Figure 5.With𝜙=10%no considerable separation was present, although the highest fraction of
particles still was located at the null-Dean-drag planes.
3.4 Curved channels with nonspherical particles
Motivated by the low separation efficiency observed for algae in experiments [113] the same approach was applied for the
separationof nonspherical particles. When designing separationdevices, the experiments are usually conducted with arti-
ficial polystyrene spherical particles due to ease of manufacture and availability. When applied to real cells, the separation
efficiency of these devices tends to be markedly smaller [23,113,146].
These simulations were conducted with Reb=100 and R∕H=80, corresponding to the case that previously resulted
in a single focusing position in the cross-section with spherical particles. The nonspherical particles were modeled as
described in the previous section, without transformation of the surface mesh. Ellipsoidal shapes were used with their
main axes, termed a,b,andc. Three shapes were considered, spherical (a=b=c=dp∕2), oblate (a=b<c), and prolate
(a<b=c). To allow the comparison between shapes, all particles were given the same volume, and the aspect ratio of
the nonspherical particles was fixed to 3. A concentration of 𝜙=0.27%was considered, equivalent to 4 particles within a
domain of L∕H=3.
In each of the three simulations conducted with the different shapes, one of the four particles was selected to illustrate
a typical path. For convenience, this was a particle close to the outer wall. The resulting trajectories with the respective
focusing position are shown in Figure 5. The simulations reveal that different focusing positions are obtained for different
particle shapes. Although the path of a single particle is shown, particles started at different positions of the cross-section
move to the same focusing position. This leads to small particle trains. It is clear, that the nonspherical particles simu-
lated focus at symmetric positions located at or near the null-Dean-drag planes. As a result, the nonspherical particles
focus closer to the center of the channel in radial direction (𝜁), the prolates even more than the oblates. This may explain
the decrease in separation efficiency observed in experiments. When analyzing the migration velocity, it was observed
that the nonspherical particles exhibit a slower migration velocity within the inner half of the channel (𝜁<1), compared
to the spheres.
A further step was made by considering a swarm of poly-disperse and polymorph particles. This was motivated
by the experiments of Warkiani et al. [146], where curved channels were used to separate blood samples containing
puffed red blood cells (RBC) infected with Malaria from white blood cells (WBC) and platelets. Flexible red blood cells
are not present, so that the solution consists basically of the blood plasma and its rigid content comprised of infected
RBCs, WBCs, and platelets. Based on the values provided in the cited reference, the WBC cells were modeled as large

FRÖHLICH  . 13 of 26
FIGURE 6 (A) Particle position in the cross-section and secondary velocity for simulation with poly-disperse solution. (A) PDF of
large spheres, (B) PDF of medium oblates, (C) PDF of small oblates
spheres with dp∕H=0.22, the RBC cells were modeled as medium sized oblates, where a∕H=b∕H=3.5c∕H=0.09,
with deq∕H=0.14, and the platelets where modeled as small oblates with a∕H=b∕H=1.8c∕H=0.04. A total volume
fraction of 𝜙=5%was used in the simulations. The results in Figure 6show that the large spheres focus in the inner half
of the cross-section, along the null-Dean-drag planes, while the medium-sized oblates focus closer to the middle of the
channel, as observed for the nonspherical particles at lower concentration discussed above. The small oblates that repre-
sent the platelets are smaller than the separation threshold of dp∕H>0.07 and, therefore, are dispersed over the channel.
This simulation shows how the shape-based separation leads to different focusing position in a single device with a simple
geometry and illustrates the capabilities of the method for complex situations.
4BUBBLES
4.1 Motivation, physical characteristics, and modeling
Due to surface tension small bubbles in a fluid remain spherical as the curvature of the surface is high. In contaminated
liquid, surfactants accumulate at the bubble surface which may be represented in good approximation by a no-slip condi-
tion [7,19]. Hence, small bubbles can duly be treated as light solid particles of constant shape. The review of Ern et al. [26]
juxtaposes the trajectories of heavy and light particles and their path instabilities highlighting similarities and differences.
Numerical simulations are particularly precious when experiments are difficult. An example is the case for bubbles in
liquid metal, an issue relevant for metallurgy. Here, bubble surfaces are contaminated with oxides, so that, again, a no-slip
condition is appropriate. Liquid metals also offer the possibility to apply Lorentz forces via external electric and magnetic
fields. Under typical conditions this requires the solution of a Poisson equation for the electric potential and a solenoidal
condition for the electric current. This can be implemented observing that bubbles are transparent for magnetic fields and
nonconductive for electric fields [118]. The strength of the magnetic field is measured by the dimensionless interaction
parameter Nrelating magnetic forces to inertial forces.
4.2 Individual bubbles and bubble chains in liquid metal
The study of single bubbles in water is a classical topic with extensive literature while single bubbles in liquid metal have
received less attention. A review of some experiments and own simulations on this topic has been provided in [30],so
that only some key features are reported here.

14 of 26 FRÖHLICH  .
(A) (B) (C) (D)
FIGURE 7 Simulation of bubbles in liquid metal. (A) Instantaneous wake of an argon bubble with diameter Deq =4.6mm in eutectic
GaInSn at an arbitrary instant in time. Two iso-surfaces of the vertical vorticity component are shown, one for a positive, the other for the
same but negative value. (B) Same visualization of the corresponding simulation with a vertical magnetic field of strength N=1. (C)
Simulation of a bubble chain in a container of height 30Deq, just before a coalescence event at the position of the arrow. Gray scale
corresponds to magnitude of instantaneous fluid velocity in the center plain of the container. (D) Same simulation just after the event.
Pictures (A) and (B) from [30], pictures (C) and (D) from a video created by S. Schwarz
The flow around a bubble rising in quiescent liquid metal and its motion without and with the action of a magnetic
field was addressed by Schwarz and Fröhlich [118] describing the modification of the wake by Lorentz forces, as seen
in Figure 7A,B. The vortex structures in the wake are straightened which results in a straighter path and an increased
rise velocity for rise velocities above a certain threshold [153]. Gaudlitz and Adams [32,33] performed similar studies for
bubbles with higher departure from the spherical shape using a hybrid level-set particle method.
A further step is constituted by the simulation of bubble chains [117]. Here, a vertical magnetic field has the same
impact on the bubble trajectories as for a single bubble and, therefore, also on the interaction of bubbles through their
wakes, so that the entire chain is altered in its behavior. Furthermore, the magnetic field also impacts on the liquid in
regionswithout bubbles and reduces turbulent fluctuations, so that both effects contribute to a substantialmodification of
the entire flow. In these simulations the bubble shape was modeled with axisymmetric spherical harmonics, the collision
model of Heitkam et al. [43] was employed, as well as the coalescence model of Schwarz et al. [121]. The latter provides
a very realistic account of the coalescence process and is illustrated in Figure 7C,D by a configuration with the bubble
chain in the center of the container and no magnetic field [69]. The numerical results reported in this reference agree
very well with the corresponding experiments.
What makes this type of simulation particularly costly is the spread of time scales between the turbulence in the
bubble wake and the duration of a revolution of the container vortex which is the time scale relevant for obtaining average
flow data. The advantage of an IBM treating the bubble as an entity with a global model for the shape is the substantial
robustness this brings about, so that extreme density ratios can be handled.
4.3 DNS of bubble swarms and use for RANS models
The next step of complexity is constituted by disperse bubbly flows, a regime with comparatively low gas fraction, which
is of substantial relevance for process engineering [91]. Such cases were investigated in an upward vertical plane channel
flow of water assuming spherical bubbles at different void fractions and sizes. [108,109] Figure 8A gives an impression of
these simulations by a case featuring two bubble size classes. The different radii were chosen such that the small bubbles
have a tendency to accumulate near the walls, while the large ones accumulate in the center region, which was shown to
result from turbophoresis [108]. As the shape of the bubbles was imposed this migration is not a result of deformability
as observed in other studies.
The DNS data obtained from these simulations allowed to provide deeper physical understanding by investigating the
fluid velocity spectrum to extract the most relevant characteristic frequency [83], as well as the anisotropy of the velocity
fluctuations in its dependency on length scale [82].

FRÖHLICH  . 15 of 26
(A) (B) (C)
(D) (E)
FIGURE 8 Simulation of disperse bubbly flows in an upward vertical plane channel flow. (A) Swarm of bidisperse bubbles: 1.07%vol
large bubbles, 1.07%vol small bubbles with Dl∕Ds=1.47, resulting in Nl=456,Ns=1440 bubbles in the domain. Small bubbles in dark, large
bubbles in light gray. Color plots show instantaneous vertical velocity component in a vertical and a horizontal wall-to-wall plane. (B) Void
fraction in a monodisperse flow containing only the smaller bubbles of (A) at a global void fraction of 2.14%vol with comparison of DNS data
and result of a RANS simulation employing the newly generated BIT model. (C) Different contributions to the balance equation for the TKE
form the same computations. (D) TKE itself, obtained from DNS and RANS, with two additional curves resulting from models of the
literature, label TH with model of [129], label PC with model of [99]. (E) Comparison of source terms in the 𝜔−equation from the same
computations. Pictures (B)–(E) from [83]
These simulations also allowed to evaluate all terms in the balance equation for the turbulent kinetic energy (TKE)
[111]. This is not trivial, since essential contributions come from the regions at the bubble surfaces which are deli-
cate to evaluate. Once the individual contributions determined it was possible to create an improved model for the
bubble-induced turbulence (BIT) in a k−𝜔Euler–Euler RANS approach [83]. As an example, Figure 8B–E provides com-
parative data from the DNS of a monodisperse swarm and the corresponding k−𝜔Euler–Euler simulation employing
the new model term. The new expression was later on integrated in the baseline model of Lucas et al. [81] and applied to
various test cases of bubbly flows with good results [79].
4.4 Densely packed bubbles
Rising bubbles can accumulate at an upper wall or free surface generating a dense particle cluster. For small bubbles this
is so-called wet foam with comparatively large fluid volume fraction, as the densest packing of monodisperse bubbles
with perfect ordering results in about 74%vol void fraction. A related simulation setup is shown in Figure 9A with bubbles
modeled as light spherical particles, as discussed above. A constant downward drainage velocity through the bubble
cluster is a further parameter.
It is well known that crystalline arrangements form under such a condition. Astonishingly, however, local FCC struc-
tures are encountered much more frequently than local HCP clusters. The particular situation of the gradual generation
of the foam allowed to discover a simple, so far unknown geometric explanation for this phenomenon which is relevant
in many fields [41].
Figure 9C illustrates that the pressure drop over the accumulated cluster increases with drainage velocity until the
flowis strong enough to makebubblesfloat freely,so that pressure dropequals buoyancy.Application of a horizontal mag-
netic field increases the mobility of the bubbles and the pressure drop with the square of the magnitude of the magnetic
field [43].
A further step is accomplished by applying a horizontal electric field and a perpendicular horizontal magnetic field
to a container with liquid metal. The resulting Lorentz force counteracts gravity and can reduce or even revert the rise

16 of 26 FRÖHLICH  .
(A) (B) (C)
FIGURE 9 Bubble agglomeration in a downward drainage flow with horizontal magnetic field. (A) Configuration with periodic
boundary conditions in horizontal direction, (B) snapshot during the agglomeration, (C) pressure drop as a function of drainage velocity
without and with magnetic field. Pictures from [30]
(A) (B)
FIGURE 10 Bubble cluster under periodic shear. (A) Bubble positions at an arbitrary instant in time with color corresponding to
horizontal velocity of each bubble. (B) Horizontal velocity of the bubbles at an instant with maximum shearing velocity. Pictures from [42]
velocity of bubbles. An intricate issue, however, is that the insulating bubbles disturb the electric field creating particular
secondary currents. While a single bubble may still float, several bubbles interact with each other generating a permanent
unsteady motion [44]. With an appropriately generated magnitude of the external fields a swarm of bubbles canbe created
which floats instead of rising. It was then investigated how this can be developed further by simulating the phenomenon
in a cylindrical container with electrodes along the walls [46].
The last example of densely packed bubbles reported here is related to foam rheology and concerns the response of
a dense sample of spherical bubbles of diameter 1.1–1.3mm in water to static and periodic shear. Experiments on this
topic would be extremely difficult, since in laboratory gravitation results in drainage making the setting anisotropic and
inhomogeneous. Furthermore, wall effects would introduce possible slip and particular spatial structure. Creating a con-
figuration suitable to investigate this topic reliably and at manageable cost constitutes an achievement [42]. An important
quantity in this circumstance is the osmotic pressure resulting from the tendency of an emulsion or a foam to take in pure
liquid through a membrane. If a certain level of pressure is applied to the membrane, the osmotic pressure, this can be
suppressed [101], so that the foam experiences this pressure throughout. In the setup devised the membrane was replaced
by judiciously constructed forces applied to certain bubbles that mimic the osmotic pressure in an infinitely extend foam.
With steady shear an increase of osmotic pressure and a decrease of gas fraction was observed. The yield stress was
found to increase by orders of magnitude and was described with appropriate fits. Figure 10A shows the configuration
with oscillatory horizontal shear applied at the top and the bottom in opposite direction varying the osmotic pressure and
the frequency of oscillation. Figure 10B displays the resulting bubble velocity and highlights the existenceof shear bands
at the top and the bottom. Evaluation of the data allowed to characterize the role of the fluid inertia enclosed between the
bubbles.

FRÖHLICH  . 17 of 26
(A) (B)
FIGURE 11 Three-phase flow with phase-resolved bubbles and point particles. (A) Snapshot of vertical velocity in a cut located in the
center region of the flow. Bubble surfaces are not displayed, point particles not to scale. (B) Contribution of the shear rate to the collision
frequency. Pictures from [68]
5THREE-PHASE FLOW WITH BUBBLES AND PARTICLES
In a number of situations bubbly flows contain additional particles. This is the case with flotation used to remove inclu-
sions from liquid steel and to separate valuable minerals from gangue components in a mine. The inclusions in the former
case are tiny particles, actually drops, lighter than the melt and orders of magnitude smaller than the bubbles. With the
same techniques as used above this provides the possibility to combine PR-DNS with PM-DNS for the inclusions. In
Kroll-Rabotin et al. [68], a setup like the one in Figure 8A was supplemented with point particles in one-way coupling
(Figure 11). These require an own data structure and the solution of own EOMs with closures for forces such as drag,
lift, and so forth. Their trajectories in the turbulent flow agitated by the rising bubbles were computed to determine the
number and the conditions of collision between inclusions. These are prone to modify the size distribution of the inclu-
sions by generating larger aggregates which are easier to separate from the melt. It is actually the limited number of such
collisions which is the bottleneck for the removal of inclusions. In the cited work the DNS data were used to generate sta-
tistical information about the collisions of the type shown in Figure 11B. These allow to provide expressions for collision
kernels to be used in macro-scale steady or unsteady RANS simulations of the entire ladle.
6SEDIMENT TRANSPORT
6.1 Motivation and physical characteristics
The transport of heavy particles in a liquid along a horizontal wall is an important class of problems, encountered with
sediment in rivers, lakes, oceans, but also in technical installations, such as pipelines for hydraulic conveying, and so
forth. Reliable simulation of these processes is key to predicting and controlling natural influences, as well as technical
measures. Among others, bedload transport of sediment is a central factor for the development of bedforms of rivers.
Bedload transport can be decomposed into three different stages: entrainment of particles, movement in the flow, and
deposition, and it is the complex interplay between driving and attenuating mechanisms which governs each of them, as
reviewed in [22,149]. Beyond the Reynolds number of the bulk flow and the shear Reynolds number, the Shields number
Sh =u2
𝜏∕[(𝜌p−𝜌f)gdp],whereu𝜏is the friction velocity, is a decisive quantity. It relates the mean fluid shear stress at the
bottom, 𝜏w=𝜌u2
𝜏, potentially mobilizing a particle, to the gravitational forces retaining a particle in place.
An overview over numerical approaches for sediment transport has recently been compiled by Vowinckel [137]. Clas-
sical simulation approaches generally treat the flow separately and employ a specific model equation for the transport
of bedload concentration [104]. More recent methods are based on transport equations for the mean flow and the parti-
cle concentration over the entire domain [14]. All these approaches treat the macro-scales and need closures for erosion,
transport, and deposition. Such models have been developed experimentally in the past, but with technical difficulties
and related uncertainties. Over the last decade, highly resolved simulations have come into play to fundamentally investi-
gate sediment transport on the micro- and meso-scale. Point particle methods are difficult to apply because they still need
closures for particle-fluid interaction in a situation with very dense particle loading where this is delicate. Hence, PR-DNS
are required to provide fundamental understanding under well controlled conditions, albeit at high computational cost.

18 of 26 FRÖHLICH  .
FIGURE 12 Instantaneous particle distribution with 13500 mobile particles on top of a rough bed made of another 13500 spheres
with the same diameter. Iso-surfaces of fluid fluctuations in blue u′∕Ub=−0.3, particles in yellow: fixed, white: up<1.5u𝜏, black: up>
1.5u𝜏. From left to right the Shields number is, respectively, 0.75, 1.18, and 1.82 times the critical value. Images from [138,141]
A typical setup of such a simulation features a quadrilateral computational domain with periodic boundary conditions
in streamwise and spanwise direction, as shown in Figure 12. The bottom is a smooth wall or a rough wall made of fixed
particles, while the upper boundary represents a flat free surface. In such a PR-DNS flow depth has to be limited for cost
reasons, so that characteristic numbers have to be chosen carefully to capture the desired effects. A further difficulty of
this type of configuration is that the time scale for the particle arrangement is much longer than characteristic fluid time.
This makes simulations expensive.
6.2 Simulations of sediment transport with spherical particles
A large body of literature exists on the simulation of rough surfaces composed of particles, such as [126],toinvestigate
the resulting vortical structures. As a preliminary step to erosion, Chan-Braun et al. [13] investigated the unsteady fluid
forces acting on fixed particles to provide information on the unsteady flow features potentially mobilizing them.
Mobile spherical particles traveling over a rough bed of spherical particles were simulated by Vowinckel et al.[140] in
a large domain. Small to moderate particle loading was considered using the ACM collision model shown to be very well
adapted for this setup [62]. Even with domains of that size a resolution of individual particles with 22 points per diameter
could be achieved.
Sample results are provided in Figure 12 addressing the effect of varying the nominal, a priori determined Shields
number from 0.75 to 1.18 to 1.82 times the critical value Shc=0.034 for this range of particle Reynolds numbers.
The case with low Shields number in Figure 12 was analyzed by Vowinckel et al. [139] to elucidate which events
mobilize the particles in such a setting. The data show that mobilization is created by strong sweep events, but
that—unexpectedly—collisions with traveling mobile particles constitute a decisive trigger, preceding practically all
events in this case, which is of relevance to understanding mobilization.
Beyond the fact that for larger Shields number more particles are entrained it is obvious that spanwise clusters form
with higher mobility. Less apparent in these figures is that between the spanwise clusters streamwise clusters, so called
ridges, tend to occur at a distance of about the water depth, as discussed in [138,140] and recently investigated by Scherer
et al. [114]. At reduced particle loading the structure of the underlying rough wall may impact on the transport of particles
over the roughness elements. This was investigated in Jain et al. [56] by disturbing the regularity of the fixed particles and
was found to increase the erosion rate, but with quantitative not qualitative consequences forthe particle motion.
Kidanemariam and Uhlmann in a series of works [64,65] also addressed pattern formation on a sediment bed of
monodisperse spherical particles of thickness ≈25Dand investigated the formation of dunes. The same method was used
to simulate further configurations, like oscillating flow, in subsequent publications.
An example of how PR-DNS results can be postprocessed and used for analysis is provided by double-averaging such
data [94,142,143]. This approach yields information about the impact of particle clusters on fluid statistics and can then
be used for statistical turbulence modeling.
6.3 Sediment with monodisperse nonspherical particles
In recent studies of the present authors the impact of particle shape on particle transport and statistics of continuous
and disperse phase was investigated. Four shapes were considered, sphere, oblate, prolate and an intermediate ellipsoid,

FRÖHLICH  . 19 of 26
(A) (B)
(C) (D)
(E)
(F)
FIGURE 13 Snapshots of the bedforms obtained with the different particle shapes viewed from the top at an arbitrary instant in time:
(A) Spherical particles, (B) prolate particles, (C) oblate particles, and (D) Zingg ellipsoids, (E) wall-normal mean streamwise velocity, (F)
wall-normal mean angular velocity in spanwise direction. The color scale in (A)–(D) corresponds to the elevation of the particle center.
Pictures from [53]
termed Zingg ellipsoid. This was done first with small particle loading [52] then with large loading [53] at a bulk Reynolds
numberof Reb=3010and Reb=3432, respectively,employing the hard-sphere collisionmodelofJain et al. [51].Figure13
shows representative results.
The essential finding is that the particle shape substantially modifies all phases of transport—erosion, movement, and
deposition—and as a result the pattern formation. It turns out that spherical particles assume large spanwise rotation.
Oblates are hindered in this respect due to their shape, Figure 13F, so that the particles exhibit a preferential orientation
depending on their shape [52]. As a result, spherical particles travel faster on top of a densely packed sediment and gen-
erate lower bottom shear stress with the same bulk flow. In contrast, oblate particles due to their shape tend to generate
a looser packing (not shown here) and hover above the sediment bed at lower velocity (Figure 13E). This yields reduced
deposition with less tendency to pattern formation compared to prolates and Zingg ellipsoids (Figure 13A–D). A detailed
analysis of the different jumping motion as a function of particle shape in these simulations was also conducted [50].
Zhang et al. [152] reported a similar study on ellipsoidal particles but with smaller domains and somewhat different
conditions.
Concerning particle–particle interaction during deposition, the effective restitution coefficient depends on the dry
restitution coefficient of the solids, but also on the surrounding fluid, that is, the Stokes number. Here, the particle shape
plays an important role creating an additional dependency on the instantaneous orientation. For example, an oblate
particle with an aspect ratio of b∕c=2 colliding with wall at an angle 45◦may settle even at a Stokes number of 75,
whereas a spherical particle would rebound with an effective restitution coefficient of 0.7 [51]. This is further complicated
by the arrangement of the resting particles possibly attenuating the rebound. At the same time, the collision may mobilize
resting particles [135], with an influence of the shape as well.
Due to the shape-dependent interactions with the surrounding liquid, as well as different arrangement of the immo-
bile particles, different particle shapes create different bottom shear stress with the samebulk velocity. The shear stress,
however, is a central quantity in modeling of such flows, for example when using the Shields criterion for mobilization.
6.4 Sediment with polymorph and polydisperse particles
Apart from nonspherical shape, an important feature of natural sediment particles is their size distribution. In a further
simulation a natural sand sample was reproduced by appropriately choosing 4 representative ellipsoidal shapes and 7
size classes with the corresponding frequency [54]. Hence, the particle sample combines different shapes (polymorph)
and different sizes (polydisperse) and was subjected to the same conditions as in Figure 13. This is the first simulation to
represent natural sediment in a PR-DNS at such a high level of realism.

20 of 26 FRÖHLICH  .
(A) (B) (C) (D)
FIGURE 14 Simulation with polymorph sediment. (A) Instantaneous snapshot particle distribution and flow field. Mobile particles
colored according to their diameter as shown in the top left scale, back plane shows instantaneous streamwise fluid velocity. (b) Mean
sediment transport rate over the Shields number. Crosses: Experimental data of Meyer-Peter and Müller [86] for natural gravel. Dotted line:
Corresponding empirical formula. Symbol: Present simulation. (C) Wall normalprofiles of horizontally averaged volume fraction of each
particle size class, 𝛼s,withs=Deq ∕Dav at t=0. (D) The same for t≈6000Dav∕Ub
A snapshot of the moving sediment is shown in Figure 14 with particles of different size colored from black to white.
Large spanwise structures with preferential accumulation of large particles are seen, as well as small particles at higher
elevations. Particles of different sizes segregate vertically in a moving sediment bed, as known from nature and exper-
iments, with small particles moving deeper and larger ones pushed upwards. [29] Whether this is due to the Brazil
nut effect or different reasons presently is a matter of research. This behavior was indeed observed, as highlighted in
Figure 14C,D. To sufficiently capture the phenomenon this simulation had to be conducted for 6000 bulk time units. A
recent study of the effect was conducted by Rettinger et al. [103], but with spherical particles. The simulation data with
polydisperse sediment were also employed to describe the propagation of dunes in this context [55]. Further analyses will
be reported elsewhere.
7CONCLUSIONS
Phase-resolving DNS for multiphase flows in recent years has become a very active field of research, providing insights
into phenomena which could not be addressed otherwise. Various approaches are employed depending on the specific
phenomena to be resolved, and algorithms are continuously being improved. The present text compiles an overview
over important issues, mostly based on own work, regarding both improvements of the numerical method, as well as
contributions of such analyses in their respective application field.
It is important to observe that freely moving bodies are algorithmically much more challenging than fixed bound-
aries. For the latter larger effort can be spent on creating an optimal grid, for example, using local grid refinement.
For moving objects this becomes more costly, so that in many approaches a homogeneous isotropic background grid is
employed. With moving objects, however, grid locking and oscillatory forces may occur and must be avoided by the use
of appropriate algorithms, such as the smoothed delta function employed in many IBMs. With the methods discussed
in this article no such issues were observed by the authors. Reformulation of the coupling between phases allowed to
significantly improve the IBM by a semi–implicit approach with better stability, now allowing arbitrary density ratios
between disperse and continuous phase.
The interaction between moving objects, for example by collisions, also poses a difficult task when the coupling with
the fluid solver is considered. The neighbor recognition algorithms, the identification of surface distance and collision
points, as well as the treatment of analytically discontinuous forces introduced by such collisions was addressed using
both soft and hard sphere models for rigid bodies of different shapes. These models, extended for multiple simultaneous
collisions, allow the simulation of densely packed spherical and nonspherical particles, as encountered in the case of
sediment transport, for example.
Another extension is the use of spherical harmonics for representing the particle shape, which can then be extended
to flow-dependent deformation [70,120]. This not only drastically increased the complexity of the model but also requires
a robust account of the coupling to deal with the constant changes of the particle shape.

FRÖHLICH  . 21 of 26
PR-DNS provide fundamental physical understanding of very complex phenomena and allow to conduct controlled
numerical experiments by using idealized boundary conditions or by switching on and off certain components of the
physical model. The price to pay is high computational cost and algorithmic complexity. On the other hand, it provides
very detailed access to both fluid fields and particle behavior in environments where this would be impossible in an
experiment, such as for micro-channels or liquid metal flows, for example. The data obtained by such simulations also
helps to address specific problems, such as the migration behavior of nonspherical particles in curved micro-channels.
A second major role of these simulations is to develop closures for meso-scale and macro-scale simulations with several
examples highlighted in the sections above. For such purposes techniques of machine learning may be employed with
benefit, as done by Tajfirooz et al. [128], for example, and many similar studies.
The phase-resolving DNS of the type presented here still poses several challenges in what concerns the development
of algorithms and the modeling of complex physical effects. By construction, the algorithms are heterogeneous, as they
combine field-based data of the continuous phase with very localized data for the disperse phase. Load balancing becomes
an issue when the disperse phase is distributed heterogeneously over the domain, possibly requiring specific frameworks.
Also, the optimal use of heterogeneous computer architecture, like a combination of CPUs and GPUs, for example, is an
active area at the frontier of computer science and physics.
The advantage of the IBM is its simplicity, versatility and easy applicability for moving objects. The downside is a
reduced order of the approximation around the phase boundary. Inward retraction of forcing points is claimed to solve
the issue [11] but without mathematical proof being available. Here, further research should improve the understanding.
Second order can be obtained by particle-conforming overset grids [144], so far realized for spherical particles. Another
approach is the cut-cell technique [115]. Higher order methods for the task have been devised as well, such as the XDG
method of Kummer [73], involving substantial algorithmic complexity, however. Other approaches such as the VOF
method and further methods have also been developed and employed for the task. The number of different algorithms
coupling fluid and particles in PR-DNS is large. There does not seem to be any best method as such. As often, the most
suitable method depends on the physical problem considered, the desired output, and the desired accuracy.
Extensions beyond the mechanical coupling between fluid and solid particles have been and are pursued by the com-
munity. Non-Newtonian fluids can be treated without modifying the coupling strategy. Adhesive forces between particles
can be introduced without further change. Consideration of deformable particles, drops or bubbles is nontrivial, with the
subsequent challenge of describing coalescence and break up with physical accuracy. Another direction is energy trans-
port, where IBM techniques can be employed directly to force the temperature or its gradient at the surface of a particle,
and so on [110]. The number of studies on PR-DNS grows rapidly, and broader applications are being developed.
ACKNOWLEDGEMENTS
The research described was undertaken in collaboration with highly motivated students and coworkers, as apparent from
the cited publications. All of them are acknowledged for stimulating discussions over the years. Jochen Fröhlich acknowl-
edges funding by DFG through the projects ESCaFlex (634058), FLOTINC (625209), and Sediment (570058). Thomas
E. Hafemann was funded through the ESF group CoSiMa (project 100231947). Ramandeep Jain acknowledges funding
through a scholarship of the State of Saxony (No. L-201535). ZIH at TU Dresden is gratefully acknowledged for support
and for providing computing time. Open access funding enabled and organized by Projekt DEAL.
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