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A sharp MLS penalty immersed finite element method for fluid-structure
interaction of highly deformable slender body in turbulent flow
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DOI: 10.1080/19942060.2023.2300451
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A sharp MLS penalty immersed finite element
method for fluid-structure interaction of highly
deformable slender body in turbulent flow
Ehsan Akrami, Mathieu Specklin, Rafael Torrecilla Rubio, Robert Connolly,
Ben Breen, Stefan Berten, Mark Kehoe, Abdulaleem Albadawi & Yan Delaure
To cite this article: Ehsan Akrami, Mathieu Specklin, Rafael Torrecilla Rubio, Robert Connolly,
Ben Breen, Stefan Berten, Mark Kehoe, Abdulaleem Albadawi & Yan Delaure (2024) A sharp
MLS penalty immersed finite element method for fluid-structure interaction of highly
deformable slender body in turbulent flow, Engineering Applications of Computational Fluid
Mechanics, 18:1, 2300451, DOI: 10.1080/19942060.2023.2300451
To link to this article: https://doi.org/10.1080/19942060.2023.2300451
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ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
2024, VOL. 18, NO. 1, 2300451
https://doi.org/10.1080/19942060.2023.2300451
A sharp MLS penalty immersed finite element method for fluid-structure
interaction of highly deformable slender body in turbulent flow
Ehsan Akrami a,b , Mathieu Specklin c , Rafael Torrecilla Rubioa , Robert Connollyb , Ben Breenb , Stefan Bertend ,
Mark Kehoeb , Abdulaleem Albadawib and Yan Delaure a
a School of Mechanical and Manufacturing Engineering, Dublin City University, DCU Water Institute, Dublin 9, Ireland; b Product Development
Department, Sulzer Pump Solutions Ireland Ltd., Wexford, Ireland; c Arts Et Metiers Institute of Technology, CNAM, LIFSE, HESAM University,
Paris, France; d Sulzer Ltd., Winterthur, Switzerland
ABSTRACT
ARTICLE HISTORY
This paper presents a new computational approach to simulate challenging fluid-structure interactions (FSI) between fluids and slender deformable structures. Key innovations address limitations of
standard immersed boundary methods, including spurious forces, stability at low density ratios, and
accuracy at high Reynolds numbers. The method couples a sharp interface immersed boundary technique with detached eddy simulation turbulence modelling to enable precise FSI for high Reynolds
number flows. A strong coupling partitioned algorithm stabilized by Aitken relaxation significantly
enhances stability for low density ratios down to 1. A moving least square compact support domain
approximation reduces spurious oscillations from moving geometries while providing second-order
accuracy. Adaptive mesh refinement imposes jump conditions on slim deformable bodies and minimizes grid leakage. The proposed method is evaluated on three conventional FSI benchmarks and
four experimental cases, confirming its robustness, accuracy, and stability in low and significantly
high Reynolds numbers. A more complex fluids engineering case is considered last to test the solution under challenging conditions with fast moving solid boundaries and fast flowing fluid. A thin
deformable membrane is forced through a submersible centrifugal pump under standard operating
conditions. The solution is shown to produce a stable solution with good collision handling ability.
Received 3 May 2023
Accepted 25 December 2023
1. Introduction
Fluid-Structure Interaction (FSI) is a Multiphysics phenomenon widely found in nature and engineering across
a broad range of areas, including, for example; biological flows with blood flow around heart valves, vocal folds
in Larynx, aquatic locomotion, flying birds and insects,
aircraft and spacecraft aerodynamics or fluttering and
buffeting of structures including bridges, buildings and
wind turbines. Many studies have been dedicated to the
development of reliable models for the simulation of such
FSI problems. The intrinsic complexity of this type of
coupled structural and fluid systems, means, however,
that no single method has emerged as a universal solution. In particular, it has proven difficult to model the
dynamics of slender bodies subjected to large nonlinear
deformations. There are several factors that affect the stability of the solution, but density difference between the
fluid and the immersed solids has been found to be the
key limiting factor in the present study.
KEYWORDS
Computational fluid
dynamics; finite element
method; fluid structure
interaction; sharp interface
immersed boundary method;
moving least square; pump
blockage
Generally, three main numerical simulation techniques have evolved for FSI problems: Meshless, bodyconforming grid and non body-conforming grid methods. Meshless methods rely on a particle-based approach
and have been successfully applied for the study FSI with
isotropic or composite structures (Gotoh et al., 2021;
Khayyer et al., 2022). Arbitrary Lagrangian-Eulerian
solutions rely on a body-conforming mesh. Solid deformations can be handled via grid-stretching when deformations remain moderate but require computationally
expensive remeshing in cases of larger deformations
(Jiao & Heath, 2004; van Loon et al., 2007). In addition, an extra set of equations is essential to model
the grid motion and resolve the boundary movement.
On the other hand, by resolving the immersed boundary, ALE solutions allow for higher accuracy modelling
of the fluid-solid interface region, but very few solutions have been published in cases of slender deformable
bodies (Förster et al., 2006; Namkoong et al., 2005;
CONTACT Ehsan Akrami
ehsan.akrami2@mail.dcu.ie, ehsan.akrami@sulzer.com
School of Mechanical and Manufacturing Engineering, Dublin City
University, DCU Water Institute, Stokes Building, DCU Glasnevin Campus, Collins Avenue Extension, Dublin 9, D09 DD7R, Ireland;
Product Development
yan.delaure@dcu.ie
School of
Department, Sulzer Pump Solutions Ireland Ltd., Whitemill Industrial Estate, Wexford Y35 YE24, Ireland; Yan Delaure
Mechanical and Manufacturing Engineering, Dublin City University, DCU Water Institute, Stokes Building, DCU Glasnevin Campus, Collins Avenue Extension,
Dublin 9, D09 DD7R, Ireland
© 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits
unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow
the posting of the Accepted Manuscript in a repository by the author(s) or with their consent.
2
E. AKRAMI ET AL.
Sawada & Hisada, 2007). The alternative Immersed
Boundary Methods (IBM) (Peskin, 1972) rely on nonconforming grids. The fluid flow is solved on an Eulerian
mesh while immersed solid boundaries are accounted for
in the fluid flow equations by a momentum source term.
The immersed solid structure is modelled on a Lagragian grid which can move over the Eulerian grid. The
interaction between grids requires that data is mapped
via some form of interpolation. The method is highly
flexible and has been widely used. Penalty methods are
one form of IBM in which the immersed structure is
considered as a porous media with a very large permeability (Arquis & Caltagirone, 1984). Angot et al. (1999)
have reported that the penalty formulation can satisfy
the Dirichlet boundary condition in the incompressible
Navier-Stokes equation. A massive penalty method was
first proposed by (Y. Kim & Peskin, 2007), and a solution
involving higher-order of spatial discretization has been
introduced by (Introïni et al., 2014). Employing an IBM
to simulate FSI problems has several limitations connected with the accuracy of the boundary layer around
immersed object, spurious forces in moving boundaries
problems, low-density ratio of the solid to fluid (W.
Kim & Choi, 2019), and high Reynolds (Re) numbers.
Incorporating recent advancements, (Proskurov et al.,
2022) explores efficient fan tone shielding simulations
for unconventional engine installations. (Adeeb & Ha,
2022) investigated the impact of streamwise displacement on tandem oscillating bluff bodies using a GPUbased immersed boundary – lattice Boltzmann approach.
Furthermore, (Hong et al., 2021) introduced a ghost-cell
IBM for finite and zero thickness bodies in large CFL
numbers.
Methods for information projection between the fluid
and solid domains can be classified into sharp and diffuse
interface methods. The diffuse interface methods spread
the effect of immersed solid on the fluid over a finite
thickness layer. The coupling is not intended to resolve
hydrodynamic stresses at the immersed boundary but
rather impose a coupling condition based on fluid velocity. The diffuse interface can improve stability not only
by decreasing the stiffness of the system matrix system
but also by damping spurious force oscillations typically
caused by the change in a cell state between fluid or solid
as an immersed interface leaves or enters a cell. This,
however, comes at a cost, since the method blurs the
fluid-solid interface and cannot resolve the jump condition in fluid’s velocity and pressure cannot be captured
across the immersed body and is at most second order
accurate for very low Reynolds number. The lack of precise interface definition of the immersed body surface
also makes it difficult to implement turbulence wall modelling. In contrast, second-order of accuracy is possible
with a sharp interface IBM by resolving the jump condition in the velocity and pressure fields. In the sharp
interface context, the jump conditions at the immersed
boundaries are reproduced by local velocity and pressure
reconstructions. An additional challenge, especially with
slender bodies, comes with potential large grid size differences between fluid and immersed structures which
causes data leakage and effectively create discontinuous
or porous-like bodies, and has been linked to instabilities (Saadat et al., 2018). Nestola et al. (2019) explored
a Variational transfer (L2 projection) technique between
fluid and solid components, resulting in the development
of a modular and adaptable coupling approach utilizing a piecewise affine mesh structure. (Hsu et al., 2015)
used isogeometric analysis to discretise both the structure and fluid domain for bioprosthetic heart valve using
through design. There is evidence that a grid size ratio
of approximately two will induce such leakage (Liu et al.,
2006). Adaptive Mesh Refinement (AMR) schemes help
to address issues of computational cost and accuracy by
allowing local grid refinement to control, for example,
the grid size ratio. The embedding of multiple grids can
be used to activate AMR selectively. In addition, several
treatments have been proposed to alleviate the spurious force oscillations expected with moving or deforming fluid-structure interactions. One of the proven techniques for reducing the spurious force relies on Moving Least Square (MLS) data mapping techniques (Haji
Mohammadi et al., 2019) in sharp interface context.
Two primary coupling schemes have evolved depending on whether the immersed boundary conditions are
calculated from the previous or the current time step. The
coupling is called explicit or Loose whenever the boundary condition is determined from the previous time step.
Loose coupling is a straightforward, efficient approach
since only one sub-iteration is needed for marching in
the time, and it is a sequentially staggered algorithm. This
efficiency comes at a cost. The kinematic and dynamic
quantities balance cannot be satisfied on the interface
where spurious energy is generated. The fluid effectively
acts as an extra mass on the immersed body resulting in
an added-mass effect (Causin et al., 2005). The stability
of the solution has been shown to strongly depend on
the density ratio of the solid and the fluid. The lower the
body relative density ratio, the more sensitive the solution
is (Piperno et al., 1995), and below a certain threshold,
the structural solver experiences a stiff system of equations and convergence can quickly fail (Kassiotis et al.,
2011). Uhlmann (2005) shows that loose coupling can
remain stable down to a density ratio of 1.2 in problems of rigid body motion. The kinematic and dynamic
balances at the interface, particularly with high acceleration in large deformation FSI remains difficult to resolve.
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
Implicit coupling, generally referred to as strong coupling uses interfacial data from the current time step and
requires iterative corrective steps. Strong coupling techniques can reduce the energy imbalance at the interface
and lead to a stable and robust solution in low-density
ratio, however, the computational cost increases (W. Kim
& Choi, 2019).
Another critical limitation of IBM relates to its inability (without very significant costs) to tailor the mesh to
the underlying flow features at the fluid solid interface.
This is particularly important at high Reynolds numbers
when the suitability of turbulent models become intricately dependent on the mesh ability to resolve turbulent features, including transition in the boundary layer
(Georgi Kalitzin & Iaccarino, 2003). In high Reynolds
flows, errors in velocity and displacement calculation can
lead to unrealistic structure deformation and even drastic distortion of the deformable structure grid (Zhang,
2017) when the velocity and displacement are calculated by field interpolation. Further developments have
occurred in order to implement the turbulence modelling in IBM, such as Reynolds Averaged Navier Stokes
(RANS) (Georgi Kalitzin & Gianluca Iaccarino, 2002),
Large Eddy Simulation (LES) (Georgi Kalitzin & Gianluca Iaccarino, 2002; Ma et al., 2019; Roman et al., 2009),
Detached Eddy Simulation (Specklin & Delauré, 2018).
Gilmanov et al. (2015) introduced a Curvlinear IBM
method for turbulent flow using LES to model thin shell
structures which was subsequently adapted to model
bicuspid aortic valves by Finite Element Method (FEM)
(Gilmanov & Sotiropoulos, 2016). In a recent study,
(van Noordt et al., 2022) introduced a hybrid central
upwind flux reconstruction scheme aimed at enhancing both accuracy and stability in wall-modeled Large
Eddy Simulation (LES) within high-speed flows. Precise IBM with wall-resolved simulation requires adequate
wall grid refinement to capture the viscous sublayer. This
can create significant challenges with practical engineering problems at high Re number when the viscous length
scale becomes much smaller than the integral length
scale.
The research presented in this article addresses several significant challenges in simulating the interaction of arbitrary thickness bodies, particularly slender
deformable bodies, with high Reynolds incompressible
fluid flows featuring low-density ratios. The key challenges motivating our approach include overcoming
issues arising from the low interface resolution inherent to IBM, reducing spurious force oscillations, handling low-density ratio and high Reynolds number flows,
and achieving accurate predictions of solid behaviour. To
tackle these challenges, we’ve developed a comprehensive
methodology that combines the Imersed Finite Element
3
Method (IFEM) to account for mass and volume occupancy of the body within the fluid using a 3D quadratic
Finite Element Model with nonlinearity in both geometrical and material aspects. The deformation, strain,
2nd Piola-Kirchhoff, and Cauchy stress of the deformable
body are obtained from this FEM model. Additionally,
our IBM fluid solver adopts a sharp interface penalty
approach integrated into the OpenFOAM open-source
FVM fluid solver, featuring the PISO-SIMPLE pressurevelocity coupling algorithm and DES turbulence modelling (Specklin & Delauré, 2018). To enhance stability
and minimize spurious force oscillations, the Eulerian
grid was adapted by an AMR with arbitrary levels of
local refinement. A MLS compact support domain (Haji
Mohammadi et al., 2019) with a slender body jump
condition was included to map fluid stresses on the
Lagrangian solid points and reconstructs solid velocity
in the fluid domain within the sharp interface framework. Both approaches were adopted to mitigate the issue
of volume leakage commonly associated with IB methods. Importantly, our approach incorporates a Penalty
method with a sharp IFEM and MLS approximation,
marking the first time such a combination has been
used to enhance stability in low-density ratio scenarios
and minimize expected spurious force oscillations associated with partitioned coupling. The structural and fluid
solvers are coupled through a fixed-point strong coupling approach, and the convergence rate of the coupling
is accelerated using an Aitken technique with underrelaxation factor optimization (Irons & Tuck, 1969). The
suitability and effectiveness of the solution presented are
assessed through comparisons with three benchmarks
and experimental measurements for additional test cases.
The paper is organized into three main parts. In
section 2, a comprehensive description of the theoretical and computational framework is presented. Section
2.1 is dedicated to the fluid model. The Penalty IBM
implementation of the fluid solver is introduced before
the velocity correction at the interface, the FVM discretization and finally, the IBM treatment of turbulence.
The governing equations of the deformable body and
Mooney-Rivlin model, and the FEM spatial and Newmark temporal discretization are discussed in section 2.2.
The Strong FSI algorithm is explained in section 02.3
with specific emphasis placed on the MLS data mapping
approach. The contact model and AMR technique are
presented after that. Section 2 concludes with a flowchart
to illustrate the entire framework of the FSI developed
algorithm. Numerical results from the 7 test cases are
discussed in section 3. Cases considered are the two conventional Turek Hron cases as well as additional forced
oscillations tested for the present study. The simulation
of a deformable body forced through a centrifugal pump
4
E. AKRAMI ET AL.
is considered to provide for more challenging conditions
with strong interaction between the deformable slender
body and fast-moving boundary walls. This final case is
presented in Section 3.5. Appendix A provides further
mathematical details of the FEM discretization used in
section 2.2.
2. The mathematical model
2.1. Fluid governing equation
f
The computational domain is defined by = t ∪
f
db
rb
db represent
3
rb
t ∪ t ⊂ R where t , t and t
the spaces occupied by the fluid, rigid body, and the
f
deformable body at time step t, respectively. trb = t ∩
f
db
db
rb
t and t = t ∩ t are the interfaces between the
fluid and rigid body and between the fluid and the
deformable body at time t, respectively.
The three-dimensional incompressible Navier Stokes
equations describe the dynamics of the fluid on an Eulerian Grid x ∈ .
∂u
ρf
(1)
+ u · ∇u = −∇p + μ∇ 2 u + fibm
∂t
∇ ·u=0
(2)
where ρf is the fluid density, t the time, u the fluid
velocity, p the static fluid pressure, and μ the fluid
dynamic viscosity. The fluid Cauchy stress tensor is
given by σf (u, p) = −pI + 2με(u), in which ε(u) is the
strain rate tensor defined as ε(u) = 1/2(∇u + ∇uT ).
rb + f db is a source term introduced to impose
fibm = fibm
ibm
the solid boundary conditions for both rigid (rb) and
deformable bodies (db). This forcing term defines a
Penalty method with a velocity reconstruction approach
and is able to predict accurate pressure distributions
around the solid object. This is crucial for correct modelling of the dynamic response of the deformable body.
The Penalty method has been introduced by (Arquis &
Caltagirone, 1984) and improved by (Specklin & Delauré,
2018) by considering a sharp interface approach with
a second-order velocity reconstruction at the interface.
In this approach, the solid domain is considered as a
porous media with a small permeability K (0 < K 1). The IBM forcing term with a second-order velocity reconstruction at fictitious boundaries is defined by
Equation (3). The subscript (i)b is defined such that i = r
and i = d represents the rigid body and the deformable
body, respectively.
ν
(i)b
∗ ν
fibm
= χ(i)b (u − u(i)b ) + χ(i)
(u − u∗(i)b ) i = r, d
K
K
(3)
χ(i)b is the characteristic function that marks the
location of solid boundaries embedded within the fluid
domain, which is defined in Equation (4), u(i)b represents the velocity of the immersed body. On the other
∗ locates the very first layer of the inner fluid
hand, χ(i)b
˜ (i)b
cells in the immersed body as illustrated in Figure 1
t
by the red colour IB cells. The modified velocity of the
immersed body u∗(i)b is defined by Equation (6) and trans˜ (i)b
by a linear interpolaferred to the cells located in t
tion. Here d1 and d2 stand for the distance between the
interface and the P cell centre, and between the interface and an arbitrary point ϕ with uϕ velocity as shown
in Figure 1. The selection of the parameter K for the
simulation was determined through a sensitivity analysis of the drag coefficient for a cylinder undergoing
inline oscillations at a maximum Reynolds number of
100, as referenced (Specklin, 2018). It was observed that
as K exceeded 0.1, there was a noticeable delay in drag
response, whereas values as low as K = 10−2 gave rise
to high-frequency oscillations, signalling the onset of
instabilities. For the current study, a prudent choice was
made, setting the penalty coefficient at K = 10−6 . This
value was found to be suitable for preventing instabilities across all tested conditions. The selection of K in this
manner ensures stable and reliable simulation results in
accordance with the specified conditions.
∗
χ(i)b
(x, t)
if
if
x ∈ t
f
x ∈ t
i = r, d
(4)
1
0
if
˜ (i)b
x∈
t
else.
i = r, d
(5)
d1 + d2
d1
u(i)b,n − uϕ i = r, d
d2
d2
(6)
=
u∗(i)b =
(i)b
1
0
χ(i)b (x, t) =
2.1.1. Fluid FVM predictor–corrector solution
The fluid flow equations are solved with a FV approach
and the PIMPLE pressure velocity coupling of the
open-source library OpenFOAM (Jasak et al., 2007).
This three-step segregated solver combines the SIMPLE
(Semi-implicit method for pressure linked equations)
and the PISO (pressure implicit with split operator). The
initial momentum predictor step calculates the intermediate velocity from the semi-discretized momentum:
ai ũi = H(ũ) − ∇pt
(7)
where ai is the matrix coefficient for cell i, ũi the estimate
of the intermediate velocity, ∇pt the pressure gradient
from the previous time step t. H(ũ) combines the transient terms, the contribution from the neighbouring cells,
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
um+1
f
=
H(um )
ai
H(um )
1
m
−
ai
ai ∇p
if
else
5
db
i ∈ rb
t ∪ t
(11)
The term H(um ) is recalculated followed by an iterative increment, advancing from m to (m + 1) after completing the velocity correction step for um+1 . Once a
predetermined convergence criterion has been met, the
velocity and pressure are updated for the next time step
with ut+ t = um+1 and pt+ t = pm+1 . Further details
on the algorithm may be found in Specklin and Delauré
(2018).
Figure 1. Immersed Boundary approach. The blue line illustrates
the actual immersed body surface. The Red dashed line specifies
the immersed body limits. stands for fluid domain with • at the cell
centre. defines the penalization domain except for the first inner
layer with ♦ at the cell centre. shows the first inner layer of the
penalization domain called IB cells with at the cell centre.
other source terms and IBM force fibm :
H(ũ) =
aj ũj +
j
=
u(i)b
u∗(i)b
uti
Uib
−
, Uib
t
K
(i)b
(i)b
˜t
if cell in t − (i)b
˜t
if cell in , i = r, d (8)
Here t stands for the time step. The follow on pressure
corrector step enforces mass conservation by combining the continuity equation (∇˙ · u = 0) and Equation
(7). The resulting Poisson equation for pressure correction is derived using the Gauss theorem to discretize the
gradients and divergence terms:
1
H(um )
m
Sf ·
(∇p )f =
Sf ·
(9)
ai f
ai
f
f
f
Sf stands for the outward normal vector of the surface
f for cell i, and the m superscript shows the PISO loop
counter with the initial condion (m = 0) of um = ũ and
pm = pt . The velocity at the cell centre and flux at the surfaces F m+1 are updated in the final momentum corrector
step by returning to the momentum equation.
F m+1 = Sf · um+1
f
H(um )
1
m
= Sf ·
−
(∇p )f
ai
ai f
f
(10)
2.1.2. Pressure correction at the immersed surface
The Sharp interface method adopted requires pressure
correction at the exact immersed surface location to prevent the propagation of local mass conservation error
and associated spurious pressure inside the immersed
body into the fluid domain. The correction led to more
accurate pressure in the vicinity of the immersed body
and was shown in (Specklin & Delauré, 2018) to achieve
second-order convergence in velocity in the vicinity of
the immersed surface. This influence of the immersed
boundary is treated by removing the neighbour cell contribution to the pressure correction equation and by
imposing the pressure value at the virtual point to satisfy
∂p
∂n (i)b = 0(i = r, d) at immersed boundaries. Further
information on implementation details are provided in
(Specklin & Delauré, 2018).
2.1.3. Turbulence modelling
A hybrid turbulence modelling approach has been
adopted to combine a scale resolving Large Eddy Simulation (LES) approach in the core of the flow domain
and a Reynolds Average Navier-Stokes (RANS) solver for
wall modelling. The Detached Eddy Simulation (DES)
(Spalart, 1997) adopted incorporates a Modified SpalartAllmaras (MSA) (Allmaras & Johnson, 2012) model to
resolve the near-wall turbulence, which has been altered
to sense themmersed moving boundaries. A secondorder turbulence viscosity νt reconstruction imposes
the necessary condition at wall surfaces by following a
penalty formulation:
νt = ν̃fν1
⎧
⎨f = χ 3
ν1
3
χ 3 +cν1
⎩χ 3 = (ν̃/ν)3
Dν̃
1
= P − D + (∇ · (ν + ν̃)(∇ ν̃) + cb2 (∇ ν̃)2 )
Dt
σ
(12)
(13)
6
E. AKRAMI ET AL.
∗
χ(i)b
χ(i)b
∗
+
(ν̃ − ν(i)b ) +
(ν̃ − ν(i)b
), i = r, b
Kν̃
Kν̃
(14)
Here, P and D are the wall production and destruction terms, respectively. cν1 , cb2 and σ . are model constants, and the reader is referred to the original reference
∗
(Allmaras & Johnson, 2012) for details. ν(i)b and ν(i)b
represent the turbulent viscosity and corrected turbulent
viscosity at the immersed surface, respectively.
The turbulence model requires another modification
to comply with IBM LES/RANS transition criteria. The
hybrid model switches to the MSA method when dw ≤
and back to the LES model when dw > where and
dw represent the average grid distance and the minimum
distance to a wall, respectively. The distance either to a
wall or immersed body should be considered to reflect
the correct switch. dw∗ = min(dw , ψ) takes the distance to
an immersed body into account, while ψ is the distance
to a surface of an immersed body.
2.1.4. Numerical simulation of the fluid
The Gauss Linear method is employed to calculate
the gradient and divergence terms. For the momentum
flux calculation, the second-order Linear Upwind Stabilized Transport scheme is used (Weller, 2012) since
this approach minimizes pressure oscillations in LES
modelling (Lysenko et al., 2014). The Laplacian terms
are expressed by the Gauss Linear Limited Corrected
method, whereby the surface area vector Af = k +
is decomposed into the cell to neighbour vector
and a face parallel component k to implement a
deferred correction for cell non-orthogonality. The nonorthogonal correction factor is applied here if φf · ≥
ψ(∇φf · k), taking ψ = 0.33. The Navier-Stokes temporal term is discretised by the Backward Euler implicit
method.
2.2. Deformable solid governing equations
The transient balance of linear and angular momentum is
expressed in an Update Lagrangian (UL) framework. The
nonlinear behaviour of the deformable body is expressed
using Equation (15). The UL method describes the reference state of the deformable body using the previous
db
converged state X ∈ db
t− t where t− t is the domain
occupied by the deformable body at time t − t. x ∈
db
t describes the current state of the deformable body
db
from its latest domain db
t . The initial state X0 ∈ 0 is
also required to calculate the deformation gradient tensor
F = ∂x/∂X0 and its Jacobian J = det(F). The mapping
db
ϕ(Xt− t , t) : db
t− t → t links the reference and current states. The following equation governs the response
of deformable solid.
∇ · ∂σdb + ρdb Bv + Bt = ρdb
∂ 2d
∂t 2
(15)
where σdb is the symmetric Cauchy stress tensor, ρdb
the density of the deformable body, Bv and Bt are the
external volumetric force and surface traction, respectively. Displacement is defined as d(Xt− t , t) = X − X0 .
A hyperelastic material model describes the constitutive equation of the deformable solid by calculating the
Second Piola-Kirchhoff stress S by a strain-energy function density as S = ∂W/∂E, where W is the energy
function. E express Green Lagrange strain defined as
E = (F T F − I), where I is a second rank unity tensor.
The Second Piola-Kirchhoff and the Cauchy stress are
related through σdb = J −1 FSF T . The Cauchy-Green tensor is defined as C = F T F. Mooney-Rivlin model is used
frequently to predict the behaviour of the rubber-like
material as a compromise between simplicity and accuracy (N.-H. Kim, 2014). In the present study, rubber
is assumed to be homogenous and isotropic omnidirectional. The following equation represents the twoparameter Mooney-Rivlin model implemented in this
study.
k
W = A10 (J1 − 3) + A01 (J2 − 3) + (J3 − 1)2
2
(16)
A10 and A01 are the model parameters dependent on the
material and are obtained by a curve fitting from several tensile tests (N.-H. Kim, 2014). k is the bulk modulus
proposed to keep the deformable body near incompressible. the J1 , J2 and J2 respectively, indicate the first, the
second and the third reduced invariants. The FSI solver
implemented uses the Mooney-Rivlin model to obtain
the Second Piola-Kirchhoff stress follows.
S = A10 J1,E + A01 J2,E + k(J3 − 1)J3,E
(17)
The material Stiffness tensor S defines the derivation
of stress with respects to strain. In Mooney-Rivlin model
this derivation is calculated as follows:
D=
∂S
= A10 J1,EE + A01 J2,EE + k(J3 − 1)J3,EE
∂E
+ kJ3,E ⊗ J3,E
(18)
The definitions of the other variables used in Equations (16)–(18) are provided in the appendix. In addition,
more detail about this model and parameter calculations
can be found in (Wriggers, 2008).
2.2.1. Deformable body FEM discretization
A Finite Element method has been adopted to discretize
the governing equations for the deformable body. The
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
solution implemented relies on 20-nodes hexahedral elements associated with 20 shape functions via an incomplete quadratic polynomial on each element for threedimensional problems, and 9-points quadrilateral elements are used for two-dimensional simulations. A Full
Newton–Raphson iterative method using Taylor’s expansion has been implemented to deal with large deformations and the nonlinear constitutive terms since it
has the fastest convergence rate for a single evaluation
term (Zienkiewicz & Taylor, 2005). The following formulation is obtained by performing the Galerkin FE discretization and the Newton–Raphson nonlinear solver on
Equation (15).
M t+ t d̈(i+1) + t C(i)t+ t ḋ(i+1)
(i) t+ t (i+1)
+ (t KL(i) + t KNL
)
d
= t+ t R(i) + t+ t Fc(i) − t F (i)
(19)
In this formulation, i indicates the Newton–Raphson
iteration number where nodal acceleration, velocity and
Displacement at time t + t are described by t+ t d̈(i+1) ,
t+ t ḋ(i+1) and t+ t d(i+1) respectively. M, t C(i) , t K (i) and
L
t K (i) stand for mass, damping, tangential linear stiffness,
NL
and tangential nonlinear stiffness matrices, respectively,
at time t. t+ t R(i) is the external loads applied at time
t + t inducing fluid pressure, viscous term and buoyancy forces. t+ t Fc(i) describes the contact force when a
collision happens between the deformable body and a
rigid media, and t F (i) internal force equivalent to the element stresses at time t. These parameters are defined as
follows:
M=
t+ t (i)
R
db
e
edb
t
NTdb t+ t fS dedb
db
e
(i)
KNL
=
t (i)
F
t (i)
C
=
N T t+ t fB(i) ddb
e
t (i) T t
db
e
db
e
db
e
(21)
D(i)tt B(i) ddb
e
(22)
t (i) T t (i) tt (i)
G
σdb G ddb
e
(23)
t (i) T t (i)
B
σ̂db ddb
e
(24)
B
(i)
= cM M + cK (t KL(i) + t KNL
)
(25)
where db
e is the domain of each element, N the shape
functions, Nedb the shape functions of the element side
(i)
(i)
the present study, the external surface pressure is equal to
the fluid induced stresses, and the body load is calculated
from the buoyancy effect. The fluid stress term is approximated on each Gauss point of the surface element by MLS
and integrated by the quadrature rule. t B(i) and t G(i) are
the linear and nonlinear derivation matrices, respectively.
t σ̂ (i) is a vector of the Cauchy stress of the deformable
db
body element. cM and cK are the damping matrix coefficients and indicate the contribution of the mass matrix
and stiffness matrix. The term (ρdb − ρf ) reflects the
buoyancy effect acting on the deformable body sinking
in the fluid. More information about the formulation and
procedure for calculating all terms can be found in the
appendix and (Bathe, 2006).
2.2.2. Contact model
Complex deformations can occur as the immersed solid
interacts with solid boundaries, in particular fast-moving
walls and special care has been required in developing
a stable collision model. Grid search in contact models is computationally expensive and usually the most
time-consuming part of such models. To simulate the collision of an IB body, (Albadawi et al., 2019) introduced
a Liquid Film contact model between a zero-thickness
deformable cloth and solid walls, which overrides the
position immersed nodes which approach the immersed
surface within a user-specified distance. This correction
approach proved unstable with the current coupling. A
modified version of the contact force proposed by (Borazjani, 2013) has been developed and adopted instead in
the present study. The modification replaces the direction of the contact force defined as the normal vector in (Borazjani, 2013) by the direction of frictionless
reflection inspired by the principle of linear momentum
conservation after the collision and defined by:
r = ḋ − 2(ḋ · n̂)n̂
e
(i)
KL =
(20)
(i)
=
+
t
(ρdb − ρf )N T Nddb
e
surfaces, t+ t fS and t+ t fB are the external surface tension and body load acting on the element, respectively. In
7
(26)
where r is the reflected velocity after collision, ḋ velocity vector before collision and n̂ is the normal vector of
the rigid surface. The contact force is calculated when the
deformable body approaches a solid boundary whether
it is from a rigid immersed body or a wall boundary
within a specific threshold ε according to the following
formulation:
⎧
⎪
s>ε
⎨0
t+ t R(i) |
t+ t (i)
t+
t
(i)
−
K
s/|
Fc = |
R |e
·r 0<s≤ε
⎪
⎩ t+ t (i)
R | − Ks) · r
s≤0
(|
(27)
where K is the contact stiffness coefficient, which is taken
as 10N/m (Borazjani, 2013). The variable s represents the
perpendicular distance between the node and the wall
8
E. AKRAMI ET AL.
unconditional stability in the Newmark scheme (Bathe,
2006). These parameters are used to compute the incremental displacement, velocity and the acceleration of the
deformable body using a modified tangential stiffness
matrix and a residual vector form Equations from (28)
to (33).
t
K̂ (i)t+
t
K̂ (i) =
t+ t (i)
R̂
Figure 2. Definition of the contact force direction as the reflection direction inspired from frictionless elastic collision models.
t
d(i+1) = t+ t R̂(i)
1
α
t2
M+
δ
α t
(28)
(i)
(i)
+ t KL + t KNL
(29)
= t F (i) − t+ t R(i) − t+ t d̈(i+1) M
− t C(i)t+ t ḋ(i+1)
t+ t (i+1)
d
(30)
= t+ t d(i) + t+
t
d(i+1)
(31)
1 t+ t (i+1) t (i)
1 t (i)
d̈
ḋ
=
(
d
− d )−
α t2
α t
1
−
(32)
− 1 t d̈(i)
2α
δ t+ t (i+1) t (i)
δ
t (i+1)
=
d
− d )−
ḋ
(
− 1 t ḋ(i)
α t
α
t δ
− 2 t d̈(i)
−
(33)
2 α
t+ t (i+1)
face, and it can take both positive and negative values.
A negative s value indicates penetration, prompting the
contact model to exert additional forces to reposition
the deformable body outside the surface. To improve the
grid search efficiency, a characteristic function was also
adopted. The contact model starts searching nodes only
˜ db
if a cell near the rigid body turns into t which means
the distance between the deformable and the rigid body
is less than the characteristic size of a fluid cell. In addition, the search is limited to a control region enclosing
the deformable body Figure 2.
In addition, stability in the contact model can depend
on the threshold ε but also the local grid size, which
is controlled through adaptive grid refinement. ε is set
in this study at a fixed value of 0.0004 m. Decreasing
ε delays the activation of the contact model to avoid
activation too early when no physical effect should be
felt by the immersed object. Smaller values of however,
increase applied contact force and can trigger instability,
in particular when the object does penetrate the surface. The threshold chosen in this study was selected
from a sensitivity analysis to strike a balance between
preventing excessive penetration, which could lead to
numerical instability, and ensuring accurate modelling
of contact interactions. To further enhance stability, an
adaptive grid refinement technique was included in the
fluid domain surrounding the deformable body. This was
used to ensure that a sufficient number of FVM cells to
resolve the flow domain at scales that match the chosen
threshold.
2.2.3. Temporal discretization
The implicit constant-average Newmark method (Newmark, 1959) has been implemented in the deformable
body solver due to its versatility and stability. Specifically, and parameters have been selected, which leads to
t+
The FEM structural solver has undergone rigorous
validation at each stage of its development. This validation included comparison against established benchmarks and simulations obtained from commercial packages. These comparisons have been omitted from the
present paper to focus on the core contributions to the
state-of-the-art.
2.3. FSI coupling
The dynamic behaviour of an FSI problem is induced by
the interplay of the solid and the fluid on each other,
described by two main principles: kinematic condition
and the balance of normal stresses. The following equations describe these conditions:
∂d
on db
∂t
(34)
J −1 FSF T = σf n̂ on db
(35)
u=
The fluid stresses at the interface between the fluid
and the deformable body are responsible for its motion.
They are described by using a velocity reconstruction
method. In this approach, the kinematic condition and
the balance of stresses are satisfied by transferring the
velocity of the deformable body to the fluid domain by
(i)b
a penalty method, which defines fibm from Equation (3)
and by transferring the Cauchy stresses from the fluid to
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
9
the boundaries of the solid and specifying the boundary conditions via the compact support domain MLS
approximation. The data transfer needed to satisfy the
FSI conditions at the interface has been implemented
within a strong coupling partitioned approach. The Newton–Raphson method (Gerbeau & Vidrascu, 2003) and
the Fixed-point method (Deparis et al., 2003; Küttler &
Wall, 2008; Küttler & Wall, 2009) are two prevalent methods for solving the nonlinearity of the energy balance at
the interface. The Newton–Raphson methods have relatively fast convergence of quadratic order but involve
additional computational cost with the calculation of
a Jacobian matrix. The Fixed-point approach is more
straightforward to implement but suffers from a slower
convergence rate. Using the Aitken relaxation optimization method can enhance the convergence order (Irons
& Tuck, 1969), and the technique has been implemented
successfully in IBM-FSI for large deformation problems
(Borazjani, 2013). The fixed-point coupling method has
successfully decreased the spurious force and increased
the solution stability in low-density ratio problems using
an inner loop to converge the residual of FSI coupling.
The convergence condition for this iterative solution r is
defined at the iteration in terms of the point displacement
velocity by:
due to grid size difference (Griffith & Luo, 2017), and it is
valid only for the uniform Cartesian fluid grids. (Vanella
& Balaras, 2009) introduced the application of MLS in
IBM diffuse interface approach by using the method
introduced by (Uhlmann, 2005). IBM MLS treatment
has been extended to sharp interface models by (Haji
Mohammadi et al., 2019). It has been shown that not only
MLS can improve the velocity reconstruction accuracy
significantly in comparison to previous force regulation
methods, but it also leads to second-order accuracy for
pressure and viscous force calculations exerted by the
fluid flow on the immersed structure, which is vital in FSI
problems. The MLS also proved effective in reducing spurious force oscillations due to the stabilizing effect of the
gradient smoothing. The versatility of the MLS method,
which is designed to construct approximation from support domains made of unorganized distributed source
points, makes it a good candidate.
A 3-D compact support domain version of the MLS
method has been developed in this study with a thirdorder spline weight function is as follows:
Rk = t+ t ḋj− t+ t ḋj∗
where f̃ (X ) is the approximated value at sample point
X , n, number of data points, fI the value at the data point
XI . In which, φI (X ) is the shape function of point Ith at
point as defined X by the following equation:
(36)
In which, t+ t ḋj is the updated velocity of the deformable
body after solving and t+ t ḋj∗ is a temporary variable
containing the velocity of the deformable body at stage
j without updating the grid. In general, direct updating
of the variables leads to instability in the strong coupling.
This issue is resolved by introducing an under-relaxation
factor in the Fixed-Point method as follows:
t+ t = t+ t +
ḋj+1
ḋj ωRj
(37)
Although the relaxation factor can be a fixed value, the
Aitken method (Irons & Tuck, 1969) proposes an adaptive factor to increase the convergence rate of the strong
coupling solution as follows
(Rj−1 ) (Rj − Rj−1 )
f̃ (X ) =
|Rj − Rj−1 |
2
(38)
φI (X )fI
(39)
I=1
φI (X ) =
m
PJ (X )[A−1 (X )B(X )]JI
(40)
J=1
where P is the vector of basis functions constructed by the
sample point component coordinates. A second-order
monomial vector has been considered as basis functions
with m components for present study as follows:
PT (X ) = [1 x y z x2 y2 z2 xy xz yz]
(41)
The moment matrix A(X ) is defined as
T
ωj+1 = −ωj
n
A(X ) =
n
W(X − XI )P(XI )PT (XI )
(42)
I=1
2.4. Moving least square approximation
The first publication on an IBM simulation reported the
pioneering work of (Peskin, 1972), which relied on a Discrete Delta Function (DDF) to diffuse the influence of a
thin deformable structure on a fluid flow resolved on an
Eulerian grid. This DDF-based method is known to suffer
from low order of accuracy, slow force convergence (Griffith & Patankar, 2020), grid leaking across the interface
B(X ) is a matrix in the aforementioned formula as
follows:
BI (X ) = W(X − XI )I = 1, . . . , n,
(43)
The choice of the weight function is not a critical
issue in MLS since the order of the approximation is
not affected by this function and only manipulates the
smoothness of the approximation (Cleveland & Loader,
10
E. AKRAMI ET AL.
1996). On the other hand, if the weight function is compatible with the compact support domain, calculating the
approximation is more efficient due to the sparsity of the
A(X ) (Fasshauer, 2007). A cubic spline weight function
is compactly supported and provides moderate smoothness. This weight function has been embedded in MLS
for this research as follows.
⎧
2
2
3
⎪
for s ≤ 12
⎨ 3 − 4s + 4s
W(s) = 34 − 4s + 4s2 − 43 4s3 for 12 < s ≤ 1 (44)
⎪
⎩
0
for s > 1
where S is the non-dimensional distance between sample
point X and data points XI defined s = ||X − XI ||/rw ,
in which rw stands for support domain size, i.e. length of
radius for spherical support domain.
Equation (41) requires at least ten fluid nodes to
perform the MLS mapping. A multiple-layer solution
has been adopted to avoid generating singular moment
matrices. Increasing the size of the compact support
domain can help ensure that sufficient fluid nodes are
marked to avoid singularity. Choosing a large support
domain however, has detrimental effects. It increases
the computational cost as well as smoothes the gradient around the deformable body. An adaptive size that
detects the minimum suitable support domain has been
implemented. This is achieved by increasing the domain
size incrementally and iteratively from a minimum using
the determinant of the moment matrix to gauge the quality of the chosen fluid nodes. Although this adaptive
approach comes at a cost, it successfully prevented the
creation of singular matrices with all test cases considered. In addition, an AMR method has been added to
limit the need for support domain size increase. Several
tests have been performed. Describe the AMR setting
(number of layers and buffer size).
The MLS relies on a spherical support domain. When
its radius is larger than the thickness of the deformable
body as shown in Figure 3, it includes fluid nodes on
both sides of the immersed body. Correctly modelling
the jump condition in the definition of fluid induced
stresses at the immersed surface requires special treatment to determine whether the fluid points in the support domain are in contact with the immersed boundary or separated from it by located fold as illustrated
by Figure 3. A masks function is defined for each MLS
fluid point and set to true when in the fluid region
affecting hydrodynamic stresses acting on the deformable
immersed boundary point. Algorithm 1 interprets the
method adopted to define this masks . In this algorithm
Pf refers the fluid points, Pn are the grid points on the
deformable boundary and is a label identifying separate
surfaces making up the immersed deformable object. On
Figure 3. MLS support domain treatment. In the pink area for corresponding interface, which is coloured in red. In order to approximate quantities using MLS at point, the support domain, which is
coloured yellow, only collect fluid point in the pink area.
line 6 of Algorithm 1, Pn is the nearest solid point to Pf . d̂
is a vector showing the difference between Pn and Pf and
n̂ is the normal vector of the surface point Pn .
Algorithm 1. Determination of the location of the fluid point
regarding the deformable body interfaces
1:
2:
3:
4:
5:
6:
7:
Input ft and tdb
for Pf ∈ ft do
for interface ∈ tdb do
for Pn ∈ interface do
If d̂ = Pf − Pn has the minimum magnitude, then
If d̂ × n̂ ≥ 0 then
Set masks (interface, Pf ) = true
An inner cross product between the surface element
normal vector n̂ and the difference between surface and
fluid points d̂ is sufficient to confirm whether the MLS
source point is on the correct side while minimizing d̂ =
Pf − Pn searches for the fluid point, which is close to the
boundary point. In cases where no fluid point remains
between the surfaces of a folding object, the collision
model described in Section 2.2.2 is activated.
2.5. Adaptive grid refinement
A local dynamic mesh refinement significantly increases
the accuracy of the interaction between the deformable
body and the fluid by enhancing the resolution of the
flow domain for a better estimation of fluid stresses and
velocity field. In addition, AMR can reduce excessive
fluid-solid grid size differences and associated interpolation errors. It is used for eddy-resolving capabilities and
has been extended to use in IBM (Roma et al., 1999). IBM
and AMR have been integrated within several studies
such as (Georgi Kalitzin & Iaccarino, 2003) and (Aldlemy
et al., 2018).
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
A Finite Volume (FV) AMR library is available in
OpenFOAM called dynamicFvMesh (Jasak & Gosman,
2010a, 2010b, 2010c)with a patch-oriented approach for
refinement based on non-overlapping rectangular grids.
An extra iterative loop with an arbitrary iteration number
is required for local refinement as well as a flux correction
on the new faces as defined below:
Ff∗ = u∗f · Sf
(45)
where Ff∗ is the intermediate flux estimation and u∗f is the
interpolated velocity field at the refined surface, which is
defined as follows:
u∗f
= (I
(−1)
− nf ⊗ nf )(uf ,I ) +
Ff(−1)
|Sf |
(46)
nf
(−1)
Here express the face unit normal vector, uf ,I
and
Ff(−1)
stand for the flux of the last time step and velocity
field of cell face-interpolated, respectively. The subscript
f refers to the face-centred values. The solution of the
pressure Poisson equation is required for the next step of
AMR flux correction:
1
Ff∗ =
(∇pm )f |Sf |
(47)
ai f
The divergence-free flux is calculated by updating the
intermediate flux estimation via the pressure correction
value as follows:
1
m
∗
Ff = F f −
(∇pm )f |Sf |
(48)
ai f
In this research, AMR has been applied to the charac∗ (i = r, b) stands for data
teristic functions χ(i)b and χ(i)b
communication region of the FSI problem with two levels of refinement. According to the available resource,
two levels of refinement provide more than ten fluid
points around each deformable node with a reasonable
computational cost.
2.6. Algorithm implementation
A flowchart summarizing the overall solution is shown in
Figure 4. It is structured around five iterative loops which
are: time marching, fluid pressure-velocity coupling, fluid
pressure correction, Strong FSI coupling and a Newton–Raphson loop. Three main data streams between the
different sections of the solver exist: the fluid stresses are
passed to the FEM deformable body solver after a complete cycle of the PIMPLE loop, which includes the IBM
for the deformable and rigid bodies; and the velocity of
the deformable body is transferred to the fluid domain
11
once the Newton–Raphson loop has converged, within
the Strong FSI coupling loop. By satisfying the strong FSI
coupling condition, the solver updates the previous solution and proceeds to the next time step. This algorithm
has been developed to support parallel processing using
the Open MPI library (Gabriel et al., 2004).
While immersed boundary methods have been widely
used for fluid-structure interaction problems, challenges
remain in handling low-density ratios, high Reynolds
numbers, and spurious force oscillations. This work
presents a solution that combines several techniques to
overcome these limitations. The key novelty lies in the
integration of (i) a sharp interface immersed boundary approach for turbulence modelling using a Detached
Eddy Simulation (DES) model, (ii) a strong partitioned
coupling method stabilized by Aitken relaxation, (iii) a
moving least square mapping for information projection
between the fluid and solid, (iv) a dynamic adaptive mesh
refinement around immersed boundaries, and (v) a modified contact model based on frictionless impact mechanics. The sharp interface DES model helped better resolve
smaller turbulence scales by comparison with a RANS
model. The Aitken relaxation method combined with the
least square mapping proved necessary to achieve stable solutions in cases of low-density ratio between the
fluid and the immersed structure. The least square based
projection helped reduce spurious force oscillations associated with the moving interface. The dynamic AMR
further improved stability by minimizing grid leakage
at the interface and in case of collisions. The combined
solution proved stable for all cases studied, from the standard benchmark case in laminar flow to the pump case,
which includes highly turbulent low with fast moving
rigid surfaces which collide with an immersed object with
a low-density ratio.
3. Result and discussion
A validation of the flow solver including the sharp interface IBM implemented here, is available from (Specklin & Delauré, 2018). The present analysis focuses on
the FSI coupling. Simulations and comparisons against
three benchmark cases and two new experimental studies are included to test the accuracy of the proposed
FSI model. The three benchmark cases include small
linear deformations under hydrostatic forcing and two
laminar flows with a very low-density ratio at Re = 100
and Re = 200. Two dedicated experimental studies to
characterize the interaction between deformable bodies and fluid under forced oscillations and flow-induced
deformations are then used to consider higher Reynolds
number flows. These two test cases are referred to as
the Forced Oscillation and Flow Induced Cases. In both
12
E. AKRAMI ET AL.
Figure 4. Solver algorithm implementation. Red line: transfer of fluid stresses to the boundaries of the deformable body. Blue line:
transfer of velocities of the deformable body to the fluid. Green line: transfer of contact model data to the deformable body.
cases, tests have been performed with two rectangular
rubber membranes of thickness 3mm and aspect ratios
between the membrane length and its width AP = 3 and
AP = 158 for overall dimensions 40×120×3 [mm] and
55×87×3 [mm] respectively. In both cases, the membrane’s transient deformations have been measured using
a stereoscopic Digital Image Correlation (DIC) technique
with a system provided by Dantec Dynamics GmbH, Ulm
Germany (Maček et al., 2021). The equipment includes
two 1600×1200 pixels cameras with Gigabit Ethernet
connection with a maximum bandwidth of 1000 [Mbit/S]
and a maximum acquisition of 62 frames per second at
full resolution, lenses for Q-400 cameras with a focal
length of 17 [mm] and aperture of 1.4. Illumination was
provided by two red HILIS illuminations consisting of
48 high power LED’s emitting cold light with a single
colour optimized for 5×4 [cm2 ] area at a typical distance of 20 [cm]. Calibration has been performed using
a high-quality calibration target also provided by Dantec
Dynamics. A synchronization time box was used for data
acquisition, and image processing was performed with
the Istra 4D software (DANTEC Dynamics).
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
13
Figure 6. Comparison of transient displacement responses at the
mid-span of the elastic plate, including simulation from current
solution (green), the solutions from (Khayyer et al., 2018) (red),
and (Fourey et al., 2017) (blue) and the analytical solution (black).
Figure 5. Schematic of the hydrostatic water column on an elastic plate (drawing not to scale).
The experimental benchmarks presented here produced flow at Reynolds numbers Re = 1.35 × 104 for
the Forced Oscillation using the length of the rag as the
characteristic length and Re = 8.43 × 104 measured with
hydraulic diameter for the Flow Induced Deformation.
The simulation of the transport of a slender deformable
solid passing through a centrifugal pump for inlet flow
at Re = 1.43 × 106 is presented to illustrate the solver’s
capability under practical and challenging flow conditions. All simulations in this study have been performed
on the Kay Cluster of Ireland’s National High Performance Computer cluster, which is equipped with a 40core Intel Xeon Gold 6148 processor and 192 gigabytes
of RAM with 400 gigabytes of local SSD.
3.1. Hydrostatic water column on an elastic plate
This hydrostatic case considers the static deformation
of elastic aluminum plate subjected to hydrostatic pressure, which is depicted in Figure 5. This test case was
initially proposed by (Fourey et al., 2017) and adopted
by (Khayyer et al., 2018) in FSI – Smoothed-Particle
Hydrodynamics (SPH) studies.
The 5 [cm] thick and 1 [m] wide aluminium plate of
density 2700 [kg/m3 ], Young’s modulus 67.5 [Gpa], and
Poisson’s ratio 0.34, initially at rest, is subjected to a sudden hydrostatic pressure load. This pressure is exerted
by a 2 m high water column. The wáter density is 1000
[kg/m3 ] and the gravitational acceleration is taken as
9.81 [m/s2 ]. The theoretical solution predicts that the
static deflection at the mid-span of an aluminium plate
under a hydrostatic pressure loading is 0.0685 [mm] at
the equilibrium.
The problem is simulated with a time step of t =
1 × 10−3 [s] and grid size of x = 0.02[m] for both fluid
and solid domains. The transient maximum displacement is compared to benchmark results the ISPH-SPH
simulation of (Khayyer et al., 2018) and the SPH-FEM
simulation of (Fourey et al., 2017) as well as the analytical solution in Figure 6. The displacement is shown
to reach a steady state well before of physical time. The
The solution from the current FEM method is shown
to converge asymptotically toward the equilibrium state
with an error of against the analytical solution. Figure 7
illustrates the contours depicting fluid pressure and solid
displacement profiles at the concluding time step of the
simulation, revealing a cohesive and undistorted gradient
in the distribution of pressure.
3.2. Laminar Turek-Hron FSI benchmarks
This section considers low Re number flows and is based
on the (Turek et al., 2010; Turek & Hron, 2006) problems
consisting of deformation of an elastic solid in a channel flow, mounted in the lee of a cylinder in cross flow
as illustrated in Figure 8. The square cross-section channel has a length L = 25 [m] and height H = 0.41 [m].
The cylinder of radius r = 0.05 [m] is located at c(0.2,
0.2 [m]) where the origin of the coordinate system is
at the bottom left corner of the channel inlet. The slender elastic body has a length l = 0.35 [m] and a square
cross-section of height h = 0.02 [m]. A Reference control
point located at the end of this beam, which is initially at
A = (0.6, 0.2 [m]) is tracked through time to characterize
the deformation.
The beam is assumed to be compressible elastic and
is modelled by the St. Venant-Kirchhoff constitutive law.
14
E. AKRAMI ET AL.
Table 1. Parameter setting for Turek-Hron FSI benchmarks.
ρdb
νdb
μdb [kg/m · s2 ]
ρf [kg/m2 ]
νf [m2 /s]
Ū[m/s]
Re
[kg/m2 ]
Figure 7. Pressure of fluid and displacement of solid contour for
hydrostatic water column on an elastic plate problem at t = 1 [s]
Figure 8. Schematic of the Turek-Hron FSI benchmark. A slender
elastic beam is mounted behind a stationary cylinder (drawing not
to scale).
In this model, the second Piola-Kirchhoff stress tensor is
calculated by the following formulation.
S = λdb (trE)I + 2μdb E
(49)
where λdb and μdb are the Lamé coefficients, which are
functions of the Poisson ratio νdb and Young’s modulus.
A parabolic velocity profile of 1.5Ūy(H − y)/0.25H 2
is imposed as the inlet boundary condition defined as
u(y) = and a constant pressure condition is set at the
outlet. A zero-slip condition is imposed on fixed walls,
including the cylinder surfaces. The elastic beam is
clamped at the left-hand side using a Dirichlet boundary condition while other surfaces are modelled as an
FSI2
FSI3
10000
0.4
5.0×105
1000
0.001
1
100
1000
0.4
2.0×106
1000
0.001
2
200
FSI interface. The fluid domain is discretized by 17,230
quadrilateral cells, while the deformable body has 728
linear (P1 ) quadrilateral elements.
Turek-Hron proposes three FSI benchmark cases in
laminar flows. The present study covers FSI2 and FSI3
only for flow Reynolds numbers Re = 100 and Re = 200,
respectively. FSI3 is based on a zero-density difference
and is particularly suitable for testing the solution’s ability to handle spurious forces at low Reynolds numbers.
The physical properties and parameters of the fluid and
deformable body are summarized in Table 1.
Figure 9 shows the transient response in terms of the
displacements of the control point A for FSI2 and FSI3
in the directions and compares the solution from the
proposed FSI model against the Turek-Hron benchmark
results. The displacements in the direction in particular are in close agreement with the benchmark data for
both FSI2 and FSI3. Some differences can, however, be
noticed with the amplitude of oscillations in thedirection
in particular and to a lesser extent with the period.
Four instants in time (1 to 4) are marked on the plots
in Figure 9 (b) for the FSI2 case. The corresponding vorticity and body displacements are illustrated in Figure 10.
They refer to the state of maximum, zero, minimum and
zero displacements in the direction.
Using a sinusoidal function [A. sin(2π .f .t + ϕ) + M]
as a curve fitting function on fully developed flow for the
x and y displacements facilitate a quantitative comparison
against reference values. A, f and M represent the amplitude of deformation, the deformation frequency, and the
mean value of the deformation, respectively. The nondimensional values obtained here are compared with the
benchmark results in Table 2. The Strouhal number is
defined by St = 2fr/Ū. The comparison confirms the
accuracy of the proposed FSI solver in laminar flows
with a significantly low-density ratio by comparison with
other published results.
3.3. Forced oscillations
The first of the two experimental test cases considered involves a deformable rectangular body subjected
to rotational oscillations in a rectangular container filled
by initially stationary fluid. The dimensions of the fluid
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
15
Figure 9. Transient response of deformable body at the control point A, compared against (Turek & Hron, 2006) results: (a) xdisplacement for FSI2 case, (b) y-displacement of FSI2 case, (c): x-displacement for FSI3 case, (d): y-displacement for FSI3 case.
Figure 10. Vorticity of fluid’s contour and displacement of the deformable body’s contour at four different times marked in Figure 9 (b).
16
E. AKRAMI ET AL.
Table 2. Comparison of fitted response obtained in this study against benchmark data from (Turek & Hron, 2006), (Tian et al., 2014) and
(Bhardwaj & Mittal, 2012).
Case
FSI2: ρdb /ρf = 10Re = 100
FSI3:ρdb /ρf = 1Re = 200
source
Ax /2r
Stx
Mx /2r
Ay /2r
Sty
My /2r
Present results
(Turek & Hron, 2006)
(Bhardwaj & Mittal, 2012)
(Tian et al., 2014)
Present results
(Turek & Hron, 2006)
(Bhardwaj & Mittal, 2012)
(Tian et al., 2014)
0.11
0.12
0.03
0.03
-
0.39
0.38
0.55
0.54
-
−0.13
−0.14
−0.29
−0.27
-
0.77
0.81
0.92
0.78
0.35
0.36
0.41
0.32
0.18
0.20
0.19
0.19
0.27
0.26
0.28
0.29
0.01
0.01
0.01
0.01
-
domain are 450×480×370 [mm] and the flow Reynolds
number evaluated from the peak boundary velocity and
the body length is Re = 1.35×104 . The deformable body
is clamped at the end of a support arm, which is rotated
about the horizontal axis, so that the flow induced by
the motion is broadly perpendicular to the un-deformed
membrane. The amplitude of oscillations was set in terms
of the maximum angle of rotation of ∓5◦ forward and
backward.
Figure 11 illustrates the experimental setup of the
tank, which is filled with water to a depth of 370 [mm].
The line of sight from the two DIC cameras is at an angle
of approximately 60°. The exact position and orientation
of the cameras relative to the measured surface are not
needed. Captured images of a pre-defined target checkerboard plate are used instead to provide calibration settings through an automated calibration procedure, see
Figure 11. The calibration target was chosen to be of
a similar size to the region of interest. All images were
recorded at the maximum available frame rate and resolution while processing used facet sizes of 19 pixels with
a grid spacing of 17 pixels at 0.05 [s] intervals.
Two rectangular 3 mm thick rubber membranes of
aspect ratios AP = 3 and AP = 1.58 were modelled. The
arm and the clamp have been simulated as an immersed
rigid body moving forward and backward, inducing the
motion in the deformable body clamped at the top surface, 35 cm above the bottom of the tank. The motion
of the arm is imposed by a prescribed displacement condition derived from experimentally recorded displacements to simulate the oscillations. Strong FSI coupling
conditions have been applied to the rest of the deformable
body surfaces. The effect of free surface motion is not
modelled due to the small impact on the deformable body
dynamics and non-slip velocity conditions are applied
to the sides of the tank. An FSI time step of t = 1 ×
10−3 s has been used for all cases, giving a Courant Number below 0.8. No time step convergence has been performed. Since the FSI time step is less than the sampling
time interval from the experiments, a linear interpolation
has been adopted to calculate the boundary conditions
for the supporting arm motion. The deformable body
Table 3. Physical properties and coefficients of the FSI model.
ρ f (kg/m2 )
μ(pa · s)
ρ db (kg/m2 )
A10 (Mpa)
A01 (Mpa)
k(Gpa)
1000
1×10−3
1180
0.565
0.719
1.73
Table 4. H-grid at successive refinements for the fluid and
deformable solid adopted for the Forced Oscillation case.
Mesh reference
Fluid: Cell count
Solid: Node/element Count
M1
M2
M3
M4
17,280
628/75
135,360
2373/384
704,888
5141/864
5,579,890
8965/1536
is discretised by an incomplete quadratic polynomial
(P2 ) element with 20 nodes per element. The fluid and
the deformable solid physical properties are detailed in
Table 3.
The grid convergence analysis was performed with
the higher aspect ratio membrane (AP = 3), and the
Forced Oscillation case. A series of H-grids have been
tested under successive refinements to identify the spatial
resolution, which provides grid converged results. The
minimum size of the fluid grid tested before adaptive
refinement is 1.0 × 10−2 , 5.0 × 10−3 , 2.9 × 10−3 , 1.7 ×
10−3 m for Meshes M1, M2, M3 and M4, respectively. A
two-level AMR was also applied to the fluid meshes twice
per time step based on the proximity of fluid cells to the
solid interface measured by the characteristic functions.
A linear interpolation retrieves fluxes on divided faces by
interpolating the velocity. The pre-AMR mesh counts for
the solid and fluid grids are reported in Table 4
Simulation results are compared in Figure 12 against
experimental measurements of displacements at the middle control point in the x and z directions over a period of
4 s of physical time. A stiff behaviour is observed with the
low grid resolution showing a growing phase difference
in spite of identical prescribed displacements imposed at
the top boundary of the deformable body. This can be
attributed to the low response time in the coarse grid case.
An increase in mesh resolution provides much improved
phase and amplitude predictions. To obtain a quantitative comparison of the grid convergence and measure
the order of convergence, the fitting of a sinusoidal function is used for x and z direction displacements. Table 5
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
17
Figure 11. CAD rendering of Forced Oscillations test case, including water tank and camera setup. Insert: Experimental set-up with
illumination.
Table 5. Interpolated parameters of sinusoidal function for the
Forced Oscillation case obtained from grid convergence analysis
for deformable body AP = 3 in and directions.
Source
M1
M2
M3
M4
experiment
Ax [mm]
fx [Hz]
Az [mm]
fz [Hz]
1.16
1.97
2.32
2.44
2.43
1.37
1.39
1.42
1.43
1.43
6.48
8.33
9.27
9.28
9.83
1.38
1.41
1.43
1.43
1.43
shows the interpolated frequency and amplitude of both
displacements for experimental measurements and four
different grids.
The comparison shown in Figure 12 indicates a very
good overall agreement between the simulated displacements and experimental measurements. The maximum
differences observed at the extremum points where maximum deformations occur, are observed at time t/T =
0.71. In this case, the relative difference with the experimental result, measured using the l1 norm is 13.2% and
11.3% for the z and x displacements, respectively, but
this reduces to smaller single values typically below 3.2%
and 2.9% for z and x displacement, respectively. The
convergence of the relative differences between numerical and experimental results, taking the experimental
results as the reference, are shown in Figure 13 for the
fitted parameters. The frequencies are shown to exhibit
a second-order convergence, while a third-order convergence is observed for the x direction amplitude. On
the other hand, the amplitude of z direction is converged between grids M3 to M4. The grid convergence test result has been extended to the next problem
since only the size of the deformable immersed body is
different.
Figure 14 depicts the Von-Mises stress contour of
a deformable body AP = 3 at a full cycle of flapping.
Within this cycle, the deformable body undergoes maximum deformation and experiences a peak at t = 1.94 s
while the minimum happens at t = 2.09 s.
Results with the shorter membrane AP = 1.58 using
the M3 grid are given in Figure 15. The comparison
against experimental measurements shows a slight difference happening in the first cycle, which reduces significantly in later cycles. The maximum relative error here
is 22% and 18.5% for the x- and z-directions, respectively, at the time t/T = 0.78. This reduces to 1.6% and
0% for the x- and z-directions, respectively, when measured with two decimal place accuracy and averaged over
the subsequent five cycles.
Overall, an excellent agreement is again observed
between the numerical and experimental results, as summarized in Table 6.
The cases considered in this section were selected
on the basis of the density ratio of 1.18. As reported
in (Uhlmann, 2005), the minimum density ratio that
a loose coupling can handle is 1.2. This sensitivity has
been linked to the spurious forces caused by the IBM
18
E. AKRAMI ET AL.
Figure 12. Grid convergence analysis from the transient displacement response of the deformable body AP = 3. The computational
predictions are compared against experimental results for (a) z-direction displacement and (b) x-direction displacement.
method as the Eulerian cells experience changes from
fluid to solid and vice et versa and should be compounded by the higher velocity flow and solid transport
considered compared to the test cases of Section 3.2.
Spurious forces have been solved in the literature with
loose coupling methods, but to the author’s knowledge,
all existing solutions are problem specific. The present
solution is intended to provide a versatile solver as a
designing tool in the pump industry. The strong coupling has been found to be the most reliable method to
alleviate the spurious forces, according to the literature
review.
3.4. Flow induced deformations
The second series of experimental measurements considered the response of a rubber membrane held in a
turbulent flow. The membrane is attached to a vertical
cylindrical pole placed in the 150×150 mm test section
of a water tunnel in its symmetry plane. The membrane is
placed vertically at 0° from the channel symmetry plane.
The general FSI tunnel setup is illustrated in Figure 16.
Flow field measurements of the tunnel without
membrane and supporting pole were obtained from a
stereoscopic three components planar Particle Image
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
19
Figure 13. The absolute relative error for grid convergence analysis of deformable body AP = 3 for amplitude and frequency in
x- and z-directions.
Velocimetry (PIV) from Dantec Dynamics. The stereoscopic PIV relied on two FlowSense EO 4M pixel 15 Hz
double frame cameras equipped with 50 mm f/1.4 lenses,
527 and 532 nm filters, a pair of 3D PIV Scheimpflug
camera mounts. The planar sheet illumination was generated by a DualPower 65-15 laser generating a 532 nm
beam and right angle light sheet optics Data acquisition
and processing was performed by the Dantec Dynamic
Studio software. PIV measurements were used exclusively to characterize the flow conditions where the inlet
Figure 14. Deformable body AP = 3 at various time intervals
represented by Von-Misses stress contour.
of the computational domain is located relative to the
cylindrical support. Tests were performed at a pump
flowrate of Q = 40.5m3 /h, giving an area-averaged flow
velocity Ū = 0.5m/s. The maximum standard deviation
Figure 15. Transient displacement response for the deformable body AP = 1.58 compared against the experimental results in both xand z-directions.
20
E. AKRAMI ET AL.
Figure 16. Experimental set-up for water tunnel with DIC system. Photo image with the water tunnel on the left hand-side, the cameras
in the middle and the water tank with the DIC calibration target on the right hand-side.
for this streamwise velocity measured by PIV ranged
from 0.005 m/s to 0.006 m/s over the section occupied
by the membrane and increased to 0.01 [m/s] that is
2% of the mean velocity towards the upper wall of the
tunnel. The spanwise and vertical components of the
mean flow velocity were measured to be within [1,2]%
and [−1.5,4]% of the mean streamwise velocity, respectively. The simulation was performed assuming a uniform velocity aligned with the channel axis and a magnitude of 0.5 m/s giving a channel flow Reynolds number
Re = 8.43×104 . Measurements of the membrane deformation were taken in the same section of the tunnel
located after the flow conditioning part with the same
DIC cameras and illumination as used in the forced
oscillation case. Calibration settings and angle between
cameras were kept unchanged. The experimental setup is
shown in Figure 16. Measurement of deformations from
the pre-stressed state was extracted from the trailing edge
of the membranes along a line aligned with the mean flow
direction.
The corresponding computational model is illustrated
in Figure 17. The fluid domain dimensions in the streamwise, spanwise and vertical directions are Lx = 0.15m,
Ly = 0.15m and Lz = 1m. The vertical cylinder used to
clamp the membrane has a diameter Dc = 0.016m and
is located downstream of the inlet boundary with an offset Lc = 0.3m. The outlet boundary is located more than
Table 6. Sinusoidal parameters comparison of numerical and
experimental results of the Forced Oscillation case for deformable
body AP = 1.58 in - and - directions.
Source
Experiment
Numerical
Ax [mm]
fx [Hz]
Az [mm]
fz [Hz]
1.25
1.27
1.43
1.42
12.7
12.7
1.43
1.45
43Dc from the centre of the cylinder. The fluid domain is
discretised by a cartesian stretched block mesh consisting
of 70 × 70 × 320 = 1.568 × 106 cells. This corresponds
to a mesh that is similar to the mesh M3 tested with the
Forced Oscillation case. The deformable body is attached
to the middle of the stationary cylinder in parallel to
the fluid flow. Both AP = 3 and AP = 1.58 deformable
bodies have been modelled in this Forced Induced Deformation case. The Finite Element elements tested were
made of 3256 and 3664 P2 elements and 16,681 and
18,981 nodes, respectively. Table 3 describes the physical properties and coefficients used in this simulation.
A uniform velocity inlet Ū = 0.5[m/s] boundary condition is used on the left-hand side of the fluid domain,
while outlet boundary condition is imposed at the surface
on the right-hand side. The lateral sides and the stationary cylinder are modelled as a non-slip wall boundary
condition. A uniform distribution of turbulent kinematic
viscosity-equivalent variable (ν̃) with prescribed value
of 3 × 10−6 [m2 /s] has been applied to both inlet and
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
21
Table 7. Maximum and Mean displacement values of deformable
bodies with AP = 3 and 1.58 derived from experiments and
simulations in the Forced Induced Oscillation problem.
Maximum displacement [mm]
Mean displacement [mm]
Figure 17. Schematic of the simulation domain for the Flow
Induced Deformation test case. A uniform velocity inlet condition
is applied on the left boundary and an outlet condition is applied
on the right boundary (drawing not to scale).
outlet boundary conditions. Furthermore, the Spalding
model has been employed to simulate wall functions
at the boundary conditions adjacent to solid walls. The
deformable body is attached to the cylinder by enforcing
a Dirichlet boundary condition to be fixed at the place,
and the rest sides have strong FSI coupling boundary
conditions.
Figure 18 provides the time evolution of the magnitude of displacement measured and predicted at the middle of the trailing edge of the deformable body extracted.
The comparison for both AP = 3 and AP = 1.58 show
that, although, the FSI model follows similar trends of
displacement, differences exist. It should be noted that
a simplified inlet boundary condition was employed for
the simulations. A uniform and constant inlet velocity was imposed along with a uniform modified turbulent viscosity. This implies that the simulations did not
aim to replicate the three-dimensional velocity fluctuations at the inlet, except for their influence on turbulent dissipation imposed through the modified turbulent kinematic viscosity. The removal of inlet fluctuations is expected to reduce the maximum deflection of
the membrane, while the mean deflection should remain
unaffected. The results presented in this study indeed
demonstrate a more accurate comparison for the mean
deflection.
Instead of considering the transient dynamic response,
the assessment focuses on the maximum and mean of
the deformations over a time period of 10s. These results
are shown in Table 7. The relative difference in the mean
displacement compared to the experimental result, measured using the l1 norm, is 12.27% and 2.53% for AP = 3
and AP = 1.58, respectively. For the maximum displacement, this becomes 10.95% and 15.69% with AP = 3 and
AP = 1.58. The difference between the simulated and
measured displacements is shown to increase with the
case
AP = 3
AP = 1.58
Experimental
Numerical
Experimental
Numerical
5.35
4.77
1.56
1.78
2.92
2.88
1.41
1.32
aspect ratio and associated increase in the amplitude of
flapping. It is likely that variability in the flow velocity at
the inlet is the main reason for the discrepancy observed
here.
The comparison of the power spectral density of displacement between numerical and experimental results,
shown in Figure 19, again does show differences but
also some similarities. Figure 19 (a) indicates that both
the experimental and numerical displacement signals are
made of two primary frequency ranges for deformable
body AP = 3, [0.5 ∼ 1.4] Hz and [2.7 ∼ 3.9] Hz.
Figure 19 (b) reveals that the deformable body AP = 1.58
is oscillating at lower frequencies with most energy
within the range [0.66 ∼ 1.4] Hz.
3.5. Crossing deformable body through a pump
In this section, a more practical application is considered. Single blade impeller pumps have been specifically
designed to handle larger suspended solids and minimize
the risk of clogging by thin, flexible membranes. Despite
significant advances in anti-clogging performance, challenges remain to a large extent due to the complexity of physical processes involved in the formation of a
pump blockage. Numerical tools do offer good prospects
as design support tools for their ability to clarify these
physical processes, but here again, challenges remain.
The geometrical complexity, the high Reynolds number
flow involved, the multi-physics and multi-scale characteristics inherent to an FSI problem all contribute to
the difficulty in achieving accurate simulations. A single
blade impeller pump produced by Sulzer Ltd is studied
in this section. Its hydraulic performance has been studied previously by (Albadawi et al., 2019) using the Sharp
Interface IBM method. A diffuse interface FSI was used
in this instance to study a cloth-like deformable membrane assuming a zero-thickness and adopting a solver
based on a massless variational derivative of the elastic
energy for the deformable solid. The diffuse IBM and the
solid solver demonstrated the feasibility of achieving predictions of the thin membrane, which were shown to be
realistic. The model could not, however, correctly predict
fluid surface stresses on the immersed surface due to the
penalty-based IBM forcing. The current model addresses
22
E. AKRAMI ET AL.
Figure 18. Temporal evolution of the magnitude of displacement of the deformable body (a) AP = 3 and (b) AP = 1.58 against
experimental results in the flow-induced deformations test.
this shortcoming, allowing for an accurate behaviour prediction of the deformable body. This case is considered
here to confirm the suitability of the modelling solution
for high Reynolds cases with a low density ratio between
the flexible immersed structure in the presence of fast
moving immersed rigid surfaces.
Figure 20 provides an overview of the pump, showing the impeller inside the volute. The specific speed
of√the pump is taken as ns = 56.4 which is defined as
N Q/H 0.75 , where N,Q, and H are rotational speed
[rpm], flow rate and head rise of the pump at the best
efficiency point (BEP), respectively. The impeller with
Dimp = 0.23[m] is rotating at ω = 150.8[rad/s] at its
BEP flowrate QBEP = 0.055m3 /s. The impeller Reynolds
number is defined as Ur Dimp /ν, where Ur is the circumferential velocity of the outer side of the impeller,
which is calculated as Re = 4.66 × 106 in current problem. The impeller is modelled by the IBM model as a
rotating rigid body using a surface mesh derived from
an STL file definition. A deformable membrane has been
placed at the pump inlet to characterize its dynamics
and behaviour crossing the pump with 10 × 5 × 0.3cm.
A velocity inlet boundary condition and pressure outlet
have been imposed on the pump inlet and outlet, respectively, while the volute is considered as a non-slip wall.
The impeller boundary condition is regarded as a surface
mesh of IBM and surfaces of the deformable body are
modelled as a strongly coupled FSI-IBM model, which
can span freely in the fluid zone. Physical properties and
coefficients are taken as Table 3 for fluid and deformable
body. The gravity force is imposed by defining the gravitational force along the vertical direction, which triggers
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
23
Figure 19. Power spectral density from displacement signals comparing the experimental and numerical results for (a) AP = 3 and
(b) AP = 1.58.
the buoyancy effect on the deformable body. (Specklin,
2018) found the optimum fluid grid size of a single blade
impeller as Dimp /100 which leads to 0.002m cells size
and 2 million cells in this problem. The deformable body
has been discretised by 2800 incomplete P2 elements and
14,355 nodes with 0.0015 [m] grid size. Moreover, a twostep AMR refines fluid cells around the impeller and the
deformable body to increase the accuracy of the gradient
calculation and the interaction of the fluid and solids. The
simulation time step has been taken as t = 1 × 10−5 s
for both fluid and deformable body models in a way that
the Courant number is maintained at less than 0.6.
The non-dimensional time and non-dimensional residence time are defined as τ = t/t ∗ and τr = t/t ∗ ,
respectively, where t is the real-time and, t is the time
taken by the deformable body to pass through the volute
and t ∗ = volute net volume/flowrate is the turns-over
Figure 20. Perspective view of the submersible pump problem configuration: Single-blade impeller (blue) inside the volute
(transparent). The impeller rotates clockwise around the y-axis.
time of the pump. The non-dimensional residence time
τr represents how many times the fluid inside the pump
should change to remove the deformable body once it
enters the volute.
24
E. AKRAMI ET AL.
Figure 21. The trajectory of the middle point of the deformable body as it flows through the volute is shown with the red spheres. Key
moments are highlighted by blue points, and occur at τ1 = 0.0338, τ2 = 0.0619, τ3 = 0.1224, τ4 = 0.1322, τ5 = 0.1561, τ6 = 0.1576,
τ7 = 0.1590, τ7 = 0.1590 and τ9 = 0.3334. (a) top view, (b) side view from the outlet line of sight and (c) side view showing the length
of the outlet pipe.
The non-dimensional residence time is calculated
as τr = 0.3391 indicating that the pump can flush the
deformable body in approximately one-third of the pump
turnover time. A set of points in time have been selected
to demonstrate the ability of the model to capture key
behaviour and challenging FSI events, including collision
with fast moving impeller. These are marked on the trajectory of the centre of gravity of the membrane Figure 21
where the time interval between successive red dots is
τ = 0.0014.
The deformable body is released horizontally at the
inlet of the volute. Downstream flow experiences a rotational velocity due to the impeller rotation. As a result, the
membrane moves around the inlet pipe and collides twice
with the wall at points 1 and 2, as shown in Figure 21.
The primary and secondary collisions are depicted in
Figure 22. The mechanical response of the deformable
body is shown in terms of the von Mises stress [Pa]
used as an indicator of the yielding of a ductile body.
The strong impact and pressure gradient of fluid in the
secondary collision (point 2) causes the rotation of the
deformable body.
Thereafter, the deformable body is sucked directly into
the impeller by the low-pressure zone in the impeller eye
and experiences a two collision events with the top part
of the impeller (Point 3 and Point 4). The first impact
with the impeller (Point 3) induces a rotation of the
deformable body, and the second (Point 4) serves to align
the membrane with the horizontal plane at the top plate
of the impeller. Pressure gradients and the centrifugal
forces drive the membrane out of the impeller region. The
velocity contour plot over the middle horizontal plane of
Figure 23 illustrates the level of fluid mixing involved and
the effect on the deformable body at τ = 0.1435 while the
deformable body is leaving the impeller.
After that point, the deformable body approaches
the volute wall (Point 5) and collides with the leading
edge of the impeller at τ = 0.1576 which is depicted in
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
25
Figure 22. Collision of the deformable body with the inlet pipe of the volute and the top part of the impeller depicting points 1, 2, 3 and
4 marked in Figure 21. Contour colouring based scaled by the von Mises stress [Pa].
Figure 24 at Point 6. This strong impact induces a strong
acceleration by the deformable body, propelling it rapidly
out of the impeller.
After a complete rotation, the deformable body starts
sliding along the volute wall pulled by the flow. It collides and repeatedly slides, as illustrated in Figure 25
at Point 8. The last significant impact Happens after
passing through the throat area of the volute when the
deformable body detaches from the curved volute wall
and approaches the diffuser wall directly (Figure 25 (b)
at τ = 0.3334, Point 9). The deformable body exits the
domain after that.
This final test case has confirmed that the proposed
methodology is able to achieve stable simulations under
challenging conditions with low density ratio (ρdb /ρf =
1.18) and collision with fast moving boundaries. Developing such a robust analytical tool can help better
understand the dynamic response of a deformable body
in complex engineering applications, including for the
design of reliable wastewater pumps.
4. Conclusion
A novel robust FSI framework has been proposed to
model slender deformable bodies and address issues
arising from low-density ratios and compounded by
high Reynolds flow and spurious forces caused by fast
moving immersed solids. A second-order IBM with
a velocity reconstructed sharp penalty approach has
been adopted to capture precise gradients around the
26
E. AKRAMI ET AL.
Figure 23. The contour of fluid velocity magnitude at a cross-section of the pump at when the deformable body is pushed outside of
the impeller as the consequence of centrifugal force.
Figure 24. Interaction of the deformable body with the leading edge of the impeller depicting points 5, 6 and 7 marked in Figure 21.
immersed body, which is crucial for accurate FSI prediction. The deformable body has been modelled using a
nonlinear IFEM method by P2 elements with a Hyperelastic material model. The interaction of the fluid and
solid is coupled through a strong algorithm to reduce the
spurious forces and maximize the stability of the solution
in low density ratio. The fluid flow solver has been coupled to a modified IBM-DES turbulent model to simulate
high Re number problems. A modified compact support domain MLS data mapping approximation has been
implemented in the coupling to transfer information
between fluid flow and deformable body. The solution
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
27
Figure 25. Collision of the deformable body with the volute and leaving the pump depicting points 8 and 9 marked in Figure 21.
has been assessed by comparison against two laminar
FSI benchmark cases and four turbulent FSI experiments,
including forced oscillations and flow-induced deformations for two different aspect ratios of the deformable
body. Second or higher-order mesh convergence was
achieved with the Force Oscillation case when combining
uniform mesh refinement with AMR for local refinement
around the immersed object. The comparisons confirm
the ability of the FSI model to achieve stable and accurate simulation, including at low-density ratios and with
high Reynolds number flow. The study of a slender flexible membrane transport through a centrifugal pump
has also confirmed that stable computations are possible
with immersed deformable bodies experiencing multiple
impacts with fast moving boundaries. For this purpose, a
new contact model inspired by the elastic collision and
a frictionless contact model has been developed. This
study exhibits certain limitations that merit attention in
future research. First, the comparison of first-order statistics from Flow Induced FSI in turbulent flow showed
key similarities in the oscillatory behaviour measured by
the frequency and amplitude of motion but also some
non-negligible differences. A factor which may have contributed to these differences is the simplified inlet turbulence boundary condition used. Future investigations
should consider using detailed experimental data to specify time and spatial fluctuations at the inlet. Additionally,
while the proposed contact model has shown promise
in simulating multiple impacts with fast-moving boundaries, it would be useful to consider factors like elastic collision and frictional contact for a more realistic
simulation of solid–solid interaction with and without
lubrication.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
This research was funded by Sulzer Ltd, Enterprise Ireland’s
Innovation (EI) Partnership scheme (IP/2017/0674) and the
Irish Research Council (IRC) (EBPPG/2019/12). Computation
of all simulations has been performed on Kay cluster of Irish
Centre for High-End Computing (ICHEC) through dceng009c
and dceng010c Class C projects.
ORCID
Ehsan Akrami http://orcid.org/0000-0002-9819-7192
Mathieu Specklin http://orcid.org/0000-0001-9119-4418
Yan Delaure http://orcid.org/0000-0002-7151-9278
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This section provides further details of the Mooney-Rivlin
model, and the nonlinear FEM deformable body model presented in section 2.2.
(A1)
J1 =
−1/3
I 1 I3
(A2)
J2 =
−2/3
I 2 I3
(A3)
1/2
J3 = I 3
(A4)
I1,E = 21
(A5)
I3,E = (2 + 4trE)1 − 4E +
9
eimn ejrs Emr Ens
4
(A6)
(A7)
1
−4/3
)I1,E − I1 (I3 )I3,E
3
2
−2/3
−5/3
J2,E = (I3 )I2,E − I2 (I3 )I3,E
3
1 −1/2
J3,E = (I3 )I3,E
2
∂S
= A10 J1,EE + A01 J2,EE + k(J3 − 1)J3,EE
D=
∂E
+ kJ3,E ⊗ J3,E
−1/3
J1,E = (I3
(A8)
(A9)
(A10)
(A11)
1 (−4/3)
− I3
(I1,E ⊗ I3,E + I3,E ⊗ I1,E )
3
4 (−7/3)
1 (−4/3)
+ I 1 I3
I3,E ⊗ I3,E − I1 I3
I3,EE
(A12)
9
3
2 (−5/3)
(−2/3)
= I2,EE I3
− I3
(I2,E ⊗ I3,E + I3,E ⊗ I2,E )
3
2 (−5/3)
10 (−8/3)
I3,E ⊗ I3,E − I2 I3
I3,EE (A13)
+ I 2 I3
9
3
1 (−3/2)
1 (−1/2)
= − I3
I3,E ⊗ I3,E + I3
I3,EE
(A14)
4
2
=0
(A15)
(−1/3)
J1,EE = I1,EE I3
J2,EE
J3,EE
I1,EE
I2,EE = 41 ⊗ 1 − I
I3,EE = 4I3 C
−1
⊗C
(A16)
−1
− I3 C
−1
IC
−1
(A17)
Here shows the number of nodes per element.
⎡
N1
N=⎣0
0
t
0
N1
0
0
0
N1
N2
0
0
...
...
...
⎤
0
0⎦
Nn 3×3n
(A18)
B = t BL0 + t BL1
t
(A19)
t
BL1 = A(θ) G
⎡
N1,x
0
⎢ 0
N1,y
⎢
0
⎢ 0
t
BL0 = ⎢
⎢N1,y N1,x
⎣ 0
N1,z
N1,z
0
⎡
Appendix
S = A10 J1,E + A01 J2,E + k(J3 − 1)J3,E
I2,E = 4(1 + trE)1 − 4E
N1,x
⎢N1,y
⎢
⎢N1,z
⎢
⎢ 0
⎢
t
G=⎢ 0
⎢ 0
⎢
⎢ 0
⎢
⎣ 0
0
Nn,x =
0
0
0
N1,y
N1,x
N1,z
0
0
0
0
0
N1,z
0
N1,y
N1,x
N2,x
0
0
N2,y
0
N2,z
...
...
...
...
...
...
0
0
0
0
0
0
N1,y
N1,x
N1,z
N2,x
N2,y
N2,z
0
0
0
0
0
0
...
...
...
...
...
...
...
...
...
⎤
(A20)
0
0 ⎥
⎥
Nn,z ⎥
⎥
0 ⎥
Nn,y ⎦
Nn,x 6×3n
(A21)
⎤
Nn,x
Nn,y ⎥
⎥
Nn,z ⎥
⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
⎥
0 ⎦
0 9×3n
(A22)
∂Nn
−1 ∂Nn
−1 ∂Nn
−1 ∂Nn
= J11
+ J12
+ J13
∂x
∂ξ
∂η
∂μ
(A23)
ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS
Nn,y =
∂Nn
−1 ∂Nn
−1 ∂Nn
−1 ∂Nn
= J21
+ J22
+ J23
∂y
∂ξ
∂η
∂μ
∂Nn
−1 ∂Nn
−1 ∂Nn
−1 ∂Nn
= J31
+ J32
+ J33
∂z
∂ξ
∂η
∂μ
⎡
∂dy
∂dx
0
0
∂x
⎢ ∂x
∂dx
0
0
⎢ 0
∂y
⎢
∂dx
⎢ 0
0
0
⎢
∂y
A(θ) = ⎢ ∂dx ∂dx
∂dy
⎢ ∂y
0
∂x
∂y
⎢
⎢ ∂dx
∂dy
∂dx
0
⎣ ∂z
∂x
∂z
∂dx
∂dx
0
0
∂z
∂y
⎤
∂dw
0
0
0
0
∂x
⎥
∂dy
∂dw
0
0
0 ⎥
∂x
∂x
⎥
∂dy
∂dw ⎥
0
0
0
∂x
∂x ⎥
∂dy
⎥
∂dw
∂dw
0
0 ⎥
∂x
∂y
∂x
⎥
∂dy
∂dw
∂dw ⎥
0
0
∂x
∂z
∂x ⎦
∂dy
∂dy
∂dw
∂dw
0
∂z
∂y
∂z
∂y
Nn,z =
∂Nn
∂dx
=
dxn
∂x
∂x
−1 ∂Nn
−1 ∂Nn
−1 ∂Nn
+ J12
+ J13
=
dxi J11
∂ξ
∂η
∂μ
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(A24)
(A25)
(A26)
(A27)
∂dx
∂Nn
=
dxn
∂y
∂y
−1 ∂Nn
−1 ∂Nn
−1 ∂Nn
+ J22
+ J23
=
dxi J21
∂ξ
∂η
∂μ
∂Nn
∂dx
=
dxn
∂z
∂z
−1 ∂Nn
−1 ∂Nn
−1 ∂Nn
+ J32
+ J33
=
dxi J31
∂ξ
∂η
∂μ
⎡t (i)
⎤
0̄
0̄
σ̃
t (i)
t σ̃ (i)
σ = ⎣ 0̄
0̄ ⎦
t σ̃ (i)
0̄
0̄
⎡
⎤
0 0 0
0̄ = ⎣0 0 0⎦
0 0 0
⎡
⎤
t τ (i)
t τ (i)
t τ (i)
11
12
13
⎢ (i) t (i) t (i) ⎥
t (i)
σ̃ = ⎣t τ21
τ22
τ23 ⎦
t τ (i)
t τ (i)
t τ (i)
31
32
33
t (i)
t (i) t (i) t (i) t (i) t (i) t (i)
σ̂ = τ11 , τ22 , τ33 , τ12 , τ23 , τ31
31
(A28)
(A29)
(A30)
(A31)
(A32)
(A33)