Download An immersed-shell method for modelling fluid–structure interactions

Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
rsta.royalsocietypublishing.org
An immersed-shell method
for modelling fluid–structure
interactions
A. Viré1 , J. Xiang2 and C. C. Pain2
Research
1 Wind Energy Group, Faculty of Aerospace Engineering,
Cite this article: Viré A, Xiang J, Pain CC. 2015
An immersed-shell method for modelling
fluid–structure interactions. Phil. Trans. R.
Soc. A 373: 20140085.
http://dx.doi.org/10.1098/rsta.2014.0085
Delft University of Technology, Kluyverweg 1, 2629 HS Delft,
The Netherlands
2 Applied Modelling and Computation Group, Department of Earth
Science and Engineering, Imperial College London,
London SW7 2AZ, UK
AV, 0000-0003-0056-8213
One contribution of 17 to a theme issue
‘New perspectives in offshore wind energy’.
Subject Areas:
energy, ocean engineering, structural
engineering, fluid mechanics
Keywords:
fluid–structure interactions, immersed-body
approach, aerodynamics
Author for correspondence:
A. Viré
e-mail: [email protected]
The paper presents a novel method for numerically
modelling fluid–structure interactions. The method
consists of solving the fluid-dynamics equations on
an extended domain, where the computational mesh
covers both fluid and solid structures. The fluid and
solid velocities are relaxed to one another through
a penalty force. The latter acts on a thin shell
surrounding the solid structures. Additionally, the
shell is represented on the extended domain by a
non-zero shell-concentration field, which is obtained
by conservatively mapping the shell mesh onto
the extended mesh. The paper outlines the theory
underpinning this novel method, referred to as the
immersed-shell approach. It also shows how the
coupling between a fluid- and a structural-dynamics
solver is achieved. At this stage, results are shown for
cases of fundamental interest.
1. Introduction
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rsta.2014.0085 or
via http://rsta.royalsocietypublishing.org.
The analysis of fluid–structure interactions (FSIs) is
important in many areas, for example, to simulate
the dynamics of turbine blades, the wave impacts on
floating bodies, the blood flow through arteries or
pollutant dispersion in street canyons [1–3]. Numerical
models are attractive in this context, because they can
tackle different configurations while limiting expensive
laboratory testing. For flows past rigid and still structures, the Navier–Stokes equations are usually solved on
a mesh excluding the solid structures (defined-body (DB)
method). This technique can give accurate results, when
using appropriate discretization schemes and sufficient
2015 The Author(s) Published by the Royal Society. All rights reserved.
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
2
.........................................................
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
spatial resolution, because the structure shape and boundary conditions are represented exactly.
However, when the structure moves in the fluid domain, re-meshing is necessary. This is
the case, for example, for wind turbine rotors or floating wind turbines. In this context, the
whole structure dynamically interacts with the surrounding wind and waves. Re-meshing is
computationally expensive and might also yield highly distorted grids. In order to avoid this
problem, another approach was developed. It consists of meshing the entire domain (containing
both fluid and structures) and modelling the effect of the structures through surface or body
forces. Because the effect of the structures is modelled through an additional forcing term
in the fluid’s momentum equation, the methods can be further divided into two categories
depending on whether the force is added to the momentum equation before discretization
(continuous forcing) or after discretization (direct forcing). This methodology underpins the socalled immersed-boundary method, proposed by Peskin [2] in a continuous-forcing context for
modelling cardiac mechanics. Reviews and applications of immersed-boundary type methods are
available in the literature [4–6]. Among the variants to Peskin’s original formulation, Goldstein
et al. [7] proposed to attach virtual springs and dampers to the immersed boundary, so that
the forcing term acts as a feedback mechanism and tends to oppose any deviations from the
desired values of velocity and position of the structure. This method suffers from limitations in
the time stepping size, owing to the need to resolve the characteristic time scales of the spring–
damper system. In order to overcome this drawback, Fadlun et al. [8] directly forced the desired
velocity at the boundary nodes by modifying the entries in the matrix of the discrete momentum
equations. This direct-forcing method, however, yielded force oscillations due to the interpolation
method between the fixed Eulerian mesh and the moving boundary nodes. Uhlmann [9] solved
this issue by combining Peskin’s original formulation with the direct and explicit formulation
of the FSI force. In this method, a volume was associated with the Lagrangian points placed
on the structure boundary, hence forming a volumetric shell of one mesh width around the
structure. The Lagrangian points followed a rigid-body motion, and smooth variants of Peskin’s
Dirac function were used to transfer information between Cartesian and Lagrangian points.
Kempe & Fröhlich [10] extended this direct-forcing method to further improve the imposition of
the boundary condition at the interface. These methods are particularly attractive for modelling
particulate flows, for which numerical efficiency, particle collisions and adequate representation
of the boundary layers are important. Recently, Viré et al. [11] presented an immersed-body
method with a continuous-forcing approach based on a penalty forcing. In this context, the
fluid and structural velocities were weakly relaxed to one another in the regions covered by
the structures. This method relied on two distinct meshes used by two separate finite-element
models: one mesh covering the entire domain (fluid mesh), and another mesh discretizing solely
the structures (solid mesh). This method is a versatile way of modelling interactions between
fluids and structures, because the use of separate models allows specific numerical requirements
to be met for each material. One of the difficulties, however, is to ensure that the laws of
physics are verified at the discrete level when exchanging information between the models. The
authors proposed a novel algorithm to ensure spatial conservation of the penalty force and solid
concentration field when projected from one mesh to the other. Importantly, spatial conservation
was verified at a discrete level, and therefore was independent of the mesh resolution and the
types of mesh used in the two models. This approach was validated on flows past rigid and
deformable structures, as well as fixed actuator discs representing turbines [11,12]. Although the
immersed-body approach has proven to be successful in modelling FSIs for various applications,
it applies the coupling force across the regions covered by the structures. Thus, the action–reaction
force between fluids and structures—although very small—is not strictly equal to zero within
the structures. This is a common issue in immersed-body type methods. However, this is not
physically meaningful and can potentially affect the level of stresses computed by the structuraldynamics model. In addition, the discrete accuracy of the force acting on the structure depends
on the resolution of the solid mesh. This limits the number of structures that can be modelled
in one simulation, owing to the computational cost associated with accurately discretizing each
individual structure.
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
The fluid/ocean dynamics model ‘Fluidity’ is a finite-element open-source (LPGL) numerical
tool that solves the non-hydrostatic Navier–Stokes equations on unstructured meshes. It can
capture small-scale structures and large-scale processes simultaneously by adapting the meshes
dynamically in time [13–15]. In this work, the Navier–Stokes equations are modified to account for
the presence of immersed solids in the fluid domain. The idea underpinning the IS method is to
solve the Navier–Stokes equations over a fluid mesh covering the extended domain V = Vf ∪ Vs
(where the computational domain V contains fluids in Vf and structures in Vs ). The first step is to
fill the regions covered by the structures with the surrounding fluid, so that the fluid velocity uf
exists across the computational domain V. In this context, the continuity equation in the extended
domain V states that
V · uf = 0.
(2.1)
The second step consists of representing the effect of the structures on the flow. This is achieved
by relaxing the fluid and structural velocities to one another in a thin shell surrounding the
structures. The shell is meshed separately from the extended domain. A shell concentration field
αsh identifies, on the fluid mesh, the region occupied by the shell. It is computed by conservatively
mapping the shell mesh onto the fluid mesh, as explained in §3. The relaxation between fluid and
structural velocities is performed through a penalty force, which is non-zero in the fluid elements
where the shell concentration field is non-zero. The penalty force F f is added to the right-hand
side of the fluid’s momentum equation, which yields
ρf
Duf
= V · σf + Bf + F f ,
Dt
(2.2)
where ρf is the fluid density, σf is the total stress tensor (including the pressure contribution) and
Bf represents additional body forces (e.g. due to gravity). The penalty force is expressed as
F f = βαsh (us − uf ),
(2.3)
.........................................................
2. Mathematical formulation
3
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
The present immersed-shell (IS) method aims at improving the representation of the action–
reaction force at the fluid–structure interface. The proposed methodology has some similarities
with both the immersed-body approach [11] and the direct-forcing methods [9,10]. First, a
penalty force is added to the momentum balances of each material in order to relax the fluid
and structural velocities to one another. Second, the force acts in the regions of non-zero solid
concentrations. However, as opposed to the immersed-body method, the exchange of forces
between fluid- and structural-dynamics models is entirely done via a shell surrounding the
structures. As a result, the penalty force exactly vanishes within the structures. A better discrete
representation of the boundary layer can also be achieved. Finally, as opposed to the existing
direct-forcing methods [9,10], the shell thickness can be chosen independently of the mesh
size. The projection of fields between the Eulerian fluid mesh and the shell mesh is also done
differently, as explained herein. The paper presents the general formalism of the method and
validates it on flows of fundamental interest. Further work will focus on implementing different
wall parametrizations in the shell region, in order to tackle high Reynolds number flows that are
relevant for practical aerodynamics applications. The aim of this paper is twofold. First, it presents
the theoretical concepts underpinning the IS method. Section 2 focuses on the set of equations
solved by the coupled models, whereas §3 details the projection steps required to transfer fields
between fluid, shell and solid meshes. The sequence of steps required in the coupling approach
is summarized in §4. Second, the paper demonstrates the capabilities of the present method on
flows of fundamental interest (§5). The method is validated on flow past a fixed two-dimensional
cylinder, an oscillating cylinder and a free-falling sphere. Finally, conclusions are drawn in §6, and
a strategy is proposed to further apply the method to engineering problems, such as innovative
offshore wind energy concepts.
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
where the fluid uf and structural us velocities are expressed in the thin shell, and
is a relaxation factor that dictates how fast the fluid and structural velocities equal one another
at the interface. In equation (2.4), t is the time step, L is the local edge length and γ = le /Δ
is the ratio between the minimum edge length le of the fluid mesh around the shell and the
shell thickness Δ, when the fluid mesh does not resolve the shell. In the limiting case where the
shell is resolved (le < Δ), γ = 1 is considered. In addition, the magnitude of the relaxation factor
β is driven by viscous effects at small Reynolds numbers and inertial effects at large Reynolds
numbers. In this study, it is assumed to be driven solely by inertial effects.
The structural dynamics is solved on a separate model. The combined finite–discrete element
model ‘Y3D’ is coupled to ‘Fluidity’ and solves the deformable body equations using a finitestrain formulation [16,17]. This can compute the stresses and vibration modes of structures of any
shape and stiffness. The equations governing the dynamics of deformable structures on the solid
mesh are given by
Lds + F int = F ext + F c + F s ,
(2.5)
where ds denotes the solid displacement, L is an operator dependent on the velocity gradient,
F int stands for the internal forces, F c is the contact force when multiple structures impact on one
another, and F ext is the external force including the surface traction force and all the body forces
other than the action–reaction force exerted by the fluid. In order to account for the presence
of the surrounding fluid, the penalty force is added to the right-hand side of the structure’s
momentum balance. Similar to the force in the fluid-dynamics equations, the penalty force acting
on the structures is expressed as
F s = βαsh (uf − us ).
(2.6)
By virtue of the action–reaction principle, the forces are such that
Vf
F f dΩf = −
Vs
F s dΩs
(2.7)
at a discrete level, as explained in §3. The equations of motion are discretized using finite
elements. Time is discretized using a Crank–Nicolson scheme in the fluid-dynamics model,
and a backward Euler scheme in the structural-dynamics model. The specific discretization
used in ‘Fluidity’ and ‘Y3D’ is explained in reference [11]. The expression for the penalty force
is analogous to the discontinuous Galerkin (DG) form of the viscous stresses, as highlighted
hereafter. In the momentum balance of the fluid outlined by equation (2.2), the viscous stress
tensor σf is given by
∂uj
2
∂ui
+
− δij V · uf ,
(2.8)
σij = μij
∂xi
∂xj
3
where the Kronecker symbol has the usual meaning, i.e. δii = 1 and δij = 0 for i = j. By denoting
the velocity components by uf = (u, v, w), the viscous stress tensor expands to
∂u
2
∂v
∂w
,
σxx = μxx 2
−
+
3
∂x
∂y
∂z
σxy = μxy
∂v
∂u
,
+
∂x
∂y
σxz = μxz
∂w ∂u
,
+
∂x
∂z
(2.9)
and similarly for the other components. Figure 1a illustrates two neighbouring finite elements,
where the solution fields are discontinuous at the nodes sharing the boundary between E− and
.........................................................
(2.4)
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
ρf ν
,
t L2
β = γ max
4
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
(a)
neighbouring discontinuous
elements in two dimensions
fluid and solid
velocities in one dimension
5
velocity jump
E
+
uf
fluid element
solid element
shell region
Figure 1. Analogy between (a) the discontinuous Galerkin discretization and (b) the interface between fluid and solid. (Online
version in colour.)
E+ . The DG discretization of the viscous force on such elements is given by
∂Ni
∂Ni
∂Ni
Ni (σxx nx + σxy ny + σxz nz ) dΓ ,
for u :
σxx +
σxy +
σxz dV +
∂x
∂y
∂z
E
∂E
∂Ni
∂Ni
∂Ni
Ni (σyx nx + σyy ny + σyz nz ) dΓ ,
for v :
σyx +
σyy +
σyz dV +
∂x
∂y
∂z
E
∂E
∂Ni
∂Ni
∂Ni
σzx +
σzy +
σzz dV +
Ni (σzx nx + σzy ny + σzz nz ) dΓ ,
for w :
∂x
∂y
∂z
E
∂E
(2.10)
(2.11)
(2.12)
where Ni is the basis function at node i. After replacing the components of the stress tensor
in equation (2.10) by their values from equation (2.9) (and similarly for v and w), DG methods
commonly simplify the aforementioned equations by expressing the surface integrals as
⎫
αxx (u+ − u− ) dΓ , ⎪
for u :
⎪
⎪
⎪
∂E
⎪
⎪
⎪
⎬
+
−
αyy (v − v ) dΓ ,
for v :
(2.13)
⎪
∂E
⎪
⎪
⎪
⎪
⎪
⎭
α (w+ − w− ) dΓ ,⎪
for w :
∂E
zz
where αxx , αyy and αzz are penalization factors. Consequently, the above expressions penalize
the jump in each component of the velocity at the element boundaries. This method of coupling
the stress terms between the elements of a DG mesh is similar to the present coupling term that
penalizes the discontinuity between fluid and structural velocities in the shell region (figure 1b).
3. Spatial projections between fluid, solid and shell meshes
In the equations presented in §2, the fluid velocity and pressure are computed on the fluid mesh,
whereas the structural velocity is computed on the solid mesh. In order to couple these quantities,
and compute the penalty force on each mesh, the fields need to be projected from fluid to shell
mesh (and vice versa), and from the volumetric shell mesh to the solid surface mesh where the
fluid forces are applied. This section shows how these projections are performed. The sequence
in which the projections are done, and the fields they apply to, is explained in §4.
In order to project a field from shell to fluid mesh, the intersections between the two meshes
are first identified. This can be done using either an R-tree spatial indexing algorithm [18] or an
advancing front algorithm [19]. For each element of the shell mesh, a so-called supermesh is formed
from the set of all the intersections, following a local supermeshing method [20]. Importantly, the
order of the basis functions on the supermesh is equal to the maximum of their orders on the fluid
.........................................................
us
–
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
E
(b)
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
3¢
6
3
D
2
1
Figure 2. Sketch of the mesh associated with a shell of thickness Δ. The shaded element discretizes the solid surface.
and shell meshes. As a result, the projected field can be represented accurately on the supermesh.
The mesh-to-mesh projection is performed via a Galerkin projection. The latter is used to obtain
the shell concentration field αsh on the fluid mesh, and also to project the coupling force from shell
to fluid mesh. Given a donor mesh D and a target mesh T on a domain Ω, a Galerkin projection
on a field q from D to T ensures that
qD dV =
qT dV
(3.1)
Ω
Ω
by minimizing the L2 norm of the interpolation error [20]. In a discrete form, equation (3.1)
involves solving a linear system of equations containing a mass matrix based on the basis
functions of the target mesh and a mixed mass matrix formed of the basis functions of the target
and donor meshes. For example, the shell concentration field αsh on the fluid mesh is computed as
nf
V
j=1
(αsh )j Njf Nkf dV =
nsh
Vsh
Ui Nish Nkf dV,
(3.2)
i=1
where the shell concentration field on the shell mesh is by definition a unitary field U, and nf
(resp. nsh ) denotes the number of nodes in the fluid (resp. shell) mesh. In addition, superscripts
refer to the mesh on which the quantity is defined (‘f’ for the fluid mesh and ‘sh’ for the shell
mesh). Such a Galerkin projection ensures spatial conservation, but modifies the bounds of the
projected quantity. It can be shown that lumping the mass matrix of the target mesh bounds the
resulting solution without affecting the conservation property. The detailed projection methods
between fluid and shell meshes are identical to those used in the immersed-body method [11].
This supermeshing approach differs from the transfer method used in similar direct-forcing
methods [9,10,21]. The latter use a smoothed Dirac delta function over one mesh element around
the structure, and therefore the interpolation between Lagrangian points and Eulerian nodes is
performed via a weighted sum of regularized Dirac functions.
Once the fluid forces are known on the shell mesh, they need to be applied on the solid
surface. Similarly, the solid quantities at the solid surface need to be mapped to the shell, before
being projected to the fluid mesh. These procedures are described hereafter. The shell mesh is
directly constructed from the triangular mesh discretizing the solid surface. This is illustrated in
figure 2, where the shaded triangular element 1–2–3 lies on the solid surface. The shell is formed
by extruding this element in the normal direction over a length Δ, which is the shell thickness.
Thus, for each surface element at the solid surface, three tetrahedral elements are added to form
the shell volumetric mesh, i.e. element 1–2–3–1 , element 2–2 –3–1 and element 2 –3 –3–1 . When
projecting a field from the solid surface to the volumetric shell, the shell is assumed to be much
thinner than the characteristic length of the solid. Consequently, it is a good approximation to
consider that the projected field q at a node on the outer shell equals the field at the corresponding
node on the inner shell (q1 = q1 ). For example, the structure velocity at the solid surface is
.........................................................
2¢
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
1¢
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
Ss
where Ss denotes the surface of the structure and superscript notation identifies the mesh on
which the field lies (i.e. ‘sh’ for the shell mesh and ‘surf’ for the surface mesh of the structure). At
sh
qi Vsh,i . The discrete values
each shell nodes, the left-hand side of equation (3.3) is given by ni=1
at the nodes 1 –2 –3 , which are located on the outer shell boundary, need to be mapped on the
corresponding nodes 1–2–3 situated on the inner shell boundary. The mapping is such that the
projected field at node 1 equals q1 Vsh,1 + q1 Vsh,1 .
4. Coupling algorithm between fluid- and structural-dynamics models
This section explains in more detail the coupling mechanism between the fluid- and structuraldynamics equations. The coupling is achieved through the penalty force across the shell, which
ensures that, at a discrete level, the penalty force on the fluid mesh F f is given by
F f = βαsh (us − uf ) = Gsh−f [β(us − uf )sh ].
(4.1)
In equation (4.1), Gsh−f [q] denotes the conservative Galerkin projection from shell to fluid mesh
of a field q, as detailed in §3. The detailed sequence of steps of the IS method is summarized
hereafter. Steps (1)–(3) are done before the simulation starts, whereas the other steps are repeated
at every time step.
(1) The computational domain covering both fluids and structures is meshed by the so-called
fluid mesh, which is used to spatially discretize equation (2.2).
(2) The structural domain is meshed by the so-called solid mesh, which is used to spatially
discretize equation (2.5).
(3) A shell is constructed around the structures (§3). The shell mesh is used to exchange the
coupling term between fluids and structures. Note that the shell mesh is superimposed
on the fluid mesh, and, importantly, does not necessarily coincide with the elements of
the fluid mesh.
(4) A unitary field is projected from the shell mesh to the fluid mesh using a Galerkin
projection. This yields a shell concentration field αsh on the fluid mesh, such that
V
αsh dΩ =
dΩsh = Vsh ,
(4.2)
Vsh
where Vsh is the shell volume.
(5) Two fields are projected from fluid to shell mesh: the pressure p, and the part of the
(1)
penalty force that depends on the fluid velocity, i.e. F s = βαsh uf .
(1)
(6) The pressure p and force F s are mapped from the shell to the solid surface (§3).
(7) The structural velocity us is computed by solving the structural-dynamics equations on
the solid mesh, see equation (2.5). The surface integral of p represents the buoyancy
force and dynamic pressure forces exerted by the surrounding fluid. It is included in
the external forces in equation (2.5). The penalty force is also applied as a surface force
in the structure’s momentum balance. Subcycling is required if the time step used to
discretize the structural-dynamics equations is smaller than that used to discretize the
fluid-dynamics equations. The total penalty force is updated based on the structural
(1)
velocity at the previous time iteration i, as F s = F s − βαsh uis .
.........................................................
Vsh
7
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
constant across the shell before projecting it to the fluid model. This assumption is not made
when projecting a field from a volumetric shell to a solid surface. For the latter projection, the
condition to satisfy is
qsh dVsh =
qsurf dSs ,
(3.3)
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
The algorithm described in the previous section is applied to three cases: flow past a twodimensional cylinder, forced oscillation of a two-dimensional cylinder in a fluid at rest, and a
fully coupled problem of flow past a sphere falling under the effect of gravity.
(a) Flow past a two-dimensional cylinder
Flow past a two-dimensional cylinder is chosen as a verification/validation case owing to the
number of references available in the literature. The computational configuration is as follows.
The two-dimensional cylinder of diameter D is centred at a distance of 5D away from the inlet,
10D away from the sides, and 20D away from the outlet. It is subjected to a uniform flow
of streamwise velocity u0 , and four Reynolds numbers are considered: ReD = u0 D/ν = 40, 100,
200, 500. In all the simulations, the Courant number is fixed at CFL = 0.1. The fluid mesh is
unstructured and does not change in the course of time. The typical value of the element edge
lengths at the outer boundaries is Le = D, whereas the minimum edge length le near the cylinder
is indicated in table 1. The domain is discretized using a mixed finite-element discretization
method, where discontinuous linear polynomials are used for the velocity field (i.e. P1-DG
discretization) and continuous piecewise quadratic polynomials are used for the pressure field
(i.e. P2 discretization). A well-known difficulty in representing solid structures as immersed in an
extended computation domain is the fact that sharp fluid–structure interfaces are smeared on the
computational cells. This affects the accuracy with which forces, and hence boundary conditions,
are represented on the structure surface. A related issue concerns the computation of diagnostic
variables. In particular, the integral of the pressure and viscous forces on the bluff-body surface
is not straightforward owing to the smeared representation of the structure. Despite extensive
literature on the numerical modelling of FSIs, this post-processing difficulty has been scarcely
addressed [22,23]. This study shows that one of the main differences between DB and immersedbody type approaches is the location of the pressure extrema. In the DB approach, the maximum
pressure occurs at the cylinder boundary, which is in accordance with experimental observations.
In the IS approach, the pressure lines close in the shell as shown by figure 3. Thus, the pressure
extrema are located close to the first node of the fluid mesh where the shell concentration field
equals zero. This occurs at a distance R ≈ R + Δ + le from the cylinder centre, where R is the
cylinder radius, Δ is the thickness of the shell surrounding the cylinder and le is the edge length
of the fluid-mesh elements around the shell. This is only an estimation of R , because it further
depends on where the shell and fluid meshes intersect one another within one fluid element.
As a result, the aerodynamic forces in the IS results are obtained by integrating the pressure
and viscous forces on the surface of the fictitious cylinder of diameter D = 2R . Table 1 shows
the drag coefficient obtained using the DB and IS approaches. The results obtained by the two
methods are very close, and also match the values reported in the literature [24,25]. This is also
apparent from the time evolutions of the lift CL and drag CD coefficients at Re = 200, as shown by
figure 4. The continuous line shows the result for the IS method, circles show the solution of Rajani
et al. [26], and triangles show the solution of Meneghini et al. [27]. The agreement is acceptable,
also considering the dispersion of the results given by other studies, see for example [28]. Finally,
based on the frequency of oscillations of the drag coefficient at Re = 100, the Strouhal number was
found to equal St = fD/u0 = 0.166, which is also in agreement with the literature.
.........................................................
5. Results
8
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
(8) At the end of the structural time stepping loop, a time-averaged structural velocity ūs is
computed from the values at each time iteration i.
(1)
(9) The penalty force F f = βαsh ūs − F s is mapped from the solid to the shell mesh, as
explained in §3. The force on the shell mesh is then projected to the fluid mesh using
a conservative Galerkin projection (§3).
(10) The fluid velocity uf is computed by solving the fluid-dynamics equations on the fluid
mesh, equation (2.2).
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
9
CD
1.5
1.0
0.5
CL
0
–0.5
–1.0
152
150
154
156
158
160
tu0 /D
Figure 4. Time evolutions of the lift CL and drag CD coefficients at Re = 200. The continuous line shows the results for the IS
method, red circles show the solution of Rajani et al. [26], and blue triangles show the solution of Meneghini et al. [27]. Data are
available in the electronic supplementary material. (Online version in colour.)
Table 1. Flow past a cylinder at four different Reynolds numbers: DB and IS approaches. Time series of the drag force are
available in the electronic supplementary material.
case
ReD
approach
Δ/D
le /D
CD
defined-0.025
40
DB
—
0.025
1.61
d0025-l0.025
40
IS
0.025
0.025
1.63
defined-0.025
100
DB
—
0.025
1.42
d0025-l0.025
100
IS
0.025
0.025
1.46
defined-0.025
200
DB
—
0.025
1.39
d0025-l0.05
200
IS
0.025
0.05
1.42
defined-0.025
500
DB
—
0.025
1.43
d0025-l0.05
500
IS
0.025
0.05
1.56
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
.........................................................
Figure 3. Contours of the pressure field for flow past a cylinder for ReD = 100, using the immersed-shell method. (Online
version in colour.)
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
pressure
0.62
0.50
0.20
0
–0.20
–0.50
–0.67
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
(a)
(b)
4
3
1
F̃x 0
−1
−2
−3
−4
0
0.5
t/T
1.0
1.5 0
0.5
t/T
1.0
1.5
Figure 5. Time evolution of the non-dimensionalized in-line force acting on the oscillating cylinder: (a) various shell thicknesses
at le /D = 0.025; (b) various minimum edge lengths at Δ/D = 0.0125. Data are available in the electronic supplementary
material. (Online version in colour.)
Table 2. Values of the minimum element edge length le , shell thickness Δ (non-dimensionalized by the cylinder diameter D)
and run time (4c, four cores; 1c, one core) for the case of an oscillating cylinder in a fluid at rest.
case
Δ/D
le /D
run time (h) per one T (number of cores)
1
0.0125
0.0125; 0.025; 0.05; 0.1
9.95 (4c); 2.79 (4c); 1.26 (4c); 1.42 (1c)
2
0.025
0.025
3.10 (4c)
3
0.05
0.025
2.97 (4c)
4
0.1
0.025
2.27 (4c)
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
(b) Oscillating two-dimensional cylinder in a fluid at rest
The in-line oscillation of a cylinder in a fluid at rest is considered in this section. This is a
classical benchmark problem for studying FSIs with moving objects and is well documented in
the literature [29–31]. This test case is also chosen to show convergence of the numerical results
with increasing resolution and decreasing shell thickness. The cylinder of diameter D is centred
in the computational domain of size 55D × 35D. The domain is discretized with an unstructured
mesh, and the maximum element edge length on the outer boundaries equals Le = D. The mesh
is refined with a minimum element edge length in a region of size 7D × 4D around the cylinder.
Convergence of the results is shown by varying the minimum element edge length le , as indicated
in table 2, in order to assess the effect of the mesh resolution on the computational results. The
translational motion of the cylinder is prescribed as x(t) = −A sin(2π ft), where A is the oscillation
amplitude. The Reynolds and Keulegan–Carpenter numbers are fixed to Re = Umax D/ν = 100
and KC = Umax /fD = 5, respectively, where Umax is the maximum velocity at the cylinder and
f is the characteristic frequency of oscillation. The computations are run with a constant value
of the time step t = 0.0013T, where T = f −1 is the period of oscillation. The simulations were
run on the cx1 cluster of Imperial College London1 and the associated computational time (in
hours) per period of oscillation T is indicated in table 2. Figure 5 shows the time evolution of the
non-dimensionalized in-line force F̃x = Fx /ρDUmax acting on the oscillating cylinder. The symbols
indicate the experimental data [29], whereas the lines are the computational results obtained with
the present method. Figure 5a shows convergence of the results for decreasing values of the shell
thickness at a fixed value of the minimum edge length le /D = 0.025 around the cylinder. It is
1
http://www3.imperial.ac.uk/ict/services/hpc/facilities.
.........................................................
2
10
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
Dütsch et al.
le/D = 0.00125
le/D = 0.0025
le/D = 0.005
le/D = 0.01
Dütsch et al.
case 1
case 2
case 3
case 4
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
The last validation test case is flow past a sphere, freely falling under the effect of gravity in a
fluid at rest. In this configuration, the time evolutions of the structural velocity and forces (both
drag and Basset forces) can be computed semi-analytically by solving the equation of motion of
the sphere [32], i.e.
t
√
1 du(s)
dus
1
= (ρs − ρf )gVs + ρf ACD u2s + 6ρf R2 π ν √
ds,
(5.1)
(ms + ma )
dt
2
t − s dt
0
where ms = ρs Vs is the sphere mass and ma = Cm ρf Vs is the added mass arising from the relative
acceleration of the sphere with respect to the fluid. The added mass coefficient of the sphere is
set at Cm = 12 , which is the theoretical value in inviscid flows [33]. The first term on the righthand side of equation (5.1) is the difference between gravity and buoyancy forces. The second
term represents the drag force due to pressure and shear effects acting on the sphere, A being the
sphere cross-sectional area and CD the drag coefficient. For non-creeping flows, the latter is given
by the following empirical formula [34]:
CD =
24
0.407
(1 + 0.15Re0.681 ) +
.
Re
1.0 + 8710/Re
(5.2)
The asymptotic value of the drag force is such that it balances gravity and buoyancy forces.
The last term in equation (5.1) is the Basset force, which accounts for the time history of the
sphere velocity [35,36]. In this study, equation (5.1) is solved using a fourth-order Runge–Kutta
method and yields the time evolution of the sphere velocity. This semi-analytical solution is used
to validate the results obtained with the IS method. The sphere of diameter D is dropped in
a computational domain of size 10D × 75D × 10D. The minimum and maximum edge lengths
of the fluid mesh are, respectively, le = 0.025D (at the fluid–solid interface) and Le = 5D (at the
domain outer boundaries). The fluid mesh is regenerated at a period of six time steps, in order
to ensure that the fine resolution follows the sphere as it falls. The Courant number is fixed at
CFL = 0.2. The time step is also kept smaller than t = 10−2 , in order to avoid large initial values.
Figure 6 shows the time evolution of the sphere velocity for three values of the Reynolds
number, which is computed as Re = 2ut R/ν (ut being the terminal velocity). Figure 6 compare the
numerical results (symbols) with the semi-analytical results (continuous lines), for two values of
the solid-to-fluid density ratio: ρs /ρf = 1.5 and ρs /ρf = 4. It is shown that the numerical results
agree well with the semi-analytical results. The relative error between the terminal velocity
computed by the coupled models, unt , and the semi-analytical value uat is defined as
E = 100
unt − uat
,
uat
(5.3)
and the values in per cent are as follows: for ρs /ρf = 1.5, E = 2 (Re = 7), E = −4 (Re = 20),
E = −9 (Re = 100); whereas for ρs /ρf = 4, E = −2 (Re = 7), E = −3 (Re = 20), E = −8 (Re = 100).
The same test case was simulated with the original immersed-body method [11] and yielded
an overestimation of the terminal velocity (compared with the semi-analytical solution). The
present IS method underestimates the theoretical value of the terminal velocity, especially at large
Reynolds numbers. This difference could be explained by the way the forcing is imposed in the
momentum equations. In the immersed-body method, the volumetric force is applied across the
entire sphere. The numerical results show a uniform fluid velocity across the sphere (on the fluid
mesh), with a slight velocity overshoot upstream of the sphere and a slight velocity undershoot
.........................................................
(c) Free fall of a sphere
11
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
clear that a sufficiently small shell thickness is necessary in order to have sensible results. This is
expected given that the shell thickness modifies the structure shape, and therefore also impacts on
the flow conditions and dynamics. Figure 5b demonstrates convergence for decreasing values of
le /D at a fixed value of the shell thickness Δ/D = 0.0125. Figure 5b shows that, for a small value of
the shell thickness, the in-line force is less dependent on the mesh resolution around the cylinder.
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
1.0
(b)
rs/rf = 1.5
Re ª 100
2.0
Re ª 20
0.6
Re ª 7
Re ª 7
1.0
0.2
0
Re ª 20
1.5
0.4
Re ª 100
rs/rf = 4
0.5
0.5
1.0
1.5
2.0
time (s)
2.5
3.0
0
0.5
1.0
1.5
time (s)
2.0
Figure 6. Sphere falling under the effect of gravity in a fluid at rest, for different Reynolds numbers: (a) ρs /ρf = 1.5,
(b) ρs /ρf = 4. Symbols show the numerical results using the IS method, whereas continuous lines are semi-analytical
solutions. Data are available in the electronic supplementary material.
downstream of the sphere. By contrast, in the IS method, the volumetric force is solely applied in
the thin shell surrounding the sphere. The numerical results show a uniform fluid velocity in the
thin shell and a slight underestimation of the velocity inside the sphere. In the coupling procedure,
this will in turn affect the sphere velocity in the structural-dynamics model. Further investigation
is needed in order to understand these differences. However, despite the underestimation in the
structural velocity, the discrete forcing term of the IS method seems to better represent the flow
physics around the sphere.
6. Conclusion
This paper presents a novel numerical method for computing FSIs for a wide range of
applications. It consists of immersing the solid structures in an extended fluid domain, and
exchanging the coupling forces through a thin shell surrounding the structures. The use of
a shell is advantageous (i) to resolve or parametrize the physical phenomena taking place at
the fluid–structure interfaces and (ii) to reduce the computational cost particularly when large
structures (or a large number of small structures) are investigated. Importantly, the IS method
works with both structured and unstructured meshes, non-coinciding fluid and solid meshes, and
different discretization schemes for the fluid- and structural-dynamics equations. This versatility
is attractive for many engineering disciplines, including offshore wind energy. For example,
the study of wave–structure interactions requires high-order representations of the velocity and
pressure fields, in order to accurately model wave propagation over several wavelengths. This
study shows that the method provides accurate results on a series of benchmarked test cases.
The present analysis is limited to relatively low Reynolds numbers because of the absence of
a turbulence model and the high computational cost associated with the present simulations.
The simulation cost is due to the fact that none of the structural-dynamics models and the
coupling algorithm are currently parallelized. However, the use of a thin shell surrounding the
structure is very attractive to tackle high Reynolds numbers flows, for which boundary-layer
resolution is critical. Future research will focus on improving the representation of physical
phenomena in the IS to tackle the aero- and hydrodynamics of offshore wind turbines. In this
context, several challenges are foreseen. As this study showed, the shell has to be sufficiently
thin to get accurate numerical results. For slender structures, such as wind turbine blades, or
complex geometries in general, having a very thin shell around the structure can be numerically
challenging. Additionally, the implementation of a wall-layer model to reproduce the dynamics
of high Reynolds number flows near the fluid–structure interface is non-trivial. The case of
12
.........................................................
0.8
2.5
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
solid velocity (m s–1)
(a)
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
References
1. Viré A. 2012 How to float a wind turbine. Rev. Environ. Sci. Biotechnol. 3, 223–226.
(doi:10.1007/s11157-012-9292-9)
2. Peskin CS. 1972 Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10,
252–271. (doi:10.1016/0021-9991(72)90065-4)
3. Tseng YH, Meneveau C, Parlange MB. 2006 Modeling flow around bluff bodies and
predicting urban dispersion using large eddy simulation. Environ. Sci. Technol. 40, 2653–2662.
(doi:10.1021/es051708m)
4. Mittal R, Dong H, Bozkurttas M, Najjar FM, Vargas A, von Loebbecke A. 2008 A
versatile sharp interface immersed boundary method for incompressible flows with complex
boundaries. J. Comput. Phys. 227, 4825–4852. (doi:10.1016/j.jcp.2008.01.028)
5. Iaccarino G, Verzicco R. 2003 Immersed boundary technique for turbulent flow simulations.
Appl. Mech. Rev. 56, 331–347. (doi:10.1115/1.1563627)
6. Mittal R, Iaccarino G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261.
(doi:10.1146/annurev.fluid.37.061903.175743)
7. Goldstein D, Handler R, Sirovich L. 1993 Modeling a no-slip boundary with an external force
field. J. Comput. Phys. 105, 354–366. (doi:10.1006/jcph.1993.1081)
8. Fadlun E, Verzicco R, Orlandi P, Mohd-Yusof J. 2000 Combined immersed-boundary finitedifference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161,
35–60. (doi:10.1006/jcph.2000.6484)
9. Uhlmann M. 2005 An immersed boundary method with direct forcing for the simulation of
particulate flows. J. Comput. Phys. 209, 448–476. (doi:10.1016/j.jcp.2005.03.017)
10. Kempe T, Fröhlich J. 2012 An improved immersed boundary method with direct forcing
for the simulation of particle laden flows. J. Comput. Phys. 231, 3663–3684. (doi:10.1016/
j.jcp.2012.01.021)
11. Viré A, Xiang J, Milthaler F, Farrell PE, Piggott MD, Latham JP, Pavlidis D, Pain CC.
2012 Modelling of fluid–solid interactions using an adaptive mesh fluid model coupled
with a combined finite–discrete element model. Ocean Dyn. 62, 1487–1501. (doi:10.1007/
s10236-012-0575-z)
12. Viré A, Xiang J, Piggott MD, Cotter CJ, Pain CC. 2013 Towards the fully-coupled
numerical modelling of floating wind turbines. Energy Procedia 35, 43–51. (doi:10.1016/
j.egypro.2013.07.157)
13. Pain CC, Umpleby AP, de Oliveira CRE, Goddard AJH. 2001 Tetrahedral mesh optimisation
and adaptivity for steady-state and transient finite element calculations. Comput. Methods
Appl. Mech. Eng. 190, 3771–3796. (doi:10.1016/S0045-7825(00)00294-2)
14. Pain CC, Piggott MD, Goddard AJH, Fang F, Gorman GJ, Marshall DP, Eaton MD, Power PW,
de Oliveira CRE. 2005 Three-dimensional unstructured mesh ocean modelling. Ocean Model.
10, 5–33. (doi:10.1016/j.ocemod.2004.07.005)
15. Piggott MD, Farrell PE, Wilson CR, Gorman GJ, Pain CC. 2009 Anisotropic mesh adaptivity for
multi-scale ocean modelling. Phil. Trans. R. Soc. A 367, 4591–4611. (doi:10.1098/rsta.2009.0155)
16. Xiang J, Munjiza A, Latham JP. 2009 Finite strain, finite rotation quadratic tetrahedral element
for the combined finite–discrete element method. Int. J. Numer. Methods Eng. 79, 946–978.
(doi:10.1002/nme.2599)
.........................................................
Funding statement. A.V. is supported by the European Union Seventh Framework Programme (FP7/20072013) under a Marie Curie Career Integration Grant (grant agreement PCIG13-GA-2013-618159). A.V. also
acknowledges support from Imperial College London and its High Performance Computing Service, as well
as Université Libre de Bruxelles. C.C.P. acknowledges support from the UK Engineering and Physical Sciences
Research Council (Programme grant EP/K003976/1). The content of this paper reflects only the authors’ views
and not those of the European Commission.
13
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
wind turbines is particularly challenging, owing to their high rotational speeds, and the complex
wakes and tip vortices induced by the rotor. Hybrid subgrid-scale models will be developed for
this purpose. Finally, moving grids will be considered to further reduce the computational cost
associated with rotating turbine blades. Despite these challenges, the IS method is promising to
improve the representation of the flow physics near complex fluid–structure interfaces.
Downloaded from http://rsta.royalsocietypublishing.org/ on May 7, 2017
14
.........................................................
rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373: 20140085
17. Latham JP, Mindel J, Xiang J, Guises R, Garcia X, Pain CC, Gorman GJ, Piggott MD, Munjiza
A. 2009 Coupled FEMDEM/Fluids for coastal engineers with special reference to armour
stability and breakage. Geomech. Geoeng. 4, 797–805. (doi:10.1080/17486020902767362)
18. Guttman A. 1984 R-trees: a dynamic index structure for spatial searching. ACM SIGMOD Rec.
14, 47–57. (doi:10.1145/971697.602266)
19. Löhner R. 1995 Robust vectorized search algorithms for interpolation on unstructured grids.
J. Comput. Phys. 118, 380–387. (doi:10.1006/jcph.1995.1107)
20. Farrell PE, Maddison JR. 2011 Conservative interpolation between volume meshes by
local Galerkin projection. Comput. Methods Appl. Mech. Eng. 200, 89–100. (doi:10.1016/
j.cma.2010.07.015)
21. Griffith BE, Peskin CS. 2005 On the order of accuracy of the immersed boundary method:
higher order convergence rates for sufficiently smooth problems. J. Comput. Phys. 208, 75–105.
(doi:10.1016/j.jcp.2005.02.011)
22. Lai MC, Peskin CS. 2000 An immersed boundary method with formal second-order accuracy
and reduced numerical viscosity. J. Comput. Phys. 160, 705–719. (doi:10.1006/jcph.2000.6483)
23. Balaras E. 2004 Modeling complex boundaries using an external force field on fixed
Cartesian grids in large-eddy simulations. Comput. Fluids 33, 375–404. (doi:10.1016/
S0045-7930(03)00058-6)
24. McVicar AJ. 2012 Numerical simulations of flow–topography interaction using unstructured grids.
London, UK: Imperial College London.
25. Sheard G, Hourigan K, Thompson M. 2005 Computations of the drag coefficients
for low-Reynolds-number flow past rings. J. Fluid Mech. 526, 257–275. (doi:10.1017/
S0022112004002836)
26. Rajani BN, Kandasamy A, Majumdar S. 2009 Numerical simulation of laminar flow past a
circular cylinder. Appl. Math. Model. 33, 1228–1247. (doi:10.1016/j.apm.2008.01.017)
27. Meneghini JR, Saltara F, Siqueira CLR, Ferrari Jr J. 2001 Numerical simulation of flow
interference between two circular cylinders in tandem and side-by-side arrangements. J. Fluids
Struct. 15, 327–350. (doi:10.1006/jfls.2000.0343)
28. Singh SP, Mittal S. 2005 Flow past a cylinder: shear layer instability and drag crisis. Int. J.
Numer. Methods Fluids 47, 75–98. (doi:10.1002/fld.807)
29. Dütsch H, Durst F, Becker S, Lienhart H. 1998 Low-Reynolds-number flow around an
oscillating circular cylinder at low Keulegan–Carpenter numbers. J. Fluid Mech. 360, 249–271.
(doi:10.1017/S002211209800860X)
30. Yang J, Balaras E. 2006 An embedded-boundary formulation for large-eddy simulation
of turbulent flows interacting with moving boundaries. J. Comput. Phys. 215, 12–40.
(doi:10.1016/j.jcp.2005.10.035)
31. Yang J, Stern F. 2012 A simple and efficient direct forcing immersed boundary framework for
fluid–structure interactions. J. Comput. Phys. 231, 5029–5061. (doi:10.1016/j.jcp.2012.04.012)
32. Clift R. 1978 Bubbles, drops, and particles. New York, NY: Academic Press.
33. Nino Y, Garcia M. 1994 Gravel saltation: 2. Modeling. Water Resour. Res. 30, 1915–1924.
(doi:10.1029/94WR00534)
34. Brown PP, Lawler DF. 2003 Sphere drag and settling velocity revisited. J. Environ. Eng. 129,
222–231. (doi:10.1061/(ASCE)0733-9372(2003)129:3(222))
35. Basset AB. 1961 A treatise on hydrodynamics with numerous examples. New York, NY: Dover.
36. Mei R. 1994 Flow due to an oscillating sphere and an expression for unsteady drag
on the sphere at finite Reynolds number. J. Fluid Mech. 270, 133–174. (doi:10.1017/
S0022112094004222)