Today there is an increasing need to develop multi-physics
approaches to address complex design challenges. The numerical
methods adopted must include coupled field analyses that allow
the combined effects of the multiple physical phenomena acting
on a given system to be evaluated. One of the most interesting
multi-physics phenomena with a wide range of applications is the
interaction between a fluid and a structure. This interaction can occur
for several reasons: it may be the working principle of the system; it
may focus on creating a lightweight design for the structure; or it may
be used to refine the design.
Fluid structure interaction (FSI) plays a key role in a wide range of
engineering fields, such as automotive, aerospace, marine, civil and
biomedical. To numerically solve the interaction, the deformation
of the computational fluid dynamics (CFD) mesh is needed to
accommodate the changes in the shape of the structure. In the present
work, a radial basis functions (RBF) based mesh morphing algorithm
is used to change the CFD mesh according to the deformed shape of
the structure. The FSI approach we propose allows the mesh to be
adapted to the shape of the deformable structure by superposition of
its natural modes during the course of the CFD calculation.
The underlying notion for the proposed workflow is to compute the
fluid forces on the surface of the structure, along with the inertial
loads at each step, as modal forces to determine the amplitude of
each modal shape. By superimposing the modal shapes, the overall
deformation of the structure can be obtained at each instant and
can be imposed in the CFD model by morphing the CFD mesh. The
method is implemented to study an industrial problem: the vortexinduced
vibration of a thermowell immersed in a stream of fluid.
Thermowells are cylindrical fittings used to protect temperature
sensors (such as thermometers or thermocouples) installed in
industrial processes. In such a configuration, the fluid transfers heat
to the wall of the thermowell which, in turn, transfers heat to the
sensor. The use of a thermowell, in addition to protecting the sensor
from the effects of the pressure and chemicals of the process fluid,
allows the sensor to be easily replaced without emptying the tank or
pipes. However, thermowells are subject to potential flow-induced
vibrations generated by vortex shedding that may cause failure due
to bending fatigue. Consequently, particularly in modern applications
involving high-strength piping and high fluid velocity, the dynamics
of the system must be carefully evaluated to prepare ad-hoc
countermeasures, such as twisted square thermowells, to limit this
phenomenon. A numerical method capable of reliably reproducing
the fluid-structural coupling is therefore needed to rapidly evaluate
different designs and reduce the time to market of new products.
Modal FSI implementation
Beginning with the undeformed configuration, the flexible
components of the system are modelled and studied using a structural
modal analysis in order to extract a suitable set of eigenvalues and
eigenvectors.
The obtained modes are used to generate an RBF solution for each
shape. At this stage, the far-field conditions and rigid surfaces
must be constrained, while the FEM results must be mapped to
the deformable surfaces of the CFD domain. The RBF solutions
obtained constitute the modal basis that, suitably amplified, permits
the structural deformation under load to be represented, generating
an intrinsically aeroelastic domain. This process is known as “RBF
structural modes embedding”. To accelerate the mesh morphing
phase, the deformations associated with each modal shape are stored
thereby containing the numerical cost of the morphing process to a
small fraction (around 10%) of the cost of a single CFD iteration.
The proposed FSI modelling technique belongs to the class of weak
approaches because, for the purposes of an unsteady analysis, the
loads are considered to be fixed during each time-step. The modal
forces are calculated on the prescribed surfaces (i.e. the deformable
ones) by projecting the nodal forces (pressures and shear stresses)
onto the modal shapes. The mesh is updated at each time step
during the course of the CFD transient calculation according to the
calculated modal coordinates. The mesh morphing tool used was
RBF Morph, with Ansys Fluent for the
CFD, and Ansys Mechanical 2021 R1
for the FEM solver.
Experimental study
The industrial problem studied
concerns a vortex-induced vibration
on a thermowell immersed in a fluid
flow. The case study experiment was
measured and recorded by Emerson
Electric Co., the multinational
corporation that owns Rosemount which manufactures the thermowell
studied (www.emerson.com/en-us/asset-detail/rosemount-twistedsquare-
a-new-twist-on-thermowell-design-1800740).
The purpose of the experiment was to evaluate the flow-induced
vibrations on a traditional cylindrical thermowell design, shown
in Fig. 1. The 470.219 mm-long sensor was equipped with an
accelerometer in the tip and immersed in a flow of water inside a
152.4 mm-diameter pipe.
The water velocity ranged from 0 m/s to 8.5 m/s. The accelerometer
enabled the evolution of the tip displacement to be reconstructed.
The results gathered are summarized in Fig. 2 in terms of the mean
square root of the tip displacement as a function of the fluid velocity.
Two lock-in regions are observed: an in-line vibration lock-in region,
and a transverse vibration lock-in region. In the in-line vibration lockin
region, the maximum Root Mean Squar (RMS) tip displacement
in the direction of the flow is 2.33 mm, recorded at a fluid velocity
of 2.44 m/s. In the transverse vibration lock-in region, the maximum
RMS tip displacement in the cross-flow direction is 8.3 mm, recorded
at a fluid velocity of 6.4 m/s. The vibrations are induced by organized
vortices that shed in sheets along the axial length of the stem and
that generate alternating forces. If the shedding frequency approaches
a natural frequency of the thermowell or its half, transverse or inline vibrations are excited and a failure of the sensor may occur.
Failure conditions have been reached for the cylindrical thermowell
at velocities greater than 6.4 m/s. The purpose of this work was to
numerically capture the transverse vibration lock-in region of the
cylindrical thermowell.
Numerical analysis
The study presented in this paper was conducted by using the FSI
module included in the RBF Morph package. The FSI module allowed
to tackle the VIV analysis using the structural modes embedding
method. The workflow is sketched in Fig.3. The structural and fluid
domains each have to be modelled in appropriate locations and with
consistent units. During initialization, the structural modes calculated
by Ansys Mechanical are transferred into the Ansys Fluent CFD
model by RBF Morph. Once the modes are
incorporated, the “flexible” CFD model is able
to: adapt the shape according to the modal
coordinates, evaluate the modal forces acting
on the wetted surfaces, and evolve the time
solution of the structural modal coordinates.
The first six modes were extracted from the
FEM modal analysis and adopted to populate
the modal base adopted for the FSI analysis.
The shapes of the six modes are shown in Fig.
4 where the first, second and third shapes can
be seen bending in the two directions.
The shapes of the modes were extracted in
terms of the displacements of the FEA mesh
nodes belonging to the sensor surface and normalized with respect to
mass; then they were used for the RBF Morph configuration depicted
in Fig. 5. It is worth noting that once the configuration for one of the
modes is completed it can be easily and automatically replicated for
the entire modal base required.
The top left of Fig.5 shows the morphing domain, which is restricted
to the region where shape deformations are expected to occur; RBF
source points are created to control the morphing of fixed parts (top
right of Fig.5) and the deforming parts (bottom left of Fig.5). For
this particular case, the proximity of the
tip of the thermowell to the boundary wall
of the pipe created a challenging problem.
In fact the large displacements that the
thermowell is expected to undergo due to
the vortex induced vibrations, combined
with the need to maintain a cylindrical
shape for the pipe wall, would result in
a significant distortion of the mesh if the
nodes in the pipe wall were imposed as
fixed.
To avoid this high mesh distortion, an advanced corrective strategy
was implemented to allow nodes belonging to the pipe wall that are
close to the tip in the clearance region to slide on the cylindrical
surface. First, a “shadow” area was defined so that the portion of the
pipe surface defined by the projection of the thermowell tip would
follow the sensor tip during the morphing action.
Then, a projection of the deformed pipe surface mesh onto the
original cylindrical surface, after mesh morphing, was made. This
complex task was accomplished by combining three pre-calculated
RBF solutions.
The first two solutions allowed the shadow area to be assigned an
appropriate rotation around the axis of the pipe, and a translation in
the direction of the axis itself, in order to keep it constantly under the
tip; the third solution, based on RBF Morph’s STL-target technology,
allowed the selected nodes to be projected
onto a target surface (and thus to recover the
cylindrical shape of the pipe).
The bottom right image of Fig. 5 represents
the surface mesh around the thermowell
tip obtained after morphing by applying the
described correction procedure. Note the high
quality of morphing, the correct placement of
the shadow area, and the preservation of the
cylindrical shape of the pipe.
The fluid dynamics domain was discretized with a structured,
multiblock mesh consisting of 3.16M hexahedra. To accurately
solve the boundary layer up to the wall, the thickness of the first
cell layer was set to obtain a dimensionless wall distance (y+) of
less than one. The SST k-ω turbulence model was adopted. At the
inlet, the velocity-inlet boundary condition was set by imposing
a flow velocity of 6.4 m/s. At the outlet, a pressure condition was
set. The unsteady incompressible RANS calculation was performed
with a time-step of 10-4 s. The structural damping ratio was set to
0.041, following guidance found in the literature and a parametric
study (www.springerprofessional.de/en/analysis-of-vortex-inducedvibration-
of-a-thermowell-by-high-fid/19244678). The mesh was
updated at each time step by calculating the modal coordinates and
amplification factors of the corrective solutions.
Results
Fig. 6 shows the contours of the magnitude of velocity on a plane
perpendicular to the thermowell axis for two different flow times
corresponding to the maximum transverse displacements, in both the
positive and negative directions.
Fig. 7 illustrates the temporal evolution of side force on the
thermowell; it also shows the temporal evolution of the transverse
tip displacement. The maximum RMS transverse tip displacement of
8.304 mm shown is in good agreement with the experimental data
available. The power spectral density distributions of the two signals
(the temporal evolution of side force and transverse tip displacement)
as a function of frequency are shown in Fig. 8. A dominant frequency
of 48.8 Hz was observed for both signals, confirming the correct
acquisition of the lock-in condition.
Conclusions
The work presented focused on an FSI analysis methodology based
on the modal superposition approach. It was applied to the study
of vortex-induced vibration of a thermowell. The problem of mesh
adaptation was addressed with an RBF-based mesh morphing
technique that provided a particularly fast and robust configuration.
The configuration studied represents a particularly challenging
problem for the mesh morphing tool. The proximity of fixed and
moving boundaries, in fact, results in strong mesh distortions that
significantly limit the tolerable displacement. The morphing software
used allowed a particularly efficient set of corrective solutions to be
configured that enabled the very large displacements relative to the
dimensions to be managed. The results of the unsteady FSI analysis
conducted were compared with the experimental data and provided a
good agreement with the measurements.
This work won the AIAS 2021 Software Simulation Award.
