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=Fluid-structure interaction in turbulent flow past cylinder/plate configuration  I (First swiveling mode)=
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=  Introduction =
== Flows around bodies ==
=== Underlying Flow Regime 2-13 ===
 
= Description =
 
==  Introduction ==


A flexible structure exposed to a fluid flow is deformed and deflected
A flexible structure exposed to a fluid flow is deformed and deflected
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simulation tools are required.  
simulation tools are required.  


The long-term objective of the present research project is the coupled
The long-term objective of the research reported here is the coupled
simulation of big lightweight structures such as thin membranes
simulation of big lightweight structures such as thin membranes
exposed to turbulent flows (outdoor tents, awnings...). To study these
exposed to turbulent flows (outdoor tents, awnings...). To study these
complex FSI problems, a multi-physics code framework was recently
complex FSI problems, a multi-physics code framework was recently
developed (Breuer et al., 2012). In order to assure reliable numerical
developed (Breuer et al., 2012) combining Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) solvers . In order to assure reliable numerical
simulations of complex configurations, the whole FSI code needs to be
simulations of complex configurations, the whole FSI code needs to be
validated at first on simpler test cases with trusted reference
validated at first on simpler test cases with trusted reference
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developed is detailed. The CFD and CSD solvers were at first checked
developed is detailed. The CFD and CSD solvers were at first checked
separately and then, the coupling algorithm was considered in detail
separately and then, the coupling algorithm was considered in detail
based on a laminar benchmark. A 3D turbulent test case was also taken
based on a laminar benchmark. A 3D turbulent test case was also calculated to prove the applicability of the newly developed
into account to prove the applicability of the newly developed
coupling scheme in the context of large-eddy simulations
coupling scheme in the context of large-eddy simulations
(LES). However, owing to missing reference data a full validation was
(LES). However, owing to missing reference data a full validation was
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community for the technically relevant case of turbulent flows
community for the technically relevant case of turbulent flows
interacting with flexible structures.
interacting with flexible structures.
==  Review of previous work ==


The present study is mainly related to two former investigations  
The present study is mainly related to two former investigations  
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structural simulation and the grid adaptation of the flow
structural simulation and the grid adaptation of the flow
prediction.
prediction.
==  Choice of test case ==


Thus, in the present study a slightly different configuration is
Thus, in the present study a slightly different configuration is
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for the coupled problem) and the processing of the respective data to
for the coupled problem) and the processing of the respective data to
guarantee a reliable reproduction of the proposed test case with
guarantee a reliable reproduction of the proposed test case with
various suitable methods. A detailed description of the present test case is published in De Nayer et. al. (2013).
various suitable methods. A detailed description of the present test case is published in De Nayer et al. (2014).


The described test case FSI-PfS-1a is a part of a series of reference
The described test case FSI-PfS-1a is a part of a series of reference
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the structure deflections are completely two-dimensional and
the structure deflections are completely two-dimensional and
larger.
larger.
 
   
= Test case description =
== Description of the geometrical model and the test section ==
 
FSI-PfS-1a consists of a flexible thin structure with a distinct
thickness clamped behind a fixed rigid non-rotating cylinder installed
in a water channel (see Fig. 1). The
cylinder has a diameter D=0.022m. It is positioned in the
middle of the experimental test section with <math>H_c \operatorname{=} H/2 \operatorname{=} 0.120\,m</math> <math>(H_c/D \approx 5.45)</math>, whereas the test section denotes a
central part of the entire water channel (see
Fig. 2). Its center is located at <math>L_c \operatorname{=}
  0.077\,m</math> <math>(L_c/D \operatorname{=} 3.5)</math> downstream of the inflow
section. The test section has a length of <math>L \operatorname{=} 0.338\,m</math>
<math>(L/D \approx 15.36)</math>, a height of <math>H \operatorname{=} 0.240\,m</math>
<math>(H/D \approx 10.91)</math> and a width <math>W \operatorname{=} 0.180\,m</math>
<math>(W/D \approx 8.18)</math>. The blocking ratio of the channel is
about <math>9.2\%</math>. The gravitational acceleration <math>g</math> points in
x-direction (see Fig.~\ref{fig:rubber_plate_geom}), i.e. in the
experimental setup this section of the water channel is turned 90
degrees. The deformable structure used in the experiment behind the
cylinder has a length <math>l \operatorname{=} 0.060\,m</math> <math>(l/D \approx 2.72)</math>
and a width <math>w \operatorname{=} 0.177\,m</math> <math>(w/D \approx 8.05)</math>.
Therefore, in the experiment there is a small gap of about <math>1.5
  \times 10^{-3}\,m</math> between the side of the deformable structure and
both lateral channel walls.
The thickness of the plate is <math>h \operatorname{=} 0.0021\,m</math> <math>(h/D
  \approx 0.09)</math>. This thickness is an averaged value. The material
is natural rubber and thus it is difficult to produce a perfectly
homogeneous 2 mm plate. The measurements show that the thickness of
the plate is between 0.002 and 0.0022 m. All parameters of the
geometrical configuration of the FSI-PfS-1a benchmark are summarized as follows:
 
[[File:table3.png]]
 
[[File:FSI-PfS-1a_Benchmark_Rubberplate_geometry0001_new.png]]
 
Fig. 1: Geometrical configuration of the FSI-PfS-1a Benchmark.
 
== Description of the water channel ==
 
 
In order to validate numerical FSI simulations based on reliable
experimental data, the special research unit on FSI (Bungartz et. al. (2006, 2010))
designed and constructed a water channel (Göttingen type, see
Fig. 2) for corresponding experiments with a
special concern regarding controllable and precise boundary and
working conditions Gomes et al. (2006, 2010, 2011, 2013). The
channel (2.8 m x 1.3 m x 0.5 m, fluid volume
of 0.9 m³) has a rectangular flow path and includes several
rectifiers and straighteners to guarantee a uniform inflow into the
test section. To allow optical flow measurement systems like
Particle-Image Velocimetry, the test section is optically accessible
on three sides. It possesses the same geometry as the test section
described in Fig. 1. The structure is
fixed on the backplate of the test section and additionally fixed in
the front glass plate. With a 24 kW axial pump a water inflow of up to
<math>u_{\text{max}}=6</math> m/s is possible. To prevent asymmetries the
gravity force is aligned with the x-axis in main flow direction.
 
[[File:waterchannel_new.png]]
 
Fig. 2: Sketch of the flow channel (dimensions given in mm).
 
== Flow parameters ==
 
Several preliminary tests were performed to find the best working
conditions in terms of maximum structure displacement, good
reproducibility and measurable structure frequencies within the
turbulent flow regime.
 
[[File:inflow1.png]]
 
Fig. 3: Experimental displacements of the structure extremity versus the inlet velocity.
 
Fig. 3 depicts the measured extrema of the structure displacement versus the
inlet velocity and Fig. 4 gives the
frequency and Strouhal number as a function of the inlet
velocity. These data were achieved by measurements with the laser
distance sensor explained in Section [[UFR_2-13_Description#Laser distance sensor|Laser distance sensor]]. The
entire diagrams are the result of a measurement campaign in which the
inflow velocity was consecutively increased from 0 to <math>2.2</math> m/s. At an inflow velocity of <math>u_{\text{inflow}}=1.385 </math> m/s
the displacement are symmetrical, reasonably large and well
reproducible. Based on the inflow velocity chosen and the cylinder
diameter the Reynolds number of the experiment is equal to
<math>\text{Re}=30,470</math>.
 
[[File:inflow2.png]]
 
Fig. 4: Experimental measurements of the frequency and the corresponding Strouhal number of the FSI phenomenon versus the inlet velocity.
 
Regarding the flow around the front
cylinder, at this inflow velocity the flow is in the sub-critical
regime. That means the boundary layers are still laminar, but
transition to turbulence takes place in the free shear layers evolving
from the separated boundary layers behind the apex of the
cylinder. Except the boundary layers at the section walls the inflow
was found to be nearly uniform (see
Fig. 5). The velocity components
<math>\overline{u} </math> and  <math>\overline{v}</math> are measured with two-component
laser-Doppler velocimetry (LDV) along the y-axis in the middle of the
measuring section at  <math>x/D=4.18 </math> and  <math>z/D=0 </math>. It can be assumed that
the velocity component <math>\overline{w} </math>  shows a similar velocity profile
as <math>\overline{v} </math>. Furthermore, a low inflow turbulence level of
<math>\text{Tu}_{\text{inflow}}=\sqrt{\frac{1}{3}~\left(\overline{u'^2}+\overline{v'^2}+\overline{w'^2} \right)}/u_{\text{inflow}}=0.02 </math> is measured. All
experiments were performed with water under standard conditions at
<math>T=20^{\circ}\,C</math>. The flow parameters are summarized in the following table:
 
{| class="wikitable"
|+ Flow parameters
|----
! scope="row" | Inflow velocity
| <math>u_{\text{inflow}}=1.385\,m/s </math>
|----
! scope="row" | Flow density
| <math> \rho_f=1000\,kg/m^3</math>
|----
! scope="row" | Flow dynamic viscosity
| <math> \mu_f=1.0 \times 10^{-3}\,Pa\,s </math>
|}
 
[[File:flow_conditions.png]]
 
Fig. 5 Profiles of the mean streamwise and normal velocity as well as the turbulence level at the inflow section of the water channel.
 
== Material Parameters ==
 
Although the material shows a strong non-linear elastic behavior for
large strains, the application of a linear elastic constitutive law
would be favored, to enable the reproduction of this FSI benchmark by
a variety of different computational analysis codes without the need
of complex material laws. This assumption can be justified by the
observation that in the FSI test case, a formulation for large
deformations but small strains is applicable. Hence, the
identification of the material parameters is done on the basis of the
moderate strain expected and the St. Venant-Kirchhoff constitutive law
is chosen as the simplest hyper-elastic material model.
 
The density of the rubber material can be determined to be
<math>\rho_{\text{rubber plate}}</math>=1360 kg/m<math>^3</math> for a thickness
of the plate h = 0.0021 m. This permits the accurate modeling
of inertia effects of the structure and thus dynamic test cases can be
used to calibrate the material constants. For the chosen material
model, there are only two parameters to be defined: the Young's
modulus E and the Poisson's ratio <math>\nu</math>. In order to avoid
complications in the needed element technology due to
incompressibility, the material was realized to have a Poisson's ratio
which reasonably differs from <math>0.5</math>. Material tests of the
manufacturer and complementary experimental/numerical structure test studies (dynamic and decay test scenarios) indicate that the Young's modulus is E=16 MPa and
the Poisson's ratio is <math>\nu</math>=0.48 (a detailed description of the structure tests is available in De Nayer et. al. (2013)).
 
{| class="wikitable"
|+ Structure parameters
|----
! scope="row" | Density
| <math>\rho_{\text{rubber plate}}=1360\,kg/m^3</math>
|----
! scope="row" | Young's modulus
| <math>E=16\,MPa</math>
|----
! scope="row" | Poisson's ratio
| <math>\nu=0.48\,</math>
|}
 
= Measuring Techniques =
 
Experimental FSI investigations need to contain fluid and structure
measurements for a full description of the coupling process. Under
certain conditions, the same technique for both disciplines can be
used. The measurements performed by
Gomes et. al. (2006, 2010, 2013) used the same camera system for
the simultaneous acquisition of the velocity fields and the structural
deflections. This procedure works well for FSI cases involving laminar
flows and 2D structure deflections. In the present case the structure
deforms slightly three-dimensional with increased cycle-to-cycle
variations caused by turbulent variations in the flow. The applied
measuring techniques, especially the structural side, have to deal
with those changed conditions especially the formation of
shades. Furthermore, certain spatial and temporal resolutions as well
as low measurement errors are requested. Due to the different
deformation behavior a single camera setup for the structural
measurements like in Gomes et. al. (2006, 2010, 2013) used was not
practicable. Therefore, the velocity fields were captured by a 2D
Particle-Image Velocimetry (PIV) setup and the structural deflections
were measured with a laser triangulation technique. Both devices are
presented in the next sections.
 
=== Particle-image velocimetry ===
 
A classic Particle-Image Velocimetry (Adrian, 1991) setup consists of a single camera obtaining two
components of the fluid velocity on a planar surface illuminated by a
laser light sheet. Particles introduced into the fluid are following the
flow and reflecting the light during the passage of the light sheet.
By taking two reflection fields in a short time interval <math>\Delta</math>t, the most-likely displacements of several particle groups on an
equidistant grid are estimated by a cross-correlation technique or a
particle-tracking algorithm. Based on a precise preliminary
calibration, the displacements obtained and the time interval
<math>\Delta</math>t chosen the velocity field can be calculated. To prevent
shadows behind the flexible structure a second light sheet was used to
illuminate the opposite side of the test section.
 
The phased-resolved PIV-measurements (PR-PIV) were carried out with a
4 Mega-pixel camera (TSI Powerview 4MP, charge-coupled device (CCD)
chip) and a pulsed dual-head Neodym:YAG laser (Litron NanoPIV 200)
with an energy of 200 mJ per laser pulse. The high energy of
the laser allowed to use silver-coated hollow glass spheres (SHGS)
with an average diameter of <math>d_{\text{avg,SHGS}}</math>=10~µm and
a density of <math>\rho_{SHGS}</math> = 1400 kg m<math>^{-3}</math> as tracer
particles. To prove the following behavior of these particles a
Stokes number Sk=1.08 and a particle sedimentation velocity
<math>u_{\text{SHGS}}=2.18 \times 10^{-5}~\text{m/s}</math> is calculated
With this Stokes number and a particle sedimentation velocity which is
much lower than the expected velocities in the experiments, an eminent
following behavior is approved. The camera takes 12 bit pictures with
a frequency of about 7.0 Hz and a resolution of 1695 x 1211 px with respect to the rectangular size of the test
section. For one phase-resolved position (The processing of the phase-resolved fluid velocity fields
involving the structure deflections is described in
Section [[UFR_2-13_Description#Generation of Phase-resolved Data|Generation of Phase-resolved Data]].) 60 to 80
measurements are taken. Preliminary studies with more and fewer
measurements showed that this number of measurements represent a good
compromise between accuracy and effort. The grid has a size of 150 x 138 cells and was calibrated with an average
factor of 126 <math>\mu</math> m/px}, covering a planar flow field of
x/D = -2.36 to 7.26 and y/D = -3.47 to ~3.47 in the middle of the test section at
z/D = 0. The time between the frame-straddled laser
pulses was set to <math>\Delta</math> t=200 <math>\mu</math>s. Laser and camera were
controlled by a TSI synchronizer (TSI 610035) with 1 ns
resolution.
 
=== Laser distance sensor ===
 
Non-contact structural measurements are often based on laser distance
techniques. In the present benchmark case the flexible structure shows
an oscillating frequency of about 7.1 Hz. With the
requirement to perform more than 100 measurements per period, a
time-resolved system was needed. Therefore, a laser triangulation was
chosen because of the known geometric dependencies, the high data
rates, the small measurement range and the resulting higher accuracy
in comparison with other techniques such as laser phase-shifting or
laser interferometry. The laser triangulation uses a laser beam which
is focused onto the object. A CCD-chip located near the laser output
detects the reflected light on the object surface. If the distance of
the object from the sensor changes, also the angle changes and thus
the position of its image on the CCD-chip. From this change in
position the distance to the object is calculated by simple
trigonometric functions and an internal length calibration adjusted to
the applied measurement range. To study simultaneously more than one
point on the structure, a multiple-point triangulation sensor was
applied (Micro-Epsilon scanControl 2750, see
Fig. 6). This sensor uses a matrix of CCD
chips to detect the displacements on up to 640 points along a laser
line reflected on the surface of the structure with a data rate of 800 profiles per second. The laser line was positioned in a
horizontal (x/D = 3.2, see
Fig. 6(a)) and in a vertical
alignment(z/D = 0, see
Fig. 6(b)) and has an accuracy of
40 µm}. Due to the different refraction indices of air,
glass and water a custom calibration was performed to take the
modified optical behavior of the emitted laser beams into account.
 
[[File:structure_sensors_scancontrolonly0001_new.png]]
 
Fig. 6: Setup and alignment of multiple-point laser sensor on the flexible structure in a) z-direction and b) x-direction.
 
= Numerical Simulation Methodology =
 
The applied numerical method relies on an efficient partitioned
coupling scheme developed for dynamic fluid-structure interaction
problems in turbulent flows (Breuer et al, 2012). The fluid flow is
predicted by an eddy-resolving scheme, i.e., the large-eddy simulation
technique. FSI problems very often encounter instantaneous
non-equilibrium flows with large-scale flow structures such as
separation, reattachment and vortex shedding. For this kind of flows
the LES technique is obviously the best
choice (Breuer, 2002). Based on a semi-implicit scheme the LES code
is coupled with a code especially suited for the prediction of shells
and membranes. Thus an appropriate tool for the time-resolved
prediction of instantaneous turbulent flows around light, thin-walled
structures results. Since all details of this methodology were
recently published in Breuer et al, 2012, in the following only a
brief description is provided.
 
== Computational fluid dynamics (CFD) ==
 
Within a FSI application the computational domain is no longer fixed
but changes in time due to the fluid forces acting on the structure.
This temporally varying domain is taken into account by the Arbitrary
Lagrangian-Eulerian (ALE) formulation expressing the conservation
equations for time-dependent volumes and surfaces. Here the filtered
Navier-Stokes equations for an incompressible fluid are solved.  Owing
to the deformation of the grid, extra fluxes appear in the governing
equations which are consistently determined considering the space conservation law (SCL) (Demirdzic 1988 and 1990, Lesoinne, 1996). The SCL is expressed
by the swept volumes of the corresponding cell faces and assures that
no space is lost during the movement of the grid lines. For this
purpose the in-house code FASTEST-3D (Durst et al, 1996a, b)
relying on a three-dimensional finite-volume scheme is used. The
discretization is done on a curvilinear, block-structured body-fitted
grid with collocated variable arrangement. A midpoint rule
approximation of second-order accuracy is used for the discretization
of the surface and volume integrals. Furthermore, the flow variables
are linearly interpolated to the cell faces leading to a second-order
accurate central scheme. In order to ensure the coupling of pressure
and velocity fields on non-staggered grids, the momentum interpolation
technique of Rhie and Chow (1983) is used.
 
A predictor-corrector scheme (projection method) of second-order
accuracy forms the kernel of the fluid solver. In the predictor step
an explicit three substep low-storage Runge-Kutta scheme advances
the momentum equation in time leading to intermediate
velocities. These velocities do not satisfy mass conservation. Thus,
in the following corrector step the mass conservation equation has to
be fulfilled by solving a Poisson equation for the
pressure-correction based on the incomplete LU decomposition method
of Stone (1968). The corrector step is repeated (about 3 to 8
iterations) until a predefined convergence criterion (<math>\Delta{m} <
{O}(10^{-9})</math>) is reached and the final velocities and the
pressure of the new time step are obtained. In Breuer et al (2012) it
is explained that the original pressure-correction scheme applied
for fixed grids has not to be changed concerning the mass conservation
equation in the context of moving grids. Solely in the momentum
equation the grid fluxes have to be taken into account as described
above.
 
In LES the large scales in the turbulent flow field are resolved
directly, whereas the non-resolvable small scales have to be taken
into account by a subgrid-scale model. Here the well-known and most
often used eddy-viscosity model, i.e., the Ssmagorinsky (1963) model
is applied. The filter width is directly coupled to the volume of the
computational cell and a Van Driest damping function ensures a
reduction of the subgrid length near solid walls.  Owing to minor
influences of the subgrid-scale model at the moderate Reynolds number
considered in this study, a dynamic procedure to determine the
Smagorinsky parameter as suggested by Germano et al (1991) was omitted and
instead a well established standard constant <math>C_s = 0.1</math> is used.
 
The CFD prediction determines the forces on the structure and delivers
them to the CSD calculation. In the other direction the CSD prediction
determines displacements at the moving boundaries of the computational
domain for the fluid flow. The task is to adapt the grid of the inner
computational domain based on these displacements at the
interface. For moderate deformations algebraic methods are found to be
a good compromise since they are extremely fast and deliver reasonable
grid point distributions maintaining the required high grid
quality. Thus, the grid adjustment is performed based on a transfinite
interpolation (Thompson et al., 1985). It consists of three shear
transformations plus a tensor-product transformation.
 
== Computational structural dynamics (CSD) ==
 
The dynamic equilibrium of the structure is described by the momentum
equation given in a Lagrangian frame of reference. Large deformations,
where geometrical non-linearities are not negligible, are allowed
(Hojjat et al, 2010). According  to preliminary structure considerations, a total Lagrangian
formulation in terms of the second Piola-Kirchhoff stress tensor and
the Green-Lagrange strain tensor which are linked by the
St. Venant-Kirchhoff material law is used in the present study.
For the solution of the governing equation the finite-element solver
Carat++, which was developed with an emphasis on the prediction of
shell or membrane behavior, is applied. Carat++ is based on several
finite-element types and advanced solution strategies for form finding
and non-linear dynamic
Problems (Wüchner et al, 2005; Bletzinger et al, 2005; Dieringer et al, 2012). For the
dynamic analysis, different time-integration schemes are available,
e.g., the implicit generalized-<math>\alpha</math> method (Chung et al, 1993). In the
modeling of thin-walled structures, membrane or shell elements are
applied for the discretization within the finite-element model. In the
current case, the deformable solid is modeled with a 7-parameter shell
element. Furthermore, special care is given to prevent locking
phenomena by applying the well-known Assumed Natural Strain (ANS) (Hughes and Tezduyar, 1981; Park and Stanley, 1986) and Enhanced Assumed Strain (EAS) methods (Bischoff et al., 2004).
 
Both, shell and membrane elements reflect geometrically reduced
structural models with a two-dimensional representation of the
mid-surface which can describe the three-dimensional physical
properties by introducing mechanical assumptions for the thickness
direction. Due to this reduced model additional operations are
required to transfer information between the two-dimensional structure
and the three-dimensional fluid model. Thus in the case of shells, the
surface of the interface is found by moving the two-dimensional
surface of the structure half of the thickness normal to the surface
on both sides and the closing of the volume (Bletzinger et al., 2006).. On
these two moved surfaces the exchange of data is performed
consistently with respect to the shell theory (Hojjat et al., 2010).
 
== Coupling algorithm ==
 
To preserve the advantages of the highly adapted CSD and CFD codes and
to realize an effective coupling algorithm, a partitioned but
nevertheless strong coupling approach is chosen. Since LES
typically requires small time steps to resolve the turbulent flow
field, the coupling scheme relies on the explicit
predictor-corrector scheme forming the kernel of the fluid solver.
 
Based on the velocity and pressure fields from the corrector step, the
fluid forces resulting from the pressure and the viscous shear
stresses at the interface between the fluid and the structure are
computed. These forces are transferred by a grid-to-grid data
interpolation to the CSD code Carat++ using a conservative
interpolation scheme (Farhat et al, 1998) implemented in the coupling
interface CoMA (Gallinger et al, 2009). Using the fluid forces provided via
CoMA, the finite-element code Carat++ determines the stresses in the
structure and the resulting displacements of the structure. This
response of the structure is transferred back to the fluid solver via
CoMA applying a bilinear interpolation which is a consistent scheme
for four-node elements with bilinear shape functions.
 
For moderate and high density ratios between the fluid and the
structure, e.g., a flexible structure in water, the added-mass effect
by the surrounding fluid plays a dominant role. In this situation a
strong coupling scheme taking the tight interaction between the fluid
and the structure into account, is indispensable. In the coupling
scheme developed in Breuer et al (2012) this issue is taken into
account by a FSI-subiteration loop which works as follows:
 
A new time step begins with an estimation of the displacement of the
structure. For the estimation a linear extrapolation is applied taking
the displacement values of two former time steps into account.
According to these estimated boundary values, the entire computational
grid has to be adapted as it is done in each FSI-subiteration
loop. Then the predictor-corrector scheme of the next time step is
carried out and the cycle of the FSI-subiteration loop is
entered. After each FSI-subiteration first the FSI convergence is
checked. Convergence is reached if the <math>L_2</math> norm of the displacement
differences between two FSI-subiterations normalized by the <math>L_2</math> norm
of the changes in the displacements between the current and the last
time step drops below a predefined limit, e.g. <math>\varepsilon_{FSI} =
10^{-4}</math> for the present study. Typically, convergence is not reached
within the first step but requires a few FSI-subiterations (5 to
10). Therefore, the procedure has to be continued on the fluid
side. Based on the displacements on the fluid-structure interface,
which are underrelaxated by a constant factor $\omega$ during the
transfer from the CFD to the CSD solver, the inner computational CFD
grid is adjusted. The key point of the coupling procedure suggested in
Breuer et al (2012) is that subsequently only the corrector step of
the predictor-corrector scheme is carried out again to obtain a new
velocity and pressure field. Thus the clue is that the pressure is
determined in such a manner that the mass conservation is finally
satisfied. Furthermore, this extension of the predictor-corrector
scheme assures that the pressure forces as the most relevant
contribution to the added-mass effect, are successively updated until
dynamic equilibrium is achieved. In conclusion, instabilities due to
the added-mass effect known from loose coupling schemes are avoided
and the explicit character of the time-stepping scheme beneficial for
LES is still maintained.
 
The code coupling tool CoMA is based on the
Message-Passing-Interface (MPI) and thus runs in parallel to the
fluid and structure solver. The communication in-between the codes is
performed via standard MPI commands. Since the parallelization in
FASTEST-3D and Carat++ also relies on MPI, a hierarchical
parallelization strategy with different levels of parallelism is
achieved. According to the CPU-time requirements of the different
subtasks, an appropriate number of processors can be assigned to the
fluid and the structure part. Owing to the reduced structural models
on the one side and the fully three-dimensional highly resolved fluid
prediction on the other side, the predominant portion of the CPU-time
is presently required for the CFD part. Additionally, the communication
time between the codes via CoMA and within the CFD solver takes a
non-negligible part of the computational resources.
 
== Numerical CFD Setup ==
 
For the CFD prediction of the flow two different block-structured
grids either for a subset of the entire channel (w'/l=1) or for
the full channel but without the gap between the flexible structure
and the side walls (w/l=2.95) are used. In the first
case the entire grid consists of about 13.5 million control volumes
(CVs), whereas 72 equidistant CVs are applied in the spanwise
direction. For the full geometry the grid possesses about 22.5 million
CVs. In this case starting close to both channel walls the grid is
stretched geometrically with a stretching factor 1.1 applying in
total 120 CVs with the first cell center positioned at a distance of
<math>\Delta</math>y/D=1.7 x 10<math>^{-2}</math>.
 
 
[[File:Benchmark_FSI-PfS-1_full_and_subset_case.jpg]]
 
Fig. 7: X-Y cross-section of the grid used for the simulation (Note that only every fourth grid line in each direction is displayed here).
 
 
The gap between the elastic structure and the walls is not taken into
account in the numerical model and thus the width of the channel is
set to w instead of W. The stretching factors are kept below 1.1 with the first cell
center located at a distance of <math>\Delta</math>y/D=9 x 10<math>^{-4}</math> from
the flexible structure. Based on the wall shear stresses on the
flexible structure the average y<math>^+</math> values are predicted to be below
0.8, mostly even below 0.5. Thus, the viscous sublayer on the
elastic structure and the cylinder is adequately resolved. Since the
boundary layers at the upper and lower channel walls are not
considered, no grid clustering is required here.
 
On the CFD side no-slip boundary conditions are applied at the rigid
front cylinder and at the flexible structure. Since the resolution of
the boundary layers at the channel walls would require the bulk of the
CPU-time, the upper and lower channel walls are assumed to be slip
walls. Thus the blocking effect of the walls is maintained without
taking the boundary layers into account. At the inlet a constant
streamwise velocity is set as inflow condition without adding any
perturbations. The choice of zero turbulence level is based on the
consideration that such small perturbations imposed at the inlet will
generally not reach the cylinder due to the coarseness of the grid at
the outer boundaries. Therefore, all inflow fluctuations will be
highly damped. However, since the flow is assumed to be sub-critical,
this disregard is insignificant. At the outlet a convective outflow
boundary condition is favored allowing vortices to leave the
integration domain without significant disturbances
(Breuer, 2002). The convection velocity is set to
<math>u_\text{inflow}</math>.
 
As mentioned above two different cases are considered. In order to
save CPU-time in the first case only a subset of the entire spanwise
extension of the channel is taken into account. Thus the computational
domain has a width of w'/l=1 in z-direction and the
flexible structure is a square in the x-z-plane. In this case a
reasonable approximation already applied in Breuer et al (2012) is to
apply periodic boundary conditions in spanwise direction for both
disciplines. For LES predictions periodic boundary conditions
represent an often used measure in order to avoid the formulation of
appropriate inflow and outflow boundary conditions. The approximation
is valid as long as the turbulent flow is homogeneous in the specific
direction and the width of the domain is sufficiently large. The
latter can be proven by predicting two-point correlations, which have
to drop towards zero within the half-width of the domain. The impact
of periodic boundary conditions on the CSD predictions are discussed
below.
 
For the full case with w/l=2.95 periodic boundary conditions
can no longer be used. Instead, for the fluid flow similar to the
upper and lower walls also for the lateral boundaries slip walls are
assumed since the full resolution of the boundary layers would be
again too costly. Furthermore, the assumption of the slip wall is
consistent with the disregard of the small gap between the flexible
structure and the side walls discussed above.
 
== Numerical CSD Setup ==
 
Motivated by the fact that in the case of LES frequently a domain
modeling based on periodic boundary conditions at the lateral walls is
used to reduce the CPU-time requirements, this special approach was
also investigated for the FSI test case. As a consequence, there are two different
structure meshes used: For the CSD prediction of the case with a
subset of the full channel the elastic structure is resolved by the
use of 10 x 10 quadrilateral four-node 7-parameter shell elements. For the
case discretizing the entire channel, 10 quadrilateral four-node shell
elements are used in the main flow direction and 30 in the spanwise
direction.
 
On the CSD side, the flexible shell is loaded on the top and bottom
surface by the fluid forces, which are transferred from the fluid mesh
to the structure mesh. These Neumann boundary conditions for the
structure reflect the coupling conditions. Concerning the Dirichlet
boundary conditions, the four edges need appropriate support modeling:
on the upstream side at the rigid cylinder a clamped support is
realized and all degrees of freedom are equal to zero. On the opposite
downstream trailing-edge side, the rubber plate is free to move and
all nodes have the full set of six degrees of freedom. The edges which
are aligned to the main flow direction need different boundary
condition modeling, depending on whether the subset or the full case
is computed:
 
For the subset case due to the fluid-motivated periodic boundary
conditions, periodicity for the structure is correspondingly assumed
for consistency reasons. As it turns out, this assumption seems to
hold for this specific benchmark configuration and its deformation
pattern which has strong similarity with an oscillation in the first
eigenmode of the plate. Hence, this modeling approach may be used for
the efficient processing of parameter studies, e.g., to evaluate the
sensitivity of the FSI simulations with respect to slight variations
in model parameters shown in a sensitivity study. For
this special type of support modeling, there are always two structure
nodes on the lateral sides (one in a plane z=-w/2 and its twin in
the other plane z=+w/2) which have the same load. These two nodes
must have the same displacements in x- and y-direction and their
rotations have to be identical. Moreover, the periodic boundary
conditions imply that the z-displacement of the nodes on the sides are
forced to be zero.
 
For the full case the presence of the walls in connection with the
small gap implies that there is in fact no constraining effect on the
structure, as long as no contact between the plate and the wall takes
place. Out of precise observations in the lab, the possibility of
contact may be disregarded. In principle, this configuration would
lead to free-edge conditions like at the trailing edge. However, the
simulation of the fluid with a moving mesh needs a well-defined mesh
situation at the side walls which made it necessary to tightly connect
the structure mesh to the walls (the detailed representation of the
side edges within the fluid mesh is discarded due to computational
costs and the resulting deformation sensitivity of the mesh in these
regions). Also the displacement in z-direction of the structure nodes
at the lateral boundaries is forced to be zero.
 
== Coupling conditions ==
 
For the turbulent flow a time-step size of <math>\Delta t_{f} = 2 \times
10^{-5}s~(\Delta t_{f}^{\ast} = 1.26 \times 10^{-3})</math> in
dimensionless form using <math>u_\text{inflow}</math> and D as reference
quantities) is chosen and the same time-step size is applied for the
structural solver based on the generalized-<math>\alpha</math> method with the
spectral radius <math>\varrho_\infty</math>=1.0, i.e, the Newmark standard
method. For the CFD part this time-step size corresponds to a CFL
number in the order of unity. Furthermore, a constant underrelaxation
factor of <math>\omega</math>=0.5 is considered for the displacements and the
loads are transferred without underrelaxation. In accordance with
previous laminar and turbulent cases in Breuer et. al. (2012) the FSI
convergence criterion is set to <math>\varepsilon_{FSI} = 10^{-4}</math> for the
<math>L_2</math> norm of the displacement differences. As estimated from previous
cases Breuer et. al. (2012) 5 to 10 FSI-subiterations are required to
reach the convergence criterion.
 
After an initial phase in which the coupled system reaches a
statistically steady state, each simulation is carried out for about
4 s real-time corresponding to about 27 swiveling cycles of the
flexible structure.
 
For the coupled LES predictions the national supercomputer
SuperMIG/SuperMUC was used applying either 82 or 140 processors for
the CFD part of the reduced and full geometry,
respectively. Additionally, one processor is required for the coupling
code and one processor for the CSD code, respectively.
 
= Unsteady results =
 
In order to comprehend the real structure deformation and the
turbulent flow field found in the present test case, experimentally
and numerically obtained unsteady results are presented in this
section.
 
A high-speed camera movie of the structure deflection is available at: http://vimeo.com/59130974.
 
Figure 8 shows experimental raw signals of dimensionless displacements from a point located at a
distance of 9 mm from the shell extremity in the midplane of the test
section. In Figure 8a) the history
of the y-displacement <math>U_y^* = U_y / D</math> obtained in the
experiment is plotted. The signal shows significant variations in the
extrema: The maxima of <math>U_y^*</math> vary between 0.298 and 0.523 and the
minima between -0.234 and -0.542. The standard deviations on the
extrema are about <math>\pm 0.05~(\pm 12 \%)</math> of the mean value of the
extrema). Minor variations are observed regarding the period in
Figure 8a). Figure 8b)
and 8c) show the corresponding
experimental phase portrait and phase plane, respectively. The phase
portrait has a quasi-ellipsoidal form. The monitoring point trajectory
plotted in the phase plane describes an inversed 'C', which is
typical for the first swiveling mode. The cycle-to-cycle variations in
these plots are small. Therefore, the FSI phenomenon can be
characterized as quasi-periodic.
 
[[File:unsteady_LDT.png]]
 
Fig. 8: Experimental raw signals of dimensionless displacements from a point in the midplane of the test section located at a distance of 9 mm from the shell extremity.
 
 
Figure 9 is composed of eight
images of the instantaneous flow field (streamwise velocity component)
experimentally measured in the x-y plane located in the middle of the
rubber plate. These pictures constitute a full period T of the FSI
phenomenon arbitrarily chosen. As mentioned before, the shell deforms
in the first swiveling mode. Thus, there is only one wave node located
at the clamping of the flexible structure. At the beginning of the
period (t = 0) the structure is in its undeformed state. Then, it
starts to deform upwards and reaches a maximal deflection at t
= T / 4. Afterwards, the shell deflects downwards until its
maximal deformation at t =3T/4. Finally the plate deforms
back to its original undeformed state and the end of the period is
reached.
 
As visible in Fig. 9 the flow is
highly turbulent, particularly near the cylinder, the flexible
structure and in the wake. The strong shear layers originating from
the separated boundary layers are clearly visible. This is the region
where for the sub-critical flow the transition to turbulence takes
place as visible in
Fig. 9. Consequently, the flow in
the wake region behind the cylinder is obviously turbulent and shows
cycle-to-cycle variations. That means the flow field in the next
periods succeeding the interval depicted in
Fig. 9 will definitely look
slightly different due to the irregular chaotic character of
turbulence. Therefore, in order to be able to compare these results an
averaging method is needed leading to a statistically averaged
representation of the flow field. Since the FSI phenomenon is
quasi-periodic the phase-averaging procedure presented above is ideal
for this purpose and the results obtained are presented in the next
section.
 
 
[[File:unsteady_PIV.png]]
 
Fig. 9: Experimental unsteady flow field (x-y plane located in the middle of the rubber plate).
 
Prior to this, however, it should be pointed out that very similar
figures as depicted in Fig. 9 could
also be shown from the numerical predictions based on LES. Exemplary
and for the sake of brevity, Fig. 10 displays the
streamwise velocity component of the flow field in a x-y-plane solely
at t=3T/4. As expected the LES prediction is capable to
resolve small-scale flow structures in the wake region and in the
shear layers. Furthermore, the figure visualizes the deformed
structure showing nearly no variation in spanwise direction.
 
[[File:unsteady_LES.png]]
 
Fig. 10: Experimental unsteady flow field (x-y plane located in the middle of the rubber plate).
 
= Generation of Phase-resolved Data =
 
Each flow characteristic of a quasi-periodic FSI problem can be
written as a function <math>f=\bar{f}+\tilde{f}+f'</math>, where <math>\bar{f}</math>
describes the global mean part, <math>\tilde{f}</math> the quasi-periodic part
and <math>f'</math> a random turbulence-related
part (Reynolds et. al., 1972; Cantwell et. al., 1983). This splitting can also be written
in the form <math>f = <f> + f'</math>, where <math><f></math> is the
phase-averaged part, i.e., the mean at constant phase. In order to be
able to compare numerical results and experimental measurements, the
irregular turbulent part f' has to be averaged out. This measure is
indispensable owing to the nature of turbulence which solely allows
reasonable comparisons based on statistical data. Therefore, the
present data are phase-averaged to obtain only the phase-resolved
contribution <math><f></math> of the problem, which can be seen as a
representative and thus characteristic signal of the underlying FSI
phenomenon.
 
The procedure to generate phase-resolved results is the same for the
experiments and the simulations and is also similar to the one
presented in Gomes et. al. (2006). The technique can be split up into
three steps:
 
* '''Reduce the 3D-problem to a 2D-problem''' - Due to the facts that in the present benchmark the structure deformation in spanwise direction is negligible and that the delivered experimental PIV-results are solely available in one x-y-plane, first the 3D-problem is reduced to a 2D-problem. For this purpose the flow field and the shell position in the CFD predictions are averaged in spanwise direction.
 
* '''Determine n reference positions for the FSI Problem''' - A representative signal of the FSI phenomenon is the history of the y-displacements of the shell extremity. Therefore, it is used as the trigger signal for this averaging method leading to phase-resolved data. Note that the averaged period of this signal is denoted T. At first, it has to be defined in how many sub-parts the main period of the FSI problem will be divided and so, how many reference positions have to be calculated (for example in the present work n = 23). Then, the margins of each period of the y-displacement curve are determined. In order to do that the intersections between the y-displacement curve and the zero crossings (<math>U_y</math>=0) are looked for and used to limit the periods. Third, each period <math>T_i</math> found is divided into n equidistant sub-parts denoted j.
 
* '''Sort and average the data corresponding to each reference Position''' - The sub-part j of the period <math>T_i</math> corresponds to the sub-part j of the period <math>T_{i+1}</math> and so on. Each data set found in a sub-part j will be averaged with the other sets found in the sub-parts j of all other periods. Finally, data sets of n phase-averaged positions for the representative reference period are achieved.
 
The simulation data containing structure positions, pressure and
velocity fields, are generated every 150 time steps. According to the
frequency observed for the structure and the time-step size chosen
about 50 data sets are obtained per swiveling period. With respect to
the time interval predicted and the number of subparts chosen, the
data for each subpart are averaged from about 50 data sets. A
post-processing program is implemented based on the method described
above. It does not require any special treatment and thus the
aforementioned method to get the phase-resolved results is
straightforward.
 
The current experimental setup consists of the multiple-point triangulation
sensor described in Section [[UFR_2-13_Description#Laser distance sensor|Laser distance sensor]] and the
synchronizer of the PIV system. Each measurement pulse of the PIV
system is detected in the data acquisition of the laser distance
sensor, which measures the structure deflection continuously with 800
profiles per second. With this setup, contrary to Gomes et. al. (2006),
the periods are not detected during the acquisition but in the
post-processing phase. After the run a specific software based on the
described method mentioned above computes the reference structure
motion period and sorts the PIV data to get the phase-averaged
results. A more detailed description of the phase-averaging method is available in De Nayer et. al. (2013).
----
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Latest revision as of 12:10, 12 February 2017

Fluid-structure interaction in turbulent flow past cylinder/plate configuration I (First swiveling mode)

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Flows around bodies

Underlying Flow Regime 2-13

Description

Introduction

A flexible structure exposed to a fluid flow is deformed and deflected owing to the fluid forces acting on its surface. These displacements influence the flow field resulting in a coupling process between the fluid and the structure shortly denoted fluid-structure interaction (FSI). Due to its manifold forms of appearance it is a topic of major interest in many fields of engineering. Based on enhanced numerical algorithms and increased computational resources numerical simulations have become an important and valuable tool for solving this kind of problem within the last decade. Today FSI simulations complement additional experimental investigations. A long-lasting vision of the computational engineer is to completely replace or at least strongly reduce expensive experimental investigations in the foreseeable future. However, to attain this goal validated and thus reliable simulation tools are required.

The long-term objective of the research reported here is the coupled simulation of big lightweight structures such as thin membranes exposed to turbulent flows (outdoor tents, awnings...). To study these complex FSI problems, a multi-physics code framework was recently developed (Breuer et al., 2012) combining Computational Fluid Dynamics (CFD) and Computational Structural Dynamics (CSD) solvers . In order to assure reliable numerical simulations of complex configurations, the whole FSI code needs to be validated at first on simpler test cases with trusted reference data. In Breuer et al. (2012) the verification process of the code developed is detailed. The CFD and CSD solvers were at first checked separately and then, the coupling algorithm was considered in detail based on a laminar benchmark. A 3D turbulent test case was also calculated to prove the applicability of the newly developed coupling scheme in the context of large-eddy simulations (LES). However, owing to missing reference data a full validation was not possible. The overall goal of the present paper is to present a turbulent FSI test case supported by experimental data and numerical predictions based on the multi-physics code developed. Thus, on the one hand the current FSI methodology involving LES and shell structures undergoing large deformations is validated. On the other hand, a new turbulent FSI benchmark configuration is defined, based on the specific insights into numerical flow simulation, computational structural analysis as well as coupling issues. Hence, the present study should provide a precisely described test case to the FSI community for the technically relevant case of turbulent flows interacting with flexible structures.

Review of previous work

The present study is mainly related to two former investigations of Turek and Hron (2006, 2010) and Gomes et al. (2006, 2012) on vortex-induced fluid-structure interactions. The well-known 2D purely numerical laminar benchmarks of Turek and Hron (2006, 2010) developed in a collaborative research effort on FSI (DFG Forschergruppe 493) consists of an elastic cantilever plate which is clamped behind a rigid circular cylinder. Three different test cases, named FSI1, FSI2 and FSI3 are provided, complemented by additional self-contained CFD and CSD test cases to check both solvers independently. These test cases were also used to validate the solvers applied in the present study (Breuer et al., 2012). In order to close the gap of complementary experimantel and numerical data, a nominally 2D laminar experimental case was provided by Gomes et al. (2006, 2013) and Gomes (2011). Here, a very thin metal sheet with an additional weight at the end is attached behind a rotating circular cylinder and mounted inside a channel filled with a mixture of polyglycol and water to reach a low Reynolds number in the laminar regime. Experimental data are provided for several inflow velocities and two different swiveling motions could be identified depending on the inflow velocity. Owing to the thin metal sheet and the rear mass the accurate prediction of this case is demanding. There are also turbulent FSI benchmarks involving 2D structures: in Gomes et al. (2010) a rigid plate with a single rotational degree of freedom was mounted into a water channel and experimentally studied by particle-image velocimetry (PIV). This study also presents the first comparison between experimental data and predicted results achieved by the present code for a turbulent FSI problem. As another turbulent experimental benchmark, the investigations of Gomes et al. (2010, 2013) and Gomes (2010) have to be cited: the same geometry as in Gomes et al. (2006) was used, but this time with water as the working fluid leading to much higher Reynolds numbers within the turbulent regime. The resulting FSI test case was found to be very challenging from the numerical point of view. Indeed, the prediction of the deformation and motion of the very thin flexible structure requires two-dimensional finite-elements. On the other hand the discretization of the extra weight mounted at the end of the thin metal sheet calls for three-dimensional volume elements. Thus for a reasonable prediction of this test case both element types have to be used concurrently and have to be coupled adequately. Additionally, the rotational degree of freedom of the front cylinder complicates the structural simulation and the grid adaptation of the flow prediction.

Choice of test case

Thus, in the present study a slightly different configuration is considered to provide in a first step a less ambitious test case for the comparison between predictions and measurements focusing the investigations more to the turbulent flow regime and its coupling to a less problematic structural model. For this purpose, a fixed cylinder with a thicker rubber tail and without a rear mass is used. This should open the computation of the proposed benchmark case to a broader spectrum of codes and facilitates its adoption in the community. Strong emphasis is put on a precise description of the experimental measurements, a comprehensive discussion of the modeling in the numerical simulation (for the single field solutions as well as for the coupled problem) and the processing of the respective data to guarantee a reliable reproduction of the proposed test case with various suitable methods. A detailed description of the present test case is published in De Nayer et al. (2014).

The described test case FSI-PfS-1a is a part of a series of reference test cases designed to improve numerical FSI codes. A second test case FSI-PfS-2a is described in Kalmbach and Breuer (2013). The geometry is similar to the first one: A fixed rigid cylinder with a plate clamped behind it. However, this time a rear mass is added at the extremity of the flexible structure and the material (para-rubber) is less stiff. The flexible structure deforms in the second swiveling mode and the structure deflections are completely two-dimensional and larger.


Contributed by: G. De Nayer, A. Kalmbach, M. Breuer — Helmut-Schmidt Universität Hamburg (with support by S. Sicklinger and R. Wüchner from Technische Universität München)


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References


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