A SIMULATION FRAMEWORK FOR TESTING FLOW CONTROL
STRATEGIES
Marek Gayer
∗
, Milan Milovanovic and Ole Morten Aamo
Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU)
O.S. Bragstads plass 2D, N-7491, Trondheim
Norway
e-mail: marek.gayer@itk.ntnu.no, milan.milovanovic@itk.ntnu.no, aamo@ntnu.no
ABSTRACT
We present a framework dedicated for simulation and con-
trol of incompressible fluid flows. The simulation is based
on the Navier-Stokes model with possible control actua-
tion from a customizable control module. We describe
components of our simulation architecture: Computational
Fluid Dynamics (CFD) code VISTA featuring a Navier-
Stokes solver, Utility library with shared functionality for
customizable Flow Control modules and the configuration
system allowing to define separate simulation cases. Flow
Control modules are responsible for performing control
calculations, reading flow fields values, actuation, storing
results for post-processing in each time step. By creating
new Flow control modules, we can simulate different flow
control behaviour and strategies. The simulation solution
and it’s components are realized in C++ using VISTA and
Diffpack API. We present a case study based on this frame-
work, where the objective is to suppress vortex shedding
around cylinders in the 2D space domain by feedback con-
trol based module with kernel coefficients pre-calculated in
MATLAB using Ginzburg-Landau model.
KEY WORDS
Flow control simulation, Navier-Stokes, Vortex shedding.
1
Introduction
1.1
Flow control
Flow control involves controlling a flow field using passive
or active devices in order to bring on desired changes in the
behaviour of the flow. For instance, laminar flow, which is
characterized by parallel layers of fluid moving in a very
regular and deterministic way, is associated with consider-
able less drag, or friction, at wall-fluid interfaces, than its
counterpart, turbulent flow, which is characterized by small
scale velocity components that appear to be stochastic in
nature. Usually, laminar flows are unstable, and will un-
less controlled evolve into turbulent flows. Common con-
trol objectives for flows include: Delaying or advancing
transition from laminar to turbulent flow, suppressing or
enhancing turbulence, preventing or provoking separation
∗
This work was carried out during the tenure of an ERCIM "Alain
Bensoussan" Fellowship Programme.
and preventing or provoking certain flow regimes for mul-
ticomponent flows.
This paper presents a Computational Fluid Dynam-
ics (CFD) simulation framework dedicated for simulation
of flow control based on Navier-Stokes model and using
customizable Flow control modules, allowing to implement
general flow controllers.
1.2
Vortex shedding simulation and control
Dynamics of flow around cylinder has been studied for
many years. Modelling and simulation of the cylinder wake
behaviour are proposed in various experiments and models
in [1], [2], [3], [4], [5] and [6]. For flows past submerged
obstacles with sufficiently large Reynolds numbers, vortex
shedding forms. A popular model flow for studying vor-
tex suppression by means of open-loop or feedback con-
trol, is the flow past a two-dimensional circular cylinder,
as sketched in Fig. 1 where the so-called von Kármán vor-
tex street is visualized using passive tracer particles. Var-
ious simple feedback control configurations, such as [7]
and [8] were proposed for successful suppression of vortex
shedding in numerical simulations. A simple model, based
on the Ginzburg-Landau equation with appropriate coeffi-
cients, has been found to well model the dynamics of vortex
shedding in [9], [10] and [11]. Systematic control designs
for finite dimensional approximations of the model from
[11] were developed in [12], [13] for the linearized model,
and in [14] for the nonlinear model. In [15] and [16],
boundary control laws based on the non-discretized, lin-
earized Ginzburg-Landau equation were rigorously derived
for the state feedback and boundary output feedback cases,
respectively. They were shown to effectively attenuate vor-
tex shedding for the nominal case with Ginzburg-Landau
as the system plant as well as the design model. In the test
case study presented in this paper, the state feedback pro-
posed in [15] is applied to the Navier-Stokes solver.
2
Simulation framework overview
The major parts of the framework are a CFD software code
named VISTA with a Navier-Stokes solver, Utility library
Figure 1. Vortex shedding from a cylinder visualized by
passive tracer particles.
Figure 2. Our simulation framework architecture overview.
containing general functionality common for flow simula-
tion and control modules and a customizable Flow control
module. The architecture is schematically depicted on Fig-
ure 2.
2.1
CFD code for solving Navier-Stokes model
Our simulation framework is based on the Navier-Stokes
model. As the basis for our CFD simulations of the flow
control we use VISTA, developed by SINTEF Applied
Mathematics
1
. VISTA is a general purpose, extendible
CFD code using the finite element method and solving wide
range of problems. The core of this code is a finite ele-
ment method based solver of incompressible Navier-Stokes
equation:
ρ
∂u
∂t
+ ρ(u · ∇u) + ∇p − μ∇u = f
(1)
∇ · u = 0
where ρ is the density of the fluid, u is the velocity
vector with velocities in x and y direction, p is the pres-
sure, μ is the dynamic viscosity and f an external body
force. The boundary conditions for the equation can be set
separately for each simulation cases.
The VISTA is written in C++ and uses Diffpack nu-
merical package [17] for simulation based on the finite el-
ement method. Beside the Navier-Stokes solver, it has also
1
http://www.sima.sintef.no/
functionalities for reading grids, obtaining simulated values
from the domain and setting Dirichlet boundary conditions
in subsequent time steps, reading configuration of simula-
tion cases (VISTA menu system) and store some results in
files. We will describe these possibilities later.
2.2
Utility library
The Utility library efforts to provide general functionality
shared by customizable Flow control modules. This code
is realized using programming VISTA API and Diffpack
[18]. The vector and matrix variables for data-structures
are provided as Diffpack real and complex array classes, al-
lowing us to use some basic mathematical operations over
these arrays (e.g. adding, multiplication), bound checking
as well as possibility for passing these data types as argu-
ments of C++ methods and functions. Utility library de-
fines additional routines over basic and limited work with
these structures provided by Diffpack. Utility library con-
tains routines for reading input files and writing output data
sets. Vector and matrices files are represented as saved
MATLAB script files with a special syntax format, con-
taining assignments for vector items. Such files with this
syntax can be both read from and written to Diffpack data
types using Diffpack API and we have a script to generate
these files in MATLAB.
2.3
Flow control modules
A Flow control module allows realization of a simulation
flow controller. At the beginning of a simulation, a module
is responsible for the initialization and reading of eventual
input values related to a control case. It is responsible for
processing an algorithm for a concrete controller computa-
tion in each time step. A module is responsible for applying
the computed values back to the Navier-Stokes model. A
module during each time step also updates various output
files with read and computed values, which can be used for
monitoring and post-processing. For realization some of its
functionalities, it uses the Utility library. A sample of such
module will be described in detail further. All Flow control
modules are implemented as C++ inherited classes. Some
inherited methods are invoked from VISTA, such as on ini-
tialization, each time step and on finishing the simulation.
By creating new Flow control modules, we can simulate
different flow control behaviour and strategies. We plan to
enrich the functionality of the Utility library during devel-
opment of such modules.
2.4
Input parameters in the VISTA menu system
Various input parameters for each of the independent sim-
ulation cases are specified in a structured text file for-
mat, which is parsed and accessed by VISTA and is called
VISTA menu system. This allows Vista CFD configura-
tion system to set various variables defining the simula-
tion case. The menu system is also accessed by our Utility
library and Flow control modules, since custom variables
can be defined and read from the menu system. This way,
we can for instance define the Reynolds number used dur-
ing simulation, starting time and length of simulation, time
steps, locations of files containing mesh and control coef-
ficients, controller fade in period, etc. Further, parameters
controlling creation of VTF data sets
2
and allowing even-
tual restarting of the simulation from an already simulated
state (or using a saved state as the basis for a simulation
with modified parameters) can be defined as well. There
are also additional plain text files referred and used in the
VISTA menu system. In these files, we can define bound-
ary conditions and points on which characteristics will be
extracted.
2.5
Computational grids
General purpose grid generators can be used to produce
computational grids for VISTA, since the VISTA code sup-
ports meshes in Diffpack and VTF formats. During devel-
opment of our simulation framework, including the later
described test case study, we have used the grid generator
GRIDDLER
3
to generate finite element meshes in the VTF
format.
GRIDDLER supports general block-structured
meshes in 2D and 3D. The vertices, curves, surfaces and
blocks are specified in an input file. The file name of the
grid is specified in the VISTA menu system and is automat-
ically read by VISTA.
2.6
Results storage and post-processing
The characteristics from the computed flow field in whole
time and space domain are stored in VTF format using
GLview Express Writer API
4
, embedded in VISTA and
also some files are produced by our Utility library in plain
text files, which can be imported in MATLAB. We post-
process these results by using both the MATLAB and
GLview Inova
5
.
In GLview Inova, we can easily render animations
from VTF files or review simulation progress of flow con-
trol, by visualizing various vector and scalar properties
such as pressure, vorticity and velocities. During and af-
ter finishing of the simulation, we can monitor in the MAT-
LAB various parameters: applied control values, velocities,
pressures, force drag and lift and many others parameters
for 2D and 3D plots. This functionality is maintained by
availability of our MATLAB scripts processing the stored
data sets.
Each of simulation cases is using different directories
for storing it’s results, thus keeping order in the file system
and offering possibilities of easy migrations and backups.
Various characteristics are stored in separate files to allow
2
http://www.ceetron.com/en/vtf_format.aspx
3
Developed by Karstein Sørli at SINTEF ICT Applied Mathematics,
http://sourceforge.net/projects/griddler/
4
http://www.ceetron.com/en/glview_express1.aspx
5
http://www.ceetron.com/en/glview_inova1.aspx
Figure 3. Computational grid with zoom around cylinder.
quick loading of subsets of output parameters in simulated
case, both in post-processing as well as during simulation
monitoring.
3
Case study of a simulation and control of
vortex shedding around cylinders
This case study is realized within our simulation frame-
work. It provides simulation of vortex shedding around
cylinders in 2D. It uses custom boundary conditions as well
as the mesh for the VISTA Navier-Stokes solver. Fig. 3
shows the grid generated by GRIDDLER, which we use
for testing. A Flow control module has been implemented
based on the control law developed in [15].
3.1
Our boundary conditions for Navier-Stokes solver
We assume free stream flow behaviour, flow at the bound-
aries is not affected by the cylinder. The following bound-
ary conditions for Eq. 1 can be then assumed:
σ · n = 0
on
Γ
1
(2)
u · n = 0
on
Γ
2
∪ Γ
4
(3)
τ · σ · n = 0
on
Γ
2
∪ Γ
4
(4)
u = u
in
on
Γ
3
(5)
u = u
c
(t)
on
Γ
5
(6)
where the boundaries are denoted by
Γ
i
, and are de-
picted in figure 4. Furthermore
u
in
is the inlet velocity
vector at
Γ
3
and
u
c
(t) is the velocity vector on the cylin-
der boundary
Γ
5
due to control actuation. Besides,
n and
τ are the unit outer normal and tangent vectors, while σ is
the stress tensor given by:
σ = −pI + μ∇u
(7)
Since the VISTA uses a continuous projection method
for solving the Navier–Stokes equations, where the com-
putation of the velocity and pressure is split, the boundary
conditions for the numerical scheme looks slightly differ-
ent. The boundary condition on the outlet is:
Γ
1
x
y
−8
0
24
−8
8
D=1.0
Γ
4
Γ
2
Γ
3
Γ
5
Figure 4. Computational domain with boundaries.
p = 0,
∇u · n = 0
on
Γ
1
(8)
and a zero gradient condition is used for the pressure
on the remaining part of the boundary
∇p · n = 0
on
Γ
2
∪ Γ
3
∪ Γ
4
∪ Γ
5
(9)
The inflow velocity
u
in
is chosen by the user before
the simulation is started, while the velocity
u
c
(t) is com-
puted by the controller as the simulation proceeds.
3.2
State feedback model using Ginzburg-Landau
based kernels
In [11] a simplified model of the computation of cylinder
wake dynamics was suggested in terms of the Ginzburg-
Landau equation.
∂A
∂t
= a
1
∂
2
A
∂ ˘
x
2
+ a
2
(˘x)
∂A
∂ ˘
x
+ a
3
(˘x) A
+ a
4
|A|
2
A + δ (˘
x − 1) u,
(10)
where A is a complex-valued function of one spatial vari-
able,
˘x ∈ R, and time, t ∈ R
+
. The boundary conditions
are A (±∞, t) = 0. The control input, denoted u, is in
the form of point actuation at the location of the cylinder,
and the coefficients a
i
, i = 1, .., 4, were fitted to data from
laboratory experiments in [11]. δ denotes the Dirac distri-
bution.
In [15, 16] a simplified form of Ginzburg-Landau
equation was considered. The resulting system is given by
∂A
∂t
= a
1
∂
2
A
∂ ˘
x
2
+ a
2
(˘x)
∂A
∂ ˘
x
+ a
3
(˘x) A
(11)
for
˘x ∈ (0, x
d
), with boundary conditions
A (0, t) = u(t) and A (x
d
, t) = 0,
(12)
where A : [0, x
d
] × R
+
→ C, a
2
∈ C
2
([0, x
d
]; C) , a
3
∈
C
1
([0, x
d
]; C), a
1
∈ C, t ∈ R
+
is time, and u : R
+
→ C
is the control input. a
1
is assumed to have strictly positive
real part. The boundary conditions are A (±∞, t) = 0. The
control input, denoted u, is in the form of point actuation
at the location of the cylinder, and the coefficients a
i
, i =
1, .., 3, were fitted to data from laboratory experiments in
[11].
In [15], stabilizing state feedback boundary control
laws for system (11)–(12) were derived based on the back-
stepping methodology [19, 20, 21, 22]. The state feedback
controller designed in [15] takes the form
u(t) =
x
d
0
1
x
d
[k
1
(
x
d
− ˘x
x
d
) − ik
c,1
(
x
d
− ˘x
x
d
)]A(˘x, t)
× exp
1
2a
1
˘x
0
a
2
(τ)dτ d˘x
(13)
where i denotes the imaginary unit, and k
1
and k
c,1
are the
controller kernels which can be computed as the solution
of a certain hyperbolic partial differential equation. Further
details can be found in [15]. The controller kernels are pre-
calculated in our MATLAB codes.
3.3
Connecting the state feedback controller with
Navier-Stokes solver
A (˘
x, t) in Eq. 13 may represent any physical variable (ve-
locities
(u, v) or pressure p), or derivations thereof, along
the centerline of the 2D cylinder flow. The choice will
have an impact on the performance of the Ginzburg-Landau
model. In our case, we associate A with the transverse fluc-
tuating velocity v (˘
x, ˘
y = 0, t) which seems to be a particu-
larly good choice [23]. As pointed out in [13], the model is
derived for Reynolds numbers close to the critical Reynolds
number for onset of vortex shedding, but has been shown
to remain accurate far outside this vicinity for a wide va-
riety of flows. For the interior flow, we define two curves
originating at the actuation slots and converging at the flow
centerline as they stretch into the 2D flow domain (see left
of Fig. 5). Transverse velocity is measured along these
curves, denoted v
l
(˘x, t) and v
u
(˘x, t) in left of Fig. 5, and
related to A as
Re(A) := v(˘
x, t) =
1
2
(v
l
(˘x, t) + v
u
˘x, t)) .
(14)
The imaginary part of A is reconstructed based on modes
of the Ginzburg-Landau equation having the form
A(˘
x, t) = Φ(˘
x)exp(iκ˘
x − iωt).
(15)
We approximate Im(A) by shifting Re(A) in space corre-
sponding roughly to
Δ˘x = T U/4, where T is the shedding
period and U is the streamwise velocity. Having obtained
A as described, the control input is computed using Eq. 13.
With the real part of A representing transverse velocity, the
control actuation which is a Dirichlet boundary condition at
the actuation end in the Ginzburg-Landau model, can be re-
alized in the CFD code by jets on the cylinder surface. We
-0.5
0
0.5
1
1.5
2
-1
-0.5
0
0.5
Figure 5. Curves along which transverse velocity is mea-
sured.
x
y
Figure 6. Suction/blowing actuation slots.
do this by employing two slots configured as shown in right
of Fig. 6, close to the separation layers for effectiveness
[24]. In our case, the actuation is done by using blowing
and suction slots at
±110º from the stagnation point with a
span of
15º. The velocity profile of the jet is set according
to:
u
c
(α, t) · n = u(t)sin(
π
α
span
(α
N
i
− α
start
)).
(16)
We recalculate values of u(t) and apply them each
time step to the cylinder slots. This is effectively providing
a suitable control parameter for the jet. Besides, α
N
i
is
the angle of the actual node in the mesh, while α
start,end
are the start and end angles of the jet. For the purpose of
solver stability, the actuation in all simulations is faded-
in linearly in approximately one natural vortex shedding
period (@Re = 60, T
f adein
= 6[s]).
When one slot applies suction the other applies an
equal amount of blowing, thereby maintaining the mass
balance in the system. The actuation is thus suction and
blowing whose size is given by the real part of u com-
puted from Eq. 13. In practice, the physical actuation could
be for instance micro/synthetic jet actuators, distributed on
the cylinder surface, and a possible choice for the physical
sensing could be pressure sensors distributed on the cylin-
der surface.
3.4
The Flow control module for this case
The Flow control module at the initialization reads the con-
trol kernels. During reading the controller kernels, we use
Utility library to interpolate, resize and scale dimensions of
smaller vector arrays into longer vector arrays. Each time
step, the module is responsible for processing transversal
velocities from sensing devices represented in the Navier-
Stokes solver, for recalculating state feedback controller
by evaluating Eq. 13. In each time step of the code, we
retrieve beside transversal velocities also additional char-
acteristics for the controller simulation and monitoring on
both surface of the cylinder as well as on the horizontal line
leading from the center of the cylinder, see Fig. 5. These
include longitudal velocity, vorticity, shear stress, nodal co-
ordinates and pressure.
For applying the computed control values back to the
cylinder, we set time dependent Dirichlet boundary condi-
tions along the surface points of the cylinder. The mod-
ule computes actuation values and sets boundary condi-
tions during each time step. They are passed to the Navier-
Stokes solver, in form of boundary velocities on the surface
on the cylinder, in place of suction and blowing jets and are
distributed according to Eq. 16. The values are applied
to set of singular points on the cylinder. The Flow control
module reads parameters of controller jets (such as radius,
center, span) using the VISTA menu system.
3.5
Simulation testing and results
We have tested our simulation solution with a case with
fully developed vortex shedding at Re = 60. We can vi-
sualize suppression of developing of the vortex shedding
in the cylinder of both scalar and vector fields (velocities,
pressures, etc). Vortex shedding subjects the bluff body, in
this case the cylinder, to periodic forcing, and would in any
physical system induce potentially destructive vibrations.
The controller applied in subsequent time steps effectively
attenuates the forcing. Fig. 7 shows a vorticity map of the
flow at t = 0. The von Kármán vortex street is clearly vis-
ible. Vortex shedding subjects the bluff body, in this case
the cylinder, to periodic forcing, and would in any phys-
ical system induce potentially destructive vibrations. The
controller applied in subsequent timesteps effectively at-
tenuates the forcing. Fig. 8 and Fig. 9 show vorticity maps
at t = 20s and t = 100s (when vortex shedding is com-
pletely removed). Fig. 10, 11 and 12 visualize transversal
velocities and also depict progress of gradual suppression
of vortex shedding in time after applying control. A sample
of 3D plots, monitoring pressures on the cylinder boundary,
is on 15 and 16. A sample of 2D monitoring, which is very
useful during ongoing simulation (e.g. figure is updated
each 10 seconds) is depicted on Fig. 14. During and after
finishing simulations, we are able to monitor and perform
3D visualizations of changing velocities, distance and time,
as it is on Fig. 13.
Figure 7. Vorticity map at t = 0. Fully developed vortex
shedding.
Figure 8. Vorticity map at t = 20s. Attenuated vortex
shedding after applying actuation.
Figure 9. Vorticity map at t = 100. Completely stabilized
flow.
Figure 10. Transversal velocity map at t = 0.
Figure 11. Transversal velocity map at t = 20.
Figure 12. Transversal velocity map at t = 100.
Figure 13. Suppression of velocities in distances up to 6 m
from the cylinder in time from 0 to 50 seconds.
0
5
10
15
20
25
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Real value
Demo 1 Figure
Jet peak velocity 1
Jet peak velocity 2
Controller result Ur
CFL number
Drag lift
Figure 14. Monitoring of simulation control related param-
eters.
Figure 15. 3D plot of pressures around the cylinder bound-
ary (±180º).
Figure 16. 3D plot of pressures around the cylinder bound-
ary (±180º) with substraction of static pressure values.
4
Implementation and portability notes
To run the simulations, our primary focus are installations
on remote multiprocessor Linux servers, so that we can
eventually compute general, usually complex 3D simula-
tions flow control cases and in future with parallel process-
ing. We use SSH tunnelling to allow mapping of remote
disk storage’s as local drives.
While this option offers advantages over local instal-
lations, namely in terms of performance, we also found this
option on the other hand often impractical namely due to
lack of development tools on the server and certain prob-
lems with queuing system and performance of transferring
simulated data sets over the network. Therefore, during
development of the framework and for our 2D case study
described in the paper, we had preferred a local installation.
VISTA CFD code currently runs only on Linux, and is
dependent on the Diffpack library. To overcome the porta-
bility limitation, as well as difficult installation of both the
VISTA and Diffpack, we decided to use VmWare virtual
machines with Debian Etch Linux as the guest the operat-
ing system for the base for our local installation. In our
case, this allows us to make experiments and run simu-
lations in one operating system, since we use MATLAB,
GLview Inova and other applications in Windows. For
sharing data between simulation environment and host op-
erating system in VmWare and other applications, we use
network drives on Windows servers and using Samba to
access them in Linux.
5
Conclusion
In this paper, we have presented a simulation framework
for general incompressible fluid flows control. Our pro-
posal features the CFD code VISTA computing 2D flow
around the cylinder using Navier-Stokes solver and a cus-
tomizable Flow control module. In each time step a Flow
control module is responsible for reading and writing state
values from Navier-Stokes solver, recalculating controller
code actuation values, and applying them back to the flow
field. With different Flow control modules, we can sim-
ulate various flow control problems and strategies. The
Utility library offers general functionality for these mod-
ules and helps to generate results data for 2D and 3D post-
processing. Results are monitored using MATLAB scripts,
while VISTA provides ability to automatically store VTF
files for animation rendering.
We have presented a case study based on our simu-
lation environment allowing control of vortex shedding in
flows around cylinders with a state feedback controller, re-
moving vortex shedding for the Reynolds number Re =
60. The simulations can be potentially repeated for higher
Reynolds numbers, under the condition of providing re-
computed kernels for state feedback controller.
For the future work, we plan to extend our simulation
design and codes to work with the output feedback prob-
lem, which was solved for the Ginzburg-Landau equation
in [16], develop simulation cases and Flow control mod-
ules for 3D control of fluid flows, and implement parallel
processing.
6
Acknowledgment
This work was supported by the BVV program at NTNU
(Program on Computational Science and Visualization).
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