Reduced order models for wake control with a spinning...

Reduced order models for wake control with a spinning cylinder

A Gronskis1, J D’Adamo1,2, A Cammilleri1 and G Artana1,2

1 Lab. Fluidodinámica - Facultad de Ingeniería - Universidad de Buenos Aires -Argentina2 CONICET - Argentina

E-mail: [email protected]

Abstract. We study the formulation of reduced order models for a circular spinning cylinder in laminar vortex shedding regime. Proper orthogonal decomposition techniques based on the snapshots generated from velocity fields and a Galerkin projection are used to generate a low dimensional representation of the infinite dimensional system described by the Navier Stokes equation. This kind of problem has generally been studied numerically, without taking into consideration the constraints associated to physical experimental conditions or under the assumption that actuations do not alter significantly the time averaged velocity field. We propose a reduction technique that enables to obtain a control function based on the mean velocity field deviations from the unperturbed reference flow. This function is educed with a Gram Schmidt construction and gives rise to an extended base in which the velocity fields of the perturbed modes are expanded. The analysis allows for easy determination of a suitable model to describe a large range of the control parameter space, without needing to recalculatethe coefficients that define the system for new operating conditions.

1. IntroductionProgress in controlling the wakes produced by bluff bodies can lead to significant benefits in different areas like energy or transport, that have close links to environmental protection.

When the geometric shape of an aerodynamic body is fixed and passive flow control devices have been fully exploited or are found to be inappropriate, the study of active flow control problems becomes essential as it may lead to additional improvements in aerodynamic performance.

An important area of flow control research is dedicated to the analysis of vortex sheddingphenomena behind bluff bodies, as it has associated problems related to drag, noise, fluid induced vibration and mixing.

In laboratory environments, the wake of circular cylinders has been particularly studied, being considered a benchmark problem representative of the possibility to effectively control wakes produced by more complex bluff bodies.

Among the different devices that can be used to alter the characteristics of the wake dynamics, the mechanical devices that introduce either a steady rotation or rotary oscillations of the cylinder around its axis have probably concentrated the larger number of research efforts.

Other studies have also tried to imitate the effect of a rotating wall through localized momentum injection, using in this case either synthetic jets technology or the so called plasma actuators technology (see for instance [1]). Particularly this last kind of device seems to open in the future the possibility to develop more complex strategies for flow control as it does not involve any moving partsand flow excitation is possible in a large range of frequencies.

One of the drawbacks of these devices is that numerical simulations of the actuation they produce,remain in general a rather difficult task. As a consequence, in order to test tools for flow control

X Meeting on Recent Advances in the Physics of Fluids and their Applications IOP PublishingJournal of Physics: Conference Series 166 (2009) 012016 doi:10.1088/1742-6596/166/1/012016

c© 2009 IOP Publishing Ltd 1

strategies, it still seems more convenient to avoid these kinds of complexities and initially restrict to simple cases with uniform moving boundaries.

The problem can be further simplified restricting the analysis to cases where the actuationconsidered is not time dependent. In this case the characteristics of the wake flow are determined only

by two non-dimensional parameters: the Reynolds number

dURe and a rotation parameter that

represents the circumferential speed at the cylinder surface normalized by the free-stream velocity

U

d

2

(d cylinder diameter, U∞ incident flow velocity,

angular rotation velocity).

Contrary to its initial simple appearance, this actuation may largely increase the different possible flow configurations, producing at the same time alterations of the mean velocity fields and of the wake dynamics.

For instance at the laminar vortex shedding range (50<Re<200), different works have shown [2, 3] that even though at moderate values of the rotation parameter (0<γ<1.95) the non dimensional frequency of shedding (fs) remains quite constant,

2.0U

dfSt s

the mean values of the forces induced on the cylinder and their fluctuations are quite sensitive to therotation parameter’s value.

This behavior can be illustrated with typical polar graphs like that of figure 1 representing the instantaneous drag coefficient (CD) versus the instantaneous lift coefficient (CL). This kind of graphs also enables to show other phenomena: we can see that once rotation exceeds a certain threshold that depends on the Reynolds number (i.e. γ ~ 1.95 for Re ~ 180), vortex shedding fully disappears giving rise to a steady flow regime.

More recently different authors [2, 4] reported that when exceeding a higher second threshold of the rotation parameter (γ ~ 4.2 for Re ~ 180) a second vortex shedding regime may re-appear with a characteristic frequency which is quite different from that of the first regime. This regime, which corresponds to very high angular velocity of the spinning cylinders, is out of the scope of this article.

a) b)

1.3 1.35 1.4 1.45 1.5-1

-0.5

0

0.5

1

CD

CL

Re=60Re=80Re=90Re=110Re=125Re=150Re=160Re=180

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-20

-15

-10

-5

0

5

CD

CL

=0

=0.25

=0.5

=0.75

=1.0

=1.25

=1.5

=1.75

=1.9

=2.0

=2.5

=3.0

=3.5

=4.0

Figure 1. Polar graphs CD vs CL: a) Non-actuated flow, b) Spinning cylinder at Re=125.


2

1.1. Reduced Order ModelingIt is possible to classify in groups the different kinds of flow control problems that one can consider,for instance control of a final state, optimization, stabilization or estimation [5].

The first kind of problem is how to determine the control action that will move the system from an initial state towards a final state. The second kind of problem studies how to find the optimal set of input parameters that maximizes an indicator of the system performance. In the study of the third kind of problem one tries to determine a control input that will keep the state of the system in aneighborhood of some solution. The last one concerns the reconstruction of the system given its incomplete or noisy data.

Usual tools of control theory can only be applied when it is possible to have at one’s disposal asimple model expressed by a set of ordinary differential equations (reduced order model (ROM)),capable of giving accurate representations of the “essential” dynamical behaviour of the flow.

This set may be determined using snapshots of experimental results (numerical or physical experiments) and considering a representation of the flow fields expressed by a finite set of a n-orthonormal spatial base functions )(xiΦ , each with an associated time dependent coefficient ai(t). To

obtain a consistent model with the observations, the velocity field expansion

)()(),(0

xxu ii

n

i

Φtat

(1)

has to satisfy the boundary conditions. To determine the base it is possible to use Karhunen Loeve (KL) decomposition (or Proper

orthogonal decomposition (POD)), where the spatial modes and temporal coefficients may be calculated from a singular value decomposition of the experimental data matrix by standard techniques [6]. This technique is optimal in the sense that it enables to obtain the set of functions that better approximates energy in comparison to any other base. The modes are ordered with decreasing energy and from the initial set of modes (equal to the set of snapshots) a number of modes may be finally retained fixing an energetic or spectral gap truncation criterion.

As the spatial modes are orthonormal, a Galerkin projection of the Navier Stokes equation under the hypothesis of incompressible flow reduces the system to an ODE system of the kind [6]

kj

s

kijk

s

ji

s

jiji

i aacalidt

da

111

(2)

This system is a reduced order model (POD ROM) and the coefficients involved depend on the basis functions and pressure field. The influence of the pressure field is in general included with other contributions in the independent term. For the case of wake flows there is evidence that this term will have negligible effects.

The POD ROM largely simplifies the complexity of the NS equation but it presents different deficiencies.

One of them is to provide reliable predictions away from the reference orbit from which the empirical model was extracted. To successfully capture the flow dynamics of different states the basisfunctions used to build the model have to be broad enough to span a hyperplane (Karhunen Loevespace) that includes all relevant flow structures associated to other states of interest.Even if this is satisfied the coefficients of the set of equations described by (2) may drastically change for the different states.

It is clear that is always possible to restrict the validity of the POD ROM to the neighborhood of the reference state and consider adaptive methods in which the bases are regularly updated when the effectiveness of the existing POD ROM fails to represent accurately the flow for the new control parameter values [7, 8, 9]. However, in order to define flow control strategies, it results more attractive


3

to determine the POD functions, and consequently the coefficients of equation (2), once for all without the need of any recalculation.

Another drawback of POD ROM is the difficulty to account for changes in the boundary conditionssuitably. This turns to be crucial when the interest is focused in developing flow control strategies based on boundary condition alterations. How to adequately include the effect of changes of a control parameter in the set of equations (2) may also represent a problem in this kind of formulation.

In this work we are interested in the study of these problems. In the next paragraphs we will give a brief summary of the background of our work.

1.2 Basis generationIn general, determining the coefficients of equation (2) using snapshots of a limit cycle associated to a stationary and periodic flow configuration (i.e. the one produced in a non actuated flow or when a continuous or regular periodic excitation is imposed to the cylinder), leads to a poor choice of the basis functions that may cancel out the benefits of the control.

To improve the ability of the model to represent different flow conditions, it is possible to generate the basis using additional snapshots. For instance, including snapshots of transient flows approaching a periodic flow configuration or limit cycle [7, 8].

Other authors [7, 9], have considered the basis generation with data sets associated to flow conditions with excitations producing rich transient configurations. The snapshots can be issued for instance from an ad-hoc time-dependent excitation following a varying-frequency sinusoidal function (chirp). By doing this, the long term and transient predictions of the POD ROM showed an improvement. However, a lack of robustness of this procedure was observed. The improvement not only depends on the nature of the excitation but also on the flow conditions which were applied when the forcing started [7, 9].

Another approach proposed in [10] or [11] adds to the original POD basis special modes called non-equilibrium modes. These non-equilibrium modes can be particular modes known to play a major role in the description of the flow dynamics not evidenced by the original decomposition (stability eigenmodes for example) or deviation modes (also named shift modes); they allow the representation of a missing dynamic mean field correction.

The continuous mode interpolation technique [12] proposes a scheme in which the basis of POD is issued from data corresponding to two close operating conditions. Between them, a POD basis is deduced considering an interpolated kernel of the Fredholm eigenproblem whose eigensolutions are associated to the spatial modes of the basis. This strategy requires in general an important number of reference states and it may have additional problems since these modes do not necessarily keep orthonormality [12].

The so called sequential proper orthogonal decomposition might also be used for generating a basis [13]. With this approach, the ensemble of snapshots correspond to different stationary and periodic flow states. Centroidal Voronoi Tessellations [14] may provide in this case a criterion to undertake the sampling of the different reference states. This kind of strategy has been applied to analyze instabilities and bifurcations in a driven cavity flow [13]. To our knowledge it has not been used togenerate POD ROM with goals associated to flow control.

1.3 Representation of actuation When a flow state is altered by some forcing, it is desired to have a POD ROM system able to describethe new physical situations by tuning some variables associated to the actuation. To incorporate the variable input control in the model, different approaches have been proposed: the “interpolation method”, the “penalty method” and the “control function method” [9].

In the “interpolation method” the temporal coefficients of the system of equations (2) are linearly interpolated between two close operating conditions. This approach has been used in combination withthe basis interpolation method in order to achieve flexible modes that can change at different flow conditions [13].


4

In the “penalty method” approach, the non-actuated POD basis is considered and non-homogenous boundary conditions are imposed in a weak implicit sense. This method presents as a drawback a parameter for which appropriate values cannot be determined a priori. Also it leads to a formulation in which appear pressure terms that have to be evaluated at the boundaries of the flow domain. That evaluation may be difficult to perform in many cases.

In the “control function method” [7, 9] the boundary actuation is incorporated in the system addingin (2) a term of the form

s

iiic tatt

0

)()(),( (x)xxu (3)

where φc(x) is an arbitrary divergence free control function satisfying the homogenous boundary conditions.

There is no specific way to prescribe the control function. One usual way is to associate it to the flow field produced when the cylinder spins in a quiescent fluid (numerically this is achieved imposing at the cylinder surface a steady rotation with Λ=1 and assuming on the outer boundaries of the flow domain null velocity (u=0) [9]). Using this approach the POD ROM is generated in the usual way from a snapshot set of a modified velocity field:

xxuxur ckkk ttt )(),(,

The spatial modes of the φi(x) thus generated have zero velocity on the cylinder surface and expansion (3) automatically satisfies the boundary condition of the problem. Thus, the system (2) canbe rewritten as [7]

2

1

111

i

N

jiijii

kj

s

kijk

s

jj

s

jiji

i

gafet

d

aacalidt

da

(4)

Some problems may arise when the control function is obtained in this way.First, the effect of the control mode results quite localized around the cylinder vanishing rapidly

due to the absence of a convective free-stream flow in the formulation. As a consequence, changes produced by actuation in the far wake are more difficult to reproduce.

Besides, this kind of formulation of the control function has in general been restricted to cases where the data are issued from numerical simulations. In these cases the modes are generated withdata that at close proximity of the boundary are relatively reliable. Unfortunately, when the experimental data are produced with real physical experiments the situation may depart from the “ideal” numerical case.

For instance, for experiments undertaken with the PIV techniques it results quite difficult to obtain information with high spatial definition in the wake and at regions close to the boundaries in the same snapshot. In addition, at close vicinity of the surface the error in the measurements may increasebecause of improper seeding of the flow in this region, or due to illumination problems like reflections or shadowing of the light sheet produced by the body itself.

Finally, most authors have used this approach in situations where the actuation did not significantlyalter the mean flow field. The coefficients of the system of equations (4) depend not only on the characteristics of the basis function but also on the mean velocity field. A model that is not capable to


5

adapt to these changes will result only well suited to describe cases in the neighborhood of thereference state.

An alternative to remediate this is to generate a control mode based on the difference of the mean values of non-actuated flow conditions (um0(x)) and an actuated reference case (umact(x)). The control function is then associated to a displacement vector defined as:

xuxux m0mact )(c (5)

This kind of approach has been proposed recently in some works [15] and explored for plasma actuated flows [16, 17].

It is in the scope of this article to develop accurate POD ROM of wake cylinders flows altered by the steady rotations of the cylinder axis. We focus our efforts on determining suitable strategies to generate the basis of the velocity field expansion and control functions.

Our study concerns the use of empirical models derived from spatio-temporal data issued from numerical simulations in which the limitations imposed by the constraints of “real” experimental conditions of laboratory tests are considered.

Next sections are organized as follows: in section 2.1 we describe the procedure used to obtain the velocity fields of the tests under study. In section 2.2 we describe the method to obtain the reduced order model and how we include in this model a control function that enables to describe the alterations of the wake produced by the rotation of the cylinder. The section 3 is dedicated to thediscussion of results. A summary with the more important achievements of our work is proposed at the end of the manuscript.

2. MethodsOur POD ROM is fed with numerical experiments of the flow around a circular cylinder spinning at constant speed.

The Reynolds range under analysis (50<Re<180) corresponds to the laminar vortex shedding regime [18] where flow remains two dimensional and 3D effects can be disregarded [19]. The range of the control parameter of interest is 0<γ<2.

2.1. Numerical techniqueAn incompressible code [20, 21] was used to perform the DNS studies. This code solves the continuity and momentum equations which are discretized using a finite difference method on a uniform Cartesian grid. Spatial derivatives are approximated with a sixth-order, compact-centered, finitedifference scheme, while temporal derivatives are evaluated using a third-order low-storage Runge-Kutta method. The boundary condition on the cylinder surface is imposed using the virtual boundarytechnique.

Mass and momentum conservation principles are represented by the Navier-Stokes equations, which have the following form for an incompressible fluid

0 u (6)

fuuωu

2υp

t(7)

where υ is the kinematic viscosity, p(x,t) the pressure field and ω(x,t) the vorticity field. The externalvolumetric force field f(x,t) is used here to impose the boundary condition on the cylinder surface (Ω0). Figure 2 shows a schematic view of the boundary conditions over the inflow, lateral and outflow regions of the domain.


6

Figure 2. Computational domain and boundary conditions imposed.

2.1.1. Spatio Temporal Discretization. The simulations were performed using 1081×1081 grid points in the x and y directions respectively. The domain size was Lx=30D, Ly=30D, and the centre of the cylinder was located at Xc=15D, Yc=15D. The grid resolution was 36 points per diameter. We have selected these values taking into account the results obtained in previous simulations with this code [20, 21] and others [22].

Numerical simulations were performed until the flow reached a fully developed state, in which all the flow characteristics were analysed.

2.1.2. Simulation Results. A comparison between the results obtained with the code described in this work and those of other research efforts is shown on figures 3 and 4 [4, 19]. In figure 3 we illustrate the effect of varying the rotation parameter γ on the time averaged value of the pressure drag, total drag, pressure lift and total lift coefficients and compare with results obtained by Kang et al. [3]. The figure 4 shows how the code reproduces changes in Strouhal number as a function of Reynoldsnumber and as a function of the rotation parameter. Results are compared with the works of Williamson [18] and Kang et al. [3], respectively.


7

a) b)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

CD

med

med p

CDmed

(DNS,Re=125)

CDmed

(Kang,Re=100)

CDmed

(Kang,Re=160)

CDp (DNS,Re=125)CDp (Kang,Re=100)CDp (Kang,Re=160)

0 0.5 1 1.5 2-6

-5

-4

-3

-2

-1

0

CL m

ed

med p

CLmed

(DNS,Re=125)

CLmed

(Kang,Re=100)

CLmed

(Kang,Re=160)

CLp (DNS,Re=125)CLp (Kang,Re=100)CLp (Kang,Re=160)

Figure 3. Effect of changes in the rotation parameter on the time averaged values of a) pressure drag(CDp) and total drag (CDmed) coefficients and b) pressure lift (CLp) and total lift (CLmed) coefficients.Kang refers to results appearing in [3]. DNS refers to present simulations.

a) b)

60 80 100 120 140 160 1800.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

Re

St

=0

Present SimulationWilliamson

0 0.5 1 1.5 20.15

0.16

0.17

0.18

0.19

0.2

St

Kang , Re=100DNS , Re=125Kang , Re=160DNS , Re=180DNS , Re=200

Figure 4. Strouhal number as a function of Control Parameters: a) Reynolds number; Williamson refers to results appearing in [18]. b) Rotation parameter; Kang refers to results appearing in [3]. DNSrefers to present simulations.

An analysis of these figures reveals that the code satisfactorily reproduces the values of theseparameters as well as their trend when the control parameters are changed.

2.2. POD ROM and Control Function In recent articles [23, 24] the importance of the time averaged flow as the basic state that can accountfor stability properties of wakes under forcing conditions has been emphasized. The dynamics of instabilities in the system is proposed to be described by the free wake Landau equation and anadditional term that modifies the linear growth rate and results proportional to the mean flow correction generated by the forcing. In [11-13] a scheme is proposed where the dynamics of an empirical Galerkin system to describe the flow around a cylinder is determined considering a shift (or zero) mode. This mode accounts for the difference existing between a state considered as base flowand the time averaged flow that takes place when the natural instabilities of the base flow develop.


8

Taking this into account, it seems rather natural, to try to determine a control function considering in the Karhunen Loeve space a correcting vector pointing from the time averaged flow of the unperturbed case (um0(x)) towards the mean of the perturbed state under actuation. The control mode is then associated to the mean flow modifications produced by actuation as stated in (5).

As we consider a steady rotation, the weighing (temporal) coefficient that affects spatial function φc

may be here directly expressed as a function Γ that depends on the rotation parameter but is independent of time

s

iiic tat

1

)()()(),( xxuxxu m0 (8)

The new phase-space is constructed considering an ortho-normalized vector associated to the mean field correction with the following procedure of construction (Gram Schmidt) that assures orthogonality of all modes. The procedure is in some sense similar to the sequential proper orthogonal decomposition (SPOD) proposed by other authors [14, 24, 25]. In our case it may be laid out as follows:

Step 1: For the unperturbed state (γ=0) we calculate the mean field um0(x) and compute the relevant POD eigenmodes that describe the fluctuations of the flow. The ensemble of these modes is a root basis Ψ0 = (φ1 , φ2 , φ3 ,…, φn) that is truncated in n modes under a given criteria (energetic or spectral gap).

Step 2: We compute for a perturbed flow taken as reference (i.e. γ=1) a new basis Ψ1 (first extending basis) obtained as the relevant POD modes of the residual field (ur) of the perturbed velocity field projection onto Ψ0

n

ii

ki

kk tatt1

)()(),(),( xxuxuxu m0r

where the temporal coefficients are determined as

)(),()( xxuxu m0 ikk

i tta

If the first temporal coefficient exhibits a constant value, the first eigenvector of the residualfield is associated to the time averaged correction vector (control function mode) φc(x). Thus the extending basis is composed by an ensemble of the fluctuating modes Ψ1 = (φn+1 , φn+2 , φn+3 ,…, φs) that is truncated in (s-n) modes under a given criteria (energetic or spectral gap).

Step 3: For the other perturbed states γ we calculate projection of the velocity field onto the modes of the truncated basis Ψ0+Ψ1 and φc(x) as

)(),()( xxu ikk

i tta )()( xxum c

The residual velocity field that remains unresolved after this projection is

s

iiic tatt

1

)()()(),(),( xxuxxuxu m0r


9

We evaluate the validity of the model to represent the flow considering some criteria based on the residual velocity field.

By construction, the first mode of the extending basis Ψ1 is associated to an ortho-normalized meanfield correction or displacement vector, from the time averaged flow fields of the unperturbed case to the mean field of the perturbed state. The number of modes s that composes the whole basis depends on the truncation criteria adopted and the range of the control parameter analysed.

The range of the control parameter γ in which the use of this control function will be able to give accurate results can be determined considering the range in which the eigenvectors with significant constant eigenvalues do not appear. If this happens, it means that is not possible to describe the mean flow with the control function correction.

However, it is always possible to compute an additional control function and a new extending basis Ψi. This new basis can be obtained applying again the POD technique over the residue of the perturbed velocity field projection. Thus, an enhanced range can be achieved by increasing the number of control functions. For instance, if two control functions of the parameter control space are included we can formulate

s

iiicc tat

12211 )()()()(),( xxuxxxu m0

The first control function is associated to the first constant eigenvector of the first extending basis while the second one is again associated to the first constant eigenvector of a second extending basis.Both control functions are orthonormal to each other by construction. If necessary, this procedure can be extended to include a larger number of control functions.

It should be noted that as the actuation for the spinning cylinder increases we get closer to the Hopf bifurcation state. In this sense it is expected that new dynamical states for high values of the rotation parameter will be less rich in modes than those for the lower values. As a consequence, the addition of just a limited number of modes will be required to obtain a good description of the flow with the PODROM.

The coefficients affecting the control functions correspond to the value of the time averaged velocity field projection onto the control functions, and one possible way to determine them uses the interpolation between the known states.

The Galerkin projection onto NS equation with two control function leads to the following system

2

1

2

1

2

1 1

2

1

111

m lmllmi

ll

N

jiliji

l

lli

kj

s

kijk

s

jj

s

jiji

i

G

aFEt

D

aaCaBAdt

da

l

(9)

where


10

ml

lll

llll

ll

cciilm

cjijciij

cicicii

cii

kjiijk

jijijiij

iiii

φφφG

φφφφφφF

φφφφφφE

φφD

φφφC

φφφφφφB

φφdxpφA

,,,

,,,

,,

,,,

,

2

2

Re

1

Re

1Re

1,

0m0m

0m0m

0m2

0m0m

uu

uu

uuun

In the independent term Ai, the effect of pressure for wake flows is usually neglected. However, the loss of symmetry as a consequence of actuation may increase the error contributed by this term and careful analyses should be undertaken before deciding its elimination. Eventually, a system based on a spatio-temporal decomposition for pressure and velocities, like the one used for the velocity in equation (1), may be proposed [9].

If the effect of the pressure is not so important for a given Reynolds number, the coefficients are calculated once and for all regardless of the changes in the values of the rotation parameter.

One of the advantages of this approach is that one can have a system of equations representing the flow that easily includes the effect of the actuation.

On the other hand the system is of higher dimension than the one obtained with conventional POD ROM as it includes a larger number of modes than in the more traditional approaches.

3. ResultsAs an example of control by spinning the cylinder in the laminar vortex shedding regime, we analyze the case for a fixed Re=125 and different values of the non dimensional rotation parameter 0<γ<2. The POD modes were obtained with snapshots equispaced in time during about 20 vortex shedding periods.

3.1. Derivation of the Control functionsThe mean velocity field for the non actuated flow is represented in figure 5.

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

Re=125 , =0

Figure 5. Time averaged flow for γ =0 and Re=125.


11

We calculated two extending basis: Ψ1 associated to values of the control parameter γ=1.0 and the other one Ψ2 for γ=1.9, which is quite close to the value where the vortex shedding phenomena disappears. Results of the mean flow corresponding to these values of the rotation parameters are given in figures 6 and 7.

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

Re=125 , =1

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

Re=125 , =1.9

Figure 6. Time averaged flow for γ=1.0 and Re=125.

Figure 7. Time averaged flow for γ=1.9 and Re=125.

The set of modes of the extending basis were determined with the Gram Schmidt construction and resulted in a first mode with an almost constant temporal coefficient, followed by other modes with temporal coefficients exhibiting a periodic behavior.

The first eigenmodes are thus ascribed to the control functions. They enable to obtain the time averaged flow for different perturbed cases (figures 6 & 7) from the time averaged flow of the unperturbed case (figure 5).

The characteristics of the first and second control function modes are provided by figures 8 and 9,respectively.

For cases corresponding to other values of the control parameters the fields were projected onto the mean of the unperturbed state and onto the correction functions. In figure 10 we represent for different values of the control parameters the energy associated to the time averaged component of flow, and the residual value associated to the fraction that can not be captured.


12

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

control function mode I (=1.0)

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

control function mode II (=1.9)

Figure 8. Control function mode φc1(x), Re=125. Figure 9. Control function mode φc2(x), Re=125.

0 1 20

2

4

6

8

10

Mea

n ki

netic

ene

rgy

1st control mode2nd control modeNon captured

Figure 10. Energy of the flow associated to the control functions and residual mean energy not captured, Re=125.

In the proximity of the value of γ where vortex shedding ceases, the vector correction introduced byφc1(x) is not sufficient to define the space of the perturbed time averaged flow, and supplementary corrections are required to generate the appropriate subspace.

The correction function φc2(x) enables to introduce the suitable corrections onto the mean field to extend the model into the range of interest of the rotation parameter.

The energy associated to the mean velocity field of the unperturbed state (not shown in this figure) attains a constant value about one order of magnitude above those represented in the figure.

3.2. Fluctuating modes of the root Basis Ψ0

The cartography of the unperturbed modes associated to the basis Ψ0 (figure 11) resulted similar to the one exhibited in [10].

The modes can be grouped by pairs and the temporal coefficients of these modes present a quasi sinusoidal shape, both modes of the pair exhibiting the same frequency but with a difference in phase.

The first pair of modes has a frequency that is in agreement with the vortex shedding of the flow. For the following pair of modes, the frequency increases monotonically. These frequencies areapproximately a multiple of the fundamental Strouhal frequency.


13

3.3. Fluctuating modes of the extending basisThe eigenvectors Ψ0+Ψ1+Ψ2 determine a unique subspace, that is common for all actuated states in the range of the rotation parameter analysed (0<γ<2.0).

A series of charts of the fluctuating modes of the extending basis is given in figures 12 & 13.

a) b)

0 1 2 3 4 5 6 7 8 9

-3

-2

-1

0

1

2

3

x/D

y/D

mode 1 , root basis

0 1 2 3 4 5 6 7 8 9

-3

-2

-1

0

1

2

3

x/Dy/

D

mode 2 , root basis

c) d)

0 1 2 3 4 5 6 7 8 9

-3

-2

-1

0

1

2

3

x/D

y/D

mode 3 , root basis

0 1 2 3 4 5 6 7 8 9

-3

-2

-1

0

1

2

3

x/D

y/D

mode 4 , root basis

Figure 11. First fluctuating modes of the root basis, Re=125: a) Mode 1, b) Mode 2, c) Mode 3, d) Mode 4.


14

a) b)

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 1 , base I

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 2 , base I

c) d)

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 3 , base I

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 4 , base I

Figure 12. First fluctuating modes of the first extending basis, Re=125: a) Mode 1, b) Mode 2, c) Mode 3, d) Mode 4.

As we can see, the modes of the extending basis enable to break the symmetry of the uncontrolled basis, allowing for the representation of tilted vortex shedding, characteristic of spinning cylinder flows.

The temporal coefficients of the extending basis can also be grouped by pairs and present a quasi sinusoidal shape, both modes of the pair exhibiting the same frequency but with a difference in phase.

In figure 14 we plot, the fluctuating energy of the flow captured by the fluctuating modes of the basis, and the residual value that is associated to the fraction that can not be represented using the eigenvectors of the basis Ψ0+Ψ1+Ψ2.

When a threshold of 95% of fluctuating kinetic energy of the flow is adopted as a criterion to truncate the set of number of modes to be retained, each extending basis adds 4 extra modes to the original basis. With this criterion, the whole basis (uncontrolled + extending basis) is composed of 12modes.


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a) b)

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 1 , base II

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 2 , base II

c) d)

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 3 , base II

0 2 4 6 8

-3

-2

-1

0

1

2

3

x/D

y/D

mode 4 , base II

Figure 13. First fluctuating modes of the second extending basis, Re=125: a) Mode 1, b) Mode 2, c) Mode 3, d) Mode 4.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-2

10-1

100

101

Flu

ctu

ating K

inetic e

nerg

y

mode 1 Root Basismode 2 Root Basismode 3 Root Basismode 4 Root Basismodo 1 1st Ext Basismodo 2 1st Ext Basismodo 3 1st Ext Basismodo 4 1st Ext Basismodo 1 2nd Ext Basismodo 2 2nd Ext Basismodo 3 2nd Ext Basismodo 4 2nd Ext BasisNon Captured

Figure 14. Fraction of the energy of the flow associated to the fluctuating modes, Re=125.


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3.4. Weighing function for the control function The mean flow of any state may be obtained by suitably adjusting the weighing coefficients Γ. In figure 15 we represent the functions Γ1 and Γ2 of equation (10) versus the parameter of rotation γ. Thevalue is obtained considering the mean of each state and the correction introduced by the flow. The parameter Γ1 can be approximated as a linear function of γ

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

I

II

I fit

II fit

Figure 15. Coefficients Γ vs γ, Re=125.

The coefficient Γ2 also exhibits an almost linear dependency with the rotation parameter, in the range where it is defined (1.0<γ<2.0)

iii nm

Fitting coefficients for each range of validity are m1=1.538, n1=0.08787, m2=1.048 and n2=-1.084.It is also clear that the influence of the considered reference state to define the first extending basis

is not as crucial as it results in the range of rotation parameter where vortex shedding occurs; the mean velocity lies in a subspace of dimension 2 or 3.

The curve of figure 15 also enables to infer that the kinetic energy accumulated in these correcting modes (control function modes) depends quadratically on γ.

The fraction of the total energy accumulated on other modes (fluctuating modes) decreases with the increase of the control parameter until it disappears when vortex shedding ceases.

3.5. Fidelity of the model representationAn evaluation of the quality of velocity field reconstruction can in principle be estimated by the

residual energy that is not captured by the truncated model. This kind of measure may hide the ability of the model to represent an arbitrary set of snapshots. It just quantifies point to point the difference between the modulus of the velocity field and the reconstructed one. Important differences of velocity field vectors associated to errors in the velocity vectors directions can not be captured by this measure.

To overcome this difficulty in [24] different alternatives have been suggested, but in our case itseems adequate to consider the Barron’s angular error in the velocity field defined as [26]

n

22

11

1arccos

1

truerec

truerec

uu

uu


17

where urec is the reconstructed field, utrue is the field obtained by DNS simulation and Ω agrees with the number of nodes of the discretization domain.

In figure 16 we show typical pictures of instantaneous angular distance fields for two differentvalues of the control parameter.

a) b)

Figure 16. Instantaneous field of Barron’s angular error for Re=125 and two control parameter values:a) γ=0.5, b) γ=1.5. Errors are in radians.

As one could expect, the largest differences are observed in the region where vortex sheddingoccurs. In the regions where the dynamics is not so strong, it does not appear to be any significant difference.

The Barron´s angular error, however, results quite small and the deviation angle in the worst casereaches about 5º. In figure 17 we show the evolution of this instantaneous measure with time. We can see that this distance always remains quite small and a fluctuation of the same frequency of the vortex shedding phenomena occurs for all cases.

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3x 10-3

Time

dis

tance

Spatial average of the distance

=0

=0.25

=0.5

=0.75

=1

=1.25

=1.5

=1.75

=1.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3x 10

-3

dist

ance

time averaged of the spatial mean of the distance

Figure 17. Temporal evolution of the distance at different values of γ for Re=125. Angles are in radians.

Figure 18. Time averaged distance at different values of γ for Re=125. Angles are in radians.


18

In figure 18 the time averaged value of this measure is presented. We can observe that the model slightly degrades its performance for the values of the control parameter that are far from the reference values considered to define the root and extending bases.

4. ConclusionsIn this article we have analysed several tools regarding the use of empirical models that are bound tosimplify the formulation of optimal control strategies for actuations produced on the boundary of external flows.

We have initially concentrated our analysis on the suitable inclusion of functions in the surrogate model that accounts for the effect on the velocity fields of a steady rotation of the cylinder around its axis. This function was educed with a Grahm Schmidt construction that gives rise to an extended basis in which the velocity field is expanded.

The control functions enable to overcome some limitations associated to the constraints of real experimental conditions data, especially in those cases where good accuracy of the velocity fields near the surface is required.

Another result of this procedure is that for a spinning cylinder at a laminar vortex shedding regime it is possible to satisfactorily reconstruct the mean velocity fields with only three vectors provided the Reynolds number is kept constant.

The obtained POD ROM is flexible enough and coefficients are calculated once and for all at the beginning of the process. This is of special interest for optimization processes where the coefficients of POD ROM have to be updated, particularly when one considers values of the control parameter that are far from the reference value.

Extension to cases where both control parameters, the Reynolds number and rotation, may vary independently or simultaneously are now under study.

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Fluids 11 (11) 3312–21[4] Stojkovic D, Breuer M and Durst F 2003 On the new vortex shedding mode past a rotating

circular cylinder Physics of Fluids 15 (5) 1257–61[5] Protas B 2008 Vortex Dynamics Models in Flow Control Problems Nonlinearity 21 No 9 R203–

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Thesis University of Trier[9] Bergmann M and Cordier L 2008 Optimal Control of the circular cylinder wake by trust region

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dimensional models for the transient and post-transient cylinder wake J. Fluid Mech. 497335

[11] Tadmor G, Gonzalez J, Lehmann O, Noack B, Morzynski M and Stankiewicz W 2007 Shift modes and transient dynamics in low order, design oriented Galerkin Models AIAA paper0111 (45th AIAA Meeting, Reno, 2007)

[12] Morzynski M, Stankiewicz W, Noack B, King R, Thiele F and Tadmor G 2007 Continuous mode interpolation for control-oriented models of fluid flows Active Flow Control ed. RKing (Berlin: Springer Verlag) pp 260–78


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[13] Christensen E, Sorensen J and Brons M 1993 Low dimensional representation of early transition in rotating fluid flow Theoret. Comput. Fluid Dynamics 5 259–67

[14] Burkardt J, Gunzburger M and Lee H C 2006 Centroidal Voronoi Tessellation-Based Reduced-Order Modeling Of Complex Systems SIAM J. Sci. Comput. 28 No 2 459–84

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[16] D’Adamo J 2007 Modelos Reducidos Para el Control de Flujos con Actuadores EHD PhD Thesis University of Buenos Aires

[17] Gronskis A, D'Adamo J, Artana G, Cammilleri A and Silvestrini J 2008 J. Electrost. 66 1–7[18] Williamson C H K 1996 Vortex dynamics in the cylinder wake Annu. Rev. Fluid. Mech. 28

477–539[19] Zdravkovic M 1994 Flow Around Circular Cylinders (Oxford: Oxford University Press vol 2)[20] Lamballais E and Silvestrini J 2002 Direct numerical simulation of interactions between a

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[25] Jorgensen B, Sorensen J and Brons M 2003 Low dimensional modeling of a driven cavity flow with two free parameters Theoret. Comput. Fluid Dynamics 16 293–317

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AcknowledgementsThis work has been supported by ANPCYT grant PICT 38070 and UBACYT grant number I017.


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