Low order modelling of the flow around a confined square cylinder.

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Summary

In the present work, the possibility of reconstructing the flow field around a confined square cylinder starting from a few external measurements and using a low-order model has been investigated. The low-low-order model is obtained by a projection of the Navier-Stokes equations on a base built by using the Proper Orthogonal Decomposition on a snapshot database obtained by previous numerical simulations. Then a calibration procedure is performed, i.e. some of the terms of the POD low-order model are calibrated in order to obtain the best fit of the POD mode coefficients with the exact values, i.e. the projactions of the database snapshots on the POD modes. The equations derived from the measurements, which in our application are shear-stress measurements, are obtained by a Least-Square approach and then they are coupled with the low-order model by using a pseudo-spectral decomposition in time and carrying out an additional Least-Square procedure. This procedure has been called Kalman-like technique. Two analysis have been carried out: one two-dimensional, at Reynolds number 150 and one three-dimensional, at Reynolds number 300; for the latter one two different cases have been considered.

An analysis of the time variation of the measured quantities and of the recon-struction values has been first carried out to set the location of the sensors. A sensitivity analysis has also been performed to investigate the effects of the varia-tion of the sensor number and of the number of retained modes on the Least-Square problem.

The Kalman-like technique has been validated inside the calibration interval for the 2D and 3D cases. Then, chosen the retained POD modes, the Kalman-like proce-dure has been carried out for the two-dimensional and the three-dimensional cases outside the calibration interval. In the three-dimensional case the technique has been applied for two different low-order models, obtained from two POD databases of different size. It has been seen that the captured energy by the POD modes

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depends on this size and consequently also the accuracy of the model. In the three-dimensional case also some velocity sensors have been used, in order to increase the accuracy of the procedure results.

In all the considered 3D cases, the Kalman-like procedure yields a significant improvement of both the accuracy and the stability of the flow dynamics predicted by the low-order model.

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Chapter 1 Introduction

In control and optimization applications in fluid dynamics, in which a large number of flow computations is required, numerical simulations based on the dis-cretization of the Navier-Stokes equations are in most cases unaffordable, due to the huge computational requirements. Thus, low-order models, in which the number of degrees of freedom of the problem is drastically reduced are needed. The Proper Orthogonal Decomposition, introduced in (Lumley [1967]), is often used to replace the actual Navier-Stokes equations for control purpose.

In the present work the possibility of costructing an accurate low-order model of the flow around a confined square cylinder is analyzed. This is a configuration interesting for fluid-dynamics application, indeed bluff-bodies may be used in lami-nar channel flows for enhancing transport and mixing. Moreover, when bluff-bodies with separation points fixed by the geometry are considered, this configuration is also interesting for experimental devices (e.g. vortex flowmeters).

The wake of unconfined square cylinders at low Reynolds number has exstensively been studied. Only a few studies have investigated the flow around a square cylinder symmetrically positioned between parallel walls, (Galletti et al. [2004], Breuer et al. [2000], Li and Humphrey [1995], Mukhopadhyay et al. [1992], Davis et al. [1984]). These studies are all two-dimensional. In (Camarri et al. [2005]) also a

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1 – Introduction

three-dimensional flow around a confined square cylinder has been studied, by using a Navier-Stokes solver, in order to analyze the 3D vortical structures in the wake. The capabilities of a new procedure, aimed at obtaining a low-order model starting from POD, are analized here using the database in (Camarri et al. [2005]), both in 3D and 2D.

Indeed, one application often targeted is the control of vortex shedding (Graham et al. [1998]); the benefits of such technology would range from reduced fatigue on materials and lower moise emission, to flight of thick-wing airship. Even though examples of vortex shedding control built on low-order models and solid optimi-sation grounds recently appeared (Bergmann et al. [2005]), they are limited to two-dimensional flows. Many studies have investigated the possibility of modelling a moderately complex flows with a POD low-dimensional model (Ma & Karniadakis [2002]), (Galletti et al. [2004]),(Galletti et al. [2005]); but a few studies have per-formed an analysis of the potentiality of the low-order modelling for more complex three-dimensional flows (Buffoni et al. [2006]). Moreover, also if a calibration also if based on a pseudo-spectral method introduced in (Galletti et al. [2005]) is used, it has seen that outside the calibration interval, non-periodic phenomena being present in the flow, the low-order model does not give an accurate prediction of the flow dynamics.

In this work, a procedure of matching between a POD low-order model and some external datas is proposed for the 2D and 3D configurations. The interest of this procedure is twofold. First, it can be useful to correct the behaviour of the low-order model outside the calibration interval for flows more complex than 2D periodic ones. Indeed, as said previously, for such flows the POD model alone, outside the calibration interval, gives a dynamics poorly accurate or even unstable. Second, the proposed procedure is interesting in control applications, also for “simple” flows, since the POD model must be applied to configurations slightly different from that from which it was derived.

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1 – Introduction

In Chapter 2, the POD and the calibration process, used to obtain the low-order model, are described. Then, in Chapter 3, the technique used to match the information given by some external measurements and the equations of the low-order model, called Kalman-like technique, is shown. In particular the equations given by some shear-stress sensors and some velocity sensors are used to build a system of equations through a Least-Square approach. Then, this system is coupled with the low-order model, using a pseudo-spectral method.

After having briefly described the domain configurations for the two-dimensional and the three-dimensional cases and the Navier-Stokes solver used to obtain the flow snapshots, the database, on which the POD-Galerkin model is based, is presented in Chapter 4.

An analysis of the sensitivity to the number and location of measurements is per-formed in Chap.5 and the sensor configurations for the different cases is chosen. Finally, the Kalman-like technique is applied in Chap.6. The three-dimensional ap-plication is divided in two cases, characterized by different POD databases, and thus by different quantity of energy reconstructed with the same number of modes. For the model obtained from the larger database also some velocity sensors are used. Finally, the results are analyzed and compared with the prediction of the low-order model.

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Chapter 2 Reduced order model

In this chapter the Proper Orthogonal Decomposition is described. In particular the procedure to build a POD base is described. Thus, in order to obtain the low-order model, the procedure of projection of Navier-Stokes equation on the POD base is explained. Finally, the procedure of calibration of the POD-Galerkin model is reported.

2.1 Description of the POD-GALERKIN model

2.1.1 Proper Orthogonal Decomposition

Let U(1)_,U(2)_{, . . . ,U}(N ) _{, a set of data obtained, for instance, in a numerical}

simulation. Each term U(k) _{represents a snapshot of the whole velocity field at the}

k− th instant, that is:

U(k)=              u(x1,tk) , v(x1,tk) , w(x1,tk) u(x2,tk) , v(x2,tk) , w(x2,tk) ... , ... , ... u(xM,tk) , v(xM,tk) , w(xM,tk)             

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where u, v and w are the velocity components in the x, y and z direction respectively. The term M indicates the number of grid points and it can be very high, for example, in the three-dimensional case studied in this research, is more than 6 millions. We can define the space L as follows :

L = SP ANU(1),U(2), . . . ,U(N )

The purpose of the POD is to find a subspace of L that represents the best approx-imation of the same L. First, we have to define a unit norm vector φ which has the

same structure of the U(k) _{vectors :}

φ =              φ1(x1) , φ2(x1) , φ3(x1) φ1(x2) , φ2(x2) , φ3(x2) ... , ... , ... φ1(xM) , φ2(xM) , φ3(xM)              ,_{k φ k= (φ,φ) = 1}

This vector φ has to be determined so that the projection of the space L on it is as largest as possible; to this goal, we have to maximize this functional :

J =

N

X

k=1

U(k),φ2

The following symbols will be used :

u(xj,tk)≡ u1jk v(xj,tk)≡ u2jk w(xj,tk)≡ u3jk φh(xj)≡ φhj              , j = 1, . . . ,M ; k = 1, . . . ,N ; h = 1,2,3

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together to the Einstein notation. Thus the problem of maximization of the func-tional J can be rewritten as follows :

“ Find the vectors φ that maximize the functional:

J1 = φhjuhjkulikφli

under the constraint:

φh_jφh_j = 1 .”

A classical method of solution of this problem is that based on the Lagrange multi-pliers. This method can be used here because the only constraint of the problem is independent of the functional. In this way, the initial problem is replaced by another

one, in which we have to calculate the extrema of this new functional J2 without

constraints :

J2 = φhjuhjkuikl φli− λ(φhjφhj − 1).

In this functional a new variable λ appears, which is the Lagrange multiplier; there is only one multiplier because the initial problem has only one constraint.

A difficulty is encountered at this point: indeed to compute the functional J2,

we have to work with the eigenvalues and the eigenvectors of a very large matrix,

that is uh

jkulik, of dimensions [M·M]. If, as in the present work, the grid used for the

numerical simulation is made by a huge number of points, the computation becomes very heavy.

To avoid this problem we proceed in accord with the method of snapshots pro-posed by Sirovich (Sirovich L. [1987]) , whose central idea is to express the vectors

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φ =

N

X

n=1

bnU(n) (2.1)

By the substitution of equation 2.1 in the functional J2 we obtain :

φh_j = uh_jkbk ⇒ J2(λ,b1, . . . ,bN) = bk uhjkuhjr ul_irul_isbs− λ bkuhjkuhjrbr− 1 .

To find the extrema of this functional we have to impose that Frechet derivatives with respect to all the unknowns vanish. Considering the derivatives with respect

to the coefficients bk, we have :

∂J2 ∂bk ≡ limα→0 J2 λ,b1, . . . ,bk+ αebk, . . . ,bN − J2(λ,b1, . . . ,bN) α After some manipulations, we have :

∂J2 ∂bk = ebk uh_jkuh_jrul_irul_imbm+ bnuhjnuhjrulirulik − λuh_jkuh_jmbm+ bnuhjnuhjk = = ebk uh jkuhjrulirulim2bm− 2λuhjkuhjmbm Thus, we obtain : ∂J2 ∂bk = 0 _{⇐⇒ u}h jkuhjrbm = λ bm

And, by imposing that the derivative of the functional with respect to the Lagrange multiplier λ vanishes, we have :

∂J2

∂λ = bku

h

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∂J2

∂λ = 0 ⇐⇒ bku

h

jkuhjrbr = 1

This second equation is obviously equal to that we obtain imposing that the vector φ has an unitary norm.

Then, in conclusion, we can say that, to find the extrema of the functional J2, we

have to solve this eigenvalue problem :

Rb= λb (2.2)

where :

R = uh_jkuh_jr ≡ TIME CORRELATION MATRIX

[R] = N· N    N = number of snapshots N M b = [b1,b2, . . . ,bN]T

We can note that the Sirovich method gives a matrix R of dimension [N · N]

in-stead of [M · M], leading thus to a significant reduction of the dimension of the

problem. Moreover R is symmetric and positivly definite; thus, it is characterized

by a complete set of orthonormal eigenvectors {f1, . . . ,fN} and by a set of positive

eigenvalues{λ1, . . . ,λN}. By imposing that : bs= fs √ λs , s = 1, . . . ,N

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derivative of J2 with respect to λ vanishes. Then these eigenvectors maximize the

functional J2. Furthermore, the eigenfunctions{φ1, . . . ,φN}, wich following equation

2.1 can be written as :

φh_jk = uh_jnbnk , j = 1, . . . ,M ; k = 1, . . . ,N ; n = 1, . . . ,N ; h = 1,2,3.

are the POD modes.

Finally it is possible to check that the POD modes are indeed orthonormal :

(φk,φs) = φhjkφhjs= brkujrh uhjqbqs = brkλsbrs = λs frk √ λk frs √ λs = δks r λs λk = δks

The istantaneous velocity field can now be approximated by a linear combination of

Nr POD modes, as following :

u (x,t)_≈

Nr

X

n=1

an(t) φn(x) (2.3)

Let us go back to the original goal of this procedure: to represent a space L of large dimension by a subspace of reduced dimension. To this aim, the less energetic POD modes, that is those corresponding to the smaller eigenvalues of the time correlation matrix, are neglected. In practice we estimate the energy contribution to the first

Nr modes as following :

Nr

X

i=1

λi

Then, if, for instance, we want to capture the 99% of the entire energy of the velocity

field, Nr is given by the relation :

PNr

i=1λi

PN

i=1λi

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It has been observed in the literature that, to capture a fixed energy level, Nr has

to increase as the Re increases. This agrees with the fact that, as the Re increases, the amount of the energy present in the smaller scales also increases and, thus, more POD modes are needed.

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2.1.2 POD-Galerkin model

   ∇ · U = 0 ∂U ∂t + (U · ∇)U = −∇P + 1 Re4U (2.4)

Considering the Navier-Stokes equations 2.4, and substituting to the velocity vector

U (x,t) = _{{u(x,t), v(x,t), w(x,t)} the linear combination 2.3, since the POD modes}

are divergence free by construction, we can write, by using again the Einstein nota-tion :    ∂ ∂tar(t)φr+ (ai(t)φi· ∇) aj(t)φj =−∇p + 1 Re4 (ai(t)φi) u (x,0) = ar(0)φr

Then, by projecting these equations on the POD modes through the Galerkin projection we obtain :    ( ˙al(t)φl,φr) + ((φi· ∇) φj,φr) ai(t)aj(t) =− (∇p,φr) + _Re1 (4φi,φr) ai(t) (u (x,0) ,φr) = al(0) (φl,φr) (2.5)

Since the POD modes are orthonormal vectors, these equations can be semplified as following :

  

˙ar(t) + Bijrai(t)aj(t) =− (∇p,φr) + _Re1 Dirai(t) , r = 1, . . . ,Nr

ar(0) = (u (x,0) ,φr) , r = 1, . . . ,Nr

where :

Bijr = (φi· ∇φj,φr)

Dir = (4φi,φr)

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The pressure term− (∇p,φr) must be treated separately; since the POD modes are

divergence-free, we can write :

− (∇p,φr) =− Z Ω ∇p · φrdΩ =− Z Ω ∇ · (pφr) dΩ≡ − Z ∂Ω pφr· ndσ

The pressure term is transformed in a surface integral; to compute it, it is sufficient to know the values of pressure and velocity on the boundary of the domain. In the case of this research, the domain is represented by a square cylinder confined by two parallel walls; then, on the surfaces of the cylinder and on the walls of the conduct

the pressure term is zero, the φr being identically zero; moreover in the

three-dimensional case also the boundary surfaces with z = constant are present, however the integral pressure terms becomes zero, having periodic boundary conditions on these surfaces; thus :

− (∇p,φr) = Z Si pφr· idΓi− Z So pφr· idΓo where :

Si = conduct entry section

So = conduct exit section

i = unit versor oriented as the axis of the conduct.

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2.1.3 POD decomposition for the fluctuating velocity field

We can define an average velocity field u(x) and to decompose a generic snapshot u(x,t) as :

u(x,t) = u(x,t) +_eu(x,t)

with _{eu the fluctuating part of the flow.}

Considering the M snapshots the average field is :

u(x) = 1 M M X k=1 uk(x)

Defined akr = ar(t), the projection of the average velocity field on one of the M

POD modes gives :

(u,φi) = 1 M M X k=1 M X r=1 akr(φr,φi) = 1 M M X k=1 aki

And using the classical POD decomposition :

u = M X r=1 crφr Thus we have : cr = 1 M M X k=1 akr = < ar >

Thus, the POD coefficients cr of the average field are the average of the coefficents

of the snapshots.

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of fields, which are the fluctuating part of the flow; and they are not linearly-indipendent, since their sum is equal to zero by construction. Thus, if we carry out

the POD with this new set, the new time correlation matrix eR has ker = M − 1,

and therefore an eigenvalue of the problem is zero. Thus we have :

euk =

M_X−1

r=1

eakrφer

Where eakr and eφr are obtained by solving the eigenvalue problem 2.2 with the

new matrix eR. From now we denote eu(x,t) as u(x,t), to semplify the notation,

remembering that this is the fluctuating velocity field and not the complete one. The application of the Sirovich method and the development of the POD-Galerkin model can be carried out for the set of fluctuating fields, all the assumptions being valid. This leads to the same equations obtained in the Sec.2.1.2, namely 2.5.

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2.2 Description of the calibration process

We describe this procedure for the previously obtained Pod-Galerkin model, see section 2.1.2. To simplify the treatment, we model, as proposed in (Galletti et al. [2005]), the pressure term in this way :

− (∇p,φr) + C

0

r = Cr0 ,

and the viscous term as:

C_ir1 =₋Dir

Re + C

1 ir

where C0 and C1_ir are added in order to model the interaction of the unresolved

modes with the resolved ones. Then the model becomes:

   ˙ar(t) = Cr0+ Ckr1 ak(t)− Bksrak(t)as(t) = fr(a1, . . . ,aN,Cr0,Ckr1 ) ar(0) = (u (x,0) ,φr) (2.6)

Exploiting the orthogonality of the POD modes, we compute the inner product

between the i− th snapshot of the velocity field at time ti and the r− th POD

mode, obtaining the “exact” value for the coefficient ar(t) at the time ti, that is:

aex_r (ti) = (u (x,ti) ,φr)

This can be done for each of the N snapshots of the velocity field, obtaining a

discrete set of N values for each coefficient ar(t) given by :

(t1,aexr (t1)) , . . . , (tN,aexr (tN))

To achieve a continuous set of values, we interpolate each of the discret set of

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Now, we can start the calibration procedure: it consists in determining the

matrices C1

ir and Cr0 in order to have that the coefficients ar(t), obtained by solving

the system (2.6), are as close as possible to the corresponding reference onesbar(t).

To this purpose we have to minimize the following functional:

J =R₀T PNr

r=1(ar(t)− bar(t))2dt, t1 = 0, tN = T

Obviously, the constraint is represented by the system 2.6. Using the technique of the Lagrange multipliers, this problem can be replaced by another one, which consists in computing the extrema of this functional without constraints:

J2 = Z T 0 Nr X r=1 (ar(t)− bar(t))2dt + Z T 0 λk ˙ak(t)− Ck0− Csk1 as(t) + Bspkas(t)ap(t) dt where: λk = Lagrange multipliers , k = 1, . . . ,Nr

Therefore, we have to impose at the Frechet derivatives of J2(ar(t),br(t),Cr0,Cir1)

with respect to all the variables vanish. After some manipulations, we obtain the following optimality problems :

   ˙ak(t)− Ck0+ Csk1 as(t)− Bspkas(t)ap(t) ar(0) = (u(x,0)φr) Direct problem (2.7)    −˙br(t) = [Crk− (Blrk+ Brlk)al(t)] bk(t)− 2[ar(t)− ˆar(t)] br(T ) = 0 Adjoint problem (2.8)    RT 0 br(t)dt = 0 RT 0 ak(t)br(t)dt = 0 Optimality conditions (2.9)

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Then, we can say that the calibration procedure consists in solving the subproblems 2.7 , 2.8 and 2.9. Note that all the subscripts that are present in these equations go

from 1 to Nr.

To solve this problem, we discretize it by a pseudo-spectral method, which allocates

Nt discretization points along the time axis in accordance with the Gauss-Lobatto

distribution ti = T /2(1− ξi) with ξi = cosπ(i− 1)(Nt− 1) and i = 1, . . . ,Nt. The

functions a (t) , b (t) and ˆa (t) are sampled at the collocation points, i.e. air = ar(ti)

, bir = br(ti) and ˆair = ˆar(ti). An interpolation is performed to retrieve the values

of the functions away from the nodal points ti, and more precisely :

ar(t)≈ Nt X i=1 ψi(ξ) air br(t)≈ Nt X i=1 ψi(ξ) bir ˆ ar(t)≈ Nt X i=1 ψi(ξ) ˆair

with ξ = 1_{− 2t/T and ψ}i(ξ) are the Lagrangian interpolating polynomials based on

the nodes ξi.

The time derivative of the first two interpolated functions at the nodal values are then : ˙ar(ˆti)≈ − 2 T Nt X j=1 dψj(ξ) dξ ξi ajr= Nt X j=1 Dijajr (2.10) ˙br(ˆti)≈ − 2 T Nt X j=1 dψj(ξ) dξ ξi bjr= Nt X j=1 Dijbjr

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to : Dij =− 2 T                                ci cj (−1)j+1 ξi−ξj j 6= i −1 2 ξi 1−ξ2 i j = i6= 1,Nt 2(Nt−1)2+1 6 j = i = 1 −2(Nt−1)2+1 6 j = i = Nt (2.11) with c1 = cNt = 2 and c2 =· · · = cNt−1 = 1 .

The optimality conditions can be rewritten in terms of the interpolated functions as follows : Z T 0 ak(t)bi(t)dt≈ Nt X i=1 Nt X j=1 aikIijbjr Z T 0 bi(t)dt ≈ Nt X i=1 Nt X j=1 Iijbjr

where the integrals :

Iij =

Z T

0

ψi(ξ)ψj(ξ)dξ with i,j = 1, . . . ,Nt

are calculated by means of Legendre quadrature. Finally, by virtue of 2.10 and 2.11, we can discretize all the equations of the problems and we obtain the following system :

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a1r = ar(0) r = 1, . . . ,Nr Dijajr− Cr0− Clr1ail+ Blsrailais = 0 i = 2, . . . ,Nt r = 1, . . . ,Nr Dijbjr+ Crs1 bis− (Blrs+ Brls)ailbis− 2(air − ˆair) = 0 i = 1, . . . ,Nt− 1 r = 1, . . . ,Nr bNtr= 0 r = 1, . . . ,Nr 1iIijbjr = 0 r = 1, . . . ,Nr aikIijbjr = 0 k = 1, . . . ,Nr r = 1, . . . ,Nr

where 1i is an array of ones of dimension Nt. This system is of 2NtNr + Nr+ Nr2

algebraic equations in the 2NtNr+ Nr+ Nr2 unknowns.

The problem obtained is solved by the Newton method: being C0

i and Ci1 the

ar-bitrarily chosen initial matrices, we compute the vectors a (t) and b (t) by solving the direct and the adjoint problems. Then, we substitute these in the optimality conditions: if the equality is satisfied within a given tolerance, then we stop the

pro-cedure, else we modify the previously used C0 _{and C}1 _{matrices with a term which}

is function of the Jacobian of the system f (Jac); in this way, we obtain new vectors

a (t) and b (t) from the direct and adjoint problems, and so on until the convergence is reached.

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Chapter 3 Kalman-like technique

In this section the basic idea of the proposal tchnique is described. In particular, the issue is how a velocity field can be reconstruct starting from some external measurements used to impose a further constraint to the POD model. Indeed the POD coefficients are computed from a system of algebraic equations obtained by coupling the low-order model equations with a set of equations achieved by the measurements.

3.1 Description of the Kalman-like method

3.1.1 Least-Squares approach

In order to develop a Kalman-like method we have to define how the measure-ments can be introduced in the numerical method. We assume to have some

shear-stress sensors, considered for simplicity unidirectional sensors τxy, and some hot-wire

velocity sensors, considered unidirectional too, measuring the longitudinal velocity component u.

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for the shear-stress measurements, because it is obvious that they have many ad-vantages with respect to the hot-wire sensors. The main one is that they do not introduce any perturbation in the flow unlike the velocity sensors. Moreover, piezo-metric shear-stress sensors, for instance, well are generally suited for industrial or aeronautical applications.

Figure 3.1. Shear-stress sensor.

The experimentally measured value at a particular time tk and at a given point,

defined by the position arrays xτ

z and xuz, has the following expression:

τ? e(xτz,tk) = µ∂u(x τ z,tk) ∂y ue(xuz,tk) = u(xuz,tk)

for shear-stress and velocity respectively.

Since in our application the viscosity µ is constant, let us define:

τe(xτz,tk) =

τ?

e(xτz,tk)

µ

wich will be used instead of τ?

e in the following.

Note that in general, the location at wich shear-stress measurements are carried out

(xτ

z) is different from that of the velovity measurements (xuz).

From the Snapshot method of Sirovich defined in the previous chapters, we know

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Φn(x) through the coefficients ar(tk), wich represent the projection of the velocity

field at time tk on the r− th mode :

u(xz,tk) = u(xz) +

Nr

X

r=1

ar(tk)Φxr(xz)

where u(xz) is the time averaged velocity at point xz.

By imposing the equality between the measured quantities and the reconstructed ones, we obtain: τe(xτz,tk) = τxy(xτz) + PNr r=1ar(tk) ∂Φx r(xτz) ∂y + τ N r ue(xuz,tk) = u(xuz) + PNr r=1ar(tk)Φxr(xuz) + uN r (3.1) where τ

N r and uN r are the errors in the reconstruction with Nr modes of

shear-stress and velocity respectively, and where τxy(xτz) is the average shear-stress value,

obviously defined as :

τxy(xτz) =

∂u(xτ

z)

∂y

Therefore, by using the information obtained from Nτ

z and Nzu measurements, the

reconstruction errors τ

N r and uN r can be minimized through a Least-Square Method.

In particular we define a functional J, for each given measurement time tk, wich is

sum of the residuals:

J(tk) = 1 2 Nτ z X z=1 τe(xτz,tk)− ∂u(xτ z) ∂y − Nr X r=1 ar(tk) ∂Φx r(xτz) ∂y !2 + +1 2 Nu z X z=1 ue(xuz,tk)− u(xuz)− Nr X r=1 ar(tk)Φxr(xuz) !2 where_{xτ

z, z = 1, . . . ,Nzτ} and {xuz, z = 1, . . . ,Nzu} are the points at wich

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Thus, the minimum of the functional J(tk) : min ar(tk)J(tk) = 1 2 Nτ z X z=1 τe(xτz,tk)− ∂u(xτ z) ∂y − Nr X r=1 ar(tk) ∂Φx r(xτz) ∂y !2 + +1 2 Nu z X z=1 ue(xuz,tk)− u(xuz)− Nr X r=1 ar(tk)Φxr(xuz) !2 wich implies ∂J ∂ar = 0 for r = 1..Nr.

Thus, Nr equations are obtained in the Nr unknowns ar for each measurement time

tk as follows : a1 Nτ z X z=1 ∂Φx 1(xτz) ∂y ∂Φx r(xτz) ∂y + a2 Nτ z X z=1 ∂Φx 2(xτz) ∂y ∂Φx r(xτz) ∂y + . . . + + an Nτ z X z=1 ∂Φx r(xτz) ∂y 2 + . . . + aNr Nτ z X z=1 ∂Φx Nr(x τ z) ∂y ∂Φx r(xτz) ∂y + a1 Nu z X z=1 Φ1(xuz)Φxr(xuz)+ + a2 Nu z X z=1 Φ2(xuz)Φxr(xuz) + . . . + an Nu z X z=1 Φx_r(xu_z)2+ . . . + aNr Nu z X z=1 ΦNr(x u z)Φxr(xuz) = = Nτ z X z=1 (τe(xτz,tk)− τ(xτz)) ∂Φx r(xτz) ∂y + Nu z X z=1 (ue(xuz,tk)− u(xuz)) Φxr(xuz) with r = 1, . . . ,Nr. (3.2)

The solution of the previous system of equations gives the Nr POD coefficients, for

a chosen set of measurements at the time tk. A more compact form of this system

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A an = b (3.3) where Aij =hPN τ z z=1 ∂Φx i(xτz) ∂y ∂Φx j(xτz) ∂y + PNu z z=1Φxi(xuz)Φxj(xuz) i i,j = 1, . . . ,Nr; bi =hPN τ z z=1(τe(xτz,tk)− τ(xτz)) ∂Φx i(xτz) ∂y + PNu z z=1(ue(xuz,tk)− u(xuz)) Φxi(xuz) i i = 1, . . . ,Nr;

Note that the time dependence is only in the array b, thus the matrix A of the

Least-Square Problem is the same for each measurement instant tk if the position

of the sensors does not change.

Furthermore, the matrix A and the array b can be obtained from the system of

Nτ

z · Nzu equation in the Nr unknowns 3.2 through the following matrix product:

MT_{M a} r = MTm Mij = ∂Φx j(xτi) ∂y i≤ N τ z Mij = Φxj(xui) Nzτ < i≤ Nzu mi = (τe(xτi,tk)− τ(xi)) i≤ Nzτ mi = (ue(xui,tk)− u(xui)) Nzτ < i≤ Nzu

Therefore A = MT_{M and b = M}T_{m, and A is a symmetric invertible matrix of}

dimension Nr · Nr and b is an array of dimension Nr. Thus the Nr reconstruction

coefficients at time tk can be obtained from the measurements through the

Least-Square approach as follows:

a0r(tk) = A−1b(tk)

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on the condition number of the matrix A k(A). This number indeed is an index of the stability of the linear system; if the condition number is high the problem is ill-conditioned and a small perturbation of the terms of A or b causes a significant

change in the solution a0

r. The condition number of a matrix A is defined as the

product between the norm of the matrix and the norm of the inverse A−1 _:

k(A) =kA−1kkAk

where k · k denotes a suitable norm. Considering a perturbation of the right-hand

side, as for instance calibration sensor error, we now consider the equation:

A(a0_r+ δa0_r) = b + δb

from wich:

δa0_r = A−1δb

by the usual properties of norms, it follows that :

kδa0

rk ≤ kA−1kkδbk.

Furthermore, we have that:

kbk = kAa0 rk ≤ kAkka0rk and, thus: _a10 r ≤ kAk 1_b

Finally, multiplying the equations (3.2) and (3.3) we get: δa 0 r a0 r ≤ kA−1kkAk δb_b and using the definition of the condition number:

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δa 0 r a0 r ≤ k(A) δb_b . (3.4)

The equation 3.4 shows that the condition number is indeed an index of the sen-sitivity of the solution to data perturbations and, in particular, that, for a given perturbation at the data, the error in the solution has an upper limit that depends on the condition number. Thus, ill-conditioned problems with large condition

num-bers may have significant solution errors, δa0

r, also for little data perturbations, δb.

Furthemore, the matrix A is a real symmetric matrix and it is:

AT = A

and thus considering the spectral norm _kAk2 defined as

kAk2 = (maximum of eigenvalues of ATA)1/2

we get that :

kAk2 =|λmax|

where λmax is the largest eigenvalue of the matrix A.

In addition, note that if λ is an eigenvalue of A, λ−1 is an eigenvalue of A−1, thus,

since: kA−1_k 2 = (maximum of eigenvalues of A−1)1/2 we get: kA−1k2 = 1 |λmin|

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From the previuos equations we obtain that:

k(A) =kA−1kkAk = |λmax|

|λmin|

and so we have a direct connection of the condition number with the largest and smallest eigenvalues of the matrix A:

k(A) = |λmax|

|λmin|

The larger it is the difference between these eigenvalues the larger it is the con-dition number and thus the perturbations on the solutions may be significant. In the present case, the matrix A depends only on the derivatives of the x

compo-nent of POD modes δΦ

x r

δy and, using also hot-wire unidirectional sensors, on the x

components of modes, Φx_{. Thus, the stability of the least square approach depends}

only on the characteristics of the Nr modes at the measurements points and, thus,

it is indipendent of time. It is important to note that with this procedure we are minimizing the sum of residuals between the measured values and the reconstructed ones, thus, if we have only shear-stress sensors, or if the number of shear-stress sen-sors is much larger than that of the hot-wire sensen-sors, we are minimizing a functional

J which is mainly the sum of the residuals derived from the τxy values. Therefore,

also if we have a small reconstruction error on the τxy values, it is possible, as we

will see in the following, that the error on the coefficients a0

r, evaluated with respect

to the projection of the actual velocity field on the Nr POD modes, see chapter two,

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3.1.2 Description of Kalman-like technique

If the previously described least-square approach is applied for Nk time instants,

the following Nk systems of equations are obtained :

a0

r(ti) = A−1b(ti) i = 1, . . . ,Nk, r = 1, . . . ,Nr

The idea of the Kalman-like technique is to couple this set of equations with the POD low-order model to impose an additional constraint to be satisfied, this could also stabilize the dynamics of the low-order model which, as it will be shown after-ward, tends to be unstable.

The low-order model equations obtained in chapters two, using the Einstein notaion, can be written as :

˙ar(t) = Cr0+ Ckr1 ak(t)− Bksrak(t)as(t) r = 1, . . . ,Nr (3.5)

The problem 3.5 is a non-linear system of Nr differential equations in the Nr

un-knowns ar(t), where, of course, Nr is a suitable number of modes. To solve this

system a discretization in time is carried out using a pseudo-spectral collocation

method. Chosen a time interval t = t1, . . . ,t1+ T , a Gauss-Lobatto distribution at

collocation points is considered and, for a fixed number of points Nt, this leads to

ˆ

tk = T /2 (1− ξk)

with

ξk = cos π(k− 1)/(Nt− 1) k = 1, . . . ,Nt

Therefore the coefficients ar(t) are computed at the Nt collocation points and the

following notation is introduced :

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Then a Lagrangian interpolation is used and the functions ar(t) are expressed as

linear combination of some polynomials, as follows :

ar(t)≈

Nt

X

k=1

Ψk(ξ) akr

where ξ is a continous function of time defined in analogy with the dicrete ξk,

ξ = (1− 2t/T ), and Ψk(ξ) are the Lagrangian interpolating polynomials at the

nodes ξk having the following expression :

Ψk(ξ) =

j=NYt

j=1j6=k

ξ_{− ξ}k

ξk− ξj

Note that Ψk(ξ) = δjk, being δjk the Kronecker δ.

Furtermore, the time derivative of the interpolating functions at the collocation nodes are given by :

˙ar(ˆti)≈ − 2 T Nt X k=1 dΨk(ξ) dξ ξi akr

in which the terms of the differentation matrix−2

T

PNt

k=1 dΨk(ξ)

dξ |ξi = Dik are known

(Canuto et al. [1988]), as previously shown in chapter two (2.11). Finally, the

low-order model system is discretized in time and becomes a non-linear system of Nr·Nt

coupled equations in the Nr· Nt unknowns akr :

Dkjajr− Cr0− Clr1akl+ Blsrailais = 0 k = 1, . . . ,Nt, r = 1, . . . ,Nr (3.6)

To couple this system with the equations derived by the least-square approach,

not-ing that the τxy(xτz,tk) and u(xuz,tk) are evaluated at instants tk generally different

from the ˆtk chosen as collocation points, it is necessary to perform an other

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U

t

t1 t2 t1 + T

Measurements Collocation Nodes

Figure 3.2. Estrapolating nodes

To this aim a spline interpolation, using cubic polynomials, is developed and the Nt

values of τxy and u respectively extrapolated, as shown in the example in Fig.3.2,

to obtain :

ˆ

τe(xτz,ˆtk) k = 1, . . . ,Nt, z = 1, . . . ,Nzτ

ˆ

ue(xuz,ˆtk) k = 1, . . . ,Nt, z = 1, . . . ,Nzu

Now, the equations for the Least-Square approach can be sampled on the collocation

times ˆtk, as follows :

ˆa0_r(ˆtk) = A−1ˆb(ˆtk) (3.7)

with r = 1, . . . ,Nr, k = 1, . . . ,Nt, and where the matrix A is the same as previously,

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and ˆue(xuz,ˆtk) is different from the array b evaluated at the measurements instants tk.

Therefore, the two systems of equations, 3.6 and 3.7 (the unknowns akr and ˆa0r(ˆtk)

are the same), must be both satisfied. Note that the first one that is a non-linear system.

In order to solve at once these two systems, a linearization of the low-order model

is necessary. To this aim, we assume to known an initial solution aI _{around wich}

the non-linear term of the system 3.6 is linearized. This leads to

Dkjajr− Cr0− Clr1 akl+ (BlsraIil) ais = 0 k = 1, . . . ,Nt, r = 1, . . . ,Nr

Then, we obtain a linear system of 2(NrNt) algebraic equations in the NrNt

un-knowns which can be written in matrical form as follows :

FI_{a = B} _(3.8)

where the unknown array a has dimension [NrNt], B is an array [2(NrNt)] and FI

is a rectangular matrix of [2(NrNt)· (NrNt)], given by :

FI =           D− C1_{+ (Ba}I r) A 0 . . . 0 0 A . . . 0 . . . 0 0 . . . A          

with the matrices D,C1_,(BaI_{) which are known for the POD approach (see Chap.2)}

and A obtained by the Least-Square approach.

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The terms of the right hand side array of system 3.8 B, have the following expression : B =        C0 ˆb . . . ˆb       

The number of equations in the system 3.8 is larger than the number of unknowns, thus, it can be solved by a Least-square method. To do it, the terms of the right

and left hand sides of 3.8 are premultiplied by the matrix FT

I :

F_IT FIa = FIT B

Finally, if we introduce the following notations, KI _{= F}T

I FI and BKI = FITB, we

obtain the Kalman-like linear system

KIa = B_KI (3.9)

where KI _{is [N}

rNt· NrNt], BKI is [NrNt] and a is the unknown vector. Note that in

this system also the right hand side depends on the initial solution used to linearize

the low-order model. Solving the system 3.9 the coefficients at the Nt collocation

points ti are obtained.

a = KI −1BKI

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make a new linearization of the dynamic model. Thus, an iterative procedure can be performed to find a converged solution of the Kalman-like system, indipendent of that used to linearize the low-order model.

The condition of convergence is defined as following : defined the errors in norm on

the coefficients ar(tk) at the iteration i-th in respect to the projection a∗r(tk) of the

Nk velocity fields on the POD modes (see chapter two) as following :

e(ar)i = PNk k=1(ar(tk)i−a∗r(tk)) 2 PNk k=1a∗r(tk)2 r = 1, . . . ,Nr.

where the projection a∗r(tk) can be calculated only if the Nk snapshots are

previ-ously obtained and ar(tk) are the values of coefficients on the snapshot instants

extrapoled after an interpolation, being the solution of system 3.9 computed at collocation times; we can define the relative difference between the errors of two susequent iterations as following :

∆e(ar)i = |e(ar)

i_−e(a r)i−1|

e(ar)i r = 1, . . . ,Nr.

where e(ar)i is the error on the r-th coefficient at the i-th iteration.

The convergence condition chosen is :

∆e(ar)i ≤ 10−4 r = 1, . . . ,Nr.

thus the iterative procedure reaches the convergence only when the errors on all the

Nr coefficients in the time interval fixed T are constants with a relative tolerance of

10−4_.

Finally the solution of Kalman-like system aK _{is achieved (see Fig.3.3).}

This procedure can be performed in any time interval and in particular it can be used to integrate, with the constraint of the external measurements, the low-order model in an interval different from the calibration one.

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Chapter 4 Application to the flow around a

confined cylinder

In this chapter the application of the POD-Galerkin model to the flow around a square cylinder confined between two parallel walls is described. In particular the Navier-Stokes solver ( AERO ), used to obtain the POD database, the domain configuration and simulationparameters are described.

4.1 Description of the code used to generate the

database

To obtain the snapshots, used as database for the construction of the POD-Galerkin model, the code AERO is used, wich is a 3D solver of the Navier-Stokes equations for compressible fluid based on a mixer finite-volume/finite-element dis-cretization in space applicable to unstructured grids (Farhat et al. [1999]). The adopted scheme is vertex centered, that is all the degrees of freedom are located at the vertices. P1 Galerkin finite elements are used to discretize the diffusive terms.

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through the rule of medians. The convective fluxes are discretized on this tessella-tion, that is in terms of fluxes through the common boundaries shared by neighboring cells.

The Roe scheme (Roe [1981]) is adopted for the numerical evaluation of the

convec-tive fluxes_{F :} ΦR(Wi, Wj, ~n) = F (Wi , ~n) +F (Wj, ~n) 2 − γsP −1_{|P R|} Wj − Wi 2 (4.1) where : • ΦR_(W

i,Wj,~n) = numerical approximation of the flux between the i-th and the

j-th cells;

• Wi = solution vector at the i-th node;

• Wj = solution vector at the j-th node;

• ~n = outward normal to the cell boundary;

• R (Wi,Wj,~n) = Roe matrix;

• P (Wi,Wj) = Turkel-type preconditioning term, introduced to avoid accuracy

problems at low Mach numbers (Guillard and Viozat [1999]). Note that, since it only appears in the upwind part of the numerical fluxes, the scheme remains consistent in time, and can thus be used for unsteady flow simulations;

• The γsparameter multiplies the upwind part of the scheme, and thus it permits

a direct control of the numerical viscosity, leading to a full upwind scheme for

γs= 1, and to a centered scheme when γs = 0.

The spatial accuracy of this scheme is only first order. The MUSCL linear recon-struction method (Monotone Upwind Schemes for Conservation Laws), introduced by Van Leer [1977], is employed to increase the order of accuracy of the Roe scheme. This is obtained by expressing the Roe flux as a function of the reconstructed values

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of W at the cell interface: ΦR_(W

ij, Wji, ~nij), where Wij is extrapolated from the

values of W at nodes i and j. A reconstruction using a combination of different families of approximate gradients (P1-elementwise gradients and nodal gradients evaluated on different tetrahedra) is adopted, which allows a numerical dissipation made of sixth-order space derivatives to be obtained (Camarri et al. [2004]). For the integration in time, in this code an implicit time marching algorithm is used, based on a second-order time-accurate backward difference scheme.

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4.2 Test-case description

4.2.1 Domain configuration

The flow around a square cylinder symmetrically positioned between two parallel walls is considered here; this configuration is sketched in Fig.4.1

Figure 4.1. Computational domain

The ratio between the cylinder edge length L and the distance between the walls H is L/H = 1/8. The incoming flow is a laminar Pouiseuille flow directed in the x direction and the considered Reynolds numbers, based on the maximum velocity of the incoming flow and on L, range between 100 and 300. Thus, the other

dimen-sions are: Lin/L = 12 and Lout/L = 20. For 2D simulations, the spanwise length

adopted is Lz/L = 0.6, and it has been systematically checked that the simulated

spanwise velocity was negligible. For the 3D simulations, the spanwise length of the

domain is Lz/L = 6. This choice has been made following the indications given

in (Sohankar et al. [1999]) and (Saha et al. [2003]) for the numerical study of the three-dimensional wake instabilities of a square cylinder in an open uniform flow. Note that no indication is available in the literature for the case here considered.

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Non-reflective inflow and outflow boundary conditions are imposed, and at the in-flow the Poiseuille in-flow is assumed to be undisturbed. Periodic boundary conditions are imposed in the spanwise direction and no-slip conditions are forced at the cylin-der and at the parallel walls.

In order to carry out grid convergence tests the domain is discretized using differ-ent grids. In particular, in the two-dimensional simulations grid convergence tests have been carried out using three grids ( GR1, GR2, GR3 ) mainly differing for the spatial resolution in the proximity of the cylinder. All these grids are unstructured and made of tetrahedrical elements. The grid GR4, used for the 3D simulations, has been built by replicating the grid GR1 (see Tab.4.1) ten times in the spanwise direction. The details of all these grids are reported in Tab.4.1.

nodes on the

grid total number of nodes cylinder perimeter Lz/L

GR1 7.5_{· 10}5 ₂₅₀ _0.6

GR2 6.6· 105 ₂₁₀ _0.6

GR3 6.0· 105 ₁₇₀ _0.6

GR4 6.6_{· 10}6 ₂₅₀ _6.0

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4.2.2 Simulation parameters

Since we are interested here to incompressible flows the simulations have been carried out by assuming that the maximum Mach number of the inflow profile is M = 0.1. This value allows compressibility effects to be reasonably neglected and does not imply serious problems for the numerics. Moreover the preconditioning term (see Sec.4.1) is usedto increase the accuracy of the results. Indeed it was observed that the preconditioner leads to a more accurate value of the pressure coefficient near the stagnation point in the upwind face of the cylinder, improving the mean value of the drag coefficient. Conversely, the time fluctuations of the force coefficients were insensitive to the preconditioner. Concerning the numerical

viscosity, the upwind parameter γs is set to γs= 1.0 on the nodes within a distance

equal to 0.1L from the cylinder and γs= 0.1 in the rest of the domain. This choice

ensures the stability of all the simulations carried out, and, at the same time, allows the preconditioner to be particularly effective in the proximity of the cylinder.

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4.3 Definition of the 2D database and of the

POD-Galerkin model

We examine, first, the two-dimensional case, for wich the domain configuration is described in the Sec.4.2.1 and sumarized in Tab.4.2 (with reference to Fig.4.1).

parameter value description

L cylinder edge length

L/H 1/8 blockage ratio

Lin/L 12 inflow length

Lout/L 20 outflow length

Lz/L 0.6 spanwise length

Table 4.2. Domain geometry values

The grid used is GR1 described in Sec.4.2.1. The distribution of the elements is choosen to have the maximum resolution in the proximity of the cylinder. The

number of grid nodes is approximately 7.5· 105 _{(758015). The boundary conditions}

are defined, as explained in Sec.4.2.1.

The Reynolds number, based on the maximum velocity of the incoming flow and on L, is Re = 150 . Obviously, in the two-dimensional simulations only a 2D vortex shedding can be described. Indeed then the vortex shedding process, after a short phase of transient, reaches rapidly a constant frequency, as it can be seen by

ana-lyzing the time variation of the lift coefficient CL, shown in Fig.4.2.

The total simulation time is about 166 seconds, and 4500 snapshots have been recorder. For the POD database, a time interval of 13.80 seconds is choosen, span-ning about two vortex shedding periods; in particular, 94 snapshots have been used, with a ∆t of 0.147 seconds, starting from t = 138.6 s, where the flow is completely developed, and thus the amplitude of the lift coefficient variation is constant. In the Fig.4.3 and Fig.4.4 the isocontours of the u-component and of the v-component of the velocity are shown for a snapshot of the database.

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Figure 4.3. Isocontours of the u-component of the velocity (t = 157.88)

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The POD modes are obtained from this two-dimensional snapshots database, after the evaluation and the subtraction of the time averaged field.

Then, to develop the low-order model, we used the first six modes, which capture about the 99.74% of the total energy of each field, as shown in Fig.4.5.

152 154 156 158 160 162 164 166 168 94 95 96 97 98 99 100 time

Reconstructed energy (in percents)

nm=2 nm=4 nm=6 nm=20

Figure 4.5. Reconstructed energy for 2D POD (%)

The calibration procedure, performed to obtain the final form of the POD-Galerkin model, is carried out by using the same time interval and thus the same set of flow

velocity fields, and by employing Nt = 41 collocation points , which is sufficient

to calibrate also the highest frequency given by the sixth mode. In Tab.4.3 all the parameters used to obtain the POD-Galerkin 2D-model are summarized.

Re 150 Reynolds number

T 13.80 s POD database time interval

NSN 94 Number of snapshots

∆SN 0.15 s Snapshots time step

Ti 138.60 s Database starting time

Nr 6 Number of modes

TCAL 13.80 s Calibration interval

Nt 41 Number of collocation points

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4.4 Definition of 3D database and POD-Galerkin

model

The three-dimensional domain is also described in Sec.4.2.1, and its main fea-tures are summarized in Tab.4.4.

L cylinder edge length

L/H 1/8 blockage ratio

Lin/L 12 inflow length

Lout/L 20 outflow length

Lz/L 6 spanwise length

Table 4.4. Domain geometry values

The used grid is GR4, described in the Sec.4.2.1, obtained by replicating the grid GR1 ten times in the spanwise direction. The total number of the grid nodes is more than 6 millions, precisely 6505397.

The boundary conditions are defined as in the two-dimensional case, see Sec.4.2.1. The Reynolds number is Re = 300; at this Reynolds number, the three-dimensional structures are present in the wake, as shown Fig.4.6, where the values of the max-imum and minmax-imum spanwise velocity in the field are reported as an indicator of the occurrence of 3D phenomena in the flow, together with the lift coefficient. The effects on the aerodynamic forces are already sgnificant when the vortex shedding phenomenon begins to take place; thus, one may say that vortex shedding forms as three-dimensional.

The total simulation time is about 442 seconds, for a total of 9235 snapshots recorded. From those two different set of snapshots are extracted, to obtain two different databases: the first one contains about two vortex shedding periods while the latter one contains about eight periods of vortex shedding. The first one, called short-database, ranges a time period of 15.4 seconds, starting from t = 382.24 s,

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Figure 4.6. Time variation of the lift coefficient CL

where the flow may be considered developed, also if, being a three-dimensional flow, the flow structures are not perfectly periodic, as in the two-dimensional case. The considered snapshots, with a ∆t of 0.347s, are 45, and on this database the short-POD modes are obtained.

The low-order model is developed using 20 modes, which capture about the 98.65% of the total energy, as shown in Fig.4.7.

380 385 390 395 400 405 410 50 55 60 65 70 75 80 85 90 95 100 time

Re=300, POD from t=382.24 to t=397.61

N_r=4 N_r=8 N_r=20 N_r=40

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The calibration procedure, performed to obtain the final form of the POD-Galerkin model, is carried out by using the same time interval and thus the same set of

flow velocity fields, and with Nt = 121 collocation points, wich is sufficient to

cal-ibrate in a good way also the highest frequency given by the last considered POD mode. In Tab.4.5 all the parameters used to obtain this POD-Galerkin 3D-model are summarized.

Table 4.5. POD and calibration parameters

To build the second database, called “long database”, a time interval of 52.40 seconds, about eight vortex shedding periods, is used, starting from t = 360.23 s, where the flow may be considered developed. The snapshots considered, with a∆t of 0.347 s, are 151, and from this long-database the long-POD modes are obtained. The low-order model is developed by retaining the first 20 modes, which capture approximately the 90.23% of the total energy, as shown in Fig.4.8.

Note that with 20 modes of this second long-POD, inside the POD-database, a lower percentual of energy is captured than that captured with 20 modes of the short-POD. However, outside the database interval, more energy is captured in this case, between 58% and 67%, while in the two period’s POD, outside the database interval, an amount of energy between 52% and 57% is captured.

The calibration procedure, is also in this case carried out by using the same time

interval and thus the same set of flow velocity fields, and with Nt= 121 collocation

points. In Tab.4.6 all the parameters used to obtain this second POD-Galerkin 3D-model are shown.

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360 370 380 390 400 410 420 430 440 50 55 60 65 70 75 80 85 90 95 100 time

Re=300, POD from t=360.23 to t=412.64 N_r=4 N_r=8 N_r=20 N_r=60

Figure 4.8. Reconstructed energy for 3D long-POD (%)

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Chapter 5 Definition of the sensor number

and location

In this chapter the application of the Least Square approach, described in chapter 3, to the domain configurations defined in the previous chapter is shown. In particular, the criteria for choosing the sensor number and location are described for the two-dimensional and for the three-two-dimensional cases. To set the position and the number of the sensors we carry out a sensitivity analysis in the two-dimensional case and then we proceed to validate the choice in the three-dimensional case.

5.1 Sensitivity to sensor position in the two-dimensional

case

In order to identify the best choice for the sensor position, we have to remember the principal aim of this work, i.e. to reconstruct a whole velocity field starting from a few external measurements. Thus, in order to capture as much as possible infor-mations about the actual velocity field, the measurements should be located where the time variation of the considered quantity is significant. Hence, an analysis of

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the varation in time of τxy is performed.

In Fig.5.1 the values of τxy, on the whole upper wall, are shown in five different

instants during a vortex-shedding period. The lower is assumed to be symmetric.

Note that for x < 0, upstream the cylinder, τxy is nearly constant, while downstream

the cylinder there is a time variation of τxy, due to the passage of the vortices. This

is clearly visible in Fig.5.2 where the maximum variation of τxy, |∆τmax|, during a

period is sketched, on the upper wall. It is possible to observe two zones of large

variation, the first in the interval x [4_{÷ 6] and the other near the end of the}

com-putational domain, x [16÷18]. The large time fluctuations in the first zone are due

to the passage of vortices, while those in the second are probably due to interaction phenomena between the wall vorticity and the vortices.

The interval x [4_{÷ 5] is chosen to place a distribution of shear-stress sensors on}

each wall; in particular, a sensor for each node has been used and this gives a distri-bution of 206 ideal sensors (sensors are considered punctual). The sensor position, is sketched in Fig.5.3.

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−15 −10 −5 0 5 10 15 20 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 τ on Upper wall X τ (a) t = 138.59 s −15 −10 −5 0 5 10 15 20 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 τ on Upper wall X τ (b) t = 140.22 s −15 −10 −5 0 5 10 15 20 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 τ on Upper wall X τ (c) t = 142.00 s −15 −10 −5 0 5 10 15 20 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 τ on Upper wall X τ (d) t = 143.78 s −15 −10 −5 0 5 10 15 20 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 τ on Upper wall X τ (e) t = 145.41 s

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−150 −10 −5 0 5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ∆τ on Upper wall X ∆τmax

Figure 5.2. _|∆τmax| during one vortex-shedding period

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The Least-Square approach, see Sec.3.1.1, is carried out with this sensor distribu-tion by using different numbers of POD modes. It is obvious that if the Least-Square problem, singularly taken, is too ill-conditioned, it introduces an undesired signifi-cant noise in the Kalman-like system. Thus, a compromise is needed; on one hand, it is suitable to have as less as possible sensors, on the other hand it is impossible to excessively decrease the number of sensors, otherwise the Least-Square problem becomes too ill-conditioned, and the error on the solution too large.

In Fig.5.4, Fig.5.5 and Fig.5.6 the solution of the Least-Square problem (blue), using two, four and six POD modes respectively, evaluated inside the POD time interval, (94 snapshots), with the chosen sensors configuration of Fig. 5.3, are shown together with the projection of the velocity fields on the POD modes (solid line).

138 140 142 144 146 148 150 152 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Least−Square Coefficients t (sec) a1 a* LS 138 140 142 144 146 148 150 152 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Least−Square Coefficients t (sec) a2 a* LS

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138 140 142 144 146 148 150 152 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Least−Square Coefficients t (sec) a1 a* LS 138 140 142 144 146 148 150 152 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Least−Square Coefficients t (sec) a2 a* LS 138 140 142 144 146 148 150 152 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Least−Square Coefficients t (sec) a3 a* LS 138 140 142 144 146 148 150 152 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Least−Square Coefficients t (sec) a4 a* LS

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138 140 142 144 146 148 150 152 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Least−Square Coefficients t (sec) a1 a* LS 138 140 142 144 146 148 150 152 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Least−Square Coefficients t (sec) a2 a* LS 138 140 142 144 146 148 150 152 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Least−Square Coefficients t (sec) a3 a* LS 138 140 142 144 146 148 150 152 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 Least−Square Coefficients t (sec) a4 a* LS 138 140 142 144 146 148 150 152 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Least−Square Coefficients t (sec) a5 a* LS 138 140 142 144 146 148 150 152 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Least−Square Coefficients t (sec) a6 a* LS

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Note that the first two coefficients have the same frequency of vortex shedding, since this phenomenon is mainly captured by the first and second POD modes. When only two modes are used, the LS (Least-Square) approach gives a perfect agreement with the actual time behavior of the POD coefficients.

As shown in Figs.5.5 and 5.5, fixed the sensor distribution, when the number of modes used in LS problem is increased also the error on the coefficients increases; or, better, by introducing new modes in the problem, we introduce also new errors which are higher in norm than those obtained with less mode. This phenomenon is shown in a better way in Fig.5.7, where the norm of the relative errors on the coefficients are sketched as a function of the number of used modes, until ten modes. The norm of the relative error on the i-th coefficient is defined as :

e(ai) =

PNSN

n=1(ai(n)− a∗i(n))2

PNSN

n=1 a∗i(n)2

where NSN is the number of snapshots, ai(n) is the i-th coefficient evaluated by the

LS approach for the snapshot n and a∗

i(n) is the projection of the snapshot n on the

i-th POD mode. When a mode is added, the introduction of new errors higher than the precedent ones is clear, also if the errors on the precedent coefficients decreases. This it is true for all the modes except that for the mode 10, wich can be however comparable to noise; indeed the fluctuating energy captured with eight modes is already equal to 99.89%. Another peculiar behavior is the one of mode 5, wich is, as we will see in the following, a “not-observable” mode. Indeed, by analyzing the condition number of the LS system matrix, it largely increases passing from four used modes to six; this may be considered as an indication of the introduction of “bad information” in the system, that becomes very ill-conditioned, as shown in Tab.5.1.

Thus, in one hand we have that the errors on the coefficients ai increases, together

with the condition number of the matrix A, in the other hand, as said in Sec.3.1.1,

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2 3 4 5 6 7 8 9 10 −7 −6 −5 −4 −3 −2 −1 0

1 Error on Least−Square Coefficients

Number of Modes e(log10) a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

Figure 5.7. Error on LS coefficients as a function of the number of modes used in

the procedure L = 1 2M 9.60 4M 1.15_{∗ 10}2 6M 1.95_{∗ 10}7 8M 1.96∗ 107 10M 2.83∗ 108

Table 5.1. Condition Number of the matrix of the Least-Square problem as a

function of the number of modes used in the procedure

Least-Squares approach is performed to minimize the residual of the τxy (Sec.3.1.1),

and, if the number of used modes increases, the captured energy also increases, and the residual, being more accurate the reconstruction of the field, may decrease. This is shown in Tab.5.2; indeed, for 4 modes and 6 modes the residual is the same, also if there are new higher errors on the coefficients, as shown in Fig.5.7.

Thus, to explain the role of modes 5 and 6 in the numerical system, we can perform another kind of analysis. Indeed, chosen a time instant t, since the matrix of the Least-Squares problem, A, is real symmetric square matrix, it can be written as :

A = U ΛUT

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e(τ ) 2M 3.15_{∗ 10}−6 4M 4.54_{∗ 10}−7 6M 4.54∗ 10−7 8M 7.31∗ 10−8 10M 1.26_{∗ 10}−7

Table 5.2. Residual of τ (averaged on one period) as a function of the number of

modes used in the procedure

diagonal matrix whose elements are the eigenvalues of A. Thus, the system 3.3 can be written as :

U ΛUTan = b

By extracting the i-th equation and using the Einstein notation, we obtain :

uj_iλjujkak= bi (5.1)

where uj_i is the i-th component of the j-th eigenvector, λj is the j-th eigenvalue, ak

is the k-th coefficient at the fixed time t and, obviously, bi is the i-th component of

the vector b.

Thus, if the components of an eigenvector, for instance the i-th, are all zero except one, for instance the w-th one, and the other eigenvectors have this component very small, Eq.5.1 becomes :

uw

i 2λwai = bi (5.2)

that this means the eigenvalue w-th is associated only with the i-th eigenvector, and thus the i-th coefficient is defined by this equation.

Thus, if λw is a very small eigenvalue, a little perturbation on the right hand side δbi

produces a large error on the computed coefficient ai. Therefore, it is more probable

to have an error on this coefficient than on the others.

In the analysed problem, using six modes, the fifth and the sixth eigenvectors have all the components approximately equal to zero, except those associated with the two smallest eigenvalues, as shown in Fig.5.8. Also by using ten modes, the fifth

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and sixth eigenvectors have a component associated with smaller eigenvalue higher than the others,as shown in Fig.5.9; in particular the fifth eigenvector has the eigth component higher than others and the sixth eigenvector the tenth. Thus also using ten modes the fith and the sixth eigenvector are associated with two very small eigenvalues. This leads to the great increment of the condition number, passing from four modes to six modes (see Tab.5.1) and it leads, consequently, to the possibility

of having high error on the corresponding coefficients. The phenomenon of the

“non osservability” of the fifth and sixth mode can be also explained in a more practical way. Indeed, by noting that the matrix A of the problem 3.3 is formed by the derivative of the POD modes and noting that if a POD mode has a very small derivative at the measurements points, a row and a column of the matrix have terms very small, thus, for the same reasons as in the previous analysis, the error on the corresponding coefficient can be high.

The derivative of the first six modes, computed in the measurements interval

x [4_{÷ 5] are shown in Fig.5.10. Note that the derivative of the fifth and sixth}

modes are indeed much smaller than those of the other modes.

In order to have an additional support to the previous analysis, we may position the sensors in an interval in wich the ratio between the derivative of the fifth and sixth modes and that of the others is higher than in the previous configuration.

For instance, the interval for x [11.5_{÷ 12.5] is used to position 206 sensors. The}

derivative of the modes are shown in Fig.5.11. As it can be seen, the ratio be-tween the derivatives of the fifth and sixth modes and the derivative of the first two modes is higher than in the previous case. This leads to a smaller condition

number, with Nr = 6, Nr number of retained modes, as shown in Tab.5.3; but the

condition number is higher than previous, when only the first two modes are used, being now their derivatives smaller. Also, by sketching the errors on the coefficients, see Fig.5.12, obtained until to ten retained modes, note that, in accord with the per-formed analysis, the error on the first two coefficients is higher than in the previous

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Figure 5.8. Eigenvector components of the Least-Square problem (six modes)

case, while the errors on the fifth and sixth modes are smaller. Moreover, the error on all the coefficients decreases also when the 10-th mode is added, because in this zone the small scales of the flow motion, are more developed; thus the derivatives of the modes representative of these small scales, for instance the tenth one, are

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Figure 5.9. [5◦-10◦] Eigenvector components of the Least-Square problem (ten modes)

higher than in the zone near the cylinder, where the small scales are not yet very developed. Summarazing, the best seems to choice, set the sensor location in the

zone x [4_{÷ 5] and to use 6 modes, because the error on the first two coefficient is}

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4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

0.12 Modes Derivative on Upper wall

X δΦy DPhi1 DPhi2 DPhi3 DPhi4 DPhi5 DPhi6

Figure 5.10. Modes derivatives on the upper wall into the measurements interval

11.5 11.6 11.7 11.8 11.9 12 12.1 12.2 12.3 12.4 12.5 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06

0.08 Modes Derivative on Upper wall

X δΦy DPhi1 DPhi2 DPhi3 DPhi4 DPhi5 DPhi6

Figure 5.11. Modes derivatives into the interval x [11.5÷ 12.5]

energetic view point, because they capture the 94.78% of the fluctuating energy.

x[11.5_{÷ 12.5] (nodes = 206) x[4 ÷ 5] (nodes = 206)}

2M 25.85 9.60

4M 25.85 1.15∗ 102

6M 6.11_{∗ 10}5 _1.95_{∗ 10}7

Low order modelling of the flow around a confined square cylinder.

Summary

Contents

Chapter 1

Introduction

Chapter 2

Reduced order model

2.1

Description of the POD-GALERKIN model

2.1.1

Proper Orthogonal Decomposition

2.1.2

POD-Galerkin model

2.1.3

POD decomposition for the fluctuating velocity field

2.2

Description of the calibration process

Chapter 3

Kalman-like technique

3.1

Description of the Kalman-like method

3.1.1

Least-Squares approach

3.1.2

Description of Kalman-like technique

Chapter 4

Application to the flow around a

confined cylinder

4.1

Description of the code used to generate the

database

4.2

Test-case description

4.2.1

Domain configuration

4.2.2

Simulation parameters

4.3

Definition of the 2D database and of the

POD-Galerkin model

4.4

Definition of 3D database and POD-Galerkin

model

Chapter 5

Definition of the sensor number

and location

5.1

Sensitivity to sensor position in the two-dimensional

case