tu-berlin intern report

8/10/2019 TU-Berlin Intern Report

1/23

A

INVEST

REDUCT

MODAL AN

AMA

In Partial F

Bachelor of

P

Department of FBerlin Inst

P

Indian Instit

PROJECT REPORT ON

GATION OF MOD

ON OF FLOWS W

NON-MODAL GR

By

SAHAI (Roll No 08010359)

lfillment of the Award of the Degree of

echnology, Mechanical Engineeri

oject Work Carried Out at

luid Mechanics & Engineering Aco itute of Technology, Berlin - 10623

Under the Guidance of

rof. Dr. Joern Sesterhenn

Dr. Julius Reiss

Submitted To

te of Technology, Guwahati 7810

MAY-JULY (2011)

L

TH

WTH

g

stics

9


2/23

I would like to express my deepest gratitude to Prof. Joern Sesterhenn for accepting me

as summer intern at the Department of Fluid Mechanics and Engineering Acoustics. His

constant support and guidance not only allowed me to explore the different facets of the

study assigned for the study but also, constantly expand on my overall understanding of

the subject and academics in general. I would also like to thank Dr. Julius Reiss, who was

instrumental in making this period of study possible for me and assisted me through

every set of obstacles I encountered while realizing this brilliant learning opportunity. His

assistance during the actual period of study was even more crucial and allowed me to

gain firsthand knowledge of implementing reduced order modeling through well-defined

and clear objectives.

I would also like to thank Flavia Cavalcanti Miranda, Mathias Lemke, JensBrouwer, Oliver Henze and Mario Sruka whose help has been invaluable in overcoming

difficulties that I managed to hit at every step of my study. I have constantly pestered

every living soul at the department and it was their unwavering patience that allowed me

to rectify multiple flaws and grapple with challenging concepts that I encountered all

along. I would also like to mention the generosity I have received from everyone around

me and the ease with which I could assimilate into the group. The satisfaction I received

from working at the Department of Fluid Mechanics and Engineering Acoustics can only

be matched by the fun I had with the rest of the team during the period of my stay. I

would find this the most apt opportunity to mention that I would very much look forwardto a similar opportunity at TU Berlin in the future and engage in exhilarating research in

Fluid Mechanics while never missing out on all the fun banter that goes around the place.

I am also deeply grateful to Indian Institute of Technology, Guwahati and the

Honda Motor Company for facilitating this entire experience and translating my desire to

explore the different dimensions of the field of aeronautics and fluid mechanics into a

tangible opportunity to build on my theoretical background (if limited) in the subject by

working on challenging real world applications and problems.

Amal Sahai

Junior Undergraduate Student

Department of Mechanical Engineering

Indian Institute of Technology, Guwahati - 781039

India


3/23

1 B .................................................................................. 1

2 .......................................................................... 1

2.1 A ......................................................................................... 12.2 () .................................................. 2

2.2.1 ......................................................................... 2

2.2.2 1: ...................................................... 3

2.2.3 2: ............................................... 4

2.3 B .................................................................................. 5

2.4 B ............................................................................................ 6

2.4.1 B ......................... 6

2.4.2 2: : ............................. 8

3 ........................................................................................ 10

3.1 ............................................................................................. 10

3.2 ......................................................................................... 11

3.3 .................................................. 12

3.4 ................................................ 13

3.4.1 B ................................ 14

3.4.2 B ........................ 15

4 ............................................................................ 17

5 AAB .............................................................. 18

6 B ................................................................................................... 19


4/23

2-1: 10 B .................................................................. 3

2-2: A & A

........................................................... 4

2-3:

........................................................................ 5

2-4: 10 B .............. 8

2-5: A & A

B ................................................................. 9

2-6: A & A

B ................................................................. 9

2-7:

B .............................................................. 10

31: ()

= 500

( = 20, 60, 100) ............................. 11

3-2: B

( = 15, 75) ............... 14

3-3: B

= 5 ............................ 15

3-4: = 5 A . 16


5/23

1

1

The following study has been conducted as a part of my summer internship for the period May-

July, 2011 at the Institute of Fluid Mechanics and Acoustics, TU-Berlin. It is aimed at

investigating different techniques for model reduction with applications in fluid mechanics

problems. Additionally, an attempt has been made to explore the suitability of applying such

techniques for calculating optimal starting disturbances for achieving non-orthogonal growth at

sub-critical Reynolds numbers in case of channel flow.

2

This section aims to explore the different techniques and present the underlying principles with

for achieving model reduction. It also presents the efficacy of these techniques at representing

non-modal growth satisfactorily.

2.1

The application of different principles of dynamical systems and control theory is a considerable

challenge in the field of fluid mechanics because of the complex nature of many of the governing

equations which are high-dimensional and non-linear. Model Reduction is aimed at obtaining

low dimensional approximations of the full systems in order to circumvent the problem of

analyzing the full flow problem directly.

The problem of obtaining a reduced model is tackled initially through the use of two

traditional techniques and eventually by a final method that combines elements from the last two

approaches to obtain satisfactory results. The method of Proper Orthogonal Decomposition

(POD) and the accompanying Galerkin projection that has seen widespread use, involves

projecting the full system onto empirically obtained base functions. POD can yield unpredictable

results based on the empirical data and these error may persist even near stable equilibrium

points.

The next approach is referred to as balanced truncation largely eliminates the limitations

faced by POD, and provides for error bounds that are close to the lowest error possible for any

reduced-order model. Although very effective, this approach has seen limited application on

problems pertaining to fluid systems due to computational difficulties it suffers at high

dimensions.

The final method, referred to Balanced Proper Orthogonal Decomposition, combines

ideas from both the aforementioned techniques. It provides for balanced truncations with

computational cost similar to POD. This technique aims at overcoming several limitations


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2

primarily regarding the computational costs at large dimensions and the unbalanced truncating of

states (poorly observable but strongly controllable and vice-versa) by incorporating a different

methods of snapshots.

2.2

()

POD, also referred to as principal component analysis has been used for a long time to generate

low-dimensional models of fluid systems. It involves the creation of subspace Vr of fixed

dimensionr, given a set of data that lies in a vector space V, such that the error in projecting onto

the subspace is minimized. Also, it would be assumed that for a fluid problem the equations have

been suitably discretized so that V has finite dimension (for a finite difference simulation, this

would simply be the number of grid points times the number of flow variables).

Suppose we have a set of data given by x(t) belongs to , with 0 < t < T. We seek aprojection of fixed rank r, that minimizes the total error 2.2(a)

The matrix is now introduced:

2.2(b)where * denotes the transpose. R being symmetric, positive-semidefinite would generate

eigenvalues that are real and non-negative. The eigenvectors may be chosen to beorthonormal. The optimal subspace of dimension r is spanned by . The vectors are also referred to as POD modes.

2.2.1

To generate POD modes for the given problem, eigenvalue problem needs to be solved. Incase of a fluid mechanics problem, this could exceed . Thus, it is a much more efficient togenerate snapshots of the system at discrete times (m in number) and then, reduce the original

eigenvalue problem to an eigenvalue problem. Thus, the integral is transformed into asum:

2.2(c)where are quadrature coefficients. The data is assembled into a matrix:

2.2(d)


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The sum is then written as . Thus, this reduced eigenvalue is then solved, 2.3(e)with the eigenvectors may be chosen to be orthonormal. The POD modes are then givenby . In matrix form, with , and , thisbecomes: 2.4(f)This problem is much more efficient when the number of snapshots m is smaller than the number

of states n.

2.2.2

1

POD modes are used to approximate the stream wise and vertical components of velocity for an

undisturbed flow. Additionally, the kinetic energy of the full system and the reduced system is

compared in order to explore the possibility of applying POD modes to the problem of reducing

different types of fluid systems. It should be noted that a sub-critical Reynolds number 1000 and

a CFL 0.5 for all subsequent simulations.

2-1: 10


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POD modes perform satisfactorily to a certain extent and successfully enable a representation of

an original grid size of (with velocity as the only variable used from thesnapshots) using a reduced system size of 200 with 20 modes. The errors encountered can easily

be marginalized using a convenient scale factor.

2.2.3

2

A similar system as above is taken but is given an initial disturbance in addition to the initial

conditions for base flow in the form of velocity peak at a single grid point. This disturbance then

propagates through the channel and affects the development of the flow in time. Additionally, the

kinetic energy of the full system and the reduced system is again compared in order to verify the

suitability of this approach.

2-2: &


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2-3:

A similar system as above with the same size of the reduced system and the number of POD

modes yields unsatisfactory results. POD involves a high degree of time averaging of the

snapshots and therefore, a large amount of information is lost while representing a disturbance

that is evolving over time and space.

2.3

Balanced Truncation is a method of model reduction for stable, linear input-output systems.

Consider a stable linear input-output system where symbols carry their usual meaning:

2.3(a) 2.3(b)

The controllability and observability Gramians are symmetric positive-semidefinite matrices and

are defined by:

2.3(c) 2.3(d)

These are usually computed by solving Lyapunov equations:

2.3(e)


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2.3(f)The controllability Gramian measures to what degree each state is excited by an input (that ittakes a smaller input to drive the system from rest. The Gramian is positive-definite if andonly if all states are reachable with some input . Conversely, the observability Gramian measures to what degree excites future outputs. States which excite larger output signals arecalled more observable, and in this sense are more dynamically important than states that are

less observable. The objective is to find a coordinate system in which the Gramians for

controllability and observability are equal and diagonal. Using a coordinate transformation

like , one obtains the Gramians in the new coordinate system: 2.3(g) 2.3(h)In order for them to be equal:

2.3(i)with the Hankel singular values . Solving Lypanov equations is very expensive for high-dimensional systems and empirical Gramians can be computed using data from numerical

simulations. This can be possible by computing the state responses to unit impulses for the

forward system (Controllability Gramian) and the adjoint system (Observability Gramian). But

this method would if done directly would require integrations of the forward system and theadjoint system, where is the number of outputs. Thus, this method is not feasible when thenumber of inputs is large, for instance if the output is the full state.

2.4

The Balanced POD approach aims at obtaining an approximation of the balanced truncation that

has acceptable computational costs for larger systems. The method combines elements of the

previous two approaches and presents a technique for computing the balancing transformation

directly from the snapshots of empirical Gramians, without needing to compute the Gramians

themselves. This is followed by an additional output projection method to enable tractable

computation even when the number of outputs is large.

2.4.1

The controllability and the observability Gramians can be factored as:

2.4(a)


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where and are matrices, but X and Y may be rectangular, with differentdimensions. X and Y are simply the data matrices from the forward and adjoint system with

snapshots stacked as columns of a matrix (given at discrete times) and the integral for the

Gramian becomes a quadrature sum.

In the method of snapshots, the balancing modes are then computed by forming thesingular value decomposition (SVD) of the matrix :

2.4(b)

Where is invertible and its order is equal to the rank of . Then, the balancingtransformation and the inverse transform can be defined as: 2.4(c)

where and . In case , then the columns of form the first columnsof the balancing transformation and the rows of form the first columns of the inversetransformation. The considerable advantage through the technique of snapshots is the enormous

savings made due when the number of number of snap shots is much smaller than .Reduced order models can be obtained be obtained by transforming to balanced coordinates.

Also, there is no need to transform all of the states:

2.4(d)

where are states to be retained and are states to be truncated. Then, the transformedequations are:

2.4(e) 2.4(f)Thus, to compute a reduced-order model of order r, all we need is the first r columns of T and the

first r rows of S.


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2.4.2

2

In order to achieve clear comparative data between different model reduction techniques, a

similar channel configuration, in terms of original & reduced system size and the number of

modes used, is utilized for balanced truncation.

2-4: 10


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2-5: &

2-6: &


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2-7:

Balanced Truncation as evident from simulations overcomes the various shortcomings of POD

method and can be used effectively to represent non-modal growth with respect to capturing the

evolution of a non-orthogonal growth initiated by an optimal disturbance.

3

3.1

We begin with a base flow that is steady or time varying and compute it with the help of direct

numerical simulations. The geometry of the flow would be restricted to a channel flow case,

similar to the ones used for applying the model reduction techniques. In case an infinitesimal

perturbation

is added to the base flow U, the development of this perturbation can be described

by the following linearized Navier-Stokes equations:

3.1(a) 3.1(b)

:


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1. In case there are solutions that grow without bounds, the base flow U is linearly unstable.

2. Even when the base flow is linearly unstable, there can be the possibility of bounded

solutions that exhibit large transient growth before eventually decaying. In this case, the flow

is linearly stable but has local regions of convective stability, or is susceptible to non-linear

instability, or both.

An illustration of point 2 is provided in figure 3-1 which illustrates the evolution of optimal

two-dimensional disturbances to steady two-dimensional flow over a backward-facing step. At

the inflow Reynolds number = 500, the flow is asymptotically stable. However the flow has

localized convective instability and hence exhibits large linear transient growth of suitable

disturbances.

31: () 500

( 20, 60, 100)

3.2

Equations (3.1) define a linear operator which evolves perturbations forward in time: 3.2(a)


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We can evaluate the linear stability of the base flow through the use of this forward evolution

operator. In case the flow is periodic with period T, we can evaluate the eigenvalue problem by

setting time t to T in the following manner:

3.2(b)Classic linear stability of the base flow is then determined from the dominant eigenvalues of

, i.e. the of largest modulus. If there are any eigenvalues with , then thereexists exponentially growing solutions of (3.1) and hence, the base flow is linearly unstable.

Conversely, if every eigenvalue satisfies , then every solution of (3.1) eventually decaysto zero and the flow is linearly stable. Generally, would represent a bifurcation point.

3.3

This section explores situations in which despite the base flow being linearly stable,perturbations exhibit substantial transient response due to regions of local convective instability.

Owing to the non-orthogonality of the eigenmodes of , which arise due to the asymmetry of theconvective terms in the Navier-Stokes equations, the dynamics of interest may not be of the form

of an exponential function of time multiplying a fixed modal shape rendering the eigenvalue

problem in (3.2(b)) of no direct relevance. Instead a better way to characterize the dynamics

would be through a computation of the singular vectors and values of .A step towards quantifying the size of a perturbation would be the choice of a suitable

norm. A physically meaningful choice would be the total energy of the perturbation field over

the flow domain. The transient growth can then be measured as the perturbation energy at time normalized to its initial energy. We can then state the following considering normalized initial

perturbations:

, 3.3(a)

The evolution operator can be used to write: 3.3(b)

where represents the adjoint evolution operator to .For the case of the transient growth problem, the objective would be to calculate the

perturbations which lead to maximal or near maximal growth. The eigenfunction corresponding

to the dominant eigenvalues of will be dictated by the largest possible growth. Lettingand denote the eigenvalues and normalized eigenfunctions, we have:


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3.3(c)Thus, , which denotes the maximum growth obtainable at time would be: 3.3(d)

The process of finding dominant eigenvalues of is equivalent to finding the largestsingular values of , and this has a direct physical interpretation. The eigenfunction in(3.3(a)) provides an initial perturbation which generates a growth over time . Defining to be the normalized perturbation at time evolved from initial condition , wehave:

, 3.3(e)where . This is nothing other than the singular value decomposition of , with being the left and being the right singular vectors respectively (real and orthonormal as well).Also, with both and being real and non-negative.

3.4

The dominant eigenvalue and the corresponding eigenvector for achieving non-orthogonal

growth are achieved by running power iteration by repeatedly pre-multiplying to anintial disturbance vector generated through random noise. Scaling is done to a convenient factor

after every iteration for the resultant variable.

A non-linear DNS solver is used for the forward computation and depending on the

choice of for which optimal growth is being obtained, the number of time steps are varied. Forevery iteration of the eigenvalue calculation, the disturbance vector is added to the base flow and

the forward solution obtained. The base flow is then subtracted from the full disturbed flow in

order to obtain the evolution of only the perturbation for the period . The effect of pre-multiplying is obtained by running the adjoint solver with initial conditionsbeing , with the code linearized around the base flow. Scaling at the end of everycomplete solution in the forward and backward direction becomes even more crucial since we

have used a non-linear DNS solver which would cause the non-linearties to start dominatingwhile calculating the evolution of the disturbance in case the initial disturbance is large

compared to the base flow.

A steady base flow used for this entire set of calculations which is initialized with

parabolic profile for the streamwise velocity component, zero normal velocity component and

uniform pressure and entropy. A choice of Reynolds number = 1000 is made to ensure the base is


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linearly stable with localized instabilities allowing for transient growth. A careful choice of grid

size is made in this regard to ensure under-resolution doesnt affect the absolute instability of the

flow and alters results.

3.4.1

The initial batch of simulations for achieving non-orthogonal growth was carried out with

periodic boundary conditions in the x-direction for both the forward and adjoint solvers. Besides

the obvious ease while defining the boundary conditions, this allowed the base flow to develop

completely and become steady much faster. It was expected that in case of the disturbed flow,

suitable non-orthogonal growth would be achieved quickly as well.

As figures clearly illustrate, when the u and v components of the disturbance were evaluated over

a period of time, certain structures which were well defined and periodic in nature were obtained.

3-2: ( 15, 75)

The energy of the disturbance as it evolved over time also showed a consistent decay mired in

oscillations. The results thus, obtained were not satisfactory and in spite of attempts to work with

different time periods for optimization, the results were largely similar. The periodicity caused


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the perturbations that would leave the domain to re-enter and interfere with the further

development of the flow causing anomalies in the results obtained.

3.4.2

The second set of simulations was carried out with non-periodic boundary in the x-directionwhile retaining the adiabatic wall boundary condition in the y direction. The left boundary (x

minus) was defined as fixed in time while the right boundary (x positive) was defined as non-

reflecting for the forward solver. In case of the adjoint equation, both the left and right boundary

(x minus & positive) were defined as non-reflecting. Additionally, the velocity was made zero at

all boundaries to ensure homogeneous boundary conditions and avoid possible complexities.

As illustrated in figures non orthogonal growth was indeed achieved for . The low valuesof amplification for the energy of the disturbance can be explained because of the low value of being used for the first run. Higher values of would yield greater growth that resemble Figure:3-1 more closely.

3-3: 5


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3-4: 5

A key consideration that should be incorporated into simulations for optimal growth

computations is that the velocity perturbations should not have significant amplitude at the

outflow boundary. If a perturbation reaches the outflow boundary with non-negligible amplitude

it is thereafter washed out of the computational domain and the corresponding perturbation

energy is lost to the computation. Thus, when simulations were run with the same domain size as

utilized for , non-orthogonal growth was not achieved because disturbances whichpreviously did not have enough time to flow out were getting washed out for larger values of .The domain should be of sufficient size so as to ensure that the velocity perturbations reach the

outflow boundary with negligible energy. In practice, it is crucial that the computational domain

for such calculations be much larger when compared to stability calculations.


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4

The viability of various model reduction techniques that have been have been put forth in

contemporary literature were investigated through section 2 with the eventual aim of applying

them to solving non-orthogonal growth problems. Of the various simulations conducted,

Balanced POD was found to be adept at representing time varying disturbances evolving through

the flow domain with the minimum set of modes and for reduction to the lowest possible order.

POD involves a high degree of averaging across the time domain using the snap shots obtained

for discrete time intervals and therefore, perturbations cannot be fully represented using the

reduced model obtained through POD satisfactorily. Although it must be added at this point that

POD did perform well while reducing base flow simulations, but again it should be stressed that

this would be inadequate for application to representing a channel system that has been given an

optimal disturbance for initiating non-orthogonal growth.

In the next section, an attempt has been made to achieve non-orthogonal growth for sub-

critical Reynolds number for a full system. This has been achieved by calculating optimal growth

for maximizing energy of the disturbance for a definite time. Simulations were initially carried

out using periodic boundary conditions in the streamwise direction but were not successful and

resulted in periodic structures being created in the disturbed flow as it evolves over time. Also,

the energy of the disturbance decayed, albeit while oscillating without any transient growth being

achieved. Non-periodic boundary conditions have resulted in more successful simulations but

require larger domains so as to ensure that no significant perturbations get washed out of the

computational domain.

Hence, this report has laid the foundation of successfully applying model reduction

techniques to this particular problem by gauging the performance of different model reduction

techniques on similar systems. Also, non-orthogonal growth has been achieved to a certain

degree on full systems thereby, paving the way for a reduced model to achieve similar results

being computed. Also, the various MATLAB programs that have been created during the course

of study can be easily tweaked to obtain results for a multitude of systems and therefore, can be

extremely helpful while executing routines for the aforementioned objectives.


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5

I. Balanced_modes.m: Creates balanced modes (the direct and inverse transformation

matrix for a given data set using the method of snapshots.

II. Balanced_reductionchannel.m: Carries out simulations for a reduced model of channel

flow and compares it with the full system simulations.

III. balanced_Truncation.m: Performs Balanced Truncation using modes from (I) to compute

the reduced model.

IV. Barkley_algo.m: Computes eigenvectors and eigenvalues using a Krylov subspace, the

nomenclature is a reference to the use of this algorithm in Barkley(2007).

V. check_channel.m: Carries simulations using optimal disturbances for achieving non-

orthogonal growth and then presents the evolution of the disturbance for channel flow.

VI. check_mixinglayer.m: Performs a similar operation as in (V), albeit for a mixing layer.

VII. controlmatrix.m: Computes the full system matrix using simulation data from the DNS

solver.VIII. MatrixCheck.m: Checks the nature of the direct and adjoint full system matrix (for

symmetry).

IX. non_orthogonal_growth.m: Computes the optimal disturbance for different values of time

for achieving non-orthogonal growth.

X. NSM_uPODmodes.m: Computes the POD modes using the method of snapshots with

the stream-wise velocity component as the only variables.

XI. NSM_uvPODmodes.m: Computes the POD modes using the method of snapshots with

velocity as the only variables.

XII. NSM_uvpPODmodes.m: Computes the POD modes using the method of snapshots with

velocity and pressure as the variables.

Additionally, it must be added that all programs created during the course of this study

are highly modular and can be applied to a host of different problems revolving around similar

principles with minor modifications. Proper commenting has been provided at all points of

interest to expedite the same.


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B, ., B, . ., & , . . (2007). A .

, 231: 120.

, . ., , B. ., & , . (2009).

A, B .

, ., , A., , ., & , . (2008).

A.

, . . (2005). , B .

.

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