FLUCOME 2019 - FLUCOME 2019 - On interpretation of spatiotemporal data decomposition · 2019. 5....

15th International Conference on Fluid Control, Measurements and Visualization

27-30 May 2019, Naples, Italy

Paper ID: 34 1

On interpretation of spatiotemporal data decomposition

Václav Uruba1,2*, Pavel Procházka1

1Institute of Thermomechanics of the Czech Academy of Sciences, Dolejškova 5, Praha 8, Czech Republic

2University of West Bohemia, Faculty of Mechanical Engineering, Department of Power System Engineering,

Universitní 8, Plzeň, Czech Republic

*corresponding author: [email protected]

Abstract Decomposition of spatiotemporal data representing dynamics of turbulent velocity field is applied very

often. There are various types of data decompositions. Classical methods are based on energy budged as the Proper

Orthogonal Decomposition or more general Bi-Orthogonal Decomposition. Those methods maximize kinetic

energy i.e. variance of velocity covered by the low order modes. However, different strategy could be used, as

stability of oscillating modes, Dynamic Mode Decomposition or Oscillation Pattern Decomposition methods are

the examples. The evaluated modes suffer of not clear physical interpretation. In the paper some of aspects of

evaluated modes is addressed. The example of wake behind circular cylinder in cross-flow is shown.

Keywords: turbulence, decomposition, POD, OPD, wake, circular cylinder

1 Introduction

A turbulent flow is characterized by presence of highly dynamical strongly 3D coherent structures, mostly

vortices. The corresponding spatio-temporal data is at our disposal only recently both from mathematical

modeling and experiments. Simulation data are provided by DNS or LES methods. Experimental data could

be obtained using time resolved PIV method, as shown e.g. in [7].

Typical example can be von Kármán vortex-street in a circular cylinder bluff body wake. More or less regular

system of moving structures verge into the next stage of instability, resulting in fully developed turbulence

with its typical attributes, as is randomness or fractal structure, nevertheless keeping the main features of the

preceding stage – presence of traveling vortical structures – see e.g. [5]. Such structures are not obvious from

the instantaneous snapshots, but they are objectively present. To study this type of flow, the decomposition

methods could be used.

2 Decomposition methods

Turbulent fluid flows behave in characteristic patterns, known as modes. In a recirculating flow, for example,

a hierarchy of vortices can be detected, a big main vortex driving smaller secondary vortices, and so on. Most

of the motion of such system can be realistically described using only a few of those patterns – modes – see

e.g. [1]. From a purely mathematical point of view, similar modes can be extracted from the governing

equations using eigenvalue decomposition.

Decomposition methods are based on the idea of representing the fluid-system state by a state vector in a state

space. The state vector is defined in a proper way and can have very high dimensionality (typically in the order

of thousands). In the case of velocity data it consists of properly ordered velocity components from all

measured points in space, subjected to experimental investigation. Time is typically chosen as independent

variable, but in principle, a spatial variable can replace it. More details see e.g. [11].

The Proper Orthogonal Decomposition (POD) method has applications in almost any scientific field where

extended dynamical systems are involved. Recently, the POD has been widely used in studies of turbulence.

Historically, it was introduced in the context of turbulence by Lumley in 1967 (see [4]) as an objective

definition of flow objects what were called big eddies in past and which is now widely known as coherent

structures.

The POD method is statistical one, but there also exists a deterministic variant of POD, called Bi-Orthogonal

Decomposition (BOD), suggested by Aubry et al. in 1991 (see [2]). Since the BOD method provides besides

topology also time-related information, it makes sense applying it on “time-resolved” data only, and it performs

decomposition in both time and space domains.

A BOD mode consists of three parts, dividing the information content into three domains. These domains



Paper ID: 34 2

customarily have their specific names: spatial mode information is named “Topos”, temporal mode information

is named “Chronos”, and global amplitude as the energy mode. The BOD approach uses the same strategy for

spatial modes evaluation as POD, based on energy content optimization. The extra time information provides

instantaneous amplitude of the given mode. Toposes are closely related to the POD modes, but unlike them

the Toposes have good physical meaning even for a non-stationary process representing a transient

phenomenon.

Exact physical interpretation of BOD (as well as POD) modes is difficult. This situation results from the non-

physical condition of orthogonality of modes or decorrelation, which mathematical in nature and could be

hardly interpreted in physics. The direct consequence of such condition is mode non-localization in spectrum

space of frequencies and thus every Chronos is characterized by a broadband spectrum, physically indicating

contributions from all possible frequencies. This means that it is generally impossible to associate the certain

mode and it’s Chronos with a single particular frequency. The other reason is that different modes do not

necessarily represent independent processes, because decorrelation does not generally mean independence. It

is known that geometries corresponding to two phases of deterministic periodical system shifted by a quarter

of period are uncorrelated in the sense that their covariance vanishes, but the states are fully dependent,

characterizing a single deterministic process. This situation is typical if any periodical process is involved,

then two different BOD modes are connected with two phases of the periodical process.

For localization of a multivariate process in the frequency domain, different method should be used. The

method based on stability assessment include Dynamic Mode Decomposition method (see [6]) and Oscillating

Pattern Decomposition. This methods differ by mathematical methods used for implementation.

Oscillating Pattern Decomposition (OPD) method is based on PIPs and POPs approaches introduced by

Hasselmann 1988 [3] in the field of climatology, details of its implementation see [10].

The OPD can be applied on time–resolved data only. It is based on stability analysis of the mean flow. Any

fluctuation of the flow is considered to be a kind of perturbation, which can either grow exponentially in time

(if the flow is unstable), or decay (if the flow is stable). This concept complies with Lyapunov stability theory,

applied on the mean flow, however application is close to the finite-time concept. As the method is based on

the long-time statistics, the mean growth of any pattern should be negative only (i.e. the mean flow is stable),

otherwise the structure exceeds the flow-field boundaries in a finite time.

Each OPD mode consists of a unique eigenmode and eigenvalue, both represented generally by complex

numbers. The complex eigenmode represents a structure or pattern moving in space in a cyclic manner. The

structure can be typically a wave or travelling vortex propagating in space. Complex eigenvalue contains

information on the cyclic phenomenon frequency and decay in time – so called e-fold time. This knowledge

has a unique feature: it allows for the prediction of system development in time, in context with the given

mode. For typical flows with a convective velocity, structures in the form of propagating waves are fairly

common.

The OPD method has several advantages. The unambiguous definition of a single frequency connected with a

given mode allows for assessment of aerodynamic forces and noise and identification of their sources. The

knowledge of exact frequency and location of the fluctuating pressure is very important for the definition of

the interaction between the flow and an elastic body, as responses including resonance can be predicted.

3 Example: cylinder in cross-flow

As an example, the canonical case of unsteady wake behind a circular cylinder in cross-flow is presented.

Reynolds number based on the cylinder diameter was 4 815, this means that the wake was fully turbulent. The

PIV method was used, 4000 snapshots with frequency 2 kHz were acquired resulting in acquisition time 2 s.

Details on instrumentation see [9].

Coordinates and velocities are to be presented in dimensionless form, the cylinder diameter (15 mm) and

incoming velocity (5 m/s) were used.

In Figure 1a there is an example of instantaneous velocity field characterized by vector field with added vector-

lines (in green). The mean velocity vector field is shown in Figure 1b (vector-lines in blue). Presence of

vortices forming von Kármán vortex street is visible. However the vortices are dynamical in nature and their

structure is close to turbulent. The averaging process erased the individual vortices leaving only two big

structures in symmetrical configuration.



Paper ID: 34 3

(a) (b)

Fig. 1 Velocity vector fields, (a) instantaneous and (b) mean

The back-flow close to the axis of symmetry in the region x < 4 is clearly visible. Symmetrical time-mean field

is confirmed by mean velocity modulus (mod) and Turbulent Kinetic Energy (TKE) distributions in

Figures 2a,b.

(a) (b)

Fig. 2 Distributions of (a) mean velocity modulus and (b) Turbulent Kinetic Energy

Minimum of the velocity modulus could be detected in the position x = 3, while fluctuation energy maxima

are split into two parts located in transversal coordinate y = +-0.5 behind the cylinder edges and x 4.

4 Results of decompositions

The decomposition methods procedures involve subtraction of the time-mean velocity field as the first step.

Then, the fluctuating energy TKE is explained.

5 Proper Orthogonal Decomposition

The POD procedure is applied on the data resulting in 4000 POD modes. The variant using BOD procedure

have been applied, the results are mode energy, Topos and Chronos.

First Energy Fraction and Accumulated Energy are to be shown. The Energy Fraction is the relative fraction

of the TKE contained in individual mode supposing that the total TKE is 1. The first POD mode contains

21.5% of the total TKE, the other modes are in Energy Fraction descending order. The Accumulated Energy

evaluate the sum of all modes from number 1 up to the mode in question, so this value is growing towards 1.

The energy distribution situation shows Figure 3. Please note, that both scales are logarithmic.



Paper ID: 34 4

Fig. 3 Energy Fraction and Accumulated Energy of POD modes.

Next, the POD modes are to be presented, both Topos and Chronos, but for selected modes only.

The first two most energetic modes 1 and 2 represent a single periodical process, obviously as explained in

[8]. The Toposes represent vortices moving in streamwise direction and Chronoses the time evolution shifted

by a quarter of period. The process itself is close to be a single frequency. The modes 1 and 2 are shown in

Figures 4 and 5, respectively.

(a) (b)

Fig. 4 POD mode 1, Energy Fraction 21.5%, (a) Topos and (b) Chronos.

(a) (b)


The following POD mode 3 represents Energy Fraction 8.4%. The Topos shows velocity pulsations in the

streamwise direction. The Chronos contains broad-band frequency process obviously with low frequency



Paper ID: 34 5

contribution, order of unites Hz, but random.

(a) (b)


The other modes are populated by vortices as a rule. The POD mode 4 in Figure 7 consists of a couple of big

contra-rotating vortices in distance about 4 behind the cylinder and several smaller vortices distributed

irregularly in close wake.

(a) (b)


The POD mode 5 in Figure 8 contains saddle point in position x = 4.

(a) (b)


Many higher order modes contain rows of contra-rotating vortices as the POD mode 11 (Figure 9), 20

(Figure 10) and 50 (Figure 11). The higher order mode, the less Energy Fraction and smaller vortices.



Paper ID: 34 6

(a) (b)


(a) (b)


(a) (b)


Even higher order modes have been evaluated. In Figures 12 and 13 the POD modes 100 and 1000 are

presented. These modes are very random both in time and space, populated by small vortices and the Energy

Fraction is almost negligible.



Paper ID: 34 7

(a) (b)


(a) (b)


The POD mode 1000 is close to be homogeneous both in space and time.

6 Oscillation Pattern Decomposition

The OPD procedure was applied on POD data. The modes with Energy Fraction higher than 0.5% were taken

into account.

Each OPD mode is represented by frequency, e-folding time characterizing typical life or time-scale of the

mode, periodicity as e-folding time and mode period ratio expressing the typical time-life in terms of periods,

and complex topology.

Fig. 14 Frequencies and periodicities of the OPD modes



Paper ID: 34 8

The process spectrum is to be shown first in Figure 14, frequency and periodicity. The modes with highest

periodicity are the most stable as their time-life. The OPD modes are ordered with descending e-folding time.

The highest periodicity exhibits the OPD mode number 1, the corresponding frequency is 48.68 Hz. The

Strouhal number could be evaluated using the cylinder diameter (15 mm) and in-flow velocity (5 m/s). Its

value is here Sh = 0.146, far from expected value 0.2. However the topology is unexpected as well. The mode

1 is shown in Figure 15. The OPD modes topology shows traveling waves, this will be demonstrated on the

case of the OPD mode 1. The real and imaginary parts of the OPD mode topology represent situation in the

phase angle 0 and /2 respectively. The process continues by angle with negative real mode part topology

and 3/2 with negative imaginary part topology. The process finishes in phase angle 2 by the real part

topology repeating again in descending amplitude.

(a) (b)

Fig. 15 OPD mode 1, f = 48.68 Hz, Sh = 0.146, e = 170.64 ms, p = 8.3, (a) real and (b) imaginary parts.

This dominant mode shows merging of the contra-rotating vortex couple in original position x = 2.4 for the

real part (phase 0) into a single big vortex in position x = 3 for the imaginary part (phase /2), next moving to

position x = 3.8 (phase /2) and finally x = 4.7 (phase 3/2).

Please note that the OPD mode 1 contains predominantly POD modes 1 and 2 (see Figures 4 and 5).

The next example with high periodicity is number 7 in Figure 16. This mode is characterized by traveling

contra-rotating vortical couples, relatively good regularity, but high frequency, Strouhal number about 0.44.

(a) (b)


The OPD mode 9 is characterized by the expected Strouhal number 0.19, however the vortex street topology

is not perfectly regular. It exhibits some features of turbulence – see Figure 17.

0 /2 3/2



Paper ID: 34 9

(a) (b)


The other OPD modes presented here are similar in topology, which is represented by the row of contra-

rotating vortices. However the frequencies vary from Strouhal number 0.236 for the OPD mode 12 (Figure 19),

Sh = 0.483 for the OPD mode 10 (Figure 18) up to Sh = 0.821 for the OPD mode 13 (Figure 20).

(a) (b)


(a) (b)




Paper ID: 34 10

(a) (b)


The OPD modes with higher number of vortices correspond to higher value of frequency, keeping the

progressive velocity of the vortex street approximately constant.

7 Conclusions

The presented spatiotemporal data decompositions produce modes which are in general a certain linear

combinations of instantaneous snapshots.

Two decomposition methods are demonstrated on the canonical case of wake behind circular cylinder: POD

and OPD. Both methods are able to represent dynamics of the flow. While the POD method shows topology

only without clear link to flow dynamics in terms of frequency spectrum, the OPD method evaluates associate

frequency and stability of the embedded pseudo-periodical process.

The POD modes are pulsating in nature. Topology of the POD mode is represented by the Topos, the pulsation

time-evolution is given by the Chronos and amplitude is defined by the mode energy.

The OPD modes represent travelling waves topology in two phases of the period.

Acknowledgement

This work was supported by the Grant Agency of the Czech Republic, projects Nos. 17-01088S and GA19-

02288J.

References

[1] Adrian R J, Christensen K T, Liu Z C (2000) Analysis and interpretation of instantaneous turbulent

velocity fields. Exp. Fluids, vol. 29, pp. 275–290.

[2] Aubry N, Guyonnet R, Lima R (1991) Spatiotemporal Analysis of Complex Signals: Theory and

Applications. J. Stat. Phys., vol. 64, pp. 683–739.

[3] Hasselmann K (1988) PIPs and POPs: The Reduction of Complex Dynamical Systems Using Principal

Interaction and Oscillation Patterns. Journal of Geophysical Research, vol. 93, D9, pp. 11.015-11.021.

[4] Lumley J L (1967) The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and

Radio Wave Propagation; Tatarsky, V.I., Yaglom, A.M., Eds.; Nauka: Moskva, Russia, pp. 166–178.

[5] Pope S B (2000) Turbulent Flows. Cambridge University Press, Cambridge, UK.

[6] Schmid P J (2010) Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech.,

vol. 656, pp. 5–28.

[7] Tropea C, Yarin A L, Foss J F eds. (2007) Handbook of Experimental Fluid Mechanics; Springer: Berlin,

Germany.

[8] Uruba V (2019) Energy and Entropy in Turbulence Decompositions. Entropy, vol. 21, issue 2, art.no. 124.



Paper ID: 34 11

https://doi.org/10.3390/e21020124

[9] Uruba V, Procházka P, Skála V (2018) On the structure of the boundary layer under adverse pressure

gradient on an inclined plate. J. Phys. Conf. Ser., vol. 1101, 012047.

[10] Uruba V (2015) Near Wake Dynamics around a Vibrating Airfoil by Means of PIV and Oscillation Pattern

Decomposition at Reynolds Number of 65 000. J. Fluids Struct., vol. 55, pp. 372–383.

[11] Uruba V (2012) Decomposition methods in turbulence research. EPJ Web Conf., vol. 25, 01095.

https://doi.org/10.3390/e21020124

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