Turbulence Hwork5 Real

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Statistical turbulence analysis of flow past a square cylinder

Homework No. 5 - author: Ernest Odhiambo

Presented to: Professor R.F. Huang

2012/05/15

Department of Mechanical Engineering, National Taiwan University of Science and Technology, No.

43,Sec.4,Keelung Rd.,Taipei,106,Taiwan,R.O.C

Abstract

The present work reports on the statistical details obtained by hotwire anemometry, for the 2D flow past a square cylinder. The mean,periodic and random phenomena are quantified statistically, alongside the fundamental frequency at three different Reynolds numbers.

Data is extracted at one fixed location for both the free stream and wake regions, to mimic homogeneity of turbulence.

Keywords: Square cylinder; Statistical turbulence; Vortex shedding; Lagrangian integral time scale; Taylor micro time scale

1. Introduction

Bluff bodies have received enormous attention in

engineering research, largely due to the sheer copious

existence of the prevailing flow features commonly

occurring in offshore, aerodynamic and other structures

associated with the built environment. The flow around a

square cylinder is representative of the patterns expected

when the boundary layer separates from a bluff body as a

result of adverse pressure gradients. Additionally, the

dominance of vortices invariably leads to vortex shedding

and turbulence in the wake, even at modest Reynolds

numbers [1]. Other related unsteady rara avises including

the separation bubble [] and shear-layer instability [2-huang], have been studied.

An array of data employing statistical tools to decipher

flow visualization results has been published. These have

shed light on the various time, length and velocity scales,

thereby enabling a better understanding of the relationship

between the coherent vortex and the incoherent

turbulence structures.

1.1 Time series

The time series plots are a record of the transient field.

In the case where the mean value of the measurable

instantaneous field (u) can be assumed stationary,

ensemble averaging is avoided and the time averaged

instantaneous quantities are derived according to equation

1.

(1)

The outcome of equation 1 allows the computation of

mean values in the time domain. Clearly, the averaging in

time over the turbulent fluctuations must then be 0.

Time series experimental results from others.square

cylinder

Thermo-Fluid

dynamics-Lab

Experiment.

Turbulence


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1. 2 Probability density function(PDF)

The probability density distribution possesses all the

ingredients that show which amplitude range the

measurable field moves in time at a specific measuring

location. Thus the PDF (f(u)), presents data in the

amplitude domain. The stationary (mean) and the

dynamic (variance) components of the flow are

mathematically related with f (u) by equations 2 and 3

respectively [Durst].

3.1.3 Probability (Cumulative) distribution function

Include formula here of integration of PDF

Show how to perform numerical integration ie the steps in

excel

F(- < E


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1. 3 Autocorrelation

1. 3.1 Lagrangian time scale

1. 3.2 Taylor micro time scale

1. 4 Power spectrum

The power spectrum also referred to as the spectral

density function enables the representation of the

frequency content of a signal. The mathematical

formulation of this statistical tool is displayed in equation5.

()

()

(5)

Essentially the bracketed expression is the mean

square value of the signal (), which is the originalsignal (), filtered around the frequency with abandwith of Normalisation of the integral as achievedthrough the same bandwidth. Hence the data properties

are presented in a frequency domain.

Key details observed from such a spectrum show (i)

the fraction of the signal fluctuations occurring in a given

frequency band (ii) (for a random signal), the range of

frequencies over which oscillations occur and (iii) the

frequency with the maximum power density. In their

efforts, Sushanta et al [] show that the power spectra plots

for flow around square cylinders with different angles of

tilt, in the near wake region, unveil a broadening behavior

for angles between 30 and 45 degrees. They explain that

this may be a result of vortex dislocation and diffusion as

documented by Williamson []. Though not indicated on

their power spectra figures, Shun et al [Flow patterns

vortex shedding behavior square cylinder], closemeasurement of the spectra slope at higher frequencies,

give a slope of -5/3, confirming the isotropy of the small

length scales turbulence structures. Applying their

spectra data, Sarioglu et al [] determined that the vortex

shedding frequency was inversely proportional to the

wake width, but that the shedding intensity was dependent

on the stationary aspect of the vortex sheet.

1. 5 Fundamental frequency

1. 6 Motivation

Aside from being a partial requirement for completing

the course in turbulence, the overwhelming driver for

carrying out this excise, has to be the first hand

appreciation of the immense usefulness of statistical tools

in their exposition of turbulent fluid flow phenomenon,

rather than consigning them to being mere mathematical

jargon. In the subsequent sections, the experimental data

is synthesized using the statistical concepts outlined in

this section and the outcome of results compared with a

limited number of similar experimental work. Further an

attempt is also made to answer two pertinent questions

[Durst] (i) how do local turbulent fluctuations of the


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velocity components vary around their corresponding

mean values? (ii) how are neighbouring turbulent

fluctuations of the velocity components correlated with

one another, and what is the physical significance of these

correlations?

2. Experiment

2.1 Experimental setup

The power spectrum also referred to as the spectral

density function enables the representation of thefrequency content of a signal. The mathematical

formulation of this statistical tool is displayed in equation

1.

() { ()}(1)

Essentially the bracketed expression is the mean

square value of the signal (), which is the originalsignal (), filtered around the frequency with abandwith of Normalisation of the integral as achievedthrough the same bandwidth.

Key details observed from such a spectrum show (i)

the fraction of the signal fluctuations occurring in a given

frequency band (ii) (for a random signal), the range of

frequencies over which oscillations occur and (iii) the

frequency with the maximum power density. In their

efforts, Sushanta et al [] show that the power spectra plots

for flow around square cylinders with different angles of

tilt, in the near wake region, unveil a broadening behavior

for angles between 30 and 45 degrees. They explain that

this may be a result of vortex dislocation and diffusion as

documented by Williamson []. Though not indicated on

their power spectra figures, Shun et al [Flow patterns

vortex shedding behavior square cylinder], close

measurement of the spectra slope at higher frequencies,give a slope of -5/3, confirming the isotropy of the small

length scales turbulence structures. Applying their

spectra data, Sarioglu et al [] determined that the vortex

shedding frequency was inversely proportional to the

wake width, but that the shedding intensity was dependent

on the stationary aspect of the vortex sheet.

1. 4 Fundamental frequency

1. 5 Motivation

Aside from being a partial requirement for completing

the course in turbulence, the overwhelming driver for

carrying out this excise, has to be the first hand

appreciation of the immense usefulness of statistical tools

in their exposition of turbulent fluid flow phenomenon,

rather than consigning them to being mere mathematical

jargon. In the subsequent sections, the experimental data

is synthesized using the statistical concepts outlined in

this section and the outcome of results compared with a

limited number of similar experimental work.


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3. Results and discussion

The outcome of the experiment is presented and

analysed in the following. Raw data was processed by the

program DataPro.

3.1 Function generator

The data from the function generator is useful invalidating the application of the data processing code, and

can also be used as a guide for interpreting the real data.

3.1.1 Time series data

As evident from figure 1a the transient data for the

function generator shows a smooth sine wave as expected.

A quick observation of the graph indicates an upper and

lower limit of the voltage E as 2.05 and -1.797 volts

respectively. A rough estimate of the mean value (which

is later validated from the PDF graph) would then be

() . An approximatevalue for the frequency of generation can also be

Fig 1a Time series for sign wave generator Fig 1b PDF for sign wave generator

Fig 1c Autocorrelation for sign wave generator Fig 1d Power spectrum for sign wave generator


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obtained, since from the graph the peak to peak time span

is 0.02513, giving a frequency of approximately 39 Hz.

Again this value is to be validated by the analysis of the

power spectrum function. The standard deviation of a sign

wave (RMS value) is given by (amplitude) / (2)1/2

. In this

particular case the standard deviation would be

RMS = 1.9235 / (2)1/2

= 1.36

The value obtained from the plot (1.36) is close to the

one provided by DataPro (1.327001).

3.1.2 PDF data

The PDF data shows a normalized (standardized)

distribution density. The data provides the amplitude of

the sign wave confirming the values estimated by the time

series data of section 3.1.1. From the PDF graph the

amplitude of the voltage fluctuation has a magnitude of

approximately 2, within a minimal % error. The graph

also displays two peaks, which is be used as a benchmark

for confirming any sinusoidal phenomenon that may be

present in the flow.

- mention turbulence intensity (how to find from graphs)

and calculate it.

3.1.3 Probability (Cumulative) distribution function

Include formula here of integration of PDF

Show how to perform numerical integration ie the steps in

excel

F(-2 < E


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3.2 Free stream

The data from the free stream is based on flow just

ahead of the cylinder. The Reynolds number for this flow

according to the mean flow provided by DataPro is

calculated below:

Re =

3.2.1 Time series datafree stream

As evident from figure 1a the transient data for the

function generator shows a smooth sine wave as expected.

A quick observation of the graph indicates an upper and

lower limit of the voltage E as 2.05 and -1.797 volts



() . An approximatevalue for the frequency of generation can also be

Fig 2a Time series for free stream, Re = 49412 Fig 2b PDF for free stream, Re = 49412

Fig 2c Autocorrelation, free stream, Re = 49412 Fig 2d Autocorrelation, free stream, Re = 49412

(maximum lag 10 sec) (4 < lag < 6.5)


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Fig 2e Power spectrum, free stream, Re = 49412

3.2.2 PDF datafree stream

Two peaks sinuiosadal??????As evident from

figure 1a the transient data for the function generator

shows a smooth sine wave as expected. A quick

observation of the graph indicates an upper and lower

limit of the voltage E as 2.05 and -1.797 volts



() .3.2.3 Probability (Cumulative) distribution functionfree

stream

3.2.4 Autocorrelation data

The PDF data shows a normalized (standardized)

distribution density. The data provides the amplitude of

the sign wave confirming the values estimated by the time

series data of section 3.1.1. From the PDF graph the

amplitude of the voltage fluctuation has a magnitude of

approximately 2, within a minimal % error. The graph

also displays two peaks, which is be used as a benchmark

for confirming any sinusoidal phenomenon that may bepresent in the flow.

3.2.5 Power spectrum data

0

0.2

0.4

0.6

0.8

1

1.2

-5 0 5 10

F

u'

Figure 2f Graph of F(-0.4 < u'< 0.4) vs. u' for

free stream


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3.3 WakeRe =19024

The data from the free stream is based on

flow just ahead of the cylinder. The Reynolds number for

this flow according to the mean flow provided by DataPro

is calculated below:

Re =

3.3.1 Time series datawake, Re = 19024

As evident from .figure 1a the transient data for

the function generator shows a smooth sine wave as

expected. A quick observation of the graph indicates an

upper and lower limit of the voltage Eas 2.05 and -1.797

volts respectively. A rough estimate of the mean value

(which is later validated from the PDF graph) would then

be () . Anapproximate value for the frequency of generation can

also be

Fig 3a Time series for wake, Re = 19024 Fig 3b PDF for wake, Re = 19024

Fig 3c Autocorrelation for wake, Re = 19024 Fig 3d Power Spectrum for wake, Re = 19024


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3.3.2 PDF datawake, Re = 19024

Two peaks sinuiosadal??????As evident from

figure 1a the transient data for the function generator

shows a smooth sine wave as expected. A quick

observation of the graph indicates an upper and lower

limit of the voltage E as 2.05 and -1.797 volts



() .3.2.3 Probability (Cumulative) distribution functionfree

stream

0

0.2

0.4

0.6

0.8

1

1.2

-5 0 5 10

F

u'

Figure 3e Graph of F(-0.4 < u'< 0.4) vs. u'

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Turbulence Hwork5 Real