Module 2 :Lecture 4 : Correlation Functions and Intensity

The lecture contains

Correlation FunctionsIdeas about eddy sizeIntensity of Turbulence or Degree of Turbulence

Module 2 :Lecture 4 : Correlation Functions and Intensity

Correlation Functions

Turbulent motion is by definition: EDDYING MOTION

Figure 4.1

A statistical correlation can be applied to fluctuating velocities in turbulence. A high degree ofcorrelation exists between the velocities at two points in space, if the distance between the points issmaller than the diameter of the eddy. Converse is also true.

Consider a random variable (velocity) at two points, separated by a distance r. An Euleriancorrelation tensor at the two points can be defined.

In other words, the dependence between the two velocities at two points is measured by the timeaverages of the products of the quantities measured at two points. The correlation of the components of the turbulent velocity of these two points is defined as

The non-dimensional form of the correlation

The value of R (r) of unity signifies a perfect correlation of the two quantities involved and theirmotion is in phase. Negative value of the correlation function implies that the time averages of thevelocities in the two correlated points have different signs.

Figure 4.2

The Figure (4.2) shows typical variations of the correlation R with increasing separation r .

To describe the evolution of a fluctuation , we need to know the manner in which the value of at different times are related. For this purpose the correlation function between the values of

at different times is chosen and is called autocorrelation function.

Module 2 :Lecture 4 :

Ideas about eddy size

For a moderate velocity ( ~ 100 m/s), smallest eddy will be of the order of 0.5 mm In gases under atmospheric conditions, mean free path is the range of 10 -4 mm Turbulent frequencies vary between a few Hz and 10 kHz Molecular collision frequency ~ 5 × 10 9 Hz (air)

Therefore turbulence is within continuum!

Large eddies:

Small eddies:

ν : dimension L2 / T ; dimension ( L2 / T 3 )

Dissipation

Velocity scale

Intensity of Turbulence or Degree of Turbulence

For isotropic turbulence,

(used in practice, even if the turbulence is not isotropic!)

Module 2 :Lecture 5 : Kolmogorov Hypothesis and Energy Cascade

The lecture contains

Kolmogorov Universal Law for the Fine Structure

Energy Cascade

Kolmogorov Length Scale

Kolmogorov's First Hypothesis

Kolmogorov's Second Hypothesis

Module 2 :Lecture 5 : Kolmogorov Hypothesis and Energy Cascade

Kolmogorov Universal Law for the Fine Structure

The sizes of the eddies are called scales

a. Large eddies (energy containing eddies) having length scales, say l . For a turbulent channelflow, l~ D, where D is the width of the channel.

b. Small eddies (energy dissipating eddies) having length scales , called Kolmogorov length

scale.

Each length scale is associated with its corresponding time scale. Smallest length scale has thesmallest time scale.

Scales smaller than Kolmogorov scale do not exist. Molecular viscosity dissipates small scaleenergy into heat.

Module 2 :Lecture 5 : Kolmogorov Hypothesis and Energy Cascade

Energy Cascade

The large eddies break down into smaller eddies. Smaller eddies break down into still smallereddies, and so on

Big whorls have little whorls

Which feed on their velocity

And the little whorls have lesser whorls

And so on to viscosity

At very small scales, the viscous forces become dominant dissipate energy. Equilibriumdistribution of energy is brought about.

The small scales can be estimated from the situation when the energy is dissipated into heat

Turbulent dissipation rate, , having the dimension

Kinematic viscosity, having the dimension

Module 2 :Lecture 5 : Kolmogorov Hypothesis and Energy Cascade

The Kolmogorov micro scales are formed using the dissipation rate per unit mass and the Kinematicviscosity ν .

Length scale,

Time scale,

Velocity scale,

We define as Reynolds number

~ ~ ~

where velocity scale, L integral length scale and λ Taylor microscale

Let us estimate Kolmogorov's length scale

Imagine a channel flow

Figure 5.1

Assume Isotropy, i.e.,

~ ~

Kolmogorov Length Scale

~

Let us take

U = 10 m/s, D =1m, Re = 6.5 × 10 5 , Tu = 0.2

= 0.38mm

Module 2 :Lecture 5 : Kolmogorov Hypothesis and Energy Cascade

Kolmogorov's First Hypothesis

At sufficiently high Reynolds numbers, there is a region of high wave numbers, where the turbulenceis statistically in equilibrium and uniquely determined by the parameters and ν . This state ofequilibrium is universal.

At this state, the turbulence is statistically homogeneous and locally isotropic.

Consequence of becoming statistically homogeneous:

Between any two points and are same

Equilibrium: The total energy supply is practically equal to total dissipation

Figure 5.2

Module 2 :Lecture 5 : Kolmogorov Hypothesis and Energy Cascade

Kolmogorov's Second Hypothesis

At sufficiently high Reynolds numbers, the statistics of the motions of scale l in the range le >>l>>have a universal form that is uniquely determined by , independent of .

Length scale l correspond to wave number,

So in the second hypothesis ke<<k<<kd,

Spectrum indicates how the turbulent kinetic energy is distributed among the eddies of differentsizes. From the second hypothesis, the spectrum is

The region in which this is valid is called the inertial subrange of universal equilibrium.

In this range of wave numbers, the inertial forces play a dominant role, while the viscous forcesbecome insignificant.

The universal function has to be determined experimentally. To be noted <<1.

Figure 5.3

Module 2 :Lecture 6 : Probability Density Functions and Averaging

The lecture contains

Introduction

Probability density function

Averaging used in the analysis of turbulent flows

Module 2 :Lecture 6 : Probability Density Functions and Averaging

Introduction

In this part of the text, for compactness, Cartesian coordinates and components

will be used in the analysis. In specific examples, the clearer notation x, y, z, and u, v, w

will be used

The summation convention will be used so that summation signs will not be written explicitly but willbe implied by repeated indices; in scalar products or vectors.

and the divergence operator

Notations:

Inst mean fluctuatingThis text (usually) u

Alternative -1 u <u>Alternative -2 u UPope (2000) U <U> u

Tennekes& Lumley (1972) U u

Module 2 :Lecture 6 : Probability Density Functions and Averaging

Probability density functions and Statistical Quantities

When a physical variable, say u can take a wide range of values, it is worthwhile using a notationthat distinguishes between the variable itself, u , and its possible values, say v . The probability thatthe variable u takes a value between v and v + dv is

where f(u) is the probability density function of u . This statement is the definition of PDF, f . The setof all possible values of v is called the sample space. Since the probability that u has a value, anyvalue at all must be 1

where represents integration over the whole of sample space.

Given the probability density function of u , the mean of any function ψ of u is

where the integral is performed over the entire sample space. The averaging is weighted by howoften each value v occurs. The mean of variable u itself is

The fluctuating part of u is

Also to be noted

The variance of u is

and the standard deviation of u is the square root of the variance is

which is the rms value of fluctuations. More generally, the nth order central moment of u is

The variance is second-order central moment. The skewness and flatness are normalized higher-order central moments. The skewness of the probability density function u is

and the flatness or kurtosis is

While deriving Reynolds average NS equation or energy equation for turbulent flows, we shall neednumber of such averaging operations:

where c is a constant

where c is a constant

where c is a constant

or

or

The normal distribution function

The probability density function of the normal distribution is

All higher- order moments of the normal distribution can be expressed in terms of the two lowest-

order moment, and

Since the PDF is symmetric, f(-v) = f(v) , so the skewness disappears. The fourth order central

moment is

So the flatness is 3.

Task

By making use of the normal distribution as the probability density function, show that the flatness is 3

We know,

Therefore

put

or or

Now,

or,

Therefore, flatness or Kurtosis =

Module 2 :Lecture 6 : Probability Density Functions and Averaging

Averaging used in the analysis of turbulent flows

The complete sample space and its associated joint probability density function are theoreticalconstructions. When faced with measured or numerically simulated data practical methods ofaveraging have to be used.

A general definition of the time average is

This is the convenient form of average often used in practical measurements of statistically stationaryflows, i.e., flows in which averaged quantities are independent of time. In statistically time-dependentflows, T needs to be chosen to be much greater than all the time scales of the fluctuations and muchless than the time scale on which U varies with t , if possible.

The spatial average is defined in a similar way to the time average. For example,

This expression will be most convenient if is expected to be independent of x3=z . Spatialaverages, which exploit spatial symmetries, are particularly useful in numerical simulations.

Higher-order statistics, such as variances and co variances, can also be calculated from timeaverages and spatial averages. For example,

In principle, statistics based on time averages implicitly provide an approximation to the probabilitydensity function of statistically stationary turbulent flows. Similarly statistics based on spatialaverages provide an approximation to the probability density function of statistically homogeneousturbulent flows.

The ensemble average

Consider an ensemble, { u(n) ; n = 1, . . . ,N} , obtained by carrying out the same experiments in Nidentical apparatuses or more realistically by repeating the same experiment N times. N should belarge. The ensemble average is defined by,