Basic definitions of the

Chapter 10 Basic definitions of the

statistical theory of turbulence

10.1 Reynolds averaging

Reynolds proposed to represent any stochastic quantity in turbulent flow as the sum of its averaged part and fluctuation. For instance, this representation applied for velocity components reads

D Nu x Cu 0 x I u y

D Nu y Cu 0 y I u z

D Nu z Cu 0 z I (10.1) The Reynolds averaged velocities are

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The averaged fluctuation is zero

u 0 x;y;z D0

The root of the averaged square of fluctuations is called root mean square, or r.m.s.. The quantity averaged twice is equal to quantity averaged once u D u. If the turbulence process is statistically unsteady (for instance r.m.s is changed in time), the definition of the Reynolds averaging (10.2) is not applicable and s hould be extended using the concept of ensemble averaging. Within the en- semble averaging the stochastic process is repeated N times from the initial state. The turbulent quantity u is measured at a certain time t N times. The ensemble averaged quantity is then:

10.2 Isotropic and homogeneous turbulence

The turbulence is isotropic if r.m.s of all three velocity fluctuations are equal

(10.3) The turbulence parameters are invariant with respect to the rotation of the

u 0 2 0 2 0 x 2 Du y Du z

reference system. The turbulence is homogeneous in some fluid volume if all statistical parameters are the same for all points in this volume, i.e.

u 0 2 .E x/ D u 0 x;y;z 2 x;y;z .E xCE r/ . This equality can be written for all statistical moments. The turbulence parameters are invariant with respect to the trans- lation of the reference system.

10.3 Correlation function. Integral length

The product of two fluctuations is the correlation. The product of two fluc- tuations at two point separated by the distance Er is the correlation function:

(10.4) If i D j the correlation function R ii is reffered as to the autocorrelation

ij .E x; E r/ D u i .E x/u j .E xCE r/

function. In homogeneous turbulence R ii depends only on the separation:

ij .E r/ D u i .E x/u j .E xCE r/

ED E ECE

The coefficient of the autocorrelation function is changed between zero and one:

A sample of the autocorrealtion function coefficient for scalar fluctuation f 0

Figure 10.1: Autocorrelation function coefficient for scalar fluctuation at three different points A; B and C across the jet mixer. [47].

at three different points A; B and C across the jet mixer is shown in Fig. 10.1. In measurements presented in Fig. 10.1 the scalar f is the concentration of the dye injected from the nozzle (see Fig. 10.1, low picture, right). The change of the function has a certain physical meaning. Let us consider the autocorrelation function with respect to the point C (blue line):

f .r C ;r A / is negative. It means the increase of the quantity f at the point C (f 0 .r C />0 ) is followed by the decrease of this quantity at the point A (f 0 .r A /<0 ) . This is true in statistical sense, i.e. the most probable consequence of the increase f .r C / is

the decrease of f .r A / .

The correlation function and autocorrelation function can be written not only for spatial separation but also for separation in time. For example, the autocorrelation temporal function of the u i fluctuation is

The integral of the spatial autocorrelation functions

is the integral length. The integral lengths are estimations of the size of the largest vortex in flow. A sample of the integral length of the scalar field f along the jet mixer centerline (Fig. 10.1, right)

D=2

L f .x/ D

f .r/dr

D=2

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is the integral time length. Coefficients of the autocorrelation function of the axial velocity fluctuations for the free jet are presented in Fig. 10.3. The line 1 corresponds to the autocorrelation function calculated along the jet boundary line (shown by the blue line in Fig. 10.3, right). The line 2 corresponds to the autocor- relation function calculated along the jet axis. Oscillating character of the dependency R uu

indicates the presence of vortex structures arising at the jet boundary and attaining the jet axis. The distance between zero points is roughly the vortex size.

10.3.1 Some re lations in isotropic turbulence

In the isotropic turbulence u 02 D u 0 l .x/u 0 l .x/ D u 0 t .x/u 0 t .x/ D ::: the autocor- relation function can be represented in the form [13]

Figure 10.2: Distribution of the integral length of the scalar field along the jet mixer centerline. [47].

Figure 10.3: Autocorrelation functions in free jet flow. [46].

u 0 l .x/u 0 l .x C r/

f .r/ D

0 u (10.14)

l .x/u 0 l .x/

is the autocorrelation of the longitudinal velocity calculated in longitudinal direction. For instance, the autocorrelation function of the u x fluctuation calculated in x direction,

u 0 .x/u x 0 x .x C r/

f .r/ D

u 0 x .x/u 0 x .x/

or the autocorrelation function of the u y fluctuation calculated in y direction:

u 0 y .y/u 0 y .y C r/

f .r/ D

u 0 .y/u y 0 y .y/

The function f .r/ is the same in both cases (10.15) and (10.16). The au- tocorrelaton function g.r/ is calculated for transversal velocities along any direction

u 0 .x/u t 0 t .x C r/

g.r/ D

u 0 t .x/u 0 t .x/

For instance, the autocorrelation function of the u x fluctuation calculated in y direction,

u 0 x .y/u 0 x .y C r/

g.r/ D

u 0 x 0 .y/u x .y/

or the autocorrelation function of the u y fluctuation calculated in x direction:

u 0 y .x/u 0 y .x C r/

g.r/ D

u 0 .x/u y 0 y .x/

The function g.r/ is the same in both cases (10.18) and (10.19). The ve- locities components used in the previous definitions are illustrated in Fig.

10.4. u i is the velocity pulsation vector, u il is its projection on the direction connecting two points (i.e. longitudinal direction), u it is its projection on the transversal direction. Products like u it u it are the correlations between points 1 and 2.

Figure 1

0.4: Illustrations of velocities used in calculations of the longitudinal

f and transversal g autocorrelations.

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Figure 10.5: Illustration of the autocorrelation functions f and g and Taylor microscales.

The following relations are valid between g and f in the isotropic homoge- neous turbulence (see [13]):

1 @f

gDfC r

2 @r

Typical form of f and g is shown in Fig. 10.5. The change of the sign of g function is due to the continuity equation of the velocity field. The integral length calculated using f is twice as large as that calculated using g.

10.3.2 Taylor microscale

Until Kolmogorov derived his estimations for vortices in 1941, it has been thought that the minimum vortices arising in the turbulent flow have sizes estimated by Taylor. Let us consider the parabola fitted to the autocorre- lation function at point r D 0. The parabola intersects the horizontal axis

Taylor and called as the Taylor microscale. The Taylor microscale can be calculated through the second derivative of the autocorrelation function at r D 0. The Taylor series of f in the vicinity of the point r D 0 is

f .r/ D 1 C

.0/r C O.r 4 /

2 @r 2

The parabola fitted to the curve f .r/ at r D 0 intersects the horizontal axis at the point:

f D @ 2 f (10.22)

@r 2 .0/

Similar relations can be derived for the transversal autocorrelation

g D @ 2 g (10.23)

@r 2 .0/

Re D u 0 (10.24) which characterizes the state of the turbulence in the flow.

10.3.3 Correlation functions in the Fourier space

Any continuous function can be represented in the Fourier space:

The new function O f.E k; t / is then known as the Fourier transform and/or the frequency spectrum of the function f . The Fourier transform is also a reversible operation:

The Fourier transformation can be also written for the correlation function:

Very often one uses one dimensional correlation functions defined as

(10.28) Proof of t he formula (10.28) The inverse Fourier transform of the function R ij .r 1 ; 0; 0/ :

ij .k 1 ; 0; 0/ D

ik r

R ij .r 1 ; 0; 0/e 1 1 dr 1 (10.29)

The inverse transformation reads:

ik R r

ij .r 1 ; 0; 0/ D

ij .k 1 ; 0; 0/e 1 1 dk 1 (10.30)

The general definition is

iE kE R r ij .r 1 ;r 2 ;r 3 /D ˚ ij .k 1 ;k 2 ;k 3 /e dk 1 dk 2 dk 3 (10.31)

From the last formula we have:

R .r ; 0; 0/ D ˚ .k ;k ;k /d k dk e ik 1 r ij 1 1 ij 1 2 3 2 3 dk 1 (10.32)

Comparison of (10.32) with (10.30) results in the desired formula

ij .k 1 ; 0; 0/ D

˚ ij .k 1 ;k 2 ;k 3 /d k 2 dk 3 (10.33)

10.3.4 Spectral density of the turbulent kinetic energy

According to the definition of the correlation function

R ij .E r/ D u 0 x/u i 0 .E j .E xCE r/

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ED E ECE

the total turbulent kinetic energy is

The quantity

Z E.k/ D ˚ ii .E k/d E k

jE k j

is the spectral density of the turbulent kinetic energy. Physically it is the energy on the sphere k D 2 pk

1 Ck 2 2 Ck 2 3 in the Fourier space. The total turbulent kinetic energy is then:

TKE D E.k/d k

10. 4 Structure fu nctions

10. 4.1 Probability density function

We take the definitions from Wikipedia: In probability theory, a probability density function (pdf), or density of a continuous random variable, is a func- tion that describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variables density over the region. The probability density function is non negative everywhere, and its integral over the entire space is equal to one.

10. 4.2 Structure fu nction

Kolmogorov introduced the structure function of q order for any stochastic function. For instance, for the longitudinal velocity along the longitudinal direction (see Fig. 10.5) it reads:

Figure 10.6: Kurtosis of the structure function for the concentration of the scalar field obtained in the jet mixer. [47].

q .l/ D h.u 2l u 1l / i

2 . If the p.d.f. of the structure function is described by the Gaussian function

2 DS

1 .x

p:d:f:.x/ D p

e (10.39)

In reality, the most of the turbulence parameters are not Gaussian. The deviations from the Gaussian turbulence is characterized by the kurtosis Kurt and skewness Sk. The kurtosis

h.u 2l u 1l / 4 i

Kurt D

.h.u 2l u 1l / 2 i/ 2

is three for the Gaussian turbulence. Big values of the kurtosis means that the p.d.f. distribution of the structure function S 1 .l/ D hu 2l u 1l i is very

is big, it means that the field of the stochastic field is very intermittent. Big differences are possible even if the separation between two points l is small. The p.d.f. function has long tails in this case. A sample of kurtosis for the scalar structure function S.x/ D f .x C r/

f .x/ is given in Fig.10.6. The skewness

.h.u 2l u 1l / 2 3=2

is zero for the Gaussian process. For the isotropic turbulence the skewness of the derivative

@u is equal to 0:5. Physically it means that negative values of the derivative l

@l

are more probable than positive ones. Please prove that the skewness (10.42) is the skewness of the structure functions of the first order S 1 .l/ D hu 2l u 1l i calculated at l ! 0.

Exercise 1 . Calculate the Reynolds averaged values of the time dependent signals

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Exercise 2 . Find the Reynolds stresses for the isotropic turbulence.

Exercise 3 . Calculate the turbulent kinetic energy of the isotropic turbulence, if r 33 D1

Solution:

k D 3=2

Exercise 4 . Relation between longitudinal autocorrelation function

u l .x/u l .x C r/

f .r/ D u 2 l .x/

and energy density E.k/is given by formula [13]

2 2 sin kr

f .r/ D 2 E.k/k r .

cos kr/d k

kr

Energy density of the isotropic decaying turbulence is described as E.k/ D k 4 ExpŒ k 2  . Calculate

longitudinal autocorrelation function f , integral length and Taylor microscale

for the isotropic decaying turbulence. Solution:

2 2 sin kr

f .r/ D

E.k/k r .

cos kr/d k D

kr

2 sin kr

=4 u l

2 ExpŒ k k r .

cos kr/d k D

kr

LD

f .r/dr D

Exercise 5. Calculate the probability density function of the time dependent signal

time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 signal