Limitation of K-41. Kolmogorov theory K-62

11.6 Limitation of K-41. Kolmogorov theory K-62

The strongest and simultaneously the most questionable assumption of the Kolmogorov-41 is: Dissipation rate is an universal constant for each turbulent flow. Already in 1942, during a scientific seminar the Nobel price laureate

Figure 11 .9: Three main methods of turbulent flows modelling.

Figure 11.10: Vortex structures resolved by different models. Landau noted, that the dissipation rate is a stochastic function, it is not

constant. We consider the consequences of the neglect of this fact. According to the Kolmogorov - Obukhov law the structure function of the q-th order

(11.11) has the following asymptotic behaviour at small l

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Fig. 11.11 shows that the predictions of Kolmogorov and Obukhov deviate from measurement data. The reason of the discrepancy is the physical phe-

nomenon called the intermittency. The intermittency is caused by the pres- ence of laminar spots in every turbulent flows even at very high Reynolds numbers.

Figure 11.11: Power of the structure function. Experiments versus prediction of Kolmogorov and Obukhov. [48].

After the deviation between the K-41 and measurement was documented, Kolmogorov tried to improve his theory. New Kolmogorov theory called as K-62 was published in 1962. New theory is based on two following assumptions:

.ln" a/ 2

P ." 2 l / D ce l2

aD lnN"

l D ln.L= l/

(11.14) q D q.1 q/

Unfortunately, various experiments showed later that the second assumption is proved to be wrong.

11 .6.1 Exercises

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

u.t / D

cos 2 t; u.t / D sin t;

u.t / D

0; t > 1:

So lution:

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

Exercise 3 . Calculate the turbulent kinetic energy of the isotropic turbu- lence, if r 33 D 1

Solution:

k D 3=2

Exercise 4 . Relation between longitudinal autocorrelation function

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

f .r/ D

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

4 k 2 ExpŒ k  . Calculate longitudinal autocorrelation function f ,

integral length and Taylor microscale

for the isotropic decaying turbulence.

So lution:

2 2 f .r/ D sin kr

E.k/k r .

cos kr/d k D

ExpŒ k 2 k . cos kr/d k D =4 u 2