L ehrstuhl für Modellierung und Simulation Statistical theory of the isotropic turbulence (K-41 theory) 2. Kolmogorov theory Lecture 3 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

2 Kolmogorov Theory K41 Andrey Nikolaevich Kolmogorov was a Soviet Russian mathematician, preeminentSovietmathematician in the 20th century, who advanced various scientific fields (among them probability theory,probability theory topologytopology, intuitionistic logic, turbulence,intuitionistic logicturbulence classical mechanicsclassical mechanics and computationalcomputational complexitycomplexity). ( „Every mathematician believes he is ahead over all others. The reason why they don't say this in public, is because they are intelligent people“ UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

3 Physical model beyond the K41 Most important physical processes are Transfer energy from large scales to small ones Transfer energy from large scales to small ones Dissipation of the energy in samll vortices Dissipation of the energy in samll vortices Two parameters are of importance: kinematic viscosity and dissipation rate The size range is referred to as the universal equilibrium range UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

4 Hypothesis of local isotropy Macroscale of the flow, characteristic velocity of macrovortices Kolmogorov‘s hypothesis of local isotropy At sufficiently high Reynolds numbers, the small -scale motion with scales are statistically isotropic. Directional information is lost. The laws describing the small-scale motion are universal. UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

5 Theory of Kolmogorov (1941) K-41 In every turbulent flow at sufficiently high Reynolds number, the statistics of the small-scale motions have a universal form that is uniquely determined by kinematic viscosity and turbulent energy dissipation rate UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

6 Kolmogorov scale, time and velocity UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

Some useful estimations 7 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

8 Distribution of Komogorov scale in jet mixer at Re=10000 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

9 The strongest and simultaneously the most questionable assumption of the Kolmogorov-41: Dissipation rate is an universal constant for each turbulent flow. Comment of Landau (1942): The dissipation rate is a stochastic function, it is not constant. UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

10 Inertial subrange In every turbulent flow at sufficiently high Reynolds number, there is the range of scales l which are small compared with L, however they are large compared with Since the vortices of this range are much larger than Kolmogorov‘s vortices, we can assume that their Reynolds numbers are large and their motion is little affected by the viscosity UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

11 Inertial subrange form UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

12 Interpretation of different subranges Kolmogorov‘s law: UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

13 Power law spectrum of Kolmogorov UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

14 Experimental confirmation Compensated energy spectrum for different flows UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

15 Statistische Auswertung: räumliches Energiespektrum in der J-Mode Measurement of the energy spectrum performed by the LTT Rostock (2007) Concentration of injected liquid UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

16 x/DKolmogorov (-0.03) The slope is between -5/3 and -2 Estimation of the Kolmogorov power in LTT Rostock measurements UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

17 Classification of methods for turbulence modelling RANS Semi-empirical modeling Inertial subrange Large energy containing structures Dissipation range LES Universal modelling DNS UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

18 Classification of methods for turbulence modelling 50 mm Resolution 300 µ 2D 2.72 mm 2.08 mm Kornev N., Zhdanov V. and Hassel E.(2008) Study of scalar macro- and microstructures in a confined jet. Int. Journal Heat and Fluid Flow, vol. 29/3. Kornev N., Zhdanov V. and Hassel E.(2008) Study of scalar macro- and microstructures in a confined jet. Int. Journal Heat and Fluid Flow, vol. 29/3. RANS Semi empiric Model Inertial subrange Large energy containing structures Dissipation LES Universsl Model. DNS Large vortices Middle vortices Small vortices UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

19 Kolmogorov - Obukhov law: Structure functions Kolmogorov - Obukhov law UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

20 Intermittency Discrepancy between measurement and the prediction from the Kolmogorov- Obukhov theory for the exponent of the structure function. The reason of the discrepancy: Intermittency (presence of laminar spots in every turbulent flows even at very high Reynolds numbers). UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

21 Kolmogorov theory K62 Assumption 1 Assumption 2 Lognormal law of Kolmogorov- Obukhov This assumption is proved to be wrong Probability density function distribution for the dissipation rate: UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION