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L ehrstuhl für Modellierung und Simulation Statistical theory of the isotropic turbulence (K-41 theory) 1. Basic definitions of the statistical theory of turbulence Lecture 3 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Basic definitions. Reynolds averaging 2 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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3 Basic definitions. Correlation function Correlation function Correlation function in homogeneous turbulence Autocorrelation function Integral length Autocorrelation temporal function Integral time UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Samples 4 50 mm Resolution 300 µ 2D A B C A B C Typical form of the autocorrelation coefficient. Scalar turbulence PLIF Measurements of the LTT Rostock Physical meaning of sign change UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Samples 5 50 mm Resolution 300 µ 2D Typical distribution of the integral length along the jet mixer. Scalar turbulence PLIF Measurements of the LTT Rostock UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Typical autocorrelation coefficient along the jet 6 (from Ginevsky et al. (2004) Acoustic control of turbulent jets. Springer) UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION 1 along upper border of nozzle at x/D=0.5 2 along the jet axis at x/d=3.0 autocorrelation coefficient of the longitudinal velocity

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Isotropic turbulence 7 UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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8 Taylor longitudinal length Taylor transverse length Taylor Reynolds number Isotropic turbulence UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Correlation function in Fourrier space 9 Usually it is possible to measure only the One dimensional spectral Function UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Proof 10UNIVERSITÄT ROSTOCK | LEHRSTUHL FÜR Modellierung und SIMULATION

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Spectral density of the kinetic energy 11 E(k) dk is the contribution of oscillations with the wave numbers k<k<k+dk to the kinetic energy of the turbulent motion. to the kinetic energy of the turbulent motion. E(k) is the density of the kinetic energy depending on wave numbers. The dependence E(k) isreferred to as the energy spectrum UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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