# CHAPTER 3: THE RANDOM NATURE OF TURBULENCE

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CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 U n m s 1 15 10 5 0 0 20 40 n Sketch of the value U n of the random velocity variable U on the nth repetition of a turbulent flow experiment Figure 3 1 CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 20 a 0 x t 20 20 b x t 0 20 40 c 20 x t x t 0 20 40 0 10 20 30 40 50 60 70 80 90 100 t Time histories from the Lorenz equations Eq 3 1 a x t from the initial condition Eq 3 2 b x t from the slightly different initial condition Eq 3 3 c the difference x t x t Figure 3 2 CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 B V a O V b C V b Va O Vb Sketch of the sample space of U showing the regions to the events a B U Vb and b C Va U Vb Figure 3 3 CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 a F V 1 P C F V b F Va Va 0 Vb V Vb V f V b Va 0 Sketch of a the CDF of the random variable U showing the probability of the event C Va U Vb and b the corresponding PDF The shaded area in b is the probability of C Figure 3 4 CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 F V a 1 b a b V a b V f V 1 b a The CDF a and the PDF b of a uniform random variable Eq 3 39 Figure 3 5 CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 F V 1 a 0 1 V f V b 1 e 0 V The CDF a and PDF b of an exponentially distributed random variable Eq 3 40 Figure 3 6 CHAPTER 3 THE RANDOM NATURE OF TURBULENCE Turbulent Flows Stephen B Pope Cambridge University Press 2000 c Stephen B Pope 2000 F V 1 0 a 0 159 0 023 3 2 1 0 1 2 3 V f V 1 2 2 b 3 2 1 0 1 2 3 V The CDF a and PDF b of a standardized Gaussian