U (n) (m s ) 5 5 2 4 n Figure 3.: Sketch of the value U (n) of the random velocity variable U on the nth repetition of a turbulent flow experiment.
(a) x (t) 2-2 (b) (c) x (t) 2-2 4 2 x (t) x (t) -2-4 2 3 4 5 6 7 8 9 Figure 3.2: Time histories from the Lorenz equations (Eq. (3.)): (a) x(t) from the initial condition Eq. (3.2) (b) ˆx(t) from the slightly different initial condition Eq. (3.3) (c) the difference ˆx(t) x(t). t
B (a) O b C (b) a O b Figure 3.3: Sketch of the sample space of U showing the regions to the events (a) B {U < b }, and (b) C { a U < b }.
(a) F( ) P(C) = F( b ) F( a ) a b (b) f( ) a b Figure 3.4: Sketch of (a) the CDF of the random variable U showing the probability of the event C { a U < b }, and (b) the corresponding PDF. The shaded area in (b) is the probability of C.
(a) F( ) a b (b) b a f( ) a b Figure 3.5: The CDF (a) and the PDF (b) of a uniform random variable (Eq. (3.39)).
F () (a) λ λ f () eλ (b) λ Figure 3.6: The CDF (a) and PDF (b) of an exponentially-distributed random variable (Eq. (3.4)).
. F () (a).59.23-3 -2-2 3 f () (2π) -/2 (b) -3-2 - 2 3 Figure 3.7: The CDF (a) and PDF (b) of a standardized Gaussian random variable.
. F y (y) (a) σ y 2 =5 2 2. 2 3 4 y 2. f y (y) (b). σ y 2 =5 2 2. 2 3 4 y Figure 3.8: The CDF (a) and PDF (b) of the lognormal random variable Y with Y = and var(y ) = 2, 2 and 5.
. F() (a) σ y 2 =5 2 2. 2. f() 2 3 4 (b) σ y 2 =5 2. 2. 2 3 4 Figure 3.9: The CDF (a) and PDF (b) for the gamma distribution with mean µ = and variance σ 2 = 2, 2 and 5.
(a). F( ) p a b (b) f() p p a b Figure 3.: The CDF (a) and the PDF (b) of the discrete random variable U, Eq. (3.69).
. F () (a).5.63-5 -3-3 5 /π f () (b) (2π) - -5-3 - 3 5 Figure 3.: The CDF (a) and PDF (b) for the Cauchy distribution (Eqs. (3.79) and (3.8)) with c =, w =.
f() h -a b Figure 3.2: Sketch of the standardized PDF in Exercise 3.3.
2 4 2 2 4 2 Figure 3.3: Scatter plot in the - 2 sample space of samples of the joint random variables (U, U 2 ). (In this example U and U 2 are jointly normal with U = 2, U 2 =, u 2 =, u 2 2 = 5 6, ρ 2 = / 5.)
2 2 Figure 3.4: The - 2 sample space showing the region corresponding to the event {U <, U 2 < 2 }.
2 2b a b 2a Figure 3.5: The - 2 sample space showing the region corresponding to the event { a U < b, 2a U 2 < 2b }, see Eq. (3.87).
2 5 5 a = b = 5 Figure 3.6: Scatter plot of negatively correlated random variables. ( U =, U 2 =, u 2 = 2, u 2 2 = 2, ρ 2 = 2/3).
f 2 (, 2 ) = b =5 = a =.2. 5 5 5 2 Figure 3.7: Joint PDF of the distribution shown in Fig. 3.6, plotted against 2 for = a = and = b = 5.
2 4 2 2 4 2 Figure 3.8: Scatter plot and constant probability density lines in the 2 plane for joint-normal random variables (U, U 2 ) with U = 2, U 2 =, u 2 =, u 2 2 = 5 6, ρ 2 = / 5.
6 5 U(t) 4 3 2 2 4 6 8 Figure 3.9: Sketch of sample paths of U(t) from three repetitions of a turbulent flow experiment. t
Figure 3.2: Sample paths of five statistically stationary random processes. The one-time PDF of each is a standardized Gaussian. (a) A measured turbulent velocity. (b) A measured turbulent velocity of a higher frequency than that of (a). (c) A Gaussian process with the same spectrum as that of (a). (d) An Ornstein-Uhlenbeck process (see Chapter 2) with the same integral timescale as that of (a). (e) A jump process with the same spectrum as that of (d). CHAPTER 3: THE RANDOM NATURE OF TURBULENCE (a) 2 2 2 3 4 5 6 7 8 9 t (b) 2 2 2 3 4 5 6 7 8 9 t (c) 2 2 2 3 4 5 6 7 8 9 t (d) 2 2 2 3 4 5 6 7 8 9 t (e) 2 2 2 3 4 5 6 7 8 9 t
4..8 U 3. 2..6 var(u).4..2.. 5 Figure 3.2: Mean U(t) (solid line) and variance u(t) 2 of the process shown in Fig. 3.9. t
(a) and (c) ρ(s) -6-4 -2 2 4 6 s (b) ρ(s) -6-4 -2 2 4 6 s (d) and (e) ρ(s) -6-4 -2 2 4 6 s Figure 3.22: Autocorrelation functions of the processes shown on Fig. 3.2. As the inset shows, for processes (a) and (c) the autocorrelation function is smooth at the origin.
E(ω) - -2-3 (b) -4-5 -6-7 (a),(c) (d),(e) -8-2 - 2 Figure 3.23: Spectra of processes shown on Fig. 3.2. ω
(a) 3 2 t 2 3 (c) 3 2 t 2 3 Figure 3.24: Sample paths of Fig. 3.2. Ü(t) for processes (a) and (c) shown on
x 2 h x b x 3 Figure 3.25: Sketch of a turbulent channel flow apparatus.
U/U ref..8.6.4.2...5..5. y/h Figure 3.26: elocity profiles measured by Durst et al. (974) in the steady laminar flow downstream of a symmetric expansion in a rectangular duct. The geometry and boundary conditions are symmetric about the plane y =. Symbols:, stable state ; stable state 2;, reflection of profile about the y axis.