Turbulent Flows. U (n) (m s 1 ) on the nth repetition of a turbulent flow experiment. CHAPTER 3: THE RANDOM NATURE OF TURBULENCE

Transcription

1 U (n) (m s ) n Figure 3.: Sketch of the value U (n) of the random velocity variable U on the nth repetition of a turbulent flow experiment.

2 (a) x (t) 2-2 (b) (c) x (t) x (t) x (t) 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

3 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 }.

4 (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.

5 (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)).

6 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)).

7 . F () (a) f () (2π) -/2 (b) Figure 3.7: The CDF (a) and PDF (b) of a standardized Gaussian random variable.

8 . F y (y) (a) σ y 2 = y 2. f y (y) (b). σ y 2 = 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.

9 . F() (a) σ y 2 = f() (b) σ y 2 = Figure 3.9: The CDF (a) and PDF (b) for the gamma distribution with mean µ = and variance σ 2 = 2, 2 and 5.

10 (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).

11 . F () (a) /π f () (b) (2π) Figure 3.: The CDF (a) and PDF (b) for the Cauchy distribution (Eqs. (3.79) and (3.8)) with c =, w =.

12 f() h -a b Figure 3.2: Sketch of the standardized PDF in Exercise 3.3.

13 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.)

14 2 2 Figure 3.4: The - 2 sample space showing the region corresponding to the event {U <, U 2 < 2 }.

15 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).

16 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).

17 f 2 (, 2 ) = b =5 = a = Figure 3.7: Joint PDF of the distribution shown in Fig. 3.6, plotted against 2 for = a = and = b = 5.

18 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.

19 6 5 U(t) Figure 3.9: Sketch of sample paths of U(t) from three repetitions of a turbulent flow experiment. t

20 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) t (b) t (c) t (d) t (e) t

21 4..8 U var(u) Figure 3.2: Mean U(t) (solid line) and variance u(t) 2 of the process shown in Fig t

22 (a) and (c) ρ(s) s (b) ρ(s) s (d) and (e) ρ(s) s Figure 3.22: Autocorrelation functions of the processes shown on Fig As the inset shows, for processes (a) and (c) the autocorrelation function is smooth at the origin.

23 E(ω) (b) (a),(c) (d),(e) Figure 3.23: Spectra of processes shown on Fig ω

24 (a) 3 2 t 2 3 (c) 3 2 t 2 3 Figure 3.24: Sample paths of Fig Ü(t) for processes (a) and (c) shown on

25 x 2 h x b x 3 Figure 3.25: Sketch of a turbulent channel flow apparatus.

26 U/U ref 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.