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Chapter 12

Q. 12.4

Consider the autocorrelation function R_{11}(τ)=\overline{u^{2}_{1}}/(1+ω^{2}_{o}\tau^{2}) for the random variable u_{1}(t). What are the integral (Λ_{t}) and Taylor (λ_{t}) length scales, and the spectrum (S_{e}) of u_{1}(t)?

Step-by-Step

Verified Solution

The integral scale can be found directly by substituting the given autocorrelation function into (12.18) and evaluating the integral using an integration variable \beta=\omega_{o}\tau:

\Lambda_{t}\equiv\overset{\infty}{\underset{0}{\int}}r_{11}(\tau)d\tau=(1/R_{11}(0))\overset{\infty}{\underset{0}{\int}}R_{11}(\tau)d\tau,                     (12.18)

\Lambda_{t}=\frac{1}{R_{11}(0)}\overset{+\infty}{\underset{0}{\int}}R_{11}(\tau)d\tau=\frac{1}{\overline{u^{2}_{1}}}\overset{+\infty}{\underset{0}{\int}}\frac{\overline{u^{2}_{1}}}{1+\omega^{2}_{o}\tau^{2}}d\tau=\frac{1}{\omega_{o}}\overset{+\infty}{\underset{0}{\int}}\frac{d\beta}{1+\beta^{2}}=\frac{1}{\omega_{o}}[\tan^{-1}\beta]^{+\infty}_{0}=\frac{\pi}{2\omega_{o}} .

The Taylor scale can be found from (12.19). Here, r_{11}(\tau)=1/(1+\omega^{2}_{o}\tau^{2}), which can be expanded around τ = 0 to find: r_{11}(\tau)=1-\omega^{2}_{o}\tau^{2}+\cdot\cdot\cdot. Thus, [d^{2}r_{11}/d\tau^{2}]_{\tau=0}=-2\omega^{2}_{o}, so (12.19) implies

\lambda^{2}_{t}\equiv-2/[d^{2}r_{11}/d\tau^{2}]_{\tau=0}       (12.19)

\lambda_{t}\equiv(-2/[d^{2}r_{11}/d\tau^{2}]_{\tau=0})^{1/2}=\frac{1}{\omega_{o}}.

As expected, the integral length scale is larger than the Taylor length scale. In high-Reynolds number turbulence, the ratio \Lambda_{t}/\lambda_{t} can be much greater than that found here. The spectrum S_{e}(\omega) is obtained from (12.20):

S_{e}(\omega)\equiv\frac{1}{2\pi}\overset{+\infty}{\underset{-\infty}{\int}}R_{11}(\tau)\exp\left\{-i\omega\tau\right\}d\tau.        (12.20)

S_{e}(\omega)\equiv\frac{\overline{u^{2}_{1}}}{2\pi}\overset{+\infty}{\underset{-\infty}{\int}}\frac{\exp\left\{-i\omega\tau\right\}}{1+\omega^{2}_{o}\tau^{2}}d\tau=\frac{\overline{u^{2}_{1}}}{2\omega_{o}}\exp\left\{-\frac{|\omega|}{\omega_{o}}\right\},

where the integral is readily evaluated using complex contour integration techniques.