By Jack B. Evett, Cheng Liu
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Extra info for 2500 Solved Problems in Fluid Mechanics and Hydraulics (Schaum's Solved Problems Series)
16) and averaging gives u(t + ∆t)2 = r2u(t)2 + s2 ξ 2 + 2r s u(t)ξ(t). 18) The last term is zero, as already explained, so the conditions u(t + ∆t)2 = u(t)2 = u2 and ξ 2 = 1 give s= (1 − r2)u2 . 16) is multiplied by u(t) and averaged, an equation for the auto-covariance at the small time separation ∆t is obtained: u(t + ∆t)u(t) = ru2. 19) It can be seen from this that r is the correlation coefficient of u with itself at two times. 19) r is only a function of ∆t. The precise form of R(τ ) is determined by the function chosen for r.
Since ξ(t) is chosen completely independent of u(t), they are uncorrelated: ξ(t)u(t) = ξ(t) u(t) = 0 × 0. 16) ∗ The Lagrangian approach is to describe fluid motion in a frame moving with fluid elements; the more familiar, Eulerian approach adopts a fixed frame. 26 MATHEMATICAL AND STATISTICAL BACKGROUND an example of a Markov chain. 17) for this stochastic process. 16). If u(t) is a statistically stationary process then its variance should not depend on time; in particular u(t + ∆t)2 = u(t)2 = u2.
I1122 = I1133 = I2233, etc. 35) these are all equal to A. 35). ‡ § The transformation matrix for reflection in the y − z plane is E = diag[−1, 1, 1]. If the reflection invariance is not added, then the skew symmetric tensor εijk is isotropic. This tensor is defined by εijk = 0 if any of i, j or k are equal, εijk = −εjik and ε123 = 1. 2 35 Tensor Functions of Tensors; Cayley-Hamilton Theorem Second moment closure modeling requires that certain unknown tensors be represented by functions of anisotropy and identity tensors.
2500 Solved Problems in Fluid Mechanics and Hydraulics (Schaum's Solved Problems Series) by Jack B. Evett, Cheng Liu