Question #239711

QUESTION 8 Given B 2 31 2 21 1 32 find the eigenvalues of B and an eigenvector for B corresponding to 1. (10) [10] QUESTION 9 A function f(x) is defined by: 2 fx x x , Determine the Fourier expansion of the periodic function f(x) with period 2 . (12) [12]

Expert's answer

(8)

In order to determine the eigenvectors of a matrix, we must first determine the eigenvalues. Substitute one eigenvalue λ into the equation *A* **x** = λ **x**—or, equivalently, into ( *A* − λ *I*) **x = 0**—and solve for **x**; the resulting *nonzero* solutons form the set of eigenvectors of *A* corresponding to the selectd eigenvalue. This process is then repeated for each of the remaining eigenvalues.

[Note that these equations are not independent. If they *were* independent, then only ( x _1, x _2)^ T = (0, 0)^ T would satisfy them; this would signal that an error was made in the determination of the eigenvalues. If the eigenvalues are calculated correctly, then there *must*be nonzero solutions to each system.

(9)

The pseudospectral **Fourier expansion** method was used for both DNS and LES. To generate the DNS data, 256, 256, and 256 grid points were used in the *x, y* and *z* directions, respectively. Periodic boundary conditions were imposed in the three directions. The size of the computational domain was *2π* in each direction, the kinematic viscosity, ν, was set equal to 0.00014, and the time interval, Δ*t*, was set equal to 0.0005. The angular velocity for the rotation, Ω, was chosen equal to 10.0. Rotation was applied to the DNS data generated with Ω= 0.0 at *t*= 2.0 when the initial transient period elapsed. The value of *Re*_{λ} (Reynolds number based on the Taylor microscale and root-mean-square value of the velocity fluctuation) at the initial instant (*t* = 2.0) was 72.2. The initial values used in the LES computations were those obtained by filtering the DNS data at *t* = 2.0, where 32, 32 and 32 grid points were used, respectively, in the *x, y*, and *z*directions. The same computational domain size was used, and the kinematic viscosity, ν, was set equal to that used for DNS. The calculated results were directly compared with the filtered DNS results.

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