Linearisation and Transformation to Frequency Domain

The governing flow equation is transformed to the frequency domain by assuming that the prescribing motion of grid points is harmonically oscillating about their steady-state positions

x = x + 3?{xeJaJt} (4)

and the resulting unsteady cell-averaged perturbations are also harmonically oscillating about its steady-state values

U = U + 5R{UeJaJt}. (5)

The discretised time derivative of the fbw solution, R (Eqn. 3) is linearised by performing a first-order Taylor expansion about the steady-state flow values, steady-state grid position, and zero grid velocity.

Note that it is assumed that the time-derivative of the fbw solution at the steady-state condition,

R = R(U, x, 0) = 0. (8)

Substituting Eqs. 7, 4, 5 into Equation 6 and dropping the e]wt term gives the following result

AR(U, x + x, 0) + j AR(U, x, ш x) (9)

where

The right hand side of the linearised fbw equation can be easily determined by calculating the change in R from the steady-state due to perturbing the grid by the amplitude of the grid motion and also calculating the change in R due to setting the grid velocity equal to the amplitude of the maximum velocity of the grid motion. The partial derivative of R with respect to the ft>w solution required on the left hand side of the equation is equal to

The Jacobians required are calculated numerically and the Jacobian of the fhx calculation includes the action of MUSCL and the limiter. Note that some of the flix schemes (AUSMDV and the exact Riemann solver) used here are not strictly differentiable for some input ft>w states. However, for the cases to be presented it was possible to obtain accurate solutions using numerically calculated derivatives of these fluxes.