Computational Aeroacoustics: A Wave Number Approach by Christopher K. W. Tam

By Christopher K. W. Tam

Computational Aeroacoustics (CAA) is a comparatively new learn region. CAA algorithms have built quickly and the equipment were utilized in lots of components of aeroacoustics. the target of CAA isn't just to strengthen computational equipment but in addition to take advantage of those how you can resolve sensible aeroacoustics difficulties and to accomplish numerical simulation of aeroacoustic phenomena. by means of examining the simulation info, an investigator can verify noise new release mechanisms and sound propagation procedures. this is often either a textbook for graduate scholars and a reference for researchers in CAA and as such is self-contained. No earlier wisdom of numerical equipment for fixing PDE's is required, in spite of the fact that, a normal figuring out of partial differential equations and uncomplicated numerical research is thought. routines are integrated and are designed to be a vital part of the bankruptcy content material. furthermore, pattern machine courses are integrated to demonstrate the implementation of the numerical algorithms.

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The governing equations are 1 ∂p ∂u + =0 ∂t ρ0 ∂x 1 ∂p ∂v + =0 ∂t ρ0 ∂y ∂p + γ p0 ∂t ∂u ∂v + ∂x ∂y = 0, where γ is the ratio of specific heats. Upon eliminating u and v, it is easy to find that the governing equation for p is ∂2 p ∂2 p + 2 , ∂x2 ∂y ∂2 p = c2 ∂t 2 c2 = γ p0 . ρ0 (A) y incident wave reflected wave x 0 Consider plane acoustic waves propagating at an angle θ with respect to the y-axis incident on a wall lying on the x-axis. The incident wave is given by iω pincident = Ae− c (x sin θ+y cos θ+ct ) .

Beam equation 4 ∂ 2φ 2∂ φ + μ = 0. ∂t 2 ∂x4 45 46 Finite Difference Scheme as Dispersive Waves c. Linearized Korteweg de Vries equation for water waves ∂φ ∂ 3φ + μ 3 = 0. ∂t ∂x d. Linearized Boussinesq equation for water and elastic waves 2 4 ∂ 2φ 2∂ φ 2 ∂ φ − a − μ = 0. ∂t 2 ∂x2 ∂x2 ∂t 2 Because these equations are linear with constant coefficients, they can be readily solved by Fourier-Laplace transform. The Fourier-Laplace transforms of these equations are (a) (ω2 − c2 α 2 − μ2 )φ˜ = H1 (α, ω).

Now divide the width into N subdivisions and the breadth into M subdivisions so that x = W/N and y = B/M. Suppose a second-order central difference is used to approximate the x and y derivatives. This converts the partial differential equation into a partial difference equation, as follows: uℓ+1,m − 2uℓ,m + uℓ−1,m uℓ,m+1 − 2uℓ,m + uℓ,m+1 ∂ 2 uℓ,m = c2 + , 2 2 ∂t ( x) ( y)2 where ℓ, m are the spatial indices in the x and y directions (ℓ = 0, 1, 2, . . , N; m = 0, 1, 2, . . , M). The boundary conditions for the finite difference equations are at ℓ = 0, ℓ = N, m = 0, m = M, uℓ, m = 0.

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