Mathematical Physics - Volume II - Numerical Methods

5.9 Tensile Instability

179

When perturbations ¯ x = x + ˜ x are introduced, the smoothing function values change as: W ( ¯ x I − ¯ x J , h ) = W (( x I + ˜ x I ) − ( x J + ˜ x J ) , h )) (5.68) with ˜ x I = u I and ˜ x J = u J equation (5.69) can be rewritten as W (( x I + ˜ x I ) − ( x J + ˜ x J ) , h ) = W (( x I − x J )+( u I − u J ) , h ) . (5.69) Using Taylor’s series expansion yields: W ( ¯ x I − ¯ x J , h ) = W ( x I − x J , h )+ ∆ xW ′ ( x I − x J , h ) (5.70) where ∆ x = u I − u J . Similarly, the derivative of the kernel function in equation (5.70) can be approximated as: W ′ ( ¯ x I − ¯ x J , h ) = W ′ ( x I − x J , h )+ ∆ xW ′′ ( x I − x J , h ) . (5.71) Hence W ′ ( ¯ x I − ¯ x J , h ) − W ′ ( x I − x J , h ) =( ˜ u I − ˜ u J ) W ′′ ( x I − x J , h ) = = ˜ W ′ ( x I − x J , h ) . (5.72) Subtracting equation (5.60) from (5.67) yields: m I ¨˜ u I = − ∑ J ∈ S m J ρ 0 J W ′ I ( ¯ x I − ¯ x J , h )( σ J F J + σ J ˜ F J + ˜ σ J F J )+ = + ∑ J ∈ S m J ρ 0 J W I ( x I − x J , h ) σ J F J . (5.73) Equation (5.73) after rearranging becomes: m I ¨˜ u I = − ∑ J ∈ S m J ρ 0 J − W ′ I ( ¯ x I − ¯ x J , h )+ W I ( x I − x J , h ) σ J F J − = − ∑ J ∈ S m J ρ 0 J ( ¯ x I − ¯ x J , h ) ( σ J ˜ F J + ˜ σ J F J ) . (5.74) And after substituting equation (5.72) into (5.74) one gets: m I ¨˜ u I = − ∑ J ∈ S m J ρ 0 J h ˜ W ′ I ( ˜ x I − ˜ x J , h ) σ J F J + W ′ I ( ¯ x I − ¯ x J , h ) σ J ˜ F J + ˜ σ J F J i . (5.75) To perform the Von Neumann stability analysis a Fourier form of perturbation was substi tuted into the linearised momentum equation

i ( ¯ ω t + kX )

˜ u = u 0 e

(5.76)

where k is wave number and ω is frequency.