Viscous Linear Instability of an Incompressible Round Jet T.M. ...

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2.3 Stability Equations The fluctuating variables are assumed to Fourier decompose into complex components typically of the form ˆθ(r,φ,z,t) = Re[θ(r)e i(αz−ωt+nφ) ] where α is the complex wavenumber, ω is the real radian frequency (spatial stability analysis) and n is the azimuthal mode number. Substitution for the variables û, ˆv, ˆw and ˆp gives: Continuity Equation r - Momentum Equation iαu + v ′ + v + inw r r = 0, (2.13) iα( Ū − c)v + p′ = 1 v R ′′ + v′ r − α 2 + (n2 + 1) r2 2n v − i w , (2.14) r2 φ - Momentum Equation in 1 iα( Ū − c)w + p = w r R ′′ + w′ r − α 2 + (n2 + 1) r2 2n w + i v r2 z - Momentum Equation iα( Ū − c)u + Ū ′ v + iαp = 1 u R ′′ + u′ r − α 2 + n2 r2 u where c = ω/α and primes denote differentiation with respect to r. 7 (2.15) (2.16)
In order to solve the linear system we must define the boundary conditions. According to Morris[3] these boundary conditions at the centreline of the jet depend on the mode n. Also, in the far field, the boundary conditions are that θ → 0 when r → ∞. In particular for each mode we must imple- ment the following boundary conditions once our system has been defined and before the eigenvalue problem is solved. u(0) = p(0) = 0, n = 0, v(0) = w(0) = 0, n = 1, v(0) + iw(0) = 0, n = 1. 8 ⎫ ⎪⎬ ⎪⎭ (2.17)
Page 1 and 2: Viscous Linear Instability of an In
Page 3 and 4: INSTITUTE OF SOUND AND VIBRATION RE
Page 5 and 6: 1 Introduction The main objective o
Page 7 and 8: 2 Theory In this section we develop
Page 9: 2.2 Linearized Navier-Stokes Equati
Page 13 and 14: 2.5 Asymptotic solutions of the sta
Page 15 and 16: 2.5.2 Far away from the axis of the
Page 17 and 18: 3 Numerical Scheme The equations de
Page 19 and 20: to r = rs, compute detF at rs and f
Page 21 and 22: method converges well for low Reyno
Page 23 and 24: 4.2 Critical Reynolds Number To obt
Page 25 and 26: 5 Summary • The linear stability
Page 27 and 28: References [1] V.K. Garg and W.T Ro
Page 29 and 30: Laplacian of a Vector, ∇ 2 V ∇
Page 31 and 32: From the above expressions for F an
Page 33: Components w ′ i w ′ 1 = − 1

Viscous Linear Instability of an Incompressible Round Jet T.M. ...

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