Lecture Dec 10, '93 3D Boundary Layer A) Complex Instability B) Transition Mechanisms Mick, AGARD-R-709 June 1984. Reading: R.S.R. 342-344, Saric 1989 Annual Review 1989 A> Recall, for equilibrium flows, we can add small perturbations, linearize and incompressible, inviscid NS \[\frac{D\vec{u}}{Dt}=-\nabla p+\frac{1}{Re}\nabla^{2}\vec{u}\] \[\nabla\cdot\vec{u}=0\] p = p0 + p' Subst. and neglecting higher powers gives linearized PDE for disturbance Asymptotic Disturbance: \[\vec{u}=\vec{\hat{u}}(y)e^{i(\alpha x+\beta z-\omega t)}\] \[\hat{p}=\hat{p}(y)e^{i(\alpha x+\beta z-\omega t)}\] iα + iβ general wave number vector -iωû + iαÛU + iβŴW + U'û = -iαp + 1/Re [D² - α² - β²]û -iωũ + iαŨU + iβŴW + U'v = -iαp + 1/Re[D² - α² - β²]û -iωŵ + iαŨŵ + iβŴW + W'v = -iβp + 1/Re[D² - α² - β²]ŵ -iω(αũ + βŵ) + i(αŨU + βŴW) + U'v + W'v = -i(αp + βp) + 1/Re[D² - α² - β²](αũ + βŵ) Add (1) + (3) we get. iα'(U' - c)v + V'v = -iα'p + 1/Re[D² - α'²]v iα'(U' - c)v + V'v = -iα'p + 1/Re(D² - α'²)v 2nd order equation for v with 2 boundary conditions 30 inputs: U(y), W(y), Re Inputs: α, β, ω or α, β, ω specified (x-spatial) (z-spatial) Eigenvalue problem: if α, β specified (temporal) if ω, β specified (spatial) 2D result is W = 0 that X aligned with ye Simplest is U(ye) = uc W(ye)=0 Fit in the framework of G = 0 approximations also known as Barler Transition Mech: Gortler, not necessarily: Gortler a1 < 0 (T-S wave) β1 < 0 (crossflow wave) Calculations: For Uod - Res > Rent; a1 < 0 promoted by inflexion U(y) (optimum σ) For Uod - Res < Rent; β1 < 0 promoted by inflexion W(y) (always!!) Example: Johnston W W = sin(πy) tanφ = (1-y²) tanp u" + (1-y²) tan²p sin² - (1-y²) tanp

16.13 Aerodynamics of Viscous Fluids: Lecture 36 Notes

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