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- W1634580374 abstract "We analytically solve the linearized Navier-Stokes equations for streamwise-invariant sinusoidal shear flow. We characterize transient energy growth for this system, a mechanism which may trigger nonlinear effects that lead to sustained turbulence. This includes numerically calculating perturbations which give optimal initial and total energy growth for large enough truncations to capture the behavior of the full system. We also numerically determine Reynolds number scalings and find optimal wavenumbers for maximum transient energy growth. Sinusoidal Shear Flow • Turbulence is found both experimentally and in numerical simulations for values of the Reynolds number well below the value at which the laminar state loses stability [1]. • Transient growth can significantly amplify small perturbations to the laminar state which can trigger nonlinear effects that lead to sustained turbulence. x streamwise y wall normal z spanwise d d 2Lz d 2Lx laminar profile Figure 1: Geometry of Sinusoidal Shear Flow, in which incompressible fluid between two freeslip walls experiences a time-independent body force which varies sinusoidally in the wall-normal direction. • Non-dimensionalized LNSE for streamwise-invariant Sinusoidal Shear Flow ∂u′ ∂t = −v′ dy + 1 Re ( ∂2 ∂y2 + ∂2 ∂z2 ) u′, (1) ∂v′ ∂t = − ∂y + 1 Re ( ∂2 ∂y2 + ∂2 ∂z2 ) v′, (2) ∂w′ ∂t = − ∂z + 1 Re ( ∂2 ∂y2 + ∂2 ∂z2 ) w′, (3) Reynolds number Re = Uod 2ν , laminar profile U(y) = ( √ 2 sin(πy/2), 0, 0). • Continuity equation ∂yv ′ = −∂zw′ (4) • Boundary conditions v|y=±d2 = 0, ∂yu|y=±d2 = ∂yw|y=±d2 = 0 Periodic boundary conditions in x and z directions with lengths Lx and Lz, respectively. • Pressure-free equations ∇2 ′ ∂t = 1 Re ∇4v′ ∂ ∂t ω′ y = − dU dy ∂v′ ∂z + 1 Re ∇2ω′ y ∂2v′ ∂y∂z = − 2w′ ∂z2 ∂ ∂z ω′ y = ∂2u′ ∂z2 (5) where ω′ y = ∂zu′. • To obtain a set of ordinary differential equations, we expand the velocities as Fourier modes and perform a Galerkin projection onto the modes. Dominant Structures" @default.
- W1634580374 created "2016-06-24" @default.
- W1634580374 creator A5077998932 @default.
- W1634580374 creator A5084757263 @default.
- W1634580374 date "2006-11-20" @default.
- W1634580374 modified "2023-09-26" @default.
- W1634580374 title "Transient Growth for the Linearized Navier-Stokes Equations" @default.
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