Numerical Simulation of N-S equations in Cylindrical Coordinate

Numerical Simulation of N-S equations in Cylindrical Coordinate

Numerical Simulation of N-S equations in Cylindrical Coordinate Kyongmin Yeo Introduction Flows in Annular pipe

Chung & Sung 2005 Centrifugal Instability Basic Instability pattern in geophysical flows Engineering Application

Couette flow Flow around cylinder Marcus & Lee 1998 Governing Equation Continuity equation

Momentum equation Spatial Discretization Spectral Method in and z directions Spectral Element Method in r direction Discretized equations Temporal Discretization: Nonlinear step

Higher-order dual splitting scheme Nonlinear step Pressure step Galerkin approximation

Pressure Step Pressure boundary condition Velocity correction Viscous step

Galerkin approximation Fourier Transform 3/2 rule is used for dealiasing 2-dimensional backward transform

complex to complex transform in direction complex to real transform in z-direction cpu 1 2 3 cpu 1

2 3 Computation Mesh -2 -1 0

Z 4 2 1 -2 -1

X 0 0 2 2 1 Y

2 0 Z 1 5 0 0

0.5 1 Y 1.5 2 X

6 Centrifugal Couette Flow Taylor-Couette Instability Azimuthal Vorticity at Re=500 Azimuthal Vorticity Azimuthal Vorticity at Re=1000

Azimuthal Vorticity Couette flow with Moving wall sin(t ) Re = 400 Z X

Y 3 V3 2 1 0 -2

-2 -1 -1 Y 0 0 1

1 2 2 X Oscillating Wall sin( t )

Stokes Layer (2 / )1/ 2 ~ Re 1/ 2 Z - Vorticity at Re = 400 & 800 Flow around a cylinder Y Z 10

Stokes flow solution X 10 5 0 0 -5

-5 -10 -10 -5 0 X

5 10 2 0 V3 Y Y

5 -10 -5 0 X 5

10 Flow around a cylinder at Re = 10 Flow around a cylinder at Re = 200 Thank you!

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