Solving the Taylor problem with horizontal viscosity

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Solving the Taylor problem with horizontal
viscosity

• Pieter C. Roos Water Engineering Management,
  University of Twente
• Henk M. SchuttelaarsDelft Institutie of Applied
  Mathematics, TU Delft
• NCK days 2008, Deltares, Delft, 27-28 March 2008

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Contents

• Motivation and goal
• Background inviscid Taylor problem
• Viscous Taylor problem
• Results
• Open channel modes
• Viscous Taylor solution
• Conclusions
• Outlook

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1. Motivation and goal

• Understand morphodynamics of tidal basins
• Tool process-based model for tidal flow
• Smooth flow field required ? add horizontal
  viscosity
• Arbitrary box-type geometries ? Taylor problem

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2. Background inviscid Taylor problem

• Semi-infinite rectangular basin of uniform depth
• No-normal flow BC
• Inviscid shallow water eqs.
• Incoming Kelvin wave

Co-tidal and co-range chart
Tidal current ellipses
Source Taylor (1921)
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2. Background inviscid Taylor problem

• Semi-infinite rectangular basin of uniform depth
• Solution as superposition of open channel modes
• Kelvin Poincaré waves
• Collocation method
• Amphidromic system and tidal current ellipses

Co-tidal and co-range chart
Tidal current ellipses
Source Taylor (1921)
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2. Extending inviscid Taylor problem

• Semi-infinite rectangular basin of uniform depth
• Solution as superposition of open channel modes
• Extension to arbitrary box-type geometries
• Problems for flow field at reflex angle-corners
• Remedy add viscosity

?(x,y,t)
Tidal current ellipses
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3. Viscous Taylor problem

• Geometry and boundary conditions
• Free surface elevation ?, depth-averaged flow
  (u,v)
• No slip at closed boundaries (u,v)0
• Incoming Kelvin wave from x8

y?
Kelvin wave
B
Uniform depth H
x?
x0
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3. Viscous Taylor problem

• Geometry and boundary conditions
• Linearized shallow water equations at O(Fr0)
• g?x ut fv ?uxxuyy
• g?y vt fu ?vxxvyy
• ?t Hux Hvy 0
• Acceleration of gravity g, Coriolis parameter f,
  water depth H, horizontal viscosity ?

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3. Viscous Taylor problem

• Geometry and boundary conditions
• Linearized shallow water equations at O(Fr0)
• Solution method
• Find viscous open channel modes
• Write solution as a superposition of these modes
• Use collocation method to satisfy no slip BC at
  x0

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4. Results open channel modes

• General form ?(x,y,t) Z(y)exp(i?t-kx) c.c.
• Angular frequency ?, (complex) wave number k
• Transverse structure Z(y) Z1e-ay Z2e-ßy
  Z3eay-B Z4eßy-B
• Solvability condition from BCs at y0,B ? k, a,
  ß, Zj

y?
B
Uniform depth H
x?
x0
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4. Open channel modes
inviscid
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4. Open channel modes
inviscid
viscous
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4. Open channel modes
viscous

• Viscous Kelvin and Poincaré modes
• Boundary layers at y0,B
• Interior structure similar to inviscid case
• Viscous dissipation, slight decrease in length
  scales

y?
B
Uniform depth H
x?
x0
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4. Viscous Kelvin mode
viscous
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4. Viscous Poincaré modes
viscous
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4. Viscous Poincaré modes
viscous
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4. New modes
viscous
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4. Viscous Taylor solution

• Truncated superposition of open channel modes
• Incoming Kelvin wave and 2N1 reflected modes
• Use collocation method to satisfy no-slip BC at
  x0
• N1 points where u0 and N points where v0

y?
Kelvin wave
x?
x0
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4. Viscous Taylor solution
viscous
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5. Conclusions

• Taylor problem has been extended to account for
  horizontally viscous effects
• No-slip condition at closed boundaries
• Solution involves viscous open channel modes
• Viscous Kelvin and Poincaré modes
• A new type of mode arises, responsible for the
  transverse boundary layer at x0

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6. Outlook

• Details of collocation method
• Residual flow and higher harmonics
• Nonlinear M2-interactions at O(Fr1) ? M0, M4
• Geometrical extension of viscous model
• To arbitrary box-type geometries ? smooth flow
  field
• Applications artificial islands, inlets,
  obstructions