1. Solving the Taylor problem with horizontal viscosity
Pieter C. Roos Water Engineering & Management, University of Twente
Henk M. Schuttelaars Delft Institutie of Applied Mathematics, TU Delft
NCK days 2008, Deltares, Delft, March 2008

2. Contents Motivation and goal Background: inviscid Taylor problem
Viscous Taylor problem
Results
Open channel modes
Viscous Taylor solution
Conclusions
Outlook

3. 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

4. 2. Background: inviscid Taylor problem
Co-tidal and co-range chart
Semi-infinite rectangular basin of uniform depth
No-normal flow BC
Inviscid shallow water eqs.
Incoming Kelvin wave
Tidal current ellipses
Source: Taylor (1921)

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

6. 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
ζ(x,y,t)
|u|(x,y)
Tidal current ellipses

7. 3. Viscous Taylor problem
Geometry and boundary conditions
Free surface elevation ζ, depth-averaged flow (u,v)
No slip at closed boundaries: (u,v)=0
Incoming Kelvin wave from x=+∞ y↑
Kelvin wave B
Uniform depth H
x=0 x→

8. 3. Viscous Taylor problem
Geometry and boundary conditions
Linearized shallow water equations – at O(Fr0)
gζx + ut – fv = ν[uxx+uyy]
gζy + vt + fu = ν[vxx+vyy]
ζt + [Hu]x + [Hv]y = 0
Acceleration of gravity g, Coriolis parameter f, water depth H, horizontal viscosity ν

9. 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 x=0

10. 4. Results: open channel modes
General form: ζ(x,y,t) = Z(y)exp(i[ωt-kx]) + c.c.
Angular frequency ω, (complex) wave number k
Transverse structure: Z(y) = Z1e-αy + Z2e-βy + Z3eα[y-B] + Z4eβ[y-B]
Solvability condition from BCs at y=0,B  k, α, β, Zj y↑ B
Uniform depth H
x=0 x→

11. 4. Open channel modes
inviscid

12. 4. Open channel modes
inviscid
viscous

13. 4. Open channel modes Viscous Kelvin and Poincaré modes
Boundary layers at y=0,B
Interior structure similar to inviscid case
Viscous dissipation, slight decrease in length scales y↑ B
Uniform depth H
x=0 x→

14. 4. Viscous Kelvin mode
viscous
ζ(x,y,t)
u(x,y,t)
v(x,y,t)

15. 4. Viscous Poincaré modes
ζ(x,y,t)
u(x,y,t)
v(x,y,t)

16. 4. Viscous Poincaré modes
ζ(x,y,t)
u(x,y,t)
v(x,y,t)

17. 4. New modes
viscous
ζ(x,y,t)
u(x,y,t)
v(x,y,t)

18. 4. Viscous Taylor solution
Truncated superposition of open channel modes
Incoming Kelvin wave and 2N+1 reflected modes
Use collocation method to satisfy no-slip BC at x=0
N+1 points where u=0 and N points where v=0 y↑
u=0
v=0
Kelvin wave
x=0 x→

19. 4. Viscous Taylor solution
ζ(x,y,t)
u(x,y,t)
v(x,y,t)

20. 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 x=0

21. 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