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, 27-28 March 2008

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

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

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)

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)

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

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→

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

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

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→

4. Open channel modes inviscid

4. Open channel modes inviscid viscous

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→

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

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

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

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

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→

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

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

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