This document describes an immersed boundary method for modeling bank retreat in Delft3D. The method uses a Cartesian grid to represent migrating bank lines without interpolating cell data or deforming the grid. Details of the numerical approach are presented, including cut-cell treatment for continuity equations and ghost cell treatment for momentum equations. Results are shown for test cases including an annular geometry and Friedkin river erosion test, demonstrating the ability to model bank migration. Next steps include making the method available in the open-source Delft3D code and improving the flooding and drying algorithm.

1. An Immersed Boundary Method for
Modeling Bank Retreat in Delft3D
Delft Software Days 2015
5 November 2015
Aukje Spruyt1, Bert Jagers1, Alberto Canestrelli2,
Willem Ottevanger1
1 Deltares, Delft, Netherlands
2 PennState University, USA / Tidal Forecasting and
Early Warning Center (ICSPM), Venice, Italy

2. How to represent a meandering river?
s
n
Boundary-fitted
curvilinear co-ordinates:
Bank lines on a
rectangular grid:

3. How to represent migrating bank lines?
Adaptive boundary-fitted
curvilinear coordinates
bank
erosion
Disadvantages:
• interpolation of all cell data to new
positions  artificial smoothing
• what about hydraulic structures?
• cell deformation  orthogonality
issues  enhanced truncation
errors

4. How to represent migrating bank lines?
Immersed boundaries
(on curvilinear grids)
bank
erosion
Advantages:
• no interpolation of cell data to
new positions; hydraulic
structures stay where they are
• no time-varying cell deformation
• curvilinear grid might still be
used to minimize grid size and
hence computer time and disk
space

5. How to represent the bank lines?
Lines (Spruyt et al, 2011):
• well defined bank line
• challenges: cut-offs & parallellization
• no flow outside lines
Deep area fraction Fd(i,j):
• dynamic reconstruction of bank lines
• dual bed properties if 0<Fd(i,j)<1
• average properties used during flood

6. How to represent migrating bank lines?
Immersed boundaries
on Cartesian mesh
Advantages:
• no interpolation of cell data to
new positions; hydraulic
structures stay where they are
• no time-varying cell deformation
• easier algorithms for bank line
reconstruction and boundary
treatment at the banks
bank
erosion

7. Immersed Boundary Method
Numerical approach:
• Cut-cell treatment for the continuity equations
• Ghost cell treatment for the momentum equations
• Simple bank-erosion model for the movement of bank lines

8. Immersed Boundary Method
Details:
• Canestrelli et al, 2015.
Int. J. Numer. Meth.
Fluids
DOI: 10.1002/fld.4180

9. Immersed Boundary Method
Results for a 2D annular test case
Also interesting
for various flow-
only applications
Round pillar

10. Simple meandering river

11. Friedkin erosion test
Q = 0.111•10-3 m3/s
i = 0.001
W = 0.05 m

12. Friedkin erosion test – Results

13. Friedkin erosion test – Results

14. A few more settings … time to simplify

15. Next steps
• Make developments available in the standard open-source
Delft3D code (by DSD 2016) & simplifying the use
• Improve flooding-and-drying algorithm
• Apply the method to a realistic river bank erosion case.
• Investigate the implementation of “bank accretion” (how to
transition a cell from low/channel to high/bank – elevation
threshold, vegetation development, …).
• Apply the method to delta simulations.
?
High
Low
Low or High?