This document discusses the use of sub-grid sampling (SGS) in 2D hydraulic modeling software to eliminate sensitivity to mesh orientation and size. SGS improves upon conventional approaches that assign single elevations to cells and faces. It describes benchmark tests showing SGS improves accuracy across rotated meshes and mesh sizes. Real-world case studies in Australia and Wales demonstrate SGS provides consistent results for different mesh resolutions. In conclusion, SGS has the potential to improve flood modeling and analysis.

1. Mesh Orientation and Cell Size
Sensitivity in 2D SWE Solvers
Duncan Kitts (Duncan.kitts@bmtglobal.com)
Greg Collecutt
Shuang Gao
Phillip Ryan
Bill Syme
RiverFlow 2020

2. Eliminating Grid Sensitivity in Fixed Grid Solvers
Introduction
 Understanding flood risk increasingly
important
 2-Dimensional hydraulic modelling software
commonly used
 High resolution DTM data becoming
available but computational constraints limit
it’s use
 2D Computational Cell > DTM Cell

3. 2D Hydraulic Modelling
Conventional Approach to assigning cell elevations
• Current Cell Volume and Face Schema
• Cell volume based on single cell centre elevation (or
average)
• Cell face based on elevation of single cell side
elevation point
• Need a grid cell size to reflect underlying
topography
Storage of a flat
square cell
ZU
ZV
ZC
Rectangular
channel to next cell
Rectangular
channel to next cell

4. TUFLOW HPC
Conventional Approach to assigning cell elevations
Cell volume elevation point
Cell face elevation point
Cell Volume / Face Schema

5. TUFLOW HPC
Conventional Approach to assigning cell elevations
Cell Volume / Face Schema
Cell volume elevation point
Cell face elevation point

6. TUFLOW HPC
Cell Resolution Effects
Topography 1m Grid Cell 20m Grid Cell

7. Topography
20m Cell Size with single
cell elevation value
20m Cell Size with sub-grid
sampling

8. TUFLOW HPC
With SGS
Cell volume / elevation curve
Cell face width / elevation curve
Sub-Grid Cell Volume / Face
Schema

9. SGS Benchmarking
Rectangular Channel Test
Numerical experiment set-up
• A rectangular channel with length of 1000m and width of 100m
• Flow rate = 100 m3/s
• Depth = 1 m
• Slope = 0.0009
• Manning’s n = 0.03
Flow
L = 1000 m
W = 100m
Uniform flow of U = 1m/s, d = 1m
Theoretical water level and energy
slope using Manning’s equation
d = 1m

10. Rotated Channel Test
TUFLOW HPC
15° 30° 45°0°
H [m] H [m] H [m] H [m]
The more mis-aligned the mesh is with the flow
direction, the poorer the simulation output
compared to those expected theoretically.

11. Rotated Channel Test
TUFLOW HPC with SGS
15° 30° 45°0°
H [m] H [m] H [m] H [m]
With SGS, simulated output compares well to
theoretically expected output with sensitivity to
mesh orientation eliminated.

12. Mesh Size Sensitivity
TUFLOW HPC
50m
H [m]
With a large mesh size, not a great comparison
with theoretically expected outputs.

13. Mesh Size Sensitivity
TUFLOW HPC
50m25m
H [m]
With a large mesh size, not a great comparison
with theoretically expected outputs.

14. Mesh Size Sensitivity
TUFLOW HPC
50m25m10m
H [m]
As the mesh size is refined, start to converge on
theoretical outputs.

15. Mesh Size Sensitivity
TUFLOW HPC
50m25m10m5m
H [m]
As the mesh size is refined, start to converge on
theoretical outputs.

16. 5m
Mesh Size Sensitivity
TUFLOW HPC SGS
H [m]
With SGS, good convergence with theoretical
outputs….

17. 5m
Mesh Size Sensitivity
TUFLOW HPC SGS
10m
H [m]
With SGS, good convergence with theoretical
outputs….

18. 5m
Mesh Size Sensitivity
TUFLOW HPC SGS
10m25m
H [m]
With SGS, good convergence with theoretical
outputs….for range of mesh cell sizes.

19. 5m
Mesh Size Sensitivity
TUFLOW HPC SGS
10m25m50m
H [m]
With SGS, good convergence with theoretical
outputs….for range of mesh cell sizes.
Eliminates sensitivity to mesh size.

20. Rectangular Channel Test
Deep Sided Channels Unaligned to Mesh
Mesh not aligned
with deep banks
• Distorts streamlines
• Artificial energy losses;
steepens gradient
Traditional Solutions
• 1D channel with cross-section
(time-consuming; full 2D solution compromised)
• Flexible mesh (quadrilaterals aligned with banks)
• Much finer gridded mesh (much longer run times)
• or…

21. Rectangular Channel Test
Deep Sided Channels Unaligned to Mesh
In conclusion:
SGS
• Conforms with Manning’s equation at all orientations
• Resolves limitation of using gridded meshes along deep sided
channels
• Removes requirement for flexible mesh, higher resolution or 1D
representation.
• Significantly reduces mesh sensitivity
• More accurate flood models and outputs



22. SGS Benchmarking
U-Bend Flume Test – Experiment Set-up
• Flume experiment conducted by De Vriend (1978)
Flow
H along outer bank
centre line
inner bank
Flow
• Flow rate = 0.189 m3/s
• Downstream H = 0.18 m
• Manning’s n estimated to be 0.0115 ~ 0.0125
R = 4.25m
W = 1.7m

23. U-Bend Flume Test
TUFLOW HPC 2019 without SGS
• TUFLOW HPC Defaults
H [m]
Use of fixed grid causes edge effects
representing curved wet-dry boundary.

24. U-Bend Flume Test
TUFLOW FV (Flexible Mesh)
• TUFLOW FV
H [m]
Flexible mesh can be aligned with
wet-dry boundary…although can be
time consuming to do so.

25. U-Bend Flume Test
TUFLOW HPC without SGS vs TUFLOW FV
TUFLOW HPC TUFLOW FV
H [m]

26. U-Bend Flume Test
TUFLOW HPC with SGS
• TUFLOW HPC with SGS
H [m]
Partially wet cells/faces smooth out the flow field
SGS approaches eliminates the edge
effects at wet-dry boundary.

27. U-Bend Flume Test
TUFLOW HPC with SGS vs TUFLOW FV
TUFLOW HPC with SGS TUFLOW FV
H [m]
As good as a well-designed
flexible mesh model!!

28. TUFLOW HPC
34cm
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-Without SGS
17cm10cm5cm
H [m]
• Slightly higher than experimental data due to
fixed grid “saw-tooth” effects.

29. TUFLOW HPC
34cm
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-Without SGS
17cm10cm
H [m]
• Slightly higher than experimental data due to
fixed grid “saw-tooth” effects.

30. TUFLOW HPC
34cm
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-Without SGS
17cm
H [m]
• Slightly higher than experimental data due to
fixed grid “saw-tooth” effects.
• Edge effects worsen with increasing mesh
size.

31. TUFLOW HPC
34cm
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-Without SGS
H [m]
• Slightly higher than experimental data due to
fixed grid “saw-tooth” effects.
• Edge effects worsen with increasing mesh
size.

32. TUFLOW HPC with SGS
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-With SGS
H [m]34cm17cm10cm
Good agreement with experimental data…
5cm
H [m]

33. TUFLOW HPC with SGS
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-With SGS
H [m]34cm17cm10cm
Good agreement with experimental data…
H [m]

34. TUFLOW HPC with SGS
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-With SGS
H [m]34cm17cm
H [m]
Good agreement with experimental data across
range of mesh sizes.

35. TUFLOW HPC with SGS
U-Bend Flume Test – Mesh Size Sensitivity
TUFLOW HPC-With SGS
H [m]34cm
Good agreement with experimental data across
range of mesh sizes.
H [m]

36. U-Bend Flume Test
Conclusions
H [m]
In conclusion:
Sub-Grid Sampling...
• Resolves limitation of using gridded meshes along curved channels
• Removes requirement for flexible mesh or higher resolution.
• Significantly reduces mesh sensitivity
• More accurate flood models and outputs

37. Real World Applications
South Johnstone River Catchment
South Johnstone River Case Study
• Whole catchment model
• Direct rainfall
• Historic major event calibration
20km

38. Real World Applications
South Johnstone River Catchment
Flow Calibration
SGS provides more comparable
results across range of mesh sizes
compared to conventional approach

39. Real World Applications
Plynlimon Catchment
Plynlimon Catchment
• Research catchment in mid Wales managed
by Centre of Hydrology, Bangor
• Heavily instrumented with long rainfall and
flow records
• Soils/land cover well mapped
• High resolution LiDAR data for majority of
catchment
Plynlimon TUFLOW Model
• Direct rainfall
• Moderate event

40. Real World Applications
Plynlimon Catchment Application
Plynlimon Catchment Model
• Direct rainfall
• Moderate event
• Compare sensitivity of mesh size to
observed flow data (red line)
With conventional approach:
• Excellent cell size convergence
• Spread with mesh resolution
TUFLOW conventional approach
shows excellent cell size convergence
but show mesh dependency

41. Real World Applications
Plynlimon Catchment Application
Plynlimon Catchment Model
• Direct rainfall
• Moderate event
• Compare sensitivity of grid size to observed
flow data (red line)
With conventional approach:
• Excellent cell size convergence
• Spread with mesh resolution
With SGS approach:
• Limited variation with mesh resolution
SGS provides near identical results
across range of mesh sizes

42. Conclusions
• Sub-Grid Sampling has potential to eliminate sensitivity to both:-
• Mesh Orientation
• Mesh Size
• Benefits seen at range of scales and applications:-
• Whole of catchment modelling
• Urban environments
• Easier Calibration and uncertainty analysis
• Long term continuous simulation can be run rapidly:-
• Natural Flood Management
• Sustainable Urban Drainage Systems
• Improved Forecasting