This document outlines a final project to model open channel flow into a sloped channel with a side basin using the shallow water equations and an upwind numerical scheme on a staggered grid. The geometry, governing equations, numerical method, initial and boundary conditions are presented. Results are shown for the velocity field, vorticity, and free surface oscillations in the basin. Observations include vortex shedding at the basin entrance and resonant transverse oscillations in the basin with a Strouhal number of 1.1.

1. CIVE 572
Final Project
Daniel Robb

2. Outline
• Describe the problem
• Present the theory
• Show the results

3. Problem Statement
Use the upwind scheme and a staggered
grid to calculate open channel flow into
a sloped channel with a side basin.

4. Geometry
L
W
b
b
L b W
4m 1m 2m

5. Geometry
So
1
L
So
0.001

6. Given Parameters
• Initial water depth d = 0.05 m
• Friction coefficient cf = 0.008
• Slope So = 0.001

7. Governing Equations
2D Shallow Water Equations
∂ζ ∂qx ∂qy
∂t + ∂x + h ∂x =0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2
∂qy ∂F vuh ∂F vvh ∂ζ 1 1
∂t + ∂x + ∂y = −gh ∂y + ghSy − 2 cf v(u2 2
+v ) 2
qi qj
Fij =
h

8. Numerical Scheme
Upwind - First Order Accurate
ui = ui + O(∆)
ˆ if ui > 0
ˆ
ui = ui+1 + O(∆)
ˆ if ui < 0
ˆ
ui ui
ˆ ui+1 if ui > 0
ˆ
ui ui
ˆ ui+1 if ui < 0
ˆ

9. Initial Conditions
In the basin:
• Constant water depth: d = 0.05 m
• No initial velocity: u = 0, v = 0

10. Boundary Conditions
At the inlet and outlet:
• Periodic boundary conditions
Along the walls:
• Non-penetrating

11. Periodic BCs
h(0,j) h(1,j) h(imax,j) h(imax+1,j)
h(0, j) = h(imax , j)
h(imax+1 , j) = h(1, j)
qx (0, j) = qx (imax , j)
qx (imax+1 , j) = qx (1, j)
qx(0,j) qx(1,j) qx(imax,j) qx(imax+1,j)

12. Uniform Flow Depth
yo
So
1

13. Governing Equations
2D Shallow Water Equations
∂ζ ∂qx ∂qy
∂t + ∂x + h ∂x =0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2
∂qy ∂F vuh ∂F vvh ∂ζ 1 1
∂t + ∂x + ∂y = −gh ∂y + ghSy − 2 cf v(u2 2
+v ) 2
qi qj
Fij =
h

14. Governing Equations
2D Shallow Water Equations
∂ζ ∂qx ∂qy
∂t + ∂x + h ∂x =0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2
∂qy ∂F vuh ∂F vvh ∂ζ 1 1
∂t + ∂x + ∂y = −gh ∂y + ghSy − 2 cf v(u2 2
+v ) 2
qi qj
Fij =
h

15. Governing Equations
Momentum in x-direction
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2

16. Governing Equations
Momentum in x-direction
0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2

17. Governing Equations
Momentum in x-direction
0 0 0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2

18. Governing Equations
Momentum in x-direction
0 0 0 0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2

19. Governing Equations
Momentum in x-direction
0 0 0 0 0
∂qx ∂F uuh ∂F uvh ∂ζ 1 2 2 1
∂t + ∂x + ∂y = −gh ∂x + ghSx − 2 cf u(u +v ) 2

20. Governing Equations
Momentum in x-direction
1 2
ghS0 = 2 cf u

21. Governing Equations
Momentum in x-direction
1 2
ghS0 = 2 cf u
2ghS0
u= cf
2S0
Fr = cf
F r = 0.5

22. Free Surface

23. Free Surface (Side Basin)

24. Velocity Field

25. Velocity (Magnitude)

26. Vorticity Field
×
ω=∇ u
∂v ∂u
ω= −
∂x ∂y

27. Vorticity Field

28. Basin Resonance
A
1
3
B B’
2
A’

29. Basin Resonance
2Γ k=0
Tn = √
(k + 1) gd
k=1
k=2
Ref: Sorensen, R. M. (2006). Basic coastal engineering, Springer Verlag.

30. Transverse Section (A-A’)
Height (m)

31. Height vs. Time At (1)

32. Height vs. Time At (1)

33. Height vs. Time At (1) (2)

34. Longitudinal Section (B-B’)
Height (m)

35. Height vs. Time At (3)

36. Height vs. Time At (3)

37. Height vs. Time At (3)

38. Observations
• Oscillating flow mechanisms are often
described by a Strouhal Number (S)
fn L
S= U
−1
fn = 0.125s
L = 3m
U = 0.35m/s
S = 1.1

39. Observations
• Vortex shedding at the leading edge of
the basin
• Some vortices recirculate in the basin
while others are entrained by the
main flow.
• Resonant transverse flow oscillations
• As the transverse sloshing increases,
so does the vortex shedding.

40. Questions ???
Thank you for your attention