CIVE 572Final Project

Daniel Robb

Outline

• Describe the problem

• Present the theory

• Show the results

Problem Statement

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

GeometryL

b

L b W4 m 1 m 2 m

b

W

Geometry

L

So

0.001

So 1

Given Parameters

• Initial water depth d = 0.05 m

• Friction coefficient cf = 0.008

• Slope So = 0.001

Governing Equations2D Shallow Water Equations

Fij =qiqjh

∂ζ∂t +

∂qx∂x + h∂qy

∂x = 0

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

∂qy∂t + ∂Fvuh

∂x + ∂F vvh

∂y = −gh ∂ζ∂y + ghSy − 1

2cfv(u2 + v2)

12

Numerical SchemeUpwind - First Order Accurate

if

if

ui = ui +O(∆)

ui = ui+1 +O(∆)

ui > 0

ui < 0

uiui ui+1

uiui ui+1

if

if

ui > 0

ui < 0

Initial Conditions

In the basin:

• Constant water depth: d = 0.05 m

• No initial velocity: u = 0, v = 0

Boundary Conditions

At the inlet and outlet:

• Periodic boundary conditions

Along the walls:

• Non-penetrating

Periodic BCsh(0,j) h(1,j) h(imax,j) h(imax+1,j)

qx(0,j) qx(1,j) qx(imax,j) qx(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)

Uniform Flow Depth

So 1

yo

Governing Equations2D Shallow Water Equations

Fij =qiqjh

∂ζ∂t +

∂qx∂x + h∂qy

∂x = 0

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

∂qy∂t + ∂Fvuh

∂x + ∂F vvh

∂y = −gh ∂ζ∂y + ghSy − 1

2cfv(u2 + v2)

12

Governing Equations2D Shallow Water Equations

Fij =qiqjh

∂ζ∂t +

∂qx∂x + h∂qy

∂x = 0

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

∂qy∂t + ∂Fvuh

∂x + ∂F vvh

∂y = −gh ∂ζ∂y + ghSy − 1

2cfv(u2 + v2)

12

Governing EquationsMomentum in x-direction

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

Governing EquationsMomentum in x-direction

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

0

Governing EquationsMomentum in x-direction

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

0 0 0

Governing EquationsMomentum in x-direction

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

0 0 0 0

Governing EquationsMomentum in x-direction

∂qx∂t + ∂Fuuh

∂x + ∂Fuvh

∂y = −gh ∂ζ∂x + ghSx − 1

2cfu(u2 + v2)

12

0 0 0 0 0

Governing EquationsMomentum in x-direction

ghS0 = 12cfu

2

Governing EquationsMomentum in x-direction

ghS0 = 12cfu

2

u =�

2ghS0

cf

Fr =�

2S0cf

Fr = 0.5

Free Surface

Free Surface (Side Basin)

Velocity Field

Velocity (Magnitude)

Vorticity Field

�ω = �∇× �u

ω =∂v

∂x− ∂u

∂y

Vorticity Field

Basin Resonance

3

1

2

A

A’

B’B

Basin Resonance

Tn =2Γ

(k + 1)√gd

Ref: Sorensen, R. M. (2006). Basic coastal engineering, Springer Verlag.

k = 0

k = 1

k = 2

Transverse Section (A-A’)H

eigh

t (m

)

Height vs. Time At (1)

Height vs. Time At (1)

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

Longitudinal Section (B-B’)H

eigh

t (m

)

Height vs. Time At (3)

Height vs. Time At (3)

Height vs. Time At (3)

Observations• Oscillating flow mechanisms are often

described by a Strouhal Number (S)

S = fnLU

fn = 0.125s−1

L = 3mU = 0.35m/sS = 1.1

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.

Questions ???

Thank you for your attention