2.2 THREE APPROACHES TO SOLVE FOR NORMAL … · 27/02/2003 · • With the assumptions made, the Shallow Water equations simplify to ... • Applying the Bernoulli equation between

CE 344 - Topic 2.2 - Spring 2003 - February 27, 2003 9:07 pm

p. 2.2.1

2.2 THREE APPROACHES TO SOLVE FOR NORMAL (UNIFORM) FLOW DEPTH IN A WIDE RECTANGULAR CHANNEL

Derivation of Normal Flow Depths by Applying the Shallow Water Equations

• Consider the stretch of channel shown:

• The geoid is defined at z=0, the free surface (water-air interface) at z= , and the bottom(water-sediment interface) at z=-h.

η


p. 2.2.2

• We assume the following:

• Steady state flow

• A wide channel of flow depth over a channel bed with slope

• Steady flow in the cross channel averaged variables in the x (almost identical to the s)direction. Therefore

• Uniform flow in the x-direction: constant in the x-direction and

• No cross channel gradient or flow: ;

• No applied free surface stress

• Flow depth is constant and set to along the entire section ofinterest.

∂∂t---- 0→

d0 θ0

u 0>

u∂u∂x------ 0=

∂η∂y------ 0= v 0=

τxs

0; τys

0==

H η h+= H d0=


p. 2.2.3

• We now consider the Shallow Water equations:

(2.2.1)

(2.2.2)

(2.2.3)

(2.2.4)

(2.2.5)

∂η∂t------

∂ uH( )∂x

---------------∂ vH( )

∂y--------------- 0=+ +

∂u∂t------ u

∂u∂x------ v

∂u∂y------ g

∂η∂x------– +=+ +

1H----

∂∂x-----

τxxt m⁄

ρ---------- u

2–� ��

zd

h–

η

�1H----

∂∂y-----

τyxt m⁄

ρ---------- uv–� ��

z1H----

τxs

ρ----

1H----

τxb

ρ-----–+d

h–

η

�+

∂v∂t----- u

∂v∂x----- v

∂v∂y----- g

∂η∂y------– +=+ +

1H----

∂∂x-----

τxyt m⁄

ρ---------- uv–� ��

zd

h–

η

�1H----

∂∂y-----

τyyt m⁄

ρ---------- v

2–� ��

z1H----

τys

ρ----

1H----

τyb

ρ-----–+d

h–

η

�+


p. 2.2.4

• With the assumptions made, the Shallow Water equations simplify to

(2.2.6)

(2.2.7)

(2.2.8)

• We are going to assume that the lateral dispersion/diffusion terms simplify as follows:

• No cross channel depth averaged currents or variations in cross channel currents;

(2.2.9)

• Since all the laminar-turbulent-depth averaged shear stress terms will all be depen-dent on gradients in the mean flow field in the x and y directions (e.g. like the EddyDispersion relationships in equations (1.5.73) through (1.5.75), the shear stresses inthe xx, xy and yy planes/directions are negligible.

0 0=

0 g∂η∂x------–

1H----

∂∂x-----

τxxt m⁄

ρ---------- u

2–

� ��

z1H----

∂∂y-----

τyxt m⁄

ρ---------- uv–� ��

z1H----

τxb

ρ-----–d

h–

η

�+d

h–

η

�+=

01H----

∂∂x-----

τxyt m⁄

ρ---------- uv–� ��

z1H----

∂∂y-----

τyyt m⁄

ρ---------- v

2–

� ��

zd

h–

η

�+d

h–

η

�=

v v 0= =


p. 2.2.5

• With these assumptions the governing equations simplify to:

(2.2.10)

(2.2.11)

(2.2.12)

• Thus the free surface gradient (gravity) term balances the bottom friction term:

(2.2.13)

• We note that for steady uniform flow:

(2.2.14)

• Also we showed that for uniform steady flow:

. (2.2.15)

0 0=

0 g∂η∂x------

1H----

τxb

ρ-----––=

0 0=

g∂η∂x------

1H----

τxb

ρ-----–=

∂η∂x------ free surface gradient = = Sw–

Sw S0=


p. 2.2.6

• Thus

(2.2.16)

• The x-direction momentum equation becomes:

(2.2.17)

• Re-arranging

(2.2.18)

• Noting that and that the total water column remains constant at the normal depth:

(2.2.19)

∂η∂x------ S0–=

gS0–1H----

τxb

ρ-----–=

τxb ρgHS0=

γ ρg≡H d0=

τxb γd0S0=


p. 2.2.7

• Now we must apply a constitutive relationship to relate bottom stress to the depth aver-aged flow field.

(2.2.20)

• Thus the momentum statement becomes:

(2.2.21)

• Solving for

(2.2.22)

• For a Darcy-Weisbach bottom friction closure:

(2.2.23)

• Thus

(2.2.24)

τxb

ρ----- cfu

2=

gS0d0 cfu2

=

u

ugd0S0

cf--------------=

cf

fDW

8---------=

u8gd0S0

fDW------------------=


p. 2.2.8

• In terms of flow rate per unit width:

(2.2.25)

• Given a flow per unit width , the normal flow depth can be computed as:

(2.2.26)

• Thus from a force balance perspective, we can characterize steady uniform flow asfollows:

• For a given flow rate per unit width, , the normal depth will have a velocityassociated with it which causes sufficient boundary shear resistance together withbottom roughness to balance the gravity forcing component.

qx ud0

8gd0S0

fDW------------------d0= =

qx

d0

qx2fDW

8gS0---------------� ��

23---

=

qx d0


p. 2.2.9

Derivation of Normal Flow Depth by Applying the Bernoulli Equation

• We consider the same stretch of channel with steady uniform flow as in the previousderivation between 2 sections.

• Based on these assumptions in the previous section listed on page 2.2.2:

(2.2.27)

(2.2.28)

• Also, we note that pressure at the free surface is atmospheric:

(2.2.29)

d1 d2 d0= =

u1 u2 u= =

p1 p2 0= =


p. 2.2.10

• Applying the Bernoulli equation between sections 1 and 2 along a streamline on the freesurface:

(2.2.30)

(2.2.31)

(2.2.32)

• Dividing by L, the distance between points 1 and 2 in the x-direction

(2.2.33)

• We note that

(2.2.34)

p1

γ----- z1

u12

2g------

p2

γ----- z2

u22

2g------ hL1 2–

+ + +=+ +

0 zb1d0

u2

2g------+ + + 0 zb2

d0u

2

2g------ hL1 2–

+ + + +=

hL1 2–zb1

zb2–=

hL1 2–

L------------

zb1zb2

–

L-------------------=

S0

zb1zb2

–

L-------------------=


p. 2.2.11

• Thus

(2.2.35)

• If we consider the head loss formula for pipe flow (a constitutive-type relationship):

(2.2.36)

• We now modify this relationship to apply to open channel flow by replacing by theHydraulic Diameter :

(2.2.37)

• For a wide open channel . In this case since the depth is the normaldepth.

(2.2.38)

hL1 2–

L------------ S0=

hL1 2–

fDWu2Ls

Dpipe2g--------------------=

Dpipe

DH

hL1 2–

fDWu2Ls

DH2g--------------------=

DH 4d= DH 4d0=

hL1 2–

fDWu2Ls

8d0g--------------------=


p. 2.2.12

• Substituting for into our simplified Bernoulli equation:

(2.2.39)

• Noting that cos , we solve for .

(2.2.40)

• This solution is identical to that obtained from the Shallow Water equation derivation.

• This is logical since both the Shallow Water equation and the Bernoulli equationwere derived from the Navier-Stokes equation.

• Compatible constitutive relationships were used in both the Shallow Water andBernoulli equations. Both were derived from formulae developed for pipe flow.

• Thus, from a mechanical energy perspective, we can characterize steady uniform flow asfollows:

• The drop in potential energy due to the fall in elevation of the channel bottom mustbe exactly consumed by the energy dissipation through bottom friction and turbu-lence.

hL1 2–

fDWu2Ls

8d0g--------------------

1L--- S0=⋅

θ0LLs----- 1≅= u

u8gd0S0

fDW------------------=


p. 2.2.13

Derivation of Normal Flow Depth by Applying Conservation of Momentum to a Fi-nite-Sized Control Volume

• We consider a finite sized control volume oriented along the channel bottom:

• Again we consider the same stretch of channel with steady uniform flow as in theprevious 2 derivations using the same assumptions listed on page 2.2.2.

• Let’s apply momentum conservation in integral form in the s-direction:

(2.2.41)∂∂t---- usρ V us ρU n⋅ Ad( ) Ts Ad

cs�� Bsρ Vd�

cv��+=

cs��+d�

cv��


p. 2.2.14

• The vanishes since the problem is steady state.

• The net momentum flux into the CV equals and assuming cross sectionally averagedflow:

(2.2.42)

(2.2.43)

(2.2.44)

• Noting that where W = the width of the channel and = normal flow depth:

(2.2.45)

∂∂t----

usρU n⋅ A usρus Ad

A1

��–=d

A1

��

u– s2ρ Ad

A1

��=

u– s2ρA1=

A1 d0W= d0

usρU n⋅ A u– s2ρ=d d0W

A1

��


p. 2.2.15

• The net momentum flux out of the CV equals

(2.2.46)

(2.2.47)

(2.2.48)

• Since :

(2.2.49)

• Total surface forces in the s-direction consist of pressure on both sides.

(2.2.50)

(2.2.51)

usρU n⋅ A usρus Ad

A2

��=d

A2

��

us2ρ Ad

A2

��=

us2ρA2=

A2 d0W=

usρU n⋅ A us2ρ=d d0W

A2

��

F1 s–

ρgd1A1

2------------------=

F2 s– ρgd2A2

2------------------–=


p. 2.2.16

• Noting that

(2.2.52)

(2.2.53)

where W=width of channel (2.2.54)

• Thus pressure forces are:

(2.2.55)

(2.2.56)

• A total frictional surface force in the s-direction also exists:

(2.2.57)

(2.2.58)

d1 d2 d0= =

A1 A2 d0W= =

F1 s–

ρgd02W

2-----------------=

F2 s– ρgd0

2W

2-----------------–=

Ff s– τ0LsPw–=

Ff s– τ0Ls W 2d0+( )–=


p. 2.2.17

• A total body force in the s-direction consists of

(2.2.59)

where (2.2.60)

• Thus

(2.2.61)

• Substituting into our momentum statement:

(2.2.62)

• This leads to

(2.2.63)

FB s– ρgALs θ0sin=

A Wd0=

FB s– ρgWd0Ls θ0sin=

0 us2ρd0W us

2ρd0Wρgd0

2W

2-----------------

ρgd02W

2-----------------– τ0Ls W 2d0+( )– ρgWd0Ls θ0sin+=–+

τ0Ls W 2d0+( ) ρgWd0Ls θ0sin=


p. 2.2.18

• Noting that sin and re-organizing:

(2.2.64)

• Again for a wide channel

(2.2.65)

• Also for modest slope channels ; thus the momentum conservation statementsimplifies to:

(2.2.66)

• Again this solution is identical to that obtained from the Shallow Water equations.

• To complete the solution, we would substitute in the appropriate constitutive relation-ship for and relate and and a friction factor.

• Thus we clearly see that we can use a variety of approaches, all rooted in the same basicphysics and using related constitutive relationships, to solve the problem.

θ0 S0 θ0cos=

τ0 γS0 θ0

Wd0

W 2d0+--------------------cos=

W d0»

Wd0

W 2d0+--------------------

Wd0

W---------- d0=≅

θ0 1≅cos

τ0 γd0S0=

τ0 us u≅ d0 to S0

Documents

2.2 THREE APPROACHES TO SOLVE FOR NORMAL … · 27/02/2003 · • With the assumptions made, the Shallow Water equations simplify to ... • Applying the Bernoulli equation between