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Surface Water Equations
Continuity (NS)
u v w


0
x y z
Kinematic Boundary Conditions
dz2
z2
z2
z2
 wz 2 
 uz 2
 vz 2
r
dt
t
x
y
dz1
z1
z1
z1
 wz1 
 u z1
 v z1
f
dt
t
x
y
Surface Water Equations
Integrate continuity equation over depth, term by term
1
z 2  z1
1
z2  z1
u
1 u z2 u
z x dz  z2  z1 x z z1  x
1
z2
v
1 v z2 v
dz

zz 
z y
1
z

z

y
y
2
1
1
z2
Surface Water Equations
Third term… (need KW boundary conditions)
1
z2  z1
z1   z 2
z1 
 z2 z1   z 2


u

u

v

v
  z2
  z2
r
z1
z1
 t
t  
x
x  
x
x 


f

Regrouping, letting h = z2 - z1 with velocity constant with depth
h  u
h   v
h 
 h
 u    h  v   r  f
t  x
x   y
y 
h (hu) (hv )


r f
t
x
y
Surface Water Equations
Momentum (NS x-direction)
 u
u
u
u 
p
   u  v  w      2 u  Fx
x
y
z 
x
 t
Term-by-term integration
First:
1
u
u
  dz 
z 2  z1 z1 t
t
Second:
2
1
u 2
u 2

dz 
z2  z1 z1 x
x
z2
z
Surface Water Equations
1
uv
uv

dz 
z2  z1 z1 y
y
z2
Third:
Fourth: need kinematic boundary conditions
1
uw
1
z2

dz 
uw z
1
z2  z1 z1 z
z2  z1
z2
 z 2
  z1

z 2
z 2
z1
z1
1
u 
 u z2
 v z2
 r   
 u z1
 v z1
 f 
z 2  z1  t
x
y
x
y
  t

Surface Water Equations
Left side of momentum equation becomes:
 hu hu 2 huv
 


 ur  uf
x
y
 t



Surface Water Equations
In terms of shear stress, the right side is written
p  xx  yx  zx




 Fx
x
x
y
z
Assume horizontal shear components are small
p  zx


 Fx
x
z
Surface Water Equations
The first term is the unbalanced pressure
force; when vertically averaged:
1
h 
  gh 
h
x 
(hydrostatic?)
Surface Water Equations
The third term is the gravitational force:
z1 
1
  gh

h
x 
Surface Water Equations
The second term must be vertically integrated:
1
z2  z1
 zx
1
dz

 zx
z z
h
1
z2
z2
z1
  xz1
Shear stress at the water surface is zero
Surface Water Equations
Combining and multiplying by depth:
 h z 
 gh  1    xz1
 x x 
Surface Water Equations
Combining all terms, the x-direction momentum
equation for overland flow is
 hu hu 2 huv
 


 ur  uf
x
y
 t

 h z 
   gh  1    xz1
 x x 

Similarly, the y-direction equation is
 hv huv hv 2

 h z1 
 


 vr  vf    gh 
   yz1
x
y
 y y 
 t

Surface Water Equations
With Some substitutions:
p = hu , q = hv
ql = r – f
S fx   xz1 
S fy   yz1 
Sox   z1 x
Soy   z1 y
Surface Water Equations
The equations become:
h p q
+ + - ql  0
t x y
p   p 2 g h 2    pq 
p
 +   - gh( S ox - S fx ) + ql = 0
+  +
t x  h
2  y  h 
h
q   q2 g h 2    pq 
q


+  +
+   - gh(S oy - S fy ) + ql = 0

t y  h
2  x  h 
h
Surface Water Equations
Friction Slope terms: Darcy-Weisbach
1/2
f p ( p 2 + q2 )
S fx =
3
8g
h
2
2 1/2
f q( p + q )
S fy =
3
8g
h
Surface Water Equations
Darcy-Weisbach continued… for laminar flow:
f = Ko
Re
Re =
So:
( p2 + q2 )1/2

K
o p
S fx =
8g h3
K
o q
S fy =
8g h3
Surface Water Equations
Mannings:
n 2 p( p 2  q 2 ) 0.5
S fx 
h10 / 3
n 2 q( p 2  q 2 ) 0.5
S fy 
h10 / 3
Surface Water Equations
Vector (compact) notation:
U G(U)  H(U)
+
+
= S(U)
t
x
y
T
U = [h, p, q ]
2
2
p gh pq T
G(U) = [ p, +
, ]
h
2 h
2
2
q gh qp T
H(U) = [q, +
, ]
h
2 h
p
q T
S(U) = [ql , ghSox  S fx  - ql , ghSoy  S fy  - ql ]
h
h
Surface Water Equations
Alternate Derivation: conservation of mass and
momentum using Reynold’s Transport Theorem
Continuity:
d
dt
 d   V  dA  0
cv
cs
Momentum:
d
Vd   VV  dA   F

dt cv
cs
Surface Water Equations
1-D St. Venant equations: conservation form
Q A

0
t x
1 Q 1   Q 2 
h

  g

 g ( S ox  S fx )  0
A t
A x  A 
x
local acceleration, convective acceleration,
unbalanced pressure force, gravity force, and
friction force
Surface Water Equations
1-D St. Venant Equations: non-conservation form
h
h
u
u h
0
t
x
x
u
v
h
v g
 g ( Sox  S fx )  0
t
x
x
Surface Water Equations
When can the kinematic wave approximation be used?
In general:
• steep slopes
• uniform flow
• no backwater effects
Surface Water Equations
For 1-D overland flow on a plane - kinematic wave number:
S0 L0
K
h0 F02
KF02  5
and F0 < 2
Woolhiser and Liggett (1967)
Surface Water Equations
For 1-D plane or channel flow
1
( S0 ) 0.5  3
n
ql (1 / n ) 3 ( S 0 ) 0.5
 0 .7
2
g
Hager and Hager (1985)
Surface Water Equations
Wave Celerity (speed)
Kinematic waves occur when there is a unique
relationshhip between flow depth and discharge:
general form:
A  Q 
from Manning’s:
 nP
A   0.5
 S0
2/3
3/ 5

 Q 3 / 5

Surface Water Equations
differentiate and sub into continuity
Q
 1  Q 
 Q 
  ql
x
 t 
the total derivative of discharge is
Q
Q
dQ 
dx 
dt
x
t
or
Q dt Q dQ


x dx t
dx
Surface Water Equations
from this we see that:
dQ
 ql
dx
discharge increases with
lateral inflow
and
dx
1

dt Q  1
kinematic wave celerity
Surface Water Equations
Is the KW celerity equal to mean velocity? - no.
in a wide rectangular channel, u = Q/h, and
Q Q (uh)
u


uh
 ck
A h
h
h
substituting Manning’s equation:
ck  u  h
  1 2 / 3 0.5 
2
5
h
S

u

u

u

f 
h  n
3
3

Surface Water Equations
Dynamic wave celerity
cd  gh
dx
 u  cd
dt
dx
 u  cd
dt