Open channel flows

Solution of the system of PDE-s for open channel flow
The 1D equation of continuity for open channel flow is
∂A
∂(
Av )
+ = 0
∂t
∂x
. (1)
∂A
Here A = A(y) [meaning that = 0 ] * is the cross section of the channel filled with water, y
∂x
denotes the water depth, v is the mean velocity of water, t is time, x is the space coordinate
along the channel.
Important: the shape of the channel does not change along the channel: A doesn’t depend
explicitly on x.
The equation of motion is
∂v
∂v
∂y
+ v + g
∂t
∂x
∂x
∂z
gn
= −g
−
∂x
R
2
4
3
h
v v
(2)
as the pressure p – whose gradient is the source of the 3 rd term – means the hydrostatic
overpressure at the bottom of the channel, p = ρgy. R h is the hydraulic radius of the wetted
channel section and n is the roughness parameter in Manning’s formula. By multiplying the
equation with A we get
⎡
2
∂v
∂v
∂ gy ∂z
gn
⎤
A
∂t
+ Av + A
∂x
( )
∂x
= −A⎢g
+
⎢ ∂x
⎣ R
4
3
h
v v⎥
.
⎥
⎦
By adding the v-times of the above equation of continuity (1) to the left hand side – this may
be done as we add zero to the equation – we get
( )
⎡
2
∂v
∂A
∂(Av)
∂v
∂ gy ∂z
gn
⎤
A + v + v + Av + A = −A⎢g
+ v v⎥
.
4
∂t
∂t
∂x
∂x
∂x
⎢ ∂x
3 ⎥
⎣ Rh
⎦
The 1 st and 2 nd and the 3 rd and 4 th terms, respectively may be united by the rule of the
differentiation of a product
( Av) ∂(Avv)
∂( gy)
∂
∂t
+
∂x
+ A
∂x
⎡
⎢
∂z
gn
= −A
g +
⎢ ∂x
⎣ Rh
2
4
3
⎤
v v⎥
.
⎥
⎦
∂A
The last step is to ad gy = 0 to the left hand side
∂x
2
2
2
( Av) ∂(Av
) ∂( Agy) ∂( Av) ∂(A{ v + gy} ) ∂z
gn denoted
+ + = +
= −A⎢g
+ v v⎥
= A[ J − J ]
∂
∂t
∂x
∂x
∂t
∂x
⎡
⎢
⎣
∂x
R
4
3
h
⎤
⎥
⎦
s
R
.

So the final version of the equation of motion in the so called conservation form is
2
( Av) ∂(A{ v + gy} )
+
= [ − ]
∂
∂t
∂x
A J s
J R
. (3)
Here Js is the gravitational driving force of motion, J R is the breaking force of fluid friction.
Naturally Eq. (1) has also the conservation form and this is natural as it just means
conservation of water along the channel in time.
It is worth of mentioning that Eqs. (1) and (3) may be written together in the compact vector
form
∂ ⎛ A ⎞ ∂ ⎛ Av ⎞
⎞
{ } ⎜ ⎛ 0
⎜ ⎟ + ⎜
2
⎟ =
∂ ⎝ ⎠ ∂ ⎝ A v + gy ⎠ ⎝ A[ J − ] ⎟⎟ .
t Av x
s
J R ⎠
In this equation or in Eqs. (1) and (3) three variables occur: A, y, v. To find their values one
needs a third equation. This will be the very simple equation of state (similar to barotropy or
especially isentropy for gases):
dA = B . (4)
dy
B is the width of the free water surface at the actual water depth y.
B
y
A
Fig. 1 Geometrical data of a channel with circular cross section
Eq. (4) will now be substituted into Eq (1).
∂A
∂(
Av )
+ =
∂t
∂x
dA
dy
∂y
+ v
∂t
dA
dy
If we divide this equation by B and we consider that
continuity equation with the embedded equation of state
∂y
∂v
⎛ ∂y
∂y
⎞ ∂v
+ A = B⎜
+ v ⎟ + A = 0 .
∂x
∂x
⎝ ∂t
∂x
⎠ ∂x
2 gA
a =
B
we have the final form of the
∂y
∂y
+ v +
∂t
∂x
gA
gB
∂v
∂x
=
2
∂y
∂y
a
+ v +
∂t
∂x
g
∂v
∂x
= 0
. (5)
It is important to note that the above formula of the wave speed is unbounded if the channel is
filled completely because in this case the free surface width B → 0 and so a → ∞. There is a
simple model helping us to overcome this difficulty. The top of the channel surface mustn’t

e closed. In the model, there is slot between two parallel vertical walls attached to the top of
the channel. The distance B s between the walls is determined so that
gA Ereduced
a = = a
full
= is equal to the desired wave speed for the completely filled
B
ρ
s
channel. Naturally there is a small additional water-filled cross section between the walls but
it is negligible; y means the pressure head in the completely filled channel in this case.
B S
y
slot
Fig. 2 Model of the completely filled circular channel
to give the correct pressure wave speed
In the literature there are different methods to solve the system of differential equations in the
red or green frames:
• method of characteristics
• Lax-Wendroff scheme
• implicit method
Method of characteristics
In this case equations (2) and (5) give the base. The slope of the characteristics is
dx
= v ± a
(6)
dt
The upper sign gives the C + the lower one the C – characteristic line. The actual section of the
grid of characteristics with the old (t L = t R ) and new time level (t P ) and three points L, R and
P is shown below
t
C -
C +
t P
P
t L
L
R
Fig. 3 Piece of the characteristic mesh
x

The first source term on the right hand side of Eq. (2) is a function of space only. The
nonlinear term in the second source term v v may be discretised as a geometric mean between
the old and new velocities
v
L
vP
on the C + and
R
vP
v on the C – characteristic line section.
By multiplying Eq. (2) with a = const., Eq.(5) by g = const., adding these equations and using
the notations for the source terms J S and J R respectively we get
( av) ∂( av) ∂( gy) ∂( gy) ∂( gy) ∂( av) + v + a + + v + a = a( J J )
∂
∂t
∂x
∂x
This equation may be summarized for the C + characteristics as
∂( gy + av)
∂
( )
( gy + av)
d
+ v + a
gy av a J
∂t
∂x
= dt
+ =
Similarly for the C – characteristics subtracting the equations results in
∂
( gy − av)
+
∂t
∂x
∂x
s −
( ) ( J )
s −
∂
( ) ( gy − av ) d
v − a = ( gy − av) = − a( J J )
R
∂t
∂x
dt
Here we have used the differential equations (6) of the characteristic lines. The discretized
forms of the above ordinary differential equations along the two characteristic line segments
are
gy
( )
⎛
2
+ a
⎞
Lv
− gy
L
+ aLvL
⎜ z
P
− z
L gn ⎟
= aL
⎜
− g − vL
v
−
−
⎟
, (7)
4
t
P
t
L
xP
xL
3
⎝
Rh,L
⎠
P
s −
R
.
.
R
.
gy
−
P
aRv
t
−
P
P
( gy − a v )
− t
R
R
R
R
= −a
R
⎛
⎜ z
⎜
− g
x
⎝
P
P
− z
− x
R
R
gn
−
4
R
2
3
h,R
v
R
⎞
⎟
v
⎟
, (8)
⎠
P
The two bold parameters be computed step by step from the two equations (7)
and (8). As it is clearly seen from Fig. 3 the grid points at the new time level are calculated
through equation (6) by discretizing it
xP
− xL
xP
− xR
xL
+ xR
vL
+ vR
+ aL
− aR
= vL
+ aL
and = vR
− aR
giving xP
= + ∆t
.
∆t
∆t
2
2
vP yP
xP
− xL
Now ∆t too can be computed as ∆ t = . It is advisable to keep the smallest value of
vL
+ aL
∆t-s and interpolate the depths y, areas A and velocities v at this new time level onto the
original equidistant space coordinates. Otherwise the naturally growing grid would be more
and more deformed.
and
can
Lax-Wendroff scheme
This method was originally developed for computing gas dynamics (compressible) flows.
Peter Lax is of Hungarian origin. It is a two-steps-in-time method (Hirsch [1991]). The
intermediate time (∆t/2) is the half of the full time step (∆t).

P
L
R
∆t
∆t/2
i-1
i
i+1
∆x
∆x
Fig. 4 Grid points of the Lax-Wendroff scheme
In the figure ∆x is the fixed distance between the main grid points. At the intermediate time
level ∆t/2 wetted areas A are computed in points L and R using the continuity equation (1) in
the conservation form. For A L the discretized form of (1) is
Ai
−1
+ Ai
AL
−
2 vi
Ai
− vi−
1Ai
−1
+
= 0
∆t
∆x
2
from which
Ai
−1
+ Ai
∆t
vi
Ai
− vi−
1Ai
−1
AL
= −
,
2 2 ∆x
A R is computed in a similar way.
Ai
+ Ai
+ 1 ∆t
vi+
1Ai
+ 1
− vi
Ai
AR
= −
.
2 2 ∆x
We see that the first step is an explicit scheme forward in time and central in space.
To complete the time step one will need y L and y R too. These values may me received in case
of a known channel shape, e.g. a circle through a purely geometric way or using the equation
∂A
∂y
of state (4). The time derivative of the area is then = B and this can be substituted into
∂t
∂t
Eq. (1) giving
yi−
1
+ yi
y −
B
−1
+ B
L
i i 2 vi
Ai
− vi−
1Ai
−1
+
= 0
2 ∆t
∆x
2
in case of point L and a similar formula results for the intermediate point R. These equations
can again be rearranged for the depths y L and y R .
Naturally, the velocities v L nd v R are computed from the conservation form (3) of the equation
of motion.
Ai
−1vi
−1
+ Ai
vi
ALvL
−
2
2
2 A ( ) ( ) 1( 1 1
) ( )
i
vi
+ gyi
− Ai
1
vi
1
gy A
i 1 i
J
s,i
− J
R ,i
+ Ai
J
s ,i
− J
− −
+
− − − −
R ,i
+
=
∆t
∆x
2
2
And a similar approximation of Eq. (3) holds for the section i – i+1 to give (Av) R = A R v R .

If one had computed (Av) L and (Av) R at the intermediate time level the velocities v L and v R
could be received by a simple division by the already known A-values.
The completing half time step is now – again for the area A P – straightforward using Eq. (1).
This step is explicit and central difference scheme for both time and space.
AP
− Ai
vR
AR
− vL
AL
+
= 0 .
∆t
∆x
After rearranging we have the new area A P :
vR
AR
− vL
AL
AP
= Ai
− ∆t
.
∆x
The water depth y P can again be computed from either the direct geometrical relation or by
employing equation of state (4). Finally one uses again the equation of motion in conservation
form to get (Av) P and then v P .
It is well known that explicit schemes are not unconditionally stable. One has proved that the
proper selection of the time step ∆t is
⎪⎧
∆x
⎪⎫
∆ t = min⎨
⎬,
i = 1,...,I
.
⎪⎩ ai
+ vi
⎪⎭
Here I denotes the number of grid points in space. This step size ∆t is the same as suggested
in case of the method of characteristics to interpolate at.
Implicit method
The time steps will decrease and are getting small if the channel is filled with water (for
example in case of a tidal wave reaching the top of the channel) as the wave speed will
increase very much and for stability the above defined ∆t mustn’t be exceeded. Implicit
methods, however, allow much larger time steps helping to save computer time.
The implicit method means that all water depths y j and velocities v j are computed
simultaneously at the new time level. If there are I nodes along the channel then there are 2I
unknowns. The space-time coordinate system is split into “final volumes” with dimensions ∆t
and ∆x. The equations in conservation form (1) and (3) must be integrated over each of these
I-1 finite volumes in space and in time. As there are two equations for each volume – one
continuity and one equation of motion – there are 2(I-1) equations altogether. The two
missing equations are prescribed values of depths and/or velocities at the two boundaries of
the channel or some algebraic relations between these flow parameters.
The finite volume is shown below
t+∆t
∆t
t
∆x
x j x j+1
Fig. 5 Finite volume of the implicit method

Let’s see first the integration of Eq. (1) over the above finite volume. The first term is
integrated first over the time and then over the space. The second term is integrated first over
the space then over the time. The time integral of the derivative of A with respect to time is
the difference of the A values during the time increment, A(t + ∆t) – A(t). A similar rule is
applied for the second term.
0 =
x j+
1
x j
t+∆t
∫ ∫
t
⎡∂A
∂( vA ) ⎤
⎢
+ dtdx =
t x ⎥
⎣ ∂ ∂ ⎦
=
x j+
1
∫
x j
x
j+
1
∫
x j
⎛
⎜
⎝
t+∆t
∫
t
∂A
⎞
dt ⎟dx
+
∂t
⎠
t+∆t
( A( t + ∆t
) − A( t )) dx +
( vA ) −( vA ) )dt
∫
t
t+∆t
∫
t
j+
1
⎛
⎜
⎜
⎝
x
j+
1
∫
x j
∂( vA ) ⎞
dx⎟dt
=
∂x
⎟
⎠
Now Eq. (4) can be applied, A(t + ∆t) – A(t) = B(t)·∆y, where ∆y denotes the depth change
during ∆t. The first integral in the second row can now be approximated by the integral mean
low as
B ∆ + ∆
j
y
j
B
j+ 1
y
j+1
∆x .
2
The second integral is approximated as ∆t times the arithmetic mean of the product (Av) over
the time interval ∆t which can be written as
( vA ) + ( vA ) ( vA ) + {( vA ) + A ∆v
+ v ∆A}
t
t+∆t
t
t t t
( vA ) ∆t
=
=
=
t+
2 2
2
.
∆v
∆A
∆v
∆y
= ( vA ) + A + v = ( vA ) + A + v B
t t
t
t t
t t
2 2
2 2
In the last term again the change of the wetted area is expressed with the width of the surface
times the change of depth.
After substituting these approximations the final form of the discretized continuity equation is
B
B
A t A t
i
j+
1
j∆
j+
1∆
( ∆ x − v
j∆t
) ∆y
j
+ ( ∆x
+ v
j+
1∆t
) ∆y
j+
1
− ∆v
j
+ ∆v
j+
1
=
j j j+
1 j+
2
2
2 2
This is an inhomogeneous equation with four unknowns:
• ∆v j velocity change at the upstream end of the finite volume
• ∆v j+1 velocity change at the downstream end of the finite volume
• ∆y j depth change at the upstream end of the finite volume
• ∆y j+1 depth change at the downstream end of the finite volume.
j
( v A − v A ) ∆t
The same unknowns appear in the integral of the equation of motion.
Eq. (2) has an alternative conservation form containing partial derivatives only with respect to
time or to space on the left hand side:
2
2
∂v
∂ ⎛ v ⎞ ∂z
gn
+ ⎜ gy ⎟ = −g
− v v
4
t x
+
2
(9)
∂ ∂ ⎝ ⎠ ∂x
3
Rh
The double integral of this equation on the final volume is
1
.

x
j + 1
x
j
=
t+∆t
∫ ∫
x
x
t
j + 1
∫
= −g
2
2
⎡ v ⎤
⎛ v ⎞
( gy )
x
x
⎢ ∂ +
( gy )
v
⎥
j + 1 t+∆t
t+∆t
⎜ j + 1 ∂ + ⎟
∂
⎛ ∂v
⎞
⎢ +
2
⎥dtdx
= ⎜ dt ⎟dx
+ ⎜ 2
dx⎟dt
t x
t
=
⎢ ∂ ∂
∫ ∫
x
x j ⎝ ∂
∫ ⎜ ∫
⎥
t ⎠
∂ ⎟
t x j
⎢
⎜
⎟
⎣
⎥⎦
⎝
⎠
t+∆t
2
2
⎛ v
v ⎞
dx + ∫
⎜(
gy )
j
( gy ) ⎟
+
+ 1
− +
j
dt =
t ⎝ 2
2 ⎠
⎛
⎞
⎜
⎟ v + v v + v
j+
1 j ⎜ 4
4 ⎟
2 ⎜
3
3
Rh
R ⎟ 2 2
⎝ j h j+
1 ⎠
( v( t + ∆t
) − v( t ))
j
2
2
1 gn gn
j j+
1 j j+
1
( z − z ) ⋅ ∆t
− + ⋅ ⋅ ⋅ ∆x
⋅ ∆t
The right hand side has already been approximated and integrated. The second row of the
above equation still contains definite integrals. Again they can be approximated by the
integral mean value rule.
x j + 1
x j + 1
∆v
j
+ ∆v
j+
1
For example ∫ ( v( t + ∆t
) − v( t )) dx = ∫ ∆vdx
=
∆x
.
2
x j
One has to notice that ∆v j , ∆y j appears not only in the equations of the finite volume (j; j+1)
but also in that of (j-1; j).
Finally together with the boundary conditions the number of equations is just the same as the
number of unknowns. Through an appropriate ordering of the equations it can be reached that
all nonzero coefficients stand in the main diagonal and the two lower and upper neighbouring
diagonals of the matrix M of the linear equation system. Such a matrix is called
pentadiagonal. There are efficient separation methods (M = L·U) for inverting this type of a
matrix; L denotes a lower triangular matrix and U is an upper triangular matrix. They can be
inverted very easily.
x j