Parameter Identification of Manning Roughness Coefficient Using ...

˙ ξ + ξ ,i u i + ξu i,i = 0, (2)
˙η + Λu i,i =0, (3)
where u i , g, ξ and η are the water velocity,the gravitational
acceleration,the water elevation and the
bed elevation,respectively. The coefficient of kinematic
eddy viscosity ν,f and Λ are expressed as
follows;
ν = k l
6 u ∗ ξ, f = u ∗
ξ , Λ= q s
(1 − e ) √ ,
u k u k
where k l , u ∗ , e and q s are the Kalman constant,the
friction velocity,the porosity of sand and the bedload
quantity,respectively. The friction velocity u ∗
is expressed as follows;
√
u ∗ = gn2 uk u k
,
ξ 1/3
where n is the Manning roughness coefficient.
The bed-load quantity q s is determined by a relation
between the tractive force and the limit tractive
force. The tractive force τ 0 is caused by the water
flow. The limit tractive force τ c is expressed as follows;
τ c = σg( ρ s − ρ w ) d m ,
where σ, ρ s , ρ w and d m are the shields parameter,the
density of water,the density of sand and
the sandy average particle size,respectively. The
Meyer-Peter Müller equation is used as an equation
so as to express the bed-load quantity.
If τ 0 >τ c ,
√
1 1
q s =8
ρ w g ( ρ s − ρ w ) ( τ 0 − τ c ) 3 2 .
If τ 0 ≤ τ c , q s =0.
3. Spatial Discretization
3.1 Bubble Function Element
As for the spatial discretization,the interpolation of
bubble function element for the velocity,the water
elevation and the bed elevation fields as shows in
Fig.2 is used. The bubble function element is expressed
as follows;
u i = Φ 1 u i1 + Φ 2 u i2 + Φ 3 u i3 + Φ 4 ũ i4 ,
ũ i4 = u i4 − 1 3 ( u i1 + u i2 + u i3 ),
ξ = Φ 1 ξ 1 + Φ 2 ξ 2 + Φ 3 ξ 3 + Φ 4 ˜ξ4 ,
˜ξ 4 = ξ 4 − 1 3 ( ξ 1 + ξ 2 + ξ 3 ),
where φ e is the bubble function of C 0 continuous
and Φ α ( α = 1 ∼ 4 ) is the bubble function element.
3
s
1
Fig.2
4
2
Bubble Function
Element
r
1.0
s
ω 2
ω 1
ω 3
r
0.0
1.0
Fig.3 Subdivision of
Element
The bubble function is defined by using the
isoparametric coordinates ( r, s ) as shows in Fig.3.
Three triangles ( ω 1 ∼ ω 3 ) is divided at the barycenter
point. The bubble function ofC 0 continuous can
be considered on each sub-triangle as the following
equation;
⎧
⎨ 3(1− r − s ) in ω 1
φ e = 3 r in ω
⎩
2
3 s in ω 3 .
3.2 Improved Bubble Element
The bubble function is capable of eliminating the
barycenter point by using the static condensation.
The discretized form derived from the bubble
function element is equivalent to those from the
SUPG[2]. Therefore,the stabilized parameter which
is derived from the bubble function element is expressed
as follows;
〈 momentum equation for shallow water flow 〉
τ eBui =
〈φ e , 1〉 2 Ω e
A −1
e
1
∆t ‖ φ e ‖ 2 Ω e
+ 1 2 [(ν +˜ν)2 ‖ φ e,j ‖ 2 Ω e
−f ‖ φ e ‖ 2 Ω e
] ,
〈 continuity equations
for shallow water flow and sand 〉
〈φ e , 1〉 2 Ω
τ eBη =
e
A −1
e
1
∆t ‖ φ e ‖ 2 Ω e
+ 1 2 [˜ν ‖ φ e,j ‖ 2 Ω e
] ,
where ˜ν is the stabilized control parameter. From
the criteria for the stabilized parameter in the
SUPG,an optimal parameter can be given as follows;
〈 momentum equation for shallow water flow 〉
η = Φ 1 η 1 + Φ 2 η 2 + Φ 3 η 3 + Φ 4 ˜η 4 ,
˜η 4 = η 4 − 1 3 ( η 1 + η 2 + η 3 ),
Φ 1 = 1 − r − s, Φ 2 = r, Φ 3 = s, Φ 4 = φ e ,
19
τ eBui =
( 1
2 τ −1
es + α ∆t
) −1
,
[ ( ) 2
τes −1 2 | U i |
=
+
h e
( 4ν
h 2 e
) 2 ] 1
2
,