The Saint Venant equations March 7, 2018
Contents 1 Notation
2
2 The various forms 2.1 Conservation form . . . . . . . . . . 2.2 Non-conservation form . . . . . . . . 2.3 Characteristic form . . . . . . . . . . 2.4 Equivalence between the three forms
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2 2 3 3 3
3 Simplied forms 3.1 Simpli ed form #1: the diusive wave approximation . . . . . . . . . . . . . . . . . . . 3.2 Simpli ed form #2: the kinematic wave approximation . . . . . . . . . . . . . . . . . . .
3 3 4
4 Discretization 4.1 What is done in most commercial packages . . . 4.2 Numerical trick #1: monotone conveyance curves 4.3 Numerical trick#2: Preissmann's slot . . . . . . 4.4 Numerical trick#3: Abbott's slot . . . . . . . . .
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4 4 5 5 5
5 Drill sessions 5.1 Geometry . . . . . . . . . . . . 5.2 Equivalence between solutions . 5.3 Kinematic wave . . . . . . . . . 5.4 Diusive wave . . . . . . . . . .
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1
Notation
See Figure 1 and Table 1. z
zs
h
b
W(z) zb
A
Figure 1: Notation. Symbol A b Cv h RH W (z) S0 Sf zb zs χ
Meaning Cross-sectional area Top width, b = W (zs ) Conveyance Water depth, h = zs − zb Hydraulic radius Width at the elevation z b Bottom slope, S0 = − ∂z ∂x Friction slope / energy grade line Bottom elevation Free surface elevation Wetted perimeter
French term Section en travers Largeur au miroir Débitance Profondeur Rayon hydraulique Largeur entre berges à l'altitude z Pente du fond Pente de frottement / pente de la ligne d'énergie Cote du fond Cote de surface libre Périmètre mouillé
Table 1: Notation. Main relationships:
zs
A=
W (z) dz
(1a)
zb
b = W (zs )
(1b)
dA = bdh
(1c)
A χ
(1d)
RH =
2
The various forms
Important note: the equations are derived for a prismatic channel. Non-prismatic channels induce extra terms in the right-hand-side member of the momentum equation.
2.1 Conservation form By de nition, the conservation form is an equation in the form ∂t u + ∂x f = s
2
(2)
For the Saint Venant equations u=
A Q
"
, f=
2
Q A
Fp = ρ Sf =
|u| u
=
4/3
K 2 RH
Q +
# Fp ρ
, s=
0 gA (S0 − Sf )
zs
(3b)
g (zs − z) W (z) dz zb
|Q| Q 4/3
K 2 A2 R H
=
n2 |u| u 4/3
n2 |Q| Q
=
(3c)
4/3
A2 RH
RH
2/3
2/3
1/2
Q = KARH Sf
(3a)
1/2
= Cv Sf
2/3
⇐⇒ Cv = KARH =
ARH n
(3d)
with K Strickler's coe cient, n = 1/K Manning's coe cient.
2.2 Non-conservation form An equation involving only the derivatives of u (eliminate f from the equations). For Saint Venant: ∂f = ∂t u + A∂x u = s, A = ∂u c2 =
0 c2 − u 2
1 2u
∂Fp gA = ∂ (ρA) b
(4a) (4b)
2.3 Characteristic form A set of ordinary dierential equations (remove the ∂ operators, replace them with total derivatives). ∂t w + Λ∂x w = s0 ⇐⇒
dx dwk = sk for = λk (k = 1, 2) dt dt
(5a)
Λ = diag (λ1 , λ2 ) , dw = K−1 du, s0 = K−1 s
(5b)
λ1 = u − c, λ2 = u + c
(5c)
For a rectangular channel: w1 = u − 2c, w2 = u + 2c For a triangular channel: w1 = u − 4c, w2 = u + 4c
2.4 Equivalence between the three forms When the ow solution is smooth (dierentiable), the conservation, non-conservation and characteristic forms are strictly equivalent. Consequently, knowing (A, Q) or (h, u) or (w1 , w2 ) is exactly the same: there are one-to-one correspondences between these three sets of variables.
3
Simplied forms
3.1 Simpli ed form #1: the diusive wave approximation Assumptions. The DW is obtained from the SV equations under the following assumptions: the ow is very slow, so |u| c ⇔ |Fr| = uc 1
the inertial terms are negligible: ∂t Q is neglected.
3
The equation. In the momentum equation ∂t Q + c2 − u2 ∂x A + 2u∂x Q = gA (S0 − Sf )
(6)
the inertial terms are neglected, then the governing equations become (7a) ∂x A = (S0 − Sf ) b (7b) The derivatives of A are eliminated by dierentiating the 1st equation wrt x and the second wrt t: S0 ∂t b − ∂t (bSf ) + ∂xx Q = 0 (8) Since b = b (A), Sf = Sf (A, Q), bSf = bSf (A, Q). Expand the derivatives: ∂t A + ∂x Q = 0
S0
then
∂b ∂ (bSf ) ∂ (bSf ) ∂t A − ∂t A − ∂t Q + ∂xx Q = 0 ∂A ∂A ∂Q
(9)
∂ (bSf ) ∂ (bSf ) ∂b − ∂x Q − ∂t Q + ∂xx Q = 0 − S0 ∂A ∂A ∂Q
(10)
∂t Q + V ∂x Q = D∂xx Q
(11a)
Rearrange V =
∂(bS ) ∂b S0 ∂A − ∂Af ∂(bSf ) ∂Q
=
∂b S0 ∂A
−
∂(bSf ) ∂A
∂S
b ∂Qf
−1 ∂Sf D= b ∂Q
(11b) (11c)
3.2 Simpli ed form #2: the kinematic wave approximation Assumptions. The KW is a stronger approximation than the DW: the ow is very slow, so |u| c ⇔ |Fr| = uc 1 the inertial terms are negligible: ∂t Q is neglected, the bottom slope is much larger than ∂x h (thin water lm owing over steep topography). This corresponds to the assumption of uniform ow. The equation. The momentum equation reduces to Sf = S0 ⇐⇒ Q = Q (A, x)
Then ∂t A + ∂x Q = 0 ⇐⇒
4
∂A ∂t Q + ∂x Q = 0 ⇐⇒ ∂t Q + λ∂x Q = 0 ∂Q ∂Q λ= ∂A
(12) (13a) (13b)
Discretization
4.1 What is done in most commercial packages Most of them use implicit techniques (the terms are estimated using the unknown variables). This leads to more stable solutions (sometimes at the expense of accuracy). The non-conservation form of the equations is solved because it is easier with implicit techniques. Some of the terms in the momentum equation are dropped willingly because they tend to make the solutions unstable, especially near critical conditions. Dry beds and full sections are usually accommodated for using a number of numerical tricks.
4
4.2 Numerical trick #1: monotone conveyance curves In closed cross-sections (e.g. pipes in urban drainage networks), the conveyance is not monotone. In a circular section, the maximum conveyance is reached for h u 0.93 D. In a rectangular cross-section, the conveyance drops suddenly as the free surface reaches the lid. This results in physical instability and subsequent pipe saturation. In commercial packages, this instability is eliminated mainly for commercial reasons (unstable simulations are not well-percieved by software users, even though it may re ect a physical reality). The conveyance curve is thus made arti cially monotone.
4.3 Numerical trick#2: Preissmann's slot In closed cross-sections, the top becomes zero as the free surface elevation reaches the lid. This width and solution instability. This problem is eliminated by introducing results in an in nite speed c = gA b a narrow slot on the top of the cross-section so that b does not become zero.
4.4 Numerical trick#3: Abbott's slot Most software packages have problems dealing with dry beds, because in natural cross-sections b = 0 when h = 0. Software developers have introduced an arti cial slot in the bottom so that a small water depth (and therefore a small discharge) is maintained even when the bed is dry.
5
Drill sessions
5.1 Geometry 1. Work out the formulae A (z), b (z),
Fp ρ
(z), c (z), Cv (z) for
a rectangular cross-section, a triangular cross-section a trapezoidal cross-section a circular cross-section (urban drainage pipe) a rectangular conduit
2. Does a channel shape exist, such that c (z) is the same for all z ?
5.2 Equivalence between solutions 1. For a rectangular channel of width b, give the expressions (a) of (w1 , w2 ) as a function of (A, Q) (b) of (A, Q) as a function of (w1 , w2 ) 2. Same questions for a triangular channel.
5.3 Kinematic wave Work out the formula λ (h) for the kinematic wave approximation in a rectangular channel, assuming that the wide channel approximation is applicable (h b). Compare it with the two wave speeds of the Saint Venant equations. Can the kinematic wave approximation account for backwater eects ?
5.4 Diusive wave Work out the formulae for D (h) and V (h) in the diusive wave approximation, assming a rectangular channel and the wide channel approximation. Compare V with the wave propagation speeds of the Saint Venant equations. 5
