CHAPTER 6

ROUTING FUNCTION - PART 2HYDRAULIC APPROACH

 

The lumped models to evaluate the flow rate at a desired location placed on a stream channel

were presented in Chapter 5. An alternative to using such a lumped model is to adopt a

distributed flow routing model that allow the flow rate and water level to be simultaneously

derived as functions of space and time in the channel system. This type of model gives a more

realistic approximation of the actual unsteady, nonuniform nature of flow propagation in a river

and much more information on the flow evolution. At the same time, the distributed flow routing

models can be used to simulate the runoff (a distributed process) over a watershed, to obtain the

flow hydrograph at the upstream section of a river reach. As all the distributed models, any

distributed flow routing model is formed by partial differential equations.

6.1. Saint-Vénant equations

For many practical applications, the velocity spatial variation within the flow cross section can be

ignored, so that the flow process can be approximated as varying in only the longitudinal

direction of the stream. This is the major assumption used to derive the Saint-Vénant equations,

which describe the one-dimensional unsteady open channel flow.

Some others hypotheses are necessary as well. In their original form (B. de Saint-Vénant, 1871),

these are as follows:

the velocity is constant across any section perpendicular to the longitudinal axis,

the streamline curvature is small and vertical accelerations are negligible, so that the

hydrostatic pressure prevails,

the effects of boundary frictions and turbulence can be accounted for through resistance

relationships analogous to those used for steady state flow,

the average channel bottom slope is small so that the cosine of the angle it makes with

the horizontal may be replaced by unity.

For the sake of clarity, various textbooks (Cunge, Holly and Verwey, 1980; Chow, Maidment and

Mays, 1988; Popa, 1997 etc.) underline some obvious assumptions as for example: the fluid is

incompressible and of constant density; the channel bed is fixed (without the scour or deposition

effects); the cross sections are of arbitrary shape and may vary along the channel axis (but in a

limited range); etc.

Under these assumptions, the basic Saint-Vénant equations may be obtained by several

approaches (see Strelkoff, 1969; Cunge a.o., 1980; Graf and Altinakar, 1996; etc.). This material

follows the Chow a.o. (1988) presentation, because of its generality and simplicity.

Continuity Equation

Consider an elemental control volume (c.v.) of length dx in a river. Though its spatial size is fixed,

the water level may vary within it, in time.

The continuity equation for an unsteady variable-density flow through this c.v. can be written as

(see Chow et al., 1988, or Popa, 1998, for Reynolds transport theorem applied to the extensive

property mass of fluid):

(6.1)

where:

 c.s

.specifies the surface of the elemental c.v.

  dA elemental area of c.s.

  V local velocity of stream

Figure 6.1 shows three views of this elemental reach of channel.

Figure 6.1. Definition sketch for derivation of unsteady flow equations.

Using the sign convention as in the Reynolds transport theorem (i.e. "-" for inflows and "+" for

outflows in/from c.v.), the last term for mass balance in equation 6.1 is:

(6.2)

where:

  Q flow discharge that enters into c.v. by the upstream section surface (u.s.)

 ∂

Q/∂x

rate of flow change along the stream axis, so that the outflow by the downstream

section surface (d.s.) becomes (Q + (∂Q/∂x)dx)

 q is accepted as a lateral inflow per unit length of channel, entering the c.v. lateral

surface (l.s.) by runoff over the banks, exfiltration from groundwater etc.

Assuming an average cross-sectional area A, the volume of the c.v. is dV=Adx, so that the rate

of change of mass stored within c.v. (first term in equation 6.1) can be written as:

(6.3)

where the partial derivative takes into account the time dependence of the water level.

By substituting equations 6.2 and 6.3 into 6.1, accepting the water density as constant, and

dividing through by  , the conservation form of the continuity equation results as:

(6.4)

which is valuable for a prismatic (with constant cross-sectional shape and bed slope along the

channel) or a nonprismatic waterway.

Momentum Equation

Momentum is the product of mass and velocity, and momentum flux through the flow section is

the product of the mass flow rate and velocity. By writing the Newton's second law as follows:

(6.5)

one can state that the sum of forces applied to the c.v. must be equal to the rate of change of

momentum stored within c.v. plus the net outflow of momentum across the c.s.

Concerning the last term in equation 6.5 and by similitude with equation 6.2 one obtains:

(6.6)

where Vq is the q's velocity component along the stream direction (0 - for perpendicular lateral

inflow).

The time rate of change of momentum stored in c.v. is found using that dV =

A.dx and  , i.e.:

(6.7)

For the sake of simplicity, only the following forces, applied to the c.v., are considered:

- Fg - the water weight component along the river bed (gravity force), given by:

(6.8.a)

because of assumption that α is small (i.e.  , the channel bottom

slope).

- Ff - frictional forces, including the boundary friction and turbulence effects. Following Chow

(1959), Chow a.o. (1988) etc., Ff is expressed as:

(6.8.b)

where the friction slope Sf is computed by some resistance equations used for steady flow

conditions.

- FPu , FPd - pressure forces, exerted as hydrostatic forces on the upstream and downstream

section surfaces, respectively.

Using the notations from Figure 6.1 where the elemental cross-sectional flow surface b.dy is

placed at the y elevation from the channel bed and has an immersed depth of (h-y), the total

hydrostatic pressure force on the upstream section surface is:

(6.8.c)

where:

 b(y

)local width at y elevation

  h the water depth

The hydrostatic force on the downstream section surface is obtained (Chow et al., 1988) as:

(6.8.d)

where   may be computed using the Leibnitz rule for differentiation of an integral, i.e.:

(6.8.e)

- FPb - pressure force due to the nonprismatic shape of the channel, i.e. the force exerted by

banks, in relation with the rate of change in width of the channel, ∂b/∂x, within c.v.

Therefore, this contribution may be expressed as:

(6.8.f)

Some others possible influences (as for example the wind shear force against the water free

surface, the local resistance due to the abrupt variation of the channel cross-section, etc.) can be

modelled as well, but are ignored in this presentation.

The sum of these forces along the x-direction is obtained with equations 6.8.a to 6.8.f as:

(6.9)

After substitution of the terms from equations 6.6, 6.7 and 6.9 into equation 6.5 and dividing

through by  , the conservation form of the momentum equation becomes:

(6.10)

if the lateral inflow is accepted as perpendicular to the main stream direction.

Because A = A(h) and dA = B.dh, the time derivative ∂A/∂t from equation 6.4 can be replaced

by B.∂h/∂t, so that the continuity equation is:

(6.11)

The two dependent variables into the above Saint-Vénant equations are the water depth h, and

the flow discharge Q. For many applications it may be preferable to use a different pair of flow

variables as principal ones (Q and z - the water surface elevation above a datum, for major

rivers; h and V - the flow velocity, for a canal flow, etc.). The corresponding forms of the Saint-

Vénant equations are derived by some simple mathematical manipulations.

As h = (z - zb) and ∂zb/∂x = -S0, the space-derivative ∂h/∂x from equation 6.10 can be written:

∂h/∂x = ∂z/∂x + S0. Due to the fixed bed assumption (zb- constant in time), the time-derivative

∂h/∂t = ∂z/∂t and the Q-z pair of Saint-Vénant equations is:

(6.12)

Replacing Q = VA in the time-derivative one obtains successively:

(6.13.a)

Concerning the space-derivative ∂Q/∂x, this becomes:

(6.13.b)

where the last term in equation 6.13.b is due to the arbitrary cross-sectional shape and vanish for

a prismatic canal. The V-h continuity equation (6.11) is then:

(6.14)

Because Q2/A = V2A , the derivative ∂(Q2/A)/∂x can be written as:

(6.15)

Using equations 6.13.a and 6.15.a into the moment equation, rearranging the terms and

accepting q = 0, one obtains the V-h momentum equation as:

(6.16)

where the continuity equation (6.14) was applied between the brackets { }.

Without the lateral inflow and for a rectangular cross-section (when A = Bh) the two V-hSaint-

Vénant equations are as follows:

(6.17)

and named the nonconservation form.

The friction slope Sf is expressed as in steady state conditions, usually by a Manning or a Chezy

type relation, i. e.:

Manning: 

Chezy: 

(6.18)

where:

  n Manning roughness coefficient

  R hydraulic radius (= A/P with P: the wetted perimeter)

  K conveyance factor

  C Chezy resistance coefficient

The appropriate values of Manning's n and Chezy's C are empirical parameters linked to the river

bed composition and the flow regime. These values are often accepted as calibration parameters

for a given routing model.

6.2. Classification of distributed routing models

Any of the above sets of Saint-Vénant equations is usually called a dynamic wave modeland

represents the most complete one-dimensional distributed routing model for flow propagation

along the channels. Such a model includes into the momentum equation all the terms significant

for the flowing process, namely:

the gravity force contribution - proportional to the bottom slope S0;

the friction force term - proportional to the friction slope Sf;

the pressure force effect - proportional to the change in water depth along the stream

∂h/∂x;

the convective acceleration term - which describes the change in momentum, induced by

the velocity change ∂V/∂x (or ∂Q/∂x) along the channel, and

the local acceleration term - describing the momentum change due to the velocity time-

variation ∂V/∂t (or ∂Q/∂t).

Because of the last three contributions, a dynamic wave model may perform a realistic simulation

of the backwater effects induced by a local flow perturbation (in water level or discharge), under

subcritical flow conditions. These backwater effects are important when the river slope is small.

A simpler model - named the diffusion wave model - neglects the acceleration terms but

incorporates the pressure force effect into the momentum equation. If the inertial terms are really

negligible, this model is a good alternative, being capable of representing the backwater

influence of tributaries, dams, etc.

The simplest distributed model is the kinematic wave model, which neglects the pressure and

inertial terms, so that the momentum equation becomes S0 = Sf , assuming the balance between

the friction and gravity forces. Such a situation may be accepted in rivers with a sufficiently steep

slope and without backwater effects.

Indifferently of the momentum equation form, any distributed routing model must include the

continuity equation. A summary of this classification is shown in Table 6.1.

Table 6.1. Distributed flow routing models based on the Saint-Vénant equations.

During the natural flood propagation, both the kinematic and dynamic wave type of motion may

be present along the stream. Generally, the kinematic type appears where the channel slope is

greater, into the supercritical flow conditions. On the contrary, the dynamic wave motion prevails

on the river reaches having a small bed slope.

In both cases, a wave (a variation in flow discharge or water surface elevation) travels along the

stream with the wave celerity. This celerity depends on the wave type and may be different from

the mean water velocity V. Also, while a kinematic wave is propagated only in the downstream

direction, the dynamic wave front can travel both upstream and downstream from the disturbance

position.

For a channel of arbitrary cross-section, the dynamic wave celerity is given as:

(6.19)

The dynamic wave moves downstream with the velocity V + cd, and in the upstream direction

with the velocity V - cd. To propagate upstream a certain perturbation the condition is that V< cd,

therefore to have a subcritical flow regime (a Froude number Fr = V/cd<1). If this condition exists

along the channel during the flood event, a dynamic wave model is preferable. In an opposite

situation, the dynamic wave model is incapable to transmit the backwater effects and the

kinematic wave model becomes profitable by its simplicity.

6.3. Analytical solution of the kinematic wave model

In particular circumstances (rectangular cross-section, large width, without lateral inflow) the

kinematic wave model allows an analytical solution, which may be useful in practical applications.

The momentum equation of this model Sf = S0 can be expressed by Manning's equation (6.18) as

a single-valued relationship between the wetted area A (or water depth h) and the flow

discharge Q, at a given spatial position x. Indeed:

(6.20)

where  ;  , and R = A/P has been used.

Then, the continuity equation (6.4)   can be written with Q as dependent variable,

in the form:

or (6.21)

Dividing a differential change in Q, i.e.:

- by dt, one obtains:

- by dx, one obtains:

(6.22)

By comparing equations 6.21 and 6.22, it can be seen that along the lines in the x-t plane having

the equation:

(6.23)

a constant discharge (dQ/dt = 0) will be propagated.

Such a line is called a characteristic line, and equations dx/dt = dQ/dA = ck and dQ/dx = 0 are the

characteristic equations for the kinematic wave model.

The term ck is the kinematic wave celerity. An observer moving at this velocity along the stream

would see a constant discharge. Because dA = B.dh, the kinematic wave celeritycan also be

written in terms of the water depth h as:

(6.24)

For a wide, rectangular canal and accepting  , one obtains successively:

so that the kinematic wave celerity can be expressed in terms of the flow discharge as:

(6.25)

where   is a constant coefficient for a given canal reach.

The solution for Q(x,t) requires knowledge of the initial conditions Q(x,0) (i.e. the values of the

flow discharge along the stream at the beginning of the analysed time period) and the boundary

condition Q(0,t) (i.e. the inflow hydrograph as a function of time, at the upstream section of the

given channel).

To compute the outflow hydrograph at the downstream end of the channel placed at the

distance L from the upstream section (i.e. the function Q(L,t)), the equation of the characteristic

line:

will be used by solving this relation as:

By integration:

or(6.26)

where td(Q) is the time after which a discharge Q appearing in the upstream section at

timetu(Q) will be passed through the downstream end of the channel.

Equations 6.26 and 6.25 constitute an analytical solution because they allow determining

thetd time for any entering (Q, tu) values. This model preserves the peak flow value along the

channel reach, but deforms the inflow hydrograph shape. If the lateral inflow q ≠ 0,

then Qand ck vary along the characteristic lines that become some curved lines, and the peak

value will be modified.

6.4. Numerical solution of the distributed routing models

Saint-Vénant equations are partial differential equations that, in general, must be solved by

numerical methods. These methods can be applied on the original Saint-Vénant equations, or for

solving a mathematically transformed set of equations, called the characteristic form of Saint-

Vénant equations. For example, the V-h pair (6.17) is equivalent to the characteristic form:

(6.27)

In any variant, many finite-difference methods are available, but the characteristic form may be

also treated by some characteristic methods.

The last category exploits the fact that the two partial differential equations (6.27) can be

replaced by two pairs of two ordinary differential equations as follows:

and (6.28)

The flow variables V(x,t) and h(x,t) can then be found by integrating the first equation of each

pairs in 6.28, along the corresponding characteristic curve in the x-t plane. A detailed description

of characteristic methods may be found in Popa (1997).

In finite-difference methods, calculations are performed on a grid placed over the interesting flow

domain into the x-t plane. This computational grid is defined by some equal or variable space

and time steps, Δx and Δt, respectively. A network of discrete points is thus obtained and the

flow variables Q, z (or Q, h; V, z; V, h) are derived only for this finite number of grid points. Figure

6.2 shows a typical computational grid. The spatial positions of the grid points are denoted by

index j and the time moments by index i.

Figure 6.2. Finite-difference computational grid.

As principle, finite-difference methods transform governing partial differential equations into a set

of algebraic (linear or nonlinear) finite-difference equations, which are solved to allow the values

of flow variables in a grid point or in all grid points on a time line. These finite-difference

equations are derived by approximating the time and space derivatives with some finite-

difference expressions. Not only the derivatives, but also the other terms into the Saint-Vénant

equations must be defined in a certain manner.

Concerning the space derivative of a continuous function f(x,t) at time moment ti and space

position xj on the grid in Figure 6.2, this can be approximated as:

 a forward difference approximation;

 a backward difference relationship;

 a central difference expression,

(6.29)

in which fji represents the value of f at grid point (xj ,ti).

In an analogous manner, the time derivative may be defined in several different ways as, for

example:

; 

(6.30)

Usually, the value of the function f(x,t) is accepted at grid point (xj,ti) as fji , but some different

approximations can also be decided.

A finite difference method must employ a certain type of finite-difference scheme. These

schemes are grouped in two major classes: explicit and implicit finite-difference schemes.

Explicit schemes are those in which the flow variables at any point j and time level i+1 may be

computed using only known data at a few adjacent points on the time line i. These schemes do

not lead to a system of algebraic equations, but rather to only two finite difference equations for

each grid point (xj,ti+1). By solving the two equations the unknown values of flow variables are

obtained and then, the computation proceeds to the next grid point along the time line i+1.

In implicit schemes, finite-difference expressions used to approximate the space and/or time

derivatives at grid point (xj, ti+1) include the unknown values of flow variables at a few adjacent

points on the time line i+1. Consequently, a system of algebraic equations is produced for a

given time line i+1 and by solving this system, all unknown values are simultaneously determined

at time level i+1.

Replacing the continuous original problem with an integration over a discrete computational grid

introduces numerical errors into the results. A finite-difference scheme is stable if such errors are

not amplified during computation from one time level to the next. The numerical stability depends

on the size of the time and space steps and on some flow characteristics. The Courant condition:

(6.31)

is a necessary but insufficient condition for stability of an explicit scheme.

Therefore, any explicit scheme is conditionally stable, the Courant condition requiring to work

with small time steps as compared with the physical phenomena evolution. Despite their

computational simplicity, the explicit methods are seldom used in river modelling for reasons of

this stability restriction.

On the other hand, the implicit schemes may appear more complicated, but can generally be

made unconditionally stable for large computation steps and with little loss of accuracy.

Another distinction among schemes belonging to the same class is related to the way in which

the non derivative terms (such as Sf(Q,h), B(h), A(h) etc.) are discretized. Because these terms

are functions of dependent variables, their treatment induces a linear or nonlinear feature of the

finite-difference equations.

A lot of finite-difference schemes will be presented in more details within the next sections.

6.5. Explicit finite-difference scheme for dynamic wave model

The simplest explicit scheme approximates the partial derivatives at grid point (xj,ti+1) in terms of

the values at adjacent points (xj-1,ti), (xj,ti) and (xj+1,ti) (Figure 6.3) as follows:

;(6.32)

where f ≡ Q (or V), z (or h), etc.

Figure 6.3. Derivative approximation into the simplest explicit scheme.

However, each particular problem requires more explanations.

Consider the pair Q-h of Saint-Vénant equations 6.10 and 6.11, which may be also written as:

(6.33)

where:

  Fr local Froude number ( )

  Sf expressed by Manning's equation ( , with  )

Using the notations:

 ;  ;(  ),(6.34)

and the following finite-difference approximations:

; ; ;(6.35)

the two finite-difference equations corresponding to equation 6.33 are obtained as:

(6.36)

First equation (6.36) gives an explicit relationship for the unknown hji+1 value, and the second for

the unknown Qji+1 value, i.e.:

(6.37)

At the boundary ends, the characteristic forms are used. For the Saint-Vénant equations (6.33),

these are as follows:

(6.38)

In equation 6.38 the superior sign operates is applicable at the downstream end and will be

approximated by some backward finite-difference expressions, while the equation with inferior

sign is valuable at the upstream end and its discretization appeals to forward finite-difference.

For upstream boundary, one obtains:

(6.39.a)

which must be completed with a supplemental relationship between the two unknown

variables Q0i+1 and h0

i+1 (usually the inflow hydrograph Q0 = fu(ti+1) ).

At downstream boundary, one obtains:

(6.39.b)

which is solved together with a relation as QNi+1 = fd(hN

i+1) (where the function fd may be the rating

curve of this section, for example). By index N is marked the last spatial grid point, placed at xN =

L.

This scheme is obviously a linear one, because all terms in equations 6.37 and 6.39 are

evaluated using the known values at time level i (excepting the unknown flow

variables Qji+1and hj

i+1 from xj at current time level i+1).

6.6. Numerical solution of kinematic wave model

First form (6.21) of the kinematic wave model is used to illustrate an explicit linear scheme on a

finite-difference cell as show in Figure 6.4.

A backward finite-difference is used to approximate the space derivative, while the time

derivative is usually expressed at the same xj+1 spatial position. In order to obtain a linear

scheme, the nonderivative term αβQβ-1 must be evaluated for a known Q value, which here is

accepted as mean between the two diagonal values Qji+1 and Qi

j+1.

Figure 6.4. Finite-difference cell for linear kinematic wave solution.

Consequently, the finite-difference form of equation 6.21 is:

(6.40)

and the unknown Qj+1i+1 results as:

(6.41)

Starting with the inflow hydrograph value Q at the time level ti+1, the computation sequentially

proceeds from upstream towards downstream grid points, along the current time line i+1.

However, this scheme supposes that within the coefficient α, the wetted perimeter P remains

constant. On the other hand, by using Q instead of A into the time derivative ∂A/∂t, relative

computational errors are decreased. Indeed, taking the logarithm of equation 6.20, i.e.

,

and differentiating, one obtains:

,

where β is 0.6 for the Manning's equation. It follows that the estimation error in Q would be

magnified by 1/0.6 ≈ 1.67 if the cross-sectional area A is used as dependent variable.

A different form of the linear kinematic wave model is due to Cunge (1969) and

namedMuskingum-Cunge method. The finite-difference cell for this scheme is shown in Figure

6.5.

Figure 6.5. Finite-difference cell for the Muskingum-Cunge method.

The kinematic wave equation:

(6.42)

is replaced by a finite-difference equation around a discrete point as P in the x-t plane. This point

is located midway between the i-th and (i+1)-th time lines, and uncentred within the space

step Δx by a weighting coefficient θ (0 ≤ θ ≤ 1).

Finite-difference form of equation 6.42 appears here as:

(6.43)

with backward finite-difference for the space derivative, averaged between the time levels, and

with finite-difference for the time derivative weighted averaged between the space positions.

Rearranging equation 6.43 for the   solution, one obtains:

(6.44)

where Ci the coefficients are:

; ;

(6.45)

and K = Δx/ck, the sum of these coefficients being equal to the unity.

By comparing equations 6.45 with the hydrologic Muskingum model (see Chap. 5), a formal

similitude may be noted despite the different theoretical bases.

Cunge showed that when K and θ are taken as constant, equation 6.44 is an approximate

solution of the kinematic wave model, while if

 and   ,

where ck corresponds to Q and B, then equation 6.44 can be considered an approximate solution

of the diffusion wave for a channel with S0 constant. He further demonstrated that for numerical

stability it is required that 0<θ<0,5. Time step Δt must be chosen as Δt ≤ K.

A simple, nonlinear kinematic wave scheme of the model (equation 6.42) with a lateral inflowq ≠

0, gives a form as relation 6.43 for the finite-difference equation but having  (ck)m instead of ck and

1/2(qji+1+qj

i) as right-hands side. Here qj corresponds to the space step Δx, and(ck)m is given by:

(6.46)

where:

  m an iteration index

  χ a weighting coefficient (0 ≤ χ ≤ 1)

  ck given by equation 6.25 in terms of Q

The unknown   is then derived at the m-th iteration step as:

,

where:

;  ;

; 

(6.47)

The iterative process can be started with

and followed until a convergence criterion as, for example:

is fulfilled. The error criterion is chosen at a suitable little value.

6.7. Implicit finite-difference scheme for dynamic wave model

Because of their favourable computational properties the implicit schemes are most widely used

in flood propagation along the channel.

Firstly, a six-point linear implicit scheme is presented in connection with the Q-h pair (6.33) of

Saint-Vénant equations, and finally a four-point nonlinear implicit scheme for natural river

conditions (Q-z pair) is detailed.

Figure 6.6. Finite-difference cell for six-point implicit scheme.

As shown in Figure 6.6, the flow variables from six grid-points are used to approximate the

derivative terms in equations 6.33, at the xj space position, by the following expressions:

(6.48)

The weighting coefficient θ must be 0.5 < θ ≤ 1 to ensure the unconditional stability of the

scheme.

Excepting the friction slope Sf, the nonderivative terms are replaced by   .

To take into account the nonlinearity of Sf in respect to the dependent variables Q and h, a Taylor

series expansion of this term is accepted as:

(6.49)

where   and  .

But

 and (6.50)

Note that (as in Section 6.5) the form  allows that the sign of the friction losses to be

according to the local flow direction at any time moment.

Using these approximations into the dynamic wave model (6.33) the following two algebraic,

linear, finite-difference equations are obtained at spatial position xj :

(6.51)

where Cj coefficients depend only of the known solution at the time level i. These equations

contain the six unknown dependent variables Q and h at xj and the two adjacent points xj-

1and xj+1.

For a computational grid having N spatial position (therefore 2N unknown flow variables at any

time level), a system of 2(N-2) algebraic, linear equations is obtained by writing the two

equations 6.51 at all interior grid points. The four remaining necessary equations are derived

using the characteristic forms (6.38) (two equations), and the boundary conditions, respectively

(also two equations). This algebraic, linear equations system is solved by an appropriate

numerical method to give all unknown values Q and h at the current time leveli+1.

The four-point nonlinear implicit scheme is presented in connection with the following dynamic

wave model:

(6.52)

The term β is known as the momentum (or Boussinesq) coefficient. It accounts for the

nonuniform distribution of velocity into the cross-sectional area of the channel and his value

ranges from 1.01 (straight prismatic channel) to 1.33 (river valley with floodplains). The last term

into the momentum equation (6.52) introduces the frictional resistance of wind against the free

surface of water. The wind velocity is W, in a direction at angle αw to the water velocity and cw is a

friction drag coefficient.

This dynamic wave model can also be used to simulate the flood propagation within a branch-

network system of channels and the four-point implicit scheme lightens such a task.

Consider a channel segment defined by the two cross-sections placed at xj and xj+1positions and

having the length Δxj as shown in Figure 6.7. The various terms from equations 6.52 are

approximated in a point as P, centred in space and placed in time according to a weighting

factor θ.

Figure 6.7. Finite-difference cell for four-point implicit scheme.

The time and space derivatives are expressed, respectively, as follows:

(6.53.a)

(6.53.b)

(excepting ∂A/∂x, where θ =1).

The nonderivative terms are approximated by:

(6.53.c)

where χ is a weighting factor, similar to θ.

For θ > 0, the scheme is of implicit type and if θ > 0.5, it is unconditionally stable. The χcoefficient

ranges from 0 to 1; if χ=0, the f value is obtained exclusively from previous time step quantities,

while χ =1 produces a fully forward approximation for f.

If χ ≠ 0, a nonlinearity appears due to any f(t) value, because of its dependence on the unknown

flow variables at (i+1)-th time line. Together with the nonlinear term   from the friction slope,

this aspect imposes an iterative procedure on each time step. Into the first iteration one

accepts  , while for the next iterations equation 6.53.c is replaced by:

(6.53.d)

with m an iteration index.

The term  is expressed as  , where equation 6.53.c is used for Q(P) and

respectively, equation 6.53.d for  .

The two Saint-Vénant equations (6.52) are then replaced by the following finite difference

equations:

where:

;

; ;

; ; ;

(6.54)

The above coefficients are all evaluated from known values at the previous time step or previous

iteration on time step.

Defining a two-component vector of state at the i-th cross-section as:

,

one may write equations 6.54 as a matrix transformation equation for the j-th segment:

(6.55)

where the four elements of the matrix are:

; ;

; ;

and the two elements of the uj matrix are:

;

Through successive application of the segment-transformation equation, a branch transformation

equation can now be obtained as:

(6.56)

where the branch-transformation matrices Ub and ub are given by:

(6.57)

Here a branch is a river reach divided into a number of computational segments Δxj, j = 1..........,

N-1 as in Figure 6.8.a.

Figure 6.8. A single branch (a) and a network of branches (b) system.

If the flow problem refers to a single branch system, the two algebraic nonlinear equations

corresponding to the matrix equation (6.56) are to be completed with the two boundary condition

equations. The resulted system of equations is solved to give the four unknown Qand z values

into the extreme 1 and N cross-sections, at the current iteration on the time linei+1. All

intermediary flow variables value can then by derived using, successively, the segment-

transformation equation (6.55). The iterative process is repeated until a convergence criterion is

fulfilled.

If the flow problem appears in a network of branches, as in Figure 6.8.b, the branch-

transformation matrix equations are written for each branch and the six corresponding algebraic

equations must be completed with the boundary conditions at the three external junctions, plus

the compatibility conditions at the internal junction (i.e. the flow discharges balance and the

equality of the water surface levels). By solving the resulted system of equations, the unknown

discharges and stages at the six extremities of the branches are derived and the computation is

then resumed as for a single-branch case.

This weighted, four-point nonlinear implicit scheme is recommended (Schaffranek, Baltzer,

Goldberg, 1981) because of its inherent computational efficiency, stability and versatility with

respect to the application of external and internal boundary conditions.

6.8. Final remarks

Hydraulic approach to the routing function is justified by the complexity of flow propagation

process through a river or a network of rivers.

Even if the hydrological interest in the routing problem can be satisfied with some lumped type

models, safety reasons for any riverine project create the necessity for more accurate

computation of flow variables.

The knowledge of flood water level is needed because this level delineates the flood plain and

determines the required height of structures such as levees and bridges. Most lumped routing

and kinematic wave models assume a single-valued function for the discharge-stage

relationship, as into the Chezy form (6.18):

(6.58)

where the bed slope S0 replace the friction slope. During the flood event, this assumption is not

realistic. If the form of the momentum equation (6.17) is solved for Sf and this expression

replaces into equation 6.58, one obtains a more complex discharge-stage relationship:

(6.59)

Visually, the difference is shown in Figure 6.9., where the looped rating curve corresponds to

equation 6.59, i.e. to the distributed routing models as dynamic or diffusion type waves.

Figure 6.9. Discharge-stage relationship into the lumped and the dynamic wave models.

For a given level, the discharge is usually higher on the rising limb of the flood hydrograph than

on the recession limb. Inversely, for a given discharge, the water level on the recession limb

overtakes the values corresponding to the uniform flow conditions, and to the rising limb,

respectively. These differences can be important and only the dynamic (or diffusion) wave model

allows an accurate simulation.

However, the practical application of a distributed routing model is rather a difficult matter,

especially for a natural river system.

 

Bibliography

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Chow, V. T., D. Maidment, and L. Mays. 1988. Applied Hydrology. McGraw-Hill Book Company,

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Cunge, J. A. 1969. On the subject of a flood propagation method (Muskingum method). J.

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Cunge, J. A., F. M. Holly, and A. Verwey. 1980. Practical aspects of computational river

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