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Chapter 6One-Dimensional Modeling of Flowsin Open Channels
Dariusz Gąsiorowski, Jarosław J. Napiórkowskiand Romuald Szymkiewicz
Abstract In this chapter, modeling of the unsteady open channel flow usingone-dimensional approach is considered. As this question belongs to the well-knownand standard problems of open channel hydraulic engineering, comprehensivelypresented and described in many books and publications, our attention is focused onsome selected aspects only. As far as the numerical solution of the governingequations is considered, one can find out that essentially there are no problems withprovided accuracy. Usually, the implementation of the Saint Venant equations (i.e.the full dynamic wave model) for any case study is successful as long as the basicassumptions introduced during their derivation are fulfilled. Otherwise, somecomputational difficulties can occur. For this reason we would like to draw thereaders’ attention only to such situations when special computational tricks orsimplification of the governing equations should be applied.
Keywords 1D open channel flow � Numerical solutions � Simplified equations �Linear approximation � Lumped parameter simulators
D. Gąsiorowski � R. SzymkiewiczFaculty of Civil and Environmental Engineering, Gdańsk University of Technology,Narutowicza 11/12, 80-233 Gdańsk, Polande-mail: [email protected]
R. Szymkiewicze-mail: [email protected]
J.J. Napiórkowski (&)Institute of Geophysics, Polish Academy of Sciences, Ks. Janusza 64,01-452 Warsaw, Polande-mail: [email protected]
© Springer International Publishing Switzerland 2015P. Rowiński and A. Radecki-Pawlik (eds.), Rivers—Physical, Fluvialand Environmental Processes, GeoPlanet: Earth and Planetary Sciences,DOI 10.1007/978-3-319-17719-9_6
137
6.1 One-Dimensional Unsteady Open Channel FlowEquations
Open-channel unsteady flow may be considered as a process of propagation of longwaves with small amplitudes in shallow water. The accelerations and velocities invertical direction are negligibly small in relation to the accelerations and velocitiesin horizontal directions. Domination of one component of the velocity vector is atypical attribute of the open channel flow. However, this dominated flow velocityvector component varies over wetted cross-section of a channel, so it is a functionof 3 spatial co-ordinates and time. Usually, the flow velocity vector componentrelated the longitudinal channel axis is averaged over wetted cross-section.Consequently, 2 spatial independent variables are eliminated. This feature allows usto consider the flow process as spatially 1D phenomenon. Similarly to the othertypes of water flows, the governing equations are derived starting from two prin-ciples of conservation:
• mass conservation law,• momentum conservation law.
For the first time it was carried out in 1871 by Barré de Saint Venant to describe theprocess of propagation of the flood waves (Chanson 2004). Assuming that:
– motion of water is gradually varied,– channel bed slope is small,– flow is considered as a one-dimensional process,– velocity distribution over cross-section is uniform,– pressure is governed by the hydrostatic law,– friction slope is estimated as for the steady flow,– lateral inflow does not influence the momentum balance.
The following system of the partial differential equations has been derived
@A@t
þ @Q@x
¼ 0: ð6:1Þ
@U@t
þ U@U@x
þ g@H@x
� g � sþ g � S ¼ 0: ð6:2Þ
where:
t time,x space co-ordinate,A wetted cross-sectional area of the channel,Q discharge,U flow velocity averaged over wetted cross-section,H flow depth,g acceleration due to gravity,
138 D. Gąsiorowski et al.
s channel bottom slope,S friction slope.
In the literature, these equations are called the system of Saint Venant equations.Equation (6.1) is the continuity equation derived from the mass conservation
principle, while Eq. (6.2) is a dynamic equation derived from the momentumconservation principle. It should be added that the system of Saint Venant equationscan be expressed in many forms depending on the assumed dependent variables,geometry of the channel and other factors taken into account. All of these forms aresystematically reviewed; see, for instance, Singh (1996). Moreover, theirs deriva-tion can be performed in various ways. It seems that the most consistent approach isto derive the unsteady open channel flow equations from general equations ofhydrodynamics, i.e. from the Reynolds averaged Navier-Stokes equations. In suchapproach, the effects of introduced simplifications and assumptions can be easilyfollowed. However, derivation of the governing equations for unsteady flow can becarried out in a simplified manner, by balancing the fluxes and forces acting on theconsidered control volume with a priori accepted assumption of uniform velocityflow distribution over a channel cross-section. Next, when the governing equationsare derived, an additional factor is introduced to correct the momentum. Moreover,in particular circumstances, apart from the factors taken by Saint Venant intoconsideration, some additional ones can be included into the continuity anddynamic equations. Consequently, more developed forms of both equations occur:
@A@t
þ @Q@x
¼ q: ð6:3Þ
@U@t
þ b � U @U@x
þ g@H@x
� g � sþ g � S ¼ aq � H Wj jW � 1
q@Pa
@xþ lD
q@2U@x2
ð6:4Þ
where:
q lateral inflow,β corrective parameter,ρ water density,W component of the wind velocity acting along channel axis,a empirical coefficient characterizing friction between water surface and
atmosphere,Pa atmospheric pressure,μD coefficient of longitudinal dispersion of the momentum.
The factor β which corrects the momentum (Abbott 1979):
b ¼ 1U2 � A
Z ZA
u2 � dA ð6:5Þ
6 One-Dimensional Modeling of Flows in Open Channels 139
is called the Boussinesq coefficient (Chanson 2004). It represents the ratio of theactual momentum and the momentum calculated with the averaged velocity U.
Equation (6.4) is a more general form of the dynamic equation for 1D unsteadyopen-channel, gradually varied flow. It contains all significant components whichcan determine the flow process. Namely, at the right hand side of Eq. (6.4) thesucceeding terms represent the following processes: wind action on the watersurface, influence of the spatial variation of the atmospheric pressure and diffusivetransport of the momentum. The only process omitted in the derivation is the lateralwater inflow distributed along the channel.
In practical applications it is more convenient to use the flow rate and the waterstage instead of flow depth and velocity as the dependent variables. The equationswith new variables can be obtained using the following relations:
H ¼ h� Z; ð6:6Þ
s ¼ �@Z=@x; ð6:7Þ
Q ¼ U � A ð6:8Þ
where:
h water stage with regard to assumed datum,Z bed elevation with regard to assumed datum.
If all the terms at the right hand side are neglected and, additionally, if we take intoconsideration the fact that the wetted cross-sectional area is a function of the waterstage, i.e. A = A(h) while the water stage is a function of time h = h(t), then thefollowing well-known version of the unsteady flow equations is obtained:
@h@t
þ 1B@ Q@x
¼ qB
ð6:9Þ
@Q@t
þ @
@xb � Q 2
A
� �þ g � A @h
@xþ g � A � S ¼ 0 ð6:10Þ
where B is channel width at the water surface level whereas the friction slope S isestimated as for the steady flow usually using the Manning formula, i.e.
S ¼ n2
R4=3
Qj jQA2 ð6:11Þ
where R is hydraulic radius of a channel cross-section while n is the Manningroughness coefficient. Equations (6.9) and (6.10) constitute the most popularmathematical model of the unsteady open channel flow, applied in many profes-sional codes widely disseminated among the engineers.
140 D. Gąsiorowski et al.
6.2 Initial and Boundary Conditions
Both continuity and dynamic equations constitute the system of first orderquasi-linear partial differential equations. Its solution is searched in the followingdomain:
0� x� L and t� 0
where L denotes the length of considered channel reach. To this order, the followinginitial-boundary problem is formulated: determine the functions h(x, t) and Q(x, t)which satisfy the Eqs. (6.9) and (6.10) in the solution region and which simulta-neously satisfy the auxiliary conditions imposed over its limits. The system of SaintVenant equations has two families of the characteristics. As both families are real,its type is classified as hyperbolic. The run of characteristics is related to the flowregime. For supercritical flow both positive and negative characteristics have asimilar slope, whereas for the subcritical flow their slopes have opposite signs.Assume subcritical (tranquil) flow over the considered channel reach. For such kindof flow the run of characteristics is displayed in Fig. 6.1. Remember that for thehyperbolic equations the following rule is valid: at any limit the number of imposedadditional conditions must be equal to the number of the characteristics which enterthe solution domain from this limit. Therefore, for the assumed subcritical flow thefollowing additional information must be prescribed (Fig. 6.1):
• initial conditions
h x; t ¼ 0ð Þ ¼ hi xð Þ and Q x; t ¼ 0ð Þ ¼ Qi xð Þ for 0� x� L;
• boundary conditions
h x ¼ 0; tð Þ ¼ ho tð Þ or Q x ¼ 0; tð Þ ¼ Qo tð Þ for t� 0
t
xx = 0 x L =
+
+
+−
−
−
0
solution domain
Fig. 6.1 Two families of the characteristics of the Saint Venant equations for the subcritical flow
6 One-Dimensional Modeling of Flows in Open Channels 141
and
h x ¼ L; tð Þ ¼ hL tð Þ or Q x ¼ L; tð Þ ¼ QL tð Þ for t� 0:
where the functions hi tð Þ;Qi tð Þ; ho tð Þ orQo tð Þ and hL tð Þ orQL tð Þ are known.
As far as practical implementation of the system of unsteady flow equations isconsidered, the initial conditions, i.e. the functions hi(x) and Qi(x) for0 ≤ x ≤ L must be computed. To this order, the steady gradually varied flow overconsidered open channel reach is assumed at t = 0. This kind of flow is governed bythe equations resulting from Eqs. (6.9) and (6.10). When the time derivatives areneglected, these equations can be reduced to the following ones:
dQdx
¼ q; ð6:12Þ
ddx
hþ a � Q2
2g � A2
� �¼ � n2 � Q2
R4=3 � A2: ð6:13Þ
Since Eq. (6.10) has been reformed to the mechanical energy equation then,instead of the momentum correction using the corrective factor β, the correction ofenergy using the factor α should be done. To obtain the water surface profilehi(x) and the flow rate distribution Qi(x) along considered channel reach of length L,the ordinary differential Eqs. (6.12) and (6.13) are solved numerically with thefollowing initial conditions: Qi x ¼ Lð Þ ¼ Q0 and hi x ¼ Lð Þ ¼ h0, where Q0 and h0are initial values of the flow rate and the water stage.
Note that numerical integration of Eqs. (6.12) and (6.13) using the implicittrapezoidal rule leads to the well-known step method (Chow 1973), commonlyapplied in hydraulic engineering for the steady, gradually varied flow. Usually thestep method is derived directly from the principle of energy conservation appliedfor two neighboring cross-sections lying along channel axis, i.e. for a discretesystem.
Another important problem is dealing with possibility of determination of therequired boundary condition at the downstream end x = L. Proper formulation ofsolution problem for the Saint Venant equations requires to impose at the down-stream end either the function hL(t) or the function QL(t). Unfortunately, usually thisrequirement cannot be fulfilled. Consequently, instead of one of these functions, arelation between them is imposed, i.e. QL tð Þ ¼ f hL tð Þð Þ. Depending on the actualcircumstances, such a relation can take the form resulting, for instance, from:
• formula for discharge over a weir if such a weir exists at the downstream end,• condition for the critical flow if free outflow from a channel exists,• mass balance written for the reservoir if the considered channel enters the
reservoir.
Very often, when the downstream end is an ordinary channel cross-section, as arelation between both functions the rating curve can be imposed. However, this
142 D. Gąsiorowski et al.
should be realized carefully since the rating curve is an appropriate relation betweenthe water stage and the discharge for the steady uniform flow only. Consequently,in the case of unsteady flow such a condition affects the computed results in thevicinity of the downstream end. To reduce influence of such inappropriate condi-tion, the considered channel reach should be artificially extended so that theimproper condition acts far from the controlled channel cross-section.
6.3 Numerical Solution of the 1D Unsteady Open ChannelFlow Equations
The open channel unsteady flow equations can be solved using the numericalmethods only. This means that we expect to obtain the approximated solution. At thevery beginning to this order, the method of characteristics has been applied. Itsdetailed description for flow with free surface was given by Abbott (1979). Someinconveniences related to the non-linearity of the Saint Venant equations andnon-uniform grid points, very troublesome in the natural channels caused that thismethod has been given up. It was replaced by various schemes of the finite differ-ence method, which appeared to be the most suitable for solving the 1D partialdifferential equations. During more than last 40 years, many algorithms have beenproposed. Their review is given by Cunge et al. (1980) and Singh (1996). Amongthem one can find the explicit and implicit schemes. Some of them use the staggeredgrid while others non-staggered one. The available schemes represent differentapproximation accuracy. Many years of applications provided valuable experienceswhich allowed to distinguish the most interesting methods for solution. It seems thatthe most willingly applied is the finite difference implicit four point scheme knownas the box scheme (Liggett and Cunge 1975: Cunge et al. 1980). One can find outthat this scheme is a basic numerical tool for solving the unsteady flow equations. Itis highly appreciated by the users for the following features (Cunge et al. 1980):
• scheme uses non staggered grid which allows computing of both unknownfunctions at the same node,
• scheme relates the variables coming from both neighboring nodes, xj and xj+1,which allows to use non-uniform grid point, i.e. variable space interval;
• scheme represents variable order of accuracy controlled by two involvedweighting parameters;
• scheme ensures exact solution of the system of linear wave equations forappropriately selected values of the space and time step, i.e. Δx and Δt,
• - scheme is implicit and absolutely stable so that for the reason of numericalstability no restriction for the value of time step exists,
• scheme allows to introduce the imposed boundary conditions in a very simplemanner.
6 One-Dimensional Modeling of Flows in Open Channels 143
Let us remember briefly the idea of box scheme and its most important numericalproperties. For numerical solution of the Saint Venant equations using the finitedifference box scheme, the solution domain is covered with a grid of dimensionsΔx × Δt as presented in Fig. 6.2. In a single mesh, containing 4 nodesj; kð Þ; jþ 1; kð Þ; j; k þ 1ð Þ and jþ 1; k þ 1ð Þ, approximation of all derivatives andfunctions is carried out at point P using the linear interpolation between the men-tioned nodes. Its position is determined by two weighting parameters, ψ and θ, bothranging from 0 to 1.
Approximation of the partial differential equations in each mesh provides thesystem of non-linear algebraic equations, which in each time step is completedusing the imposed boundary conditions and next it is solved using the iterations.
Numerical stability analysis carried out using the Neumann method for thefollowing system of linear wave equations
@H@t
þ H0@U@x
¼ 0; ð6:14Þ
@U@t
þ g@H@x
¼ 0; ð6:15Þ
where:
H flow depth,U flow velocity,H0 averaged flow depth considered as constant,
obtained by simplification of Eqs. (6.1) and (6.2), shows that the box scheme isabsolutely stable on the condition that:
k+1
Δx
ψΔx
θΔtΔt
(1 )− θ Δt
(1−ψ)Δx
P
t
k
j j +1
x
Fig. 6.2 Mesh applied by thefinite difference implicit boxscheme
144 D. Gąsiorowski et al.
w� 1=2 ð6:16Þ
h� 1=2 ð6:17Þ
Very interesting conclusions result from accuracy analysis performed for the samesystem using the modified equation approach (Fletcher 1991). Replacing of allnodal values of both functions H and Q in algebraic equations approximating thepartial differential ones provides the following system of modified linear waveequations:
@H@t
þ H0@U@x
¼ m1@2H@x2
þ m2@2U@x2
þ e1@3H@x3
þ e2@3U@x3
þ . . .; ð6:18Þ
@U@t
þ g@H@x
¼ m1@2U@x2
þ m3@2H@x2
þ e1@3U@x3
þ e3@3H@x3
þ . . .; ð6:19Þ
The coefficients ν1, ν2, ν3 and ε1, ε2, ε3 associated with the higher order derivativesoccurring at the right hand side of the modified Eqs. (6.18) and (6.19) havenumerical roots. They are given as:
m1 ¼ g � H0 � Dt h� 12
� �; ð6:20Þ
m2 ¼ H0 � Dx 12� w
� �; ð6:21Þ
m3 ¼ g � Dx 12� w
� �; ð6:22Þ
e1 ¼ g � H0 � Dx � Dt2
1� w� hð Þ; ð6:23Þ
e2 ¼ H0 � Dx26
ð3h� 2ÞC2r þ ð3w� 1Þ� �
; ð6:24Þ
e3 ¼ g � Dx26
ð3h� 2ÞC2r þ ð3w� 1Þ� �
: ð6:25Þ
where Cr is the Courant number
Cr ¼ffiffiffiffiffiffiffiffiffiffiffiffig � H0
pDt
Dx: ð6:26Þ
6 One-Dimensional Modeling of Flows in Open Channels 145
Note that all terms with derivatives of higher order occurring at the right hand sideof Eqs. (6.18) and (6.19) were omitted while approximating the linear waveequations using the box scheme. The coefficients ν1, ν2, ν3 are associated with thedissipative properties of the box scheme while the coefficients ε1, ε2, ε3 are asso-ciated with the dispersive properties.
The derived system of modified equations allows concluding on all numericalproperties of the box scheme. Note that for θ = 1/2, ψ = 1/2 and Cr = 1 the boxscheme provides exact solution of the linear wave equations. In such a case, allterms at the right hand sides of Eqs. (6.18) and (6.19) vanish. Note as well as thatassuming ψ = 1/2 the numerical diffusion generated by the scheme is stronglylimited. In such a case the modified equations become
@H@t
þ H0@U@x
¼ m1@2H@x2
þ e1@3H@x3
þ e2@3U@x3
þ . . .; ð6:27Þ
@U@t
þ g@H@x
¼ m1@2U@x2
þ e1@3U@x3
þ e3@3H@x3
þ . . .; ð6:28Þ
with
m1 ¼ g � H0 � Dt h� 12
� �; ð6:29Þ
e1 ¼ g � H0 � Dx � Dt2
12� h
� �ð6:30Þ
e2 ¼ H0 � Dx212
1� C2r
� �; ð6:31Þ
e3 ¼ g � Dx212
1� C2r
� �: ð6:32Þ
This version of box scheme (with ψ = 1/2) is known as the Preissmann scheme,usually applied for numerical integration of the Saint Venant equations. Note that inpractical application the value of coefficient of numerical diffusion can reach theorder of a couple thousand m2/s. Fortunately, its influence becomes remarkableonly when the steep waves occur. Typical role of this term is to suppress spuriousoscillations occurring in the numerical solution. To this order θ > 1/2 should beapplied. Cunge et al. (1980) recommend θ = 2/3.
More detailed information on the numerical solution of the Saint Venantequations using both the finite difference method and the finite element method isgiven, for instance, by Cunge et al. (1980), Singh (1996), or Szymkiewicz (2010).
146 D. Gąsiorowski et al.
6.4 Extension of Application of the 1D Unsteady OpenChannel Flow Equations
The system of Saint Venant equations has been derived with the following fun-damental assumptions: the open channel flow is a gradually varied one, the channelhas compact cross-sections, the free surface of water exists and the flow depth isstill greater than zero. On condition that these assumptions are valid, the resultsprovided by the system usually are very satisfying. However, in engineeringpractice it sometimes happens that some of the above-mentioned assumptionslocally and temporarily are not preserved. To overcome the difficulties occurringwhile applying the Saint Venant equations in such circumstances, some particularapproaches are used. As a matter of fact, in such situations we try to apply the SaintVenant equations beyond the region of their validity. Let us recall some of them.
Locally the flow profile in an open channel can vary so rapidly that in fact adiscontinuity occurs. A very steep wave can occur in many circumstances. Forinstance, the steep waves can be generated by the operation of hydro-power plant,when the turbines are started or stopped suddenly or caused by sudden opening orclosing of a gate. The extreme shock wave occurs in an open channel, when a damseparating different water levels is suddenly removed (break dam problem).
For those types of steep waves, the system of Saint Venant equations is formallynot valid, since in its derivation the vertical acceleration has been omitted. On theother hand, in hydraulic engineering, when the main subject of interest is flowconsidered in large scale, the local untypical but complicated phenomena can be lessimportant and their internal structure may be neglected. For instance, length of thehydraulic jump is small comparing with the spatial dimension of practically appliedgrid. Consequently, it is spatially reduced to a point in which the water surface isdiscontinuous. The Saint Venant equations are capable to reproduce such a case offlow on condition that they are expressed in particular, so-called conservative form.This means that the forms of equations have to satisfy the integral form of conser-vation principles from which they were derived. Note that the system of SaintVenant equations can be written in integral form as well as in differential form(Chanson 2004; Cunge et al. 1980). Both representations are equivalent as long assmooth solutions are considered. The difference becomes noticeable when discon-tinuities occur. The second required condition concerns the numerical methodapplied. It is assumed that the standard schemes of the finite difference method basedon the Taylor series expansion can break down near discontinuity arising in thesolution (LeVeque 2002). An appropriate method should ensure required solutionaccuracy. First of all, it is expected that the considered conservation principles willbe satisfied on the numerical grid. This requirement is satisfied particularly exactlyby the finite volume method based on the equations written in integral form.However, to solve successfully some flow problems of shock wave type one can usethe finite difference methods as well. The flow of shock wave type means that weconsider flow with large gradients rather than containing pure discontinuity. In sucha case the computation should be carried out very carefully. First of all, as it was
6 One-Dimensional Modeling of Flows in Open Channels 147
stated previously, the Saint Venant equations must be expressed in the conservativeform. Secondly, their solution needs a particular numerical approach. There areknown three possible techniques (Cunge et al. 1980).
In the first approach, called the shock fitting method, the problem of disconti-nuity propagation and the unsteady flow between the discontinuities is split andconsidered separately. Traveling discontinuities are analyzed using the method ofcharacteristics, whereas for solution of the Saint Venant equations describing flowin the regions limited by discontinuities, the standard methods are used.
The second approach, called pseudoviscosity method, requires introduction of anartificial diffusive term into the dynamic equation. This term, having smoothingproperties, allows us to control the solution near discontinuity. Its particular formrelating intensity of generated diffusion with the wave steepness ensures that thisextra term acts strongly only locally, being insignificant far away from the dis-continuity. This approach is applicable for non-dissipative schemes (Potter 1973;Cunge et al. 1980).
The last possible approach bases on the solution using the dissipative methods.In this approach it is assumed that the process of smoothing can be ensured by thenumerical diffusion generated by the applied scheme. Controlling artificial diffusionone can obtain an acceptable solution. To this order, one can apply for instance thePreissmann scheme which is able to generate numerical diffusion.
The disadvantage of the presented approaches is that they base on the differentialequations. This can be avoided by using the finite volume method. The finitevolume method has very important feature—it ensures very good conservation ofthe transported quantity since it uses the flow equations in the integral form. Thefinite volume method appears to be an effective tool for solving the propagation ofshock waves, particularly for extreme auxiliary conditions. However, it deals withrather idealized situation, when the homogenous equations are solved.Unfortunately, the method cannot be simply implemented for more realistic cases.The presence of source terms in the form of bed and grade line energy sloperequires applying non-trivial approaches to overcome occurring difficulties.Detailed description of the method and its implementation for solution of the 1Dshallow water equations is given by LeVeque (2002), among others.
Very often the Saint Venant equations are solved for a storm sewer network.Such a system is constituted by closed conduits where the flow is usuallyfree-surface (Fig. 6.3a).
However, every now and then, after heavy rain, the conduits can be locally filledto the top and pressurized flow occurs. Since it happens only temporarily one canapply the so-called “Preissmann slot” (Cunge et al. 1980) shown in Fig. 6.3b.Owing to this concept it is possible to continue the computation using the samemathematical model, i.e. the system of Saint Venant equations. When the waterlevel in conduit reaches its top, then the further increasing pressure does not causean increase of the cross-section parameters. They are still constant, corresponding tothe entirely filled conduit. The assumed width of slot should be relatively small(after Cunge et al. (1980)—of the order of 0.01 m) so that the mass balance remainsnot affected.
148 D. Gąsiorowski et al.
A similar idea can be also applied in open channels with a temporarily dry bed.This can arise similarly, in a storm sever network or in the channels of wateringsystems. We have to remember that the equations of unsteady flow are valid forpositive values of depth. In such a case one can apply a concept opposite to the“Preissmann slot”, called “Abbott slot” (Abbott and Basco 1989). The slot goesdown from the level of bottom, as presented in Fig. 6.3c. This technique allows usto continue the computations even if a channel bed becomes temporarily dry.
In hydraulic practice, compound channel cross-sections are encountered, forexample, if the river has flood plains as presented in Fig. 6.4. When the water stageexceeds the level of flood plain, the flow process becomes more complicated, sinceit takes place in parts of the channel having different hydraulic properties. If 1DSaint Venant equations are used to model the unsteady flow, they should bemodified and special treatment of the cross-section is needed. Usually in channelcross-section two parts playing different roles in flow process are distinguished (seefor instance Abbott and Ionescu 1967). The first one, called active, is connectedwith the main channel and it is involved in the dynamic equation. The second partof section is called inactive, since it serves as a reservoir, which stocks the wateronly. This part which together with the active part constitutes the whole area ofsection is involved in the continuity equation.
Such an approach requires modification of the original Saint Venant equations.Although many attempts are still undertaken for modeling flood wave propagation in
h
h
datumh
(a) (b) (c)
Fig. 6.3 Flow in closed conduit: a with free surface; b pressurized; c with dry bottom
AAin
Ainac
Fig. 6.4 Channel of compound cross-section
6 One-Dimensional Modeling of Flows in Open Channels 149
the rivers with adjacent large flood plains it seems that nowadays such approachesbecame less interesting. Actually, such a case of flow should be considered rather asa 2D process with the river bed being partially and temporarily dry. It can bemodeled with one of the widely available computer codes for 2D unsteady flow.Expectations that 1D model will be able to reproduce accurately being in fact 2Dprocess, is not founded. The problems dealing with propagation of the flood waveover flooded area next to a river are discussed by Moussa and Cheviron (2015).
6.5 Simplified Open Channel Flow Equations
The system of Saint Venant equations, referred to as the dynamic wave model,requires relatively large number of data representing channel and initial-boundaryconditions. The difficulties in acquiring such data cause that instead of the fullSaint-Venant equations, simplified models are often used for modeling of the floodwave propagation in open channels. Such models require less input data and thenumerical algorithms applied to their solution are simpler and more effective than inthe case of the dynamic wave model. The simplified models are obtained byomitting less important terms in the dynamic equation of Saint Venant system.Analysis performed by Cunge et al. (1980) and Henderson (1996) shows that thereare physically founded reasons to neglect the terms of less significance in thedynamic equation. There are known two simplified models: the kinematic wavemodel and the diffusive wave model. Both models are based on the original con-tinuity equation and on the appropriately simplified dynamic equation.
The kinematic wave model is derived by elimination of the inertial and pressureterms in dynamic equation and simultaneously the continuity equation is taken inthe unchanged form (6.1). According to this simplification, the kinematic wavemodel is constituted by the following system of equations:
@A@t
þ @Q@x
¼ 0 ð6:33Þ
s ¼ S ð6:34Þ
where:
s slope of bottom,S slope of the energy line (friction slope).
Equation (6.34) is a simplified dynamic equation describing the steady uniformflow where the channel bed slope is the same as the slope of energy grade line. Thekinematic wave model is usually reduced to a single differential transport equationwith regard to a single unknown function only with the flow rate as the dependentvariable. Using the Manning formula for the steady flow, Eq. (6.34) is rewritten as:
150 D. Gąsiorowski et al.
A ¼ a � Qm ð6:35Þ
with
a ¼ 1
s1=2n�p2=3
� �3=5; ð6:36aÞ
m ¼ 35: ð6:36bÞ
If we assume a wide and shallow channel with constant value of wetted perimeter p,then Eq. (6.35) can be differentiated with respect to time. Substituting the result ofdifferentiation into the continuity Eq. (6.33) yields the kinematic wave model in theform of advection equation:
@Q@t
þ C Qð Þ @Q@x
¼ 0: ð6:37Þ
where C(Q) is the kinematic wave celerity given as:
C Qð Þ ¼ 1a � m � Qm�1 ð6:38Þ
The second simplified form of Saint-Venant equations—the diffusive wavemodel—is obtained by omitting the inertial force in the dynamic Eq. (6.2). Thus,this model is constituted by the continuity Eq. (6.1) and the simplified dynamicEq. (6.2), which can be rewritten as:
@H@x
þ Qj jQk2
� s ¼ 0 ð6:39Þ
where the coefficient k, called conveyance, is a function of the cross-sectionalparameters only:
k ¼ 1nR2=3 � A ð6:40Þ
Similarly to the previously obtained kinematic wave model, the system ofEqs. (6.33) and (6.39) can be reduced to a single differential equation. The deri-vation of the diffusive wave equation with the flow rate as dependent is performedby differentiation of both above-mentioned equations in order to eliminate thederivatives of the water depth (function H(x,t)). Rearranging of obtained relationswith additionally assumed constant width of channel gives the diffusive waveequation in the following final form:
6 One-Dimensional Modeling of Flows in Open Channels 151
@Q@t
þ C Qð Þ @Q@x
� D Qð Þ @2Q@x2
¼ 0 ð6:41Þ
where C is the kinematic wave celerity and D is the coefficient of hydraulic dif-fusivity given, respectively, as:
C Qð Þ ¼ Qk � B
@k@H
; ð6:42aÞ
D Qð Þ ¼ k2
2B Qj j ð6:42bÞ
For wide and shallow channel with constant value of the wetted perimeter and forthe slope of energy line replaced by the bottom slope, the kinematic wave celerityC given by Eq. (6.42a) coincides with Eq. (6.38), whereas the hydraulic diffusivity(6.42b) becomes (Eagelson 1970):
D Qð Þ ¼ Q2B � s ð6:43Þ
The diffusive wave model (6.41) is an advection-diffusion transport equation.The theory of the kinematic wave was given by Lighthill and Whitham (1955)
whereas the diffusive wave theory was presented by Hayami (1951). The propertiesand applicability of both above-mentioned models have been discussed by manyauthors, e.g. Miller and Cunge (1975), Woolhiser and Liggett (1967), Ponce, Li andSimmons (1978) and Ponce (1990). It is worth to add that the kinematic anddiffusive wave models are frequently formulated with respect to the discharge andto the water surface elevation as the dependent variables. However, other forms ofboth models are also possible. A comprehensive review focusing on the differentforms of these models as well as their applicability for open channel flow andfloodplain problems can be found in Singh (1996).
The dynamic wave model as well as its simplified forms—kinematic wave modeland diffusive wave model—constitute the distributed models where all dependentvariables occurring in equations are the functions of time and space. 1D flood routingcan be also carried out using the Muskingum model where the variability in space iscompletely eliminated. This model is derived from the storage equation
dVdt
¼ Qj � Qj�1 ð6:44Þ
where:
V volume of water stored by channel reach of length Δx,Qj inflow entering a channel reach through the upstream cross-section xj,Qj+1 outflow entering a channel reach through the downstream cross-section xj+1,
152 D. Gąsiorowski et al.
obtained by spatial integration of the continuity Eq. (6.33) over a channel reach oflength Dx ¼ xjþ1 � xj (Fig. 6.5).
Equation (6.44) requires an additional relationship between the storage V, inflowQj and outflow Qj+1, which for the Muskingum model has the following form(Chow, Maidment and Mays 1988):
V tð Þ ¼ K X � Qj þ 1� Xð ÞQjþ1� � ð6:45Þ
where the parameter X is dimensionless, while the parameter K has the dimension oftime and represents the travel time of wave propagating between two cross-sectionsof channel reach. Substitution of Eq. (6.45) into storage equation yields thewell-known Muskingum model:
XdQj
dtþ 1� Xð Þ dQjþ1
dt¼ 1
KQj � Qjþ1� � ð6:46Þ
While using the Muskingum model, the channel reach is commonly represented bya cascade of M − 1 reservoirs rather than by a single reservoir. In this case, thetransformation of flood wave by each reservoir is described by Eq. (6.46), where theindex of cross-section takes the values j = 1, 2, 3,…, M.
It appears that Eq. (6.46) constitutes a separate family of the simplified models.As a matter of fact, the Muskingum model should be considered as a semi-discreteform of the kinematic wave equation (Szymkiewicz 2002). For the first time thekinematic character was noticed by Cunge (1969). Using an accuracy analysis, heproved that an attenuation in solution is caused by numerical diffusion which iscontrolled by the parameter X and a space interval Δx. Cunge (1969) proposed sucha value of the parameter X which allows to generate the numerical diffusion equal tothe hydraulic diffusivity present in the linear diffusive wave Eq. (6.41).Consequently, an equivalence of both models with regard to the solution is satisfiedfor the following condition:
X ¼ 12� Q2B � s � C � Dx ð6:47Þ
x xj j+1
Δ x
Qj+1
Qj+1
Qj
Qj
Fig. 6.5 Interpretation of the lumped models—the volume of water V stored by a channel reach oflength Δx between the cross-sections xj and xj+1
6 One-Dimensional Modeling of Flows in Open Channels 153
Usually this version of the Muskingum method is called the Muskingum-Cungemodel (Chow, Maidment and Mays 1988).
The simplified flood routing models can be written as a system of equations or asa single transport equation. In the first case, the problem of physical interpretation israther simple. The equations constituting the simplified model represent the massand momentum conservation principles, respectively. The troubles arise when thesimplified model is reformed to a single transport equation. As it is obtained fromtwo equations representing different principles of conservation then the followingquestion seems to be relevant: which principle of conservation is represented by thefinal single transport equation? It appears that no problem exists in the case whenlinear version of the simplified equations is considered. In such a case, both massand momentum conservation principles are satisfied perfectly. However, very oftenthe non-linear simplified transport Eqs. (6.37) and (6.41) with C(Q) and D(Q) areused for flood routing. The idea of such approach is to increase the reproductionaccuracy of flood wave propagation. Unfortunately, in such a case new challengingproblems usually occur.
The non-linear partial differential equations can be used in a conservative(divergent) or non-conservative (non-divergent) form. Application of adequateconservative form of differential equation causes that the corresponding equationsatisfies the integral form of conservation principle from which it was derived.Consequently, the conservation form allows to avoid the balance errors.Unfortunately, the standard way of derivation of the kinematic wave Eq. (6.37) andthe diffusive wave Eq. (6.41) provides to the non-conservative forms, which doesnot guarantee the mass conservation. The numerical calculations performed byGąsiorowski and Szymkiewicz (2007) indicate that the non-conservative formsgenerate in numerical solution of the diffusive wave equation mass balance errorwhich can achieve a significant value.
Let us begin with consideration of the kinematic wave Eq. (6.37). It appears thatit is possible to derive another non-linear but conservative form. To this order, thesteady flow Eq. (6.35) is directly substituted into the continuity Eq. (6.33).Assuming that parameter a is constant, one obtains (Gąsiorowski and Szymkiewicz2007):
a@Qm
@tþ @Q
@x¼ 0 ð6:48Þ
In order to show the conservative and non-conservative features of non-linearequations, the kinematic wave Eq. (6.37) written in the non-conservative form isintegrated with regard to x over considered channel reach of length L bounded byupstream (x = 0) and downstream (x = L) ends:
@
@t
ZL0
Qdx ¼ C � Qjx¼0�C � Qjx¼LþR ð6:49Þ
154 D. Gąsiorowski et al.
R ¼ZL
0
Q@C@x
dx ¼ZL0
1a � mQ1�m @Q
@xþ 1a � mQ2 @Q
�m
@x
� �dx ð6:50Þ
Similar integration over considered reach performed for the conservative form ofthe kinematic wave Eq. (6.48) yields:
a@
@t
ZL
0
Qmdx ¼ Qjx¼0�Qjx¼L ð6:51Þ
Comparing both integral equations, (6.49) and (6.51), a significant difference isapparent. From Eq. (6.51) it results that the time variation of the total quantity Qm
stored over considered channel reach is caused only by the net advective fluxthrough upstream and downstream ends of channel. Thus, the form of Eq. (6.48) isconservative and represents the mass conservation principle. However, in the caseof Eq. (6.37) besides the net flux through the channel endpoints there is also anadditional unbalanced term R. This term results from non-conservative form of thenon-linear equation and causes that the integral Eq. (6.49) does not represent aglobal conservation law. As a consequence, the kinematic wave equation written inthe non-conservation form (6.37) cannot ensure the mass conservation in numericalsolution. The form of relation (6.50) indicates that the mass balance errors dependon the spatial derivative of the flow rate @Q=@x Thus, this error achieves a sig-nificant value for rapidly varied waves with a strong gradient. Moreover, it is worthto add that the balance errors resulting from improper form of non-linear kinematicwave equation can be amplified by the numerical diffusion generated in the solution(Gąsiorowski 2013). Consequently, these errors depend also on the numericalparameters of the numerical algorithm applied to the solution of non-linearequation.
As far as the diffusive wave equation is considered, it is impossible to obtain itsadequate conservative form with the flow rate satisfying the mass conservationprinciple (Gąsiorowski and Szymkiewicz 2007). This is caused by the way ofstandard derivation of the diffusive wave equation, which leads to the non-divergentform of diffusive term containing the second derivative:
D Qð Þ @2Q@x2
ð6:52Þ
The resulting form of this term cannot be transformed to its divergent form, andconsequently the mass balance errors cannot be eliminated from numerical solution.
Similar difficulties with respect to the conservative properties, as for the kine-matic wave equation, can be observed in the case of the non-linear Muskingummodel as well. When instead of constant parameters K and X they are related to thesolution, then a variable-parameter Muskingum model is obtained. Such approach
6 One-Dimensional Modeling of Flows in Open Channels 155
has been presented, for example, by Ponce and Yevjevich (1978), Tung (1984),Ponce and Chaganti (1994) and Mohan (1997). According to the studies performedby Tung (1984), Mohan (1997), Tang, Knight and Samuels (1999a, b), Perumal andSahoo (2008), Perumal and Price (2013), if the non-linear storage relation or thevariable parameters are applied in the various forms of the Muskingum model, thenthe numerical solution is frequently affected by mass balance errors. It seems thatthe reason of this fact is associated with an improper non-conservative form ofdifferential equations. Therefore, the derivation and application of the variableMuskingum model without an analysis of its conservative forms can lead to invalidresults. The conservative aspects with respect to the form of the non-linear equa-tions of Muskingum model were reported by Gąsiorowski (2009) as well as byReggiani, Todini and Meißner (2014).
More information on the conservative aspects of the non-linearadvection-diffusion equations is comprehensively discussed for instance byGresho and Sani (1998).
6.6 Linear Approximation of the One-Dimensional OpenChannel Flow Equations
In estuarine hydraulics, the aim is to predict levels or velocities at various points inthe channel, given the variation of water level at the downstream end. In problemsof flood routing, the aim is to predict the level and/or the flow at the downstreamend of the channel when given the level or the flow at the upstream end. Inhydrologic flood routing, only the upstream boundary condition is properly spec-ified and the downstream boundary is either ignored or crudely approximated. Bystudying the linearized St Venant equations for a finite channel reach with aproperly defined boundary condition at each end, a basis for the analysis of theerrors involved in the solution due to inadequate specification of one of theboundary conditions can be provided.
The problem of flood routing involves the prediction of the hydrograph of flowQ(t), or flow depth H(t) (or velocity U(t), or area A(t)) at the selected channelcross-sections. The problem involves the solution of the St. Venant Eqs. (6.1, 6.2)subject to a given initial condition and two appropriate boundary conditions.
Since the system of equations is a hyperbolic one, two boundary conditions arerequired and for the case of tranquil flow (i.e. the Froude number less than 1) one ofthese boundary conditions must be prescribed at each end of the reach. No ana-lytical solution is available and the problem must be solved either by simplificationof the equations or by same method of numerical approximation as that described inSects. 6.3 and 6.5.
The complete non-linear set of Eqs. (6.1, 6.2) can be solved analytically byconsidering variations from a steady state trajectory (Dooge and Napiórkowski1987, Napiórkowski and Dooge 1988).
156 D. Gąsiorowski et al.
The second Eq. (6.2) can be written in terms of the same dependent variables asthe first (Q and A) by substituting Q x; tð Þ=A x; tð Þ for the velocity U x; tð Þ in Eq. (6.2)and grouping the terms to obtain the set of equations (Dooge et al., 1982):
@A@t
þ @Q@x
¼ 0 ð6:33Þ
1� F2� �
gAB@A@x
þ 2QA
@Q@x
þ @Q@t
¼ gA s� Sð Þ ð6:53Þ
where F is the Froude number defined by
F2 ¼ Q2BgA3 ð6:54Þ
To compute the linearized second-order equation we make use of expansionof nonlinear terms in Eq. (6.53) in a Taylor series around the uniform steady state(Qo, Ao)
Q x; tð Þ ¼ Qo þ Q0 x; tð Þ þ eQ x; tð ÞA x; tð Þ ¼ Ao þ A0 x; tð Þ þ eA x; tð Þ ð6:55Þ
and limitation of this expansion to the first order incrementsQ0 x; tð Þ; A0 x; tð Þ; eQ x; tð Þ; eA x; tð Þ represent the higher-order terms (that is, theerror of the linear approximation)
@A0
@tþ @Q0
@x¼ 0
1� F20
� �gho
@Q0
@xþ 2uo
@Q@x
� @Q@t
¼ gAoð� @S@Q
Q0 � @S@A
A0Þð6:56Þ
with (see Dooge and Napiórkowski 1987)
@S@Q
¼ 2sQo
;@S@A
¼ �2cksQo
; ck ¼ � @S@A
=@S@Q
¼ dQdA
; uo ¼ Qo
Ao; ho ¼ Ao
Bo
ð6:57Þ
In Eq. 6.57 ck is the kinematic wave speed as given by Lighthill and Whitham(1955). For convenience, we may define a parameter m as the ratio of the kinematicwave speed to the average velocity of flow, m ¼ ck=uo. The parameter m is afunction of the shape of channel and of the area of flow. For a wide rectangularchannel with Manning friction m is always equal to 5/3. For shapes of channel otherthan wide rectangular, m will take on different values.
6 One-Dimensional Modeling of Flows in Open Channels 157
Making the necessary substitution to eliminate the dependent variable A’(x, t) inEq. 6.56 and leaving a single dependent variable Q’(x, t), one gets a second orderpartial differential equation for the perturbation Q’(x, t) from the steady uniformreference area Qo.
1� F20
� �gho
@2Q0
@x2� 2uo
@2Q@x@t
� @2Q@t
¼ gAoð� @S@A
@Q0
@xþ @S@Q
@Q0
@tÞ ð6:58Þ
For any given channel, the relative error due to linearization varies with theinflow hydrograph being routed and with the choice of reference conditions for thelinearization. Equation (6.58) can be generalized to a large number of choices forthe dependent variable. A perturbation flow potential can be defined as the function/0ðx; tÞ whose partial derivative with respect to x gives the minus perturbation fromthe reference area and partial derivative with respect to time t gives the perturbationfrom the reference discharge (see Deymie 1935; Dooge et al. 1987a),@/0=@x ¼ �A0ðx; tÞ; @/0=@t ¼ Q0ðx; tÞ. It can be shown that perturbation flowpotential /0ðx; tÞ and any linear function of the perturbation flow potential /0ðx; tÞrepresent a solution of Eq. (6.58), e.g. area of flow A′(x,t), average water depth H′(x,t), average velocity U′(x,t), the Froude number F(x,t), etc.
The solution of (6.58) for an upstream boundary condition and a semi-infinitewide rectangular channel with Chezy friction was discussed in Lighthill andWhitham (1955), and Dooge and Harley (1967). The two-point boundary problemin which both an upstream and a downstream boundary condition are taken intoaccount has been reported in the hydrologic literature, e.g., by Dooge andNapiórkowski (1987), Napiórkowski and Dooge (1988) and Moramarco et al.(1999). In this section the basic case of tranquil flow is considered (Froude numberless than one), in which Q’(x,t) is prescribed both at the upstream boundary x = 0and at the downstream boundary x = L. The problem is to solve Eq. (6.58) subject tozero initial condition and subject to boundary conditions:
Q0 0; tð Þ ¼ Qu tð Þ; Q0 L; tð Þ ¼ Qu tð Þ ð6:59Þ
The solution can be sought in terms of the Laplace transform. Equation (6.55)when transformed to the Laplace transform domain becomes:
1� F20
� �gho
d2 �Qdx2
þ �2uopþ gAo@S@A
� �d�Qdx
� p2 þ gAo@S@Q
p
� ��Q ¼ 0 ð6:60Þ
where parameter p is the complex number frequency, �Q x; pð Þ is the Laplacetransform of Q’(x, t). Equation (6.60) is a second-order homogeneous ordinaryequation, so the solution can be written in the general form:
�Q x; pð Þ ¼ C1 exp k1 pð Þx½ � þ C2 exp k2 pð Þx½ � ð6:61Þ
158 D. Gąsiorowski et al.
where λ1 and λ2 are the roots of the characteristic equation for Eq. (6.60) and aregiven by:
k1;2 ¼ epþ f �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiap2 þ bpþ c
pð6:62Þ
The parameters a, b, c, e, f are given in terms of hydraulic variables by thefollowing relationships:
a ¼ 1
gho 1� F2o
� �2 ; b ¼ 2suoho
1þ m� 1ð ÞF2o
1� F2o
� �2 ; c ¼ msho
� �2 1
1� F2o
� �2e ¼ uo
gho
11� F2
o; f ¼ m
s
ho 1� F2o
� � ð6:63Þ
Having determined the parameters C1(p) and C2(p) from the boundary condi-tions, one can write �Q x; pð Þ in terms of the hu(x,s) and hd(x,s) which may be definedas the Laplace transforms of the responses of the channel
�Q x; pð Þ ¼ hu x; pð ÞQu pð Þ þ hd x; pð ÞQd pð Þ ð6:64Þ
where the downstream and upstream linear channel responses are given by
hu x; pð Þ ¼ exp epþ fð Þx½ �sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiap2 þ bpþ c
pL� xð Þ
h isinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiap2 þ bpþ c
pL
h i ð6:65Þ
hd x; pð Þ ¼ exp epþ fð Þ L� xð Þ½ �sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiap2 þ bpþ c
px
h isinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiap2 þ bpþ c
pL
h i ð6:66Þ
After the inversion of Eq. (6.64) to the time domain, the original function Q’(x, t)in the time domain is determined from the corresponding boundary conditionsthrough the relationship (Dooge and Napiórkowski 1987):
Q x; tð Þ ¼ hu x; tð ÞQu tð Þ þ hd x; tð ÞQd tð Þ ð6:67Þ
The transfer functions hu x; tð Þ and hd x; tð Þ have two distinct parts (Dooge andNapiórkowski 1987):
hu x; tð Þ ¼ h1u x; tð Þ þ h2u x; tð Þ; hd x; tð Þ ¼ h1d x; tð Þ þ h2d x; tð Þ ð6:68Þ
6 One-Dimensional Modeling of Flows in Open Channels 159
The first part of the transfer function due to upstream input, which may betermed as the head of the wave, is given by
h1u x; tð Þ ¼X1n¼0
exp �2nLa1 � a2xð Þd t � nto � x=c1ð Þ
�X1n¼1
exp �2nLa1 þ a3xð Þd t � nto � x=c2ð Þð6:69Þ
where
a1 ¼ b=2ffiffiffiffia;
pa2 ¼ a1 � f ; a3 ¼ a1 þ f ;
c1;2 ¼ u1;2 ¼ uo �ffiffiffiffiffiffiffiffiffigho;
pto ¼ L=c1 � L=c2
ð6:70Þ
The behaviour of the first part of the solution is shown in Fig. 6.6. It can be seenthat the head of the wave moves downstream at the dynamic speed c1 (along thefirst characteristic, see Fig. 6.1) in the form of a delta function of exponentiallydeclining volume proportional to exp �a2xð Þ. At x = L the delta function is reflectedwith inversion of sign and then is propagated upstream at the speed c2 (along thesecond characteristic, see Fig. 6.1) and with a heavier damping factorexp �a3 L� xð Þ½ �, then is reflected again at x = 0 to move in a downstream directionand so on until the volume of the head of the wave becomes negligible.
The second part of the solution, which may be termed the body of the wave, isgiven by:
Fig. 6.6 Head of the wave reflections indicating the direction and speed of travel, intervalbetween reflections and the rate of damping
160 D. Gąsiorowski et al.
h2u x; tð Þ ¼X1n¼0
exp �b1t þ b2xð Þh 1=c1 � 1=c2ð Þ 2nLþ xð Þ
� I1 2hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � nto � x=c1ð Þ t þ nto � x=c2ð Þp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � nto � x=c1ð Þ t þ nto � x=c2ð Þp 1 t � nto � x=c1ð Þ
�X1n¼0
exp �b1t þ b2xð Þh 1=c1 � 1=c2ð Þ 2 nþ 1ð ÞLþ xÞ½
� I1 2hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � nþ 1ð Þto � x=c2½ � t þ nþ 1ð Þto � x=c1½ �p� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � nþ 1ð Þto � x=c2½ � t þ nþ 1ð Þto � x=c1½ �p 1 t � nþ 1ð Þto � x=c2½ �
ð6:71Þ
where I1 is a modified Bessel function of the first kind, 1[t] is a unit step function,and the remaining parameters are given by:
b1 ¼ b=2a b2 ¼ f � be=2a; h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2=4� acÞ
p=2a ð6:72Þ
The solution is in the form of an infinite series which seems too complicated forpractical application in river flow forecasting. However, due to heavy damping,only the first few terms of the two transfer functions would normally be required,and the polynomial approximation of the first order modified Bessel function(Abramowitz and Stegun 1965) is sufficiently accurate and can be easily calculated.
As in the case of the head of the wave, the body of the wave is subject tosuccessive reflection at both downstream and upstream boundaries but moves anddissipates more slowly than the head of the wave.
The flow at any intermediate point in the reach is determined by the upstreamboundary condition Qu(t) and the downstream boundary condition Qd(t) in accor-dance with Eq. (6.67). It is clear that if the value of Qd(t) is very much larger thanQu(t) then this boundary condition will have a dominant influence on conditionsthroughout most of the reach. In many cases, however, the two boundary conditionsare of the same order of magnitude. Accordingly, it is instructive to check how theposition of the downstream boundary conditions affects the shape of the impulseresponses hu(x,t) and hd(x,t). For the purpose of illustration, a broad rectangularchannel with Manning friction (m = 5/2) is considered. The results are represented interms of dimensionless independent variables defined with the help of the bottomslope s, the depth ho, and the velocity uo, for the steady uniform reference conditions:
x0 ¼ xsho
; t0 ¼ tuosho
ð6:73Þ
Figures 6.7 and 6.8 illustrate the effect of the position of the downstream controlon the shape of the body of the wave for F = 0.2 and F = 0.8, respectively, for twovalues of the length factor: x′ = 1 (a short channel), and x′ = 20 (a long channel) andthe selected locations of the downstream boundary L0 ¼ x0 þ Dx, namelyDx 2 0:1; 0:3; 0:5; 1:0; 1½ �.
6 One-Dimensional Modeling of Flows in Open Channels 161
The behaviour shown in Fig. 6.7 can be explained in terms of successivereflections shown in Fig. 6.6. For values of t’ less than L0
c01 � L0 � x0ð Þ c02
onlythe first term in the first sum in Eq. 6.66 and the first term in Eq. 6.68 differ fromzero. For values of t′ greater than L0
c01 � L0 � x0ð Þ c02
the first term in the secondsums in both equations comes into play because of the first reflection of the wave bythe zero downstream boundary condition at t0 ¼ L0
c01. For values of t’ greater than
t0 ¼ L0 c01 � L0
c12 the second term in the first sums in both equations, (6.66) and
(6.68), becomes effective because of the reflection by the upstream boundarycondition. Each reflection brings a new term into effect at a time appropriate to theposition in the channel.
Even Fig. 6.7 for a short channel shows only one reflection and thus confirmsthat only the leading terms in the infinite series are significant.
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
short channel x'=1, F=0.2
t'
upst
ream
tran
sfer
func
tion
abcde
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
long channel x'=20, F=0.2
t'
upst
ream
tran
sfer
func
tion
abcde
(a) (b)
Fig. 6.7 Upstream transfer function for a reference Froude number Fo = 0.2, and selectedlocations of the downstream boundary L
0 ¼ x0 þ Dx, a Dx ¼ 0:1; b Dx ¼ 0:3; c Dx ¼ 0:5;
d Dx ¼ 1:0; e Dx ! 1
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
short channel x'=1, F=0.8
t'
upst
ream
tran
sfer
func
tion
a
b
cd
e
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
long channel x'=20, F=0.8
t'
upst
ream
tran
sfer
func
tion
abcde
(a) (b)
Fig. 6.8 Upstream transfer function for a reference Froude number Fo = 0.8, and selectedlocations of the downstream boundary L
0 ¼ x0 þ Dx, a Dx ¼ 0:1; b Dx ¼ 0:3; c Dx ¼ 0:5,
d) Dx ¼ 1:0, e Dx ! 1]
162 D. Gąsiorowski et al.
The solution for the special case of the downstream movement of the floodwaves in a semi-infinite channel is well known (Lighthill and Whitham 1955;Dooge and Harley 1967). The general solution for a semi-infinite channel waspublished by Dooge et al. (1987a). This special case of the semi-infinite channel hasthe solution
hu x; tð Þ ¼ d t � x=c1ð Þ exp �a2xð Þ
exp �b1t þ b2xð Þh x=c1 � x=c2ð Þ I1 2hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � x=c1ð Þ t þ x=c2ð Þp
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � x=c1ð Þ t þ x=c2ð Þp ð6:74Þ
The properties of the impulse response for a linearized channel of any shape andany friction law were studied by Dooge et al. (1987b) using the cumulants andshape factors of the general response and the amplitude and phase spectra. It wasconfirmed that even for this very general case the average downstream movement isgiven exactly by the kinematic approximation. It was shown that for very longwaves the attenuation approaches zero, whereas for very short waves the amplitudedecreases exponentially with distance.
For the downstream transfer function, the head of the wave is given by
h1d x; tð Þ ¼X1n¼0
exp �2nLa1 � a3 L� xð Þ½ �d t � nto þ L� xð Þ=c2½ �
�X1n¼0
exp � 2nþ 1ð ÞLa1 � fL� a2x½ �d t � nto þ L=c2 � x=c1ð Þð6:75Þ
and is subject to reflection at the two ends of the reach as in the case of h1u x; tð Þ.The body of the wave in the case of downstream transfer function is given by
h2d x; tð Þ ¼X1n¼0
exp �b1t � b2 L� xð Þ½ �h 1=c1 � 1=c2ð Þ 2nLþ L� xð Þ½ �
� I1 2hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � nto þ L� xð Þ=c2½ � t þ nto þ L� xð Þ=c1½ �p� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit � nto þ L� xð Þ=c2½ � t þ nto þ L� xð Þ=c1½ �p 1 t � nto þ L� xð Þ=c2½ �
�X1n¼0
exp �b1t � b2 L� xð Þ½ �h 1=c1 � 1=c2ð Þ 2 nþ 1ð ÞLþ x½ �
� I1 2hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit þ nto � x=c2 þ L=c1ð Þ t � nto þ L=c2 � x=c1ð Þp� �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit þ nto � x=c2 þ L=c1ð Þ t � nto þ L=c2 � x=c1ð Þp 1ðt � nto þ L=c2 � x=c1Þ
ð6:76Þ
The solution is in the form of an infinite series which seems too complicated forpractical application in river flow forecasting. However, due to heavy damping,only the first few terms of the two transfer functions would normally be required,and the polynomial approximation of the first order modified Bessel function(Abramowitz and Stegun 1965) is sufficiently accurate and can be easily calculated.
6 One-Dimensional Modeling of Flows in Open Channels 163
As an illustration of the effect of the transmission a change in the value ofQd(t) at the point (L′ − x′), the change in flow due to a constant downstreamboundary Qd(t) = 1 is calculated. The backwater curve is shown in Fig. 6.9 forFo = 0.2 and Fo = 0.8. It is clear from Fig. 6.9 that the backwater effect is effectiveonly for (L′ − x′) < 1.2 for Fo = 0.2 and for (L′ − x′) < 0.5 for Fo = 0.8.
6.7 Modeling of Unsteady Open Channel Flow by Meansof Stochastic Transfer Function
Solution of the flood operating problem in the system of reservoirs requires repeatedsolving of unsteady flow equations for successively generated operation scenarios.Thus the solution algorithms applied in such a case should be maximally efficient,not only in respect of the computer capabilities requirements, but—particularlyimportant in this case—time of computations required to obtain the solution as well.
In order to facilitate the computations, lumped parameter simulators of a dis-tributed flow routing (e.g. St. Venant equations) are usually used in the multiob-jective optimization of a water reservoir management system. The simulator canadvantageously apply the Stochastic Transfer Function (STF) approach togetherwith a nonlinear transformation of variables. Experience gained by Romanowiczet al. (2004, 2006) indicated that STF models are compatible with distributed modelpredictions at cross-sections where observations are available. The STM model iscalibrated on historical data and on distributed model realizations for the parts of theriver where the observations are not available. In Romanowicz and Beven (1998), alumped model based on a Stochastic Transfer Function (STF) approach was used toupdate on-line flood inundation forecasts for the River Culm, UK.
0 0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(L'-x')
Bac
kwat
er p
rofil
e
F=0.2
F=0.8
Fig. 6.9 Backwater profiledue to a constant unit excessof downstream flow
164 D. Gąsiorowski et al.
The STS model is stochastic, enabling derivation of prediction uncertainty in astraightforward manner; therefore, it is suitable for scenario analysis of the watermanagement system under uncertain climatic conditions. Estimated probability ofwater levels at the cross-sections along the river enables the derivation of proba-bility maps of inundation at different times of the year.
At the reach scale, the discrete-time STF describes the open channel dynamicsand can be presented as:
xk ¼ B z�1ð ÞA z�1ð Þ uk�d ; yk ¼ xk þ fk ð6:77Þ
where k is a discrete time period, uk�d denotes STF model input (flow or waterlevel), xk is the underlying “true” flow or water level, is the noisy observation of thisvariable, d is a time delay, while A z�1ð Þ and B z�1ð Þ are polynomials of the fol-lowing form:
A z�1� � ¼ 1þ a1z�1 þ a2z
�2 þ � � � þ apz�p
B z�1� � ¼ 1þ b1z�1 þ b2z�2 þ � � � þ bmz�m
ð6:78Þ
in which z�1 is the backward shift operator. The additive error fk in (6.77) isusually both heteroscedastic and autocorrelated in time. It is assumed to account forall the uncertainty at the output of the system that is associated with the inputsaffecting the model, including measurement error, unmeasured inputs, and uncer-tainties associated with the model structure. The orders of the polynomials, p andm, are identified from the data during the data-based identification process. Whenflow is used as a routing variable, water levels are derived from the rating curveequation.
Stochastic transfer function model and a nonlinear transformation of modelvariables was recently successfully applied to combined reservoir management andflow routing on the Upper Narew River, northeast Poland (Romanowicz et al.2010).
References
Abbott MB (1979) Computational hydraulics-elements of the theory of free surface flow. Pitman,London
Abbott MB, Basco DR (1989) Computational fluid dynamics. Longman Scientific and Technical,Harlow
Abbott MB, Ionescu F (1967) On the numerical computation of nearly-horizontal flows. J HydrRes 5:97–117
Abramowitz M, Stegun IA (1965) Handbook of mathematical functions. Dover Publications Inc.,New York
Chanson H (2004) The hydraulics of open channel flow: an introduction, 2nd edn. Elsevier,Amsterdam
6 One-Dimensional Modeling of Flows in Open Channels 165
Chow VT (1973) Open channel hydraulics. Mc Graw-Hill, New YorkChow VT, Maidment DR, Mays LW (1988) Applied hydrology. McGraw-Hill International
Editors, New YorkCunge JA (1969) On the subject of a flood propagation computation method (Muskingum
method). J Hydraul Res 7(2):205–230Cunge J, Holly FM, Verwey A (1980) Practical aspects of computational river hydraulics. Pitman
Publishing, LondonDeymie P (1935) Propogation d’une intumescence allongée. Revue Générale de l’Hydraulique
3:112–135Dooge JCI, Harley BM (1967) Proceedings of International Hydrology Symposium, Fort Collins,
Colorado, paper no. 8, 1, 57–63Dooge JCI, Strupczewski WG, Napiórkowski JJ (1982) Hydrodynamic derivation of storage
parameters of the Muskingum model. J Hydrol 54:371–387Dooge JCI, Napiórkowski JJ (1987) The effect of the downstream boundary conditions in the
linearized St. Venant equations. Quart J Mech Appl Math 40, part 2, pp 245–256Dooge JCI, Napiórkowski JJ, Strupczewski WG (1987a) The linear downstream response of a
generalized uniform channel. Acta Geophysica Polonica 35(3):279–293Dooge JCI, Napiórkowski JJ, Strupczewski WG (1987b) Properties of the generalized linear
downstream channel response. Acta Geophysica Polonica 35(4):405–416Eagleson PS (1970) Dynamic hydrology. McGraw-Hill, New YorkFletcher CAJ (1991) Computational techniques for fluid dynamics, vol. I. Springer, BerlinGąsiorowski D (2009) Flood routing by the non-linear Muskingum model: conservation of mass
and momentum. Arch Hydro-Eng Environ Mech 56(3–4):3–19Gąsiorowski D (2013) Balance errors generated by numerical diffusion in the solution of non–
linear open channel flow equations. J Hydrol 476:384–394Gąsiorowski D, Szymkiewicz R (2007) Mass and momentum conservation in the simplified flood
routing models. J Hydrol 346(1–2):51–58Gresho PM, Sani RL (1998) Incompressible flow and the finite-element method, vol .1.
Advection-diffusion. John Wiley, ChichesterHayami S (1951) On the propagation of flood waves. Bulletin 1, Disaster Prevention Research
Institute, Kyoto University, Kyoto, JapanHenderson FM (1996) Open channel flow. Macmillan Company, New YorkLeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press,
CambridgeLiggett JA, Cunge JA (1975) Numerical methods of solution of the unsteady flow equations. In:
Mahmood K, Yevjewich V (eds) Unsteady flow in open channels. Water ResourcesPublishing, Fort Collins
Lighthill MJ, Whitham GB (1955) On kinematic waves, I: flood movement in long rivers. ProcRoy Soc London Ser A 229:281–316
Moramarco T, Fan Y, Bras RL (1999) Analytical solution for channel routing with uniform lateralinflow. J Hydr Eng ASCE 125:707–713
Moussa R, Cheviron B (2015) Modeling of floods—State of the art and research challenges. Thisissue
Miller WA, Cunge JA (1975) Simplified equations of unsteady flow. In: Miller WA, Yevjewich V(Eds) Unsteady flow in open channels. Water Resources Publishing, Fort Collins
Mohan S (1997) Parameter estimation of non-linear Muskingum models using genetic algorithm.J Hydraul Eng ASCE 123(2):137–142
Napiórkowski JJ, Dooge JCI (1988) Analytical solution of channel flow model with downstreamcontrol. Hydrol Sci J 33(3), part 6, pp 269–287
Perumal M, Price RK (2013) A fully mass conservative variable parameter McCarthy-Muskingummethod: theory and verification. J Hydrol 502:89–102
Perumal M, Sahoo B (2008) Volume conservation controversy of the variable parameterMuskingum-Cunge method. J Hydraul Eng ASCE 134(4):475–485
166 D. Gąsiorowski et al.
Ponce VM (1990) Generalized diffusion wave equation with inertial effects. Water Resour Res 26(5)
Ponce VM, Chaganti PV (1994) Muskingum-Cunge method revised. J Hydrol 163:433–439Ponce VM, Li RM, Simmons DB (1978) Applicability of kinematic and diffusion models.
J Hydraul Divis ASCE 104(3):353–360Ponce VM, Yevjevich V (1978) Muskingum-Cunge methods with variable parameters. J Hydraul
Divis ASCE 104(12):1663–1667Potter D (1973) Computational physics. Wiley, LondonReggiani P, Todini E, Meißner D (2014) A conservative flow routing formulation: Déjà vu and the
variable-parameter Muskingum method revisited. J Hydrol 519:1506–1515Romanowicz R, Beven K (1998) Dynamic real-time prediction of flood inundation probabilities.
Hydrol Sci J 43:181–196Romanowicz RJ, Young PC, Beven KJ (2004) Data assimilation in the identification of flood
inundation models: derivation of on-line multi-step ahead predictions of flows. In: Webb B,Arnell N, Onf C, MacIntyre N, Gurney R, Kirby C (eds) Proceedings of the BHS internationalconference: hydrology, science and practice for the 21st century, London, pp 348–53
Romanowicz RJ, Young PC, Beven KJ (2006) Data assimilation and adaptive forecasting of waterlevels in the river Severn catchment, United Kingdom. Water Resour Res 42:W06407. doi:10.1029/2005WR004373
Romanowicz RJ, Kiczko A, Napiórkowski JJ (2010) Stochastic transfer function model applied tocombined reservoir management and flow routing. Hydrol Sci J 55(1):27–40
Singh VP (1996) Kinematic wave modelling in water resources: surface water hydrology. Wiley,New York
Szymkiewicz R (2002) An alternative IUH for hydrological lumped models. J Hydrol259:246–253
Szymkiewicz R (2010) Numerical modeling in open channel hydraulics. Springer, BerlinTang X, Knight DW, Samuels PG (1999a) Volume conservation characteristics of the variable
parameter Muskingum-Cunge method for flood routing. J Hydraul Eng ASCE 125(6):610–620Tang X, Knight DW, Samuels PG (1999b) Variable parameter Muskingum-Cunge method for
flood routing in a compound channel. J Hydraul Res 37(5):591–614Tung YK (1984) River flood routing by non-linear Muskingum method. J Hydraul Divis ASCE
111(12):1447–1460Woolhiser DA, Ligget JA (1967) Unsteady one-dimensional flow over a plane: the rising
hydrograph. Water Resour Res 3(3)
6 One-Dimensional Modeling of Flows in Open Channels 167
