Unsteady, nonuniform free surface flow can be described by the full hydrodynamic wave (HD) equations, reflecting the conservation laws of mass and momentum. Usually, the dependent variables
of the equations (e.g., flow velocity, flow area, etc.) are averaged over the water depth.1 For
the one-dimensional case2, the governing equations are referred to as Saint-Venant equations
(de Saint-Venant, 1871; Eagleson, 1970), named after J. C. de Saint-Venant. Keeping the pressure- gradient and momentum-source/sink term, but neglecting the inertia and acceleration terms in the momentum equation of the Saint-Venant model, leads to the zero-inertia wave (ZI) approximation (Hayami, 1951). The kinematic wave (KW) model, reported on thoroughly in the classical paper by Lighthill and Whitham (1955), neglects all of the aforementioned terms, which implies that friction slope is parallel to bottom slope.
All three flow models are mathematically expressed as a system of two equations, where one (KW) or two (HD, ZI) are partial differential equations (PDEs), representing continuity (conservation of mass) and conservation of momentum. Such a system describes an initial-boundary-value problem.
This means that the solution functions of the PDEs3 are dependent on the initial values of the
considered process variables4 at a specific point in time for all points in space (expressed in the
initial condition, cf. Section 3.3), as well as the temporal evolution of the process variables, given at specific points in space (boundary conditions, cf. Section 3.3). The solution of the PDEs is, therefore, only valid for a certain domain in both the spatial and temporal dimension, spanned by the initial and boundary conditions.
3.2.1
The Saint-Venant Equations (Full Hydrodynamic Model)
Assuming no lateral losses or inflows to the flow domain, the continuity and momentum equations of the one-dimensional full hydrodynamic (Saint-Venant) model read
∂A ∂t + ∂Q ∂x = 0 (3.1) ∂h ∂x = S0− Sf− u g ∂u ∂x− 1 g ∂u ∂t (3.2)
where t: time [T]; x: longitudinal space coordinate [L]; A(x, t): wetted cross-sectional area [L2];
Q(x, t): discharge [L3T−1]; h(x, t): water depth [L]; S
0: bottom slope [−]; Sf: friction slope [−];
u(x, t): flow velocity [LT−1]; and g: acceleration due to Earth’s gravity [LT−2].
The Saint-Venant model is widely used for modeling unsteady, nonuniform free surface flows. Equations (3.1) and (3.2) form a system of quasilinear first-order partial differential equations of
the hyperbolic type5. However, coming from the more general Navier-Stokes equations (Tenman,
1 This simplification is usually referred to as shallow water approximation. 2 This means an additional averaging over the flow width.
3 For example, discharge and water depth as a function of the independent variables space and time. 4 Flow velocity and wetted cross-sectional area, as well as water depth, respectively.
5 Given is a PDE for a function ν(x, t) of two variables. Such an equation has the general form
aνxx+ 2bνxt+ cνtt+ dνx+ eνt+ fν + g = 0
and the coefficients a = a(x, t), ..., g = g(x, t) are functions of two variables. The PDE is called elliptic if ac− b2> 0,
3.2 Hydrodynamic Models
1977), it is important to review the assumptions made for the derivation of Eqs. (3.1) and (3.2), since these assumptions constrain the model validness to specific flow phenomena:
. Acceleration of the fluid in the vertical direction is neglected; the vertical pressure gradient is
approximated by a hydrostatic pressure distribution1;
. the flow is assumed to be one-dimensional; all dependent variables are averaged over the width and depth of the flow;
. bottom slope is small, such that sin(S0)≈ S0;
. the flowing liquid (water) is assumed to be incompressible; and
. the considered fluid is a Eulerian fluid, i.e., the flow exposes no or only low inner friction (viscosity). Friction slope is not dependent on flow type, turbulence, or sediment load. The relationship between friction and the hydraulic variables is expressed with a flow formula (e.g., the Manning-Strickler formula).
Especially the last assumption might restrict the validness of the Saint-Venant model applied for heavily sediment-laden flows as occurring in ephemeral rivers, which is discussed in the closing section of this thesis. Nevertheless, the listed restrictions do not endanger the model’s applicability for a wide range of flow processes on natural surfaces, as present in numerous hydraulic and hydrologic problems.
3.2.2
Zero-Inertia Approximation
As stated before, simplifications of the Saint-Venant equations only address the momentum equation (3.2). The continuity equation (3.1) remains unchanged for all discussed modeling concepts (HD,
ZI, KW). The diffusion wave or zero-inertia approximation neglects the inertia and acceleration terms of Eq. (3.2), yielding
∂h
∂x = S0− Sf (3.3)
In contrast to the full hydrodynamic model, Eqs. (3.1) and (3.3) now form a parabolic system. The ZI momentum equation cannot account for pronounced, unsteady, and nonuniform flow phenomena, since the governing terms are neglected. Nevertheless, a certain degree of nonuniformity is preserved in the momentum balance through the possible distinction of bed and friction slope. The model is an adequate substitute for the full hydrodynamic model if the acceleration and inertia terms are negligible (cf. Section 3.7.1). The portrayal of backwater effects is possible with the ZI model.
3.2.3
Kinematic Wave Approximation
The kinematic wave approximation additionally neglects the ∂h
∂x pressure-gradient term in Eq. (3.3).
Hence, bottom and friction slope can be assumed parallel, which yields the momentum equation of the kinematic wave model as
0 = S0− Sf (3.4)
In turn, together with the continuity equation, this yields a biunique relationship of discharge and stage (rating curve). Furthermore, any rating curve—expressed, e.g., by a steady and uniform flow
equation like Manning-Strickler1, Ch´ezy, or Darcy-Weisbach—which is coupled to the continuity
equation (3.1) is referred to as a kinematic wave model. As applies for the Saint-Venant model, the system of Eqs. (3.1) and (3.4) is of the hyperbolic type.
3.2.4
Other Simplifications of the Full Hydrodynamic Model
Two other simplifications of the full dynamic momentum equation are physically reasonable (Ponce
and Simons, 1977). The steady dynamic wave neglects the unsteady 1g∂u∂t term of the full dynamic
model. If the influence of the bottom slope is insignificant (e.g., for great water depths2 or for
zero-slope conditions), the S0− Sfterm can be omitted, leading to a process description applicable
for a gravity wave. These two special cases of flow models are not further regarded in this thesis.