Dimensional Analysis¶

This section introduces the scaling theory according to Barenblatt [Bar87], Barenblatt [Bar96], and Yalin [Yal71].

Mathematical Model Description¶

River hydrodynamics can be expressed by a simplified expression of the one-dimensional Navier-Stokes equations for incompressible fluids, assuming hydrostatic pressure distribution [GA11, KC08]). This results in the Saint-Venant shallow water equations as used in some hydraulic computer models (e.g., HEC-RAS or BASEMENT1D [USACoEngineeers16, VAW17]. This shallow water equation consists of five terms [JVBVdB+94]:

\[ \overbrace{\frac{1}{g} \frac{\partial u}{\partial t}}^{I} + \overbrace{\frac{u}{g} \frac{\partial u}{\partial x}}^{II} + \overbrace{\frac{\partial h}{\partial x}}^{III} + \overbrace{\frac{\partial z}{\partial x}}^{IV} = \overbrace{-\frac{u\left| u \right|}{C^2 h}}^{V = S_e} \]

The five terms can be related separately to each other for the derivation of scale factors$\lambda$. Thus, equating the scales of the terms I and II results in [DV93]:

\[ \frac{\lambda_u}{\lambda_t} = \frac{\lambda_u^2}{\lambda_l} \Longrightarrow \lambda_l = \lambda_u \cdot \lambda_t \]

where

$\lambda_u$ $\equiv$ velocity scale and
$\lambda_t$ & $\equiv$ & time scale.

Postulating that the gravity scale $\lambda_g$ is unity, the comparison of the scales of terms II and V results in:

\[ \frac{\lambda_u^2}{\lambda_l} = \frac{\lambda_u^2}{\lambda_C^2 \cdot \lambda_h} \Longrightarrow \lambda_C^2 = \sqrt{\frac{\lambda_l}{\lambda_h}} \]

where $\lambda_C$ $\equiv$ Ch’ezy roughness scale.

Similitude Concepts¶

The similarity of the Froude number in a scaled model and a prototype is achieved based on the Froude condition, which results from equating the scales of terms II and III in the above equation [DV93]:

\[ \frac{\lambda_u^2}{\lambda_l} = \frac{\lambda_h}{\lambda_l} \Longrightarrow \lambda_u = \sqrt{\lambda_h}. \]

The similarity of sediment transport is of particular interest in this study and requires that the scales of the dimensionless bed shear stress $\tau_{*}$ and of the bed load transport intensity $\Phi_b$ are unity (i.e., $\lambda_{\tau_*}$=1 and $\lambda_{\Phi_}$=1 [DV93]). With respect to the shear velocity $u_*$ = $\sqrt{\tau/\rho_f}$ = $\sqrt{\tau_*(s-1)gD}$ and the requirement of $\lambda_{\tau_*}$=1, the similarity of sediment transport is given when [JVBVdB+94]:

\[ \lambda_u^2 \approx \lambda_s \cdot \lambda_{D} \]

where

$\lambda_s$ $\equiv$ scale of relative sediment density
$\lambda_{D}$ $\equiv$ scale of grain diameter.

The similarity of unitary sediment transport (i.e., per unit width) can be verified based on the scale $\lambda_{q_b}$, which is derived from the Exner equation: $$ \frac{\partial z}{\partial t} = -\frac{1}{1-\zeta} \cdot \frac{\partial q_s}{\partial x} $$ With respect to the scale considerations above, $\lambda_{q_b}$ is derived as: $$ \frac{\lambda_l}{\lambda_t} = \frac{\lambda_{q_b}}{\lambda_l} \Rightarrow \lambda_{q_b} =\frac{\lambda_l^2}{\lambda_t} = \lambda_l^{3/2} $$

$\lambda_{q_b}$ refers to volumetric fluxes. The scale of the mass flow rate $\lambda_{\dot{q}_b}$ can be computed by multiplying the above equation by the sediment density $\rho_s$. Postulating the density scale of $\lambda_{s}$=1, the mass flow rate scale is also $\lambda_{\dot{q}_b}= \lambda_l^{3/2}$. The boundary conditions imposed by the feasibility of the laboratory experiments entail that the densities of the sediment in nature and in the model are similar (i.e., $\lambda_s$=1). Thus, the Froude similarity ($\lambda_u = \sqrt{\lambda_h}$) and the similarity of sediment transport ($\lambda_u = \sqrt{\lambda_{D}}$) require that $\lambda_{D}$=$\lambda_h$ (i.e., the same geometric scales apply to the grain diameter as well as to the water depth) [JVBVdB+94]. This condition can be considered as fulfilled in this study, as of coarse sediments in the shape of gravel are used for the experiments.

Hydro-Informatics

Dimensional Analysis¶

Mathematical Model Description¶

Similitude Concepts¶