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.