# 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 . This shallow water equation consists of five terms :

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

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 :

$\lambda_u^2 \approx \lambda_s \cdot \lambda_{D}$

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) . This condition can be considered as fulfilled in this study, as of coarse sediments in the shape of gravel are used for the experiments.