Über Datenanalyse

Traditionell wurden dimensionale Analysen verwendet, um Erkenntnisse aus verschiedenen experimentellen Setups und Umfrageumgebungen abzuleiten. Dieses Kapitel befasst sich kurz mit der Art der Daten und erklärt traditionelle Dateneinblicke mit dimensionaler Analyse.

Die Art der Daten¶

Dimensionsanalyse¶

Dieser Abschnitt stellt die Skalierungstheorie gemäß Barenblatt (1987), Barenblatt (1996) und Yalin (1971) vor.

Beschreibung des mathematischen Modells¶

River hydrodynamics can be expressed by a simplified expression of the one-dimensional Navier-Stokes-Gleichungen for incompressible fluids, assuming hydrostatic pressure distribution Kundu & Cohen, 2008Graf & Altinakar, 2011). This results in the Saint-Venant shallow water equations as used in some hydraulic computer models (e.g., HEC-RAS or BASEMENT1D U.S. Army Corps of Engineeers, 2016VAW, 2017. This shallow water equation consists of five terms Jansen et al., 1994:

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 De Vries, 1993:

wo

• $\lambda_u$ $\equiv$ velocity scale and

• $\lambda_t$ & $\equiv$ & Zeitskala.

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

Wobei $\lambda_C$ $\equiv$ Ch'ezy Rauheit Skala.

Similitude-Konzepte¶

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 De Vries, 1993:

The similarity of sediment transport is of particular interest in this study and requires that the scales of the Dimensionslose Schubspannung $\tau_{*}$ and of the bed load transport intensity $\Phi_b$ are unity (i.e., $\lambda_{\tau_*}$=1 and $\lambda_{\Phi}$=1 De Vries, 1993).

In Bezug auf die Schergeschwindigkeit $u_*$ = $\sqrt{\tau/\rho_f}$ = $\sqrt{\tau_*(s-1)gD}$ und die Anforderung von $\lambda_{\tau_*}$=1 ist die Ähnlichkeit des Sedimenttransports gegeben, wenn Jansen et al., 1994:

wo

• $\lambda_s$ $\equiv$ scale of relative sediment density

• $\lambda_{D}$ $\equiv$ scale of grain diameter.

Die Ähnlichkeit des einheitlichen Sedimenttransports (d.h. pro Einheitsbreite) kann anhand der Skala $\lambda_{q_b}$ verifiziert werden, die von Exner-Gleichung abgeleitet wird:

With respect to the scale considerations above, $\lambda_{q_b}$ is derived as:

$\lambda_{q_b}$ 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) Jansen et al., 1994. This condition can be considered as fulfilled in this study, as of coarse sediments in the shape of gravel are used for the experiments.