Double Averaging method
Note: This article will be finished with the submission of deliverable 3.3 and 3.4 in October, 2020
Contents
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Type: Method
Introduction
The Double Averaging Method (DAM) is used for hydrodynamic variables and to provide simplified equations for the hydrodynamic modeling of flow similar to the Reynolds averaged Navier-Stokes equations (RANS), which are time-averaged. Over rough river beds flow structures are highly heterogenous. Therefore, they need to be resolved in detail instead of using simplified 2D assumptions, which is common within the RANS (Nikora et al. 2001). The concept of the DAM shall overcome this problem by taking into account the averaging over a certain area in addition to the averaging over time. This area should be chosen parallel to the river bed. The approach leads then to new equations of conservation of continuity and momentum.
The main advantages of the double-averaging approach include (Nikora et al 2007 a):
- a consistent link between spatially averaged roughness parameters, bed shear stress, and double-averaged flow variables
- explicit accounting for the viscous drag, form drag, form-induced stresses, and substance fluxes as a result of rigorous derivation rather than intuitive reasoning
- the possibility for scaling considerations and parameterizations based on double averaged variables
- the possibility for the consistent scale partitioning of the roughness parameters and flow properties, such as the bed shear stress.
However, the key limitation of this method is the need of a clear separation between turbulence and the mean flow on a wide enough scale.
A detailed explanatory derivation of the Double-Averaged Equations can be found for example within Giménez-Curto & Lera, 1996; Mignot et al., 2009; Nikora et al., 2001; Nikora et al., 2007 a and b.
Application
How can the DAM be applied?
Turbulent flow is characterized chaotic - irregular in space and time. Hence, a statistical description needs to be used to define its properties. This can be done using the Reynolds-decomposition, which is a fundamental basis in characterizing turbulence: The measured instantaneous variable u is divided into the ensemble average ū (time-averaged) and the temporal fluctuation u', as illustrated in Figure 1.
Hence, the instantaneous variable is a combination of these parts. As a next step the time-averaged variable is also averaged in space, similar to the Reynolds-decomposition. This is most often done in planes zi parallel to the bed. The resulting instantaneous variable consists of the DA-value, the spatial fluctuation and the temporal fluctuation in the time-averaged flow variable:
θ(𝑥, 𝑦, 𝑧, 𝑡) = 〈θ̅ 〉(𝑧𝑖) +θ̃ (𝑥, 𝑦, 𝑧) + θ′(𝑥, 𝑦, 𝑧, 𝑡)
