Preliminary conclusion. Turbulence

The collection of various structures in nature or in engineering subjected to a flow of water or air will be extended and discussed in more details in the consequent chapters. The flows associated with them, despite their diversity, can be nevertheless united by the fact that one needs to account both for the internal flow within the permeable structure and for the external free flow over it. Deceleration of the flow within the obstructed but penetrable layer was found to depend significantly upon the closeness of the obstructions characterized by the density n, l/m3 or s, m2/m3. This fact prompts a uniform mathematical treatment of all the above-discussed different flows. It can be suggested to represent obstructions in mathematical models by individual forces Pj7 whereas their collective action on the flow can be described by a smeared (distributed) force (1.6)—(1.7) that acts within the layer but equals zero outside it. The force is discontinuous on the interface between the structure and the flow z = h, so that the interaction between the internal retarded flow and the free external one takes place. 

Many authors have successfully applied the models of a distributed force to a number of problems (see [318, 319] and the consequent chapters the general investigation of such models is presented in the Section 3.1). The structures thus affecting the flow may be equally called canopies or penetrable roughnesses. The hypothesis of the additivity of individual forces is applied to almost all canopies except, perhaps, urban canopy. With this idealization in mind, it might be spoken about easily penetrable roughnesses (EPR). 

The mathematical expression for the force acting on an individual obstruction Pi can be taken from various precise results in basic fluid mechanics. It is known to depend upon the first or second power k of the flow speed relative to the obstacle and be directed oppositely to the flow direction. While smearing all the forces in a unit 

In this Eulerian theoretical approach, a number of obstructions is considered as a continuous medium, or even media. They may have their own motion, for example, the waving of leafs in the case of vegetation canopy. In the case of droplets, their motion in the vertical direction and along the wind can be described by the following momentum equation for each size r  

The purpose of the Section 3.1 is to analyze the implications for mathematical models, each of a practical value, because of the introduction of this force term. A number of known fluid mechanics models can be generalized in such a way. Their consideration highlights most of the properties of the obstructed flows, allows the verification of this theoretical approach, and is a useful methodological tool for teaching and learning the general problem. 

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