242 Flow dimensionality Dimensionality of a flow field is a concept that often

242 flow dimensionality dimensionality of a flow

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2.4.2 Flow dimensionality Dimensionality of a flow field is a concept that often causes considerable confusion, but it is actually a very simple notion. The first thing to note is that it is not necessarily true that dimensionality of the flow field equals the geometric dimension of the container of the fluid. But at the same time
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30 CHAPTER 2. SOME BACKGROUND: BASIC PHYSICS OF FLUIDS steady transient Time Flow Property (a) well-defined average (c) Time Flow Property (b) Time Flow Property Figure 2.13: Different types of time-dependent flows; (a) transient followed by steady state, (b) unsteady, but stationary, (c) unsteady. we must recognize that all physical flows are really three dimensional (3D). Nevertheless, it is often convenient, and sometimes quite accurate, to view them as being of a lower dimensionality, e.g. , 1D or 2D. We will start with the following mathematical definition of dimension, and then provide some examples that will hopefully clarify these ideas. Definition 2.9 The dimensionality of a flow field corresponds to the number of spatial coordinates needed to describe all properties of the flow. We remark that the typical confusion arises because of our tendency to associate dimension with the number of nonzero components of the velocity field; often, coincidentally, this turns out to be correct. But it is not the correct definition, and it can sometimes lead to inaccurate descriptions and interpretations of flow behavior. We have already seen an example of 1-D flow in our discussions of viscosity. Namely, the flow between two horizontal parallel plates of “infinite” extent in the x and z directions. A similar flow, but now between plates of finite extent in the x direction, is shown in Fig. 2.14(a). (a) x y z (c) x y z (b) x y z Figure 2.14: Flow dimensionality; (a) 1-D flow between horizontal plates, (b) 2-D flow in a 3-D box, (c) 3-D flow in a 3-D box. It should be clear that if we associate u with the x direction, v with the y direction and w with the z direction, then the v and w velocity components do not depend on any coordinates; they are constant and equal to zero in Fig. 2.14(a). At the same time, u depends only on y . If we now
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2.4. CLASSIFICATION OF FLOW PHENOMENA 31 assume density and temperature are constant ( i.e. , do not depend on any spatial coordinate) then at least for a gas it is easily argued that the pressure p must also be constant. Thus, all of the main flow field variables can be completely specified by the single coordinate y , and the flow is 1D. We remark that the one-dimensionality did not arise from the fact that there was only one nonzero component of velocity. Indeed, in principle, v and w could both also depend only on y in the same (or, possibly different) manner as does u ; then all three components of the velocity field would be nonzero, but the flow would still be only 1D.
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