It is important to note that the form of the NS equations given in 343 does not

It is important to note that the form of the ns

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It is important to note that the form of the N.–S. equations given in (3.43) does not possess this property because we could change either ρ or ν (or both) in these equations thus producing different coefficients on pressure and viscous force terms, and the equations would have different solutions for the two flow fields, even for flows about geometrically similar objects. The method we will employ to achieve the desired form of the equations of motion is usually called scaling , or sometimes dimensional analysis ; but we will use the latter term to describe a specific procedure to be studied in the next section. Independent of the terminology, the goal of
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86 CHAPTER 3. THE EQUATIONS OF FLUID MOTION such an analysis is to identify the set of dimensionless parameters associated with a given physi- cal situation (in the present case, fluid flow represented by the N.–S. equations) which completely characterizes behavior of the system ( i.e. , solutions to the equations). The first step in this process is identification of independent and dependent variables, and parameters, that fully describe the system. Once this has been done, we introduce “typical values” of independent and dependent vari- ables in such a way as to render the system dimensionless. Then, usually after some rearrangement of the equations, the dimensionless parameters that characterize solutions will be evident, and it is these that must be matched between flows about two geometrically similar objects to guarantee dynamic similarity. We will demonstrate the scaling procedure using the 2-D incompressible continuity and N.–S. equations: u x + v y = 0 , (3.45a) u t + uu x + vu y = 1 ρ p x + ν ( u xx + u yy ) , (3.45b) v t + uv x + vv y = 1 ρ p y + ν ( v xx + v yy ) g . (3.45c) The independent variables of this system are x , y and t ; the dependent variables are u , v and p , and the parameters are g , the gravitational acceleration in the y direction (taken as constant), density ρ and viscosity ν , also both assumed to be constant (or, alternatively, g , ρ and μ ). In general, there must be boundary and initial conditions associated with Eqs. (3.45); but these will not introduce new independent or dependent variables, and usually will not lead to additional parameters. Thus, in the present analysis we will not consider these. We next introduce the “typical” values of independent and dependent variables needed to make the equations dimensionless. These values must be chosen by the analyst, and experience is often important in arriving at a good scaling of the equations. Here we will demonstrate the approach with a simple flow in a duct as depicted in Fig. 3.13. In this case we have indicated a typical length c U H Figure 3.13: 2-D flow in a duct. scale to be the height H of the duct, and we have taken the velocity scale to be the centerline speed U c (which is the maximum for these types of flows). This provides sufficient information to scale the spatial coordinates x and y as well as the velocity components u and v . But we still must scale time and pressure.
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