It is important to note that the form of the N.–S. equations given in (3.43) does not possessthis property because we could change eitherρorν(or both) in these equations thus producingdifferent coefficients on pressure and viscous force terms, and the equations would have differentsolutions 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 usuallycalledscaling, or sometimesdimensional analysis; but we will use the latter term to describe aspecific procedure to be studied in the next section. Independent of the terminology, the goal of
86CHAPTER 3. THE EQUATIONS OF FLUID MOTIONsuch an analysis is to identify the set of dimensionlessparameters associated with a given physi-cal situation (in the present case, fluid flow represented by the N.–S. equations) which completelycharacterizes behavior of the system (i.e., solutions to the equations). The first step in this processis identification of independent and dependent variables, and parameters, that fully describe thesystem. 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 rearrangementof the equations, the dimensionless parameters that characterize solutions will be evident, and itis these that must be matched between flows about two geometrically similar objects to guaranteedynamic similarity.We will demonstrate the scaling procedure using the 2-D incompressible continuity and N.–S.equations:ux+vy= 0,(3.45a)ut+uux+vuy=−1ρpx+ν(uxx+uyy),(3.45b)vt+uvx+vvy=−1ρpy+ν(vxx+vyy)−g .(3.45c)The independent variables of this system arex,yandt; the dependent variables areu,vandp, andthe parameters areg, the gravitational acceleration in theydirection (taken as constant), densityρand viscosityν, also both assumed to be constant (or, alternatively,g,ρandμ). In general, theremust be boundary and initial conditions associated with Eqs. (3.45); but these will not introducenew 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 makethe equations dimensionless. These values must be chosen by the analyst, and experience is oftenimportant in arriving at a good scaling of the equations. Here we will demonstrate the approachwith a simple flow in a duct as depicted in Fig. 3.13. In this case we have indicated a typical lengthcUHFigure 3.13: 2-D flow in a duct.scale to be the heightHof the duct, and we have taken the velocity scale to be the centerline speedUc(which is the maximum for these types of flows). This provides sufficient information to scalethe spatial coordinatesxandyas well as the velocity componentsuandv. But we still must scaletime and pressure.
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