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Unformatted text preview: 42 CHAPTER 2. FUNDAMENTAL LAWS dm Av n dt where is the density of the fluid and v n is the face-normal velocity across the face of area A of a control volume (see Problem 2.7.7). 2.6 The Law of Similarity The law of similarity [7, 8] enables in some situations to use a single solution to the equations of fluid motion to represent a whole family of different cases. Consider an example of a steady flow past a solid body, where the flow velocity upstream of the body is U . Consider also several cases of such flows when the body has the same shape but different sizes, L . Now, if the only fluid property affecting this process is the kinematic viscosity, , then in all these cases the distribution of velocity should be a function of space coordinates x x 1 x n , and of at least three additional parameters, ( L U ): (2.95) u i u i x L U The number of parameters 7 can be reduced by considering the dimensions of physical units in which they are measured: L length ; U length time 1 ; length 2 time 1 The only non-dimensional combination of these parameters is provided by the Reynolds number: R e LU We can non-dimensionalize other variables by scaling them with the appro- priate length and velocity scales: (2.96) x i x i L ; u i u U 7 A parameter can be looked at as just another independent variable, like space coordinate or time. However, we treat them separately, since parameters are specific for each physical law, whereas x i t are not. 2.6. THE LAW OF SIMILARITY 43 Since units dimensions should be preserved in an expression of a physical law, a law formulated in dimensionless variables can only contain dimensionless parameters. Hence the new dimensionless variables (2.96) should enter into a relation with R e only, since its the only dimensionless parameter derived from the properties of the system. Following the convention that the velocity is the dependent variable (2.95), we can write this relation as: u i f x R e Thus, using simple considerations of physical dimensions, we reduced the number of parameters from three, ( L U ) to one, ( R e ). Similar considerations allow to reduce the number of parameters in a more general case, which is proved in a so-called PI-theorem . 2.6.1 PI-Theorem Lets consider a physical law formulated for a set of N variables, X 1 X N : (2.97) X 1 X N Suppose that the law requires that each variable can be expressed in units of length, time an mass, which we call the primary dimensions : (2.98) X i length i time i mass i where X represents the dimensionality, and i i i are the powers of the respective units 8 . For example, the dimensionality of the gravity acceleration,....
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- Spring '11