Similarity Parameters

In Section 1.7, we introduced the concept of dimensional analysis, from which sprung the similarity parameters necessary to ensure the dynamic similarity between two or more different flows (see Section 1.8). In the present section, we revisit the governing similarity parameters, but cast them in a slightly different light.

Consider a steady two-dimensional, viscous, compressible flow. The x-momen­tum equation for such a flow is given by Equation (15.19a), which for the present case reduces to

In Equation (15.27), p, u, p, etc., are the actual dimensional variables, say, p = kg/m3, etc. Let us introduce the following dimensionless variables:

/ _ p

, u

, _ V

U = ——

Poo

Voo

V ~ Voo

/ p

, _ X

, У

/X = ——

У = –

P’00

c

с

where Poo, Vqo, poo, and рьж are reference values (say, e. g., freestream values) and c is a reference length (say, the chord of an airfoil). In terms of these dimensionless

variables, Equation (15.27) becomes

where M0о and Reoo are the freestream Mach and Reynolds numbers, respectively, Equation (15.28) becomes

Equation (15.29) tells us something important. Consider two different flows over two bodies of different shapes. In one flow, the ratio of specific heats, Mach number, and Reynolds number are у і, Мж, and Re^j, respectively; in the other flow, these parameters have different values, y2, M^, and Re^. Equation (15.29) is valid for both flows. It can, in principle, be solved to obtain u’ as a function of x’ and y’. However, since y, Moo, and Reoo are different for the two cases, the coefficients of the derivatives in Equation (15.29) will be different. This will ensure, if

f (x y’)

represents the solution for one flow and

и = h(x y’)

represents the solution for the other flow, that

However, consider now the case where the two different flows have the same values of y, Moo, and Reoo – Now the coefficients of the derivatives in Equation (15.29) will be the same for both flows; that is, Equation (15.29) is numerically identical for the two flows. In addition, assume the two bodies are geometrically similar, so that the boundary conditions in terms of the nondimensional variables are the same. Then, the solutions of Equation (15.29) for the two flows in terms of u’ — f(x’, v’) and и’ = /г(х’, у’) must be identical; that is,

f{(x’,y’) = f2(x, y) [15.30]

Recall the definition of dynamically similar flows given in Section 1.8. There, we stated in part that two flows are dynamically similar if the distributions of V/Vx, p! P<x>, etc., are the same throughout the flow field when plotted against common nondimensional coordinates. This is precisely what Equation (15.30) is saying—that

u’ as a function of x’ and v’ is the same for the two flows. That is, the variation of the nondimensional velocity as a function of the nondimensional coordinates is the same for the two flows. How did we obtain Equation (15.30)? Simply by saying that y, Mqo, and Reoo are the same for the two flows and that the two bodies are geometrically similar. These are precisely the criteria for two flows to be dynamically similar, as originally stated in Section 1.8.

What we have seen in the above derivation is a formal mechanism to identify governing similarity parameters for a flow. By couching the governing flow equations in terms of nondimensional variables, we find that the coefficients of the derivatives in these equations are dimensionless similarity parameters or combinations thereof.

To see this more clearly, and to extend our analysis further, consider the en­ergy equation for a steady, two-dimensional, viscous, compressible flow, which from Equation (15.26) can be written as (assuming no volumetric heating and neglecting the normal stresses)

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