Fundamental aerodynamic analogies

Aerodynamic theory and Darcy flow modeling in porous media are similar in one respect only both derive from the Navier-Stokes equations governing viscous flows (Milne-Thomson, 1958 Schlichting, 1968 Slattery, 1981). We emphasize this because the great majority of our new solutions derive from the classical aerodynamics literature, but in a subtle manner. Very often, the superficial claim is made that, because petroleum pressure potentials satisfy p/9 + 9 p/9y = 0, the analogy to aerodynamic flowfields, which satisfy Laplace s equation + S cj/Sy = 0 for a similar velocity potential, can be 

Navier-Stokes equations. There are pitfalls in the preceding reasoning while true as far as the equation is concerned, the types of elementary solutions used in applications are different. To understand why, it is necessary to learn some aerodynamics. To be sure, the Navier-Stokes equations for Newtonian viscous flows do apply to both, but different limit processes are at work. For clarity, consider steady, constant density, planar, liquid flows governed by 

u and v are Eulerian velocities in the x and y directions p and p are constant fluid viscosity and density. These equations contain three unknowns, u, V, and the pressure p. To determine them, the mass continuity equation 

The Darcy flow limit. In reservoir engineering, Equations 1-1 and 1-2, known as Darcy s equations, apply (Muskat, 1937). Historically, they were determined empirically by the French engineer Henri Darcy, who observed that the inviscid, high Reynolds number models then in vogue did not describe hydraulics problems. Darcy s laws do not follow immediately from Equations 1-21 and 1-22, but they can be derived through an averaging process taken over many pore spaces and, then, only in the low Reynolds number limit (Batchelor, 1970). If Equations 1-1 and 1-2 are substituted in Equation 1-20 and if constant viscosities and constant isotropic permeabilities are further assumed, Laplace s equation 5 p/9x -l- 5 p/9y = 0 for reservoir pressure p(x,y) follows. Now let us derive the Laplace s equation used in aerodynamics. 

The aerodynamic limit. Inviscid aerodynamics, the study of nonviscous flow, is obtained by contrast in the limit of infinite Reynolds number. In this limit. Equations 1-21 and 1-22 become 

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