Subcritical flow, small perturbation theory (Prandtl-Glauert rule)

In certain cases of compressible flow, notably in supersonic flow, exact solutions to the equations of motion may be found (always assuming the fluid to be invisdd) and when these are applied to the flow in the vicinity of aerofoils they have acquired the soubriquet of exact theories. As described, aerofoils in motion near the speed of sound, in the transonic region, have a mixed-flow regime, where regions of subsonic and supersonic flow exist side by side around the aerofoil. Mathematically the analysis of this regime involves the solution of a set of nonlinear differential equations, a task that demands either advanced computational techniques or some degree of approximation.

Compressible flaw 335

The most sweeping approximations, producing the simplest solutions, are made in the present case and result in the transformation of the equations into soluble linear differential equations. This leads to the expression linearized theory associated with aerofoils in, for example, high subsonic or low supersonic flows. The approximations come about mainly from assuming that all disturbances are small disturbances or small perturbations to the free-stream flow conditions and, as a consequence, these two terms are associated with the development of the theory.

Historically, H. Glauert was associated with the early theoretical treatment of the compressibility effects on aerofoils approaching the speed of sound, and developed what are, in essence, the linearized equations for subsonic compressible flow, in R and M, 305 (1927), a note previously published by the Royal Society. In this, mention is made of the same results being quoted by Prandtl in 1922. The significant compressibility effect in subsonic flow has subsequently been given the name of the Prandtl-Glauert rule (or law).

Furthermore, although the theory takes no account of viscous drag or the onset of shock waves in localized regions of supersonic flow, the relatively crude experimental results of the time (obtained from the analysis of tests on an airscrew) did indicate the now well-investigated critical region of flight where the theory breaks down. Glauert suggested that the critical speed at which the lift falls off depends on the shape and incidence of the aerofoil, and this has since been well-substantiated.

In what follows, attention is paid to the approximate methods of satisfying the equations of motion for an invisdd compressible fluid. These depend on the simultaneous solution of the fundamental laws of conservation and of state. Initially a single equation is desired that will combine all the physical requirements. The complexity of this equation and whether it is amenable to solution will depend on the nature of the particular problem and those quantities that may be conveniently minimized.

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