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Summary of Chapter 5
In this chapter, unsteady incompressible and compressible flows are discussed within the framework of thin airfoil and small disturbance approximations.
The fundamental aspect of unsteady, inviscid flow is the shedding of vorticity from the trailing edge of the thin profile. A Kutta-Joukowski condition still applies at the trailing edge to render the solution unique. The steady linear models of subsonic compressible and supersonic potential flows require a cut across which the potential has a jump equal to the circulation around the profile, Г =< ф >. In steady flow, the cut could be placed anywhere to make the domain simply-connected. However, in unsteadys flow, the cut represents the inviscid “wake” or slip line, downstream from the trailing edge, along the x-axis, where the potential function is discontinuous and the jump in potential is a function of time and space. The unsteady Bernoulli’s law (or generalized Bernoulli’s law in compressible flow) provides the condition along the slip line by specifying the continuity of pressure.
The example of the plunging plate in incompressible flow is simulated numerically and the results are compared with the exact solution. Care must be exerted to include
the suction force at the leading edge of the plate in order to obtain zero drag when the flow returns to steady-state. Noteworthy is the first instant in the simulation when the circulation in the plate is still close to zero and the vorticity distribution similar to that of an elliptic wing. The lift and drag are equal, resulting in a force perpendicular to the plate and their magnitude is inversely proportional to the time step.
Another example is the pitching NACA0012 at low speeds and low frequency.
Extension to the unsteady full potential equation is discussed. The unsteady transonic small disturbance equation (unsteady TSD) is derived and the magnitude of the coefficients of the time derivatives analyzed in relation to the reduced frequency. The low-frequency TSD is highlighted as it is of great importance to the study of transonic unsteady flows. Key papers are referenced for the reader interested in more in depth exposure.
