IRROTATIONAL FLOW OF AN INVISCID FLUID

This lecture is devoted to the study of irrotational plane flows of an inviscid fluid. The ultimate goal is to determine the force exerted by a uniform flow on a Joukowski wing. Needless to say the resulting flow field would be the same if the wing moves with the same velocity in a fluid at rest.

Forces exerted on a body - the wing in this case - by an inviscid fluid are always normal to the surface of the body itself and due to the pressure. The solution proceeds then through the determination of the flow and pressure field to obtain the pressure distribution on the profile and then finally the resulting force acting on the body.

It will be shown that this force is always directed at a right angle with respect to flow (lift), that it is proportional to the fluid velocity far from the body and to the circulation around the profile. The component of the force in the direction of the flow is always zero. This apparent paradox can only be solved by dropping the hypothesis of irrotational (and inviscid) flow in a region close to the profile (boundary layer) where viscosity plays a fundamental role and the flow is rotational and viscous.

Nevertheless, boundary layer theory, which will not be considered here, shows that pressure forces are transported throughout the boundary layer without being altered (if separation is absent), so that the information obtained on lift by means of the irrotational theory can be regarded as a good approximation of the actual lift of the wing.

A table of contents follows:

Elements of irrotational flow theory Some elementary notions of Fluid Mechanics are recalled to fully understand the solution method.
Examples of elementary irrotational flows Some simple analytic solutions of irrotional plane flows are shown. The superposition of these simple solutions allows for the study of more complex flows.
Flow around a cylinder The irrotional flow around a cylinder is analysed in terms of flow and pressure fields. Some animations show how the solution is affected by changing the circulation around the cylinder.
Joukowski transformation The conformal transformation due to Joukowski is able to transform the flow domain around a cylinder into the flow domain around a Joukowski wing profile. An interactive application allows the user to experiment with the parameters of the transformation creating different wing profiles.
Flow around a wing The irrotational flow around a wing is analysed in terms of flow and pressure fields. Some animations show how the solution is affected by changing the angle of attack of the wing.

A sequential browsing of the various pages is suggested, at least for first time visitors.

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	Elements of irrotational flow theory Some elementary notions of Fluid Mechanics are recalled to fully understand the solution method.
	Examples of elementary irrotational flows Some simple analytic solutions of irrotional plane flows are shown. The superposition of these simple solutions allows for the study of more complex flows.
	Flow around a cylinder The irrotional flow around a cylinder is analysed in terms of flow and pressure fields. Some animations show how the solution is affected by changing the circulation around the cylinder.
	Joukowski transformation The conformal transformation due to Joukowski is able to transform the flow domain around a cylinder into the flow domain around a Joukowski wing profile. An interactive application allows the user to experiment with the parameters of the transformation creating different wing profiles.
	Flow around a wing The irrotational flow around a wing is analysed in terms of flow and pressure fields. Some animations show how the solution is affected by changing the angle of attack of the wing.