Expert: Paul Soderman - 9/27/2011

Question
QUESTION: Sir,

D Alembert's paradox predicts zero drag because of the symmetrical pressure distribution about a vertical axis passing through the body placed in a potential flow. But, this would apply only to symmetrical bodies right? Would paradox hold for an unsymmetrical body( say a tea cup) in a potential flow?

ANSWER: Potential flow theory allows us to compute the flow about arbitrary bodies by superimposing solutions from elementary sources, sinks, and vortices.  But since no viscosity is allowed, no flow separation occurs and drag is always zero, even for a tea cup or any arbitrary body shape. The problem with the tea cup is that it is very difficult to find the proper distribution of elementary solutions and difficult to find a unique solution.  Even the flow about a cambered airfoil at angle of attack, which is much simpler than flow around a teacup cannot be solved until the circulation is defined using the Kutta condition, a kind of empirical fix for that problem.  So we really only use potential flow to get quick solutions for flow around simple bodies.  The solutions might be useful to check data or more complicated algorithms.  And we never use potential flow for drag analysis.

D'Alembert's paradox is not much of a paradox.  Real flows contain viscosity that causes flow separation on bodies and drag.  A math model that does not incorporate viscosity cannot predict drag.
Paul

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QUESTION: Sir,

I thought about your explanation and hit upon this query. Flow separation involves the concept of boundary layer ( and hence viscosity). But drag on a body can also be caused by sum of components of pressure tangential to the surface right? Is this the type of drag called form drag? So, wouldn't unsymmetrical bodies have unequal distribution of such tangential components that cause drag?

Answer
Sangeeth
Form drag is the term often given to friction plus pressure drag.  Real bodies in flow have both.  But in potential flow viscosity is not allowed so friction drag is zero.  And because potential flow is inviscid, separation is not allowed and the pressures in the flight (drag) direction cancel out.  See Low-Speed Aerodynamics by J. Katz and A. Plotkin.  They show that potential flow solutions for a sphere, for example, give the same pressures on the upwind side of the sphere as on the downwind side.  No drag.  The same is shown for a flat plate normal to a flow.  And the same is true for unsymmetric bodies such as thick, cambered airfoils. The most interesting case is a thin flat plate at small angle of attack.  Integration of pressures on the upper and lower surfaces result in a resultant force that is normal to the plate surface.  In other words, the resultant force is tilted aft of the vertical direction and can be resolved into a lift normal to the flight direction and a drag component parallel to the flight direction.  However, further analysis shows that there is a strong suction force on the leading edge that cancels the drag due to surface normal forces.  Again - no drag.  This is why potential flow can only be used to calculate lift forces or moments due to lift.
Paul