Expert: Paul Soderman - 9/24/2011

Question
Sir,
The constant in Bernoulli equation has only one value for the entire flow field in case of irrotational flow, while has values that varies with streamlines in case of rotational flows. I have seen the mathematics of it, but could you suggest any physical interpretations to this?

Answer
Sangeeth
Good question. Imagine a wing flying at uniform speed with a coordinate system attached to the wing.  If you looked far ahead of the wing you would see a velocity field that was uniform vertically and horizontally.  A uniform velocity field is irrotational, and from Helmholtz we know that a fluid initially irrotational remains irrotational as long as viscosity is not introduced.  Therefore, the sum of static and dynamic pressures in Bernoulli's equation would be constant (total pressure) along a streamline and perpendicular to a streamline; i.e., in the entire field as you stated.  This is also true for the flow around the wing except in the boundary layer and wake.

Now consider the boundary layer, which has a velocity gradient that varies from zero at the surface to a free stream value.  So du/dy is large (u and x parallel to the surface, v and y normal to the surface).  But dv/dx is very small because the vertical velocity, v, is small. Therefore, the curl V has a finite value and the flow in the boundary layer is rotational.

It can be shown that in the boundary layer the static pressure variation normal to the surface is essentially zero.  So if the static pressure is constant, but the velocity changes, the sum of static plus dynamic pressure must change along a line perpendicular to the surface.  Thus, the constant in Bernoulli's equation is only valid along a streamline not perpendicular to a streamline in a rotational flow.  
Paul