Extension of the Galilean-Invariant Formulation of Bernoulli’s Equation

Bernoulli’s equation is a famous, elegant fluid mechanics relation that, in its most popular version, relates pressure, velocity, and elevation changes along each streamline in a steady barotropic inviscid flow. Despite its simplicity and restrictive assumptions, this equation is a powerful and effective analysis tool in a variety of real-flow situations. However, the Bernoulli equation is unfortunately not Galilean-invariant, i.e., it does not satisfy the desirable property of remaining unchanged under a Galilean transformation between two inertial reference frames reciprocally moving with constant velocity. In a previous paper, the author presented a Galilean-invariant formulation of the Bernoulli equation, in which a new term is introduced in the definition of the Bernoulli constant. In the present technical note, this Galilean-invariant form of the Bernoulli relation is extended to (1) unsteady flow, (2) compressible flow, and (3) a noninertial frame of reference accelerating with constant rectilinear acceleration. Similar extensions are consolidated for the classic Bernoulli equation and treated in fundamental fluid mechanics textbooks. The use of the updated Galilean-invariant Bernoulli equation in these less-restrictive contexts is illustrated with typical engineering application examples. The results demonstrate that the extended version of the Bernoulli principle can also be applied along the same flow lines in a reference frame moving with constant velocity relative to another reference frame in which the conventional Bernoulli equation is valid, provided that the Bernoulli constant includes a suitable additional term.