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Velocity potential

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A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.

It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case, ∇ × u = 0 , {\displaystyle \nabla \times \mathbf {u} =0\,,} where u denotes the flow velocity. As a result, u can be represented as the gradient of a scalar function ϕ: u = ∇ φ   = ∂ φ ∂ x i + ∂ φ ∂ y j + ∂ φ ∂ z k . {\displaystyle \mathbf {u} =\nabla \varphi \ ={\frac {\partial \varphi }{\partial x}}\mathbf {i} +{\frac {\partial \varphi }{\partial y}}\mathbf {j} +{\frac {\partial \varphi }{\partial z}}\mathbf {k} \,.}

ϕ is known as a velocity potential for u.

A velocity potential is not unique. If ϕ is a velocity potential, then ϕ + f(t) is also a velocity potential for u, where f(t) is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable.

The Laplacian of a velocity potential is equal to the divergence of the corresponding flow. Hence if a velocity potential satisfies Laplace equation, the flow is incompressible.

Unlike a stream function, a velocity potential can exist in three-dimensional flow.

Usage in acoustics

In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential ϕ instead of pressure p and/or particle velocity u. ∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = 0 {\displaystyle \nabla ^{2}\varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=0} Solving the wave equation for either p field or u field does not necessarily provide a simple answer for the other field. On the other hand, when ϕ is solved for, not only is u found as given above, but p is also easily found—from the (linearised) Bernoulli equation for irrotational and unsteady flow—as p = − ρ ∂ φ ∂ t . {\displaystyle p=-\rho {\frac {\partial \varphi }{\partial t}}\,.}

See also

Notes

  1. Anderson, John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 978-0521669559.
  2. Pierce, A. D. (1994). Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America. ISBN 978-0883186121.

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