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In flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. [1] Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid which is why potential flow is used for various applications.
The problem of potential compressible flow over circular cylinder was first studied by O. Janzen in 1913 [4] and by Lord Rayleigh in 1916 [5] with small compressibility effects. Here, the small parameter is the square of the Mach number M 2 = U 2 / c 2 ≪ 1 {\displaystyle \mathrm {M} ^{2}=U^{2}/c^{2}\ll 1} , where c is the speed of sound .
Note that the relation of the potential to the flow velocity is unaffected by this transformation: ∇Φ = ∇φ. The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics.
For two-dimensional potential flow, streamlines are perpendicular to equipotential lines. Taken together with the velocity potential , the stream function may be used to derive a complex potential. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition , while the velocity potential ...
Streamlines for the potential flow around a circular cylinder in a uniform flow. The flow pattern is symmetric about a horizontal axis through the centre of the cylinder. At each point above the axis and its corresponding point below the axis, the spacing of streamlines is the same so velocities are also the same at the two points.
Vorticity is useful for understanding how ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a diffusion of vorticity away from the vortex cores into the general flow field; this flow is accounted for by a diffusion term in the vorticity transport equation. [9]
A flow is irrotational if the curl of is zero: = That is, if is an irrotational vector field.. A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential, with =.
Potential flow streamlines for an ideal line source. The case of a vertical line emitting at a fixed rate a constant quantity of fluid Q per unit length is a line source. The problem has a cylindrical symmetry and can be treated in two dimensions on the orthogonal pl