ABSTRACT
In this chapter, we will consider the most basic two-dimensional (2D) flows in ideal and viscous incompressible fluids. In 2D flows, the velocity field has two components which depend on two physical space coordinates and time, u = (u(x, y, t), v(x, y, t), 0). Respectively, the vorticity field has only one nonzero component, ω = (0, 0, ω(x, y, t)). This component satisfies the equation (2.21), which we will reproduce here for convenience:
Dtω ≡ (∂t + (u · ∇))ω = ν∇2ω. (6.1)
For 2D incompressible flows, one can introduce a representation of the velocity field in terms of stream function ψ(x, y, t) as follows,
u = ∇ψ × zˆ, (6.2)
or in component form u = ∂yψ, v = −∂xψ. (6.3)
In terms of the stream function the vorticity is:
ω = −∇2ψ. (6.4)
If the 2D flow is irrotational, then we also have a representation in terms of the velocity potential u = ∇φ, or
u = ∂xφ, v = ∂yφ, (6.5)
and the viscous term is automatically zero, ν∇2u = ν∇∇2φ = 0. Combining expressions (6.3) and (6.5) we have
∂xφ = ∂yψ, ∂yφ = −∂xψ. (6.6)
These relations are nothing but the Cauchy-Riemann conditions, i.e. necessary and sufficient conditions for a complex function w(z) = φ + iψ (where z = x+ iy) to be complex differentiable, that is, holomorphic. The function w(z)
Solutions
parts of it Laplace’s equation,
∇ · u = ∇2φ = 0, ∇× u = ∇2ψ = 0. (6.7)
For the velocity u = (u, v) in terms of the complex potential we have
u− iv = ∂zw, (6.8)
so that |u| =
√ u2 + v2 = |∂zw|. (6.9)
The identification of the complex potential of a 2D ideal irrotational flow with a holomorphic function had profound consequences for the rapid development of both aerodynamic theory and complex analysis, notably utilising conformal maps to obtain new fluid flow solutions out of already known and often simpler ones. This approach culminated in 1906 in Zhukovskiy’s lift theorem, stating the following.
