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The Miracle of Eddy-Free Flow: How Can Potential Flow Help Us Understand Fluid Dynamics?
In fluid mechanics, potential flow (or irrotational flow) is a way of describing the flow of a fluid characterized by the absence of vorticity in the fluid. This description usually occurs in the vanishing viscosity limit, that is, in the case of inviscid fluids, where the flow contains no vorticity. The velocity field of a potential flow can be expressed as the gradient of a scalar function called the velocity potential. Thus, the potential flow is characterized by an irrotational velocity field, which is a reasonable approximation in many applications. The irrotational property of the potential flow stems from the fact that the curl of the gradient of a scalar is always zero.
"In an irrotational flow, the vorticity vector field is zero."
In incompressible flows, the velocity potential satisfies Laplace's equations, which allows the application of potential theory. However, potential flow can also be used to describe compressible flow as well as Hele-Shaw flow. The model of potential flow is applicable to both stationary and non-stationary flow cases. The application range of potential flow is very wide, including the peripheral flow field of aerodynamic wings, ocean waves, water flow and electroosmotic flow.
Despite the advantages of potential flow, estimation of potential flow becomes inapplicable when the flow (or some part of it) contains strong vorticity effects. In those flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory cannot provide reasonable flow predictions. Fortunately, however, certain large regions of the flow can be assumed to be irrotational, which is why potential flows are widely used. For example, the assumption of potential flow is valid in cases such as flow around aircraft, groundwater flow, acoustics, and water waves.
"The characteristic of latent flow is that it is irrotational, which makes it computationally simpler."
Description and characteristics of potential flows
In potential flow or irrotational flow, the vorticity vector field is zero, that is, ω ≡ ∇ × v = 0, where v(x, t) is the velocity field and ω(x, t) is the vorticity field. Any vector field with zero curl can be expressed as the gradient of some scalar function, such as φ(x, t), which is called the velocity potential. Since the curl of the gradient is always zero, we can get v = ∇φ. The velocity potential is not unique, since an arbitrary time function f(t) can be attached to the velocity potential without affecting the associated physical quantity v.
The properties of the potential flow are such that the number of cycles Γ around any simple connected contour C is zero. This can be proved by Stokes' theorem: Γ ≡ ∮C v · dl = ∫ω · df = 0, where dl is the line element on the contour and df is the area element on any surface enclosed by the contour.
In multiply connected spaces (e.g., a contour around a solid object or a torus in three dimensions), or in the presence of concentrated vorticity (e.g., so-called irrotational or point vortices, or in smoke rings), ), the loop Γ does not need to be zero. When surrounding the contour of a self-prolongated solid cylinder, Γ = Nκ, where κ is the circulation constant, and this example belongs to a doubly connected space.
Incompressible Flow
In the case of incompressible flows, such as liquids or gases at low Mach numbers, the velocity v has zero divergence, i.e. ∇ · v = 0. At this time, assuming v = ∇φ, we can obtain that φ satisfies the Laplace equation ∇²φ = 0. Because the solutions to Laplace's equations are harmonic functions, each harmonic function represents a potential flow solution.
"In an incompressible flow, the potential flow is completely determined by its kinematics."
The potential flow actually satisfies the entire Navier-Stokes equations, not just the Euler equations, because the viscosity term is identically equal to zero. Factors that cause the potential flow to fail to satisfy the necessary boundary conditions, especially near solid boundaries, make it invalid for representing the desired flow field. If the potential flow satisfies the required conditions, then it can be a solution to the incompressible Navier-Stokes equations.
So, when potential flow makes us re-examine our basic understanding of fluid mechanics, can it bring new thoughts and inspirations?
