Potential flow theory simplifies fluid mechanics by using velocity potential and stream functions. These mathematical tools help describe fluid motion in idealized scenarios, making complex problems more manageable.
Superposition allows engineers to combine simple flow elements to model intricate situations. While potential flow has limitations, it remains a valuable approximation tool for many real-world applications in aerodynamics and hydrodynamics.
Potential Flow Fundamentals
Velocity potential and stream function
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Velocity potential (ϕ) is a scalar function whose gradient gives the velocity field
Velocity components expressed as u=∂x∂ϕ and v=∂y∂ϕ
Satisfies the Laplace equation ∇2ϕ=∂x2∂2ϕ+∂y2∂2ϕ=0
Stream function (ψ) is a scalar function whose contours represent streamlines
Velocity components expressed as u=∂y∂ψ and v=−∂x∂ψ
Also satisfies the Laplace equation ∇2ψ=∂x2∂2ψ+∂y2∂2ψ=0
Simple potential flow cases have known expressions for ϕ and ψ
Uniform flow: ϕ=U∞x and ψ=U∞y (where U∞ is the freestream velocity)
Source/sink flow: ϕ=2πmln(r) and ψ=2πmθ (where m is the source/sink strength and (r,θ) are polar coordinates)
Doublet flow: ϕ=−2πμx2+y2x and ψ=2πμx2+y2y (where μ is the doublet strength)
Flow field analysis techniques
Streamlines are curves tangent to the velocity vector at every point represented by constant ψ values
Equipotential lines are perpendicular to streamlines and have constant ϕ values
Velocity magnitude calculated as ∣V∣=(∂x∂ϕ)2+(∂y∂ϕ)2=(∂y∂ψ)2+(−∂x∂ψ)2
Stagnation points occur where the velocity is zero (u=v=0) such as at the leading edge of an airfoil or cylinder
Superposition in potential flow
Potential flow solutions can be linearly combined using the principle of superposition
Resulting velocity potential is the sum of individual potentials: ϕ=ϕ1+ϕ2+...
Resulting stream function is the sum of individual stream functions: ψ=ψ1+ψ2+...
Enables modeling complex flows by combining simple potential flow elements
Flow around a cylinder modeled by superimposing uniform flow and doublet flow
Flow around a Rankine oval obtained by superimposing uniform flow, source/sink flow, and doublet flow
Limitations of potential flow theory
Assumes inviscid, incompressible, and irrotational flow which may not hold in real-world scenarios
Does not capture flow separation, boundary layer effects, or turbulence
Limited to low-speed flows (Mach number < 0.3) due to incompressibility assumption
Not suitable for flows dominated by viscous effects (pipe flow, flow around bluff bodies)
Despite limitations, provides valuable approximations for many engineering applications
Airfoil design, hydrodynamics of ships and submarines, flow around buildings and structures