Fluid Mechanics

Topics: Fluid dynamics, Fluid mechanics, Potential flow Pages: 19 (4079 words) Published: June 8, 2013
CBE 6333, R. Levicky

1

Potential Flow
Part I. Theoretical Background. Potential Flow. Potential flow is frictionless, irrotational flow. Even though all real fluids are viscous to some degree, if the effects of viscosity are sufficiently small then the accompanying frictional effects may be negligible. Viscous effects become negligible, for example, for flows at high Reynolds number that are dominated by convective transport of momentum. Thus potential flow is often useful for analyzing external flows over solid surfaces or objects at high Re, provided the flows still remain laminar. Moreover, when the flow over a surface is rapid (high Re), the viscous boundary layer region (within which potential flow would be a bad assumption) that forms next to the solid body is very thin. Then, to a very good approximation, the presence of the boundary layer can be neglected when analyzing the potential flow region. That is, the potential flow can be assumed to follow the contours of the solid surface, as if the boundary layer was not present. When the thickness of the boundary layer is small compared to the dimensions of the object over which the potential flow is occurring, we can proceed as follows to analyze the total (potential flow + boundary layer flow) problem: i). First, determine the velocities and pressure distribution in the potential flow region, assuming that the potential flow extends all the way to any solid surfaces present (ie. neglecting the presence of the boundary layer). ii). Solve the flow inside the boundary layer using the pressure distribution obtained from the potential flow solution (i) as input. In other words, the potential flow imposes the pressure on the boundary layer (see the earlier discussion of boundary layers). At the edge of the boundary layer, the velocities are matched with those obtained from the potential flow solution (i) through the use of appropriate boundary conditions.

The Velocity Potential. In potential flow the velocity field v is irrotational. This means that vorticity = ω = ∇ × v = 0 (1)

When ∇ × v = 0 the rate of rotation of an infinitesimal element of fluid is zero. From vector calculus (see the first handout on vector analysis) we know that if a velocity field is irrotational then it can be expressed as the gradient of a "scalar potential" Φ: v = -∇Φ ∇ (irrotational flow) (2)

In equation (2), Φ is the "velocity potential." Using the definition of the ∇ operator, in cartesian coordinates: v1 = ∂Φ ∂x1 v2 = ∂Φ ∂x 2 v3 = ∂Φ ∂x3

(2a)

In cylindrical coordinates:

CBE 6333, R. Levicky

2

vr = -

∂Φ ∂r

vθ = -

1 ∂Φ r ∂θ

vz = -

∂Φ ∂z

(2b)

In spherical coordinates: ∂Φ vr = ∂r

vθ = -

1 ∂Φ r ∂θ

vφ = -

1 ∂Φ r sin θ ∂φ

(2c)

If, in addition, the flow is incompressible, then ∇•v = 0 ∇ 2Φ = 0 ∇•∇Φ = 0 what is equivalent to (3)

(irrotational, incompressible flow)

The Stream Function. The stream function ψ can be defined for any two-dimensional flow, whether the flow is irrotational or not, compressible or incompressible. Two-dimensional means that at least one of the velocity components is zero (in other words, at most two of the velocity components are nonzero). Some flow types for which the stream function is useful, and the accompanying definitions of the stream function, are: Flow in Cartesian coordinates, with v3 = 0: ∂ψ ∂ψ v1 = − v2 = ∂x2 ∂x1 Flow in cylindrical coordinates with vZ = 0: 1 ∂ψ ∂ψ vr = vθ = r ∂θ ∂r Flow in cylindrical coordinates with vθ = 0: 1 ∂ψ 1 ∂ψ vr = vZ = r ∂z r ∂r Flow in spherical coordinates with vφ = 0: 1 ∂ψ vr = - 2 r sin θ ∂θ

(4a)

(4b)

(4c)

vθ =

1 ∂ψ r sin θ ∂r

(4d)

The definitions (4a) through (4d) are joined by the requirement that the stream function automatically satisfy the equation of continuity for an incompressible fluid, ∇•v = 0. The divergence ∇•v will automatically equal zero because of the equivalence of the mixed second derivatives of the stream function. For...
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