7.3 Potential flows with plane symmetry [potential flow top]

Potential flow with plane symmetry has a number of applications, including airfoil theory and prediction of flow in shallow microfluidic devices. In both cases, the variations in one dimension (e.g., the depth of a microchannel) are assumed minor as compared to the variations in the other direction. When this is the case, the flow may be approximated as two-dimensional. For 2D irrotational flows, the velocity potential and stream function are orthogonal harmonic functions—isopotential contours are orthogonal to streamlines and these two functions can be combined and manipulated using complex algebra.

In this section, we create flow solutions from basic elements, such as uniform flows, vortexes, sources, sinks, and doublets. Sources, sinks, and doublets are components of a multipolar expansion of the flow solution, as discussed in Appendix F. First, we describe how complex algebra is used for bookkeeping, then we describe basic elements, then we describe how these elements can be superposed. This is a Green’s function approach to the solution of the governing equation.

7.3.1 Complex algebra and its use in plane-symmetric potential flow

We use complex algebra in this section to simplify the mathematics for plane-symmetric potential flows. The algebra of complex variables is convenient for describing velocity potential and stream function, for reasons analogous to those for circuit analysis with sinusoidal functions, as summarized in Table 7.1.

Complex distance Given two points in the xy-plane separated by an x-distance Δx = x₂ -x₁ and a y-distance Δy = y₂ -y₁, we can define a complex distance :

where j = . This complex distance is a complex number that contains both x- and y-distance information. The undertilde is used to denote a complex representation of two real, physical quantities. This distance can also be written in terms of one length and one angle, in a manner analogous to cylindrical coordinates:

where Δr = and Δθ = tan^-1. Here, Δr is the distance between the points, and Δθ is the angle the line from point 1 to point 2 makes with the x-axis. The value of tan^-1 is understood to fall between 0 and 2π, cosΔθ is equal to , and sinΔθ is equal to . Also, recall Euler’s formula (expjα = cosα+jsinα).

Rotation An immediate example of the utility of complex descriptions of distances in the xy-plane is the ease with with a distance can be rotated. A complex distance  can be rotated by an angle α by multiplying it by exp[jα]. To rotate a flow by the angle α, we replace  with exp[-jα] in the formulas for .

Complex velocity potential We define also a complex velocity potential :

where ϕ_v is the velocity potential and ψ is the stream function. This combines the velocity potential and the stream function into one complex function. Just as the x-direction and y-direction are orthogonal and conveniently represented with a complex variable , ϕ_v and ψ are orthogonal and conveniently represented with a complex value .

Complex velocity Define also a complex velocity :

where u and v are the velocity components in the x- and y-directions.

When a flow is centered on the origin, it is useful to describe flow in terms ofradial (u_r) and counterclockwisecircumferential (u_θ) velocities. In this case, we can equivalently define the complex velocity as

Note that u_r and u_θ are defined with respect to the origin, as are the distances r and θ. The distances Δr and Δθ, in contrast, are measured with respect to a point of interest.

Relation between complex velocity, distance, and velocity potential The relation between the complex velocity, complex potential, and complex distance is analogous to the relation = ∇ϕ_v:

where the star denotes the complex conjugate. The derivative with respect to  can also be written as

(7.20)

See Appendix G for more detailed information about complex differentiation.

Solutions to the Laplace Equation The use of complex variables has an even more elegant result—any well-behaved (i.e., suitably differentiable) complex velocity potential that is a function of  alone ( = ()) automatically satisfies the Laplace Equation and is thus a solution of the fluid mechanical equations. This mathematical concept is discussed in Section G.1.2, but is stated here as well. Any () satisfies the Laplace equation, so = , = ³, = ln, and = all satisfy the Laplace equation. This is not true if we specify any ϕ_v(x,y)—for example, ϕ_v = x², ϕ_v = y^-1, ϕ_v = x³ +y³, and ϕ_v = yxlnx all fail to satisfy the Laplace equation. The reason the complex variable technique works is because the use of a complex variable enforces a specific spatial relation between the derivatives with respect to x and y.

Thus, a Green’s function-type approach (i.e., first finding solutions to the governing equations, then combining them to solve boundary conditions) becomes simple–any complex velocity potential () is already a solution, so all that is required is to combine functions to match the intended boundary conditions, rather than actually solving the Laplace equation directly. Since the Laplace equation is linear and homogeneous, we can superpose solutions at will to match boundary conditions. Some of these solutions are described in the following sections.

7.3.2 Monopolar flow: plane-symmetric (line) source with volume outflow per unit depth Λ

A plane-symmetric source with volume outflow per unit depth Λ located at the origin is denoted by a velocity potential of ϕ_v = lnr and a stream function of ψ = θ, and leads to an exclusively radial velocity of u_r = . A source is a monopole, i.e., a point singularity that induces a flow in the field around it (see Section F.1.2).

More generally, the complex velocity potential

corresponds to the flow induced by a point source of fluid, where Λ is the source strength or volume outflow rate per unit depth [m²∕s] and  is the complex distance measured from the location of the source.

After complex differentiation of , we examine the real part to find

and examine the imaginary part to find

This corresponds to a flow radiating outward from the location of the source. The velocity magnitude scales inversely with the distance from the source (as is the case for all monopolar flows) and is equivalent in all directions. The velocity potential is

(7.24)

and the stream function is

(7.25)

Thus, the the isopotentials are concentric circles around the source, and the streamlines are lines radiating out from the source (Figure 7.2).

Plane-symmetric source located at the origin. For the special case of a source at the origin, Δx = x, Δy = y, Δr = r, and Δθ = θ. Thus ² = r² and = rexpjθ. Given that

(7.32)

we can write

(7.33)

And from = expjθ, we find

and

7.3.3 Plane-symmetric vortex with counterclockwise circulation per unit depth Γ

A plane-symmetric vortex with counterclockwise circulation per unit depth Γ located at the origin is denoted by a velocity potential of ϕ_v = θ and a stream function of ψ = -lnr, and leads to an exclusively circumferential velocity of u_θ = .

More generally, the complex velocity potential

corresponds to the flow induced by a plane-symmetric point vortex, where Γ is the vortex strength or circulation per unit depth [m²∕s] and  is measured from the location of the vortex. The two-dimensional vortex (Figure 7.3) is given by an expression analogous to that of the complex potential for a two-dimensional source, but multiplied by -j. In this sense, a vortex is a monopole with an imaginary strength.

The flow induced by a plane-symmetric vortex can be determined by differentiation of  followed by an examination of the real and imaginary parts of the result, leading to

and

This corresponds to a flow rotating counterclockwise (for positive Γ) around the location of the vortex. The velocity magnitude scales inversely with the distance from the vortex and is equivalent in all directions.

The velocity potential is given by

(7.39)

and the stream function is given by

(7.40)

From this, we see the structure of isopotentials (lines radiating from the vortex) and streamlines (concentric circles around the vortex).

Plane-symmetric vortex located at the origin. For the case of a vortex at the origin, Δx = x, Δy = y, Δr = r, and Δθ = θ. Thus ² = r² and = rexpjθ. Given that

(7.41)

we can write

(7.42)

And from = expjθ,

and

7.3.4 Dipolar flow: plane-symmetric doublet with dipole moment κ

A plane-symmetric doublet is mathematically equivalent to a source and a sink (both of equal and infinite strength) separated by an infinitesimal distance. We define the vector dipole moment of the doublet in the direction from the source to the sink. The doublet is a dipole, which is the mathematical limit of a combination of two oppositely-signed, infinitely strong monopoles separated by an infinitesimal distance (see Section F.1.2).

A plane-symmetric doublet with doublet strength or dipole moment κ located at the origin and aligned along the x-axis is denoted by a velocity potential of ϕ_v = and a stream function of ψ = -. It leads to a radial velocity of u_r = - cosθ and a circumferential velocity of u_θ = - sinθ.

The general complex velocity potential for a plane-symmetric doublet is given by

where κ is the dipole moment per unit depth [m³∕s]. Here  is measured from the location of the source, and α indicates the angle of the vector dipole moment measured counterclockwise with respect to the positive x-axis. The dipole moment thus has components of κcosα and κsinα. We characterize the flow induced by a plane-symmetric doublet by differentiation of . The real component of the result is

while the imaginary part is

This corresponds (for positive κ) to a flow emanating from a point just left of the doublet, circling around, and returning to a point just right of the doublet. The velocity magnitude scales inversely with the squared distance from the doublet (this is the case for all dipolar flows) and is equivalent in all directions.

The velocity potential for this flow is

(7.55)

while the stream function is

(7.56)

The streamlines of the doublet flow are circles tangent to the axis of the doublet, and the isopotentials are circles tangent to the normal to the axis of the doublet (Figure 7.4).

Plane-symmetric doublet located at the origin with dipole moment aligned along positive x-axis. For the case of a doublet at the origin with dipole moment aligned with positive x-axis, Δx = x, Δy = y, Δr = r, Δθ = θ, and α = 0. Thus ² = r² and = rexpjθ. Given that

(7.57)

we set α = 0, Δr = r, and Δθ = θ:

(7.58)

Since this flow is centered on the origin, cylindrical coordinates are convenient, and we look for velocity in the form of = expjθ:

(7.59)

from which we see that

and

__________________________________________________________________________________________________________________________________________________________
EXAMPLE PROBLEM 7.4: 

Consider 2D potential flow. In the domain -20 < x < 20, -20 < y < 20, plot the streamlines for a doublet located at the origin with strength 50π and dipole moment aligned in the x-direction

Solution: See Figure 7.5.

__________________________________________________________________________________________________________________________________________________________

7.3.5 Uniform flow with speed U

A uniform flow moving with speed U in the x-direction is given simply by ϕ_v = Ux and ψ = Uy. Uniform flow is the degenerate case of the Legendre polynomial solution to Laplace’s equation, where A₀ = U.

More generally, the complex velocity potential

corresponds to a uniform flow at speed U in a direction rotated α counterclockwise from the positive x-axis, i.e., a flow with x- and y-velocity components U cosα and U sinα. This flow is by definition uniform in space, so here  can be measured relative to any arbitrary point. Usually,  is measured from the origin.

Uniform Flow: velocity by complex differentiation. The uniform flow is easy to characterize by differentiation of , from which we can see that

and

The velocity potential is given by

(7.65)

and the stream function is given by

(7.66)

The streamlines for this flow are aligned at an angle α with respect to the x-axis, and isopotentials are aligned at an angle α with respect to the y-axis.

Uniform Flow: α = 0. For the α = 0 case (flow from left to right), we substitute in zero for α and find

and

and for the velocities we find

and

7.3.6 Flow around a corner [potential flow top]

Consider a geometry consisting of a wall starting at the origin and proceeding to r→∞ along a line at some angleθ = θ₀, as well as a second wall starting at the origin and proceeding to r→∞ along the x-axis (θ = 0). Consider flow that enters along the θ = θ₀ wall at speed U, runs along the surface, and exits along the x-axis, again at speed U (see Figure 7.6).

This flow is closely related to the uniform flow from the previous section; in fact, the previous section gives the solution for the case where θ₀ = π. In the domain for which 0 < Δθ < θ₀, the solution for this flow is given by

(7.71)

The corner in this flow can be located arbitrarily (by measuring  with respect to any point); further, this flow can also be rotated so that it applies to flow entering along a wall at Δθ = θ₀ +α and exiting along a wall at Δθ = α. In this general case, the complex velocity potential is given by

Here  is measured relative to the location of the corner.

The velocity potential for this flow is given by

where n = π∕θ₀, and the stream function for this flow is given by

The corner flow is commonly observed in any microdevice with a sharp corner. The Laplace equation solutions predict stagnation (zero velocity) at any sharp concave corner and infinite velocity at any sharp convex corner, as shown in Figure 7.7. In any real system, of course, corners have finite curvature. Thus, the velocities at concave corners of finite radius of curvature are low, but finite; and the velocities of convex corners of finite radius of curvature are high, but finite. If these corners had radii of curvature smaller than the Debye length, the use of the Laplace equation to describe the flow outside the boundary layer would be invalid.

7.3.7 Flow over a circular cylinder

A doublet with strength κ and a uniform flow with speed U, both aligned in the same direction, can be superposed to describe the potential flow over a circular cylinder with radiusa = :

(7.75)

This flow is observed in shallow microdevices with circular obstacles.

7.3.8 Conformal mapping

For plane symmetric potential flow, certain mapping functions make it easy to transform problems spatially in a manner that still satisfies the kinematic and mass conservation relations. Since all that is required for a plane-symmetric potential flow to satisfy the Laplace equations is that  be a function of  only, any of a variety of transforms can be useful. Rotation in the plane (Section 7.3.1) is the simplest transform, and orients flows with respect to the coordinate axes. Perhaps the most well-known of these mapping functions is theJoukowski transform (see Section G.5 in the appendix), which maps a family of conic sections. It is particularly useful when mapping results for circular objects to predict flow over elliptical or linear obstacles. Finally, the Schwarz-Christoffel transform maps the upper half of the complex plane onto arbitrary polygonal shapes and thus facilitates calculation of plane-symmetric potential flow through arbitrary polygonal channels. Given the relative ease of numerical solutions of the Laplace equation, the use of these more advanced transforms by practising engineers has become relatively uncommon; when the transforms are applicable, though,they can greatly simplify analysis.

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