Two dimensional potential flow around a rectangular pole solved by a multiple

linear regression

Eunice J. Kim1 and Ildoo Kim2, a)

1)Department of Mathematics and Statistics, Amherst College,

Massachusetts 010022)School of Engineering, Brown University, Providence,

Rhode Island 02912

(Dated: October 25, 2017)

A potential flow around a circular cylinder is a commonly examined problem in an

introductory physics class. We pose a similar problem but with different boundary

conditions where a rectangular pole replaces a circular cylinder. We demonstrate

to solve the problem by deriving a general solution for the flow in the form of an

infinite series and determining the coefficients in the series using a multiple linear

regression. When the size of a pole is specified, our solution provides a quantitative

estimate of the characteristic length scale of the potential flow. Our analysis implies

that the potential flow around a rectangular pole of the diagonal 1 is equivalent to

the potential flow around a circle of diameter 0.78 to a distant observer.

a)Electronic mail: ildoo_kim@brown.edu

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I. INTRODUCTION

Hydrodynamic similarity is one of the core concepts in fluid dynamics1–3. When a fluid

system is free of any external forces, the Navier-Stokes equation is non-dimensionalized using

a system’s characteristic length scale D and a characteristic velocity U . Then the Reynolds

number Re = UD/ν becomes the only parameter of the governing equation where ν is the

kinematic viscosity. At the inviscid limit, i.e. when ν approaches to zero, the velocity field

is simplified to v/U = F (x/D), where F is an arbitrary function.

In a certain fluid system, the determination of a characteristic length scale D can be

equivocal. The problem of von Kármán vortex streets may be one of them. A von Kármán

vortex street is staggered rows of vortices that form behind an obstructing body in a stream

of fluid4. In an effort to understand its dynamics, one of the most widely studied topics is

the relationship between the Strouhal number St = fD/U and Re, where f is the frequency

of vortex shedding5–8. In this problem, like many others in fluid mechanics, ‘the widest

dimension that faces the flow’, say W , is conventionally used as the characteristic length

scale D of the flow.

Phenomenologically, the St-Re relation has two regimes8; 1) when Re / 200, St increases

with Re, and 2) when Re ' 200, St reaches an asymptote St∞. A theoretical deliberation

suggests that the St∞ ' 0.2 is expected9, and it is observed from many experiments of

vortex streets from circular cylindrical objects4,8,10. However, when a rectangular pole is used

instead of a circular cylinder, different St∞ values are reported from both computational and

experimental studies. A few examples of computational studies include those by Sohankar

et al.11 (St∞ = 0.165), Saha et al.12 (St∞ = 0.167), Inoue et al.13 (St∞ = 0.151), and Ali et

al.14 (St∞ = 0.1600) On the experimental side, Norberg15 measured St∞ ' 0.17 when the

angle of attack is 45 degrees (see Figure 1(a)) and St∞ ' 0.13 when the angle of attack is

zero (see Figure 1(b)).

Suppose that Re specifies the properties of an input flow, and St describes the response

of the system to the input. Then, the system’s response is a function of the input, but the

response function can also vary. In the case of the vortex street problem, the response func-

tion is usually modeled with two parameters5,6,8, and these two parameters are determined

by the hydrodynamic properties of the flow such as the vorticity strength and the drag

coefficient9,16,17. Therefore, the fact that we observe different St∞ provides the link between

2

Figure 1. Two orientations of the rectangular pole with respect to the flow direction are considered.

The flow meets either (a) a point of the square pole and either side symmetrically (‘diamond’), or

(b) a face of the square pole head-on (‘square’).

the geometry of the obstacle and hydrodynamic properties18. However, the comparison be-

tween non-dimensionalized variables is no longer meaningful if our physical description of

the flow is not self-consistent; namely, our measure of D must represent the most relevant

length scale of the flow. The convention of setting D = W is trivial for the wakes behind

a circular cylinder, because there is no other length scale that is relevant. However, for

the wakes of rectangular poles, such a strict convention breaks down because the widest

dimension of the body depends on the orientation of the pole with respect to the flow.

In this paper, we suggest an alternative way of measuring D based on the potential flow

around a rectangular pole. A pure potential flow does not interact with the rectangular pole

and will not shed any vortices19. However, in real flow a boundary layer exists between the

flow and the rectangular pole, and the potential flow solution is valid outside the boundary

layer20. We assume that the thickness of such boundary layer is negligible compared to the

size of the pole itself. Therefore, vortices created and discharged from the boundary layer

can be safely approximated as singularities that are freely advected by the surrounding

potential flow. The characteristic time scale of a flow in this regime is D/U , which specifies

the shedding frequency.

To derive the characteristic length scale of the potential flow around the rectangular pole,

we first arrive at the solution of the Laplace equation as an infinite series. Next, a multiple

linear regression is used to determine the coefficient of each term in the series. Unlike the

potential flow around a circular cylinder, the exact analytic solution of the potential flow

around a rectangular pole is not known, and there may not be one. Any harmonic function

must be smooth at all points, which is not the case at the four corners of a rectangle

forming mathematical singularities. Even though the problem is soluble either numerically

or using conformal mapping, outcome of such methods requires extra steps to inspect the

3

characteristic length scale of the flow. Our contribution here is to provide an alternative

method and a solution that can be directly translated to the length scales of the flow. The

introduction of the current method also bears educational value as it helps students to digest

the concept of constructing a potential flow by adding simple elementary flows to the mean

flow. Furthermore, a parallel can be drawn to any irrotational and incompressible field such

as electrostatics.

II. ANALYTIC SOLUTION WITH A MEAN FLOW

Suppose that a stream of fluid flows in the positive x direction, and it encounters a

rectangular pole located at the origin. Let us assume that the pole is infinitely long in the z

direction, therefore the problem becomes two dimensional. In cylindrical polar coordinates,

the potential function for the flow φ satisfies the Laplace equation,

∇2φ =1

r

∂

∂r

(r∂φ

∂r

)+

1

r2∂2φ

∂θ2= 0. (1)

Using the separation of variables, φ = R(r)Θ(θ), Eq. (1) is separated into the radial and

polar equations with solutions R = r±m and Θ = exp(±imθ), where m is an integer. These

constitute the general solution φ such that

φ = UA0 ln r+U∞∑m=1

(Pmr

−m cosmθ +Qmr−m sinmθ + P ′mr

m cosmθ +Q′mrm sinmθ

). (2)

Far from the pole, the mean flow is the sole remaining component. Therefore φ = Ur cos θ

is desired as r →∞ rendering all coefficients of the diverging terms to be 0 except P ′1 = 1.

Then Eq. (2) simplifies to

φ = U

∞∑m=1

(Pmr

−m cosmθ +Qmr−m sinmθ

)+ Ur cos θ. (3)

Eq. (3) is further simplified when the object is assumed symmetric. Let us consider two

orientations of a rectangular pole where square refers to the flow meeting the pole directly

face on and diamond refers to the flow reaching the side at a 45-degree angle (see Figure 1).

In both cases, the flow potential function is symmetric about the x-axis and anti-symmetric

about the y-axis. The symmetry about the x-axis suggests that ur(r, θ) = ur(r,−θ) and

uθ(r, θ) = −uθ(r,−θ) and yields

φ(r, θ) = φ(r,−θ). (4)

4

This eliminates all sine solutions, namely Qm = 0 for all m. Similarly, the anti-symmetry

about the y-axis suggests

φ(r, θ′) = −φ(r,−θ′), (5)

where θ′ = θ − π/2. This condition implies that Pm = 0 for all even numbered m’s.

Finally we get the x-symmetric and y-antisymmetric general solution,

φ = Ur cos θ + U∑

n=1,2,...

Anr−(2n−1) cos [(2n− 1)θ] , (6)

where An = P(2n−1).

The first term in Eq. (6) represents to the mean flow far from the pole. Each term

in the summation, cos [(2n− 1)θ] /r(2n−1), represents (2n − 1) pairs of flow dipoles with

the strength of An; these coefficients are determined by the boundary condition given to a

specific problem.

III. POTENTIAL FLOW AROUND A DIAMOND

A. Boundary Equations

In this section, we consider the potential flow around the diamond configuration in which

the pole meets the flow at 45 degrees (see Figure 1(a)). In polar coordinates, the shape

is expressed as r = a/(sin θ + cos θ) for 0 ≤ θ < π2, r = a/(sin θ − cos θ) for π

2≤ θ < π,

r = a/(− sin θ − cos θ) for π ≤ θ < 3π2

and r = a/(− sin θ + cos θ) for 3π2≤ θ < 2π, where

the length of the diagonal is 2a.

In the first quadrant, the velocity component normal to the surface vn can be acquired

by taking an inner product of ~v and the surface normal n = (x + y)/√

2, and it becomes

zero by the slip boundary condition,

vn = ~v · n = ∇φ · (x+ y)√2

= vr

(r · x+ r · y√

2

)+ vθ

(θ · x+ θ · y√

2

)= 0. (7)

Substituting vr = ∂φ/∂r and vθ = (1/r)∂φ/∂θ to Eq. (7), and using r ·x = cos θ, r ·y = sin θ,

θ · x = − sin θ, θ · y = cos θ and trigonometric identities, we get

vn = 0 =U√

2− U

∑n

{An

(2n− 1)

r2n· cos (2nθ) + sin (2nθ)√

2

}. (8)

5

Using r = a/(sin θ + cos θ) and sin θ + cos θ =√

2 sin(θ + π/4), we rewrite Eq. (8) using

sines of θ:

0 = − 1√2

+∑n

An · (2n− 1) · a−2n · 2n · sin2n(θ +

π

4

)sin(

2nθ +π

4

), (9)

where 0 ≤ θ < π/2. We can do similar calculations in the second quadrant, and we get

0 =1√2

+∑n

An · (2n− 1) · a−2n · 2n · sin2n(θ − π

4

)sin(

2nθ − π

4

), (10)

where π/2 < θ < π.

Equation (10) is identical to Eq. (9) under substitution of θ = π−θ′ because the symmetry

conditions are already imposed on Eq. (6). Likewise, the boundary equations in the third

and fourth quadrants are redundant. Therefore, the consideration of Eq. (9) is sufficient to

solve for the coefficients An’s.

B. Determination of Coefficients

Now, we solve for the coefficients An’s in Eq. (9). A potential flow is scale-invariant,

therefore An’s can be specified when the scale factor a is fixed. Without loss of generality,

we set a = 1/2, equivalent to the diagonal of the rectangle being 1. Then Eq. (9) is in the

following form:

0 = − 1√2

+∑n

An ·[(2n− 1) · 8n · sin2n

(θ +

π

4

)sin(

2nθ +π

4

)](11)

where 0 < θ < π/2.

The left-hand side of Eq. (11) physically refers to normal velocity vn, which is zero at 0 <

θ < π/2, when the slip boundary condition is satisfied. The right-hand side represents the

components of a physical model in a reduced form. The expansion series in the summation

does not constitute a complete set where 0 < θ < π2, therefore, An’s are not uniquely defined

using orthogonality. We estimate the coefficients An’s rewriting Eq. (11) as

0 = − 1√2

+N∑n=1

AnXn(θ) + ε(θ), (12)

with an N -component finite series explaining a large proportion of flow and a high-order

infinite series containing deviations in the measurement of a flow. We estimate a finite

6

0 . 0 0 . 5 1 . 0 1 . 5

- 0 . 8

- 0 . 5

- 0 . 2

0 . 1

S l i p B . C . 1 - p a r a m e t e r 1 0 - p a r a m e t e r 1 0 0 - p a r a m e t e r

v n

�

�0

Figure 2. The final fitted models for diamond in Eq. (12) while varying the number of parameters

to 1, 10 and 100. As the number of parameters used in the model increases, the fit matches the

actual flow (i.e. normal velocity) more closely. As N increases, the largest θ, denoted as θ0(N),

where the model and data intersect approaches π/2.

number of An’s using a multiple linear regression, taking the left-hand-side of Eq. (12), a

theoretically valid value of 0 for 0 < θ < π/2, as observed and the model driver Xn(θ) =

(2n−1) ·8n · sin2n(θ + π

4

)sin(2nθ + π

4

)fixed for 0 ≤ θ < π/2. With a limit on the precision

of computing, we take N as large as 100 and obtain the coefficient estimates that minimize

the sum of squares of residual ε(θ) across a finite number of θ values. The number of

evaluation points on θ can be distributed equally or more heavily near the boundaries 0

and π/2. In any case, this whole procedure takes the thought experiment of measuring the

normal velocity of a potential flow at the boundary and applies a mathematical modeling

framework to estimate the relative scale of each component.

In Figure 2, a dash line represents a single parameter model, which consists of the mean

flow and the flow dipole. This effectively simplifies the flow around a diamond as that around

a circle. We can see that the single parameter model is obviously insufficient to accurately

describe the actual flow; the discrepancy between the theoretical value and the model flow is

not negligible especially for large values of θ. The dash-dot line with 5 local maxima show a

10-parameter model, and the solid line with 50 local maxima show a 100-parameter model.

As the order of multipoles increases, the discrepancy between the full and the model flows

decreases except at the singularities θ = 0 and θ = π/2.

7

The first few coefficients of the 100-parameter model are A1 = 1.53×10−1, A2 = −3.76×

10−3, A3 = 6.35×10−4, A4 = −5.65×10−5, A5 = 1.25×10−5, and etc. To check the relative

importance of these terms, we calculate the velocity field due to each term. Shown in Figure

3 are the radial velocities of the n-th order terms, at θ = 0, normalized by the first order

term. In other words, ∣∣∣∣∣v(n)x

v(1)x

∣∣∣∣∣ = (2n− 1)

∣∣∣∣AnA1

∣∣∣∣x−(2n−2). (13)

Here, x = r because θ = 0. While the contribution from the higher-order terms are not

negligible close to the pole (x ≈ 0.5) when θ = 0, they diminish quickly. At x = 1 the

second-order term gives less than 10% of the velocity field strength than the first-order

term. Further from the pole where x > 1, the dipole field is the strongest, and the potential

flow is no longer influenced by the shape of obstruction.

Away from the obstructing object (r & 2a), the potential flow around a diamond of

diagonal 1 is effectively approximated to

φdia = Ur cos θ + 0.153 · cos θ

r. (14)

The potential flow around a circle21 of diameter d0 is

φcir = Ur cos θ +d204

cos θ

r. (15)

Therefore, the potential flow around a diamond can be effectively approximated to the poten-

tial flow around a circle of diameter 0.78 (≈√

0.153 · 4). It is inferred that we overestimate

D by using the D = W convention for this case.

C. Convergence

To compare the fits of the models, we calculate the root mean square error R, which is

defined as

R =

√√√√ 1

m

m∑i=1

(1√2−

N∑n=1

AnXn(θi)

)2

, (16)

where m is the number of virtual data points we evaluated for 0 < θ < π/2. In Figure 4, we

show the rate of convergence of R as a function of N .

Another measure of the fit is θ0. We define it as the largest θ where the modeled data

intersect the ideal normal velocity of 0. It demonstrates the sharpness of the model at

8

1 2 3 4 51 E - 5

1 E - 4

1 E - 3

0 . 0 1

0 . 1

1

1 0

|v x(n)/v x(1)

|

x

n = 2 n = 3 n = 4 n = 5

Figure 3. Near the pole, the higher order multipolar fields are not negligible, but they diminish

quickly with x. At x ∼ 1, the dipolar field is 102 times stronger than the next strongest multipolar

field.

1 1 0 1 0 00 . 3 0

0 . 3 5

0 . 4 0

0 . 4 5

0 . 5 0

0 . 5 5

R

N

R

0 . 0

0 . 3

0 . 6

0 . 9

1 . 2

1 . 5

� 0 �0

Figure 4. To quantitative compare the quality of fitting, the residual sum of squares (RSS) and θ0

are plotted with respect to nmax. Both RSS and θ0 shows the improvement of the fitting quality

as more parameters are used in the model.

the singularity θ = π/2. For example, in Figure 2, we point out θ0 for the 100-parameter

model. Figure 4 shows the convergence of θ0 to π/2 as N increases in open circles. The

100-parameter model has θ0 ≈ 1.5 = 86 deg. The convergence of θ0 to π/2 seems to be at

the rate of (π/2− θ0) ∼ N−0.55.

9

0 . 0 0 . 5 1 . 0 1 . 5- 1 . 2- 0 . 9- 0 . 6- 0 . 30 . 00 . 30 . 60 . 91 . 2

S l i p B . C . 1 - p a r a m e t e r 1 0 - p a r a m e t e r 1 0 0 - p a r a m e t e r

v n

�

Figure 5. Multiple linear regression analysis for the square case. We align the model in Eq. (20)

to the true state in dots at vn = 0 and vary the number of parameters in the model.

IV. FLOW AROUND A SQUARE

Now, we turn to a square case, where the rectangular pole is rotated 45 degrees from

the diamond case discussed above (see Figure 1(b)). Let 2b be the base of the square. The

boundary of the square is expressed in polar coordinates as:

r cos θ = b, 0 ≤ θ <π

4and r sin θ = b,

π

4≤ θ <

π

2(17)

in the first quadrant. When the diagonal of the square is 1, b = 1/√

8.

Applying the boundary condition ~v · n = 0, the piecewise boundary equations are derived

for 0 ≤ θ < π/4:

vn = 0 = −U + U∑n

An ·[(2n− 1) · 8n · cos2n θ cos (2nθ)

], (18)

and for π/4 ≤ θ < π/2

vn = 0 = U∑n

An ·[(2n− 1) · 8n · sin2n θ sin (2nθ)

]. (19)

The general form of Eqs. (18) and (19) is

0 =[1−H(θ − π

4)]

(−1) +∑n

An · (2n− 1) · 8n

·{[

1−H(θ − π

4)]

cos2n θ cos (2nθ) +H(θ − π

4) sin2n θ sin (2nθ)

}, (20)

10

whereH denotes a Heaviside function. We take the left-hand side of Eq. (20) as the observed

normal velocity and the right-hand side as model inputs. We evaluate the model inputs on

an equally-spaced line 0 < θ < π/2 and obtain the coefficients An’s using the least squares

method as in the diamond case. The 1-, 10- and 100-parameter models are shown in Figure

5.

The first few coefficients of the 100-parameter model are A1 = 1.53× 10−1, A2 = 3.81×

10−3, A3 = −6.38× 10−4, A4 = −5.75× 10−5, A5 = 1.25× 10−5 and etc. In absolute values,

these are identical to the coefficients for the diamond. This suggests that the two potential

flows are rotational transformations of each other. The rotational transformation is unitary.

Hence, the characteristic length scales of two cases remain unchanged.

V. DISCUSSION AND SUMMARY

We presented a procedure to solve the potential flow around a rectangular pole. First,

we write out the solution of the Laplace equation in the form of an infinite series. Second,

we match it to the slip boundary condition assuming the components of the model and the

normal velocity at the boundary are observed. Last, we solve for the coefficients plugging

in several different θ values to the components and estimating the relative scales of each

component using the least squares method in multiple linear regression. Two orientations of

the rectangular pole have been examined; in both cases, we find that only the dipole field

survives far from the pole.

Our attempt is to provide a better understanding of the vortex streets from a rectangular

pole. In high Reynold number regime, the boundary layer thickness is negligible compared

to the size of the pole. We assume that the vortices are freely advected by the surrounding

potential flow. Then our calculation shows that the potential flow around a rectangular pole

of diagonal 1.28 (or base 0.91) has the same characteristic length scale D as the potential

flow around a circular cylinder of diameter 1. When a rectangular pole is mounted so that

one side directly faces the flow (a square), D = 1.1W where W is the widest dimension that

faces the flow. When a rectangular pole is mounted so that its two sides equally face the

flow (a diamond), D = 0.78W , which is commensurate with the potential flow around a

circle.

Our results can be applied to previously reported experimental data; for example, in a

11

three-dimensional wind tunnel15, St∞ ' 0.14 for square and St∞ ' 0.13 for diamond, and

in a two-dimensional soap film flow18, St∞ ' 0.17 for square and St∞ ' 0.16 for diamond.

These numbers can be directly compared to St∞ ' 0.2 for the wake behind a circular

cylinder; the difference comes from hydrodynamic reasons.

ACKNOWLEDGEMENT

We thank X.L. Wu for discussions.

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