arXiv:1710.08561v1 [ ] 24 Oct 2017 · PDF file · 2017-10-25conditions where a...
Transcript of arXiv:1710.08561v1 [ ] 24 Oct 2017 · PDF file · 2017-10-25conditions where a...
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: [email protected]
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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
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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
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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
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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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