Analytic solutions for three dimensional swirling strength in compressible and
incompressible flows
Huai Chen, Ronald J. Adrian, Qiang Zhong, and Xingkui Wang
Citation: Physics of Fluids (1994-present) 26, 081701 (2014); doi: 10.1063/1.4893343
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PHYSICS OF FLUIDS 26, 081701 (2014)

Analytic solutions for three dimensional swirling strength
in compressible and incompressible flows
Huai Chen,1 Ronald J. Adrian,2,a) Qiang Zhong,1 and Xingkui Wang1
1

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing
100084, China
2
School for Engineering Matter, Transport and Energy, Arizona State University, Tempe,
Arizona 85287, USA
(Received 27 April 2014; accepted 27 July 2014; published online 20 August 2014)

Eigenvalues of the 3D critical point equation (∇u)ν = λν are normally computed
numerically. In the letter, we present analytic solutions for 3D swirling strength
in both compressible and incompressible flows. The solutions expose functional
dependencies that cannot be seen in numerical solutions. To illustrate, we study the
difference between using fluctuating and total velocity gradient tensors for vortex
identification. Results show that mean shear influences vortex detection and that
distortion can occur, depending on the strength of mean shear relative to the vorticity at
C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4893343]
the vortex center. 

There are a number of point-wise vortex identification techniques based on the velocity gradient
tensor. These include the second invariant of velocity gradient tensor,1 the discriminant of the
velocity gradient tensor,2 the Hessian of pressure,3 and the swirling strength method.4 The swirling
strength method uses the imaginary part of the complex eigenvalue of the velocity gradient tensor
to visualize vortices.
Let Amn = ∂um /∂xn (m, n = 1,2,3) be the velocity gradient tensor A(x) = ∇u of 3D flow at a
point, x. The eigenvalue problem Aν α = λα ν α has three eigenvalues λα and three corresponding
eigenvectors ν α , α = 1, 2, 3. Local motion around the critical point exhibits swirling (rotation) if the
eigenvalues of A are complex.2 In this case, the gradient tensor can be decomposed as4
⎤
⎡
λr 0 0
vr vcr vci ]−1 ,
A = [
vr vcr vci ] ⎣ 0 λcr λci ⎦ [
(1)
0 −λci λcr
where λr is the real eigenvalue with a corresponding eigenvector, ν r and λcr ± iλci are the conjugate
pair of the complex eigenvalues with complex eigenvectors ν cr ± iν ci . The advantage of the swirling
strength concept is its simple physical meaning as the frequency at which fluid particles swirl around
the critical point.4
Universally, until now, λci has been evaluated in 3D flow point-wise by solving the eigenvalue
problem, numerically. While the decomposition of A is easily done numerically, numerical solution
does not provide insight into the parametric dependencies of swirling strength. Here, we present an
exact algebraic solution for λci , in both incompressible and compressible flows, whose value lies in
(1) reducing computation; and (2) providing a basis for further analysis of its properties.
The characteristic equation of the velocity gradient tensor A is given by
λ3 + Pλ2 + Qλ + R = 0,

(2)

where P = −tr(A), Q = 0.5tr(P -AA) and R = −det(A). We solve this equation using Cardano’s
method.5 Let t = λ-P/3, p = Q-P2 /3, and q = R-PQ/3-2P3 /27 to get the simpler equation
2

t 3 + pt + q = 0.

(3)

a) Electronic mail: rjadrian@asu.edu

1070-6631/2014/26(8)/081701/6/$30.00

26, 081701-1


C 2014 AIP Publishing LLC

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Phys. Fluids 26, 081701 (2014)

Putting t = ζ +η and setting ζ η = −p/3 yields a quadratic equation in ζ 3 ,
(ζ 3 )2 + q(ζ 3 ) − p 3 /27 = 0.

(4)

It is easy to solve (4) by the quadratic formula, and as a result of symmetry between ζ and η, we
can set ζ 3 = −q/2 + sqrt() and η3 = −q/2-sqrt(), where  = (q/2)2 +(p/3)3 is the discriminant.
When  is positive, Eq. (2) has a real eigenvalue and a conjugate pair of complex eigenvalues:
⎧
⎪
⎪ λ1 = ζ + η + P / 3
⎨
√
λ2 = βζ + β 2 η = − (ζ + η) / 2 + P / 3 + i 3 (ζ − η) / 2 ,
(5)
⎪
⎪
√
⎩
λ3 = β 2 ζ + βη = − (ζ + η) / 2 + P / 3 − i 3 (ζ − η) / 2
where β is the primitive cube root
√ of unity. In accordance with Eq. (1), λr = ζ + η + P/3,
λcr = −(ζ + η)/2 + P/3, λci = 3 (ζ − η) /2. It is easy to prove that ∀ζ , η ∈ R, λci > 0.
When the flow is incompressible, P is identically zero. The characteristic equation of velocity
gradient tensor A reduces to λ3 + Qλ + R = 0, so p = Q and q = R. The expression for λci is still,
√
(6)
λci = 3 (ζ − η) /2,
but,
λr = ζ + η,

(7a)

λcr = − (ζ + η) /2,

(7b)

where
ζ =

3

−R / 2 +

(R / 2)2 + (Q / 3)3 ,

(8a)

η=

3

−R / 2 −

(R / 2)2 + (Q / 3)3 ,

(8b)

and
R = −det( A),

(9a)

Q = 0.5 tr(− AA).

(9b)

To illustrate the utility of the analytic solution, we now consider the effects of using either
the total velocity field or the fluctuating field to identify vortex filaments. Both approaches have
been used in the literature, and there is uncertainty concerning which is better. We consider first the
general case of 3D, fully developed, incompressible turbulent flow and then the specific case of a
Burgers vortex in a mean shear flow, the Burgers vortex representing the fluctuating flow.6
The (x, y, z) coordinates of the fully developed flow are taken to be the streamwise, wall-normal
and spanwise directions, respectively, and the mean shear is ∂U/∂y. The corresponding mean,
fluctuating and total velocity components are (U, V, W), (u, v, w), and (U (y) + u, v, w). Then, the
gradient tensors of total and fluctuating velocity in 3D flow are
⎡ ∂u ∂(U + u) ∂u ⎤
⎢ ∂x
⎢
⎢ ∂v
At = ⎢
⎢ ∂x
⎢
⎣ ∂w
∂x

∂y
∂v
∂y
∂w
∂y

∂z
∂v
∂z
∂w
∂z

⎥
⎥
⎥
⎥,
⎥
⎥
⎦

(10a)

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Phys. Fluids 26, 081701 (2014)

FIG. 1. Profiles of rms and relative deviations of the tensor invariant terms affected by mean shear. (a) RMS of Qf , term1,
and Qt , normalized by (ν/u 2τ )2 ; (b) relative deviation between σ Q f and σ Q t ; (c) rms of Rf , term2, and Rt , normalized by
(ν/u 2τ )3 ; and (d) relative deviation between σ R f and σ Rt .

⎡ ∂u
⎢ ∂x
⎢
⎢ ∂v
Af = ⎢
⎢ ∂x
⎢
⎣ ∂w
∂x

∂u ⎤
∂z ⎥
⎥
∂v ⎥
⎥.
∂z ⎥
⎥
∂w ⎦
∂z

∂u
∂y
∂v
∂y
∂w
∂y

(10b)

From Eq. (6), the algebraic solutions of 3D swirling strength can be calculated as
⎧

√ 
⎪


2 
3
2 
3
3
3 3
⎪
⎪
⎪
λci, f =
R f /2 + Q f /3 − −R f /2 −
R f /2 + Q f /3
−R f /2 +
⎪
⎪
2
⎨
√ 
⎪
⎪
⎪
3 3
⎪
⎪
−Rt /2 +
⎪
⎩ λci,t = 2


(Rt /2)2 + (Q t /3)3 −

3

−Rt /2 −

,

(Rt /2)2 + (Q t /3)3

where Qf = 0.5tr(−Af Af ), Rf = −det(Af ), Q t = Q f − ∂∂vx ∂U
, and Rt = R f + ∂U
∂y
∂y
Hence, the mean shear influences the 3D swirling strength through


∂v ∂U
∂U ∂v ∂w ∂v ∂w
,
ter m1 =
and ter m2 =
−
∂x ∂y
∂ y ∂ x ∂z
∂z ∂ x

 ∂v ∂w
∂ x ∂z

(11)

∂w
.
− ∂v
∂z ∂ x

and the magnitude of its effect will be significant, unless ∂U/∂y is much less than the fluctuating
gradients. It will be shown later that this condition may or may not hold, depending on the local
nature of the turbulent flow.
To quantitatively analyze the influence of mean shear on vortex detection by swirling strength,
we must assess the relative magnitudes of Qf , term1, Qt , Rf , term2, and Rt , in a well-known and
representative turbulent flow. The direct numerical simulation (DNS) of fully developed turbulent
channel flow at friction Reynolds number Reτ = 934 by Del Alamo et al.7 is used for this purpose.

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Phys. Fluids 26, 081701 (2014)

Since ter m1 and ter m2, averaged over time and homogeneous directions, vanish in fully
developed flow, we measure the magnitudes of term1 and term2 by their root mean square (rms)
values. Figures 1(a) and 1(c) show that the rms values of term1 and term2 are large enough to
potentially influence vortex identification by swirling strength in the region y+ < 50, where the
mean shear is strongest. In Figure 1(a), the rms values σ Q f and σ Q t differ clearly in the domain 2 <
y+ < 30; in Figure 1(c), the σ R f and σ Rt differ in the domain 1 < y+ < 50. Relative deviations of
these discrepancies are shown in Figures 1(b) and 1(d). The relative deviation between σ Q f and σ Q t
is larger than 2% for 1 < y+ < 30, and the largest deviation reaches about 20%. Moreover, relative
deviation between σ R f and σ Rt is larger than 5% for 0.1 < y+ < 50, and the largest deviation reaches
about 80%. It is concluded that mean shear influences vortex identification mainly in the region y+
< 50, a layer slightly thicker than the buffer layer. In the outer region of the flow, the mean shear
has little effect on vortex identification.
To assess the effects of mean shear on a well-known vortex, we consider the fluctuating velocity
field (u, v, w) of a Burgers vortex rotating about the z-direction:
⎧

 2

x + y2
1
α
⎨
(−y, x)
(u, v) = − (x, y) +
1 − exp −
.
(12)
2
2π
4ν / α
x 2 + y2
⎩
w = αz
√
Here, α is the strain-rate, ν is the kinematic viscosity, r0 = 4ν / α is the core radius (determined
by the balance of viscous diffusion
 strain), is the circulation, and the spanwise vorticity ωz at
 and
the vortex center is given by / πro2 . If > 0, the vortex rotates counter-clockwise.
Suppose that the flow field in (12) is imbedded in a background mean shear ∂U/∂y > 0. In
the parlance of turbulent vortices, a clockwise rotating Burgers vortex ( > 0) is called retrograde,
because it rotates with the mean shear; a counter-clockwise rotating vortex ( < 0) is called
prograde.8 For purposes of illustration, it suffices to analyze the values of swirling strengths of the
fluctuating and total velocity fields at the center of the Burgers vortex. They are
⎛
⎞


 2



√




| |
| | 2 + 144π 2 ν 2 ⎟
+ 144π 2 ν 2
3⎜
⎜
⎟,
λci, f (0, 0, 0) =
−
3 f (α) +
3 f (α) −
 √
3
 √
3


⎝
⎠
2
2
2
2 3πr0
2 3πr0
(13a)

√ 

!
 α 2
∂U
∂U
3
3

λci,t (0, 0, 0) =
+g
f (α) −
2
2 4π ν ∂ y
∂y

2


√ 

!
 α 2
∂U
∂U
3
3

−
−g
f (α) −
2
2 4π ν ∂ y
∂y
where
f (α) =
and


g

∂U
∂y



"
"
= ""

2

 α 3
2

+

α
2

2

∂U
∂y

− 2πr02

∂U
,
∂y

(13b)

2



+ 144π 2 ν 2 − 2πr02

− 2πr02

2πr02
"#
3
∂U ""  √
2 3πr02 .
"
∂y

(14)

(15)

By inspection of (13a), it is always true that
λci, f (0, 0, 0) > 0,
so the fluctuating velocity gradient tensor can always successfully detect a prograde or retrograde
Burgers vortex.

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Phys. Fluids 26, 081701 (2014)

FIG. 2. Probability P( ω2z | y <

∂U
∂y

(y)) as a function of wall-normal position.

In the case of the total velocity field, if
ter m3 =

2

− 2πr02

∂U
∂y

> 0,

(16)

then (13b) implies
λci,t (0, 0, 0) > 0,
and the swirling strength of the total field successfully indicates the presence of a Burgers vortex. For
prograde vortices ( < 0), (16) is inherently positive, and λci, t (0, 0, 0) is always greater than zero,
which means the mean shear has no influence on their identification. But, in the case of retrograde
vortices ( > 0), when
∂U
>
,
∂y
2πro2

(17)

Term3 is negative and λci, t is complex, contradicting λci > 0. Thus, when the background mean
shear is greater than one-half of the vorticity (ωz > 0) at the center of a retrograde Burgers vortex,
the total velocity gradient tensor will fail to identify it.
This result is disappointing, but it may or may not be an unphysical artifact of artificially
summing a mean shear and an independent Burgers vortex. In real turbulence, the vortices and
the mean flow are not really independent because vortices arise from instabilities of the base
flow, and turbulent mean flow is likely the result of averaging over many eddies. Thus, the mean
flow actually contains a contribution from the eddies, and the flow surrounding an eddy may not be
accurately represented by the mean flow. Put another way, the fluctuating field found from Reynolds’
Decomposition may not be a good representation of the eddies. This is a deep question that we will not
attempt to solve here. Consequently, application of (17) to real turbulent shear flows is problematic,
and it is, therefore, important to ask how often the mean shear might satisfy the inequality in (17)
in real turbulence. To this end we have used the same DNS channel flow data to study the relation
between ∂U/∂y and ωz /2 at the retrograde vortex centers. Unlike Wu and Christensen8 and Herpin
et al.,9 the 3D swirling strength of the fluctuating velocity gradient tensor is used to extract vortex
core centers, but the criterion λci, f (x, y, z) / λrci,msf (y) ≥ 1.5, where λrcims, f (y) is the root mean square
of λci , f at a given y position, is otherwise the same as theirs. Corresponding fluctuating velocity
fields in x-y planes are used to calculate the fluctuating vorticity ωz at retrograde vortex centers.
Cumulative distribution functions (CDFs) of ωz /2 at various wall-normal positions (10 < y+ <
(y)) is defined as the ratio of the number of centers, at
930) are calculated. Probability P( ω2z | y < ∂U
∂y
which half of the fluctuating vorticity is smaller than the mean shear, to the total sample number.
Figure 2 presents the probability as a function of wall-normal position. It is obvious that probability
is zero for y+ ≥ 50, which means in this region both total and fluctuating velocity gradient tensors
can detect the original vortices. So there is no difference between results of these two methods for y+

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081701-6

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Phys. Fluids 26, 081701 (2014)

≥ 50. However, as y+ decreases from 50 to 10, the probability increases dramatically approaching
one at y+ = 10. Hence, in the buffer layer a large proportion of retrograde spanwise vortices would
not be detected by using the total velocity gradient tensor. Fortunately, quasi-streamwise vortices
are believed to dominate the buffer layer.
In conclusion, this work presents new analytic solutions for the 3D swirling strength of compressible or incompressible flows. To illustrate the utility of these solutions, the difference between
detecting vortices using the gradient tensors of either the fluctuating velocity or the total velocity
was studied. A 3D Burgers vortex in a mean shear was used to study the difference, which depends
on the strength of mean shear relative to the vorticity at the vortex center. Vortex detection always
works well for prograde Burgers vortices, but for retrograde Burgers vortices the swirling strength
fails to detect the vortex if the mean shear is greater than one-half of the maximum vorticity in the
core. Data from DNS of channel flow at Reτ = 934 show that the swirling strengths of the total
versus the fluctuating field differ only for retrograde spanwise vortices in the region for y+ < 50,
which may explain the marked reduction in the density of retrograde vortices relative to prograde
vortices near the wall.8, 9 Above this height, the λci criterion is insensitive to using the fluctuating
field or the total field. It should be noted that in consideration of the complexity of the full range of
physical flows, the problem of which field is more appropriate for vortex identification in general
merits further research.
This work was funded by National Natural Science Foundation of China (Grant No. 51127006)
and the U. S. National Science Foundation under Grant No. CBET-0933848. H.C. also received
financial support from the China Scholarship Council (CSC). We wish to acknowledge Professor R.
D. Moser for supplying the channel DNS data, which were produced in conjunction with J. Jiménez,
J.-C. Del Álamo, and P. S. Zandonade.
1 J. C. R. Hunt, A. A. Wray, and P. Moin, “Eddies, streams, and convergence zones in turbulent flows,” in Proceedings of the

Summer Program of the Center for Turbulence Research (NASA Ames/Stanford University, USA, 1988), pp. 193–208.
2 M. S. Chong, A. E. Perry, and B. J. Cantwell, “A general classification of three-dimensional flow fields,” Phys. Fluids 2(5),

765–777 (1990).
Jeong and F. Hussain, “On the identification of a vortex,” J. Fluid Mech. 285, 69–94 (1995).
4 J. Zhou, R. J. Adrian, S. Balachandar, and T. M. Kendall, “Mechanisms for generating coherent packets of hairpin vortices
in channel flow,” J. Fluid Mech. 387, 353–396 (1999).
5 N. Jacobson, Basic Algebra I, 2nd ed. (W. H. Freeman and Company, New York, 1985), pp. 264–266.
6 The Burgers vortex is an Oseen vortex imbedded in an axisymmetric staining motion. Although the straining extends to
infinity, it is treated as a local strain (i.e., fluctuating) when Burgers vortex is used to model a small vortex in larger scale
turbulence.
7 J. C. Del Álamo, J. Jiménez, P. Zandonade, and R. D. Moser, “Scaling of the energy spectra of turbulent channels,” J. Fluid
Mech. 500, 135–144 (2004).
8 Y. Wu and K. T. Christensen, “Population trends of spanwise vortices in wall turbulence,” J. Fluid Mech. 568, 55–76
(2006).
9 S. Herpin, M. Stanislas, and J. Soria, “The organization of near-wall turbulence: A comparison between boundary layer
SPIV data and channel flow DNS data,” J. Turbul. 11(47), 1–30 (2010).
3 J.

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