XXIV ICTAM, 21-26 August 2016, Montreal, Canada
INVARIANT LOCAL FLOW TOPOLOGY IN TRANSITION INTO
A VORTEX AND PROPERTY OF ITS PREDICTION
K. Nakayama1
1
Department of Mechanical Engineering, Aichi Institute of Technology, Toyota, Aichi, Japan
Summary Local ‡ow topology speci…ed by the velocity gradient tensor and features of corresponding physical properties
are investigated in transition into a vortical ‡ow. The swirlity that indicates the uniformity and intensity of azimuthal ‡ow
follows the transition process, and it shows that this property in the transition depends on two real eigenvalues of the tensor
with the same sign, not on vorticity. Sourcity shows that the uniformity of the radial ‡ow depends on both these eigenvalues
and vorticity (shear) in the plane that becomes the swirl plane.
INTRODUCTION
The eigenvalues of the velocity gradient tensor rv have contributed greatly to categorize the local ‡ow geometry
(topology or pattern) in turbulences that is Galilei invariant. They also derived the -de…nition of a vortex [1]
that speci…es the swirling motion, and vortex de…nitions focused on the pressure minimum feature in terms of the
‡ow kinematics, i.e., the Q- and 2 - and their integrated de…nitions [2], are associated with rv and its eigenvalues.
Recently the physical interpretation of the complex eigenvalues of rv has been clari…ed, and it has derived other
invariant properties to specify the detail ‡ow geometry, e.g. swirlity, sourcity, and property of ‡ow symmetry [3].
The swirlity represents the uniformity and intensity of azimuthal ‡ow, and can be applied to the prediction of
generation of a vortex [4]. The sourcity speci…es the above characteristics of the radial ‡ow. These properties
specify not only the detail ‡ow geometry of a vortex, but also that before the generation of a vortex. The present
study investigates features of the ‡ow geometry before the generation of a vortex, and characteristics of the swirlity
and sourcity.
REPRESENTATION OF VELOCITY GRADIENT TENSOR
The local ‡ow geometry around a point xi (i = 1; 2; 3) in the velocity …eld vi (i = 1; 2; 3) can be expressed by rv,
i.e., dxi =dt = (@vi =@xj )xj , where the summation convention is applied. In this local ‡ow, the radial and azimuthal
velocities in the x1 -x2 plane, vr and v , are extracted as vr = t x
^Qr x
^= j^
xj and v = t x
^Q x
^= j^
xj, respectively, where
p
x
^ = (x1 ; x2 ) and j^
xj = xi xi (i = 1; 2), and Qr and Q denote the respective matrix of the quadratic forms. Qr
and Q are expressed as follows:
Qr =
a11
(a12 + a21 )=2
(a12 + a21 )=2
a22
;
Q =
(a11
a21
a22 )=2
(a11
a22 )=2
a12
:
(1)
The eigenvalues of these matrices specify the feature of vr and v .
Here we derive a representation of rv that has three real eigenvalues i (i = 1; 2; 3). We assume that 1 and 2
have the same sign, and set an orthonormal coordinate system xi with unit bases ei (i = 1; 2; 3) where the x1 -x2
plane, referred to as P hereafter, is the eigenplane associated with 1 and 2 , and e1 are parallel to the eigenvector
of 1 . Then A = rv (= [aij ] = [@vi =@xj ] (i; j = 1; 2; 3)) can be expressed as:
2
3
!3 !2
1
!1 5 ;
A=4 0
(2)
2
0
0
3
where !i (i = 1; 2; 3) denote components of the vorticity vector, and a12 , a13 , and a23 are expressed in terms of !i .
When rv with i becomes to have a pair of conjugate complex eigenvalues, 1 and 2 change to them, and then
P becomes the swirl plane de…ned by the complex eigenvector of them [4]. Equation (2) is a general representation
of rv with three real eigenvalues, which is expressed as an upper triangular matrix. It shows that the vorticity
component normal to P or shear in P is expressed by only one component of rv.
SWIRLITY AND SOURCITY, AND LOCAL TOPOLOGY IN A VORTEX TRANSITION
From Eq. (2), the eigenvalues
=( 1
2 )=2. We note that
p
!32 + 4 2 =2 where
(i = 1; 2; 1 < 2 ) of Q are expressed as i = !3 =2
and 2 have the di¤erent sign and the azimuthal ‡ow has both clockwise and
i
1
Fig. 1: Transition of ‡ow geometry in P and decomposed ‡ow of v (and vr in (c)) with contours of t x
^Q x
^ (and
t
x
^Qr x
^) in transition
into
a
vortex.
Through
(a)
and
(b),
vortical
‡
ow
is
generated
in
(c).
(a)
=
2,
(b)
=
1=2,
p
and (c) = 3=2 and = 3=4. (Note that the vector lengths are adjusted in respective …gures.)
counterclockwise directions in P , as shown in Fig. 1. The swirlity
=
j j < 0:
is given by
= sgn(
1
2
p
) j
1
2
j, thus
(3)
This feature is of great interest. The swirlity, that speci…es the geometrical characteristic of azimuthal (swirling)
‡ow, depends on not the vorticity component !3 but the eigenvalues associated rather with the radial ‡ow. Figure 1
shows the transition of the ‡ow geometry into a vortex followed by .
< r2 ) of Qr are expressed as ri = ( 1 + 2 )=2
p On the other hand, the eigenvalues ri (i = 1; 2; r1 p
!32 + 4 2 =2. The sourcity is de…ned as = sgn( r1 r2 ) j r1 r2 j, then in P becomes
=
1 2
!32
:
4
(4)
Equation (4) shows that is positive while !32 =4 < 1 2 , then the radial ‡ow is in‡ow in all directions in P ,
if 1 ; 2 < 0. If !32 =4 exceeds 1 2 , then the radial ‡ow has both in‡ow and out‡ow (such as Fig. 1 (c)). The
symmetry or uniformity of the radial ‡ow depends on both intensities of 1 2 and !3 . If !3 is small (and 1 ; 2 < 0),
the generated vortex may be weak. However, as is positive, this vortex exhibits e¤ective vortex stretching that
have compression (in‡ow) from all directions in P and increases vorticity.
CONCLUSION
The general representation of the velocity gradient tensor and the local ‡ow topology in the transition into a
vortex were shown. The swirlity and the generation of a vortex depend on the two eigenvalues of the tensor with
the same sign, not on the vorticity. The sourcity shows that the uniformity or symmetry of the radial ‡ow depends
on the two eigenvalues and the vorticity normal to the plane (shear in the plane) that becomes the swirl plane.
Acknowledgements. This study was supported by the 31st grant from The Nitto Foundation.
References
[1] Chong, M. S., Perry, A. E., Cantwell, B. J.: A general classi…cation of three-dimensional ‡ow …elds.Phys. Fluids A2(5):
765–777, 1990.
[2] Nakayama, K., Sugiyama, K., Takagi, S.: A uni…ed de…nition of a vortex derived from vortical ‡ow and the resulting
pressure minimum. Fluid Dyn. Res. 46: 055511, 2014.
[3] Nakayama, K.: Physical properties corresponding to vortical ‡ow geometry. Fluid Dyn. Res. 46: 055502, 2014.
[4] Nakayama, K., Mizushima, L. D.: Study of local ‡ow topology in transition into vortical ‡ow Proc. ICFD2015, 2015