report - on the identification of a vortex

8/11/2019 Report - On the Identification of a Vortex

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Report on On Identification of a vortex - Jeong & Hussain for MAE 545

The paper explains a new way of detecting a vortex in an incompressible flow and presents a contrast

study of existing (when the paper was published) ways of identification to the proposed definition.

Why is this important? - Since coherent structures are strongly associated with vortices, an

appropriately defined scheme would permit us to detect coherent structures in inhomogeneous flowsmore accurately, would result in more accurate and universal turbulence models and shed some light

on their role in turbulence.

According to the author, there are two requirements to identifying a vortex core, which are:

1. An existence of net vorticity (net circulation); therefore potential flow regions are not

included.

2. Geometry of the identified vortex core should be a Galilean invariant, i.e. it is independent of

the reference frame.

The author showcases the inadequacy of (3) intuitive measures - (a) Local pressure minimum(b)Pathline and streamline & (c)Vorticity magnitude and compares the definitions of - Chong et al.

(1990) & Hunt et al.(1988) with the proposed definition.

1. Intuitive measures:

a. Local pressure minimum - The (intuitive) reasoning for this criterion is that in a steady

inviscid planar flow, there exists a pressure minimum on the axis of a swirling motion. In

three dimensions, pressure may be minimum in all three directions or just in the in the

plane perpendicular to the axis of circulatory motion - this condition includes the former

case.

It is inadequate as:

a.

A well-defined pressure minimum can occur in an unsteady irrotational motion

which may or may not involve a vortex which is caused by an unsteady strain rate

(t).

b. In Karmans viscous pump, centrifugal force is balanced by the viscous force and

not pressure force hence a pressure minima condition wont detect the vortex.

c.In planar Stokes flow, pressure gradient is balanced only by viscous term hence

pressure minimum condition remains unsatisfied while vortices do occur.

d.Since pressure is inherently of a larger scale than vorticity (pressure is gorverned

by Poissons equation), the difference in scale makes it problematic in marking a

vortex using an isopressure surface.

b. Pathline and Streamline - Closed pathlines to detect vortices is inefficient as,

i. It fails to satisfy the condition of being a Galilean invariant.

ii. Vortices undergo many nonlinear processes like breakdown, pairing, tearing even

before the completion of a full revolution, which results in incomplete particle

paths hence no closed pathlines.

iii. Interacting regions of vortices result in regions of reconnections which result in

crooked pathlines and streamlines.

c. Vorticity Magnitude (||) - This criterion should ideally be a subset in the scheme of

identification of a vortex core but it cannot the sole criterion, as:


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i. In a shear flow, when the background shear is comparable to the vorticity

magnitude within the vortex, || fails to identify the vortex core.

ii. In planar wall-bounded flows, maxima and minima of || occur only at the wall;

the flow immediately near the wall is characterized by shear and not swirling

motion. Hence if vortex identification excludes the wall then this condition cannot

be used to detect vortex cores in a boundary layer.

iii. In free shear flows, a vorticity sheet has a large || but it isnt a vortex.

2. Previously proposed definitions:

a. Complex eigenvalues of velocity gradient tensor (Chong et al.1990):

Vortex core is a region with complex eigenvalues of velocity gradient tensor u,i.e. local

streamline pattern is closed in a reference frame moving with the point.

Eigenvalues, , of u,satisfy the characteristic equation,

3- P

2+ Q -R = 0

where, P ui,i= 0 (incompressible flow)

Q -ui,juj,i

R = Det(ui,j)

When Discriminant ()is positive, we shall get complex eigenvalues.

= (1/3Q)

3+ (

1/2R)

2 > 0

b. The second invariant of uand kinematic vorticity number Nk(Hunt et al.1988):

Eddy is defined by the positive second invariant,Q,of u, with an additional condition

that pressure be lower than the ambient value. Q is defined as,

Q -(ui,juj,i) = (||||2- ||S||2)

Where, S and are the symmetric and antisymmetric components of u respectively,

Si,j= (u2i,i- ui,juj,i) and i,j= (ui,j- uj,i)

Qvanishes at the wall hence free from the problem associated with ||definition.

Replacing the local rotation rate (||||), Kinematic vorticity number (Nk) is used to

measure the quality of rotation.

3. New Definition:

The author uses the pressure minima criteria as the starting point for the new definition.

The discrepancy in the pressure minimum definition arose because of two reasons - (i)

unsteady straining creating a pressure minimum without a swirling motion. (ii) viscous

effects, which eliminates pressure minimum in a vortical motion. Hence these two effects are

discarded.

Taking the gradient of N-S equations,

and decomposing the acceleration gradient (ai,j) into symmetric and antisymmetric parts,


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the symmetric part yields,

Since, the first and second terms represent unsteady irrotational straining and viscous effects

respectively we ignore it. Therefore, considering only S2+2vortex core is defined as a

connected region with two negative eigenvalues of S2+

2.

Say 1, 2, 3are the eigenvalues and 1 2 3, the new definition requires 2


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ii. Elliptic vortex ring

Here DNS data for an elliptic vortex ring (Husain and Hussain 1993) is considered

where two classes of vortices appear: an elliptic vortex ring with high | | and rib-like

streamwise vortices with lower || behind the elliptic ring. Since we have a large

variation of || to depict, you can understand clearly from figures (a) and (b) that ||-

definition fails to show both ribs and ring of the structure clearly in a single frame.

There is no clear demarcation between the ribs and ring in figure (a) while in figure (b)

the ribs are absent.

In contrast, 2-definition, shows both the structures clearly in figure (c).

b. Inadequacy of -definition.

i. Conically symmetric vortex

Here a proposed model for a tornado (Shtern & Hussain 1993) is considered. It

consists of a swirling jet emerging into a half-space with a source of axial momentum

and circulation, located at the origin. The flow has a conical symmetry, satisfies the full

N-S equations and has no singularity on the axis except at the origin. Since the flow has

an axial velocity in addition to an azimuthal velocity around the vortex axis (z-axis) it

exhibits strong helical motion. Considering that the flow has a conical symmetry, the

vortex core boundary should also be conical in shape.


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The -definition, as shown below in figure (b), depicts two cores as shaded region

1 and 2 which is not the correct representation of the vortex geometry. Contrastingly,

the 2-definition depiction is without a detached vortex core.

ii. Mixing Layer.

Here we consider the DNS data of a temporal mixing layer, initialized with ahyperbolic tangent streamwise velocity profile and a sinusoidal spanwise

distribution of streamwise vorticity, at the time of shear layer rollup.

Figure (a) shows a vortex core boundary based on -definition- notice that it is very

noisy. In contrast the figure (b) that is based on 2-definition, shows both the vortex

ribs and rolls structures clearly.


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Considering the contour plots in the x-z plane in figures (c) and (d), you can see the

unwanted noise and insignificant regions A and B in fig. (c) which are excluded in

fig. (d).

c.

Inadequacy of the Q-definition:

i. Conically symmetric vortex:

The Q-definition shows a narrow hollow core near the axis as otherwise

shown by 2-definition. Since fluid particles near the axis undergo almost solid-

body rotation, this exclusion is unreasonable.


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report - on the identification of a vortex

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