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Local and nonlocal conditional strain rates along gradient trajectories from various scalar fields in turbulence

Lipo Wang

Institut für Technische Verbrennung

RWTH-Aachen, Germany

TMB-2009, Trieste

2009.07.28

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Content

• Introduction: background and general analysis

• Local statistics

• strain rate along scalar gradients

• interaction of different gradient trajectories

• Non-local statistics

• a new two-point velocity difference structure function

• Conclusions

Content

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Background

Mathematically, dependent variables to describe turbulence can be:

(a) in scalar form: the passive scalar , kinetic energy …

(b) in vector form: the velocity vector , the vorticity vector …

(c) In tensor form: rate-of-strain, the stress tensor …

Only for those in (a), their presentations are independent of the choice of coordinate systems.

In the Cartesian coordinate system, vector and tensor variables consist of projections (along constant directions) in scalar form.

Scalar field variables: variables in scalar form + spatial projections of vector variables.

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Background

The scalars of interest and importance: the passive scalar, projections of the velocity and vorticity vector, kinetic energy and energy dissipation.

Question: are there any generic properties or relations among these different scalars?

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Background

Hints from previous studies:

• Tennekes & Lumley (1972): turbulence transports passive contaminants in much the same way as momentum.

• Tsinober (2001): under the action of strain tensor, local statistics of the passive vector behaves qualitatively the same as the active ones.

• Wang & Peters (2008): dissipation elements from different scalar fields evolve in a similar fashion.

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General analysis

Formally the governing equation is:

For the momentum equations ( ) : at large enough Reynolds numbers, the viscous term and pressure gradient term are relative small; the convection term and time-dependent term approximately balance.

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Observation: in

if instantaneously the magnitude of is relatively small (at high enough Re), different scalars may assume some generalities.

General analysis

• for , , therefore instantaneously small at high Re;

• for , at high Re, both the pressure gradient and are small;

• large tends to be perpendicular to the directions of largest (positive and negative) strain rates. The product of and ‘could’ be small?

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Local statistics

Laminar diffusion analysis by Batchelor (1959): locally under the action of a constant velocity field (x,y,z), the governing equation of the passive scalar becomes

Thus the gradient vector evolves from any arbitrary orientation

to

Ashurst et al. (1987): the passive vector locally tends to align with the most compressive strain direction.

Object parameter(s): for different scalars, their convection terms contain the same (turbulent) velocity; thus the velocity-related properties may statistically be similar:

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A finer resolution of x/<1 is important for diagnosing fine structures.

Numerical verification from DNS (homogenous shear flow)

Local statistics

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Local statistics

PDF of (the passive scalar) Case 2: PDF of

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Local statistics

DNS result: the mean strain rate from different scalar gradients

is an index showing the alignment between the scalar gradient vector and the principle directions of the strain rate.

Question: how will the orientation of different scalar gradient vectors behave?

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Local statistics

Orientation conditioned on the magnitude of gradients

P1: from small gradient points

P2: from large gradient points

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Local statistics

The sixth moment of the orientation PDF.

case 1 case 2

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Local statistics

Conclusion: in regions of large scalar gradients, gradient vectors tend to align with each other, while if gradients are of small magnitudes, they orientate irrelevantly.

Physically, large scalar gradients are due to compressive strain.

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Illustration of the interaction of various gradient trajectories.

Local statistics

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Local statistics, although informative, are not enough to describe turbulent fields. There is a strong need to develop knowledge about the nonlocal properties.

Nonlocal statistics

Two-point structure function conditioned along gradient trajectories?

Existing theories: Karman-Howarth equation; Kolmogorov’s structure function…

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Nonlocal statistics: theory

The passive scalar equation:

The passive gradient equation:

Along a same gradient trajectory, the two-point correlation equation of the scalar gradient:

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The scalar gradient correlation in the Cartesian frame:

Normalized from:

Nonlocal statistics: theory

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In the inertial range:

The scalar gradient correlation along gradient trajectories:

Normalized from:

Nonlocal statistics: theory

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(b) In the viscous diffusive range:

From the two-point scalar gradient correlation equation

we obtain (a) in the inertial range:

Nonlocal statistics: theory

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Turbulent signals: primary variables correlate differently from derivatives

Because derivatives are not correlated at scales >>, then

(*)

a

b

s

aa

b

b

structure of a large eddy primary variables (ui, ) derivatives

Nonlocal statistics: theory

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Consequently, in the inertial range:

(Reference: Lipo Wang, PRE 79, 046325 (2009))

Nonlocal statistics: theory

In the viscous range:

Negative velocity difference is from

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Numerical verification: the Reynolds number effect in the viscous range

Nonlocal statistics: numerical results

case 1 case 2

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Differently from the Kolmogorov 1/3 scaling in the Cartesian, the two-point velocity difference yields a linear scaling when conditioned along gradient trajectories.

Physically: extensive strain elongates gradient trajectories, therefore at large separation arclengths gradient trajectories selectively proceed through extensive strain rate regions; while compressive regions are likely to be occupied by shorter gradient trajectories.

Nonlocal statistics: theory

In the Cartesian, information can be mixed and equally partitioned into each axis to lead to a smaller 1/3 scaling.

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(JFM 608, 113-138 (2008))

Application:

at small scales dissipation elements are compressed, while stretched at large scales.

0 1 2 3 4-4

-3

-2

-1

0

1

2

x

a/a

Nonlocal statistics: application

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Numerical results of other scalars: the linear scaling holds as well

case 1 case 2

Nonlocal statistics: extension to other scalars

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The gradient correlation equation of other scalar :

At scales >>:

Nonlocal statistics: extension to other scalars

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DNS results: the effect from Re

case 1 case 2

Nonlocal statistics: extension to other scalars

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Conclusions

Conclusions:

1. Because of the relative small magnitude of the source terms, there are some generic properties with respect to the strain rate conditioned on gradients of different scalars. Similar to the passive scalar, the mean conditional strain rate of other scalars are negative as well.

2. The alignment relation of scalar gradients behaves qualitatively different in different regimes; based on which it can be expected that gradient trajectories tend to be parallel along compressive axes, while in extensive planes under the stretching action, these trajectories becomes uncorrelated.

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Conclusions

Conclusions (continued):

3. For the passive scalar, along gradient trajectories the two-point velocity difference is proportional to the separation arclength of the gradient trajectories. This property holds for other scalar in a similar way if the Reynolds number is high enough.

4. This linear scaling is qualitatively different from the classic Kolmogorov’s 1/3 scaling. Physically it can be explained by the selectivity of large gradient trajectories under the stretching action of turbulent velocity.

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The effect of the weighting factor on statistics

Differently from the Cartesian system, in which reference points are equally weighted, the weighting factor of sample points varies along gradient trajectories.

Velocity difference structure function

(a) Cartesian points (b) trajectory points

sample points