DNS and Structure-Based Modeling of Rotated Shear Flows: Implications for Accretion Disks?

University of Cyprus

DNS and Structure-Based Modeling of Rotated Shear Flows: Implications

for Accretion Disks?

S. C. Kassinos

Stanford University/University of Cyprus

ESF PESC Exploratory Workshop:

Frontiers for Computational Astrophysics

Wengen, Switzerland 26-30 September 2004

Also supported by AFOSR Grant No. F49620-99-0138


Motivation

Strongly rotating flows are challenging to turbulence models.

Most well-known models have been calibrated against 20-year-old LES!

Surprising lack of modern high resolution simulations of these flows.

Objectives

Create a modern high resolution DNS database of homogeneous turbulence that is sheared or strained in rotating frames.

Modeling

DNS results are used to validate a new type of model that was developed before the results were available.


Outline

Flow configuration

Results that could be of relevance to accretion

Direct Numerical Simulation (DNS): what are the open issues

Structure-Based Modeling: what are the open issues

Structure-Based Modeling: why is it different (better)?

Future steps

Discussion


Discussion focus

DNS: +more accurate physics – limited to low Reynolds numbers

We discuss results from Direct Numerical Simulations (DNS) and one-point turbulence modeling based on RANS

Turbulence models: +calibrated for high Reynolds numbers –often questionable physics


Flow Configurations

DNS configurations

rate shear

rate rotation frame Sf -+

counter-rotating frame

co-rotating frame


frame co-rotating frame counter-rotating

0

00

? ???

Flow physics: spanwise rotation

rate shear

rate rotation frame Sf

t t

k k kk k

ttt

decay decayexponentialalgebraic algebraic

- +


How does equilibrium vary if at all withP ?

P

1

t

turbulence thrives

turbulence dies

Flow physics: basic question


DNS Code Description

Governing equations solved in coords deforming with the mean flow to allow Fourier pseudo-spectral methods with periodic B.C.’s.

Time advance is based on a third-order Runge-Kutta method.

Aliasing errors due to periodic remeshing are removed.

Mean shear skews the computational grid, but periodic remeshing allows the simulation to progress to large total shear.

The code is implemented in Vectoral using MPI and has been ported to the ASCI Red and a 48-node Linux cluster.

Accuracy, grid independence and scalability have been tested.


Reynolds Decomposition

ijjjjiijiji uupuuu ,,,, )(3

11

pPpuUu iii ,

kikkkiii RUP

Dt

DU,,,

1

continuity: 0, iiumomentum:

averaging? jiij uuR


ijij ΩS ,

,k

2ij

ij

Rr

k

one-point model ijR

mean deformation

rate

turbulence scales

Directional intensity of velocity fluctuations

Standard Assumption (RST)


ijij ΩS ,

,k

2ij

ij

Rr

k

one-point model ijR

mean deformation

rate

turbulence scales



Is this enough information for consistent accuracy?


ijij ΩS ,

,k

2ij

ij

Rr

k

one-point model ijR

mean deformation

rate

turbulence scales



Is this enough information for consistent accuracy?

ONLY FOR SIMPLE CASES!!


What Other Information?

Most turbulent kinetic energy organized in large structures.

The statistical description of the energy-containing structures is another degree of freedom in addition to .

Velocity magnitude

5123 DNS of rotated shear flow

ijR


One-point turbulence structure tensors

1X

,1 ,0 ,0 111111 frd

1X

0 1, ,0 111111 frd

0d means eddy-alignment in the x direction.

means all velocity fluctuations organized in jetal motion in x direction.

1r

means all large-scale circulation organized in vortical motion around x direction.

1f


Importance of Structure in Dynamics

Two fields with same , but different structure have different dynamics.

No dynamical effect of rapid frame rotation.

Rapid frame rotation modifies one-point state of the turbulence.

ijR

ijR



kijijk

njniij

jninij

ΨuQ

ΨΨF

ΨΨD

,

,,

,,

Turbulent streamfunction:

0 , - , ,,, iiikkipqipqi ΨΨΨu

dimensionality

circulicity

stropholysis

describes the elongation and orientation of energy-containing eddies.

describes the distribution of large-scale circulation in the turbulence field.

contains information about the breaking of reflectional symmetry by mean/frame rotation.

Like the pressure, carries non-local information'jΨ



Near-wall streaks in fully-developed channel flow

09.0 ,84.0 ,03.0 :5.3 111111 frdy

skin friction


ijij ΩS ,

, k

ijkQ

one-point model ijkQ

mean deformation

rate

turbulence scales

Results support this as a more

fundamentally based approach

Directional intensity of velocity fluctuations and morphology of large eddies

Structure-Based Modeling (SBM) Assumption


ijk

exactterms

dQ= .... + rotational randomization model

dt

...

, , ijk ij ij ij

d

dt

Q R D F

Structure-Based Formulation


ijk

exactrotational randomization model term

2

s

1 3 (( )dQ

= .... dt

- ( ))RFij ijm mk mkk ij mj mmk ik jF DQ R DQ

Coefficients set by matching standard homogeneous flows with mean and frame rotation (shear, elliptic, axisymmetric strain+rotation, plane strain+rotation). Then validated in fully developed channel flow and rotating pipe flow.

Relative narrow range supported by match:

1 2 3 0.5 0.7 0 0.4 almost negligible

Differential SBM

, , ijk ij ij ijQ R D F


(For details see Phys. Fluids, 14(7), April 2002)

At high Re, LSE model constants are evaluated by an asymptotic analysis for decaying turbulence in stationary and rotating frames.

2 2

*

production by se

* 2

lf-stretc transfer to small hingand cross-scal scale voticity

does not vae stretching

(vanishes for 2D-2Cnish for 2D-2C

produc

)

t

( )P ij jT iC f S

dC

dt

2 2

vanishes for 2D-2C(forward/inv

ion by mean strain

erse cascade balance)

* *

,

, wh

, model const

ere

9

ant

3

s

ik kj j

i ji

T

j

P

i

k d

d

C C

r

f

f

The High Re Large-Scale Enstrophy (LSE) Equation


Results (preliminary): equilibrium P/

P

DNS

SBM

predictions using standard model eqn. (SSG, v2f, …)

SBM using the large-scale enstrophy equation agrees with DNS.

standard model seriously in error!

From a practical point of view, the most important info are the values of where crosses 1 (that we can answer). P



P



P


Results: evolution histories of structure tensors

normalized Reynolds stress normalized dimensionality normalized circulicity

Level of agreement between DNS and SBM typical for other . solid lines: DNS dashed lines: SBM using large-scale enstrophy.


Results: Reynolds stress tensor at St = 9 vs.


Results: evolution histories of structure tensors


Level of agreement between DNS and SBM typical for other . solid lines: DNS dashed lines: SBM using large-scale enstrophy.

11

22

12

33

11

12

11

12

33


Results: structure tensors at St = 9 vs.

symbols: DNS solid lines: SBM using large-scale enstrophy.



DNS configurations for MHD turbulence

frame rotation rate

shear ratef S

B

B

2(B )

ext

extshear

B

MS

20

0

(B )

ext

extu

B

LN

u

uLRm

Stuart NumberMagnetic Reynolds No.


MHD Results (preliminary): equilibrium P/

P


Evolution of Energies

2 10 50 MNRm

0.75 1.0


Production over dissipation

2 10 50 MNRm

0.75 1.0 2 2


v-field C221

v-field C221W2

v-field C221W2-B=0

Results: moderate shear timescale

25.0 2 10 1 MNRm

0 B03 f 0 B03 f 0 B03 f

BB

horizontal slabs vertical slabsstreamwise eddies


Results: scale-dependent anisotropy

Reminiscent of the observations of Cho and Lazarian (2003) in high Rm compressible MHD turbulence


But…

DNS is for low Reynolds number periodic flow in a box.

Model predictions are for high-Reynolds number limit.

Establish if possible Reynolds number dependence

Conclusion

SBM with large-scale enstrophy in excellent agreement with DNS!

Future Plans

DNS seems predicts that turbulence is suppressed for . 2 3 DNS seems predicts that MHD turbulence with spanwise B can survive .50, (1), 10, 2m mR P O N M

10243 Or bigger DNS would help! (both Re effects and eddy containment issues

Documents

DNS and Structure-Based Modeling of Rotated Shear Flows: Implications for Accretion Disks?