Dilute Polymer Solutions In Isotropic Turbulence, Part 2 Direct

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```					   Fluids Seminar, Department of Mathematical and Statistical Sciences
University of Alberta, Edmonton, AB November 20, 2008

Dilute Polymer Solutions In
Isotropic Turbulence, Part 2:
Direct Numerical Simulations
Shi Jin
Chemical and Materials Engineering, University of Alberta
Mechanical and Aerospace Engineering, Cornell University
Collaborators:
Cornell MAE
• Vaithi Thirunavukkarasu (Clear Science)
• Lance Collins
Cornell Physics
• Dario Vincenzi (CNRS)
• Eberhard Bodenschatz (Max Planck)
PSU
• Ashish Robert
• James Brasseur
Motivation: Polymer Turbulence Drag Reduction

Hoping to build to faster ship?
doubles the maximum capacity
DARPA Friction Drag Reduction Program
The flows are all turbulent.
Governing Equations of a Polymer
Solution using FENE-P Model

Symmetric
Positive definite
Polymer
Parameters                  Cij
Bounded: 0<r<L
Questions
   Is the FENE-P model accurate?
Yes, for strong turbulent flow.
   Polymeric DNS
 Keep using the Newtonian DNS grid?
Newtonian grid cannot resolve the polymer field.
 Numerical difficulty in the polymer stress tensor

   RANS: how to close the polymer equation?
   Who is right? Lumley or de Gennes?
Turbulence DNS: Spectral Method
   In Newtonian DNS, pseudo-spectral method is
the de facto numerical method due to its
exponential convergence property
   Moin and Mahesh (1999): spectral DNS ~ second
order central difference at twice the resolution
   Homogenous (Isotropic, shear): Fourier transform
   Channel Flow: Fourier + Chebyshev Transform
   Jet Flow: Chebyshev + Fourier Transform
   With FENE-P, spectral method is the first
method used.
Problems in Spectral Method
   Polymer Conformation Tensor (bounded, SPD)
   Overshoot: Ckk>L2
   Undershoot: eigen value<0
   Beris and coworks’ solution: Add artificial polymer
stress diffusivity D
   Schmidt number: Sc=ν/D
   For high molecular
weight polymers, Sc~106
   To maintain the
polymer DNS: Sc~0.1-1
   May impede stretching
   Need higher Wi to predict
the same drag reduction
Hyperbolic Governing Equation
   FENE-P Equation
   Hyperbolic: similar to the shock wave problem in
aerodynamics
   Spectral method: Gibbs phenomenon (ringings)
   Artificial diffusivity in spectral method:
   Attenuate ringings
   Reduce the jump magnitude
   Finite difference:
   maintain the jump magnitude
   satisfy the Rankine-Hugoniot conditions

New Method: adapt a finite difference
scheme to maintain overall conservation
of polymer stress
Eigen Decomposition Method
   Vaithianthan and Collins (03)
   Eigendecompose of Cij
   Advance the eigenvalues and eigenvectors
separately
   Compact finite difference method
   Maintain Ckk<L2 with implicit method
   However, positive eigenvalues is not guaranteed
   Set to 0 if found negative
   This zeroing leads significant spurious values
Problem with Eigendecomposition
   In homogenous flows, overall conservation

The New Method should satisfy
this overall conservation too!
The New Finite-Difference Scheme
   Kurganov and Tadmor (2000) method
   Developed for gas dynamics
   Second order accurate
   Maintain the positiveness of a scalar
   Much less numerical diffusive than earlier hyperbolic solvers
   Vaithianathan et al (06) adapted KT scheme
   Extend it to tensor
   Positiveness of a scalar  SPD
   Implicit stretching term  bounded
   Convective flux formula  conservation
   Slope limiting procedure to choose the gradients.
   Scalar: minmod limiter
   Tensor: choose the one satisfying SPD
Lagrangian vs Eulerian DNS
   Homogeneous shear flow: 5%
   Isotropic DNS
Numerical difficulty in the
polymer stress tensor
   FENE-P equation is hyperbolic
   Challenges are
   Positive definite
   Bounded
   Overall conservation
   Need a non-dissipative scheme
   A new method is developed to meet all the
requirement
   It is still very expensive to do: polymer simulation time is
about 10 times the Newtonian simulation time
   Improvement is needed for larger simulations
Questions
   Is the FENE-P model accurate?
   Polymeric DNS
   Keep using the Newtonian DNS grid?
   Numerical difficulty in the polymer stress tensor
   RANS: how to close the polymer equation?
   Who is right? Lumley or de Gennes?
RANS Model of FENE-P
Reynolds Averaged Navier-Stokes
   Solves the mean quantities
   Ideal for industrial applications
RANS
average

0

The difficult part is to model the stretching term Λij
Stochastic Model For Lagrangian
Isotropic Trajectory
   Isotropic Velocity Gradients in Lagrangian Frame

Greek suffix: no summation rule

   TS, TR are free to choose
   From DNS
Analytical Work
   Assumption: the velocity gradient in a Lagrangian
frame is Gaussian white noise in time:
   A model is developed analytically in the white noise
limit.
 Gaussian integration

 Functional calculus

 Perturbation method

   Extend the model to real isotropic turbulence:
 Keep the form

Non-dimensional Model in Isotropic
Turbulence in the White Noise Limit

Model:

Due to isotropy:
Validation: Simulation Results



0.7

0.6

0.5

0.4
Z

0.3

0.2         'first order model'
WeΩ=1, simulation
WeΩ=5, simulation
0.1         WeΩ=10, simulation

0.0
0.2           0.4           0.6   0.8   1.0
Ω
Model for Ω = 2.3 (Real Turbulence)
 Known:
 Propose:
20

Model Flow Simulation
y = 0.2022 x + 0.2561
15
y=0.172 x + 0.079
10
Y

5
1
0

0   20        40         60   80   100
We

Stochastic Model Flow: α= 0.198               2563 Decaying DNS: α= 0.172
Closure for Stretching Term
   In the analytical limit:
   Isotropic turbulence after coil stretch transition
1283 Decaying DNS

   Below coil stretch transition:
Closure for the restoring term
PDF of x From DNS
Below coil stretch transition

Peterlin type preaveraging (assume f (x) is a delta function):
Beyond Coil Stretch Transition
Approximate f (x) using a triangle

Mean: [1/3,2/3]            Mean: [0,1/3]
Full Closure in Isotropic Turbulence

Steady      continuity &           isotropy & homogeneity
State      homogeneity

stretching term                      restoring term

take trace
Validation with SDE Simulations

Cannot go beyond 2/3

Not smooth

For stretching term:
SDE Flow : α= 0.198
Decay DNS: α= 0.172
decays to 0 too fast
RANS: how to close the polymer
equation?
   Stretching term after coil stretch transition

   Restoring term
   Below coil stretch transition

   Intermediate Weissenberg Number

   Large Weissenberg Number
Questions
   Is the FENE-P model accurate?
   Polymeric DNS
   Keep using the Newtonian DNS grid?
   Numerical difficulty in the polymer stress tensor
   RANS: how to close the polymer equation?
   Who is right? Lumley or de Gennes?
Two Most Influential Theories
   Lumley (1969, 1973): Boundary Layer Experiments
   Explains qualitatively much of the phenomenology
   Drag reduction scales linearly with polymer concentration

   De Gennes (1986, 1990): Dynamical Energy Spectral
in Isotropic Turbulence
   Screenivasan & Whilte(2000) applied it locally in a
turbulent boundary layer and recovered much of the
phenomenology as well
   Predicts a minimal concentration, below which there is no
drag reduction---- Look for the existence of this
critical concentration in two-way coupled isotropic
polymeric DNS
The de Gennes Theory
   Trapping length scale r*: turbulence scale where
stretching of polymer begins
   Similar to Lumney arguments: it is where the flow time
scale = polymer relaxation time:
¿*=¿p 
   Note that r* is polymer concentration independent
   Another length scale r**: polymer elastic energy =
turbulent kinetic energy at r**
   For r**<r<r*, polymers are not stretched enough to
change the flow
   Scales smaller than r** are strongly affected
   Note r** scales with concentration
The de Gennes Theory (cont)
   Trapping length scale r*: turbulence scale
where stretching of polymer begins
   Another length scale r**: polymer elastic
energy = turbulent kinetic energy at r**
   If r**<η (Kolmogorov length): no polymer
effects on the flow
   So r**= η predicts a minimal concentration
   To find the critical concentration, need to
have small We~1
Concentration Study: Onset Problem
Is there a critical concentration?
Experiments by Crawford (2004)
Lagrangian Accelerations

   Bodenschatz and co-works (2006,
science)
   Shape of normalized PDF unchanged
   Acceleration variance decreases with
polymer —quantity used in onset study
Polymer Effects In Isotropic DNS

2563 Forced DNS: Newtonian Rλ=141

Large error bars, non conclusive results too!
Energy Spectrum, We=1.5

   There is significant energy transferred into polymer at large scales
   Artificial forcing at largest scales may affect polymers drastically.
   Forcing prohibits very long simulations for good statistics
Who is right? Lumley or de
Gennes?
   Too early to tell
   Maybe they are both correct
   In wall-bounded flow, the flow is always strong
enough to have r**> η, thus the critical
concentration cannot be observed
   Future plans
   Better forcing or run decaying DNS
   Larger simulations: 25635123
   Speedup polymer algorithm
   More careful about the range of concentration
Questions
   Is the FENE-P model accurate?
Yes, for strong turbulent flow.
   Polymeric DNS
 Keep using the Newtonian DNS grid?
Newtonian grid cannot resolve the polymer field.
 Numerical difficulty in the polymer stress tensor

A second order central difference scheme.
   RANS: how to close the polymer equation?
A first step for systematical RANS development.
   Who is right? Lumley or de Gennes?
Not clear. Still working on it.

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