Statistical closure for homogeneous turbulent flow of a dilute polymer solution

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Statistical closure for homogeneous turbulent
flow of a dilute polymer solution

• Shi Jin
• Dario Vincenzi, Lance Collins
• Cornell Fluid Dynamics Seminar, Nov 1, 2005

Sibley School of Mechanical and Aerospace
Engineering Cornell University, Ithaca, NY
Sponsor DARPA, ACS-PRF
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Outline

• Motivation
• Introduction
• Model performance
• Mapping of parameters
• Performance
• Spatial Resolution
• Statistical Closure
• Summary
• Future plans

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Motivation
Skin friction is the dominant drag in flow
through pipes, flow over sub-marines and ships
Alaska pipeline Drag Reduction Agent doubles
the maximum capacity
Hoping to build to faster ship?
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Objectives

• Global Objectives
• Develop a RANS model for polymer solution that
  can be used for ship hull design
• Understand the physics of polymer in turbulence
• Sub objectives
• How is the FENE-P model performance?
• DNS Newtonian grid resolution enough for
  polymer?
• RANS model statistical closure
• Polymer mixing
• Lagrangian polymer tracking in DNS

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Polymer models
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Finite Extensible Nonlinear Elastic

• FENE multi-bead-spring, nonlinear, discrete
  model
• FENE-P FENE-dumbbell with pre-averaging
• Closed equation

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Part-1 FENE-P performance

• Use Multi-bead FENE to compare with FENE-P
• Brownian Dynamics along fluid trajectories
• Parameter mapping
• Performance in isotropic turbulence
• Magnitude
• Orientation

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Brownian Dynamics of FENE

• Simulate an ensemble of polymers along given
  fluid trajectories, then do statistics
• Lagrangian frame
• One-way coupling, no feedback from polymer to
  flow
• Stochastic ordinary equations

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• Polymer stress
• Elastic Energy

• Nondimensional parameters

• Important scales
• Length
• Time

Molecular parameters B, We Parameter
Mapping How to choose the two parameters for
arbitrary length FENE chains?

• Length
• Relaxation timemore complicated

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Mapping for relaxation time

• Rouse(1953) theorylinear spring force law
• Ghosh et al.(2001)Wiest Tanner(1989)
• We proposed a new mapping

Figure shows B3000, We50 Uniaxial elongational
flow Neglecting Brownian forces
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Mapping Convergence Comparison for Isotropic
Turbulence
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Polymers in Isotropic turbulent flow

• Fluid trajectories obtained from DNS
• Multi-bead FENE computed along trajectories
• Rouse mapping and new mapping are used
• Parameters
• We10, 50, 100 or 8.62, 43.09, 86.18
• B3000, 30000
• N2, 5, 10, 20
• 100 trajectories
• Ensemble size512
• Study
• Magnitude
• Stress
• Elastic energy
• Orientation

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Magnitude
We50, B3000, Rouse mapping

• Stress magnitude
• Definition
• Elastic energy

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Orientation

• Define a polymer direction
• Only possible when polymer is stretched over 95
  of the time, it is.
• defined as V1, with
• Study the CDF of ß
• Used by Cristini, Blawzdziewicz, Loewenberg and
  Collins, JFM, (2003)

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Dumbbell rotates like a material element

• For large We
• Material element
• Polymer (FENE)

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Summary of Part-1 (FENE-P performance)

• New mapping is introduced
• Extremely good for uniaxial elongational flow
• Good for isotropic turbulent flow, especially for
  large We
• FENE-P performance is accurate in isotropic
  turbulent flow for moderate to high We
• Rotates like a material element
• 5 difference using our new mapping

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Part-2 Is the Fluid grid fine for polymers?

• Well resolved field
• Approximately local linearity
• Sub-grid values approximated by linear
  interpolation

• When sub-grid values cannot be approximated by
  linear interpolation of grid values, the grid is
  NOT able to resolve the filed

Fluid grid fine for polymer? gt Is the polymer
field on the sub-grid locally linear?
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2D sub-grid demonstration

• How to compute the sub-grid values?
• Interpolate the grid values Nogtalways linear
• Track Lagrangian trajectories which finally fall
  on the sub-grid points

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3D converging trajectories

• More trajectories
• Run backwards
• Run FENE-P along trajectories

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Measure of Linearity
Need a bounded scalar to indicate the linearity

• 3D field
• Real 2nd order tensor
• Solution
• stretch the tensor as a long vector
• Solve the multiple linear regression model using
  least squares

R2 coefficient of determination SSR sum of
squares due to regression SST total sum of
squares SSE sum of squares for error The closer
to 1 R2 is, the more linear is the tensor
field Call R2 linearity
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Linearity of the trajectories

• Simulation
• 643 Isotropic DNS
• 43 evenly distributed bunches
• 73 sub-grid points per bunch
• R?X/d1,20,100,10k,1M

Finally fall to the sub-grid points near a single
grid point
Initially randomly distributed in space
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FENE-P linearity along trajectories

• The original grid is not able to resolve the
  polymer grid while finer grids are.

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Conclusion for part-2

• For large We, Newtonian grid may not be
  sufficient to resolve the polymer field
• There is no strong dependence on We for large We,
  in agreement with Martins Afonso and Dario
  Vincenzi, JFM, (2005)

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Part-3 statistical closure

• Objective statistical closure (RANS)

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Mathematical approach

• Assume the velocity gradient is white noise in
  time
• Real flow has finite correlation time
• From Isotropic DNS
• But it is informative to see the format of the
  relationship
• Use functional calculus
• Gaussian integration by parts
• Perturbation method

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Non-dimensional models
First order model
Second order model (for isotropic turbulence
only)
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Validating the model

• Use Brownian Dynamics Simulation
• Along stochastic models for isotropic
  trajectories
• An ensemble of trajectories is simulated
• Statistics taken only for steady state

Greek suffix no summation rule
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Model validation

• Due to isotropy only consider the traces

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When O is not close to 0
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DNS (O2.3)
Forced Isotropic DNS BD along trajectories
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Summary for modeling

• For Isotropic turbulence, large We
• Use DNS to determine parameters
• More work to do
• Low We cases
• Shear flow second order model instructive

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Summary for all

• FENE-P is a good model for stretched polymers
• With correctly mapped parameters new mapping
• Good agreement in magnitude and orientation
• Spatial Resolution
• Fluid grid resolution not sufficient for polymer
• Modeling
• Isotropic

All conclusions are under the assumption that We
is moderate or large.
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Future plans

• Modeling
• Settle down the parameter for Isotropic DNS
• Found out a model for Shear flow
• DNS
• Shear flow Lagrangian polymer tracking
• Polymer mixing study