Adaptive Multigrid Solutions of Thin Film Flows over Topgraphy

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Adaptive Multigrid Solutions of Thin Film Flows
over Topgraphy

• Yeawchu Lee, Harvey Thompson Phil Gaskell
• School of Mechanical Engineering, University of
  Leeds, UK

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Contents

• 1/ Motivation
• 2/ The Lubrication Approach
• 3/ Numerical Method
• 4/ Adaptive Multigrid
• 5/ Conclusions

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1 Motivation

• Continuous thin film flows over topography have
  many important practical applications, including
• Manufacture of photographic film and ink-jet
  media
• Deposition of coatings and inks ink-jet
  printing
• Photolithographic production of printed circuits,
  displays etc
• Microfluidic devices
• Redistribution of liquid lining in lungs, blood
  oxygenation,
• Heat exchangers etc
• Hence, important to understand and predict
  associated fluid mechanical phenomena.

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1 Motivation
Problem considered here Gravity-driven flow over
well-defined topography e.g. combination of
square, circular and diamond-shaped trenches.
Relevant to photolithographic flows in
electronics sector.
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2 The Lubrication Approach

• Lubrication equations - asymptotic expansion of
  the Navier-Stokes equations in terms of ?
  H0/L0.
• Valid for small
• free surface gradients

• Scaling (Aksel (2000))
• H0 (3?Q/?g sin?) 1/3 film thickness for the
  fully developed film flow down an inclined plane,
  U0surface velocity
• L0 (?H0/3 ?g sin?) 1/3 Capillary length
• Dimensionless groups
• ? H0/L0 ltlt 1, Ca ?U0/?.

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2 Lubrication Equations
Note Equation (1) ensures mass conservation and
equation (2) gives pressure due to Capillary and
hydrostatic terms. Possible to account for
variable viscosity (due to temperature,
concentration etc) and surface tension. Here
focus on flow of water films with
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3 Numerical Method Discretised Lubrication
Equations
Finite Difference discretisation Control
Volumes centred at grid vertices. Time
integration using Crank-Nicolson.
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3 Discretised Lubrication Equations

• Non-linear lubrication equations are challenging
  to solve numerically. Usual approach is to use
  semi-implicit Alternating Direction Implicit
  methods - Time-Splitting.
• Our approach use Full Approximation Storage
  Multigrid method remove longer wavelength
  errors by relaxation on coarser grid levels.

G0 9x9 G1 17x17 G2 33x33 G3 65x65 etc
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3 Solution of the Discretised Lubrication
Equations - Multigrid

• Efficiency of the non-adaptive Multigrid
  approach
• (1) CPU time for a given number of unknowns, N,
  is O(N).
• (2) Implicit good stability
• (3) Lends itself to adaptive refinement around
  topographies

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3 Comparison with Decre and Baret (2003)
2-D Flow of Water Film over a Trench Topography
Comparison between experimental free surface
profiles and those predicted by solution of the
full Navier-Stokes and Lubrication
equations. Agreement is very good between all
data. Lubrication theory can be very accurate.
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3 Accuracy of Lubrication Apporoach for 2-D flow
over a step-down topography
Error is classified by the maximum difference
between the Navier-Stokes and lubrication profiles
Shows how error increases with Reynolds number
(Re) and topography height. Contours show
recirculations in the Navier-Stokes solutions for
Re15 (top) and Re0.15 (bottom) where errors are
respectively 13 and 8.5.
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4 Adaptive Multigrid
Motivation To capture the effects of
distributions of small, isolated topographies
efficiently use coarse grids in regions of
simple flow. Areas requiring local refinement
are identified using a Truncation Error
Analysis.

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4 Adaptive Multigrid
Error Analysis Re-write discretised lubrication
equations on the kth grid in the form where
are the unknown h and p
variables on grid k and the superscripts (n1), n
indicate values of these variables at end of
(n1)st and nth time steps respectively. Residual
measures the
error in satisfying the discretised equations on
grid level k. Relative Truncation Error where
is a Restriction Operator from Grid k to
Grid k-1. Large values of indicate local
refinement needed.


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4 Adaptive Multigrid Conservative Interpolation
Adaptive refinement proceeds by conserving
numerical flux per Control Volume (CV) area at
coarse and locally-refined regions. O grid
vertex on fine grid k ? - grid vertex on next
coarsest grid k-1 ? - ghost node at adaptive
boundary
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4 Adaptive Multigrid Conservative Interpolation
Adaptive refinement proceeds by conserving
numerical flux per Control Volume (CV) area at
coarse and locally-refined regions. For
lubrication equations flux across CV defined by
(Lee et al (2006)) where
or At boundary of
local refinement interface conserving flux per
area Enables values of h and p at ghost nodes
to be determined as Dirichlet conditions for the
adaptive solution.
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4 Adaptive Multigrid Results
Flow of thin water films of asymptotic thickness
100 µm, at constant flow rate 1.64 x 10-6 m2/s.
Capillary length LC0.78mm, N0.122 (gravity
little influence on free surface) Results
obtained using a FMG V(4,2) cycle with a 9x9
coarse grid, with finest grid levels between k2
(33x33) and k6 (513x513). Flow domain extends
over 50 Capillary lengths in each direction
39mmx39mm.
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4 Adaptive Multigrid Results
Evolution to steady state for flow over a square
trench with 3.9mm sides, 10µm depth. Global
coarse grid 33x33 refinement performed over
next two finer if
Soon after solution starts with planar interface
initial condition
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4 Adaptive Multigrid Results
Steady-state solution, local refinement over next
two grid levels performed if Shows
characteristic bow wave disturbance
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4 Adaptive Multigrid Results
Effect of Error Tolerance local refinement
performed if Decreasing e expands
local refinement regions.
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4 Adaptive Multigrid Results
Non-adaptive solutions of flow past square trench
of size 0.78mm x 0.78mm x 10 µm Smalle
r topographies require finer uniform meshes for
grid independent solutions.
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4 Adaptive Multigrid Results
Adaptive solution offers order of magnitude
reduction in CPU time compared to non-adaptive
solutions on 513x513 grids.
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4 Adaptive Multigrid Results
Practical applications require flows past
topographies of more complex shape. E.g. Flows
past 3.9mm x 3.9mm x 10µm trenches Diamond
trench Circular trench
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4 Adaptive Multigrid Results
Flow past topography combinations Uniform coarse
33 x 33 grid with two levels of grid
refinement. Two circular upstream peaks Diameter
1.95mm, height 25µm, Diamond trench 1.95mm x
1.95mm x 10µm Two circular downstream trenches
1.95mm x 1.95mm x 15µm
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4 Adaptive Multigrid Results
Free surface
Computational Grid
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4 Adaptive Multigrid Results
Industrial applications often require free
surface planarisation BUT very difficult to
achieve in 3-D flows. Possible to reduce free
surface disturbances by moving topographies
closer together, e.g.
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4 Adaptive Multigrid Results
Reduction in streamwise free surface disturbance
when moving topographies closer
together Need automatic means of
removing full 3-D disturbances
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5 Conclusions

• Continuous film flows over topography arise in
  many important applications.
• Lubrication approach can yield valuable insight
  even in cases for which not strictly valid.
• Adaptive Multigrid approach much more efficient
  than ADI methods and provides ability to
  efficiently solve flow past distributions of
  isolated topographies.
• Little experimental data is currently available.