Brian England

Objective Process of whiskey maturation & Current Efforts to

improve

Comparison Goals

Modeling nonlinear Diffusion Cylindrical Coordinates

Initial and Boundary Conditions

Methodologies and Computational Results Finite Difference

Finite Volume

Function Space

Final Comparison and Conclusion

Maturation of Whiskey Driven by two processes

Diffusion of Oak barrel goodness Short time scales

Chemical reaction Long time scales

Current Efforts Diffusion

Wooden inserts , tastes more processed and is lacking

Chemical Reactions Pressure vessels / burners to shift % yields and reaction rates

Much better and they’re improving on this front

Comparison Goal General Comparisons

Accuracy

Stability

Computational Efficiency

I will primarily deal with Computational Efficiency

Validate a methodology for utilization in future work

Modeling Nonlinear Diffusion What makes the Diffusion Nonlinear?

Changes in differential operators

Best coordinate system Ellipsoidal

Much more Complex

Cylindrical Coordinates Easier & exact solution are readily available

Cylindrical Coordinates Differential Operator Adjustments

Gradient

Divergence

Curl

Laplacian

1, ,

U U UU

r r z

1 1r zrF F F

Fr r r z

1 1, ,z r z r

rFFF F F FF

r z z r r r

2 2

2 2 2

1 1rU U UU

r r r r z

Initial and Boundary Conditions First Comparisons

Boundary Conditions

Chosen to ensure an equilibrium

Initial Condition

(0, , , ) 0U r z

( ,1, , ) cos sin2

U t z z

( , , , 1) cosU t r r

Finite Difference Approach General PDE

Centered Difference Spatial Discretizations

Forward Euler in Time

2 2 2

2 2 2 2

1 1U U U U U

t r r r r z

21, , , , 1, ,

22

i j k i j k i j kU U UU

r r

1, , 1, ,

2

i j k i j kU UU

r r

1( , ) ( ) ( , )n nUf U r U U t f U r

t

Stability The discretization

Requirement for Stability

For our PDE, the largest linearized coefficient is the most restrictive

21 1 1

2 2

2( )

n n nn n i i iU U UU U

U U tt x x

2

2

xt

2 4

2 2 2

min

1 1

2

U rt

r r

Good news (->Stability) Can get away with ~ dR = 0.05 & dt = 0.000625

Wall Clock time = 36.68 Seconds

Tolerance set to Flux/function values of 0.01

Finite Volume Method More work apriori!

Approach PDE with Gauss’ Law & Integral Form

Our PDE in Integral Form

The Integral Form allows us to represent the average value of our function

V V

UdV K U dV

t

1

V V V

U UdV UdV V UdV V

t t t V t

Final Exact Formulation

Utilizing Gauss’ Law

Making appropriate substitutions

We obtain the final form 6

1i

i

i S

U KU dS

t V

V S

FdV F dS

V S

K U dV K U dS

Finite Volume Approximations Difference in time

Differences on the boundaries

Calculation of Fluxes

1n n

Net Flux

KU U t

V

1

n n

i iU UU

r r

1 1( / 2)( ) ( / 2)( )r i i i iNet Radial

S r dr U U r dr U U

1 12rj j i

Net Azimuthal

SU U U

r

1 12Z j j iNet Axial

S U U U

Cell Volume & Surface Integrals Volume

Axial Surface

Radial Surfaces

Azimuthal Surfaces

2 21

2out inV r r z r z r

2 21

2Z out inS r r r r

S r z

rS r z

Items of Note Method is fully conservative

Only approximations lie in the derivative at the cell surfaces

When proper substitutions of the volume / surfaces is performed, it’s almost identical to finite difference

What benefits are there?

We’re still approximating derivatives

Leads to order of accuracies

Computations is about the same

Results Same scenario as finite difference

Stability – Required dt = 0.0001

1/6th the dt

There may have been corner issues

Wall Clock 359 Seconds 10 times as long

Even if the dt was matched, this method would still be slower For now

Function Space Methods Rely on Eigen function products

Coefficients are solved via inner products

Integrals are approximated by function evaluations at collocation points

Our Problem - Diffusion Equation in Cylindrical Coordinates

Choice of Eigenfunctions

Radial - Bessel functions of the first kind

Azimuthal – Trigonometric

Axial – Trigonometric

Temporal - Exponential

Exact Solution Fundamental Technique

Principle of Superposition

Handle a separate problem for each boundary and initial state for a steady state solution and transient

( , , ,1) ( , )topU t r f r

( , , , 1) ( , )botU t r f r

( ,1, , ) ( , )LatU t z f z

(0, , , ) ( , , )LatU r z f r z

General Solution Step by step “Separation of Variables”

First Time

Which Gives

Separating and solving

Letting

( , , , ) ( ) ( , , )U t r z T t r z

2( )( , , ) ( ) ( , , )

T tr z T t r z

t

( ) tT t e 2 2 2

2 2 2 2

1 10

r r r r z

( , , ) ( ) ( ) ( )r z R r Z z

2 2 2

2 2 2 2

1 ( ) 1 ( ) 1 ( ) 1 ( )

( ) ( ) ( ) ( )

R r R r Z z

rR r r R r r r Z z z

General Solution You can show that our ODE’s

Provides

General Solution – depends on our BC-s

2

2

1 ( )

( )

Z zm

Z z z

22

2

1 ( )

( )l

22 2 2

2

( ) ( )( ) ( ) 0

R r R rr r m r l R r

r r

Solution Primary Case - Initial Conditions with homogenous BC=0

l = is an integer greater than or equal to 0.

, is the nth root of the Bessel function of the first kind of order L

m is an integer greater than or equal to 0.

Finally

2

0 0 0

2

0 0 0

( , , , ) cos in2

cos2

lmn

lmn

t

lmn l

l m n

t

lmn l

l m n

U t r z A e mz J m r S l

B e mz J m r Cos l

,l nk

2

,lmn l nk m

Constants The constants are solved via

1 2 12

1 0 0

1 2 12 2 2 2

1 0 0

(0, , , ) cos sin2

cos sin2

l

lmn

l

U r z mz J m r l rdrd dz

A

mz J m r l rdrd dz

1 2 12

1 0 0

1 2 12 2 2 2

1 0 0

(0, , , ) cos cos2

cos cos2

l

lmn

l

U r z mz J m r l rdrd dz

B

mz J m r l rdrd dz

Collocation Points Azimuthal Fourier

Evenly distributed

Axial fourier Chebyshev nodal locations

Radial – Bessel Roots of m’th order Bessel function

Best used in tabular form to save on computations

2 /j N

2 1cos

2j

z

jx

N

Collocation Methods Numerous Integration techniques

Quadrature at collocation points for A and B

Standard – Gaussian Quadrature

Radial and Axial Weights

Polynomial Approximations

Azimuthal Weights

Trigonometric Approximation

We’ll use nodal locations, i,j,k, to calculate weights

1 2 1

1 0 01 1 1

( , , ) ( , , )ijk i j k

i j k

f r z rdrd dz w f r z

Separation of Variables Through some integration and appropriate initial

conditions we can state

We can now approach each integral with our approximation

1 2 12

0 0 01 0 0

12 2

0

( )cos ( )sin ( )2

l

lmn

l

Z z mz dz l d R r J m r rdr

AJ m r rdr

1 2 12

0 0 01 0 0

12 2

0

( )cos ( )cos ( )2

l

lmn

l

Z z mz dz l d R r J m r rdr

BJ m r rdr

Weight Calculation (Axial) A bit of matrix manipulation and brute force

With fixed nodes

1 inverse and done!

11

11 2 1 2

2 2 2 2 11 2 1 1

1 1 1 1 11 2 1

1 1 ... 1 1 1

... 2

... ... ... ... ... ... 1 1...

... ( 1)

...

N Nn

N N N N

N N N

N N N N

N N N

c

x x x x c

x x x x c N

x x x x c N

Weight Calculation (Azimuthal) Trigonometric Gaussian Quadrature

Maximal Trigonometric Degree of Exactness

New Quadrature Methodologies

“Trigonometric Orthogonal Systems and Quadrature Formulae with Maximal Trigonometric Degree of Exactness”

Gradimir V. Milovanovic

Easier assumption

Requires proper initial conditions

1

sin( ) ( )i i

i

n d w f

11

11 2 1 2

2 2 2 2 11 2 1 1

1 1 1 1 11 2 1

1 1 ... 1 1 1

sin sin ... sin sin 2

... ... ... ... ... ... ...

sin sin ... sin sin ( 1)

sin sin ... sin sin

N N

N N N N

N N N

N N N N

N N N

w

x x x x w

x x x x w N

x x x x w N

1

1n

Weight Calculation (Radial) Requires Bessel functions

21 1

*

0 00 0

1 1 1( )

! ( 1) 2 ! ( 1) 2 1

m mm n

n N

m m

xJ r dr J dr

m m n m m n m n

1

01

( ) ( )n i n i

i

J r dr w f r

*0 1 0 2 0 1 0 1 1

*1 1 1 2 1 1 1 2 2

*1 1 1 2 1 1 1 1 1

1 2 1

( ) ( ) ... ( ) ( )

( ) ( ) ... ( ) ( )

... ... ... ... ... ... ...

( ) ( ) ... ( ) ( )

( ) ( ) ... ( ) ( )

N N

N N

N N N N N N N N

N N N N N N N

J x J x J x J x w J

J x J x J x J x w J

J x J x J x J x w J

J x J x J x J x w J

*

N

Conclusion Function Space methods

Very time consuming apriori

Didn’t get to finish

Expectation of fast convergence

Approximation lies only with initial / boundary conditions

Exact solution for all time after (for the given approx)