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Progress Through Quality Education

  

P. R. Naren

School of Chemical & Biotechnology

  

SASTRA Deemed to be University

Thanjavur 613401

E-mail: [email protected]

at

  

One Week

National Workshop on Advances in Computational Mechanics

  

School of Mechanical Engineering

SASTRA Deemed to be University

Thanjavur 613401

  Finite Volume Method

14-Dec-2018

  Finite volume method

  Outline

  • Conservation equations and control volume

    • – Eulerian and Lagrangian framework
    • – Integral form of conservation equation

  • FVM approach

    • – Steady state diffusion equation in 1D
    • – Convective term

  • Issues with collocated grid

15-Dec-18

  Governing Equations

  • Conservation of mass

  Transport Equations Conservation of momentum • Conservation of energy •

  Concept of CV •  m

    lim   

  • Finite volume method 15-Dec-18

   Framework

  • Lagrangian
  • Eulerian

    • – Fixed reference
    • – Moving reference

  • Finite control volume
  • Infinitesimally small

  control volume

  Samimy et al., 2003

  • – Integral form
  • – Gross behaviour
  • – Differential form
  • – No discontinuity

  Finite volume method

15-Dec-18

  Advection

  V V 

   N i

  N i

  T T P P u

    u u Balance Unit Mass Unit Volume

      m  1  

    N

    m  u A i

  N N i i

    x u 

  T  

   T T

  P m u   P

    u C T P

  P P

    Q mC T 

  P  x i

   N i

  Q   C T P

   N i

     C x i i

   Finite volume method 15-Dec-18 Generic Transport Equation Transport equation for a quantity f •

    f  

    f   f  div( V ) div grad S   f

   t Accumulation + Net outflow = Net Diffusion + Net source

   Equation Specific quantity f S f  per unit mass)

  Mass balance

  1 m  Momentum u g

   P balance

  • D

    Energy balance C T k H UADT

  p R

  Species balance i x D r i i

  Finite volume method 15-Dec-18 Mass Balance

  • Mass

       d i v ( ) U

   t         u v w

  

     

        t x y z

  Finite volume method 15-Dec-18 Momentum Balance Navier Stokes •

  D U      div p div S

  M

  Dt   u

     p

     -  m  div( u ) div grad u S U

  

 

Mx

    t x Stokes

  Navier Finite volume method 15-Dec-18

  Taylor

  Numerical Techniques

  • Finite Difference • Finite Element • Finite Volume

  Finite volume method

15-Dec-18

  Geometry

  Finite volume method

15-Dec-18

  Finite volume method

15-Dec-18

  

Illustration: 1D heat Conduction

Steady state 1D heat conduction in a solid rod with ends kept at fixed temp

  Model Formulation

  • Axisymmetric

    • – q
    • – L >> D – Radial variation ignored
    • – No heat loss thro’ surrounding

  • Governing equation
  • BC: z = 0 T = T

  A

  ; z = L T = T

  B

  with constant thermo-physical properties

  Finite volume method T

A

  T B L

T

A

T B d d T d z d z

          axisymmetric

  1D approx’on

15-Dec-18

  • Analytical Solution

    • – Continuous

      •  

  B A A

      -  

  2 T T i 2 to N 1 T T T T

  T

  1 A N B

  i 1 i i 1

  T T T T z L

   

  Grids Soln

  • Numerical solution

    • – Finite Difference – Solution at discrete points
    • – Affected by Dz

  T B Finite Diff

  T B

T

A

T B

T

A

  Finite volume method

T

A

  Solution to Heat Conduction in Rod

15-Dec-18

  Typical Temperature Profile

  Finite volume method

15-Dec-18

  Issues with Finite Difference Solution discrete vs. governing equation continuous • Treatment of local source terms • Discontinuities in solution • Nature of conservation laws •

  Finite volume method 15-Dec-18 Integral Form of Conservation Equation  

  x

  

 f      

   

  Integrating over CV

  Finite volume method For steady state flow For 1D flow

    

   f   

 

 

  D f

 

    x x x x d d d u S. d x S. d x S S.d x d x d x d x f D D

     Gaub

    x d d d

u d d S d

d x d x d x f D  D  D  f  

  S dx d dx d u dx d

   f   f S grad div u div

  f

  

S grad div u div

t    

   f   f   f 

    f  

   

f

    

f

     

  f

15-Dec-18

  Integral Form of Momentum Equation               

    u  uu   vu  p u u

  •     m  m             t x y x x  x  y y  

  Integrating over CV

            

         

u uu vu p u u

    m  m dt dv dV dV   dV dV  

  • -

           

    t x y x x x y y

     

       

  D D D D D D t

  V V

  V V V   For steady state flow

  

          

   

uu vu p  u  u

   m  m dv dV dV   dV dV

  •         

    x y x x x y y

     

      

  D

  V D

  V D

  V D

  V D V   Finite volume method 15-Dec-18 FVM Approach for Heat Conduction Computational domain for • heat conduction in rod

  Temperature at node is – L average of CV around the

  T B

T

A node

  • – Boundary nodes

  Internal nodes – T

T

A

B

  Finite volume method 15-Dec-18 Finite Volume Formulation  f

     f   f  div  u  div  grad  S f

   t N

  f   f  div  u  div  grad  S f n f   d d d w P e E

  W f   

    u   S f x dx dx  dx  s S f d d  d 

   f     u dV   dV S f dV x

     dx dx dx  

  D D D

  V V

  V

15-Dec-18

  19

  Some Mathematics !!

  • Taylor Series

  2 h

  ' ''     

f  x h  f   x hf   x f   x ...

  2 !

  2 h

  ' ''

  •  

    f  x h  f   x hf   x f   x ...

  2 !

  3

  2 h

  ' "' 2 h ''

     - - f  x h   f x h  2 hf   x 2 f   x ...

    -            f x h f x h 2 f x f x ...

  3 ! 2 !

  3     - - f x h f x h

  1 h "'

    - ' f x h  2 f    x f x h 

  • ''

        f x f x ...

  •  f    x ...
  • 2

  2 h h 3 ! h

  Finite volume method 15-Dec-18

  w e

                    2 u u

  S N Finite Volume Formulation . . .

    S dV dV dx d dx d dV u dx d

  V x

  V V   

  

D

f D D

     

     f

    f      

  P E W e w s n

  V

  u u dV u dx d

  f - f  f 

  D

  • f  f 

  2 u u 2 u u

  W E P W P E

  f - f  f  f

15-Dec-18

  21 Central difference method Finite Volume Formulation . . . f   d d d f   

   u  dV   dV S dV

  f x

     dx dx dx  

  D D D

  V V

  V N

  n w P e E W s f f f S d  d   d   d 

  

  • -   

    dV      

   dx dx dx dx      

  D e w

  V

   f  f  f  f - -            

  E P P W

   - D D x x

EP PW

15-Dec-18

  22 Finite Volume Formulation P E W e w s n S

  N f

   f  f  f s a a a

  E E W W P P f

   f  f  f  f  f s a a a a a

  S S N N E E W W P P  

  S dV dV dx d dx d dV u dx d

  V x

  V V   

  D f D D

     

     f

    f

  Finite volume method

15-Dec-18

  Difficulty in pressure term discretization Checker board

   p  - -

      dV p p

  Solution? e w

    x

  D

  V  

   p p   p p  E P P W

  2

  2 N p p  

  • E W

  n  w P e E

2 W

  s Suhas V Patankar

  S Professor Emeritus Univ. of Minnesota

  Finite volume method 15-Dec-18 Summary

  • Finite volume

    • – Solution represent average over the region NOT a point

  solution

  • Finite volume approach applied to Integral form of

  Conservation equation

  • Discretization of diffusion and advective terms

    • – How to get values of advected quantities at face f

  e f w

  ?

   Other alternatives

  • – How to resolve pressure discretization ?

  Finite volume method

15-Dec-18

  Resources Chung T. J. (2002) Computational Fluid Dynamics. Cambridge University Press

  • Date A. W. (2005).
  • Fox, R. O. (2003) Computational Models for Turbulent Reacting Flows. Cambridge University

  Introduction to Computational Fluid Dynamics. Cambridge University Press

  • Press Hoffmann K. A. and Chiang S. T. (2000).
  • Engineering Education System, Kansas, USA.
  • John F. W., Anderson, J.D. (1996)

  Computational Fluid Dynamics Vol1, 2 and 3.

  Computational Fluid Dynamics: An Introduction Springer Patankar, S. (1980) Numerical Heat Transfer and Fluid Flow. Taylor and Francis

  • Ranade, V.V. (2002).
  • Academic Press, New York. Versteeg, H.K. and Malalasekera, W. (1995) An Introduction to computational Fluid Dynamics

  Computational Flow Modeling for Chemical Reactor Engineering,

  • The Finite Volume Method. Longman Scientific and Technical

  Finite volume method 15-Dec-18 Web Resources

  • http://www.cfd-online.com
  • http://en.wikipedia.org/wiki/Computational_fluid_dynamics
  • http://www.cfdreview.com/

  • • https://confluence.cornell.edu/display/SIMULATION/FLUENT

    • Learning+Modules

  • http://weblab.open.ac.uk/firstflight/forces/#
  • NPTEL

    • – Balchandra Puranik and Atul Sharma – Srinivaas Jayanthi

  Finite volume method

15-Dec-18

  Finite volume method

  Gratitude

  • Dr. Vivek V. Ranade – My Mentor Guide and Teacher

    • – iFMg - Research group at NCL, Pune

  • Audience

    • – For patient hearing and for their thirst in knowledge

15-Dec-18

THANK YOU

  A person who never made a mistake never tried anything new

  • Albert Einstein
  • 1879 -1955

  Finite volume method 15-Dec-18 Concept of staggered grid

  Finite volume method

15-Dec-18

                   u  uu   vu  p u u

     -  m  m             t x y x x  x  y y  

                     u uu vu p  u  u

  •     m  m dt dv dV dV   dV dV  

                t x y x x x y y

    D t D

  V D

  V D

  V D

  V D V             

   uu   vu  p  u  u    m  m dv dV dV  dV dV

  •  

              x y x x  x  y y

  D D D D D  

  V V

  V V

  V Finite volume method

15-Dec-18

  U Control volume

  Finite volume method

15-Dec-18

  X Momentum Equation

              

   uu   vu  p u u

     m -  m   dv dV dV dV dV

   

      

         x y x x  x  y y  

  D D D D D

  V V

  V V

  V  

   uu     dv  uu   uu  e w

  •  x

  

  D

  V   

  • ,I J

   uu   uu 

  I

  1 , J

  •    

   uu   uu   uu   uu       i 1 , J ,i J ,i J i 1 , J

  2

  2    uu   uu 

  •   i

  

  1 , J i 1 , J 

  2 Finite volume method 15-Dec-18 X Momentum Equation . . .

              

   uu   vu  p u u

     m  m   dv dV dV dV dV

  •        x y x x  x  y y

   

      

   

  D D D D D

  V V

  V V

  V  

   vu     dV  vu   vu 

  • n s

  

   y D

  V   

   vu   

  • vu

  ,i j  1 ,i j     - -

    vu   vu     vu   vu   ,i J ,i J  1 ,i J 1 ,i J

  • 2

  2     vu   vu  

  • ,i J 

  1 ,i J

  1 

  • 2

  Finite volume method 15-Dec-18

  •  

     

  Finite volume method

  I V p p dV x p

      J ,

    

    

    

   

    m

    

   

    m

    

    

  D D D D D

  V     

  V V

  V V

  dV y u y dV x u x dV x p dV y vu dv x uu

     

  • D
  •   

1 I J ,

  •  X Momentum Equation . . .

15-Dec-18

  •  

  D            

     

     

      m

     

      m 

   

    - m  

  w e

     

      m

  2 u u 2 u u x u x u

  J , J 1 i , i J , 1 i J ,

  J 1 i , i J , i J , 1 i J ,

  I X Momentum Equation . . .

  Finite volume method

  V x u x u dV x u x

    

    

     

    

   

    m

    

     

   

    m

    

    

  D D D D D

  V     

  V V

  V V

  dV y u y dV x u x dV x p dV y vu dv x uu

  • m     
  • m     

1 I J ,

  • 2 u u 2 u
  • m
  • m     

15-Dec-18

  •  

    - m 

     

     

      m

     

      m 

   

  D            

  

  s n

     

      m

  2 u u 2 u

  2 u u 2 u u y u y u

  , 1 J i J , i , 1 J i , 1 J i J , i J , i , 1 J i

  , j i , 1 j i

  X Momentum Equation . . .

  Finite volume method

  V y u y u dV y u y

    

    

     

    

   

    m

    

     

   

    m

    

    

  D D D D D

  V     

  V V

  V V

  dV y u y dV x u x dV x p dV y vu dv x uu

  • m     

  • m     
  •  

  • m

  • m     

15-Dec-18

  •  

    m

    

    

    

   

    m

    

     

   

    

    

  D D D D D

  V     

  V V

  V V

  dV y u y dV x u x dV x p dV y vu dv x uu

1 I J ,

      

  I J , i J , i p p au u a

    J ,

  Final form

  •  

  Finite volume method

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