Faculty Development Program on Computational Fluid Dynamics

  Finite Volume Method

P. R. Naren

  

School of Chemical & Biotechnology

SASTRA University

Thanjavur 613401

  E-mail: prnaren@scbt.sastra.edu

at

  

Faculty Development Program on Computational Fluid Dynamics

School of Mechanical Engineering

  

SASTRA University

Thanjavur 613401

08 June 2016

  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

8-Jun-16

  Governing Equations

• Conservation of mass

  Transport Equations Conservation of momentum • Conservation of energy •

  Concept of CV •  m   lim ^*

     

  Finite volume method 8-Jun-16 Framework

• Lagrangian
• Eulerian
• – Moving reference
• – Fixed reference
• Finite control volume
• Infinitesimally small

  control volume

  Samimy et al., 2003

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

  Finite volume method

8-Jun-16

  

Advection

  V V 

   N _i

  N i

  T T P P u u   u

  Balance Unit Mass Unit Volume 

      m

  

  N m u A   N N i _{i i}

  1    

   x   u 

  T T   T

   P m u  P

    u  C T

  P P

   P Q  m C T P

   x  i

  N i

  Q   C T P

   N i

   C   x i i

   Finite volume method 8-Jun-16 Generic Transport Equation Transport equation for a quantity

•   

  φ

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

      t

  

Accumulation + Net outflow = Net Diffusion + Net source

Equation

  S Specific quantity f G f ( per unit mass) Mass balance

1 Momentum u

  g µ ρ

   P balance

Energy balance C T k H UA T

  −∆ ∆ p

  R Species balance i x D r i i

  Finite volume method 8-Jun-16 Mass Balance

• Mass

  d i v ( ) t     

   U

  

     

u v w

t x y z

                 

  Finite volume method

8-Jun-16

  Momentum Balance

• Navier Stokes

  U D p

    div  div  S τ

  M

  Dt   u  p

     div( u )  U    div  grad u  S

    Mx  t  x

  Stokes Navier

  Finite volume method 8-Jun-16

  Taylor

  Numerical Techniques

• Finite Difference • Finite Element • Finite Volume

  Finite volume method

8-Jun-16

  Geometry

  Finite volume method

8-Jun-16

  Finite volume method

8-Jun-16

  

Illustration: 1D heat Conduction

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

  Model Formulation

• Axisymmetric
• – θ
• – 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

8-Jun-16

• Analytical Solution
• – Continuous

  T T T T z L 

   

  1 T T T T

  2 T T i 2 t o N

  T

  1 A N B

  i 1 i i 1

   

  A

    B A

• Numerical solution

  

T

A T B Finite Diff Grids Soln

  

A

T B

  

T

A

T B T

  8-Jun-16 Finite volume method

  ∆ z

  Solution to Heat Conduction in Rod

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

         Typical Temperature Profile

  Finite volume method

8-Jun-16

  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 8-Jun-16 Integral Form of Conservation Equation  

     div  u  div  grad   S

  

   



   t For steady state flow div  u  div  grad   S

     

  

  For 1D flow d d d   

      u S    

   x

  dx dx dx   Gau

  β Integrating over CV d d d 

   

  u d    d   S d 

 

   x  

     d x d x d x  

        d d d 

 

  

  u S . d x   S . d x  S S .d x

 

   x

 

     d x d x d x

 

     x x x

  Finite volume method 8-Jun-16 Integral Form of Momentum Equation   u   uu   vu  p   u    u   

                 

    t x y x x x y y          

    Integrating over CV

  

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

        dt  dv  dV   dV   dV   dV  

         

  

 t  x  y  x  x  x  y  y

          _{t V V V V V} For steady state flow

  

          

 uu   vu  p u u

    dv  dV   dV   dV   dV    

             

x y x x x y y

     V  V  V  V  V  

  Finite volume method 8-Jun-16 FVM Approach for Heat Conduction Computational domain for • heat conduction in rod

  Temperature at node is – L average of CV around the

  T

T

  B node

  A

• – Boundary nodes

  Internal nodes – T

  

T

A

B

  Finite volume method 8-Jun-16

    x

  S dx d dx d u dx d

    

       

       S dV dV dx d dx d dV u dx d _{V x V V}

      

     

       

     P E W e w s n S N

  Finite Volume Formulation      

         

    S grad div u div t

      

       S grad div u div

8-Jun-16

  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 ...

          ₃ 2 !

  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 ! ₃

  2 ! _{'    "'} 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 8-Jun-16

  P E W e w s n S N Finite Volume Formulation . . .

       

  21 Central difference method

     

         

  W E P W P E

  2 u u 2 u u

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

  

  u u dV u dx d

    S dV dV dx d dx d dV u dx d

  V

  w e

          

     

     

  



  

  V V   

  V x

8-Jun-16

  Finite Volume Formulation . . .

   d d d    u dV   dV  S dV

     

   x

     dx dx dx  

   V  V 

  V N

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

   dV            dx dx dx dx      

  e w 

  V

                       

  E P P W

      x x

  EP PW

8-Jun-16

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

         s a a a

  E E W W P P 

            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   

     

     

     

    

  Finite volume method

8-Jun-16

      w e

  V p p dV x p  

    

   

       

  2 p p 2 p p

  2 p p W E W P P E

     

   

P E

W

e

w s n S N

  Difficulty in pressure term discretization Checker board Solution?

  Finite volume method Suhas V Patankar

  Professor Emeritus Univ. of Minnesota

8-Jun-16

  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

  φ e

  φ w

  ?

   Other alternatives

• – How to resolve pressure discretization ?

  Finite volume method

8-Jun-16

  Resources Chung T. J. (2002) Computational Fluid Dynamics. Cambridge University Press • Date A. W. (2005). Introduction to Computational Fluid Dynamics. Cambridge University Press • Fox, R. O. (2003) Computational Models for Turbulent Reacting Flows. Cambridge University • Press Hoffmann K. A. and Chiang S. T. (2000). Computational Fluid Dynamics Vol1, 2 and 3. • Engineering Education System, Kansas, USA.

• John F. W., Anderson, J.D. (1996) Computational Fluid Dynamics: An Introduction Springer

  Patankar, S. (1980) Numerical Heat Transfer and Fluid Flow. Taylor and Francis • Ranade, V.V. (2002). Computational Flow Modeling for Chemical Reactor Engineering, • Academic Press, New York. Versteeg, H.K. and Malalasekera, W. (1995) An Introduction to computational Fluid Dynamics •

• - The Finite Volume Method. Longman Scientific and Technical

  Finite volume method 8-Jun-16 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

8-Jun-16

  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

8-Jun-16

THANK YOU

  A person who never made a mistake never tried anything new

• Albert Einstein
• 1879 -1955

  Finite volume method 8-Jun-16 Concept of staggered grid

  Finite volume method

8-Jun-16

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

              t x y x x x y y

             

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

  

       

dt dv dV dV dV dV    

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

          _{t V V V V V}   uu   vu  p   u    u 

        dv dV dV dV dV

            

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

   ^{V  V  V  V  V  } Finite volume method 8-Jun-16

  U Control volume

  Finite volume method

8-Jun-16

     

    

  Finite volume method

  X Momentum Equation

         

         

  J , ^{J 1 i , 1 i J , J 1 i , i J , i J , 1 i J , 1 I J , I}     

  2 uu uu uu uu

  2 uu uu 2 uu uu

   

       

       

          

     

        _{w e V} uu uu dv x uu    

    

  dV y u y dV x u x dV x p dV y vu dv x uu _{V V V V V}

    

   

   

   

   

    

    

   

   

   

    

    

       _{    }

8-Jun-16

     

        s n

  Finite volume method

      X Momentum Equation . . .

     

     

     

  , 1 J i ^{, 1 J i J , i , 1 J i , 1 J i J , i , j i , 1 j i     }

  2 vu vu vu vu

  2 vu vu 2 vu vu

         

       

          

     

  V vu vu dV y vu    

    

  dV y u y dV x u x dV x p dV y vu dv x uu _{V V V V V}

    

    

   

   

   

   

    

    

   

   

   

    

    

       _{    }

8-Jun-16

     

  dV y u y dV x u x dV x p dV y vu dv x uu _{V V V V V}

  Finite volume method

     X Momentum Equation . . .

      

  I V p p dV x p

      J ,

    

    

    

   

   

   

   

    

    

   

   

   

    

    

       _{    }

1 I J ,

8-Jun-16

  X Momentum Equation . . .

    uu   vu  p   u    u 

       

  dv  dV   dV   dV   dV  

   

      

   x  y  x  x  x  y  y _{ V  V  V  V }   _{V  }

    u  u  u        dV          

    x  x  x  x _V     e   w

  

   u  u    

         

   x  x   ^{I , J   I  1 , J}

   u  u  u  u

       i  ^{1 , J i , J   i , J i  1 , J }

     

  2

  2  u  2 u  u

     i  ^{1 , J i , J i  1 , J }

   2 

  Finite volume method 8-Jun-16

     

     

    

   

     

     

         

     

      

     

        

      

     

        

  

  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

     

     

  dV y u y dV x u x dV x p dV y vu dv x uu _{V V V V V}

   

       _{    }

    

    

   

   

   

    

    

   

      

   

   

    

    

    

  s n ^V y u y u dV y u y

     

     

8-Jun-16

    J ,

  I J , i J , i p p au u a

  Finite volume method

  Final form

    

    

    

   

   

   

   

    

    

   

   

   

    

    

       _{    }

  dV y u y dV x u x dV x p dV y vu dv x uu _{V V V V V}

      

     