NumOpOpenFOAM Tutorial Finite Volume Method, Dictionary Syntax and Implementation DetailsenFOAM_P01

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[302.044] Numerical Methodsin Fluid Dynamics

Introduction to FVMConservation Laws, Fluxes Approach,

and Discretization Strategies

Univ. Assist. MSc. Francesco Roman

[email protected]

December 11th, 2014
mailto:[email protected]:[email protected]


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ConservationLaws

DiscreteApproach

FiniteVolumeMethod1D FVM

2D FVMDiscretizationStrategy

Outline

1 Conservation Laws

2 Discrete Approach

3 Finite Volume Method1D FVM

2D FVMDiscretization Strategy

[302.044] Univ. Assist. MSc. Francesco Roman 2/21


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ConservationLaws

DiscreteApproach



IntegralDifferential ProblemsConservation Laws

Example ConvectionDiffusion EquationT,t + u T = ( T ) u = 0

T = transported quantity;u = advection velocity = diffusion coefficient.



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ConservationLaws

DiscreteApproach




Example ConvectionDiffusion EquationT,t + u T = ( T ) u = 0

T = transported quantity;u = advection velocity = diffusion coefficient.

To obtain the conservative form:

T,t + u T = ( T ) u = 0 (u T ) = u T + T u

T,t + (u T ) = ( T )



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ConservationLaws

DiscreteApproach




Example ConvectionDiffusion EquationT,t + (u T ) = ( T ) , x = I d , t 0



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DiscreteApproach





Being the PDE pointwise valid, it can be integrated overinnitesimal volumes V :

V T,t dV + V (u T )dV = V ( T )dV



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DiscreteApproach





Being the PDE pointwise valid, it can be integrated overinnitesimal volumes V :

V T,t dV + V (u T )dV = V ( T )dV Introducing the Reynolds theorem:

ddt V T dV = V T,t dV + V T u dV u = 0

ddt V

T dV +V

(u T )dV =V

( T )dV



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ConservationLaws

DiscreteApproach





Being the PDE pointwise valid, it can be integrated over

innitesimal volumes V ; Introducing the Reynolds theorem;



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ConservationLaws

DiscreteApproach





Being the PDE pointwise valid, it can be integrated over

innitesimal volumes V ; Introducing the Reynolds theorem;

Introducing the Greens divergence theorem:

V f dV =

S = V

f n dS

ddt V T dV + S n (u T )dS = S n ( T )dS



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ConservationLaws

DiscreteApproach




Example ConvectionDiffusion Equation



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ConservationLaws

DiscreteApproach




Example ConvectionDiffusion Equation Differential form:

T,t + u T = ( T ) u = 0



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ConservationLaws

DiscreteApproach





T,t + u T = ( T ) u = 0

Integral form:

ddt V T dV + S n (u T T )dS = 0



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ConservationLaws

DiscreteApproach





T,t + u T = ( T ) u = 0

Integral form:

ddt V T dV + S n (u T T )dS = 0

Generic conservation law:

ddt V UdV + S n F dS = 0



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ConservationLaws

DiscreteApproach




Generic Conservation Laws Generic conservation law:




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ConservationLaws

DiscreteApproach






Integrating the differential equation lowers the derivativeorder. Its importance is due to solutions which changeso rapidly in space that the spatial derivative does notexist (e.g. supersonic shock waves);



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Conservation

Laws

DiscreteApproach







Discontinuous functions do not have derivatives at thediscontinuity location, so the differential form is invalidthere. On the contrary, because of weaker constraints,the integral conservation law is still valid;



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Conservation

Laws

DiscreteApproach







Discontinuous functions do not have derivatives at thediscontinuity location, so the differential form is invalidthere. On the contrary, because of weaker constraints,the integral conservation law is still valid;

Reducing the order of the spatial derivative simplies thespecial treatment generally required for discontinuities.



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Conservation

Laws

DiscreteApproach







ff


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Conservation

Laws

DiscreteApproach




Generic Conservation Laws



The Navier-Stokes system, in its most general form, iswritten in terms of conservation laws:

V ,t dV + S u n dS = 0

V (u ),t dV + S (u )u n dS = V f dV S pn dS + S dS

V [(e + u 2

2 + )],t dV + S [(e + u 2

2 + )]u n dS = S q n dS S n u dS


l ff l bl


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Conservation

Laws

DiscreteApproach

FiniteVolumeMethod1D FVM2D FVMDiscretizationStrategy






I l Diff i l P bl


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Conservation

Laws

DiscreteApproach






Lower derivatives order;


I l Diff i l P bl


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Laws

DiscreteApproach







Weaker constraints for the representable solutions;


I t l Diff ti l P bl


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Conservation

Laws

DiscreteApproach








Simpler treatment for discontinuities;


I t g l Diff ti l P bl


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Conservation

Laws

DiscreteApproach









Natural treatment for Fluid Dynamics equations;


Integral Differential Problems


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Conservation

Laws

DiscreteApproach









Natural treatment for Fluid Dynamics equations; Intrinsic conservativity property for the discrete schemes;


Integral Differential Problems


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Conservation

Laws

DiscreteApproach









Natural treatment for Fluid Dynamics equations;

Intrinsic conservativity property for the discrete schemes;

Finite Volume Methods are preferred to Finite DifferenceMethods in solving problems whose solution is not sosmooth or has local discontinuities.


Conservation Laws


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Conservation

Laws

DiscreteApproach


Conservation LawsDiscrete Approach

Conservation Laws Discrete Approach




Conservation Laws


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Conservation

Laws

DiscreteApproach






Averaging the solution in each elementary volume:

U = 1V V UdV


Conservation Laws


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Conservation

Laws

DiscreteApproach






Averaging the solution in each elementary volume:U =

1V V UdV

Discretizing the ux integral:

S n F dS faces n

k F k


Conservation Laws


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Conservation

LawsDiscreteApproach






Averaging the solution in each elementary volume:U =

1V V UdV

Discretizing the ux integral:

S n F dS faces n

k F k

Conservation law discretized in space:

V dU dt

faces

n k F k


Finite Volume Method


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Conservation



Finite Volume MethodGeneral Approach


Starting point for each FVM:

V dU dt

faces

n k F k




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Conservation






V dU dt

faces

n k F k

At rst the domain is divided into computational cells wherethe shape of the cell average function U j is known;




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Conservation






V dU dt

faces

n k F k


The uxes F k are the unknowns of the FVM and they arecalculated in two steps:




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Conservation






V dU dt

faces

n k F k


The uxes F k are the unknowns of the FVM and they arecalculated in two steps: function reconstruction:

F k requires the calculation of the function values andeventually of their derivative values at cell edges;U is approximated with a polynomial whose coefficientsare determined recovering the cell averages over acertain number of cells.




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Conservation



General Approach



V dU dt

faces

n k F k

The uxes F k are the unknowns of the FVM and they arecalculated in two steps:




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Conservation



General Approach



V dU dt

faces

n k F k

The uxes F k are the unknowns of the FVM and they arecalculated in two steps: function reconstruction:

U =P

n =1

an n (x ) , {n } = P interpolation functions

V j + m UdV = U j + m V j + m , m = 0 , 1, 2,...,P 1{V j + m } = P cells surrounding the cell V j



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Conservation


FiniteVolumeMethod1D FVM2D FVM

DiscretizationStrategy

General Approach



V dU dt

faces

n k F k

The uxes F k are the unknowns of the FVM and they arecalculated in two steps: function reconstruction 1

U =P

n =1an n (x ) , {n } = P interpolation functions

P

n =1Am,n an = U j + m V j + m Am,n = V j + m n dV

1 Introducing suitable quadrature formulae in dependence on the

chosen polynomial rank, no quadrature errors are committed.[302.044] Univ. Assist. MSc. Francesco Roman 13/21


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ConservationLaws

DiscreteApproach



General Approach



V dU dt

faces

n k F k


The uxes F k are the unknowns of the FVM and they arecalculated in two steps: function reconstruction; uxes evaluation.


Finite Volume MethodG l A h


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ConservationLaws

DiscreteApproach



General Approach

Finite Volume Method 1D Case

The cell volume reduces to the width of the segment x i ;


Finite Volume MethodG l A h


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ConservationLaws

DiscreteApproach



General Approach


The cell volume reduces to the width of the segment x i ; The ux integrals reduce to evaluation of the term at thecell edges:

dU idt =

F i+1

/2 F

i 1

/2

x i



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ConservationLaws

DiscreteApproach



General Approach


n is a piecewise constant polynomial:

U i = a0




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ConservationLaws

DiscreteApproach



General Approach


n is a piecewise constant polynomial:

U i = a0

One unknown coefficient means a single constraint:

x i +1 / 2

x i 1 / 2a0 dx = U i x i a0 = U i



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ConservationLaws

DiscreteApproach



General Approach


n is a piecewise linear polynomial:

U i = a0 + a1 , = x x i

x j




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ConservationLaws

DiscreteApproach



General Approach



U i = a0 + a1 , = x x i

x j

Two unknown coefficient means a double constraint(symmetric stencil: x [x i 1 / 2 , x i +3 / 2 ] [ 1/ 2, 3/ 2]):

1 / 2

1 / 2 (a 0 + a1 )d = U i

3 / 21 / 2 (a 0 + a1 )d = x i +1 x i U i +1

a 0 = U ia 1 = U i +1 U i




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ConservationLaws

DiscreteApproach



pp



U i = a0 + a1 , = x x i

x j

Two unknown coefficients mean a double constraint(symmetric stencil: x [x i 1 / 2 , x i +3 / 2 ] [ 1/ 2, 3/ 2]);




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ConservationLaws

DiscreteApproach



pp



U i = a0 + a1 , = x x i

x j

Two unknown coefficients mean a double constraint(symmetric stencil: x [x i 1 / 2 , x i +3 / 2 ] [ 1/ 2, 3/ 2]);

Centred Scheme:

F i +1 / 2 =F (U i ) + F (U i +1 )

2




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ConservationLaws

DiscreteApproach



pp





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ConservationLaws

DiscreteApproach




n is a piecewise parabolic polynomial:

U i = a0 + a1 + a2 2 , = x x i

x j




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ConservationLaws

DiscreteApproach





U i = a0 + a1 + a2 2 , = x x i

x j

Three unknown coefficients mean a triple constraint;




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ConservationLaws

DiscreteApproach





U i = a0 + a1 + a2 2 , = x x i

x j

Three unknown coefficients mean a triple constraint;

QUICK Scheme:

F i +1 / 2 = F (U i 1 ) + 6 F (U i ) + 3 F (U i +1 )

8




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ConservationLaws

DiscreteApproach




The cell volume reduces to the width of the area of theelement;




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ConservationLaws

DiscreteApproach





The ux integrals reduce to evaluation of the term at thecell edges:




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ConservationLaws

DiscreteApproach






3 uxes for triangular elements; 4 uxes for quadrangular elements; 5 uxes for pentangular elements; ...




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ConservationLaws

DiscreteApproach






3 uxes for triangular elements; 4 uxes for quadrangular elements; 5 uxes for pentangular elements; ...

V i d U i

dt

faces

n ik F ik


Finite Volume MethodDiscretization Strategy


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ConservationLaws

DiscreteApproach



Finite Volume Method Discretization Strategy

The equations will be considered in their conservative form;


Finite Volume MethodDiscretization Strategy


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ConservationLaws

DiscreteApproach



Finite Volume Method Discretization Strategy

The equations will be considered in their conservative form;

An integral approach is a natural representation for theNavierStokes system and it is the most suitable one forFinite Volume Methods;



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NumOpOpenFOAM Tutorial Finite Volume Method, Dictionary Syntax and Implementation DetailsenFOAM_P01