University of LjubljanaFaculty of Mathematics and Physics

Seminar - 4. letnik

Finite volume method

Avtor: Janez Turk

Mentor: doc. dr. Daniel Svensek

Ljubljana, november 2011

Abstract

Finite volume method is numerical method for representing partial differential equations in formof algebraic equations. ”Finite volume” refers to the small volume surrounding each node point on amesh. In the finite volume method integral form of partial differential equation is evaluated. Termscontaining a divergence or gradient operator are converted to surface integrals, using the divergencetheorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Becausethe flux entering a given volume is identical to that leaving the adjacent volume, these methods areconservative. Method is easily formulated to allow for unstructured meshes. The method is used inmany computational fluid dynamics packages.

Contents

1 Introduction 3

2 Other methods 3

3 Motivation 4

4 Discretization of spatial domain 5

5 Discretization of differential equations 55.1 Discretization of dependent variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.2 Discretization of divergence operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.3 Dealing with the convection term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

6 Convection Differencing Schemes 76.1 Central difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.2 Upwind scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.3 Exact solution - exponential scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.4 Power series approximation for exact solution . . . . . . . . . . . . . . . . . . . . . . . . . 8

7 Staggered grid 9

8 Boundary conditions 108.1 Fixed value boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108.2 Fixed gradient boundary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

9 Calculating flow field 11

10 Example of use - ventilation of African school 1210.1 Usage of finite volume computational package . . . . . . . . . . . . . . . . . . . . . . . . . 12

10.1.1 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1310.1.2 Profile of flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410.1.3 Profile of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

10.2 Consideration of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510.2.1 Equations being solved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

11 Conclusions 15

2

1 Introduction

Finite volume method method is based on discretization of the integral form of governing equationsover each control volume. The basic quantities, such as mass and momentum, will therefore be conservedat the discrete level. The control volumes can be of a general polyhedral shape, with a variable numberof neighbors, thus creating an arbitrarily unstructured mesh. All dependent variables share the samecontrol volumes, which is usually called the collocated variable arrangement.

Figure 1: Simulation of passanger cabin ventilation

2 Other methods

It is crucial at this point to acknowledge other numerical methods for solving partial differential equationswith similar names:

Finite difference method is the most obviousmethod for solving PDEs. It is so because it is de-rived directly from calculus. Only difference betweenFDM and real calculus is finite length of infinitesi-mal length dx. Method suffices for solving most ofwell behaved equations on structured grids.

Figure 2: Finite difference discretization

Finite element method is subset of method ofweighted residuals. Along FEM the most famousmethod based on weighted residuals is well-knownGalerkin method. Both are quite similar with onemajor difference. Galerkin method uses trial func-tions stretching along whole computational domainwhich satisfies boundary conditions. FEM methoduses umbrella shaped trial functions that differ fromzero just at mesh point and surrounding cell.

Figure 3: Finite element discretization

3

3 Motivation

Wishlist of properties of numerical method:

• Separation of geometry and solver is very desirable. Computational package comes withseveral solvers for different use i.e. (heat conduction, incompressible fluid flow, turbulent flow...).User presents the computational package with geometry in suitable format and the computationcan begin. However there the separation is not complete. Boundary conditions have to be definedon geometry based on the equations in the solver.

• Physicaly realistic solution even on coarse grids leads to managable problem. For robustnumerical method there have to be as little stability criterion as possible.

• Use of unstructured mesh is the most important property of desirable method. Usage ofunstructured mesh is free from binding computational domain to coordinate axes. For example:Consider fluid flow around sphere in structured grid against unstructured mesh, both having thesame number of discretization points, as depicted in figure 3.

Figure 4: structured cartesian mesh and unstructured mesh

Finite difference method, does not offer none of properties described above. Finite element methodsuffers with conservation of physical quantities, although it fullfils first and third request. Finite volumemethod on the other hand fullfils all of the above defined behaviour. Even more it offers:

• Solution with moving boundary togetherwith conservational property offers strong toolin calculation of fluid flows in machines suchas turbines and combustion engines. Due togeometry’s independance to equations beingsolved, the full solution is obtainable. Chemi-cal, fluid flow and thermal fields of geometry,can be calculated simultaneously.

Figure 5: Simulation of combustion motor with mov-ing boundaries.

4

4 Discretization of spatial domain

Discretization of the solution domain produces a computational mesh on which the governing equationsare subsequently solved. It also determines the positions of points in space and time where the solutionis sought.

Figure 6: Typical control volume.

Control volumes do not overlap and completely fill the computational domain. A typical controlvolume is shown in picture above. The computational point P is located at the centroid of the controlvolumes. This statement can be written in mathematical manner as:

∫(x− xP )dV = 0. (1)

The control volume is bounded by a set of flat faces and each face is shared with only one neighboringCV.

Since time is a parabolic coordinate the solution is obtained by marching in time from the prescribedinitial condition. For the discretization of time, it is therefore sufficient to prescribe the size of thetime-step that will be used during the calculation.

5 Discretization of differential equations

Every differential equation must be discretized in order to be solved on the discrete grid. A genericexample of fairly complicated equation would be transport equation for some physical quantity denotedby φ. It can be always imagined as temperature or concentration.

∂φ

∂t︸︷︷︸temporal derivative

+ ∇ · vφ︸ ︷︷ ︸convection term

− ∇ ·D∇φ︸ ︷︷ ︸diffusion term

= Sφ(φ) (2)

The accuracy of the discretization method depends on the assumed variation of the function φ = φ(x)in space and time around the point P . In order to obtain a second-order accurate method, this variationmust be linear in both space and time, therefore it is assumed that:

φ(x) = φP + (x− xP ) · (∇φ)P +O(x2) (3)

5

This is also the first order of accuracy that includes variation of quantity φ inside the control volumeitself.

The Finite Volume method requires that transport equation is satisfied over the control volume VParound the point P . This is the integral form:

∂

∂t

∫VP

φ dV +

∫VP

∇ · vφ dV −∫VP

∇ ·D∇φ dV −∫VP

Sφ(φ) dV = 0 (4)

Equations spatial terms will be discretized using these identities followed from Gauss theorem:∫V

∇ · a dV =

∮∂V

a · dS (5)∫V

∇φ dV =

∮∂V

φ dS (6)∫V

∇a dV =

∮∂V

a dS (7)

5.1 Discretization of dependent variable

Discretization of dependent variable is nothing more than calculating its integral over one control volume:

∫VP

φ dV =

∫VP

[φP + (x− xP ) · (∇φ)P ] dV = φP

∫VP

dV +

[∫VP

(x− xP )dV

]· (∇φ)P = φpVp (8)

5.2 Discretization of divergence operator

Control volume is bounded by a series of flat faces, Gauss theorem can be transformed into a sum ofintegrals over all faces: ∫

VP

∇ · a dV =

∮∂V

a · dS =∑f

∫f

a · dS (9)

Taking into account linear variation of ai a second-order accurate discretized form of the Gauss’theorem is obtained:

∇ · a VP =∑f

S · af (10)

Here, the subscript f implies the value of the variable (in this case, a) in the middle of the face andS is the outward-pointing face area vector.

5.3 Dealing with the convection term

In convection term fluid velocity v and dependent variable φ are multiplied and struck with divergence.According to previous section we can write:∫

VP

∇ · vφ dV =∑f

S · vfφf (11)

In the last term, velocity vf and dependent variable φf need to be evaluated on the face of controlvolume. For now it is assumed that velocity is evaluated in points situated in the middle of face, laterit will be shown this is true and why. But dependent variable is evaluated only in centroids of controlvolumes. Values on faces must be therefore obtained with interpolation. Methods for interpolation arecalled convection differencing schemes.

6

6 Convection Differencing Schemes

The role of the convection differencing scheme is to determine the value of φ on the faces, denoted φf ,from the values in the cell centers φp and neighbor cell centers φN . In the framework of arbitrarilyunstructured meshes, it would be impractical to use any values other than φP and φN , because ofthe storage overhead associated with the additional addressing information. We shall therefore limitourselves to differencing schemes using only the nearest neighbors of the control volume.

6.1 Central difference scheme

Assuming the linear variation of φ between P and N , the face value is calculated according to:

φf = fxφP + (1− fx)φN (12)

Figure 7: Linear interpolation between adjacent cell centres.

Here, the interpolation factor fx is defined as the ratio of distances fN and PN :The differencing scheme using linear interpolation to determine the face value of φ is called Central

Differencing

6.2 Upwind scheme

Upwind difference scheme is rather of physical origin. It states, that if the velocity if big enough, thedownstream points do not affect upstream points. This scheme therefore assigns upstream value of φ toφf .

φf =

{φN for Pe > 0

φP for Pe < 0(13)

Figure 8: Upwind interpolation between adjacent cell centres.

Dimensionless quantity Pe is called Peclet number and represents ratio of velocity and diffusion. Itis given by expression: Pe = ρuL

D

7

6.3 Exact solution - exponential scheme

It has been shown that central differencing scheme is at least of second-order accuracy. Nevertheless itis utmost unstable when being dealt with high convection velocity in comparison to diffusion coefficient.Exact solution over small interval can be acquired.

the governing equation of convection problem overinterval (P,N) is:

∇ · vφ−D∇ · ∇φ = 0

With additional conditions:

φ(P ) = φPφ(N) = φN∇ · v = 0

φ′′ − Peφ′ = 0 (14)

Solving this differential equation in one dimension for x solution of φ’s profile between points P andN is obtained.

φ(x) =−φN + (φN − φP )ePex + edPeφP

−1 + edPe(15)

Figure 9: Exact solution of φ’s profile.

6.4 Power series approximation for exact solution

Exact solution is clearly the best way to interpolate face value of dependent variable. But it is rathertime demanding because calculation of exponent is time demanding for computer. In real finite volumecalculation power series approximation for exact solution are used, which reduces computational difficulty.

8

7 Staggered grid

Insight into physical partial differential equations reveals that velocity acts as cause for convection. Theconvective term of equations would always benefit from velocity being defined on the faces of controlvolume.

Figure 10: Velocity is defined on cell’s surfaces.

Continuity equation is also easily expressed with velocities defined on faces of control volume.

∂ρ

∂t= −∇ · ρv (16)

where ρ is fluid density, t is time, and v is the flow velocity vector field. If density ρ is a constant,as in the case of incompressible flow, the mass continuity equation simplifies to a volume continuityequation

∇ · v = 0 (17)

Latter equation is very simple in case of finite volume discretization method.∑f

S · vf = 0 (18)

Momentum equation for fluids expresses temporal acceleration of fluid on mesh in terms of pressuregradient.

∂v

∂t+ (v · ∇)v = η∇2v −∇ p (19)

Pressure is dependent variable with its value stored in the centres of control volumes. Pressuregradient is evaluated as difference between two adjacent control volume centers. Assigning pressuregradient to the middle of calculation interval is very physical thing to do.

Figure 11: Location of pressure and velocity points.

9

8 Boundary conditions

Boundary conditions denote behavior of solution on borders of domain. Boundary consists of faces.

Figure 12: Scheme of boundary cell.

Vector d is vector drawn from centroid of control volume P to the middle face b. dn is part of thevector d that is orthogonal to boundary face, and simultaneously parallel to face normal S

8.1 Fixed value boundary condition

The fixed value boundary condition prescribes the value of φ at the face b to φb. This has to be takeninto account during discretization on the boundary face.

Convection term∫VP∇ · vφ dV =

∑f S · vfφf

Value of φ on the boundary face is set to φb. There-fore, the term for the boundary face is:

Svbφb

Diffusion term∫VP∇ ·D∇φ dV =

∑f S · ∇φf

The face gradient at b is calculated from the knownface value and the cell center value

S · ∇φb = |S|φb−φP

|dn|

8.2 Fixed gradient boundary condition

In the case of the fixed gradient boundary condition, the dot-product of the gradient and the outwardpointing unit normal is prescribed on the boundary.

S

|S|· ∇φb = gb (20)

Convection termThe face value of φ is calculated from the value inthe cell center and the prescribed gradient.

φb = φP + |dn|gb

Diffusion termThe face gradient at b is calculated from the knownface value and the cell centre value

S · ∇φb = Sgb

10

9 Calculating flow field

Calculating values of dependent variable with given fluid flow is easy. Quite more complicated situationoccurs when it is necessary to calculate propagation of flow through time and space. There are severalprocedures and algorithms that ensure the correct solution such as PISO and SIMPLE algorithm. Theyare quite different but based on the same principle described in this section

For incompressible flows there are two governing equations. Navier-Stokes equation for momentumand continuity equation which takes care of flow’s incompressibility.

ρ

[∂v

∂t+ v · ∇v

]− η∇ · ∇v = −∇p (21)

∇ · v = 0 (22)

Calculation is being started with initial conditions, which do satisfy continuity equation. Left handside of equation is discretized. All operators are expressed as algebraic relations between mesh points.Navier-Stokes equation is being represented as a system of linear equations.

Au = b (23)

u is assembly of mesh points with stored values of v, this is not a vector in physical sense .A is matrixof algebraic relations between elements of vector u. And b is assembly of mesh points with stored valuesof −∇p or/and any explicit operator such as linearized v ·∇v term . System is solved with linear algebraroutines directly or iteratively.

Calculated values of velocity in new time step v∗t+1 are not just the correct ones (hence the ∗). Thereis no guaranty that they fulfill the continuity equation, and they actually do not. It is necessary tocorrect the velocity with that kind of pressure that would inflict obedience to continuity equation.

vt+1 = v∗t+1 −δt

ρ∇pcorrection (24)

Divergence operator applied on equation would eliminate left side of equation, because real v obeyscontinuity equation and it therefore zero.

∇ · vt+1 = 0 = ∇ · v∗t+1 −δt

ρ∇ · ∇pcorrection (25)

Finally the final equation for correction pressure is obtained. It is in form of Poisson equation

∇2pcorrection =ρ

δt∇ · v∗t+1 (26)

Correct pressure is obtained by adding guessed pressure and correction:

p = p∗ + pcorrection (27)

Correct velocity is get by equation (24)

11

10 Example of use - ventilation of African school

Faculty for architecture of University of Ljubljana has built school for children of South-african republic.Materials intended for the built were never used before by constructors. They have designed uniqueshape of windows that would presumably ventilate room better than windows used in Europe.

Figure 13: Crossection of the window.

• School is geographically located 45◦S. Al-though air is warm during the day, it is coldduring the night. Design of the school at-tempts to heat the room with the warm morn-ing air. Ventilation is considered the mainmechanism for heating as only other option isfor economical reasons open fire.

• Windows were designed with double glasseswith possibility of opening from both sides.In the morning outer windows would all beopened. Top inner window would be opened.With aid of dark metal between the windowsthe ”chimney” effect would push hot outsideair into the room.

Question of architects was: ’Would thermal airflow be sufficient to change the air in one hour beforeclass starts?’

10.1 Usage of finite volume computational package

This was problem solvable by finite volume computation package OpenFoam. Its solver BouyantPisoFoamsolves equations for thermal buoyancy, and Navier-Stokes equation for fluid flow. Furthermore it modelsturbulent flow according to method RAS.

Figure 14: Logo of openfoam software.

12

10.1.1 Mesh generation

From the real plans, geometry had to be put into computer and then discretized in order to obtainmeshed control volumes that finite volume computation package can read and calculate with.

Figure 15: Rendered wiev of plans for school.

Mesh was built with OpenFoam’s own meshing algorithm. Geometry of the project was rectangular,therefore all of the cells were rectangular. For the sake of simplicity and symmetry only one crossectionof the whole room was taken for the calculation. Boundary conditions for fluid velocity and temperaturehad to be put into mesh.

Figure 16: Computational simplification of school’s inside.

• Temperature of red patches (representing windows) was set to ambient temperature of outside. Airflow from outdoors was allowed with Inlet boundary condition.

• Temperature of black patches (representing metal between windows) was set to educated guessmade from predicted heating from the sun.

• Blue patch represents inner window and does not have any special boundary condition.

• On the side walls zero gradient boundary condition was used in order to mimic periodicity ofproblem.

13

10.1.2 Profile of flow

After completed calculation of air flow and temperature evolution during heating time, representativeair flow was taken and streamlines were calculated with ordinary integration.

Figure 17: Airflow around the school’s inside.

10.1.3 Profile of temperature

The main part of solution in which the architects were interested in, was mean temperature of air inthe classroom during time of heating. Simple integration was carried out and temperature profile wasobtained.

Figure 18: Air temperature inside school in time.

14

10.2 Consideration of solution

Building is made from 7 identical blocks. Only the first and last block have walls on the side. For thesake of simplicity and computational const, the approximation was made. Computation was run only forthe center block. Center block borders to neighbouring blocks with air, which can pass through entirelenght of the building unobstructed.

The idea of approximation was to impose periodic boundary condition to the side borders, as thiswould imply infinite number of blocks constituting the building. This is a justified approximation becauselength of the building is in fact many times greater as width or height. Another justification arrives whenno significant side air velocity is observed when solution is inspected .

10.2.1 Equations being solved

Navier-Stokes equation is for the transport of momentum with sources and sinks. The term R arisesfrom turbulence modeling with RAS method.

∂(ρv)

∂t+∇ · (ρvv) +∇ · (ρR) = −∇p+ ρg (28)

Conservation of mass provides a relation between the local rate of change of density and the degreeto which the local mass flux is acting as a source or sink.

∂ρ

∂t+∇ · v = 0 (29)

Temperature equation is in well known form, except for the κeff term. This term is turbulenceaveraged thermal conductivity.

T

∂t+∇ · (vT )−∇2κeffT = 0 (30)

Equation for the term R is given as:

R =2

3Ik − 2

µtρ

[1

2

(∇v + (∇v)T

)− 1

3tr (∇v) I

](31)

But all this is not necessary to understand in order to use the model. All that user is required toknow is basic nature of problem.

11 Conclusions

In conclusion I would like once again to point out the obvious benefits of finite volume method overother numerical methods for solving partial differential equations. Finite volume method is manifestedin ready-to-use programming packages, which are easy to use. That way users can focus on physics ofproblems, not the computational and numerical complications.

15

References

[1] Suhas V. Patankar: Numerical Heat Transfer and Fluid Flow.Hemisphere Publishing Corporation, 1981, ISBN 0-07-048740-5

[2] Hrvoje Jasak: Error Analysis and Estimation for the Finite Volume Method with Applications toFluid Flows. PhD thesis (1996),Imperial College, University of London.

[3] Janez Turk: Pretok zraka v afriski soli, elaborat za gradnjo afriske sole (2010). Fakulteta za Arhitek-turo Univerze v Ljubljani.

16