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OpenFOAM course, part 1: theoretical foundations offinite volume approachSibgatullin I.sibgat@ocean.ru,Moscow Lomonosov State UniversityMarch 10, 2016Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20161 / 20To work in CFD, one needs a solid background in both fluidmechanics and numerical analysis; significant errors have been madeby people lacking knowledge in one or the other.Estimation of numerical errors.

A qualitatively incorrect solution of aproblem may look reasonable (it may even be a good solution ofanother problem), the consequences of accepting it may be severe.Computational Methods for Fluid Dynamics.Professor Joel H. Ferziger, Dr. Milovan PerićIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20162 / 20“OpenFOAM is first and foremost a C++ library, used primarily to createexecutables, known as applications.”OpenFOAM User GuideIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20163 / 20Conservation principlesConservation laws can be derived by considering a given quantity of matteror control mass (CM) and its extensive properties, such as mass,momentum and energy.Z d+ sources + flows through boundariesφρ dΩ =dtzero 0ΩCMIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20164 / 20The conservation equation for mass:dm=0.dt(1)On the other hand, momentum can be changed by the action of forces andits conservation equation is Newton’s second law of motion:d(m~v ) X ~=F ,dt(2)where t stands for time, m for mass, ~v for the velocity, and ~f for forcesacting on the control mass.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20165 / 20CM approach is used to study the dynamics of solid bodies, where the CM(sometimes called the system) is easily identified.In fluid flows, however, it is difficult to follow a parcel of matter.

It is moreconvenient to deal with the flow within a certain spatial region we call acontrol volume (CV), rather than in a parcel of matter which quicklypasses through the region of interest. This method of analysis is called thecontrol volume approach.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20166 / 20Closer look to control volumeIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20167 / 20φ – conserved intensive property (for mass conservation, φ = 1; formomentum conservation, φ = ~v ; for conservation of a scalar, φ representsthe conserved property per unit mass) The corresponding extensiveproperty Φ can be expressed as:ZΦ=ρφ dΩ ,(3)ΩCMΩCM stands for volume occupied by the CMLHS of each conservation equation for a control volume can be written:ZZZddρφ dΩ =ρφ dΩ +ρφ (~v − ~vb ) · ~n dS ,(4)dtdtΩCMΩCVSCVΩCV is the CV volume,SCV is the surface enclosing CV,~n is the unit vector orthogonal to SCV and directed outwards,~vb is the velocity with which the CV surface is moving.For fixed CV ~vb = ~0, first derivative on the RHS becomes a local (partial).The last term is usually called the convective (or sometimes, advective)flux of φ through the CV boundary.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20168 / 20∂∂ = ∂t:If CV does not change in time vb = 0, ∂tCVZZZd∂ρφ dΩ =(ρφ) dΩ +ρφ (~v , ~n) dS =dt∂tΩCMΩCV(5)SCV(Homework: Consider “Differentiation under the integral sign”https://en.wikipedia.org/wiki/Differentiation under the integral signand describe its connection to differentiation over CM and to ReynoldsTransport Theorem.)Z ∂(ρφ) + div (ρφ~v ) dΩ(6)=∂tΩCVIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 20169 / 20Mass Conservation∂∂tφ→1Zρ dΩ + ρ~v · ~n dS = 0 .ZΩ(7)(8)SBy applying the Gauss-Ostrogradsky divergence theorem to the convectionterm, we can transform the surface integral into a volume integral.Allowing the control volume to become infinitesimally small leads to adifferential coordinate-free form of the continuity equation:∂ρ+ div (ρ~v ) = 0 .∂tIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(9)March 10, 201610 / 20Momentum Conservation∂∂tZΩφ → ~vZX~ .ρ~v dΩ + ρ~v (~v , ~n) dS =F(10)(11)STo express the right hand side in terms of intensive properties, one has toconsider the forces which may act on the fluid in a CV:body forces (gravity, centrifugal and Coriolis forces, electromagneticforces, etc.).surface forces (pressure, normal and shear stresses, surface tensionetc.);Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201611 / 20Momentum Conservation∂∂tZZρ~v (~v , ~n) dS =ρ~v dΩ +ΩZS~f ρ dΩ +ΩZ~σn dS .(12)S~f – body mass forces per unit of mass, σ~n – surface forces per unit of area.σ~n = ~σ i ni , ~σ i – surface forces per unit of area on i-th coordinate plane.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201612 / 20Momentum Conservation∂∂tZZρ~v (~v , ~n) dS =ρ~v dΩ +ΩZS~f ρ dΩ +ΩZ~σn dS .(12)S~f – body mass forces per unit of mass, σ~n – surface forces per unit of area.σ~n = ~σ i ni , ~σ i – surface forces per unit of area on i-th coordinate plane.ZZZZ∂i~ρ~v dΩ + ρ~v v ni dS =f ρ dΩ + ~σ i ni dS .(13)∂t ΩSΩSIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201612 / 20Momentum Conservation∂∂tZZρ~v (~v , ~n) dS =ρ~v dΩ +ΩZS~f ρ dΩ +ΩZ~σn dS .(12)S~f – body mass forces per unit of mass, σ~n – surface forces per unit of area.σ~n = ~σ i ni , ~σ i – surface forces per unit of area on i-th coordinate plane.ZZZZ∂i~ρ~v dΩ + ρ~v v ni dS =f ρ dΩ + ~σ i ni dS .(13)∂t ΩSΩSZ ΩZ ∂i~f ρ + ∇i ~σ i dΩ .(ρ~v ) + ∇i (ρ~v v ) dΩ =∂tΩIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 2016(14)12 / 20Momentum Conservation∂∂tZZρ~v (~v , ~n) dS =ρ~v dΩ +ΩZS~f ρ dΩ +ΩZ~σn dS .(12)S~f – body mass forces per unit of mass, σ~n – surface forces per unit of area.σ~n = ~σ i ni , ~σ i – surface forces per unit of area on i-th coordinate plane.ZZZZ∂i~ρ~v dΩ + ρ~v v ni dS =f ρ dΩ + ~σ i ni dS .(13)∂t ΩSΩSZ ΩZ ∂i~f ρ + ∇i ~σ i dΩ .(ρ~v ) + ∇i (ρ~v v ) dΩ =∂tΩ∂(ρ~v ) + ∇i (ρ~v v i ) = ~f ρ + ∇i ~σ i .∂tIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(14)(15)March 10, 201612 / 20Scalar ConservationX∂(ρφ) + ∇i (ρφv i ) =Fφ(16)∂tFφ – sources and fluxes through boundariesDiffusive transport is always present (even in stagnant fluids), and it isusually described by a gradient approximation, e.g.

Fourier’s law for heatdiffusion and Fick’s law for mass diffusion:Zdfφ =Γ grad φ · ~n dS ,(17)Swhere Γ is the diffusivity for the quantity φ.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201613 / 20Difference between solids and fluidsFluid is a substance that continually deforms (flows) under an appliedshear stress.Fluid is a substance whose molecular structure cannot resist any shearforce applied to it.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201616 / 20Difference between solids and fluidsFluid is a substance that continually deforms (flows) under an appliedshear stress.Fluid is a substance whose molecular structure cannot resist any shearforce applied to it.In ideal fluid:~σ~n = −p~n(19)Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201616 / 20Difference between solids and fluidsFluid is a substance that continually deforms (flows) under an appliedshear stress.Fluid is a substance whose molecular structure cannot resist any shearforce applied to it.In ideal fluid:~σ~n = −p~n(19)In viscous incompressible fluid: ~σ~n = ~σ i ni = σ ik ~ek niσ ik = −pg ik + 2µe ikIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(20)March 10, 201616 / 20Difference between solids and fluidsFluid is a substance that continually deforms (flows) under an appliedshear stress.Fluid is a substance whose molecular structure cannot resist any shearforce applied to it.In ideal fluid:~σ~n = −p~n(19)In viscous incompressible fluid: ~σ~n = ~σ i ni = σ ik ~ek niσ ik = −pg ik + 2µe ik(20)In viscous compressible fluid (second viscosity assumed to be 0):2ikσ = − p + µ div ~v g ik + 2µe ik3Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 2016(21)16 / 20Difference between solids and fluidsAn ideal elastic solid will deform under load and, once the load is removed,will return to its original state.

Some solids are plastic. These deformunder the action of a sufficient load and deformation continues as long asa load is applied, providing the material does not rupture. Deformationceases when the load is removed, but the plastic solid does not return toits original state.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201617 / 20Difference between solids and fluidsAn ideal elastic solid will deform under load and, once the load is removed,will return to its original state.

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