OpenFOAMslides-02 (1185932)

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Introduction to OpenFOAM, part 1: theoreticalfoundations of nite volume approachSibgatullin I.sibgat@ocean.ruMoscow Lomonosov State University19 ìàÿ 2016 ã.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.1 / 89To work in CFD, one needs a solid background in both uid mechanicsand numerical analysis; signicant errors have been made by peoplelacking 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 PericIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.2 / 89OpenFOAM is rst and foremost a C++ library, used primarily to createexecutables, known as applications.OpenFOAM User GuideIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.3 / 89Conservation principlesConservation laws can be derived by considering a given quantity of matterorcontrol mass(CM) and itsextensiveproperties, such as mass,momentum and energy.ddtZϕρ dΩ =zeroΩCMIlias Sibgatullin (Moscow University)0+ sources + ows through boundariesOpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.4 / 89The 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 ,dtwheretstands for time,mfor mass,~v(2)for the velocity, andf~ forforcesacting on the control mass.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.5 / 89CM approach is used to study the dynamics of solid bodies, where the CM(sometimes called thesystem)is easily identied.In uid ows, however, it is dicult to follow a parcel of matter.

It is moreconvenient to deal with the ow within a certain spatial region we call acontrol volume(CV), rather than in a parcel of matter which quickly passesthrough the region of interest. This method of analysis is called thevolume approach.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMcontrol19 ìàÿ 2016 ã.6 / 89Closer look to control volumeIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.7 / 89Reynolds transport theoremφ conserved intensive property (for mass conservation,conservation,φ = ~v ;for conservation of a scalar,φφ = 1;property per unit mass) The corresponding extensive propertyas:for momentumrepresents the conservedΦcan be expressedZΦ=ρφ dΩ ,(3)ΩCMΩCMstands for volume occupied by the CMLHS of each conservation equation for a control volume can be written:ddtZρφ dΩ =ΩCMddtZZρφ (~v − ~vb ) · ~n dS ,ρφ dΩ +ΩCV(4)SCVΩCV 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 xed CV ~vb = ~0, rst derivative on the RHS becomes a local (partial).The last term is usually called theconvective(or sometimes, advective) ux ofφthrough the CV boundary.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.8 / 89Reynolds transport theorem if CV is xed in spaceIf CV does not change in timeddtZZρφ dΩ =ΩCMΩCVvb = 0,∂ ∂t CV∂(ρφ) dΩ +∂t=Z∂∂t :ρφ (~v , ~n) dS =(5)SCV(Homework: Consider Dierentiation under the integral signhttps://en.wikipedia.org/wiki/Dierentiation_under_the_integral_signand describe its connection to dierentiation over CM and to ReynoldsTransport Theorem.)Z ∂(ρφ) + div (ρφ~v ) dΩ(6)=∂tΩCVIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.9 / 89Mass Conservation∂∂tZΩφ→1Zρ dΩ +ρ~v · ~n dS = 0 .(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 innitesimally small leads to adierential coordinate-free form of the continuity equation:∂ρ+ div (ρ~v ) = 0 .∂tIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(9)19 ìàÿ 2016 ã.10 / 89Scalar ConservationddtZZsφ source ofsϕ ρ dv −ϕρ dv =ΩCMZΩCV#„~qϕ · ds(10)∂ΩCV∂(ρφ) + ∇i (ρφv i ) = sφ − ∇k qϕk∂tϕ, ~q ux of ϕ through boundaries.(11)Diusive transport is always present (even in stagnant uids), and it isusually described by a gradient approximation, e.g.diusion andFick's lawFourier's lawfor heatfor mass diusion:Z~qϕ = −λ ∇ϕ ,Qϕ = −λ ∇ϕ · ~n dS ,(12)Swhereλis the diusivity for the quantityIlias Sibgatullin (Moscow University)ϕ.OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.11 / 89Momentum Conservationφ → ~v∂∂tZZρ~v dΩ +Ωρ~v (~v , ~n) dS =(13)XF~ .(14)STo express the right hand side in terms of intensive properties, one has toconsider the forces which may act on the uid 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 FVM19 ìàÿ 2016 ã.12 / 89Momentum Conservation∂∂tZZρ~v dΩ +ΩZρ~v (~v , ~n) dS =Sf~ρ dΩ +ΩZ~σn dS .(15)Sf~ 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 FVM19 ìàÿ 2016 ã.13 / 89Momentum Conservation∂∂tZZρ~v dΩ +ΩZρ~v (~v , ~n) dS =Sf~ρ dΩ +ΩZ~σn dS .(15)Sf~ 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~f ρ dΩ + ~σ i ni dS .ρ~v dΩ +ρ~v v ni dS =(16)∂t ΩSΩSIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.13 / 89Momentum Conservation∂∂tZZρ~v dΩ +ΩZρ~v (~v , ~n) dS =Sf~ρ dΩ +ΩZ~σn dS .(15)Sf~ 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~f ρ dΩ + ~σ i ni dS .ρ~v dΩ +ρ~v v ni dS =(16)∂t ΩSΩSZ ΩZ ∂i(ρ~v ) + ∇i (ρ~v v ) dΩ =f~ρ + ∇i~σ i dΩ .∂tΩIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.(17)13 / 89Momentum Conservation∂∂tZZρ~v dΩ +ΩZρ~v (~v , ~n) dS =Sf~ρ dΩ +ΩZ~σn dS .(15)Sf~ 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~f ρ dΩ + ~σ i ni dS .ρ~v dΩ +ρ~v v ni dS =(16)∂t ΩSΩSZ ΩZ ∂i(ρ~v ) + ∇i (ρ~v v ) dΩ =f~ρ + ∇i~σ i dΩ .∂tΩ∂(ρ~v ) + ∇i (ρ~v v i ) = f~ρ + ∇i~σ i .∂tIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(17)(18)19 ìàÿ 2016 ã.13 / 89Dierence between solids and uidsFluid is a substance that continually deforms (ows) under an applied shearstress.Fluid is a substance whose molecular structure cannot resist any shear forceapplied to it.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.16 / 89Dierence between solids and uidsFluid is a substance that continually deforms (ows) under an applied shearstress.Fluid is a substance whose molecular structure cannot resist any shear forceapplied to it.In ideal uid:~σ~n = −p~nIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(20)19 ìàÿ 2016 ã.16 / 89Dierence between solids and uidsFluid is a substance that continually deforms (ows) under an applied shearstress.Fluid is a substance whose molecular structure cannot resist any shear forceapplied to it.In ideal uid:~σ~n = −p~nIn viscous incompressible uid:(20)~σ~n = ~σ i ni = σ ik~ek niσ ik = −pg ik + 2µeikIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(21)19 ìàÿ 2016 ã.16 / 89Dierence between solids and uidsFluid is a substance that continually deforms (ows) under an applied shearstress.Fluid is a substance whose molecular structure cannot resist any shear forceapplied to it.In ideal uid:~σ~n = −p~nIn viscous incompressible uid:(20)~σ~n = ~σ i ni = σ ik~ek niσ ik = −pg ik + 2µeik(21)In viscous compressible uid (second viscosity assumed to be 0):σikIlias Sibgatullin (Moscow University)2= − p + µ div ~v g ik + 2µeik3OpenFOAM course 1: theory of FVM(22)19 ìàÿ 2016 ã.16 / 89Navier Stokes equationsConvective form:ρ∂~v+ v k ∇k~v∂t= −∇p + ∇k µ∇k~v +µ∇(∇ · v) + ρ~g3Conservative form:∂µ(ρv k ) + ∇i (ρv k v i ) = −∇p + µ∆~v + ρ~g + ∇(∇ · v)∂t3Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.17 / 89Dierence between solids and uidsAn ideal elastic solid will deform under load and, once the load is removed,will return to its original state.

Some solids are plastic. These deform underthe action of a sucient load and deformation continues as long as a loadis applied, providing the material does not rupture. Deformation ceaseswhen the load is removed, but the plastic solid does not return to itsoriginal state.Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM19 ìàÿ 2016 ã.18 / 89Dierence between solids and uidsAn ideal elastic solid will deform under load and, once the load is removed,will return to its original state. Some solids are plastic. These deform underthe action of a sucient load and deformation continues as long as a loadis applied, providing the material does not rupture.

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