$$$$ NAVIER NOTICE CHAT 11/09/12 21:17:21 7124 DATE 11/09/12 Voir aussi : | Navier_Stokes : Icompressible Flows | ----------------------------------- | | I Physical models | _________________ | | | The Navier_Stokes model allows to compute incompressible or near | incompressible flows. | | The flow regimes can be either transient or steady state, laminar | or turbulent, forced free or mixed convection. | | | Near incompressible flows : | | Boussinesq approximation | low Mach number approximation | | Turbulence models : | | K - Epsilon model | RNG K - Epsilon model | | Boundary conditions : | | velocity, temperature .. prescribed | total stresses (velocity + pressure) prescribed | heat flux or mass flux imposed | heat exchange | wall function for momentum, temperature or mass | | Body forces / sources or sinks | | Two fluids models | Transport of particles (aerosols or droplets) | | | II Numerical Methods | ____________________ | | | II.1 Spacial discretization | --------------------------- | | The spatial discretization is obtained by a multidimensional finite | element method, 2D (in cartesian or cylindrical cordinates) or 3D. | | One will reffers to the following elements family : | | LINE : SEG2 TRI3 QUA4 CUB8 PRI6 TET4 PYR5 | QUAD : SEG3 TRI6 QUA8 CU20 PR15 TE10 PY13 (Structural analysis) | QUAF : SEG3 TRI7 QUA9 CU27 PR21 TE15 PY19 (CFD) | MACRO : SEG3 TRI6 QUA9 CU27 PR18 TE10 (CFD) | | Only few special elements are working for the mixed velocity-pressure | formulation. | One can distinguish two classes of elements. Continuous pressure | elements and discontinuous one. | In Castem 2000 only discontinuous pressure elements are available | | | A - Quadratic elements QUAF | | P2+bulle - P1 nc (nc : non conforming) et Q2 - P1 nc | (Crouzeix-Raviart) [1] (Bercovier-Pironneau) [2] | and parent elements in 3D | | Ref : | [1] M. Crouzeix and P.A. Raviart : Conforming and non conforming | finite element methods for solving the stationary Stokes equations.| R.A.I.R.O. (7eme annee,decembre 1973,R-3,p.33a76) | | [2] M. Bercovier and O. Pironneau : Error estimates for finite element | solution of the Stkes problem in the primitive variables. | Numer. Math. 33,p.211-224, 1979. | | The convergence rates are : | --------------------------- | for the velocity O(h**3) | for the pressure O(h**2) | h beeing a mesure of the element. | | Practical aspects | ----------------- | | 1/ Construct a mesh composed of LINE or QUAD elemnts. | | 2/ Turn the elements of the mesh into QUAF elements. see CHAN operator | | 3/ Eliminate points which eventualy has been created twice. | see ELIM operator | | 4/ Create the MMODELs 'NAVIER_STOKES' associated to the meshes and | precise QUAF for the type of finite elements. | sea MODE operator | $MT = MODE MT 'NAVIER_STOKES' QUAF ; | | 5/ Create velocity and pressure fields. | The velocity being a CHPOINT VECT SOMMET | The pressure being a CHPOINT SCAL CENTREP1 | sea KCHT operator | UN = KCHT $MT VECT SOMMET (0 0 0) ; | PN = KCHT $MT SCAL CENTREP1 0 ; | | | | B - MACRO linear elements iso P2 - iso P1 nc [1],[2] | iso Q2 - iso P1 nc | | They are macro-elements divided into 4 linear elements in 2D | (8 in 3D) for the velocity | the pressure is 3xP0 in 2D 4xP0 in 3D Cf figure underneath | | u | /\ u-----u-----u | /p \ | | | | u/____\u | p | | | /\ /\ u-----u--p--u | /p \ /p \ | | | | /____\/____\ | p | | u u u u-----u-----u | | Ref : | [1] J. Boland and R.A. Nicolaides: Stability of finite elements under| divergence constraints, SIAM Jour. Numer. Analy. 20,722-730 1983 | | [2] D. Gunzburger. | Finite Element Methods for Viscous Incompressible Flows | A Guide to Theory, Practice, and Algorithms. ACADEMIC PRESS 1989 | pages 28-31 | | | The convergence rates are : | --------------------------- | for the velocity O((h1/2)**2) | for the pressure O(h) | h being a mesure of the macro-element. | h1/2 being a mesure of the linear sub element. | | Practical aspects | ----------------- | | 1/ Construct a mesh composed of LINE or QUAD elemnts. | | 2/ Turn the elements of the mesh into QUAF elements. see CHAN operator | | 3/ Eliminate points which eventualy has been created twice. | see ELIM operator | | 4/ Create the MMODELs 'NAVIER_STOKES' associated to the meshes and | precise MACRO for the type of finite elements. | sea MODE operator | $MT = MODE MT 'NAVIER_STOKES' MACRO ; | | 5/ Create velocity and pressure fields. | The velocity being a CHPOINT VECT SOMMET | The pressure being a CHPOINT SCAL CENTREP1 | sea KCHT operator | UN = KCHT $MT VECT SOMMET (0 0 0) ; | PN = KCHT $MT SCAL CENTREP1 0 ; | | | | C - Linear elements LINE Q1 - P0 [1] | | This elements are not stable. Moreover a mesh composed uniquely of | P1 - P0 triangles can lead to a blockage. The unique solution Uh | compatible with Div Uh = 0 being Uh = 0 . This element family is not | advisable. | The numerical method used in CASTEM 2000 which stabilises the elements| consists to gather the linear elements into a MACRO element. In fact | the mesh of MACRO elements is first constructed as described above. | The stabilisation is obtained by modifying the continuity equation | (Cf Brezzi, Silvester Kechkar [2]). The stabilisation at the MACRO | element level preserves the main interests of discontinuous pressure, | among them the exact mass conservation (at least) at the MACRO element | level [3]. However it appears an additional parameter Beta which is not| always easy to choose. For this reason we advise against this method | too. | | Ref : | [1] P.M. Gresho, S.T. Chan, R.L. Lee,and C.D. Upson. | A Modified Finite Element Method for Solving the Time-dependent | Incompressible Navier-Stokes Equations. Part 1: Theory. | Int. J. Num. Meth. Fluids, 4:557-598,1984. | | [2] N. Kechkar and D. Silvester: Analysis of Locally Stabilized Mixed| Finite Element Methods for the Stokes Problem. | Mathematics of Computation,58(197):1-10,1992. | | [3] H. Paillere and J.P. Magnaud : A finite Element Flow Solver For | Low Mach Number Compressible Flows. | Proc. 10th Int. Conf. on Finite Elements in Fluids, | January 5-8 1998, Tucson, Arizona. | | | | | II.2 Algorithms | --------------- | | Steady states solution using Velocity-pressure unknowns | | Transients Implicit scheme | | Transients semi explicit | implicit for the pressure | explicit for all other unknowns | | | | | A - Navier_Stokes : Isothermal steady convection | | | ro U Grad U = mu Lapl U - Grad P | | Div U = 0 | | | RV = EQEX | OPTI EF IMPL | ZONE $MT OPER NS (mu/ro) INCO UN | ZONE $MT OPER KCCT 1 (-1.) INCO UN PN | ; | | | | B - Navier_Stokes : Isothermal transient convection | | | ro dU/dt + ro U Grad U = mu Lapl U - Grad P | | Div U = 0 | | ** Implicit large time steps Euler (first order in time) | | RV = EQEX $MT | OPTI EF IMPL | ZONE $MT OPER NS (mu/ro) INCO UN | ZONE $MT OPER KCCT 1 (-1.) INCO UN PN | ZONE $MT OPER DFDT ro 'UN' dt INCO UN | ; | | ** Implicit large time steps Crank Nicholson (second order in time) | | RV = EQEX $MT | OPTI EF SEMI 0.5 | ZONE $MT OPER NS (mu/ro) INCO UN | ZONE $MT OPER KCCT 1 (-1.) INCO UN PN | ZONE $MT OPER DFDT ro 'UN' dt INCO UN | ; | | ** Semi implicit small time steps Euler (first order in time) | | RV = EQEX $MT | OPTI EFM1 EXPL | ZONE $MT OPER NS (mu/ro) INCO UN | ZONE $MT OPER DFDT ro 'UN' dt INCO UN | ; | | RVP = EQPR $MT | ZONE $MT OPER PRESSION 0. | ; | | RV.'PRESSION' = RVP ; | | | | II.3 Convection schemes | ----------------------- | | The following schemes are available : | | -------------------------------------------------------- | scheme |order in |algorithm | | | space | | | -------------------------------------|-----------------| | elements |MACRO QUAF | | | |LINE | | | -------------------------------------|-----------------| | Galerkin | 2 3 | impl/expl 2D/3D | | upwind Hughes-Brooks | 2 3 | impl/expl 2D/3D | | upwind SUPG | 2 3 | impl/expl 2D/3D | | upwind SUPGCC [1] | 2 3 | impl/expl 2D/3D | | upwind PSI [2] | 2 3 | explicit 2D | | -------------------------------------------------------- | | Ref : | [1] T.J.R. Hughes, M. Mallet and A. Mizukami. | A New Finite Element Formulation for Computational Fluid Dynamics| II. Beyond SUPG | | [2] H. Paillere. | Multidimensional Upwind Residual Distribution Schemes for the | Euler and Navier-Stokes Equations on Unstructured Grids | These Von Karman Institute June 1995 | | sea burger*.dgibi | | | | II.4 Time discretizations | ------------------------- | | The time discretization schemes available are : | | ------------------------------------------------------- | scheme |order in |algorithm | | |time | | | ------------------------------------------------------| | elements | | | | | |LINE | | ------------------------------------------------------| | Euler | 1 | impl/expl 2D/3D | EF/EFM1 | Balancing viscous tensor | 2 | impl/expl 2D/3D | | Crank Nicholson | 2 | implicit 2D/3D | | generalized Crank Nicholson 4 | implicit 2D/3D | | ------------------------------------------------------- | | sea cone.DGIBI | | | | III List of useful operators needed for a Navier_Stokes computation | ___________________________________________________________________ | | | ------------------------------------------------------------------------| | OPTION : General options for calculations | ------------------------------------------------------------------------| | ET : Enables the gathering of the properties described by | | region: meshes, fields, stiffnesses... | ------------------------------------------------------------------------| | MODELE : Definition of Navier_Stokes model | ------------------------------------------------------------------------| | KCHT : Definition des CHPOINTs pour decrire les proprietes | | : physiques les champs etc | ------------------------------------------------------------------------| | EQEX : enables to create the TABLE for the solution computation | ------------------------------------------------------------------------| | EXEC :The EXEC procedure makes it possible to chain the operations| | : in the context of a Navier_Stokes | ------------------------------------------------------------------------| | : Dicretization operators | | : Transport : convection/diffusion | | DFDT : Time derivative for a scalar | | LAPN : Laplace operator for a scalar (diffusion) | | KONV : Convection | | TSCA : Diffusion/convection/source | | FIMP : Heat (or mass) flux condition | | ECHI : Heat exchange condition | | : Navier Sokes | | DFDT : Time derivative for a vectot | | NS : Momentum equation Diffusion/convection/source | | LAPN : Laplace operator for a vector (viscous term) | | KONV : Convection | | DUDW : Penalisation operator for Div U = 0 | | KMAC : | | KMCT : | | KCCT : | | TOIM : | | : Turbulence | | NSKE : Momentum equation Diffusion/convection/source + | | : K-Epsilon model | | FPU : Wall function for the momentum | | FPT : Wall function for the temperature | | FPA : Wall function for mass concentration (aerosols) | | FILTREKE : Filter for the K and Epsilon fields | | : mass operator | | MDIA : | | FROT : | ------------------------------------------------------------------------| | : Projection operator | | ELNO : CENTRE -> SOMMET | | NOEL : SOMMET -> CENTRE | | KSOF : SOMMET -> FACE | ------------------------------------------------------------------------| | DBIT : mass flow rate through a surface (or a line in 2D) | ------------------------------------------------------------------------| | TRACE : Graphical plots | | LIST : listing | ------------------------------------------------------------------------| | SAUVER : Save | | RESTITUER: Restart of a computation | ------------------------------------------------------------------------| | | IV Examples | ___________ | | | List of examples DGIBI | ------------------------------------------------------------------------| | Comments | ------------------------------------------------------------------------| | Transport | | 15wedge.dgibi | | burgerC.dgibi | | burgerNC.dgibi | | burgerpsi.dgibi | | smithhutton.dgibi | Smith and Hutton bench mark | transport1.dgibi | | cone.dgibi | 2D transient convection of a scalar. | | Tests different time schemes | convnonlin1.dgibi | | consmasse.dgibi | | | Forced convection NS | | blasius.dgibi | | hy1.dgibi | | ccar1.dgibi | | ccar2.dgibi | | ccar3.dgibi | | tubesrc.dgibi | | couette.dgibi | | | Free convection NS | | dvisi.dgibi | | benchmark_imst.dgibi | 2D plan NS low Prandtl number | vahldavis3D.dgibi | | vahldavis.dgibi | De Vahl-Davis bench mark | villers_platten.dgibi | | vortex.dgibi | | | Radiation coupled with convection | | cvry-2D-1.dgibi | | wsgg.dgibi | | | Turbulence | | bc30.dgibi | | gridturb.dgibi | | tubturb.dgibi | | | more complex models | | linekman.dgibi | | ODWp.dgibi | | BINGHAMp.dgibi | | dynasp.dgibi | Spray (two fluids model 2D plan | mistra.dgibi | | aerosol1.dgibi | | aerosol2.dgibi | | aerosol3.dgibi | | centrif.dgibi | | ale_mecaflu.dgibi | 2D plan NS ALE 1/2 expl | basmachQ.dgibi | 2D plan NS combustion low Mach number | basmachT.dgibi | | | Post processing | | trajec.dgibi | | lignecourant.dgibi | | ------------------------------------------------------------------------| | | V Advises | _________ | | | | The elaboration of data input needs several steps : | | | construction of the mesh | define the models (discretization operators) | define the boundary conditions | initialisation | execute | |
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