Universitat Politecnica` de Catalunya PhD Thesis STABILIZED PRESSURE SEGREGATION METHODS AND THEIR APPLICATION TO FLUID-STRUCTURE INTERACTION PROBLEMS by Santiago I. BADIA RODR¶IGUEZ supervised by: Ramon Codina Rovira Barcelona December 2005 Stabilized Pressure Segregation Methods and their application to Fluid-Structure Interaction Problems Santiago I. Badia Rodr¶³guez A Angels,` Hugo, es meus pares i es meus padrins pes seu recolzament... Abstract In this thesis we design and analyze pressure segregation methods in order to approxi- mate the Navier-Stokes equations. Pressure correction methods are widely used because they allow the decoupling of velocity and pressure computation, decreasing the computa- tional cost. We have analyzed some of these schemes, obtaining inherent pressure stability. However, for second order accurate methods (in time) this inherent stability is too weak, requiring the introduction of a stabilized ¯nite element methodology for the space dis- cretization. Moreover, we have carried out a complete convergence analysis of a ¯rst order pressure segregation method. We have used a stabilization technique justi¯ed from a multiscale approach that allows the use of equal velocity-pressure interpolation spaces and convection dominated flows. A new kind of methods has been motivated from an alternative version of the mono- lithic fluid solver where the continuity equation is replaced by a discrete pressure Poisson equation. These methods belong to the family of velocity correction schemes, where it is the velocity instead of the pressure the extrapolated unknown. Some stability bounds have been proved, revealing that their inherent pressure stability is too weak. Further, predictor corrector schemes easily arise from the new monolithic system. Numerical ex- perimentation shows the good behavior of these methods. We have introduced the ALE framework in order for the fluid governing equations to be formulated on moving domains. Taking as the model equation the convection-di®usion equation, we have analyzed the blend of the ALE framework and a stabilized ¯nite element method. We suggest a coupling procedure for the fluid-structure problem taking bene¯t from the ingredients previously introduced: pressure segregation methods, a stabilized ¯nite element formulation and the ALE framework. The ¯nal algorithm, using one loop, tends to the monolithic (fluid-structure) system. This method has been applied to the simulation of bridge aerodynamics, obtaining a good convergence behavior. We end with the simulation of wind turbines. The fact that we have a rotary body surrounded by the fluid (air) has motivated the introduction of a remeshing strategy. We consider a selective remeshing procedure that only a®ects a tiny portion of the domain, with little impact on the overall CPU time. Resum En aquesta tesi hem introduijti analitzat m`etodes de segregaci¶ode la pressi¶oper aproximar l'equaci¶ode Navier-Stokes. Els m`etodes de correcci¶ode la pressi¶oestan molt estesos degut a que permeten el c`alculdesacoblat de velocitat i pressi¶o,reduint el cost computacional. Hem analitzat alguns d'aquests esquemes, obtenint estabilitat de la pressi¶oinherent als m`etodes. Malauradament, per a esquemes de segon ordre (en temps), aquesta estabilitat inherent es insu¯cient, for»cant la introducci¶ode m`etodes estabilitzats d'elements ¯nits per a la discretitzaci¶oespacial. A m¶es,hem dut a terme una an`aliside converg`enciacompleta per a un m`etode de primer ordre. S'ha fet servir un m`etode d'estabilitzaci¶obasat en el concepte de multiescales que permet l'¶usd'id`entics espais d'interpolaci¶oper velocitat i pressi¶oi fluxos amb convecci¶o dominant. Hem justi¯cat una nova classe de m`etodes a partir d'una forma alternativa de l'esquema monol¶³ticper a la resoluci¶odel fluid on l'equaci¶ode continuijtat¶esreempla»cadaper una equaci¶ode Poisson per a la pressi¶o.Aquests m`etodes pertanyen a la fam¶³liadels esquemes de correcci¶ode la velocitat, donat que ¶esla velocitat en lloc de la pressi¶ola variable extrapolada. Hem obtingut alguns resultats d'estabilitat, mostrant que l'estabilitat de la pressi¶oinherent a aquests m`etodes es massa feble. Per una altra banda, esquemes de predicci¶ocorrecci¶oson f`acilment justi¯cats amb la nova versi¶ode l'esquema monol¶³tic. L'experimentaci¶onum`ericamostra el bon comportament d'aquests m`etodes. Hem introduijtla formulaci¶oALE per poder formular les equacions de Navier-Stokes en dominis m`obils.Prenent com a equaci¶omodel la de convecci¶o-difusi¶o,hem analitzat el m`etode resultant de la conjunci¶ode la formulaci¶oALE i un m`etode d'estabilitzaci¶o. En aquest treball se suggereix l'¶usd'estrat`egiesd'acoblament apro¯tant els ingredients pr`eviament introduijts: esquemes de segregaci¶ode la pressi¶o,m`etodes d'estabilitzaci¶oi la formulaci¶oALE. L'algorisme ¯nal tendeix, amb nom¶esun bucle, al sistema (fluid- estructura) monol¶³tic. Aquest m`etode s'ha fet servir a la simulaci¶onum`ericade l'aerodin`amicade ponts, amb una bona converg`encia. Aquesta tesi acaba amb la simulaci¶od'aerogeneradors. El fet de tenir cossos rotatoris envoltats de fluid (aire) ha obligat a la introducci¶od'estrat`egiesde remallat. S'ha consi- derat un m`etode selectiu que nom¶esafecta a una petita part del domini, amb poc impacte sobre el temps total de CPU. Acknowledgments Estic molt agraijta Ramon per tot el temps que m'ha dedicat. Ha sigut una aut`entica sort poder treballar di`ariament amb un investigador tan excepcional. Les seves idees i comentaris m'han ajudat molt¶³ssim. Per damunt de la ci`encia,¶esuna gran persona, que sempre ha tingut la porta oberta i la paci`encianecess`aria. Tamb¶evull agrair a Eugenio la seva con¯an»caen jo, i haver-me donat l'oportunitat de fer feina al CIMNE. He passat molt bons moments en aquesta casa, i he treballat amb molt bona gent. Especialment, els amics del C-4, quasi tan bona gent com frikis. En un `ambit m¶espersonal, vull agrair a Angels` haver estat al meu costat tots els dies (la immensa majoria feli»cos)que he passat fent el doctorat. Me commou que no tingui cap dubte en deixar tot per anar amb jo all`aon pugui continuar la meva carrera investigadora. Tamb¶evull agrair a Hugo els bons moments que hem passat junts. Estic molt content d'haver-nos retrobat a Barcelona. I com no, vull agrair als meus pares que, des de Mallorca, sempre hagin estat fent-me costat i ajudant-me en tot el que han pogut. Per ¶ultim,vull enrecordar-me dels meus quatre avis. M'ha costat explica'ls-hi el que feia, si feia feina, si estudiava,... Ara ja podr¶erespondre que s¶³,que aquesta vegada ja he acabat d'estudiar. Encara que espero que aix`omai sigui del tot cert... Contents Introduction 13 1 Preliminaries 17 1.1 The Incompressible Navier-Stokes equation . 17 1.2 Some function spaces . 19 1.3 Finite element approximation . 22 1.4 The variational formulation of the Navier-Stokes equations . 23 1.4.1 The continuous level . 23 1.4.2 The discrete approach . 25 1.5 Time discretization . 26 2 Stabilization with Orthogonal Subscales 29 2.1 Introduction . 29 2.2 Stationary Oseen equations . 31 2.2.1 Problem statement . 31 2.2.2 The subgrid scale approach . 32 2.2.3 Orthogonal subscales . 34 2.3 Transient Oseen equations . 36 2.3.1 Discretization in time . 36 2.3.2 Subgrid scale decomposition and modeling of the subscales . 37 2.3.3 Stabilized ¯nite element problem . 38 2.4 Extension to the Navier-Stokes problem . 39 2.4.1 Temporal discretization and linearization . 39 2.4.2 Final algorithm . 40 3 Pressure Correction methods 43 3.1 Overview . 43 3.1.1 Classical projection methods . 43 3.1.2 Higher order methods . 46 3.2 Some pressure correction methods . 47 3.2.1 Monolithic time discretization . 48 3.2.2 Space discretization . 48 3.2.3 Classical pressure correction methods . 49 3.2.4 Momentum-pressure Poisson equation methods . 51 3.2.5 Predictor corrector schemes . 52 3.3 Stability of pressure correction methods . 53 3.3.1 Stability results for non-stabilized schemes . 54 3.3.2 Improved stability results for a stabilized scheme . 59 3.4 Numerical tests . 65 9 10 3.4.1 Convergence test . 65 3.4.2 Flow in a cavity . 66 3.4.3 Flow over a cylinder . 66 3.5 Conclusions . 70 4 Numerical Analysis of a ¯rst order Pressure Correction method 73 4.1 Introduction . 73 4.2 Problem statement and preliminaries . 74 4.3 Classical projection method . 78 4.4 Analysis of a semi-discrete projection-like problem . 79 4.4.1 The semi-discrete system . 79 4.4.2 The penalized semi-discrete system . 82 4.4.3 The semi-discrete projection-like system . 86 4.5 Convergence results satisfying the inf-sup condition . 90 4.6 Convergence results for a stabilized scheme using the pressure Poisson equa- tion . 95 4.7 An alternative convergence analysis under the inf-sup condition . 101 4.7.1 Introduction . 101 4.7.2 A ¯rst error estimate . 101 4.7.3 An improved error estimate . 105 4.8 Conclusions . 109 5 Velocity Correction methods based on a Discrete Pressure Poisson Equa- tion 111 5.1 Introduction . 111 5.2 Preliminaries and problem statement . 112 5.2.1 The continuous problem . 112 5.2.2 Weak form . 113 5.2.3 Discrete problem . 114 5.3 Velocity correction methods based on a DPPE . 116 5.3.1 Equivalent stabilized monolithic formulation . 118 5.3.2 An alternative form of velocity correction methods . 118 5.4 Predictor corrector methods .