Finite Element Method for Turbomachinery Flows

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SOCRATES Tachng Staff Moblty Program 2000-200 DMA-URLS Fnt Elmnt Mthod for Trbomachnry Flos Alssandro Corsn

1 SOCRATES Tachng Staff Moblty Program DMA-URLS Fnt Elmnt Mthod for Trbomachnry Flos Alssandro Corsn Dpartmnto d Mccanca Aronatca, Unvrsty of Rom "La Sapnza" BUDAPEST Unvrsty of Tchnology and Economcs - 28 Novmbr 2000

2 FEM for Trbomachnry Flos WHY FEM IN TURBOMACHINERY CFD CONTENTS () INTRODUCING THE INTEGRAL APPROXIMATION METHODS VARIATIONAL AND RESIDUAL METHODS WEAK RESIDUAL FORMULATION OF FLOW PROBLEMS FINITE ELEMENT METHOD AND DISCRETIZATION FINITE ELEMENT INTERPOLATION SPACES STABILIZED FINITE ELEMENT FORMULATIONS STABILIZATION OF CONVECTIVE LIMITS STABILIZATION OF DIFFUSIVE LIMITS Alssandro Corsn, BUTE - 28 Novmbr 2000

3 FEM for Trbomachnry Flos CONTENTS (2) FLUID MODELLING MODEL EQUATIONS FLUID DYNAMIC CLOSURES SOLUTION STRATEGY ITERATIVE SOLVER OVERLAPPING DOMAIN DECOMPOSITION METHOD Alssandro Corsn, BUTE - 28 Novmbr 2000

4 FEM for Trbomachnry Flos CONTENTS (3) TEST CASES 2D AND 3D LAMINAR&TURBULENT CLASSICAL FLOWS COMPRESSOR CASCADE FLOWS NUMERICAL STUDIES ON AXIAL FLOW FAN ROTOR AERODYNAMICS ROTOR NUMERICAL MODELLING ROTOR FLOW ANALYSIS CFD BASED DESIGN IMPROVEMENT Alssandro Corsn, BUTE - 28 Novmbr 2000

5 FEM for Trbomachnry Flos WHY FEM IN TURBOMACHINERY CFD () FEW FEM APPLICATIONS CONCERN WITH TURBOMACHINERY CFD GEOMETRIC FLEXIBILITY ON COMPLEX DOMAIN ISO-PARAMETRIC MAPPING CARTESIAN FIELD DESCRIPTION IMPLICIT UNSTRUCTURED MESHING (EVEN FREE-MESH ALGORITHM) NO STRUCTURED MULTI-BLOCK OR EMBEDDING TECHNIQUES ARE REQUIRED BOUNDARY CONDITIONS ACCURACY TRUE DIRICHLET CONDITIONS NATURAL OUTFLOW CONDITIONS PHYSICAL PERIODIC CONDITIONS Alssandro Corsn, BUTE - 28 Novmbr 2000

6 FEM for Trbomachnry Flos WHY FEM IN TURBOMACHINERY CFD (2) STABILITY AND ACCURACY, CONSISTENT PETROV-GALERKIN (PG) FORMULATIONS SU/PG (Brooks and Hghs, 982), ADVECTIVE-DIFFUSIVE LIMIT INSTABILITIES NON-OSCILLATORY COUNTERPART OF GALERKIN METHODS PS/PG (Tzdyar, 992), PURE DIFFUSIVE LIMIT INSTABILITIES CIRCUMVENTING THE BABUSKA-BREZZI CONDITION FOR Q-Q ELEMENTS XENIOS PG FORMULATION (Corsn, 996), (Borllo, Corsn and Rspol, 997) ON Q2-Q ELEMENTS IMPROVED MATRIX CONDITION CHARACTERISTIC, RAPID CONVERGENCE ACCURATE ELEMENT-WISE MODELLING OF HIGHER ORDER TERMS OF ANISOTROPIC EVM GLOBAL PHYSICAL LAWS SATISFACTION Alssandro Corsn, BUTE - 28 Novmbr 2000

7 FEM for Trbomachnry Flos THE INTEGRAL APPROXIMATION METHODS varatonal prncpl dscrtzaton qato of Raylgh -Rtz PDE nnr prodct rsdal prncpl qato of Galrkn VARIATIONAL METHODS ENERGY EQUILIBRIUM CONDITION RESIDUAL METHODS ORTHOGONAL PROJECTION CONDITION Alssandro Corsn, BUTE - 28 Novmbr 2000

8 FEM for Trbomachnry Flos Gvn a Gnrc Dffrntal Oprator L f = 0 R d Drchlt condto = g on g THE RESIDUAL METHODS Nmann condto, n = h on h Dfn to collcto of fncto S h, tral or canddat ~ W h, ght or varaton ~ APPROXIMATE th solton th TRIAL L ~ f = BUILD an nnr prodct as an ORTHOGONALITY condto ( ) = d = 0, ~ ~ Alssandro Corsn, BUTE - 28 Novmbr 2000

9 FEM for Trbomachnry Flos THE WEAK FORM OF NAVIER-STOKES PROBLEM () WEAK GLOBAL APPROACH (Hghs, 987) TRIALS and WEIGHTS gnralzd proprts of CONTINUITY and INTEGRABILITY ovr = R d APPROXIMATE collcto h th ( ) H SOBOLEV SPACE S h h = { ~ ~,~ = g } H g W h h = { ~ ~,~ = 0 } H g Alssandro Corsn, BUTE - 28 Novmbr 2000

10 FEM for Trbomachnry Flos Sqar ntgrabl fncton spac FUNCTION SPACES 2 L ( ), spac fncto sqar ntgrabl ovr : ( p, q) = pqd < q = ( q, q 0 ) / 2 Sobolv spac H k 0 2 s 2 { q L ( ) : D q L ( ), s,... k} 2 s 2 / 2 q = ( q + D q ) k 0 0 ( ) = = 2 H ( ) = L ( ), ( ) H frst drvatv sqar ntgrabl, { q H ( ) q = } ( 2 d 2 / 2 q = ( q + q x 0 ) 0 H0 ) = : 0 / 2 H ( ) rstrcton to th bondary of fncto H ( ) q = nf v, v ( ) H / 2, v = q on Alssandro Corsn, BUTE - 28 Novmbr 2000

11 FEM for Trbomachnry Flos THE WEAK FORM OF NAVIER-STOKES PROBLEM (2) Rsdal formlaton of INCOMPRESSIBLE N-S BOUNDARY VALUE PROBLEM ( ) d ( ) d = f d j, j σ j, σ j d c, = = g n = h g h 0 scond ordr trms LOWERING th ordr of DIFFUSIVE INTEGRAL (Grn-Gass thorm) ( ) d d + n d σ j, j =, σ j h σ j j Alssandro Corsn, BUTE - 28 Novmbr 2000

12 FEM for Trbomachnry Flos THE WEAK FORM OF NAVIER-STOKES PROBLEM (3) WEAK Rsdal form of INCOMPRESSIBLE N-S BOUNDARY VALUE PROBLEM ( ) d + d = f d + h d j, j d c, = = g g, σ j 0 ssntal BCs h natral BCs compact WEAK Rsdal form ( ) d + d f d d + d 0 j, j, σ j h h c, = contnty scalar qaton acts as an addtonal cotrant c as th Lagrangan mltplr of th ncomprssblty cotrant Alssandro Corsn, BUTE - 28 Novmbr 2000

13 FEM for Trbomachnry Flos FINITE ELEMENT METHOD AND DISCRETIZATION () Ho th collcton S h and W h ar composd? for ach s S h on = s g = v g + g v 0 = g Th sam approxmat fncton v cold b th BASIS for TRIAL and WEIGHTS v s calld th BASIS or SHAPE fncton Th FEM basd on sch a choc s namd th GALERKIN rsdal mthod. Alssandro Corsn, BUTE - 28 Novmbr 2000

14 FEM for Trbomachnry Flos FINITE ELEMENT METHOD AND DISCRETIZATION (2) Lt codr a doman R d and a st of nodal ponts l Th APPROXIMATE GLOBAL fncto hav a lnar POLYNOMIAL form nods = cl l= φ l ghtng fncto nods s = d l l= φ l tral fncto Th ntrpolatng SHAPE fncton φ l flflls th nodal proprts φ φ 0 m [ ] l l l = l m = Alssandro Corsn, BUTE - 28 Novmbr 2000

15 FEM for Trbomachnry Flos FINITE ELEMENT METHOD AND DISCRETIZATION (3) Lt codr a -D doman R d and a st of nodal ponts l nodal lnar shap fncto φ φ A φn+ global ntrpolatd fncto Alssandro Corsn, BUTE - 28 Novmbr 2000

16 FEM for Trbomachnry Flos FINITE ELEMENT METHOD AND DISCRETIZATION (4) Lt codr th doman sbdvson nto lmnts = nt = = nods Th GLOBAL ak rsdal formlaton cold b thn dscrtzd onto th doman Σ = g ( ρ ) j g, j d + Σ, σ j d Σ nt [ σ j n ]d + Σ c, d = Σ f d + h h h d balanc of ntr-lmnt dffsv flxs Alssandro Corsn, BUTE - 28 Novmbr 2000

17 FEM for Trbomachnry Flos Alssandro Corsn, BUTE - 28 Novmbr 2000 FINITE ELEMENT METHOD AND DISCRETIZATION (5) Lt no codr th rsdal NS problm rttn for a gnrc lmntary doman ( ) 0 d d d d d, c j j, j, j = + + σ σ ρ f n hr th ghts and th tral ar smply C 0 on ach lmnt Composng ach lmntary formlato th rsdal GLOBAL form cold b bld as ( ) Σ Σ σ Σ σ Σ ρ Σ d d d d d, c j j, j, j f n = + + hr [ ] σ σ Σ σ Σ d d d h j h j nt j n n n + =

18 FEM for Trbomachnry Flos STEPS TO A FINITE ELEMENT METHOD th approxmat rprsntaton of CONTINUOUS COMPUTATIONAL DOMAIN, as th composton of small sb-doma calld lmnts th dfnton of APPROXIMATE SOLUTION, obtand by th ntrpolaton of th nknon vals n a fnt nmbr of nodal ponts dfnd n ach lmnt sng a collcton of bass fncto th dfnton of an INTEGRAL EQUATION for ach nknon varabl, by s of a rsdal prncpl Alssandro Corsn, BUTE - 28 Novmbr 2000

19 FEM for Trbomachnry Flos Alssandro Corsn, BUTE - 28 Novmbr 2000 FINITE ELEMENT INTERPOLATION FUNCTIONS () Modlng complx gomtrs ntrodcs DISTORTIONS of th lmntary sb-doma rcall th FEM capablty of orkng on trctrd&dstortd grds from a CODING VIEWPOINT t s mandatory to ntrodc a spac corrlaton btn R d and LOGIC FRAME OF REFERENCE (ξ, η, ζ) = ς η ξ f z y x mappng or coordnat traformaton oprator

20 FEM for Trbomachnry Flos FINITE ELEMENT INTERPOLATION FUNCTIONS (2) ELEMENT-BY-ELEMENT th doman gomtry s cotrctd sng approprat traformaton polynomal oprators x( y( z( ξ, η, ς ξ, η, ς ξ, η, ς ) = ) = nl = nl = nl ) = = N ( ξ, η, ς N ( ξ, η, ς N ( ξ, η, ς )x )y )z lmnt shap fncto Th SHAPE fncto N cold b dfnd sng a drct formlaton mposng N () =, and N (j) = 0 for j th contnty of lmnt fncto along th lmnt bondars ISOPARAMETRIC lmnts s dntcal shap fncto for gomtry&dof Alssandro Corsn, BUTE - 28 Novmbr 2000

21 FEM for Trbomachnry Flos Alssandro Corsn, BUTE - 28 Novmbr 2000 FINITE ELEMENT INTERPOLATION FUNCTIONS (3) Shap fncto DERIVATIVES (frst ordr) smply dfnd n th LOGIC rfrnc = ς Φ η Φ ξ Φ ζ ζ ζ η η η ξ ξ ξ z y x x x x z y x z y x Φ Φ Φ =J z y x Φ Φ Φ z y x Φ Φ Φ =J - ς Φ η Φ ξ Φ D to th ntrc smplcty of LOGIC gomtry&dof dfnton th comptaton of ntgral trms s prformd as [ ] ς η ξ d d d J dt K KdV V =

22 FEM for Trbomachnry Flos INTERPOLATION SPACES n XENIOS () MIXED fnt lmnt approxmato spacs vlocty nods prssr&vlocty nods qadratrc approx for vlocty DOFs, lnar approx for prssr DOF Alssandro Corsn, BUTE - 28 Novmbr 2000

23 FEM for Trbomachnry Flos INTERPOLATION SPACES n XENIOS (2) Alssandro Corsn, BUTE - 28 Novmbr 2000

24 FEM for Trbomachnry Flos REASONS TO STABILIZE A FEM FORMULATION () prly advctv flo lmt, th localzaton of advctv traport of varabl along th straml prly dffsv flo lmt (Stoks flo), prssr tablty rlatd to th ncomprssblty cotrant Alssandro Corsn, BUTE - 28 Novmbr 2000

25 FEM for Trbomachnry Flos ADVECTIVE LIMIT INSTABILITY ORIGIN s not possbl to smlat strongly asymmtrc trms by s of SYMMETRIC oprators Galrkn shap fncto or qvalntly cntrd FD stncls for -D th stablty condton for nmrcal approx. of convctv trm s ϕ ϕ x <0 ϕ ϕ x + ϕ 2 x ntral stablty to ϕ classcal approach to rcovr th stablty modls th convctv trm va UPWIND FD ϕ ϕ ϕ ϕ + ϕ x ϕ + + 2ϕ = + 2 x x 2 x 2 x ϕ cntrd frst and scond ordr trms Alssandro Corsn, BUTE - 28 Novmbr 2000

26 FEM for Trbomachnry Flos NON CONSISTENT STABILIZED FEM FORMULATION Galrkn FEM cold b stablzd ntrodcng an artfcal 'balancng' dffsv trm n -D th dffsv lk scalar pnd stablzaton s optmal optmal scalar artfcal dffsvty k ~ = h ζ 2 ζ = coth( α ) α, magc fncton α = h, grd or local Pclt nmbr 2k Alssandro Corsn, BUTE - 28 Novmbr 2000

27 FEM for Trbomachnry Flos NON CONSISTENT STREAMLINE UPWIND FEM FORMULATION n mlt-dmonal cass a STREAMLINE pnd corrcton n mandatory th balancng oprator dpnds on a toral artfcal dffsvty k ~ j = k ~ j rsdal 'ntrprtaton' of stramln pnd trm, k j σ j d +, k ~ j, j d, k ~ j, j d, k ~ j, j d prtrbd ght for th convctv ntgral Alssandro Corsn, BUTE - 28 Novmbr 2000

28 FEM for Trbomachnry Flos CONSISTENT STREAMLINE UPWIND-PETROV GALERKIN FEM FORMULATION () th costncy of th stablzaton mthod s rcovrd by prtrbatng th Galrkn ghts n sch a ay that a Ptrov-Galrkn rsdal formlaton s achvd ' = + p Alssandro Corsn, BUTE - 28 Novmbr 2000

29 FEM for Trbomachnry Flos CONSISTENT STREAMLINE UPWIND-PETROV GALERKIN FEM FORMULATION (2) SUPG formlaton of Navr-Stoks problm Σ [ Σ nt ( ρ ) + f ]d + [ p ( ρ ) [ σ j j n, j ]d, h σ j h h d = 0 Σ j, j p σ j, j p f ]d or Σ ( ρ ) f ]d [ ]d ( )d = 0 [ j, j σ j, j Σ σ j n h σ j n h nt h from th ntgral problm s agan possbl to xtract th orgnal dffrntal form ρ ( σ j [ σ j, j + σ j, j f = 0 n h ) = 0 h n ] = 0 j nt Elr - Lagrang condto of rsdal FEM formlaton Alssandro Corsn, BUTE - 28 Novmbr 2000

30 FEM for Trbomachnry Flos CONSISTENT STREAMLINE UPWIND-PETROV GALERKIN FEM FORMULATION (3) th xprsson of stramln pnd prtrbaton p s p = k ~, / j j th artfcal dffsvty for mlt-dmonal cass s dfnd as k ~ = ( ) ξ + η h + ς h ξ h ξ η 2 η ς ς ξ = coth( α ξ ) α ξ η = coth( α η ) α η ς = coth( α ς ) α ς Alssandro Corsn, BUTE - 28 Novmbr 2000

31 FEM for Trbomachnry Flos THE XENIOS STREAMLINE UPWIND-PETROV GALERKIN to fndamntal changs has bn ntrodcd th rspct to SUPG formlaton (Brooks&Hghs, 982) k ~ j p, jd, j p σ d p p d p ( 2k ( j, j = + j, j +, j p, j A j, )), j d, j artfcal lmntary flxs hav to b xplctly balancd along physcal doman bonadrs p σ d = p σ d + p σ d δ p σ d j, j, j j jn j b j, j Alssandro Corsn, BUTE - 28 Novmbr 2000

32 FEM for Trbomachnry Flos STABILIZATION OF DIFFUSION DOMINATED FLOWS th rlaxng of ncomprssblty cotrant to vlocty fld s mandatory th orgnal non-costnt approach (Brzz&Ptkaranta, 984) st (, p, ), for Stoks problm a costnt stablzaton lads to a Ptrov-Galrkn rsdal mthod th prtrbd Stoks ght of th form = s s +α h 2 c, Σ [ Σ nt s, s σ [ σ j j + s f ]d + Σ α h n ]d h h d + Σ h 2 c, [ σ c j,, + f ]d d = 0 + th stablzng ntgral conta th trm α h 2 c, p, d = α h 2 ( c p, ), d α h 2 c p, n d Alssandro Corsn, BUTE - 28 Novmbr 2000

33 FEM for Trbomachnry Flos CONSISTENT PRESSURE STABILIZED - PETROV GALERKIN FEM FORMULATION a gnral stablzaton tchnq for th ncomprssblty cotrant of Navr-Stoks flos acts by ntrodcng a prtrbaton of th contnty qaton (Tzdyar, 992) Σ ρ t pspg c, [ ρ j, j σ j, j f ]d h U th stablzaton factor s t pspg = γ (R ) 2U th rlaxaton of th ncomprssblty cotrant s mad proportonal to th rror affctng th vlocty fld solton Alssandro Corsn, BUTE - 28 Novmbr 2000

34 FEM for Trbomachnry Flos COMPARATIVE ANALYSIS OF STABILIZATION EFFECTS () Borllo, Corsn and Rspol, 2000 Ptrov-Galrkn Schms ( m & ) Schms Grd Coffcnts n n m& ot t λ p λ c %( m& n ) av av 0n 0ot ( P P %(Q n ) PG C PG2 C IPG2 C PG M PG2 M IPG2 M PG F PG2 F IPG2 F ) Grds (ξ, η) C, (53 5) M, (05 3) F, (209 63) Basson and Lakhsmnarayana, th ordr artfcal dsspaton schm ε 2 ε 4 ε p FV C ε 2 and ε 4 ar rspctvlly th coffcnt for 2 nd and 4 th ordr artfcal dsspaton, ε p s th coffcnt for prssr ghtng INVISCID FLOW IN 45 TWO-DIMENSIONAL BEND Alssandro Corsn, BUTE - 28 Novmbr 2000

35 FEM for Trbomachnry Flos COMPARATIVE ANALYSIS OF STABILIZATION EFFECTS (2) p nflo p p ξ ξ ξ Comptd prssr dstrbto on bnd prssr srfac, (M) grd symbols: Galrkn solton; sold l: PG2 solton; dashd l: PG solton Alssandro Corsn, BUTE - 28 Novmbr 2000

36 FEM for Trbomachnry Flos Alssandro Corsn, BUTE - 28 Novmbr 2000

37 FEM for Trbomachnry Flos Alssandro Corsn, BUTE - 28 Novmbr 2000

A FE Method for the Computational Fluid Dynamics of Turbomachinery

A FE Method for the Computational Fluid Dynamics of Turbomachinery SOCRATES Tachng Staff Moblty Program 999-000 DMA-URLS Lctur not on A FE Mthod for th Computatonal Flud Dynamcs of Turbomachnry Alssandro Corsn Dpartmnto d Mccanca Aronautca Unvrsty of Rom La Sapnza - Octobr