Diprtimento di ingegneri idrulic e mientle SPH simultion of fluid-structure interction prolems C. Antoci, M. Gllti, S. Siill
Reserch project Prolem: deformtion of plte due to the ction of fluid (lrge displcement of the structure, fluid free surfce) Numericl technique: SPH Lgrngin: it utomticlly follows moving interfces Reltively esy to include the simultion of different mterils
Scheme of the model Structure (SPH) Fluid (SPH) Coupling conditions on the interfce: Kinemtic condition: v f = v s v nˆ = f v s nˆ (perfect fluid) Dynmic condition: σ s T s nˆ s = σ s s f nˆ f f n ˆ σ nˆ = p (perfect fluid)
Continuum equtions Dρ v + ρ Dt x i i = 0 Dv σ ρ i = ρgi + Dt x ij j (i=1,3) (i,j=1,3) continuity eqution eqution of motion ρ t x i v i σ ij g i (isotherml conditions) density time position (i-component of the vector) velocity (i-component of the vector) stress (ij-component of the tensor) grvity ccelertion (i-component of the vector)
Stte eqution nd constitutive equtions pressure devitoric stress σ ij = p δ + ij S ij (i,j=1,3) p = c ( ρ ) 2 0 ρ0 fluid solid c = ε 0 ρ 0 c 0 = k ρ 0 (ε compressiility modulus) (k ulk modulus) S ij fluid solid ε ij = S ij = 0 (perfect fluid) ds dt ij 1 vi 2 x j 1 = 2µ εij δ ij ε 3 v + x j i Ω ij ij = + S 1 vi 2 x j ik Ω jk v x j i + Ω ik S kj (µ sher modulus)
2D SPH equtions (1) 2 Eqution of stte: = c ( ρ ) p 0 ρ0 Continuity eqution: Dρ = m ( v v ) Dt W ( nd : prticle lels)
2D SPH equtions (2) Eqution of motion: fluid solid Dv Dt Dv Dt i i p = p + W m δij + g 2 2 ρ x ρ j = σ ρ W x + + + + m rtificil viscosity (Monghn nd Gingold 1983) ij 2 σ ρ ij 2 δ ij R ij rtificil stress (Gry, Monghn nd Swift 2001) i f n j g i σ ij = p δ + ij S ij S ij clculted y n implicit scheme from the incrementl hypoelstic reltion
2D SPH equtions (3) Velocity grdient: v x i j = m ρ ( v v ) i i W x j spin rte of deformtion [or: velocity grdient normlized to ccount for non uniformity in prticle distriution]
- Two sets of prticles, fluid nd solid Fluid-structure interction - A simple pproch: to consider prticles in the equtions regrdless of the fct they re fluid or solid - Interpenetrtion of fluid nd solid prticles cn e prevented using XSPH ut the interction is not well reproduced (excessive dhesion) - Definition of the fluid-solid interfce (nd norml) - Dynmic condition (ction-rection principle) - Kinemtic condition
Definition of the fluid-solid interfce Università degli Studi di Pvi Since solid prticles mintin the sme regulr sptil distriution (no frgmenttion): ( ) = = + + + + 1 1 1 1 1 1 1 1,, ˆ y x x x y y x x x x t t t ( ) x y t t n, ˆ = n d x x ˆ 2 int + = (for every solid prticle ner the interfce)
Dynmic condition p int = Ω f Ω m ρ f m ρ p W W ( x x, h) ( x x, h) int int 0.5 (constnt, in order to tke in ccount possile seprtion of the two medi) surfce term: F f s = pint Γ W F s f,* = - F f s, ( x x', h) dγ' int (liner interpoltion) (F f s /ρ dded term in the momentum eqution)
v iint Università degli Studi di Pvi * = Ω s Ω m viw ρ m W ρ s Kinemtic condition - fluid prticle * solid prticle (the nerest) - velocity of the interfce: ( x x, h) ( x x, h) int int * * - velocity distriution is ssigned to solid prticles in order to otin y SPH interpoltion (on the interfce) the interfce velocity (norml component): v n = v = t v v nint t * + d * nint ( v v ) * n d * = mx (d /d,2)
Other fetures of the code Correction of velocity: -XSPH (solid)(monghn 1989) -dissiptive correction (fluid) (Gllti nd Brschi 2000) (on the interfce prticles from oth medi hve to e included in the correction) Boundry conditions: - fluid - imginry prticles which reflect velocity - lyer of fixed prticles (2h) - solid (clmp) - lyer of fixed prticles which re clculted just like others ut with velocity equl to zero Time integrtion scheme: Euler explicit (stggered) Kernel: cuic spline (Monghn 1992)
Exmple: Elstic gte Dimensions A H L S 0.1 m 0.14 m 0.079 m 0.005 m Sluice-gte (ruer) ρ s E (Young modulus) 1100 kg/m 3 10 7 P
Simultion ε = 2 10 6 N/m 2 (compressile) ρ f = 1000 kg/m 3 K = 2 10 7 N/m 2 µ = 4.27 10 6 N/m 2 ν = 0.4 (Poisson coefficient) E=1.2 10 7 N/m 2 h/d=1.5 n P =6012 dt=8.34 10-6 s t=0 s
Comprison etween simultion nd experiment t=0 s
t=0.04 s
t=0.08 s
t=0.12 s
t=0.16 s
t=0.2 s
t=0.24 s
t=0.28 s
t=0.32 s
t=0.36 s
t=0.4 s
Free end of the plte: displcements (1) horizontl displcement 0,06 0,05 0,04 h.d. (m) 0,03 experiment simultion 0,02 0,01 0 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 time (s)
Free end of the plte: displcements (2) verticl displcement 0,025 0,02 v.d. (m) 0,015 0,01 experiment simultion 0,005 0 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 time (s)
Free surfce (1) wter level ehind the gte 0,16 0,14 0,12 0,1 y (m) 0,08 experiment simultion 0,06 0,04 0,02 0 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 t (s)
Free surfce (2) wter level 5 cm fr from the gte 0,16 0,14 0,12 0,1 y (m) 0,08 experiment simultion 0,06 0,04 0,02 0 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 time (s)
Future work To complete the normliztion of the elstic equtions with explicit tretment of oundry conditions To improve the model in order to mnge lrger deformtions To write the code in cylindricl coordintes in order to simulte xilly symmetric prolems ( flex-flow vlves, sloshing in cylindricl tnks ) To use two different sptil discretiztions for the fluid nd the solid (memrnes, hemodynmics) This reserch ws finnced y Dresser Itli