Porous Euler-Lagrange Coupling: Application to Parachute Dynamics

Transcription

1 9 th International LS-DYNA Uer Conerence Fluid/Structure Porou Euler-Lagrange Coupling: Application to Parachute Dynamic Jaon Wang, Nicola Aquelet, Benjamin Tutt Ian Do, Hao Chen, Mhamed Souli 3 Livermore Sotware Technology Corporation 7374 La Poita Road, Livermore, CA 94550, USA aquelet@ltc.com Irvin Aeropace Inc. 370 Wet Warner Avenue, Santa Ana, CA 9704, USA Btutt@irvinaeropace.com 3 Laboratoire de Mécanique de Lille UMR CNRS 807 Bd Paul Langevin Cité Scientiique Villeneuve d Acq cedex mhamed.ouli@univ-lille.r Abtract A newly developed approach or tridimenional luid-tructure interaction with a deormable thin porou media i preented under the ramework o the LS-DYNA otware. The method preented couple a Arbitrary Lagrange Euler ormulation or the luid dynamic and a updated Lagrangian inite element ormulation or the thin porou medium dynamic. The interaction between the luid and porou medium are handled by a Euler-Lagrange coupling, or which the luid and tructure mehe are uperimpoed without matching. The coupling orce i computed with an Ergun porou low model. A tet cae, the method i applied to an anchored air parachute placed in an air tream Introduction Many important engineering application uch a airbag, parachute, to name a ew, involve tranient low in deormable porou media []. In general the deormable porou media problem hould be decribed at both the micro and macro cale. The problem at the microcopic level ha a deormable keleton urrounded by one or everal luid. At the macrocopic level the olid i uually decribed by the Lame equation o linear elaticity and the luid by the Navier-Stoke equation. The irt imple model o a mechanical ytem compried o a deormable porou olid matrix illed with a luid ha been developed by Biot [] who ormulated the macrocopic equation or the eective medium. The application o aymptotic homogenization method [3,4] ha lead to theoretical jutiication o Biot equation [,5,6] along with appropriate pore problem rom which the macrocopic parameter can be computed numerically. The macrocopic equation are derived under the aumption that the olid-luid interace diplacement are mall compared to the pore ize. For a large cla o poroelatic problem it i not poible to derive macrocopic equation. In general the upcaling problem or poroelaticity medium i not eparable even when the pore are well eparated. Thi i due to the act that the 9-

2 Fluid/Structure 9 th International LS-DYNA Uer Conerence keleton can deorm arbitrarily large due to dierent parameter uch a macrocopic diplacement, preure and velocity. However it i not practical to model the low at the pore cale and undeirable to have to gather the tremendou amount o ine cale data that i required to model an entire parachute or example. Moreover preent computational reource are not able to handle low imulation o thi ize. Hence the model in thi paper decribe the eential phyical behaviour in an averaged ene at the mega cale without modelling iner cale detail. Thi aumption or thin porou media uch a parachute eem reaonable. Actually the three phae that occur during a parachute miion are deployment, inlation, and terminal decent. The ocu o thi paper i the cae when, due to problem parameter uch a macrocopic preure and velocity ield or a parachute in inlation and terminal decent phae, the deormation o the luid-olid interace i not coniderable at the pore level. Thi interace could be approximated by a rigid motion o it initial poition. Thu the porou coupling orce i computed by uing the Ergun equation [7] with a contant poroity. In thi work, the governing equation or luid and thin porou medium problem are irt ormulated together with boundary condition. Then a decription o the porou Euler-Lagrange coupling algorithm i preented. Further, thi numerical method i applied to a porou parachute Fluid Structure Interaction problem by comparing the numerical reult to experimental data. Fluid and Structure Modeling The two application decribed in thi paper involve the modeling o a channel air low interacting with a porou tructure. The luid i olved by uing a Eulerian ormulation [8] on a Carteian grid that overlap the porou tructure, while thi latter i dicretied by Lagrangian hell baed on the Belytchko-Lin-Tay ormulation [9]. In the application the porou tructure are nylon abric decribed at the macrocopic cale 3 (*MAT_FABRIC). Let Ω R, the domain occupied by the abric mehed by hell element (*ELEMENT_SHELL) baed on the Belytchko-Lin-Tay hell ormulation [9] (*SECTION_SHELL), and let Ω denote it boundary. An updated Lagrangian inite element ormulation i conidered: the movement o the thin porou medium Ω decribed by x i ( t), ( i =,,3) can be expreed in term o the reerence coordinate X i ( t), ( i =,,3) and time t: x i = x ( X, t) () i The momentum equation i given by Eq.(6) in which σ i the Cauchy tre, ρ i the denity, dv i the orce denity, dt boundary Ω : α i acceleration and n i the unit normal oriented outward at the dv ρ = div(σ ) + () dt de = σ : grad( v) +. v dt ρ (3) 9-

3 9 th International LS-DYNA Uer Conerence Fluid/Structure The Lagrangian ormulation or the abric hell i input by the ollowing keyword: *PART Canopy $ pid, ecid, mid,, *SECTION_SHELL $ ecid, elorm, *MAT_FABRIC $ mid, ro EA, EB, EC, PRBA, , 4.309e8, 0, 0, 0.4 *ELEMENT_SHELL $ eid, pid, n, n, n3, n4 The olution o Eq.()-(3) atiie the diplacement boundary condition Eq.(4) on the boundary and the traction boundary condition Eq.(5) on the boundary. Ω Ω x ( X, t) = D( t) on σ. n = τ ( t) on Ω (4) Ω (5) In the application the traction boundary i the luid-ructure interaction urace. Thu the traction i the air preure applied on the abric canopy. In thi paper the hell ormulation ued to model thi canopy i the Belytchko-Lin-Tay ormulation [9]. The Belytchko-Lin-Tay hell 4-node element i baed on a co-rotational coordinate ytem and a contitutive computation uing a rate o deormation. The embedded element coordinate ytem that deorm with the element i deined in term o our corner node. A the element deorm, an angle may exit between the iber direction and the unit normal o the element coordinate ytem. The magnitude o thi angle i limited in order to keep a plane hell geometry. In thi local ytem, the Reiner- Mindlin theory give the velocity o any point in the hell according to the velocity o midurace and the rotation o the element' iber. Then, the rate o deormation are computed at the center o the element. The new Cauchy tree are computed by uing the material model and by accounting or the incremental rotation, Δ R. Thi latter i obtained by expreing the element bae vector at t(n+) in the local ytem at t(n). Since the material rotation i equal to the rotation o the local ytem, Δ R i the identity matrix. Thi involve the Belytchko-Lin- Tay hell element i a computationally eicient alternative to the Hughe-Liu hell element. Then, the element-centered reultant orce and moment are obtained by integrating the tree through the thickne o the hell. The relation between thee orce and moment and the local nodal orce and moment are obtained by perorming the principle o virtual power with one point quadrature. Finally, uing the tranormation relation deined by the global component o the corotational unit vector derive the global nodal orce and moment. The ollowing paragraph preent the decription o the luid. 3 Let Ω R, repreent the domain occupied by air deined by *MAT_NULL, and let Ω denote it boundary. Thi domain i mehed by olid element (*ELEMENT_SOLID) baed on a Eulerian ormulation, which i a particular cae o the ALE ormulation (*SECTION_SOLID_ALE). The equation o ma, momentum and energy conervation or the 9-3

4 Fluid/Structure 9 th International LS-DYNA Uer Conerence air low in a general ALE ormulation in the reerence domain, which i here the meh are given by: ρ + ρdiv( v) + ( v w) grad( ρ) = 0 (6) t v + ρ( v w). grad( v) = div( σ ) + t ρ (7) 9-4 e + ρ( v w). grad( e) = σ : grad( v) +. v t ρ (8) where i the body orce and e i the peciic internal energy. v and w are the luid and meh velocity ield repectively. I v = w, the equation (6)-(8) give the previou Lagrangian ormulation. In the Eulerian ormulation w = 0, thi aumption eliminate the remehing and moothing proce, but doe not impliy the Navier-Stoke equation (6)-(8). Thee equation are olved by the plit approach detailed in [8],[0] and implemented in LS-DYNA. In the equation (6)-(8) ρ i the denity and σ i the total Cauchy tre given by: T σ = p. Id + μ( grad( v) + grad( v) ) (9) where p i the preure and μ i the dynamic vicoity. The preure i computed by the ideal ga law (*EOS_IDEAL_GAS): p = ρ ( Cp Cv) T (0) where Cp and Cv are the peciic heat capacitie at contant preure and volume repectively. The Eulerian ormulation i input by the ollowing keyword: *ALE_MULTI-MATERIAL_GROUP_PART $ pid 3 *PART Ambient air part $ pid, ecid, mid, eoid,,, *PART Initial air part $ pid, ecid, mid, eoid 3, 3,, *SECTION_SOLID $ ecid, elorm, aet, 4 *SECTION_SOLID $ ecid, elorm 3, *MAT_NULL $ mid, ro

5 9 th International LS-DYNA Uer Conerence Fluid/Structure,.9 *EOS_IDEAL_GAS $ eoid, cv0, cp0, cl, cq, T0, V0, 77.5, 004.5, 0, 0, 70., 0 *ELEMENT_SOLID $ eid, pid, n, n, n3, n4, n5, n6, n7, n8 The multi-material Eulerian ormulation (*ALE_MULTI-MATERIAL_GROUP_PART and elorm=) i employed in order to highlight the eect o the poroity in the application. Thu a part (pid=3) i created and the dummy material repreent the initial air around the parachute. Thi initial material i puhed out o the Eulerian grid by the ambient material (pid=), whoe the olid element occupy a layer at the inlet o the channel meh. Equation (6)-(8) are completed with appropriate boundary condition. The part o the boundary at which the velocity i peciied i denoted by Ω. The inlow boundary condition i: v g( t) on Ω = () In the application the channel low i created by an inlow condition applied on the inlet node by the ollowing keyword: *BOUNDARY_PRESCRIBED_MOTION_NODE $ nid, do, vad, lcid The traction boundary condition aociated with Eq.(4) are the condition on tre component. Thee condition are impoed on the remaining part o the boundary o Ω. σ. n = h( t) on Ω (3) In the application the traction boundary i the luid-ructure interaction urace ( Ω = Ω ) Actually thi boundary i the parachute and the traction orce i computed by the porou coupling method implemented in the ALE ormulation (*CONSTRAINED_LAGRANGE_IN_SOLID). The ollowing ection preent thi method. Fluid Porou Structure Interaction In an explicit time integration problem the main part o the procedure in the time tep i the calculation o the nodal orce. Ater computation o luid and tructure nodal orce we compute the orce due to the coupling, thee will only aect node that are on the luid - porou tructure interace. For each tructure node, a depth penetration d i incrementally updated at each time tep, uing the relative velocity v rel at the lave and mater node. For thi coupling, the lave node i a tructure meh node, wherea the mater node i not a luid meh node, it can be viewed a a luid particle within a luid element, with ma and velocity interpolated rom the luid element node uing inite element hape unction. The location o the mater node i alo computed uing the ioparametric coordinate o the luid element. I d n repreent the n penetration depth at time t = t, it i incrementally updated in Eq.(4): d n+ = d n + v n+ / rel. Δt (4) 9-5

6 Fluid/Structure 9 th International LS-DYNA Uer Conerence In Eq.(4) 9-6 v n+ / n+ / n+ / rel = v v in which the luid velocity v i the velocity at the mater node location and the tructure velocity v i the velocity at the lave node location. The n coupling act only i penetration occur, n. d < 0, where n i built up by averaging normal o tructure element connected to the tructure node. The porou coupling orce are derived rom the integration o the Ergun Equation [7] on the hell volume: ( v. ) rel n dp = a( μ, ε) vrel. n + b( ρ, ε) (5) dzˆ in which ẑ i the local poition along the iber direction o the hell element and ε i the poroity. The coeicient a ( μ, ε ) i the reciprocal permeability o the porou hell or vicou coeicient. b ( ρ, ε ) repreent the inertia coeicient. For low under very vicou condition the econd term in Eq.(5), which repreent the inertia eect drop out and the Blake-Kozeny equation or laminar low in porou media i obtained. At high rate o low it i the irt term or vicou term, which drop out and the Blurke-Plummer equation or turbulent low in porou media i obtained. For the parachute application the inertia eect hould be preponderant. The orce F derived rom Eq.(5) i applied to both mater and lave node in oppoite direction to atiy orce equilibrium at the interace coupling, and thu the coupling i conitent with the luid-tructure interace condition namely the action-reaction principle. At the tructure coupling node, we applied a orce: F = F (6) wherea or the luid, the porou coupling orce i ditributed to the luid element node baed on the hape unction, at each node i (i=,..,8 ), the luid orce i caled by the hape unction N : Where F N i i. N i i the hape unction at node i. Since F = (7) 8 i= F i = F, the action-reaction principle i atiied at the coupling interace. In the application the channel low i created by an inlow condition applied on the inlet node by the ollowing keyword: *CONSTRAINED_LAGRANGE_IN_SOLID $ lave, mater, typ, mtyp, nquad, ctype, direc, mcoup 45,, 0, 0,, $ tart, end, pac, ric, rmin, norm, normtyp, damp $ cq, hmin, hmax, ileak, pleak, lcidpor, nvent, iblock $ iboxid, ipenchk, intorc, ialeo, lagmul, pacmm, thk $ a, b, a, b, a3, b , The coupling with porou hell i deined by ctype= and the one with porou olid correpond to ctype=. In thi latter cae all the coeicient on the lat line can be deined or each direction (a and b or x-direction, a and b or y-direction and a3 and b3 or z- direction). For hell a and b are the vicou and inertia coeicient normal to the egment i

7 9 th International LS-DYNA Uer Conerence Fluid/Structure repectively. The parameter thk i a cale actor o the hell thickne. The ollowing paragraph preent the application o thi approach to a porou dik parachute in terminal decent. Porou Parachute Application A wind tunnel tet o dik-gap-band parachute deign were carried out by Cruz and al. [] in the ramework o the Mar Exploration Rover miion. In thi paper one o thee parachute named.6 Viking made in MIL-c-700 type III abric i modeled by the porou Euler-Lagrange coupling method. Firt the vicou and inertia coeicient mut be deterrmined by uing the experimental permeability curve o the MIL-c-700 type III abric [] (ee Fig.). Second the value o thee parameter are checked with a model tet. Finally the parachute model i perormed. Determination o the Vicou and Inertia Parameter A indicated in the introduction the poroity o the canopy i aumed contant. At the teady tate the air denity and dynamic vicoity are uppoed uniorm. Under thee aumption the vicou and inertia parameter in Eq.(5) are contant. The experimental permeability curve o the MIL-c-700 type III abric give the rate o low through the nylon canopy veru the preure drop. To determine the vicou and inertia parameter in Eq.(5), the Ergun theoretical permeability hould be a parabolical it o the experimental one. Thu the coeicient ax and bx were computed by olving the ollowing ytem: dp dp / e = a. v / e = a. v + b. v + b. v (8) where e = 0. 06mm i the hell thickne and the couple o point ( v, dp ) and ( v, dp ) wa choen on the experimental plot o that the Ergun equation it it a cloe a poible. The value o the vicou and inertia parameter are a = kg m. and b = 48054kg. m. Figure. EXPERIMENTAL POROSITY CURVES [] A better approach to determine thee coeicient hould be to employ the ollowing equation derived rom the Ergun theory: 50μ( ε) a = (9) 3 D ε 9-7

8 Fluid/Structure 9 th International LS-DYNA Uer Conerence b.75ρ( ε) = (0) 3 Dε vvoid 6( ε) V where ε i the poroity : ε = and D i a characteritic length deined by: D = vtotal S with V, the volume o the canopy and S, the wetted urace. However it i tricky to get the poroity o the MIL-c-700 type III abric in the literature. The ollowing application i dedicated to the validation o thee parameter. Model Tet The model tet i a channel with a contant precribed rate o air low at the inlet. The air denity 3 i.9kg. m. The channel ketched on Fig. i a Eulerian meh o 3000 cubic olid element baed on the luid ormulation decribed previouly. The irt layer o olid element on Fig. i compoed o ambient or reervoir element with a contant preure. The quare ection o the 3 channel i 00m. A deormable nylon (MIL-c-700 type III abric, ρ = kg. m, E = GPa ) hell occupie all one ection o the channel, which i located at m rom the inlet. Thi membrane i mehed by 00 Lagrangian Belytchko-Lin-Tay quare hell element. The imulation time i enough large to reach the teady tate. It i 0ec. The run take about h on a AMD Opteron Proceor 48 (CPU: GHZ, cache ize:mb) becaue o the time tep i caled down to avoid intability o the computation: Δ t = 55μ. Actually the timetep need to be adapted to prevent the run rom crah. The higher the velocity i, the lighter the abric i, the lower the time tep hould be. Figure. CHANNEL MODEL The average preure on the canopy at the teady tate i pot-treated or dierent inlow velocitie. The purpoe i to check i the porou behaviour o the abric i well modeled. The rate o air low through the deormed nylon hell enable to compute an average permeability velocity, or which the experimental and numerical preure drop through the abric are compared on Tab.. : 9-8

9 9 th International LS-DYNA Uer Conerence Fluid/Structure Inlow velocity (m/) Table. NUMERICAL AND EXPERIMENTAL PRESSURE DROPS Permeability Velocity (m/) Experimental Preure Drop (Pa) Numerical Preure Drop (Pa) Relative Error (%) % % % % % The lower the velocitie are, the larger the relative error on Tab. are. However the relative error are acceptable. Thu the Ergun equation with a = kg m. and 3 b = 48054kg. m approximate well the porou behaviour o the MIL-c-700 type III abric which make up the parachute canopy o the ollowing paragraph. Parachute Model The whole model i baed on the inormation o the article o Cruz et al. []. The geometry and dimenion o the releaed.6 Viking parachute and module model during the terminal decent are hown on Fig.3. Figure 3..6 VIKING PARACHUTE AND MODULE MODELS The canopy and module wall are mehed by Lagrangian Belytchko-Lin-Tay hell [9]. A or the model tet air low i modeled by an Eulerian meh, which repreent the wind tunnel. Thu the ection o the Eulerian grid i a quare o ide m (6eet). The rate o low i maintained by a contant velocity at the inlet or everal Mach number: 0.34, 0.9 and The air denity i.9kg. m and the ound peed i 345m.. 9-9

10 Fluid/Structure 9 th International LS-DYNA Uer Conerence t=0ec t=0.0ec t=0.ec Figure 4. FLOW OF THE INITIAL AIR AT M=0.34 Figure 4 emphaize the eect o the porou coupling on the air low at M=0.34. On thi igure the dynamic o the initial air i highlighted by a colored volume o raction. A t=0ec air urrounding the parachute and illing all Eulerian domain ha an initial velocity correponding to the contant drop velocity o the module. Then thi air i emptied out the computational luid domain however a part i trapped in a vortex between the parachute and module. The trapped air can not only ecape through the parachute vent but alo through the porou canopy. The drag orce applied on the parachute canopy i invetigated and the numerical drag coeicient at the teady tate i compared to the experimental one given by []. The experimental drag coeicient i deined in [] by Eq.(): C D F = () D / ρv S 0 where S 0 i the nominal urace and V i the velocity at the inlet o the channel. The drag orce F D i a luid-tructure interaction orce given by the databae ile, dbi. For each Mach number the drag orce ocillate more or le. Thee luctuation are due to the wake o the backhell, which perturb the teady tate o the canopy. Thu the numerical drag orce i gived on Tab. with an uncertainty like the experimental tudy. Thi range o orce i deined by the highet and lowet value reached by the luctuation. Table. NUMERICAL AND EXPERIMENTAL DRAG COEFFICIENTS Mach Number Inlow Velocity (m/) Experimental Drag coeicient Numerical Drag coeicient ± ± ± ± ± ± On Tab. the experimental and numerical reult are cloe. However the luctuation make the etimation o the teady orce tricky and they might aect the tability o the canopy. A 9-0

11 9 th International LS-DYNA Uer Conerence Fluid/Structure propective invetigation will model the tability o the parachute or which experimental data can be ound in []. Concluion Thi paper ha decribed a method to olve luid-tructure interaction problem between a thin porou media and a luid. Thi method wa ucceully applied to the porou problem o a dik parachute in terminal decent at contant velocity. The numerical and experimental reult agree well. The propective goal o thi ongoing reearch will be to implement Ergun coeicient what will depend on the poroity, denity and dynamic vicou. Reerence [] Bear, J., 97, Dynamic o luid in porou media, Dover, New York. [] Biot, M.A., 94, General theory o three dimenional conolidation, J. Appl. Phy.,, pp [3] Benouan, A., Lion, J.L., Papanicolaou, G., 978, Aymptotic analyi or periodic tructure, Studie in Mathematic and It Application, 5, North-Holland, Amterdam. [4] Zhikov, V.V., Kozlov, S.M., Oleinik, O.A. 994, Homogenization o dierential operator and integral unctional, Springer-Verlag, Berlin. [5] Sanchez-Palencia E., 98, Non-homogeneou media and vibration theory, Lecture Note in Phyic, 7, Springer-Verlag, Berlin. [6] Mei, C.C., Auriault, J.-L., 989, Mechanic o heterogeneou porou media with everal patial cale, Proc. Roy. Soc. Lond., A 46, pp [7] Ergun, S., 95, Fluid low through packed bed, Chem. Eng. Prog., 48(), [8] Benon D.J., 99, Computational method in Lagrangian and Eulerian hydrocode, Computer Method in Applied Mechanic and Engineering, 99(), pp [9] Belytchko T., Lin J., Tay C.S., 984, Explicit algorithm or nonlinear dynamic o hell, Comp. Meth. Appl. Mech. Engrg., 4, pp-5-5. [0] Hughe, T.J.R., Liu, W.K., Zimmerman, T.K., 98, Lagrangian Eulerian inite element ormulation or vicou low, J. Comput. Method Appl. Mech. Engrg.,, pp [] Cruz, J. R., Mineck, R. E., Keller, D. F., Bobkill, M. V., Wind Tunnel Teting o Variou Dik-Gap-Band Parachute, AIAA 003-9, 7th AIAA Aerodynamic Decelerator Sytem Technology Conerence and Seminar, 9 May 003, Monterey, Caliornia. [] Air Force Flight Dynamic Laboratory Technical Report (AFFDL-TR-78-5), Recovery Sytem Deign Guide, June

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