CONTROLLED FLOW SIMULATION USING SPH METHOD

Transcription

1 HERI COADA AIR FORCE ACADEMY ROMAIA ITERATIOAL COFERECE of SCIETIFIC PAPER AFASES 01 Brasov, 4-6 May 01 GEERAL M.R. STEFAIK ARMED FORCES ACADEMY SLOVAK REPUBLIC COTROLLED FLOW SIMULATIO USIG SPH METHOD Vasle ĂSTĂSESCU*, Lavna GAVRILĂ** *Mltary Techncal Academy, Bucharest, Romana** Ar Force Academy, Braşov, Romana Abstract: Smoothed Partcle Hydrodynamcs (SPH) method s a meshless numercal method, developed after 1977 and nowadays ths method s practcally valdated as flud flow s concerned. The most mportant aspects and the most dffcult problems, lke flud free surface, nteracton flud-structure any many others can be better solved and n a shorter tme too. Ths paper presents some theoretcal fundamentals of the SPH method, ncludng specfc approxmaton of aver-stokes equatons n ths numercal method and next to t, an nterestng applcaton s also presented. The applcaton s referrng to the numercal smulaton of a controlled flow, for nstance, through a gate. Ths flud flow numercal smulaton, n such condtons, has many practcally applcatons and s an mplct descrbng of the free flud surface. The authors hope that ths numercal method (SPH) to be a ftted one, n applcatons such aerodynamcs and a drect analyss of structure behavor under aerodynamc loads. These aspects are over the subect of ths paper, but n future, we wll present more detals and examples regardng usng of the SPH method n aerodynamcs. We also hope that ths paper to be useful for teachers, researchers and students nterested n SPH method. Keywords: SPH, flud flow, numercal analyss, kernel functon, smoothng length 1. ITRODUCTIO SPH (Smoothed Partcle Hydrodynamcs) method s a grddles Lagrangan technque whch comes from astrophyscs (Lucy, 1977). The method was extended to flud smulaton, especally wth free-surface (Monaghan, 199), nowadays SPH method beng also used n many scentfc felds. Appled mechancs doman s perhaps the last one, but t s ntensvely researched and sgnfcant advances have also been made.. THEORETICAL FUDAMETALS OF THE SPH METHOD The SPH method belongs to the meshless methods, so the nvestgated doman s represented by a number of nodes, representng the partcles of ths doman, havng ther materal and mechancal (mass, poston, velocty etc.) characterstcs. Each partcle represents an nterpolaton pont on whch the materal propertes are known. Practcally, depandng on the used software, the boundary condtons have to be mposed to some of partcles, accordng to the problem analyzed, lke n the case of fnte element method. Theoretcal fundamentals of ths aspect are a bt dfferent and requre more space to be presented here. The problem soluton s gven by the computed results, on all the partcles, usng an nterpolaton functon. We can say that the fundamentals of SPH theory consst n nterpolaton theory; all the behavor laws are transformed nto ntegral equatons. The kernel functon, or smoothng functon, often called smoothng kernel

2 functon, or smply kernel, gves a weghted approxmaton of the feld varable (functon) n a pont (partcle). Integral representaton of a functon, used n the SPH method starts from the followng dentty: f ( x) f ( x ) δ( ) dx (1) where f s a functon of a poston vector x, whch can be an one-, two- or threedmensonal one; δ ( ) s a Drac functon, havng the propertes: 1 x x δ( ) () 0 x x In equaton (1), s the functon doman, whch can be a volume, that contans the x, and where f ( x ) s defned and contnuous. By replacng the Drac functon wth a smoothng functon W (,h ) the ntegral representaton of f ( x ) becomes: f ( x) f ( x ) W( dx (3) where W s the smoothng kernel functon, or smoothng functon, or kernel functon. The parameter h, of the smoothng functon W, s the smoothng length, by whch the nfluence area (support doman) of the smoothng functon W s defned. smoothng functon W nstead of Drac functon, the ntegral representaton can only be an approxmaton. Ths s the reason for the name of kernel approxmaton. Usng the angle bracket, ths aspect s underlned and the equaton (3) can be rewrtten as: f ( x) f ( x ) W( dx (4) The smoothng functon W s usually chosen to be an even one, whch has to satsfy some condtons. The most mportant requrements of a kernel functon are presented below: a) the smoothng functon has to be normalzed over ts support: W ( dx 1 (5) b) the smoothng functon has to be compactly supported: W ( 0 for > kh (6) c) the smoothng functon has to be postve for any pont at x wthn the support doman: W ( 0 (7) d) the smoothng functon value has to be monotoncally decreasng wth the ncrease of the dstance away from the partcle. e) the smoothng functon value has to satsfy the Drac delta functon condton as the smoothng length approaches to zero: lm W ( δ( ) h 0 (8) Fg. 1 The support doman of the kernel functon As long as Drac delta functon s used, the ntegral representaton, descrbed by equaton (1), s an exact (rgorous) one, but usng the f) the smoothng functon value has to be an even functon (symetrc). The lterature presents dfferent smoothng functon. Theoretcally, any functon havng the propertes presented above, can be employed as SPH smoothng functon.

3 HERI COADA AIR FORCE ACADEMY ROMAIA ITERATIOAL COFERECE of SCIETIFIC PAPER AFASES 01 Brasov, 4-6 May 01 GEERAL M.R. STEFAIK ARMED FORCES ACADEMY SLOVAK REPUBLIC One of the most used smoothng functon s a cubc B-splne kernel functon, n the form gven by relaton (9), where s r/h, n s the number representng the spatal dmenson and s a constant whch has the value: /3, 10/7 π or respectvelly 1/ π, dependng on the space dmenson (1D, D or 3D) s s ;0 s < W ( s, h) ( s) ;1 s < n h 4 (9) 0; s fundamental physcal laws of conservaton, whch n SPH formulaton represents a summaton of the Lagrangan form: a) the contnuty ecuaton: t 1 m v (10) where x s a generalzed coordnate and denotes the coordnate drectons. As the densty s concerned, ths s expressed n the followng two forms, the last beng proposed for mprovng the accuracy near the free boundares or near the materal nterfaces havng a densty dscontnuty. The graphcal representaton of ths smoothng functon and ts dervatves (frst and second) can be seen n the Fgure. 1 m W (11) 1 1 m W m W (1) b) the momentum equaton: v t 1 m σ σ (13) Fg. The cubc B-splne kernel functon v t 1 σ m σ (14) 3. SPH FORMULATIO OF THE AVIER-STOKES EQUATIOS The aver-stokes equatons consst of the followng equaton set, representng three Both forms are used, the last beng more popular. The symmetrzed form of both equatons leads to reducng of the errors. c) the energy equaton:

4 de dt 1 1 m p p v μ ε ε (15) de 1 m dt 1 p p v μ ε ε (16) t 0,10 s The above forms are most used n many SPH software. The superscrpts and, used n the relatons (13) (16) are used for denotng the coordnate drectons. The others notatons are those frequently used n flud mechancs ( ε s shear stran rate, μ s the dynamc vscosty and p s the pressure). t 0,5 s 4. COTROLLED FLOW SIMULATIO USIG SPH METHOD The controlled flow s referrng to the 3 collapse of a water quantty (1 m ) through a gate wth a varable space by a law. The model s a D one and ts orgnal state s presented n the Fgure 3. t 0,50 s t 1,00 s Fg. 3 The D SPH model The water s modelled by 601 partcles wth a nodal dstance of cm. Only the weght as body force s consdered. The gate movng on vertcal drecton and ts comng back n a tme shorter than analyss perod, the water wll be n two domans, at dfferent levels. Fgure 4 presents the water evoluton at dfferent moments. An elastcplastc-hydro materal model was used wth the water charaterstc parameters. t,00 s t 3,00 s

5 HERI COADA AIR FORCE ACADEMY ROMAIA ITERATIOAL COFERECE of SCIETIFIC PAPER AFASES 01 Brasov, 4-6 May 01 GEERAL M.R. STEFAIK ARMED FORCES ACADEMY SLOVAK REPUBLIC In the fgures 6 9 the tme evoluton of two partcles s presented. The ntal poston of these partcles s shown n the Fgure 5. t 4,00 s t 5,00 s Fg. 6 UX dsplacement of those two partcles t 6,00 s Fg. 4 Flud evoluton n tme Fgure 4 presents, n the same tme, the flud state and the velocty feld of partcles. Fg. 7 UY dsplacement of those two partcles Fg. 5 Intal poston of two partcles Fg. 8 X-coordnate of those two partcles

6 Analysng the evoluton of the X any coordnates and UX and UY dsplacements, we can know where those two partcles wll be fnally. So, for ths case, the both partcles wll be n the left sde of the gate (fgure 4), where the water level s greater than the rght sde of the gate. We also can obtan, by graphcal post-processng of the results, the water level around the gate. Fg. 9 Y-coordnate of those two partcles In the fgures 6 9, partcle A s the partcle 400 and partcle B s the partcle 135, presented n the Fgure 5. Fg. 10 The evoluton n tme of the water knetc energy The water knetc energy, presented n the Fgure 10 s n connecton wth the movng law of the gate, presented n the Fgure 11, as everyone could notce and understand the curve allure descrbng the knetc energy of the water. At the end of the analyss perod (sx seconds), the knetc energy s zero, ths meanng a properly analyss tme. Fg. 11 The movng law of the gate 5. COCLUSIOS The SPH hydrodynamcs model used n ths paper can be an example for many other smlar problem, SPH method beng a very powerful method. The same problem could be solved usng a 3D model, but more computer tme would have been necesary. Practcally, SPH method s valdated as an efcent numercal method n flud mechaqncs. Our work s ntended as a recommendaton for usng the SPH method n researchng and even n educaton. Many facltes are gong to be dscovered and only the method power and the user talent can lead to unexpected results. REFERECES 1. Lu, G. R., Lu, M. B., Smoothed Partcle Hydrodynamcs a meshfree partcle method, World Scentfc Publshng Co. Pte. Ltd., 009, ISB Lu, G. R., Meshfree Methods, Moovng Beyond the Fnte Element Method, Second Edton, CRC Press Taylor & Francs Group, 010, ISB ăstăsescu, V., Ilescu,., umercal Smulaton of the Impact Problems by SPH Method, Proceedngs of IV-th atonal Conference THE ACADEMIC DAYS of the Techncal Scence Academy of Romana, Iaş, 19-0 ovember 009, vol. 1, pag , ISS: Qngwe M., Advances n umercal Smulaton of onlnear Water Waves, World Scentfc Publshng Co. Pte. Ltd., 010, ISB , ISB