Title Novi algoritam za povezivanje polja brzine i tlaka Title (english) A new pressure-velocity coupling algorithm Author Severino Krizmanić Mentor Zdravko Virag (mentor)Committee member Jurica Sorić (predsjednik povjerenstva)Committee member Zdravko Virag (član povjerenstva)Committee member Mario Šavar (član povjerenstva)Committee member Hrvoje Jasak (član povjerenstva)Committee member Senka Maćešić (član povjerenstva)Granter University of Zagreb Defense date and country 2011-06-29, Croatia Scientific / art field, TECHNICAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics 53 - Physics Abstract U radu je razvijena metoda konačnih volumena za rješavanje Navier-Stokesovih jednadžbi za slučaj laminarnog nestlačivog strujanja. Metoda se temelji na novom algoritmu za povezivanje polja brzine i tlaka. Osnovna razlika novog algoritma u odnosu na danas najčešće korišteni SIMPLE algoritam, sastoji se u tome da je u njemu jednadžba kontinuiteta zadovoljena u svakoj iteraciji, a konačnom se rješenju približava korigiranjem masenih protoka kroz stranice konačnih volumena sve dok polje gradijenta tlaka ne postane bezcirkulacijsko. Brzina konvergencije novog algoritma (primijenjena u vlastito razvijenom računalnom programu) uspoređena je s brzinom konvergencije algoritma SIMPLE (primijenjenom u komercijalnom računalnom programu FLUENT) u nekoliko tipičnih slučaja strujanja fluida pri niskim i visokim vrijednostima Reynoldsova broja, te za slučaj slobodne konvekcije pri niskim i visokim vrijednostima Rayleighova broja. Temeljem dobivenih rezultata zaključuje se da je u primjerima neviskoznog i viskoznog strujanja: (1) brzina konvergencije novog algoritma značajno veća, pri čemu korisnik ne treba zadavati podrelaksacijske faktore za brzinu i tlak o kojima bi ta brzina ovisila, kao u slučaju algoritma SIMPLE. (2) Brzina konvergencije novog algoritma praktično ne ovisi o gustoći mreže, za razliku od brzine konvergencije algoritma SIMPLE koja opada s porastom gustoće mreže. Brzina konvergencije novog algoritma blago opada s porastom složenosti slike strujanja u konačnom rješenju (pojava vrtloga, tj. odvajanja strujanja) i primjenom sheme višeg reda točnosti (zbog primjene „deferred correction“ postupka). Kod primjene sheme višeg reda, s porastom gustoće mreže brzina konvergencije raste. (3) Učinkovitost novog algoritma (u smislu utroška računalnog vremena za postizanje željene točnosti rješenja) pokazala se većom, unatoč manje optimalnoj implementaciji novog algoritma u usporedbi s implementacijom algoritma SIMPLE u komercijalnom paketu FLUENT.
U primjerima slobodne konvekcije broj potrebnih iteracija za postizanje rješenja zadane točnosti raste obzirom na slučaj neviskoznog i viskoznog strujanja, zbog nepotpune implicitnosti u obračunu uzgonskih sila. Brzina konvergencije novog algoritma bi se povećala kad bi se jednadžbe za korekcije protoka i temperature rješavale simultano. Primijećeno je da broj potrebnih iteracija raste s porastom Rayleighova broja, ali u svim slučajevima novi algoritam brže konvergira od algoritma SIMPLE.
Novi algoritam rezultira sustavom linearnih algebarskih jednadžbi s manje nepoznanica, ali je matrica sustava gušće popunjena, tako da je sa stajališta zauzeća memorije zahtjevniji. Daljnji nedostatak novog algoritma se ogleda u činjenici da ova matrica nema svojstva poznate M-matrice, kao u slučaju SIMPLE algoritma, što zahtijeva dodatna istraživanja na razvoju efikasnih iterativnih rješavača za takav tip matrice sustava.
Abstract (english) A finite volume method for solving Navier-Stokes equations in cases of laminar incompressible flows was developed. The method is based on a new-proposed pressure-velocity coupling algorithm. The fundamental difference of the new algorithm, compared to the most frequently used SIMPLE algorithm, is that in the new algorithm, the continuity equation is satisfied in each iteration. Within the new algorithm, the final solution is approached by correcting the cell face mass flux until the pressure gradient field becomes irrotational. The new algorithm (implemented in a in-house computer code) and the SIMPLE algorithm (implemented in the commercial software package, FLUENT) are compared in terms of convergence rate in few typical cases of laminar flow ranging from low to high Reynolds number, and in cases of natural convection covering the range from low to high Rayleigh number. In accordance with the obtained results, it is concluded that in the cases of inviscid and viscous flow: (1) The convergence rate of the new algorithm is significantly higher, and the user need not to prescribe pressure and velocity under-relaxation factors on which this rate would depend, as is the case with the SIMPLE algorithm; (2) The convergence rate of the new algorithm is practically independent on mesh density (mesh size), in contrast to the convergence rate of the SIMPLE algorithm, which decreases with increasing mesh density. Rate of convergence of the new algorithm slightly decreases with increasing complexity of flow pattern (the appearance of vortices and zones of flow separation) and with applying a higher order differencing scheme (due to deferred correction approach). Here, when applying the higher order scheme, the convergence rate increases with increasing mesh density; (3) The new algorithm shows better efficiency than SIMPLE, regardless to possibly suboptimal coding of the new algorithm when compared to the SIMPLE algorithm, professionally coded within the FLUENT software package.
In cases of natural convection the number of iterations required to achieve the prescribed solution accuracy is greater with respect to the cases of pure inviscid and laminar flows, due to partially explicit treatment of the buoyancy force. Here, in order to increase the convergence rate of the new algorithm, the equations for the mass flux and temperature corrections should be treated in a coupled manner. The number of iterations required in new algorithm increases with the Rayleigh number, but here also, throughout the whole range of Rayleigh number, the new algorithm shows better performance than the SIMPLE algorithm.
The new algorithm results in a linear system with less unknowns, but with a denser matrix than in the case of SIMPLE algorithm, thus it is more memory demanding. The drawback of the new algorithm is that this matrix does not possess the properties of the well known M-matrix, as in the case of the SIMPLE algorithm, which requires additional research in finding of an efficient iterative solver for such type of matrix.
Keywords
računalna dinamika fluida
metoda konačnih volumena
algoritam FLOP
algoritam SIMPLE
brzina konvergencije
nepomaknuta mreža
nestrukturirana mreža
Keywords (english)
computational fluid dynamics
finite volume method
FLOP Algorithm
SIMPLE Algorithm
convergence rate
collocated grid
unstructured grid
Language croatian URN:NBN urn:nbn:hr:235:419465 Study programme Title: Mechanical Engineering and Naval Architecture Type of resource Text File origin Born digital Access conditions Open access Terms of use Created on 2020-05-15 14:33:33
