(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate
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1 Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1 Radologcal Physcs & Advsory Dvson, BARC, Mumba, Inda Technology Development Dvson, BARC, Mumba, Inda ABSTRACT Lattce Boltzmann models for dffuson equaton are generally n Cartesan coordnate system. Very few researchers have attempted to solve dffuson equaton n sphercal coordnate system. In the lattce Boltzmann based dffuson model n sphercal coordnate system extra term, whch s due to varaton of surface area along radal drecton, s modeled as source term. In ths study dffuson equaton n sphercal coordnate system s frst converted to dffuson equaton whch s smlar to that n Cartesan coordnate system by usng proper varable. The dffuson equaton s then solved usng standard lattce Boltzmann method. The results obtaned for the new varable are agan converted to the actual varable. The numercal scheme s verfed by comparng the results of the smulaton study wth analytcal soluton. A good agreement between the two results s establshed. Keywords:Radal dffuson; lattce Boltzmann method; sphercal coordnate 1. Introducton Nowadays the Lattce Boltzmann (LB) method [1 3] has emerged as a promnent numercal technque for solvng partal dfferental equatons that model varous scentfc and engneerng problems [, 4]. LB method, whch was ntally ntroduced to solve Naver Stokes equaton [5], has been successfully appled to solve flow, heat and mass transport problems [1, 6, 7]. Dffuson equaton s a second order parabolc equaton whch s generally appears n the feld of heat and mass transport. Dffuson equaton n sphercal and cylndrcal coordnate systems has mportant practcal applcatons because most of engneerng problem are assocated wth sphercal or cylndrcal geometry. Standard LB method solves dffuson equaton n Cartesan coordnate system. To solve dffuson equaton n sphercal coordnate system, researchers have modeled the extra term that arses due to ncrease n surface area along radal drecton as source term n the LB equaton [8 10]. In ths work, the dffuson equaton n sphercal coordnate system s frst converted to dffuson equaton whch s smlar to that n Cartesan coordnate system. Ths converson s done by makng a proper substtuton of the actual varable. The dffuson equaton n the new varable s then solved usng standard LB method. The solutons of the LB equaton are then converted back to ts orgnal varable. The LB solutons are tested by solvng bench mark problems. The paper s organzed n the followng way. In Secton, one dmensonal dffuson equaton n sphercal coordnate system s converted to the form whch s smlar to dffuson equaton n Cartesan coordnate system. The LB dffuson model for Cartesan coordnate system s descrbed n Secton 3. The LB scheme s verfed and tested by solvng benchmark problem n Secton 4. Fnally, conclusons are drawn n Secton5.. Dffuson Equaton n Sphercal Coordnate System The dffuson equaton n sphercal coordnate system for a unform and sotropc materal s wrtten as t = k r r r r + 1 r snθ θ snθ θ + 1 r sn θ y ϕ (.1) Copyrght 018 Debabrata Datta et al. do: /jmss.v1.815 EnPress Publsher LLC.Ths work s lcensed under the Creatve Commons Attrbuton-NonCommercal 4.0 Internatonal Lcense (CC BY-NC 4.0). by/4.0/ 1
2 where y s the dffusve varable (t may be temperature, solute concentraton etc. ), k s called dffuson coeffcent, (r, θ, ϕ) s a pont n sphercal coordnate. One dmenson form of Eq. (.1) n radal drecton can be wrtten as (r,t) = k y(r,t) t r + k (r,t) (.) r r The second term of the rght hand sde of the above equaton s due to ncrease n surface area along radal drecton. Because of ths term standard LB method cannot be used to solve the Eq. (.). In the rest of ths Secton, Eq. (.) s modfed to the form of dffuson equaton whch s smlar to that n Cartesan coordnate system by substtutng the actual dffusve varable (y) by a new varable of the form u r,t = ry r,t (.3) The dervatves terms n the above equaton can also wrtten n term of the new varable as (r,t) r (r,t) t = 1 r y(r,t) r = 1 u(r,t) r r u(r,t) r r Usng the above equatons Eq. (.1) can be wrtten as = 1 u(r,t) r t (.4) u(r,t) u(r,t) r r (.5) + u(r,t) r 3 (.6) 1 u(r,t) = k u(r,t) r t r r k u(r,t) r + ku(r,t) r r 3 + k u(r,t) r ku(r,t) r r 3 (.7) Snce the last 4 terms of the above equaton cancel out each other, we get the modfed equaton as u(r,t) = k u r,t t r (.8) The above equaton s smlar to dffuson equaton n Cartesan coordnate system and therefore can easly be solved usng standard LB dffuson model. The LB soluton of Eq. (.8) s n terms of u(r,t), whch can be wrtten n terms of y(r,t) usng Eq. (.3). 3. Lattce Boltzmann Dffuson Model LB equaton whch governs the evoluton of partcle dstrbuton functon s a dscrete velocty Boltzmann equaton. The dscrete velocty Boltzmann equaton s solved n a unform doman of lattce nodes. The LB equaton wth Bhatnagar-Gross-Crook (BGK) collson operator can be wrtten as f r + e t, t + t = f r, t + Ω BGK r, t 3.1 Ω BGK r, t = 1 τ f eq r, t f r, t 3. where f r, t s partcle dstrbuton functon at spatotemporal coordnate ( r, t ) along th drecton, e represents partcle velocty along th drecton, Ω BGK r, t s BGK collson operator along th drecton at same spatotemporal coordnate, t s tme step, τ s relaxaton coeffcent, and f eq r, t s partcle equlbrum dstrbuton functon (EDF) along th drecton. The EDF for dffuson process s wrtten as f eq r, t = w C r, t (3.3) where w are the weghts for partcle s dstrbuton functon along th drecton. For 1-D most commonly used lattces are D1Q, D1Q3, for -D most commonly used lattces are DQ4, DQ5 and for 3-D, D3Q15, D3Q19 etc. Here for DnQm lattce n represent the dmenson of the problem and m represent the number of dscrete velocty vectors. Schematc of D1Q3 and DQ5 lattces are shown n Fg (3.1) and (3.), respectvely. The values of w for a D1Q3 lattce are 4/6 for =0 and 1/6 for =1 and.
3 Fgure 3.1. D1Q3 Lattce Fgure 3.. DQ5 Lattce The EDF defned n Eq. (3.3) follows followng constrants. f eq r, t = C r, t (3.4) e j f eq r, t = 0 (3.5) e j e k f eq r, t = e s τ 1 δ jk (3.6) where e s s called pseudo sound speed [3]. Partcle velocty drectons are connected to the neghborng lattce ponts such that there s a free streamng n between two lattce ponts. Velocty components for D1Q3 and DQ5 lattces are as gven n Eq. (3.7) and (3.8), respectvely. = = 0, = 0 th 1, = 1, 0, = 0 th 1, h 1, = 1,,3,4 (3.7) (3.8) Durng the recovery of macroscopc dffuson equaton (.8) from LB equaton (3.1 and 3.) usng Chapmann-Enskog multscale expanson technque, followng relatonshp between lattce dffuson coeffcent and relaxaton parameter (τ) s establshed [8] = h (3.9) Macroscopc partcle densty s calculated by summng over dstrbuton functons as, =, (3.10) The LB equaton (3.1 and 3.) can be solved numercally by LB algorthm whch conssts of followng two processes. Collson process: In collson process partcles dstrbuton functon relaxes towards local equlbrum dstrbuton functon and t can be descrbed by the followng equaton 䁈, + =, + Ω, (3.11) where, t + s the post collson partcle s dstrbuton functon, the values of collson operators as same as gven n Eq. (3.). Streamng process: In ths process, partcles move from one lattce pont to nearest lattce pont along the drecton of the lattce velocty. Computatonally ths process s just memory swappng. Algorthm of ths process can be wrtten as 3
4 +, + =, Addtonal bounce-back boundary condtons are mposed at obstacle stes and along boundary walls at whch partcles reverse ts drecton after collson wth obstacles or boundary walls. 4. Numercal Examples The developed LB scheme s verfed and valdated by solvng followng benchmark problems. 4.1 Dffuson of solute from the surface to the center of a sphere Ths test problem models the dffuson of solute from the surroundng envronment to a cementtous faclty of sphercal shape. Snce the surroundng meda s very large, constant supply of solute at the surface of the faclty s a reasonable assumpton. Ths process can be mathematcally modeled as a radal dffuson equaton (.) wth followng ntal and boundary condtons., = 0 = 0 = 0, = 0 =, = 0 (4.1) In terms of the new varable, (, ) as gven n Eq. (.), the above ntal and boundary condtons can be wrtten as, = 0 = 0 = 0, = 0 =, = 0 (4.) Closed form soluton of the problem can be wrtten as (, ) = 0 + =1 1 sn exp 0 (4.3) The LB smulaton s carred out n lattce unt wth 101 lattce ponts. The physcal lattce length s 0.1 m. Followng nput parameters are used n the smulaton.the concentraton profle of solute nsde the sphercal object s calculated and the results are compared wth analytcal soluton. Graphcal plots of spatal profle of solute concentraton after 365, 1000 and 3000 days are shown n the Table 1. Parameter Radus (a) of the sphere Inlet concentraton (C 0 ) Value 10 m 1.0 mg/l Dffuson coeffcent (D) m /s Smulaton tme Relaxaton parameter (τ) 1.0 Physcal lattce length (dr) 365, 1000 and 3000 days 0.1 m Table 1. Varous parameters used n the smulaton 4
5 Fgure 4.1. Spatal profle of solute concentraton after 365, 1000 and 3000 days 5. Conclusons One dmensonal radal dffuson equaton n sphercal coordnate system s solved usng standard LB equaton. The extra term n the governng radal dffuson equaton s not modeled as source term, rather the dffuson equaton s modfed usng a proper varable and the resultant equaton s exactly smlar to that n Cartesan coordnate system. The results show that the scheme s capable to smulate the radal dffuson equaton very accurately. References 1. Chen S, Doolen GD. Lattce Boltzmann method for flud flows. Annu. Rev Flud Mech. 1998; 30: Succ, S. The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, Oxford, U. K.: Oxford Unv. Press Wolf-Gladrow, D.A. Lattce-Gas Cellular Automata and Lattce Boltzmann Models: An Introducton. New York: Sprnger Benz R, Succ S, Vergassola M. The lattce Boltzmann equaton: theory and applcatons, Phys. Rep. 199; : Frsch U, Hasslacher B, Pomeau Y. Lattce-gas automata for the Naver-Stokes equaton, Phys. Rev. Lett. 1986; 56 (14): Guo ZL, Shu C. Lattce Boltzmann method and ts applcaton n engneerng, World Scentfc press Zhao CY, Da LN, Tang GH, et al. Numercal study of natural convecton n porous meda (metals) usng Lattce Boltzmann Method (LBM), Internatonal Journal of Heat and Flud Flow 010; 31 (5): Mohamad A. Lattce Boltzmann Method Fundamentals and Engneerng Applcatons wth Computer Codes, Sprnger, London Zhou JG. Axsymmetrc lattce Boltzmann method, Phys. Rev. E 008; 78: Mohamad AA. Lattce Boltzmann method for heat dffuson n axs-symmetrc geometres, Prog. Comput. Flud Dyn. 009; 9 (8):
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