An effective blob approach to unsteady thermal buoyant flows

Transcription

1 Comutational Methods and Exerimental Measurements XII 685 An effective blob aroach to unsteady thermal buoyant flows C. Golia & B. Buonomo Deartment of Aerosace and Mechanical Engineering, Second University of Nales, Italy Abstract Vortex and thermal blob methods recently conceived by the authors to solve the couled momentum and energy equations in thermal free convection unsteady roblems, is udated here, with the use of a self-otimized FMM methodology, to comute the velocity from the vorticity field. Instead of the usual O(N ) needed for full direct comutations, this aroximate method requires for N blob articles to use only O(N) oerations. The reduced comuting time allows detailed analysis of long term, large flow field for roblems having Grashoff numbers of realistic magnitude, even with the use of modern PCs. After the resentation of the method, reliminary tests are discussed to ascertain the accuracy of the results and to test the use and erformances with classical roblems of interest. Keywords: vortex and thermal blob methods, article-based lagrangian method, unsteady flows, buoyant heat transfer, thermal lume. Introduction Even if vortex blobs methods have been extensively used for forced convection viscous roblems, see [] for a wide review, scant attention has been osed on natural thermal convection roblems. Indeed vortex methods automatically assume the asymtotic vanishing of vorticity, so if ones combine vortex and thermal blobs, the resultant method will be ideally at to solve fluid buoyant/mixed roblems in free sace such as lumes, fires and geostrohic flows, etc. without having to deal with the cumbersome ressure closure roblem.

2 686 Comutational Methods and Exerimental Measurements XII The method resented in [] was based uon Lagrangian articles (vortex and heat, indeendently treated) that, as comutational elements, transort (with the velocity induced by the vorticity field) momentum and energy. Vorticity is roduced by the thermal gradient resent in the flow field in the context of validity of Boussinesque hyothesis. Heat is generated by the Thermal Boundary/Initial Conditions. Both vorticity and heat diffuse according to their transort coefficients and Lalacians. The field equations Under the limit of validity of the Boussinesque aroximation we do consider the usual Continuity, Secific Momentum (V), Secific Internal Energy (c v T) equations: V = 0 () V + t ρ ( V ) V = + ν V β g ( T T ) 0 o () T + t ( V ) T = α µ T + Φ c v ρo Note that we have discarded the dissiation term in the energy equation (3) since it is usually negligible in free convection buoyant roblems. Since we want to make use of the Helmholtz vorticity equation, we make the rotor of () to get in -D [x, y, ω = ωk = V ]: ω + t 0 T T ( V ) ω = ν ω β g g (3) x y (4) y x The velocity field needed in (3 and 4) can be reresented as a convolution integral of the vorticity field (extension of the Biot-Savart law in free sace): D ( r r' ) kω V (r,t) = dr' (5) π r r' Due to its linearity, the Lagrangian aroach to the resulting roblem (3+4) is erformed, according to [3,4], by using a slitting technique that recasts it by exlicitly searating the convection from the diffusive stes: Convective ste: Diffusive ste: dω(r) = ν ω(r) β g dt d r = V(r, t) (6) dt x T g y y T x (7)

3 Comutational Methods and Exerimental Measurements XII The blob discretization dt(r) = α T(r) (8) dt In regularized vortex blob method, the discretization of the equations (6+7+8) is made by considering N-blobs roblems where the vorticity and the temerature are reresented for a general article located in (r, t) as: N Vortex Blob r r ( r,t) ω = Γ (t) η (9) σ σ = Thermal Blob N T r r T(r, t) = Θ (t) η σ σ = (0) where: η regularized blob (cutoff) function σ = λdr blob radius dr elementary volume radius λ overla arameter ( λ ) r osition of the -article at time t Γ = ωdr vortex blob intensity of the and -article (i.e. the elementary circulation of the velocity) Θ = Tdr thermal blob intensity of the -article From (5) it follows that the velocity field can be comuted as: Vrot ( r,t) = Λ Γ t = r σ N [ y, x] r r' () () The diffusive Lalacian oerators are discretized according to the Particle Strength Exchange (PSE) method roosed by Degong and Mass-Gallic [5], i.e. for Γ : N Γ ' q Γ r r Γ = Lal () σ q= σ The temerature gradient, needed in (9) is comuted as a generalized integral reresentation [6] that is coherent with PSE: ( ) N r r q q() t () t r r' Θ +Θ Θ (,) rt = Grd (3) σ q= πσ

4 688 Comutational Methods and Exerimental Measurements XII In (), (), (3) Λ(.), Lal(.), Grd(.) are kernel functions derived according to the secific η(.) blob function used. The Integration of the convective ste is made with 3 rd order low storage (Runge-Kutta like) Williamson scheme [7]. The regridding ste The accuracy of both the convection ste as well as of the diffusive ste requires that the article must, to a certain degree, be uniformly distributed in the field. This is done, when seemed necessary, by executing a regridding ste rior the diffusive ste: a regular lattice is concetually laid over the actual disordered one, and new articles are allocated on the new mesh oints and their values is interolated from the cluster of neighborhood ones on the old disordered lattice. A 4-th order interolation kernel with continuous derivatives and that conserve the first 3 moments is here used [8]: 3 3 7u + 4u 3 0 u u+ 36 u 7u 48 u Kcc4 ( u) = 3 ( 8+ u 8 u + u ) 48 u u The regridding ste ensures accuracy of the PSE oerators, but can be keenly utilized as a tool both to clean the field from noisy too small valued blobs (cancellation of articles) as well as to udate the field by adating it to the growing volume interested by flow henomena (creation of new articles). This is articular interesting since one does not must face the roblems of the metrics and of the dynamics of moving grid required by classical formulations. The regularized Blob Functions used are of 8-th order Gaussian tyes: Blobs function: 4 6 ( z ) η( z) = 4 6 z + z z 6 e π 4 6 ( z ) Λ ( z) = z 9z + z e π 4 6 ( z ) Lal( z) = z + 0z z 3 e π 4 6 ( z ) Grad( z) = 0+ 0 z 5z + z 3 e π Velocity Kernel: ( ) Lalacian Kernel: Gradient Kernel: 4 The FMM method As said before the resulting method reduces, in ractice, to a N-bodies roblem jeoardized by the time consuming rocess of comuting the velocities for each one of the N blobs that, for direct exact case, according to (), requires O(N ) oerations. This is a very heavy burden since we want to consider roblems for long times where N grows with time.

5 Comutational Methods and Exerimental Measurements XII 689 Since many toics of alied sciences and theoretical hysics must face the equivalent roblem (making the scalar roduct of very large and articular dense matrices with a vector) many algorithms have been roose [9, 0, ]. The algorithms of Ael [9] and Barnes and Hut [0] introduce the concets of a hierarchy structure, known as N-body Treecodes; they require O(N log(n)) oerations. The Fast Multiole Method devised by Rokling and Greengard [] that requires down to O(N) oerations, was awarded of the Steel Prize as one of the ten most significant findings of the 0 th century. The FFM relies on the mix of several ideas: (i) function reresentation and factorization, (ii) sace artitioning, (iii) translation theory, (iv) error analysis and bounds, (v) data structures. In D a hierarchy quad-tree of meshes, which refine the comutational box into smaller and smaller regions, is introduced. In articular, mesh level L=0 is equivalent to the entire field containing all articles, while the generic mesh level L+ is obtained from level L by subdividing each box into four equal children boxes. The number of boxes at mesh level L is then equal to 4 L. The nearest neighbour boxes are the set of all boxes, at same level, which have any boundary oint in common. The interaction list for a given box, X, is the collection of boxes, I, at the same level as the box X whose arents are nearest neighbours of the arent of X but which are not themselves nearest neighbours of X For a given article in a given box X at lowest level, we define as near articles those in the same box or in the boxes which are nearest neighbours of X, whereas all other ones are defined as far from articles. The velocity (for unitary Kernel) is formulated in comlex form as: N i V* = Γi π z z i= j i j i and exressed in terms of a comlex otential Φ: dφ dφ u = Re, v = Im dz dz ( zj) near far i log ( zi zj) Φ =Φ +Φ = Γ N π i = j i where: Φ near is obtained by direct calculations, Φ far is obtained by multiole exansions (i.e. a comlex Laurent series, centred in the box centre, is used for all inducing articles external to the box). The comutation of Φ far requires the following stes: Evaluation of the multiole exansion for each box al level L max,

6 690 Comutational Methods and Exerimental Measurements XII For every box, starting from level L max - the multiole exansion is obtained by translating and summing the multiole exansions of the four children, the same for all other levels, Starting from level L=, the multiole exansions of all the boxes of the interaction list (if not void) are comuted, and are called local exansions, The local exansions of each father box al the level L= are then translated to the four children of the successive level; this u to the finest level where the local exansions shall reresent the otential Φ far. The R-G method deends on the following arameters: N number of the overall blobs N Lev number of level N Box number of boxes N b number of articles within each box number or harmonics in the Laurent series exansion EV =Φ z Φ z Γ : redicted overall Error Value ( ) ( ) exact FMM i i deriving from to truncation of the series. The method can be made more efficient by introducing some otimization: Direct formulation of interaction between terms of each coule of articles, Number of terms for the conversion of multile in local exansion variable with the distance between boxes, deending from the evaluation of the resulting error that can be estimated as: ε = O ( c) where c is the distance between a given box and the box of the interaction list minus and the number of terms of the series. The otimization rocess conceived here fixes all arameters in terms of N and EV. The erformance of the self-otimized FMM was analysed by considering the time needed by the comutation of the velocity in N oints by N vortices randomly ositioned in a square. Besides the value of N, the other only arameter entered is the value of the Error Value exected in the comutation. Runs were erformed by varying N from.e+4 u to 3.E+5, with EV=.E-4, all values of ractical use. Figure shows the scaled results obtained and comared with a O(N) line. The overall linear trend is self-evident. Seckles aear when there is a nonotimal transition for the number of levels imosed by the algorithm. 5 The results The otimized FMM allows analyzing roblems with large number of blob articles and longer flow time, and to discover more detailed henomena u to the level of fine turbulent scales. In the following we shall resent some classical cases of free atches as well as of fixed atches of interest. +

7 Comutational Methods and Exerimental Measurements XII 69 Performance of the otimized FMM 0 scaled CPU time 0. CPU=O(N) 0.0 e+3 e+4 e+5 e+6 Nr blobs Figure : CPU scaled time erformances obtained by the otimized FMM for Error Value of.0e The free atches 5.. Purely vorticity free atches The ellitical vortex is the classical reference case for vorticity dynamics since it was demonstrated by Kirchhoff [3] that, in the inviscid case, the atch must maintain its shae and rotate rigidly with an angular seed Ω deending solely from the value of the vorticity ω and from the value of the axis ratio τ=b/a: Ω = ω τ ( + τ) In the secific case we show the results of a viscid case with ω=, τ=0.5, dx=dy=0.0, dt=0.005, N x =N y =0, overla=.5, regrid ste=0. For this case the inviscid theory foresees a 45 rotation for t=3.534 sec. It can be seen that the rotation is ractically the same of the theoretical one for the inviscid case, at least for relatively small times, where the viscous sreading of the vorticity is small comared to the dimension of the atch. Obvious since vorticity is null outside the atch the 0-th order Moment of the vorticity are invariant and equal to zero.

8 69 Comutational Methods and Exerimental Measurements XII Figure : Ellitic vortex atch, Reωa= Purely thermal free atches The case of thermal free atches is also interesting. Here we consider T as the temerature rise with resect to the infinite. In such case both 0th order Moments of Vorticity and Temerature must be invariant. This means that vorticity is generated by the thermal gradient combined with the gravity vector, but since the total vorticity must be zero, to a ositive vorticity must corresond a negative one of the same amount. The flow field then remains symmetric u to the incoming of instabilities. Figure 3: Purely thermal initially ellitical atch, Gr=.45E+8. The results deicted in Fig.3 confirm the exectations [3]: the 0th order Moments of Vorticity is zero within machine recision, and the 0th moment of Temerature is absolutely constant equal to its initial value. According to its

9 Comutational Methods and Exerimental Measurements XII 693 initial shae the atch rises and deforms symmetrically, breaking u in dioles. For the ellitical shae we find a configuration with sex-oles at time of 3 sec. 5. The fixed atches Finally the case of fixed thermal atch simulates the lume rising from a fire. In this case Vorticity is invariant, Temerature is not. As exected, in Figure 4, we notice that initially a mushroom shaed lume is formed and rises u to a length where instability aears. According the shae of the fixed atch and to the value of the Grashoff number either symmetry is lost, either intermittence in releasing wakes, or both henomena verify. Frame 00 6 Nov 004 CONTOUR_Ellisse.4. Time=.500 Frame 00 6 Nov 004 CONTOUR_Ellisse.4 Time= Y Y X X Figure 4: Fixed thermal atch, Gr=.83E+4. 6 Conclusions The imlementation of an otimized FMM algorithm in a vortex-thermal blob code generates an effective methodology for the study and the analysis of thermal buoyant roblems of real interest. The -D simulations here resented are very romising and well erforming.

10 694 Comutational Methods and Exerimental Measurements XII Currently we are working to the equivalent 3-D formulation where the vorticity is a true vector and stretching terms aear. We are, at moment, maintaining the vorticity formulation since it should allow easy analysis for noninertial reference frames. Preliminary test are ositive both for thermal/vorticity formulation and for the otimization of the relative FMM algorithm. Details will be reorted in a forthcoming aer. References [] Cottet, G.H. and Koumoutsakos, P.D., Vortex Methods: Theory and Practice, Cambridge University Press, 000. [] Golia, C., Buonomo, B., Manca, O., Viviani, A., A Vortex-Thermal Blobs Method For Unsteady Buoyancy Driven Flows, ASME-IMECE. November 3-9, 004, Anaheim, California. [3] Chorin, A.J, Numerical Study of Slightly Viscous Flow, J. Fluid Mechanics, 57, art 4, , 973. [4] Chorin, A.J, Vortex sheet aroximation, J. Com. Phys. 7, , 978. [5] Degong, P., Mass-Gallic, S., The weighted Particle method for Convection-Diffusion Equations, art: the case of an isotroic viscosity, Maths of Comutation, 53, 88, , 989. [6] Eldredge, J.D., Leonard, A. and Colonius, T., A General Deterministic Treatment of Derivatives in Particle Methods, J. Com. Phys., 80, , 00. [7] Williamson, J.H., Low Storage Runge-Kutta Schemes, J. Com. Phys., 35,, 48-56, 980. [8] Meijering, E., Unser, M., A Note on Cubic Convolution Interolation, IEEE Trans. Imag. Process.,, 4, 003. [9] Ael, A.W., An Efficient Program for Many-Body Simulation, SIAM J. Sci., Stat. Comut., 6, 85-03, 985. [0] Barnes, J., Hut, P., A Hierarchical O(NlogN) Force Calculation Algorithm, Nature, 34, , 986. [] Rokhlin, V. Greengard, L., A fast algorithm for article simulation, J. Com. Phys. 73, , 987. [] Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press.534, 967. [3] Riccardi, G., Golia, C. Integral Balance Laws for D buoyancy driven flows. VII SIMAI., Venice 004