Optimal Control of Temperature in Fluid Flow

Transcription

1 Kawahara Lab. 5 March. 27 Optmal Control of Temperature n Flud Flow Dasuke YAMAZAKI Department of Cvl Engneerng, Chuo Unversty Kasuga -3-27, Bunkyou-ku, Tokyo 2-855, Japan E-mal : d33422@educ.kc.chuo-u.ac.jp Abstract Ths paper presents an optmal control problem of temperature usng the adjont equaton method of an optmal control theory and the fnte element method. The three dmensonal model s used to analyze the thermal flud flow, whch s descrbed by the ncompressble Naver-Stokes equaton and the energy equaton. The formulaton s based on the optmal control theory n whch the performance functon s expressed by the computed and target temperatures. The optmal value can be obtaned by mnmzng the performance functon. The fractonal step method s appled to solve the ncompressble Naver-Stokes equaton. The weghted gradent method s employed as a mnmzaton technque. Keyword : T hermal F lud F low, Optmal Control T heory, F ractonal Step M ethod, W eghted Gradent M ethod. Introducton The thermal flud flow s a phenomenon generated by non-homogenety of temperature. The thermal flud flow s dvded nto natural flud flow and forced flud flow. The Natural flud flow s caused by the dfference of the specfc gravty between warmed flud and cool flud. Forced flud flow s compulsorly caused by fan or other tems. In recent years, the optmal control of thermal flud flow s used n varous places n our lfe. For nstance, the optmal control of the thermal flud flow s appled to ar-condtonng, heatng, refrgerator and bath and so on. Therefore, the optmal control of thermal flud flow s ndspensable and very mportant problem. Especally, the optmal control n the natural flud flow s one of most-attracted recent problem. Ths s because the control brngs low cost and good for the ecology. Floor heatng system and temperature management of grass feld n the football stadum are good examples. Therefore, ths research covers the problem n the natural flud flow. The purpose of ths study s the control of temperature n the natural flud flow. The temperature s controlled usng the three dmensonal feld as a computatonal doman. The ncompressble Naver-Stokes and the energy equatons are employed for the state equaton. In the optmal control theory, the control varable that makes the optmal state can be obtaned by mnmzng the performance functon. The performance functon s composed of the square sum of dfference between computed and target temperatures. The control varable can be converged to the target temperature n case that the performance functon s mnmzed. The weghted gradent method s appled as the mnmzaton technque. The fractonal step method s appled to solve the Naver-Stokes equaton. The fractonal step method n ths research uses the ntermedate velocty that does not completely satsfy the contnuty equaton. The Crank-Ncolson method and the Galerkn method expanded by the mxed nterpolaton are appled to temporal and spatal dscretzaton, respectvely. The stablzed bubble functon s appled to velocty and temperature felds. The lnear nterpolaton s utlzed to pressure feld. Ths study presents an analyss and a control problem of thermal flud flow, n whch two numercal studes are carred out. In numercal study, thermal flud flow s analyzed as the three dmensonal model. In numercal study 2, the optmal control of thermal flud flow s performed out n the three dmensonal model.

2 2 State Equaton The thermal flud flow s descrbed by the ncompressble Naver-Stokes equaton and the energy equaton. Therefore, these equatons are employed for the state equaton. The ncompressble Naver-Stokes equaton s appled to the flud feld. The energy equaton s appled to the temperature feld. In the ncompressble flow, densty of flud s assumed constant. The Naver-Stokes equaton and energy equaton are wrtten as follows: where ν, κ and f are wrtten as follows: u + u j u,j + p, ν(u,jj + u j,j ) = f θ n Ω, () u, = n Ω, (2) θ + u j θ,j κθ,jj = n Ω, (3) ν = Pr, f = PrRa, (4) where u, p, θ and f are the velocty, the pressure, the temperature and the gravtatonal acceleraton, ν and κ are the knematc vscosty coeffcent and the dffuson coeffcent, respectvely. The Raylegh number and the Prandtl number are denoted by Ra and Pr, respectvely. The ntal condtons are gven as follows: u (t ) = û, (5) θ(t ) = ˆθ, (6) where û and ˆθ are the ntal known value for velocty and temperature, respectvely. The boundary condtons are gven as follows: u = û on Γ u D, (7) t = pδ j + ν(u,j + u j, )n j = ˆt on Γ u N, (8) θ = ˆθ on Γ θ D, (9) b = κθ, n = ˆb on Γ θ N, () θ = ˆθ cont on Γ θ C, () where subscrpts D, N and C of Γ are the Drchlet boundary, the Neumann boundary and the control boundary, respectvely. The control varable on the control boundary Γ θ C s denoted by ˆθ cont. 3 Fractonal Step Method To solve the state equaton and the adjont equaton, the Crank-Ncolson method s used for the temporal dscretzaton. Ths method s capable of takng long tme ncrement and superor n stablty. Therefore, a lot of tme cycles can be taken n the computaton. The momentum equaton, contnuty equaton and energy equaton are expressed as follows: u n+ u n t + u j u n+ 2,j + p n+, ν(u n+ 2,jj + u n+ 2 j,j ) = f θ n Ω, (2) u n+, = n Ω, (3) θ n+ θ n t + u j θ n+ 2,j κθ n+ 2,jj = n Ω, (4) u n+ 2 = 2 (un + un+ ), (5) θ n+ 2 = 2 (θn + θn+ ). (6) 2

3 The fractonal step method s appled to solve the ncompressble Naver-Stokes equaton. Ths method s a technque that s dvded nto the velocty and the pressure by dervng the pressure Posson equaton. The pressure Posson equaton s derved after obtanng the ntermedated velocty ũ. The pressure at the prevous step p n s assumed to be approxmate pressure. The pressure of the momentum equaton s replaced wth p n. Unknown velocty s replaced wth the ntermedate velocty. Then, the momentum equaton n Eq.(2) s as follows: ũ n+ u n t + u j ũ n+ 2,j + p n, ν(ũ n+ 2,jj + ũ n+ 2 j,j ) = f θ n Ω. (7) The dfference between Eq.(2) and Eq.(7) s taken, the followng equaton s obtaned. u n+ ũ n t + 2 u j(u n+,j ũ n+,j ) + (p n+, p n, ) 2 ν{(u,jj + u j,j ) n+ (ũ,jj + ũ j,j ) n+ } = f θ n Ω. (8) The pressure Posson equaton s obtaned from the transpraton of Eq.(8) consderng the contnuty equaton as follows: t(p n+, p n,) = ũ n+, n Ω. (9) 4 Spatal Dscretzaton 4. Bubble Functon Interpolaton The Galerkn method s appled to the spatal dscretzaton. The mxed nterpolaton based on the bubble functon and lnear functon s used for the spatal dscretzaton. The bubble functon nterpolaton s appled to the velocty and the temperature felds and the lnear nterpolaton s appled to the pressure feld as follows: For bubble functon nterpolaton: u = Φ u + Φ 2 u 2 + Φ 3 u 3 + Φ 4 u 4 + Φ 5 ũ 5, (2) ũ 5 = u 5 4 (u + u 2 + u 3 + u 4 ), (2) θ = Φ θ + Φ 2 θ 2 + Φ 3 θ 3 + Φ 4 θ 4 + Φ 5 θ5, (22) θ 5 = θ 5 4 (θ + θ 2 + θ 3 + θ 4 ), (23) and for lnear nterpolaton: Φ = L, Φ 2 = L 2, Φ 3 = L 3, Φ 4 = L 4, Φ 5 = 256L L 2 L 3 L 4, (24) p = Ψ p + Ψ 2 p 2 + Ψ 3 p 3 + Ψ 4 p 4, (25) Ψ = L, Ψ 2 = L 2, Ψ 3 = L 3, Ψ 4 = L 4, (26) 3

4 Fg. : Bubble functon element 2 Fg.2 : Lnear element 2 where u u 4 and θ θ 4 are the velocty and the temperature at nodes 4 of each fnte element. Φ α (α = 5) s the bubble functon for the velocty and the temperature, n whch p p 4 are the pressure at nodes 4 of each fnte element. The lnear nterpolaton functon s denoted by Ψ α (α = 4) for pressure. Uncoordnated s expressed by L. 4.2 Stablzed Form In the bubble functon, the numercal stablzaton s not enough. Therefore, the stablzed parameter whch determnes the magntude of streamlne s used for stablzaton. In equaton, the stablzed parameter τ eb can be wrtten as follows: τ eb = φ e, 2 Ω e ν φ e,j 2 Ω e A e, (27) where Ω e s element doman and < u, v >= uvdω, Ω e u 2 Ω e = uudω, Ω e Ae = dω. Ω e (28) The ntegral of bubble functon s expressed as follows: < φ e, > Ωe = Ae 6, u e,j 2 Ω e = 2A e g, g = 2 Ψ α, 2, (29) = where ν s control parameter for the stablzng acton. Ths value s determned to become equvalent to τ es usng the stablzed fnte element method. ( τ es = ) 2 2 u + h e ( ) 2 4ν h 2 e 2, u = u + u 2 + u 3, (3) where h e s an element sze. In generally, the stablzed parameter n Eq.(3) s not equal to the optmal parameter n Eq.(3). Thus, the bubble functon whch gves the optmal vscosty satsfes the followng equaton. φ e, 2 (ν + ν ) φ e,j 2 Ω e A e = τ es. (3) Eq.(32) adds stablzed operator control term only of the barycenter pont to the equaton of moton as follows: N e e= ν φ e,j 2 Ω e be, (32) where N e and be are the total number of element and the barycenter pont. In the energy equaton, ν s replaced wth κ. 4

5 5 Optmal Control Theory 5. Performance Functon In the optmal control theory, control varable can be obtaned by mnmzng performance functon. The performance functon J s composed of the square sum of dfference between computed and target temperatures. The control varable are computed to mnmze the performance functon under the state equaton and boundary condtons. Mnmzng the performance functon means that computed temperature should be as close as possble to the objectve temperature. The performance functon s wrtten as follows: J = 2 tf t Ω (θ θ obj ) T Q(θ θ obj )dωdt, (33) where θ, θ obj and Q are the computed temperature, the target temperature at the object pont and the weghtng dagonal matrx. Superscrpt T means transpose. 5.2 Adjont Equaton The Lagrange multpler method s appled to mnmze the performance functon. The Lagrange multpler method s used for the mnmzaton problem wth constrants. The extended performance functon J s expressed as follows: tf J = J u T ( u + u j u,j + p, ν(u,jj + u j,j ) f θ)dωdt t Ω tf p T (u, )dωdt t tf t Ω Ω θ T ( θ + u j θ,j κθ,jj )dωdt, (34) where u, p and θ denote the Lagrange multpler for velocty, pressure and temperature, respectvely. The extended performance functon J s dvded nto the Hamltonan H and the tme dfferentaton term as follows: J = tf where the Hamltonan H s defned as follows: H = 2 (θ θ obj) T Q(θ θ obj ) t Ω {H + (u T u + θ T θ)}dωdt, (35) u T {u j u,j + p, ν(u,jj + u j,j ) f θ} p T u, θ T (u j θ,j κθ,jj ). (36) The statonary condton s needed to mnmze performance functon. The frst varaton of J should be zero as follows, δj =. (37) Therefore, the adjont equaton and termnal condtons are obtaned. The adjont equaton and the termnal condtons are wrtten as follows: u T = H u, H p =, θ T = H θ, (38) u (t f ) =, p (t f ) =, θ (t f ) =. (39) 5

6 6 Mnmzaton Technque 6. Weghted Gradent Method The weghted gradent method s appled to the mnmzaton technque. Ths method s a technque that control varable are renewed by the statonary condton of a modfed performance functon. The modfed performance functon K (l) to whch the penalty term s added s defned as follows: K (l) = J + 2 ( θ (l+) (t) θ (l) (t)) T W (l) ( θ (l+) (t) θ (l) (t)), (4) where θ(t), W (l) and l are control varable, weghtng dagonal matrx and teraton number. To mnmze the modfed performance functon, the penalty term should be zero. To mnmze the extended performance functon s equal to mnmze the modfed performance functon. Then, the followng equaton s obtaned. K θ The control varable s renewed by the followng equaton: =. (4) θ (l+) (t) = θ (l) (t) W (l) grad(j)(l). (42) 7 Numercal Study 7. Numercal Study The numercal analyss of natural flud flow s carred out usng the three dmensonal model. The fnte element mesh and computatonal model are shown n Fgs.3 and 4. The total number of nodes and elements are 46 and 73728, respectvely. On the bottom surface, temperature at ponts s specfed to be equal to.. On the top surface, temperature s set to be equal to.. The boundary condton for velocty s non-slp condton on all the surface. The Raylegh number and the Prandtl number are set to. and.7, respectvely. The tme ncrement t s.3. The natural flud flow usng the fnte element method s carred out. The temperature of non-dmensonal tme t=.45 s shown n Fg.7. The velocty of non-dmensonal tme t=.45 s llustrated n Fg.8. It s seen that phenomenon of thermal flud flow s smulated. 3 T=. T=. Z X Y Fg.3 : Fnte Element Mesh Fg.4 : Computatonal Model 6

7 7.2 Numercal Study 2 The optmal control of temperature n natural flud flow s performed. The fnte element mesh and computaton model are shown n Fgs.5 and 6. The total number of nodes and elements are 2975 and 356, respectvely. The control and the specfed boundares are shown n Fg.6. The temperature on the specfed boundary s shown n Fg.9. The boundary condton for velocty s gven as non-slp condton on the all surface. In ths study, the Raylegh number and the Prandtl number are set to. and.7, respectvely. The tme ncrement t s.. The object pont s set as shown n Fg.6. Usng the specfed temperature shown n Fg.9, the natural flud flow s occurred. The temperature at the object pont shown n Fg. can be obtaned. To reduce the temperature at the object pont from ntal to target temperature, the optmal control s carred out. The target temperature s set to.. Ths optmal control problem s to fnd the control temperature on the control boundares so as to mnmze the performance functon. The weghted gradent method s appled as the mnmzaton technque. The tme hstory of the performance functon s shown n Fg.. The tme hstory of the control temperature at the control boundary s shown n Fg.2. The tme hstory of temperature at the object pont s shown n Fg.3. At the target pont, the temperature could be controlled to the objectve temperature. The temperatures on Y=.5 and Z=.25 are shown n Fg.4. The temperature and the velocty wthout control at non-dmensonal tme t=2. are shown n Fgs.5 and 7. The temperature and the velocty wth control at non-dmensonal tme t=2. are shown n Fgs.6 and 8. The temperature at the same alttude can be reduced by settng only one object pont as shown n shown Fg.4. The optmal control of temperature n natural flud flow can be carred out..5 Z X Y Control Boundary Z Specfed Boundary Object Pont X Y Fg.5 : Fnte Element Mesh Fg.6 : Computatonal Model 8 Concluson The optmal control of temperature n the natural flud flow s presented. The control temperature can be found so as to mnmze the performance functon usng the weghted gradent method. It s shown that the temperature at object pont can be reduced usng the control temperature. The all temperature of the same alttude s reduced by settng only one object pont. The temperature at the object pont s well controlled usng the present method. References. M.Kawahara, K.Sasak and Y.Sano, Parameter Identfcaton and Optmal Control of Ground Temperature, Int.J.N umer.m eth.f l., vol.2,pp.789-8, S.Suzuk, A.Anju and M.Kawahara, Management of Ground Temperature by Bang-Bang Control Based on Fnte Element Applcaton, Int.J.N umer.m esh.eng., vol.39, pp.885-9, J.Matsumoto, A.A.Khan, S.S. Wang, and M.Kawahara, Shallow Water Flow Analyss wth Movng Boundary Technque Usng Least-Squares Bubble Functon, Int.J.Comp.FludDyn., vol.6(2), pp.29-34, J.Matsumoto, T.Umetsu and M.Kawahara, Stablzed Bubble Functon Method for Shallow Water Long Wave Equaton,Int.J.Comp.F luddyn., vol.7(4), pp , T. E. Tezduyar, Stablzed Fnte Element Formulatons for Incompressble Flow Computatons, AdvancesnAppl.M ech., vol.28, pp.-42,992 7

8 Fg.7 : Temperature ( t=.45 ) Fg.8 : Velocty ( t=.45 )..9 Specfed Temperature.2. Wthout Control Temperature Temperature Non-Dmensonal Tme Fg.9 : Specfed Temperature Non-Dmensonal Tme Fg. : Temperature at Object Pont wthout control Performance Functon. -. Control Temperature Performance Functon 2.5 Temperature Itaraton Number Non-Dmensonal Tme Fg.: Performance Functon Fg.2 : Control Temperature at Control boundary 8

9 .2. Wth Control Wthout Control.5.4 Wth Control Wthout Control.8.3 Temperature.6.4 Temperature Non-Dmensonal Tme Fg.3 : Temperature at Object Pont X-axs Fg.4 : Temperature Dstrbuton (t=2.) Fg.5 : Temperature Wthout Control (t=.6) Fg.6 : Temperature Wth Control (t=.6) 9

10 Fg.7 : Velocty Wthout Control (t=.6) Fg.8 : Velocty Wth Control (t=.6)