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1 Avalable onlne at ScenceDrect Proceda IUTAM 5 (5 ) 34 4 IUTAM Symposum on Multphase flows wth phase change: challenges and opportuntes, Hyderabad, Inda (December 8 December, 4) Turbulent Transport Processes at Rough Surfaces wth Geophyscal Applcatons Srkanth Toppaladodd a,b, Sauro Succ c, John S. Wettlaufer a,b, a Mathematcal Insttute, Unversty of Oford, Oford OX 6GG, Unted Kngdom b Yale Unversty, New Haven, Connectcut 65-89, USA c Isttuto Applcazon Calcolo Mauro Pcone, C.N.R., Rome 85, Italy Abstract In ths numercal study we use the Lattce Boltzmann Method to nvestgate the effects of perodc and randomly rough surfaces on the turbulent transport of momentum and heat. A two-dmensonal MPI code has been developed and valdated aganst many test cases. We eamne n some detal ylegh-benárd convecton n the presence of rough walls and study the effects of the wavelength and ampltude of the roughness dstrbuton on the flow. c 5 4The Authors. Publshed by by Elsever B.V. B.V. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of Indan Insttute of Technology, Hyderabad. Peer-revew under responsblty of Indan Insttute of Technology, Hyderabad. Keywords: Turbulence, rough surfaces, sea ce;. Introducton The recent declne n the Arctc sea-ce etent has trggered debates on clmate change and ts mplcatons for humanty. A range of observatons have shown a decrease both n the areal etent and the average thckness of Arctc sea ce. For eample, n the central Arctc, the average sea-ce thckness has decreased by appromately 44 %. Gven the complety and the couplng of the varous subsystems of the Arctc clmate, there could be many factors responsble for ths declne. One of the factors whose contrbuton s relatvely poorly understood s the turbulent heat flu from the ocean to the undersde of the ce. Though the clmatologcal thckness of sea ce has been found to be very senstve to the oceanc heat flu n theoretcal models, broad scale systematc observatonal studes are challengng 3 and typcal closure schemes n Global Clmate Models (GCMs) are unrelable as can be understood from recent laboratory eperments 4. As an ntal step towards addressng ths ssue, we nvestgate the effects of a rough surface (a) wth a perodc structure and (b) whose spectral propertes are the same as that of Arctc sea ce, on a turbulent flow. Ths s done usng the Lattce Boltzmann Method (LBM) whch offers many advantages over the tradtonal Naver-Stokes solvers. One of the prmary ams of ths study s to understand how the turbulent flues are affected by the roughness of the E-mal address: john.wettlaufer@yale.edu The Authors. Publshed by Elsever B.V. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of Indan Insttute of Technology, Hyderabad. do:.6/j.putam.5.4.6

2 Srkanth Toppaladodd et al. / Proceda IUTAM 5 ( 5 ) surface, and how ths, n turn, affects the meltng/growth of the sea ce at the ce-ocean nterface. Before addressng ths challengng latter aspect of phase change, we begn by studyng a statonary rough surface.. The Lattce Boltzmann Method The Lattce Boltzmann Method s derved from Boltzmann s knetc theory as follows. Because the Boltzmann equaton descrbes the tme evoluton for a sngle partcle dstrbuton functon, varous moments of the dstrbuton functon gve the approprate macroscopc felds lke densty, velocty, and temperature. One can smplfy the Boltzmann equaton 5,6, whch s an ntegro-dfferental equaton that s quadratc n the dstrbuton functon, by replacng the ntegral wth a lnear term, known as the BGK-W collson term (named after Bhatnagar, Gross, Krook, and Welander). Although the collson term n ths appromaton s lnear, the nonlnearty s hdden n the equlbrum dstrbuton functon. In the orgnal equaton, the partcles travel n and nfnte number of drectons wth constant veloctes, but these drectons must be truncated for computaton purposes, and t was shown by Frsch, Pomeau & Hasslacher that n two dmensons a mnmum of seven velocty drectons, wth symmetry, s requred to recover the Naver-Stokes Equatons (NSE) from the Lattce Boltzmann equaton (LBE) 7. There are several advantages to solvng the LBE rather than the NSE, whch nclude: (.) The flow s weakly compressble n the LBM, and hence pressure s a local quantty. (.) The streamng operaton s eact. (3.) LBM can handle comple geometres more naturally and effcently. (4.) LBM s not just for flud flows. Physcal processes lke phase transtons can be ncluded n the full smulaton 7. (Ths s advantageous as we plan to study sea ce growth and ts nteracton wth the oceanc turbulent flow 8 ) The LBE, wth the BGK appromaton, s wrtten as: f ( + c Δt, t +Δt) = f (, t) Δt τ ( f f eq ); =,,,..., 8, () where f are the non-equlbrum dstrbuton functons, f eq are the equlbrum dstrbuton functons, s the poston vector, c are the constant partcle veloctes, Δt s the tme step, and τ s the collson tme-scale. For the present study we use the DQ9 model (D = denotng dmensons and Q = 9 denotng 9 veloctes). The form of f eq s crucal for the recovery of the NSE, whch s obtaned n the lmt of low Mach numbers, where Ma u/c s s the Mach number, u s the macroscopc flud speed and c s s the speed of sound. In the Boltzmann knetc theory f eq s a Gaussan but, n the numercal model, terms only up to O(Ma ) are retaned. The truncated form of f eq used n ths model s gven by: [ f eq = ρw + c.u + (c.u) ] u. () c s c 4 s c s Here, w are the weghts for dfferent drectons, c =Δ/Δt and c s = c/ 3. The NSE can be recovered from the LBE by carryng out a multple-scale analyss, consderng the small Knudsen number (the rato of the mean free path of a molecule to the largest length scale n the problem) lmt 5. The form of ( τ Δt ). In lattce unts Δ =Δt =, and knematc vscosty s obtaned from ths epanson, and s gven by ν = c s hence the scheme develops numercal nstabltes as τ.5 +. Fnally, once the f are calculated, the macroscopc felds are obtaned from the followng equatons: ρ = 8 f ; ρu = = 3. Valdaton 8 f c. = We have developed a two-dmensonal, MPI parallelzed LB code to study the flow effects of rough surface n two dmensons. The code has been etensvely valdated, and we dscuss some of the valdaton cases here: (3)

3 36 Srkanth Toppaladodd et al. / Proceda IUTAM 5 ( 5 ) Channel flow n two-dmensons The canoncal geometres n the study of wall-bounded turbulent flows are ppes and channels 9. Gven the smple geometry, we test our code to study transtonal flow n two-dmensonal (D) channels. The flow s forced usng a constant pressure gradent along the channel, wth a no-slp condton at the top and bottom walls and perodc condtons along ts length. In our smulatons the pressure gradent s constant, so the velocty scale s chosen to be u = p L /ρν, where p s the pressure gradent, L s the half-channel wdth, ρ and ν are the densty and knematc vscosty of the flud, and hence the pressure Reynolds number (Re p ) s defned as Re p = p L 3 /ρν. The soluton can be represented as u(, t) = U (z)+v(, t), where u(, t) s the total velocty feld, U (z) = ( z ) s the Poseulle velocty profle, s the unt vector along the channel, and v = (v, v z ) s the departure from U (z). Another quantty of nterest s the flu rate, ΔQ = v dz, where... denotes horzontal averagng. Fgure (a) shows the tme varaton of ΔQ for Re p = 935 and α =.3387, where α s the mnmal wavenumber of the mposed dsturbance. Because the pressure gradent s constant, the flow rate decreases before reachng a steady state value of.88, whch s close to.7 obtaned by Rozhdestvensky & Smakn usng spectral methods. Fgure (b) shows the the temporal behavour of (v, v z ) once the flow has reached a statstcally steady state. The results obtaned are n good agreement wth the fndngs of Rozhdestvensky & Smakn. ΔQ v t. vz t (a) Tme varaton of ΔQ for Re p = 935 and α = t (b) Tme varaton of v and v z at (, z) = (, ) for Re p = 935 and α = Fg. : Valdaton for D channel flow. Comparson wth Rozhdestvensky & Smakn who used spectral methods.. ylegh-benárd convecton n two dmensons The classcal problem of ylegh-benárd convecton nvolves the transport of heat by convecton when a crtcal ylegh number () s eceeded. Two horzontal walls are mantaned at dfferent temperatures the bottom wall s mantaned at a temperature T +ΔT and the upper wall s mantaned at a temperature T separated by a vertcal dstance H. The knematc vscosty and the thermal dffusvty of the flud are ν and κ respectvely, and β s the co-effcent of thermal epanson. The non-dmensonal parameters mportant n ths problem are the ylegh number, = gβδth3 νκ, and the Prandtl number Pr = ν/κ. Wth these boundary condtons convecton sets n when = 78 for all Pr. The heat transported to the cold wall s measured n terms of the sselt number (), whch s rato of total heat flu to the heat flu due to conducton alone. The sselt number s = k dt dz +ρc wt kδt/h, where k and c are the thermal conductvty and the specfc heat of the flud, w s the vertcal component of flud velocty,...and... denote temporal and horzontal averages. We consder three cases for valdaton: (a) In the frst case the horzontally averaged temperature profle for = 4 and Pr =.7 s compared wth results from the numercal smulatons of Lpps n fgure (a). (b) In the second case, we have compared for dfferent wth the results of Clever & Busse, who used a Galerkn method to solve for the conservaton equatons numercally, for = to 5 and Pr =.7. Ths s shown n fgure (b).

4 Srkanth Toppaladodd et al. / Proceda IUTAM 5 ( 5 ) T Present Lpps (976) Present Clever and Busse (974) =.56(/c) y (a) Horzontally averaged temperature ( T ) vs. y for = 4 and Pr =.7, compared wth the smulatons of Lpps (b) vs. for Pr =.7, compared wth the smulatons of Clever & Busse Fg. : Valdaton cases for ylegh-benárd convecton. (c) In the fnal test, we have reproduced the () scalng law from Johnston & Doerng 3, who used a Fourer-Chebyshev spectral method to smulate ylegh-bénard convecton n a cell of aspect rato of Γ=, defned as the rato of length to heght of the cell, and Pr =. The resoluton used n ther study was such that there were at least 8 grd ponts n the thermal boundary layers. We use the resolutons but wth a unform grd. The comparson s shown n fgure (3). Johnston & Doerng (9) Present Fg. 3: Comparson of () wth Johnston & Doerng Results: ylegh-benárd Convecton aganst Rough Walls. Perodc Roughness To understand the effects of roughness on thermal convecton, we frst use perodc roughness dstrbutons, and descrbe here the valdaton studes that underle a detaled eamnaton of the assocated physcs 4. Turbulent convecton over pyramdal elements 5,6,7, rectangular elements 8 and V-shaped grooves 9 have been studed both epermentally and numercally. However, the specfc effects of roughness on the scalng law dffer, wth some reportng a change only n the pre-factor 5,6,8, and others reportng a change both n the prefactor and the scalng eponent 9,7. We use a snusodal rough wall on the upper sde of the cell. For all the cases dscussed, Γ = and Pr =. The perodc roughness dstrbuton s characterzed by two length scales: wavelength (λ ) and ampltude (h ). The ampltude h h /H =. s fed for all cases, and two cases of λ λ /H =.4 and. are consdered. Fgures (4a) and (4b) show the - scalng laws for λ =.4 and λ =. respectvely. It s clear that ntroducng roughness elements nfluences both the pre-factor and the eponent n the scalng law. The scalng laws obtaned also show that wth the ncrease n λ, the planar case result ( = ) s approached. To understand the cause for the change n the scalng law, we analyze n fgures (5a) and (5b) the temperature felds for the two

5 38 Srkanth Toppaladodd et al. / Proceda IUTAM 5 (5) 34 4 cases consdered. As found n the prevous studes, there s enhanced plume producton at the rough surface due to the nteracton of the boundary layer and core flows, as seen n figure (5a). To drve ths nteracton, t s crucal that h > δt, where δt s the thckness of the thermal boundary layer, and λ has a value from an optmum range. When these condtons are met, there s an ncrease n the rate of heat transfer due to the plumes. However, as λ ncreases ths nteracton weakens, thereby leadng to a decrease n the number of plumes produced, as seen n figure (5b). Smulaton Smulaton N u.5.74 N u = (a) - scalng law for λ = (b) - scalng law for λ =.. Fg. 4: () for turbulent convecton over perodc roughness dstrbutons z z (a) Temperature field for λ =.4 and = (b) Temperature field for λ =. and = 9. Fg. 5: Temperature fields over perodc roughness dstrbutons.. ndom Roughness From the analyss of the sonar profiles of the undersde of Arctc sea ce, t s found that at hgh wavenumbers the spectral densty of roughness dstrbuton decays as k 3, where k s the wavenumber. A mathematcal functon havng the same spectral propertes as reported by Rothrock & Thorndke s used to generate the rough wall for LB smulatons, and s gven by f () = ( p )/ k p/ cos (k + φk ), (4) k= wth p = 3, where φk s an ndependent random varable unformly dstrbuted n (, π). mercal smulatons are carred out for dfferent, now defined based on the heght of channel mnus the mamum heght n the roughness dstrbuton, rangng from to 5. Because from our valdaton runs, we understand the behavor of the for these n the smooth case, the effect of roughness on the heat transfer can be easly seen here. All smulatons were carred out at Pr =.7. Fgures (6a) and (6b) show the temperature field for = 3 and = 5 respectvely. The effect of the wall roughness on the cellular structures s strkng and they are modfied apprecably relatve to the smooth-wall

6 Srkanth Toppaladodd et al. / Proceda IUTAM 5 ( 5 ) cases. There are two competng effects 8 : () By ntroducng roughness one has decreased the effectve, whch would result n a decrease n ; and () The ncreased area for heat transport can actually lead to an ncrease n. In fgure 7, the latter effect appears to domnate for the range of consdered, and there s a monotonc ncrease of the wth y 4.4 y (a) Temperature feld for = 3. (b) Temperature feld for = 5. Fg. 6: Temperature feld for dfferent and Pr = Rough wall Smooth wall Fg. 7: vs. for the rough wall for Pr = Conclusons A two-dmensonal LB code has been developed and valdated aganst the results n the lterature for D channel flow and for ylegh-benárd convecton. The latter problem s studed wth perodc and random roughness dstrbutons on the upper sde of cell. In the case of perodc rough walls, t s found that both the pre-factor and the eponent n the () scalng law are affected. The cause for ths change s the enhanced plume producton from the tps of the roughness elements. In the case of random rough walls, the large-scale flow structures are affected, and the ncreased relatve to the planar case. Fnally, the LBM s found to be a natural way to deal wth rough surfaces, because mplementng the no-slp condton s easer than n tradtonal methods. References. Kwok R, Unterstener N. The thnnng of Arctc sea ce, Phys. Today, 64, pp Maykut GA, Unterstener N. Some Results from a Tme-Dependent Thermodynamc Model of Sea Ice, J. Geophys. Res. 97, 76, pp M.G. McPhee, Ar-ce-ocean nteracton: Turbulent ocean boundary layer echange processes, Sprnger, Hultmark M, Vallkv M, Baley SCC, Smts AJ. Turbulent Ppe Flow at Etreme Reynolds mbers, Phys. Rev. Lett., 8.

7 4 Srkanth Toppaladodd et al. / Proceda IUTAM 5 ( 5 ) Chen S, Doolen GD. Lattce Boltzmann Method for Flud Flows, Ann. Rev. Flud Mech. 998, 3, pp Harrs S, An Introducton to the Theory of the Boltzmann Equaton, Dover Publcatons, Succ S, The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, Oford Scence Publcatons,. 8. Toppaladodd S, Wettlaufer JS, Succ S. Turbulent transport at rough surfaces. Bulletn of the Amercan Physcal Socety 58, (3). 9. Monn AS, Yaglom AM. Statstcal Flud Mechancs: Mechancs of Turbulence Volume, Dover Publcatons, 97.. Rozhdestvensky BL, Smakn IN. Secondary flows n a plane channel: ther relatonshp and comparson wth turbulent flows, J. Flud Mech. 984, 47, pp Lpps FB. mercal Smulatons of Three-dmensonal Benárd Convecton n Ar, J. Flud Mech. 976, 75, pp Clever RM, Busse FH. Transton to Tme-dependent Convecton, J. Flud Mech. 974, 65, pp Johnston H, Doerng CR. Comparson of Turbulent Thermal Convecton between Condtons of Constant Temperature and Constant Flu, Phys. Rev. Lett. 9,, Toppaladodd S, Succ S, Wettlaufer JS. Breakng the boundary layer symmetry n turbulent convecton usng wall geometry ( 5. Du Y.-B, Tong P. Enhanced Heat Transport n Turbulent Convecton over a Rough Surface, Phys. Rev. Lett. 998, 8(5). 6. Du Y.-B, Tong P. Turbulent thermal convecton n a cell wth ordered rough boundares, J. Flud Mech., 47, pp We P, Chan T.-S, N R, Zhao X.-Z, Xa K.-Q. Heat transport propertes of plates wth smooth and rough surfaces n turbulent thermal convecton, J. Flud Mech. 4, 74, pp Shskna O, Wagner C. Modellng the Influence of Wall Roughness on the Heat Transfer n Thermal Convecton, J. Flud Mech., 686, pp Strngano G, Pascazo G, Verzcco R. Turbulent thermal convecton over grooved plates, J. Flud Mech. 6, 557, pp Rothrock DA, Thorndke AS. Geometrc Propertes of the Undersde of Sea Ice, J. Geophys. Res. 98, 85, pp