Computation of drag and flow noise along wavy wall in turbulent flow

Transcription

1 roceedngs of 0 th Internatonal Congress on Acoustcs, ICA August 010, ydney, Australa Computaton of drag and flow nose along wavy wall n turbulent flow Huaxn Zhang (1), Kunyu Meng (1) and Yongln h (1) (1) chool of Naval Archtecture, Ocean and Cvl Engneerng, hangha Jaotong Unversty hangha, 00030, Chna AC: 43.8.RA ABTRACT In ths paper, a numercal smulaton of flow-nduced nose by the low Mach numbers, the turbulent flow wth a snusodal wavy wall s presented based on the unsteady ncompressble Naver-tokes equatons and Lghthll s acoustc analogy. Large Eddy mulaton (LE) was used to nvestgate space-tme flow feld and the magornsky sub-grd scale (G) model was ntroduced for turbulence model. Usng Lghthll s acoustcs analogy, the flow feld smulated by LE was taken as near-feld sound sources and radated sound from turbulent flow was computed by Curle s ntegral formulaton under the low-mach-number approxmaton. The relatonshp between flow nose and drag on the wavy wall s studed. Whch knd of spanwse wavy wall or streamwse wavy wall wth varous wall wave ampltude could get the effect on reducng drag and flow nose are dscussed. 1. INTROUCTION The control of the near-wall coherent structures s of great mportance for varous engneerng applcatons snce many researches show that some wall surface wth mcrogrooves, so called rblets, algned n the flow drecton presents the advantage of reducng the drag. Although the rblets are recognzed as a successful devce, at least, or the drag reducton purpose, the mechansm responsble s not fully clear and nether wall geometry optmzaton nor the effect to reduce flow nose has been made up to the present. In order to mprove dynamcal performance and reduce the acoustc nose, a very attractve ssue s whether t s possble to obtan a wall geometry that reduces a skn drag coeffcent and radated sound from turbulent flow. The wavy wall s effect of drag and nose reducton has been wdely studed by many researchers over the past few decades. Intal studes nto the drag reducton wth the use of rblets were conducted by Walsh[1] and hs colleagues at NAA Langley Research Center, whose researches were devoted to optmze the rblet sze and shape for a maxmum drag reducton. The result shows that the V-type grooves wth a heght to length rato n 5~30 has a better frctonal drag reducton n the range of 4-8%. Ths ssue attracts wdely nterests and researchers are studyng the effects of rblets on near-wall drag by experments or numercal smulatons. ark, Wallace[] sad that the flow wthn the rblet valleys s strongly domnated by vscosty. Cho[3] concluded that such drag reducton occurs only n turbulent regon, whle a drag ncrease s n varably obtaned n lamnar condtons. On the other hand, Bechert[4] whose study was focusng on experments has nvestgated more thoroughly dfferent confguratons of rblets ncludng rectangular, scalloped and sharkskn-shape rblets. Although numercal technques had lagged the experments due to the lack of the computatonal resources, t can be used for more complex confguratons when more parameters are to be dscussed such as the rblet ampltude and lengths. Flows along both spanwse and streamwse wavy walls dsplay pecular characterstcs that are not observed n the flows over a plate surface. In the case of wavy wall flows, the perodc changes of pressure gradent and of streamlne curvature generate turbulence structure dfferent from the counterpart of plate flows, but the effects to reduce drag and flow nose are not same on spanwse and streamwse wavy walls. Many studes show that the spanwase wavy wall vertcal to the stream flow drecton s effectve. On the bass of the coherent structure theory, It was proposed that the drag reducng grooved surface can not only control the spaces between two low-speed streaks to further reduce turbulent burst frequency, but also make a part or the whole quet flud n grooves avod encounterng the hgh-speed flud from the upper layer downwash and a hgher shear stress nduced by t. Thus, turbulent drag reducton can be acheved. But the research about streamwse wavy wall s very few. In ths paper, whch knd of spanwse wavy wall or streamwse wavy wall wth varous wall wave ampltude could get the effect on reducng the skn drag and the flow nose are dscussed. Ths paper s composed of four parts. ecton s devoted to a bref dscusson of the theory of the Large Eddy mulaton (LE) and the Lghthll s acoustcs analogy whch would be used n ths study. The computatonal model of the wavy wall n the turbulent flow s to be descrbed n ecton 3. ICA 010 1

2 Results of the drag and flow-nduced nose on both spanwse and streamwse wavy wall are provded n ecton 4. Also, there wll be some analyss focusng on why the wavy wall can reduce the drag and the flow nose.. Basc Theory.1 Large Eddy mulaton method The LE method parttons the turbulent flow nto two parts by the grd flterng operaton. The large-scale eddes, generated by the nstablty of the mean flow, are responsble for most of the transport of momentum, mass, and other scalars. The small-scale eddes (subgrd-scale), generated by the energy-cascade process from larger eddes, are contrbute lttle to heat and momentum transport, but have flux and feed-back effects on the large-scale flow. Accordng to these assumptons, the larger structures can be smulated drectly by the fltered N- equatons and the smaller one should be descrbed approxmately by usng the magornsky sub-grd scale model. The flterng operaton s defned by: f x, t G x, x', f( x', t) dx ' (1) where s the computatonal doman, G s the flter functon and s the flter wdth, usually we use the box type. The spatally fltered contnuty equaton and the ncompressble N- equaton can be wrtten n Cartesan tensor notaton as follows: u 0 x u 1 u t x x x p ( uu j) j j x j j () (3) where u s the large scale relatve velocty (or mean velocty) component, j uu j uu j s the subgrd scale (G) stress tensor (, j 1,,3). j The subgrd scale stress s the reflecton of the nteracton between the large eddes and the small, such as the energy and momentum transportaton and the feed-back effect on large eddes. It s always modeled by a magornsky type eddy-vscosty model as follows: 1 j 3 u u xj x j j kk T T j (4) where T s the subgrd eddy vscosty coeffcent, j s the Kronecker symbol and j s the resolved-scale stran rate tensor, n whch j 1 u u j x j x The eddy vscosty T s defned as (5) where 1/ T C s (6) j j s the magntude of j. The 1/3 flter wdth ( x, y, z), where x, y and z are the grd spacng n the x, y and z drectons, respectvely. The model coeffcent C s s the magornsky constant, n ths study, t s chosen to be 0.1. Thus, snce we have modeled the G stress j, the Large Eddy mulaton can be enforced by equaton () and (3).. Lghthll s acoustc analogy Lghthll s acoustc analogy[5] forms the startng pont for the dervaton of the flow-nduced nose predcton. The acoustc wave equaton relates the sound propagaton and the nose generaton. We can get the sound propagaton process and the change of the sound pressure at observaton ponts n far-feld approxmately from t. The Lghthll s equaton s derved from the contnuty equaton and the N- equatons n the Cartesan tensor form: T t j c 0 ' x jxj xxj where ' 0 s the densty fluctuaton, 0 s the freestream densty of flud, c 0 s the sound speed n water under undsturbed condtons. T j s the Lghthll s stress tensor whch s defned as T uu p c ( ) j j j j 0 0 pj p j j, In above equaton, p j s the stress tensor, p s the statc pressure, j s the Kronecker symbol and u u j uk j ( j ) s the vscous part of tokes x j x 3 xk stress tensor. Because the hydrodynamc and acoustc terms n equaton (7) are coupled each other, whch makes we couldn t get an accurate answer from (7), we should make some assumptons that the couplng s neglected n low Mach flow, and then the source terms n the rght sde of equaton (7) can be seen as known quanttes from the flow feld. If sold surface exsts n the sound source regon, Curle[6] derved a smple soluton for nose produced by the rgd surface movng through a quescent medum 1 np j( y, t r / c0 ) ( x, t) 0 ( d 4 xc0 x y r Tj ( y, t r / c0 ) d 3 ) xx y r j V (7) (8) (9) where x and y represent the poston vectors of the observer and the sources, respectvely, r x y, r c0 s called the delay tme and represents the tme nterval of sound wave ICA 010

3 travelng from the sound sources to the observer, n j represents the vector normal to the rgd surface over whch the surface ntegraton s taken place. The far-feld s defned by r le / M, where l e s the typcal eddy sze, M U c0 s the freestream Mach number. To smplfy equaton (9), all varables, such as, r and t can be non-dmensonalzed by a characterstc length L, and the nflow velocty U. Furthermore, f the sold surface and the unsteady flow regon are small compared wth the typcal acoustc wavelength le / M, a compact sound source can be assumed, so that the far-feld densty can be approxmated. Fnally, we obtan M x (, ) 1 (, ) 3 x t n jpj t d 4 x t y y 4 xx j 3 T (, ) 3 V j y t d y M 4 x t (10) In equaton (10), the frst term represents a dpole sound source and the second term shows a quadrupole sound source, whch respectvely represent the compact surface and volume sound radaton. The pressure fluctuatons and vscous shear tensors on the wall boundary generate sound radaton of dpole type, the Lghthll s stress tensor behave as quadrupole. In low Mach number condton, the frst term domnates, and sometmes the second term could be neglected. 3. Computatonal Model In the acoustc computaton part, the observaton pont at far-feld was chosen to be (0m, 0.5m, 0m). To get the hgher frequency parts of the flow-nduced nose, we use as the tme step. Acoustc data collecton wll be taken place after a perod of tme when the flow-feld has become statstcally steady. Table 1. pecfc characterstcs of spanwse wavy wall a(m).8375e e e e e-4 λ(m) e e-3.700e e e-3 a/λ 1/4 1/6 1/8 1/1 1/0 Table. pecfc characterstcs of streamwse wavy wall T a(m) 5.675e e e e-05 λ(m) 1.135e e e e-03 a/λ 1/00 9/800 1/80 3/00 4. Numercal Results The smulaton results of ths study can be dvded nto two parts: case of spanwse and streamwse wavy wall, whch wll be shown n 4.1 and 4., respectvely. Also, there are two steps n each case, the result of drag and the flow-nduced nose. 4.1 Case of panwse wavy wall rag results of spanwse wavy wall In ths case, n order to fnd models whch have the best effect on drag reducton, we have tred many models wth dfferent a / values and fnally found fve wavy walls as lsted n Tab. 1. We also nvestgate a plate s (called - 00) drag and nose performance as comparsons. And we obtan the drag of these 6 models as s lsted n Tab. 3: Fgure 1. Computatonal doman Fg. 1 shows the computatonal doman. An nfnte board s placed n the water. In ths study, the length of x, y and z drecton are , respectvely, where 0.05 s the boundary layer thckness of the plate. The nflow s at a rate of 5m/s n the drecton of -z and the two sdes along z drecton of the plate are assumed to be symmetry. Computaton has been made for three cases: spanwse, streamwse wavy walls and plate whch for comparson. We descrbe the wavy wall as the snusodal undulaton havng a ampltude a and wavelength. For spanwse wavy wall case, the ampltude to wavelength rato of wall undulaton a / s 1/0, 1/1, 1/8, 1/6 and 1/4. And for streamwse wavy wall case, ampltude to wavelength rato of wall undulaton a / s 1/00, 9/180, 1/80 and 3/00. The specfc characterstcs of these models are lsted n Tab. 1 and. For example, the wave curve equaton of model can be wrtten as: Table 3. rag results of spanwse wavy wall a/λ ressure Frctonal Total Total coeffcent E E E E / E E E E / E E E E / E E E-0.418E / E E E E / E E E E-03 Then, we have the relatonshp between / a and the total coeffcent as shown n Fg. : y asn xa a (11) ICA 010 3

4 Combned wth ths theory, we can explan our wavy wall results from two aspects: the wall shear stress and velocty component. The average shear stress of the plate, as s shown n Fg. 4, s the largest one (34.9 a), whle model - 05 s lower than t, followed by -01. Ths resulted n a drag dstncton and drectly explaned why -01 has the best effect. From Fg. 4 we can also see that the appearng regons of these three models surface hgh stress are dfferent, where the plat has an average dstrbuton of shear stress, and the other two s are manly concentrated n the top of the grooves, whch ndcates that there have bgger frctonal drag. Fgure. Total drag coeffcent and effect of drag reducton results of spanwse wavy wall The effect of drag reducton factor can be defned as: Fgure 4 Wall shear stress of 3 models n case of spanwse wavy (1) where and wavy are the total drag of the plate and wavy wall, respectvely. The curve of a / are drawn n Fg.. From Fg., we can explcty see that the model -05 has the worst effect of drag reducng capablty, whle -01 s the best. And as the wave ampltude to length rato a / ncreased, the effect factor become obvously more hgher. Accordng to our study, the length and ampltude of the wavy should be restrcted n a sutable range, otherwse, there wll be a dfferent result. Fgure 5 Vertcal Velocty gradents of 3 models n case of spanwse nce we have already found a group of wavy wall models that can reduce the drag, and there are some regular thngs we can fnd from Fg.. Now our queston s that why the wavy surfaces have these effects. For ths phenomenon, scentsts have proposed many knds of hypotheses whle there sn t a convnced descrpton. Bacher s quadratc-vortex theory, whch has wde-range nfluence, beleves that the nteracton between the turbulent flow and top of the mcrogrooves or rblets forms quadratc-vortexes. The quadratcvortex, as shown n Fg. 3, whch has reverse drecton to the spanwse-vortex, not only weaken the ntensty of the spanwse-vortex, but also hold the flow quet and make t have lower velocty n the grooves. Fgure 3. The quadratc-vortex Fgure 6.Velocty dstrbutons along spanwse of -00, 01 and 05 4 ICA 010

5 Fg. 5 shows the vertcal velocty dstrbuton n Y-Z plane and Fg. 6 shows the velocty dstrbuton of these three models. The plate s velocty dstrbuton along spanwse s smooth and average, whle -01 and 05 obvously express as wave-shape, whch bottom of the groove has a thck vscous layer and the velocty gradents there are very small. Ths stuaton could be seen more clearly n Fg. 5. Compared wth 00 and 01, the velocty gradents n the top of model - 01 are a bt bgger than t n the plate, but n the mddle and bottom of the groove the gradents are much lower. It means that n ths area of the wavy wall s groove there are very large quantty of the lower-velocty flud, whch extremely decreased the velocty fluctuaton and the frequency of the moment exchangng. These facts explan why the shear stress n the groove s small. Consderng about model -01 and 05, we can see that although -05 s total drag s larger than 01, the maxmum of these two models shear stresses are dfferent, wth the value of 43.1a and 47.3a, respectvely. The reason s that the locaton of the quadratc-vortex n model -05 s the lowest and the hgh-ntensty flud s momentum transportaton s the strongest, whch nduces a larger area of hghstress zone. Further more, because the slope of -05 s wall curve s small, whch means the flud near the groove bottom s blocked and the volume of quet flow s less than other wavy walls, these thngs made t have bgger drag than -01. The effect of nose reducton coeffcent s defned as wavy where, wavy are the sound pressure of the plate and wavy plate, respectvely. From these two pctures we can fnd that all the models sound pressure exsts n a reasonable range and each of them doesn t have large varaton. Although ths effect s small, we also can see the trend that as the value of a / ncreased, wll be non-strctly become hgher. The followng pctures (Fg. 8~9) show the sound pressure to frequency curve of each model. In order to make an comparson, we put -00 and 01 s sound pressure and power spectrum curve n one dagram, respectvely. From all these analyss, we can conclude that the exstence of the quadratc-vortex can hold the flow quet and make t have a low speed. It s the man reason of drag reducton. Further more, wth the top of groove s slope decreasng, the effects wth surroundng flud are becomng ntensty, and the strength of the quadratc-vortex whch s generated n the mddle and upper of groove also grows hgher, so that t brngs two aspects of nfluences: frst, whch s also the domnant one, the local shear stresses are let down because of the reducng of velocty gradents. econd, the mddle and upper flow s velocty gradents are becomng larger, whch made the shear stress goes up. nce the frst aspect s nfluence s larger, the total shear stresses are decreased and then lead the drag reduced Acoustc results of spanwse wavy wall The acoustc computaton s executed based on the results of an unsteady computaton by usng the nformaton of the flow-feld. urng ths process, we used Lghthll s acoustc analogy to smulate the numercal model. ound pressure (n db) and power densty results are shown n Fg. 7: Fgure 8 Comparson of power densty and sound pressure between -00 and 01 Compared wth 00 and 01, from Fg. 8 we can see that ther power denstes have bg dfference, especally n the hgh-frequency regon, the plate s power spectral densty can reach nearly, but under the same frequency the wavy wall of 01 s only not more than 0.4. Ths contrast express that the wavy wall has obvously effects of nose reducton than the plate. We may also see that all ther sound power spectrums are gathered n the range of more than 3000Hz, whch ndcates that the flow-nduced nose performance n our study s manly manfested as hgh-frequency. Ths ssue s consstent wth that n turbulent flow, the flow-nose are always dsplay as hgh-frequency. Fg. 9 shows the varaton of other four models sound pressure n the frequency-feld. All of them have the smlar growng tendency, and ther power densty are gathered n the range of 500Hz to 5000Hz. Combned wth Fg., we fnd that the models who has the effect of drag reducton may not have the same effect n reducng nose, but ther varaton trends are the same on the whole. Fgure 7 ound pressure and the effect of nose reducton results of spanwse wavy wall ICA 010 5

6 The man dmenson of the models n ths case s the same as t n the last smulaton, but ths tme the velocty drecton of the nflow s changed to be postve x. We have smulated many models, because the wave drecton s the same as the flow, whch makes the flow-feld unstable and lead our results of drag seem to be fluctuant. In ths case, fnally we choose four wavy models whch we called T-001, 00, 003 and 004. The specfc characterstcs of these models are lsted n Tab.. And we obtan the drag results of these 5 models as s lsted n Tab. 4: Table 4. rag results of streamwse wavy wall T a/λ ressure frctonal Total Total coeffcent E E E E /00.700E E E-04.88E / E E E E / E E E E /00.110E E E-04.57E-03 Fgure 10 Total drag coeffcent and effect of drag reducton results of streamwse wavy wall What makes us dsapponted s that, as s shown n Fg.10, all these models drag are larger than the plate s and they ddn t have the effect of drag reducton. Among these wavy walls, model T-003 has the lowest drag. Ether a / s larger or lower, they wll have a bgger drag. than the plate. Fgure 9 ound pressure to frequency curves of model -0~05 4. Case of treamwse wavy wall 4..1 rag results of streamwse wavy wall Fgure 11 Wall shear stress of 3 models n case of streamwse 6 ICA 010

7 Fgure 13 ound pressure and effect of nose reducton results of streamwse wavy wall Fgure 1 Vertcal velocty gradents of 3 models n case of streamwse Now we should fnd the reason why these models don t have the effect also from the wall shear stress and the velocty-feld. Fg. 11 shows the average, max and mn wall shear stress of T-000, 001 and 003. We found that the shear stress doesn t lke t show n the case of spanwse (Fg. 4), where the average value of T-001 s larger than the plate and other models are all lower. The wall shear stress s the reflecton of frctonal drag, so t can explan the data of frctonal drag column n Tab. 4. Fg. 1 shows the vertcal velocty gradent dstrbuton of T-000, 001 and 003. Ths pcture llustrates that why the shear stress are resulted as shown n Fg. 11. The plate s velocty s larger than T-003 s, whle t s lower than 001 s., and ths make the shear stress dfferent. From all ths analyss we know that both n the case of spanwse and streamwse, the frctonal drag could be reduced n some ranges. nce the shear stress of T-003 s lower than the plate, why ts total drag s larger on the other hand? Integrate wth Tab. 4, we wll fnd that there sn t a drect relatonshp between the frctonal resstance and the total resstance. But we also know that the total and frctonal drag have the same growng trend, and because the pressure drag part n streamwse case s more larger than t s n spanwse, whch plays an mportant role and affects the total drag. Ths fact gves us a hnt that the pressure drag mght be the reason why these models do not have drag reducton effect. Further more, for frctonal parts of the total drag, ether n spanwse or streamwse case, the wavy wall could obtan the drag reducng effect. 4.. Acoustc results of streamwse wavy wall The followng fgures (Fg.13~14) show the flow-nduced nose results of each model n the case of streamwse wavy wall: Fgure 14 ound pressure varaton by frequency results of models T 000~004 n case of streamwse ICA 010 7

8 The acoustc results of streamwse, lke t s n the drag step, also don t have the effect of nose reducton. Fg. 14 show the flow-nduced nose pressure to frequency results of streamwse wavy wall. We can fnd that each model s growng trend doesn t lke t n the case of spanwse (Fg. 8, 9). The power densty of the nose n ths step usually concentrates n the lower frequency regon than t n the spanwse case. 4.3 ummarze 5. Conclusons Numercal smulaton for drag and nose reducton has been appled to a turbulent flow along wavy wall wth varous wave ampltude. It s observed that dfferent spanwse wavy wall or streamwse wavy wall caused dfferent skn drag and dfferent flow nose. In some specal sze of spanwse wavelength, the wavy wall can reduce the skn drag and the flow nose. fferent wavelength has dfferent effect of drag and nose reducton. From the results we may make some conclusons that the spanwse wavy wall surfaces surely have the effect of drag reducton. The man reason s that the exstence of the quadratc-vortex can drop the wall shear stress down, whch leads to the reducng of frctonal drag. When the wave ampltude a and length are hold n a sutable range, as ther rato ncreased, the effect of drag reducton tends to be better. On the other hand, the spanwse wavy wall also can reduce the flow-nduced nose. It s a pty that n the streamwse case the drag reducton model ddn t appear, but what we fnd n ths case s that the relatonshp between the total drag and nose s very clearly, where ther have the same growng trend. Fgure 15 Results of spanwse wavy walls Fgure 16 Results of streamwse wavy walls Fg. 15 shows the result of spanwse wavy walls. In ths numercal smulaton, It s found that the spanwse wavy walls have both effects of drag and nose reducton, when a/λ= 1/4 the effects are largest. We also fnd that although the model whch has the effect of drag reducton may not have the ablty of nose reducton, but ther basc varaton trends are the same. And f one model has obvous effects on drag reducng, t seems to have the same ablty on the soundfeld. Fg. 16 shows that all the streamwse wavy walls have no effects of drag and nose reducton. It s means that the drag and flow nose of the streamwse wavy walls are larger than plate. Fg. 16 obvously shows that the relatonshp between the total drag and nose s very clearly, where ther have the same growng trend. These prove that we put the drag and nose together to study the reducng effects wth wavy wall s justfed. REFERENCE 1 Walsh, M. J., and Lndemann, A. M., Optmzaton and Applcatons of Rblets for Turbulent rag Reducton. AIAA aper NO ,1984. ark,.-r., and Wallace, J. M., Flow Feld Alteraton and Vscous rag Reducton by Rblets n a Turbulent Boundary Layer. In Near-Wall Turbulent Flows, , Elsever, K.-. Cho,. M. Orchard, Turbulence management usng Rblets for Heat and momentum transfer. Expermental Thermal and Flud cence, Volume 15, Issue, August 1997, ages W. Bechert, M. Bruce, W. Hage, J.G.T. Van er Hoeven, G. Hoppe, Experments on drag reducng surfaces and ther optmzaton wth an adjustable geometry, J. Flud Mech. 338(1997) Lghthll M J., On sound generated aerodynamcally II. Turbulence as a source of sound, roc Royal oc London, er A,, 1-5(1954) 6 Curle N., The nfluence of sold boundares upon aerodynamc sound, roc Royal oc London, er A, 31, (1955) 7 Bakewell H, Carey G F, Lbuha J J, et al. Wall pressure correlatons n turbulent ppe flow. U.. Navy Underwater ound Laboratory Report No.559, ougls. Henn and R. Ian ykes,large-eddy smulaton of flow over wavy surfaces,j. Flud Mech., 383, 1999, p Hang eok Cho, Kenjro uzuk,large eddy smulaton of turbulent flow and heat transfer n a channel wth one wavy wall, Internatonal Journal of Heat and Flud Flow, 6, 005, p Vlachoganns M., Hanratty T. J.,Influence of wavy structured surfaces and large scale polymerstructures on drag reducton,experments n Fluds 36,004, p ICA 010