Wall Pressure Fluctuations and Flow Induced Noise in a Turbulent Boundary Layer over a Bump

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1 Proceedngs of the rd Internatonal Conference on Vortex Flows and Vortex Models (ICVFM005) Yokohama, JAPAN, November 1 -, 005 Wall Pressure Fluctuatons and Flow Induced Nose n a Turbulent Boundary Layer over a Bump Joongnyon Km and Hyung Jn Sung Department of Mechancal Engneerng Korea Advanced Insttute of Scence and Technology 7-1 Guseong-dong, Yuseong-gu, Daeeon, , Republc of Korea Key Words: wall pressure fluctuatons, flow nduced nose, turbulent boundary layer, over a bump, longtudnal surface curvature, drect numercal smulaton ABSTRACT Drect numercal smulatons of a turbulent boundary layer over a bump were performed to examne the effects of surface longtudnal curvature on wall pressure fluctuatons (p w ) and flow nduced nose. Turbulence statstcs and frequency spectra were obtaned to elucdate the response of wall pressure fluctuatons to the longtudnal curvature and to the correspondng pressure gradent. Wall pressure fluctuatons were sgnfcantly enhanced near the tralng edge of the bump, where the boundary layer was subected to a strong adverse pressure gradent. Large-scale structures n the dstrbuton of wall pressure fluctuatons were observed to grow rapdly near the tralng edge of the bump and convect downstream. Acoustc sources of the Lghthll equatons were nvestgated n detal at varous longtudnal surface curvatures. The acoustc sources (S) were hghest near the tralng edge of the bump, where the r.m.s. wall pressure fluctuatons were greatest. The maxmum correlaton coeffcent between p w and S was located ust above the locaton of maxmum wall pressure fluctuatons. Far-feld acoustc densty fluctuatons were computed usng the Lghthll acoustc analogy. We found that the surface dpole s domnant n the total acoustc feld. The contrbuton of the volume quadrupoles to the total acoustc feld gradually ncreases wth decreasg the radus of the surface curvature (δ/r). 1. INTRODUCTION The turbulent boundary layer of a flow passng over a bump exhbts extremely complex flow characterstcs, despte the relatvely smple geometry of ths system (Baskaran et al., 1987, Webster et al., 1996, Wu and Squres, 1998). A model surface bump can be constructed from three tangental crcular arcs, as shown n Fg.1. When the flow mpnges on ths bump, the boundary layer experences a short regon of concave surface, a longer regon of convex surface, another short regon of concave surface, and then returns to the flat plate. As a result of ths geometry, the streamwse pressure gradent changes from adverse to favorable n the regon upstream of the bump apex. Downstream of the bump apex, the boundary layer s subected to an adverse pressure gradent before returnng to a favorable pressure gradent over the flat plate. Thus, f we are to desgn an effectve strategy for controllng the nose generated by such flows, a clear understandng of the effect of the longtudnal surface curvature and the assocated pressure gradent s essental. One practcal example n whch such knowledge s vtal s nose generaton caused by flow over sonar transducers mounted on shps or submarnes. The maorty of prevous studes n ths area have focused on wall pressure fluctuatons n the equlbrum turbulent boundary layers over flat plates or nsde channels (Km, 1989, Cho and Mon, 1990, Km et al., 00). Recent advances n drect numercal smulaton (DNS) have ntensfed nterest n calculatng wall pressure fluctuatons n nonequlbrum turbulent flows. Neves and Mon (1994) examned the effects of convex transverse curvature on wall pressure fluctuatons n axal flow boundary layers. They found that r.m.s. wall pressure fluctuatons decrease wth ncreasng the transverse curvature. In a study of the effects of separaton on wall pressure fluctuatons, Na and Mon (1998) observed large two-dmensonal roller-type structures nsde a separaton bubble. Recently, Km et al. (00) nvestgated the characterstcs of wall pressure fluctuatons after a sudden applcaton of wall blowng or sucton. They showed that blowng has a greater effect than sucton on wall pressure fluctuatons. However, the effects of longtudnal surface curvature and the assocated pressure gradent on wall pressure fluctuatons have yet to be establshed. The mechansm of aerodynamc nose generaton by a turbulent shear flow has been a subect of great nterest snce Lghthll frst formulated a general theory for the aeroacoustc nose (Lghthll, 195). When a flow encounters a sold body, Curle's ntegral soluton (1955) to the Lghthll equaton provdes a theoretcal framework for predctng the nose generated by the flow--body nteracton. Accordng to the extenson of Lghthll s formalsm by Powell (1960), fluctuatng veloctes produce quadrupole nose sources, whereas fluctuatng wall shear stresses take the form of dpole sources. Thus, when an nfnte surface plate boundary layer s assumed, the most mportant dpole source term, that from wall pressure fluctuatons, s elmnated by reflecton from the surface plate. However, n the present study, we found that the acoustc sources of the Lghthll equatons wthout consderaton of wall pressure fluctuatons are closely correlated wth the dstrbuton of wall pressure fluctuatons. Here we endeavor to develop a quanttatve descrpton of the

2 Fg. Comparson of the calculated mean wall pressure wth expermental data Fg. 1 Schematc dagram of the flow geometry; (a) Top vew of the computatonal doman; (b) Sde vew of the computatonal doman relatonshp between wall pressure fluctuatons and acoustc sources. The man obectve of the present study was to nvestgate the effects of longtudnal surface curvature and the assocated pressure gradent on wall pressure fluctuatons and flow nduced nose. To acheve ths, we performed DNSs of a turbulent boundary layer over a bump. Turbulence statstcs and frequency spectra of wall pressure fluctuatons were extracted usng standard technques for analyzng stochastc data. In addton, two-pont correlaton coeffcents were used to deduce the spatal structure of the wall pressure fluctuatons. Acoustc sources of the Lghthll equatons were nvestgated n detal as a functon of longtudnal surface curvature. A quanttatve statstcal descrpton of the relatonshp between wall pressure fluctuatons and acoustc sources was formulated n terms of the correlaton coeffcent. Far-feld acoustc densty fluctuatons were computed usng the Lghthll acoustc analogy.. DIRECT NUMERICAL SIMULATION For an ncompressble flow, the non-dmensonal governng equatons are u p 1 uu u + = + t x x Re x x, (1) u 0 x =, () where x are the Cartesan coordnates and u are the correspondng velocty components. All varables are non-dmensonalzed by a characterstc length and velocty scale, and Re s the Reynolds number. By ntroducng generalzed coordnates η, the velocty components u are transformed nto the volume fluxes across the faces of the cell q or q. Formulaton of the problem n terms of the contravarant velocty components, weghted wth the Jacoban J n conuncton wth the staggered varable confguraton, leads to dscretzed equatons. The transformed governng equatons are rewrtten as q + N ( q ) = G ( p ) + L1( q ) + L( q ), () t 1 1 q q q Dq = =, (4) J η η η where N s the convectve term, G (p) s the pressure gradent term, L 1 and L are the dffuson terms wthout and wth cross-dervatves, and D s the dvergence operator. More detals can be found n Cho et al. (199). The governng equatons are ntegrated n tme usng the fully mplct, fractonal-step method proposed by Cho and Mon (1994). Table 1 lsts the bump parameters used n the present work, along wth those used n prevous studes (Webster et al., 1996, Wu and Squres, 1998). In the present DNS, we specfed the bump parameters so that the streamwse dstrbuton of the computed wall pressure coeffcent for δ/r=0.015 matched well wth that of Webster et al. (1996) (Fg. ). Because the Reynolds number consdered n the present DNSs was smaller than those of prevous studes, a smaller curvature parameter (δ/r) was used n the present work. Bradshaw (197) demonstrated that a convex curvature of δ/r=0.01 sgnfcantly reduced the skn frcton coeffcent compared to that over a flat plate. Takng nto consderaton ths fndng, the bump used n the present work s expected to sgnfcantly affect the turbulent boundary layer. The surface bump has two dscontnutes n surface curvature: the concave-to-convex surface near the leadng edge and the convex-to-concave surface near the tralng edge. Note that, at both of these dscontnutes, the bump used n the present work has a value of k* = , whch satsfes the crteron k* > for the formaton of an nternal layer (Baskaran et al., 1987). Here, k*=(1/r -1/R 1 )ν/u τ where R 1 and R are upstream and downstream rad of curvature, respectvely. In the DNSs, tme-dependent turbulent nflow data were provded at the nlet based on the method of Lund et al. (1998). Usng ths approach, nstantaneous planes of velocty data were extracted from an auxlary smulaton of a spatally developng Table 1. Comparson of bump parameters

3 Fg. Streamwse dstrbuton of (a) mean wall pressure coeffcent, and (b) pressure gradent parameter turbulent boundary layer over a flat plate. A plane velocty feld near the doman ext was modfed by the rescalng procedure and rentroduced to the nlet of the computatonal doman n the nflow-generaton smulaton. The man smulaton of a turbulent boundary layer over a bump was then carred out. The convectve boundary condton mposed at the ext had the form (u /t) + c (u /x) = 0, where c s the local bulk velocty. A no-slp boundary condton was mposed at the sold wall, and the boundary condtons on the top surface of the computatonal doman were u 1 /y=0, u =0 and u =0. Perodc boundary condtons were appled n the spanwse drecton. The computatonal doman sze was L x = 40, L y = 45 and L z = 40. The nlet Reynolds number based on the nlet momentum thckness (θ 0 ) and free stream velocty (U ) was Re=00. The mesh contaned 57x97x19 ponts n the streamwse, wall-normal, and spanwse drectons, respectvely. Non-unform grd dstrbutons were used n the streamwse and wall-normal drectons, whereas a unform grd dstrbuton was used n the spanwse drecton. The computatonal grd was generated usng the drect dstrbuton control technque of Thomas and Mddlecoff (1980). The computatonal tme step used was t=0. θ 0 /U and the total averagng tme to obtan the statstcs was T avg =5000 ν/u τ, where ν and u τ are the knematc vscosty and the frcton velocty, respectvely. A streamlne-normal coordnate system (s, n, z) was used for post-processng, where the n-axs s perpendcular to the lower surface n Fg. 1(b). The correspondng velocty components n (s, n, z) are denoted (u s, u n, u z ). A normalzed streamwse coordnate, x =(x-x 0 )/L c, was also used, where x 0 corresponds to the leadng edge of the bump and L c s the bump length. In the x'-coordnate system, x =0 concdes wth the leadng edge, x =0.5 wth the bump apex and x =1.0 wth the tralng edge.. AEROACOUSTIC THEORY From the Lghthll acoustc analogy (195), the concentrated unsteady flow regon s the aeroacoustc source regon, whch can be obtaned from an ncompressble DNS. Fg. 4.Streamwse dstrbuton of (a) skn frcton coeffcent, and (b) r.m.s. wall pressure fluctuatons The densty fluctuatons due to propagaton of an acoustc wave from the aeroacoustc source regon are governed by an nhomogeneous wave equaton, whch can be wrtten n the followng form: ρ ρ T 1 + = t M x x x x (5) where ( / ) T = ρu u + δ p ρ M τ, (6) 1 u u u k τ = + δ Re x x x k. (7) Equaton (5) s a restatement of the Lghthll equaton n terms of the values relatve to the freestream values. Equaton (6) s the Lghthll stress tensor composed of three terms, and Eq. (7) s the vscous part of the Stokes stress tensor. When the reference frame s fxed on the movng sold boundary, Curle's soluton (1955) of the Lghthll acoustc analogy can be appled. Curle showed that the general soluton for the flow past a rgd surface can be wrtten as: ρ (, ) M (, ) x t 1 = n x S r (, ) M T y t Mr + xx r V P y t Mr, (8),

4 Fg.5 Sde vew of nstantaneous velocty vectors n the x-y plane for δ/r=0.00: (a) tu /θ 0 =0; (b) tu /θ 0 =0 where x s an observaton pont poston vector, y s a source pont poston vector, r = x-y, P =pδ -τ, and n s the drectonal cosne of the outward normal to the rgd surface. Powell (1960) ponted out that for flow past an nfnte plane surface one can consder a new extended flow feld obtaned by reflecton of the orgnal one n the plane y=0, whch gves an dentcal acoustc feld. Based on the result of Powell, the acoustc densty n the upper half-plane can be wrtten as: ρ ( xt, ) M 1 = x k S ( yt, Mr) k (, ) M T y t Mr + xx r τ V * * * 4 * V r * (, ) M T y t Mr + π x x r, (9) where k=1, and x* s the mage of poston x n the rgd surface x =0. Note that reflecton at an nfnte surface plate boundary layer thus elmnates the dpole source term arsng from wall pressure fluctuatons. When the dstance between observaton and source ponts s larger than the acoustc wavelength, the far-feld approxmaton can be appled. Also, when the sold body and the source regon are smaller than the typcal acoustc wavelength, the source regon s acoustcally compact. The acoustc densty at the far-feld generated from a compact source regon can be approxmated by ρ M x π x xt yt Mr k (, ) 1 = τ k (, ) M xx + t S + x x 4 * * T ( y, t Mr) x t V, (10) whch ncludes two typcal nose source functons, a surface dpole and a volume quadrupole, whch are generated from the surface of the rgd wall and the entre unsteady flow regon, respectvely. The acoustc densty at the far-feld generated by a compact source regon s rewrtten as: Fg. 6.Top vew of nstantaneous skn frcton coeffcent n the x-z plane: (a) δ/r=0.015; (b) δ/r=0.00; (c) δ/r=0.045 M xk ρ ( xt, ) 1 = D& ( ) k t Mr π x where 4 * * M xx + x x, (11) + Q&& ( ) t Mr x D& ( ) ( ) k t Mr = τ k y, t t, (1) S Q&& ( t Mr ) = T ( y, t) t, (1) V are dpole and quadrupole sources, respectvely. The resdual effect at the ext boundary s elmnated by usng the followng correctve formula derved by Wang et al. (1996), Q&& = Q&& + F&, (14) where the dots denote tme dervatves and F s the flux of Lghthll stress components (Wang et al., 1996). 4. RESULTS AND DISCUSSION 4.1 Mean Wall Pressure and Wall Pressure Fluctuatons Before proceedng further, t would be advantageous to see the varaton n the mean wall pressure along the wall. Fgure (a) shows the streamwse dstrbuton of the wall pressure coeffcent (C p ). The boundary layer ntally develops under a zero pressure gradent at the nlet, after whch the streamwse pressure gradent becomes mldly adverse over the upstream flat plate. The boundary layer then experences a short regon of concave curvature before encounterng a regon of convex curvature. The correspondng pressure gradent changes from adverse to favorable. Downstream of the bump apex, the streamwse pressure gradent s strongly adverse then changes to mldly favorable over the ext flat plate. Fgure (b) shows the dstrbuton of the non-dmensonal pressure gradent parameter P +, whch s gven by

5 Fg.8 Two pont correlaton coeffcent of wall pressure fluctuatons as a functon of streamwse spatal and temporal separatons: (a) x =0; (b) x =1/1 Fg. 7 Frequency spectra of wall pressure fluctuatons wth outer varable scalng: (a) δ/r=0.015; (b) δ/r=0.00; (c) δ/r=0.045 ν d p P + = w. (15) ρu ds τ Here u τ s the frcton velocty and the brackets ndcate an average over the spanwse drecton and tme. For δ/r=0.015, P + changes sgn at three streamwse locatons, x'=0.08, 0.5 and Thus, the computatonal doman can be dvded nto four streamwse regons accordng to the sgn of the streamwse pressure gradent. Notably, n the range of 0.78<x'<1.0, P + exceeds the value of 0.09 suggested by Patel as the threshold value above whch separaton processes occur (Patel, 1965). In ths regon, whch s the zone of strong adverse pressure gradent, the boundary layer experences ntermttent reversal and separaton n the vcnty of the wall. Ths regon of strong adverse pressure gradent expands wth ncreasng δ/r. The dstrbuton of the skn frcton coeffcent, C f =τ w /(ρu /), s shown n Fg. 4(a). Relatve to C f at the nlet, C f decreases when the boundary layer s subected to an adverse pressure gradent but ncreases n regons of favorable pressure gradent. Near the flat-to-concave transton, C f ncreases rapdly n the regon where the pressure gradent decreases. A rapd ncrease n C f also occurs near the tralng edge of the bump. As ponted out by Wu and Squres (1996), ths latter change n C f can be regarded as evdence of nternal layer generaton at a curvature dscontnuty. The locatons of zero wall shear stress are x'=0.78 and x'=1.01 for δ/r=0.00, and x'=0.68 and x'=1.06 for δ/r= Snce detachment s related to the streaky structures from upstream of the bump apex, the locaton at whch detachment occurs s more varable than that of reattachment. The streamwse dstrbutons of r.m.s. wall pressure fluctuatons (p w ) rms normalzed by the reference dynamc pressure q =ρu / are dsplayed n Fg. 4(b). Two peaks are observed for all cases. The locatons of these peaks, x'=0.07 and x'=1.0, approxmately concde wth the concave-to-convex and concave-to-flat surface transtons, respectvely. Over the downstream of the bump apex, (p w ) rms begns to ncrease near x'=0.8 (δ/r=0.00) and x'=0.7 (δ/r=0.045), where the pressure gradent parameter (P + ) exceeds the value of Fgure 5 shows nstantaneous velocty vectors n the x-y plane at the mddle of the spanwse doman at two dfferent nstants. Comparson of the two velocty vector dstrbutons shows that the nstantaneous separaton bubbles and assocated shear layers change over tme. The detachment and reattachment regons move up and downstream, ndcatng the hghly unsteady nature of the flow near the tralng edge of the bump. Fgure 6 shows contour maps of the nstantaneous skn frcton coeffcent for surface curvatures of δ/r=0.015, 0.00, and Sold and dotted lnes denote postve and negatve C f, respectvely. As the turbulent boundary layer moves through a regon of adverse pressure gradent near the tralng edge of the bump, the flow decelerates untl some reversal flow occurs. Close nspecton of Fg.6 reveals that the streaky structures dsappear after the detachment regon. The spanwse lnes of detachment are related to the streaky structures that come from upstream of the bump. Small-scale structures are generated near the tralng edge of the bump and the streaky structures are completely destroyed. To examne the spectral features of the wall pressure fluctuatons, the frequency spectra of p w are obtaned usng standard technques for stochastc data. The wall pressure fluctuatons p w (x, z, t) are Fourer-transformed n the spanwse drecton and tme. Lettng p w (x, k z, ω) be the dscrete Fourer transform of p w (x, z, t), the power spectral densty s computed by * ( k, ω ; x) pˆ ( x, k, ω) pˆ ( x, k, ) Φ = ω, (16) z w z w z

6 Fg.9 Two pont correlaton coeffcent of wall pressure fluctuatons as a functon of spanwse spatal and temporal separatons: (a) x =0; (b) x =1/1 where * denotes the complex conugate and the brackets ndcate an average over the spanwse drecton and tme. The dependence of the spectral densty on the streamwse locaton x s consdered due to the flow nhomogenety. The frequency spectra ϕ(ω; x) are obtaned by ntegratng Φ(k z, ω; x) over k z. Each spectrum presented n ths paper s normalzed such that ts ntegral s equal to the mean-square of the wall pressure fluctuatons. The spectra normalzed by the outer varables are llustrated n Fg. 7. For δ/r=0.015 (Fg. 7(a)), the spectra converge n the hgh frequency regon. For 0.4<ω θ 0 /U, the spectra decrease wth a slope of about -5. The computed pressure spectra shows a regon wth a slope of -1, whch arses from the contrbuton of motons n the logarthmc regon. The frequency spectra for δ/r=0.00 and are presented n Fg. 7(b) and (c), respectvely. Any spectra scaled for the present turbulent boundary layer over a bump do not converge n the presence of a pressure gradent. Ths s due to the fact that the dynamc pressure s no longer an mportant parameter n boundary layers wth a pressure gradent. The spatal characterstcs of the wall pressure fluctuatons are obtaned from the two-pont correlatons as a functon of the streamwse spatal ( x) and temporal ( t) separatons, Rpp ( x, t; x) = pw( x, z, t) pw( x + x, z, t + t) ( p ) ( x, z, t)( p) ( x + x, z, t + t) w rms w rms, (17) where the brackets ndcate an average over the spanwse drecton and tme. Agan, the dependence on the streamwse locaton x s consdered due to the flow nhomogenety. Contour maps of the two-pont correlaton for all cases at two streamwse locatons are presented n Fg. 8, where the contour levels are from 0.1 to 0.9 n ncrements of 0.1. To facltate comparson, the spatal separatons are normalzed by the nlet momentum thckness. The strong convectve nature of the wall Fg.10 Dstrbuton of nstantaneous acoustc source n the x-y plane: (a) δ/r=0.015; (b) δ/r=0.00; (c) δ/r=0.045 pressure fluctuatons s reflected n the concentraton of the contours nto a band. The convecton velocty of large eddes s hgher than that of small eddes, as ndcated by a slghtly hgher value of the slope x/ t. Fgure 8 shows that the wall pressure feld loses coherence as convecton proceeds and as surface curvature ncreases. Note that the contour plot shows a slghtly tlted shape at x'=1/1, ndcatng that the wall pressure fluctuatons do not proceed further downstream. Fgure 9 shows the two-pont correlaton of the wall pressure fluctuatons as a functon of the spanwse spatal and temporal separatons, Rpp ( z, t; x) = pw( x, z, t) pw( x, z + z, t + t) ( p ) ( x, z, t)( p) ( x, z + z, t + t) w rms w rms, (18) Here, the brackets ndcate an average over the spanwse drecton and tme. The contour levels are from 0.1 to 0.9 n ncrements of 0.1. Compared to the contours at x'= 0, the contours near the tralng edge of the bump are elongated along the spanwse drecton and are more separated for all cases. The spanwse extent of the wdest contour (contour level 0.1) ncreases up to x'=1/1, at whch (p w ) rms has a maxmum value. Ths ndcates that the spanwse ntegral length scale of the wall pressure fluctuatons ncreases. The results obtaned by examnng the correlaton coeffcents thus reveal a two-dmensonal structure of wall pressure fluctuatons near the tralng edge of the bump. 4. Flow Nose Source and Far Feld Nose Lghthll's equaton for acoustc densty fluctuatons can be rewrtten as, ρ 1 ρ + = S, (19) t M xx where

7 Fg.11 Contours of r.m.s. acoustc source n the x-y plane: (a) δ/r=0.015; (b) δ/r=0.00; (c) δ/r=0.045 { ρ δ ( ρ / ) τ} x x S = uu + p M, (0) s the nomnal acoustc source. Contour maps of the nstantaneous acoustc source calculated usng Eq. (0) are shown n Fg. 10. It s seen that S s small near the leadng edge of the bump and bump apex. Large acoustc sources are located near the tralng edge of the bump, where the streamwse pressure gradent s strongly adverse. The source term S becomes small agan by x'=1.5. The strength of the acoustc source ncreases wth ncreasng the radus of surface curvature δ/r. Contour maps of the r.m.s. acoustc source calculated usng Eq. (0) are shown n Fg. 11. Smlar to the behavor of the nstantaneous acoustc source (Fg. 10), the r.m.s. acoustc source S s small near the leadng edge of the bump and bump apex, and reaches a maxmum near the tralng edge of the bump. Comparson of Fgs. 11 and 4(b) ndcates that the locaton of the domnant acoustc source qualtatvely concdes wth the locaton where the wall pressure fluctuatons are greatest. Ths suggests that wall pressure fluctuatons are closely correlated wth acoustc sources. Such a correlaton between acoustc sources and wall pressure fluctuatons would call nto queston the valdty of elmnatng the dpole source term comng from wall pressure fluctuatons n the acoustc source of Lghthll s equatons (Powell, 1960). A more quanttatve descrpton of the relatonshp between wall pressure fluctuatons and acoustc sources s presented below n Fg. 1. To obtan further nsght nto the acoustc source S, profles of the acoustc source were examned (Fg. 1(a)). For the boundary layer over a flat plate (δ/r=0), S has the largest r.m.s. value near y + =0, whch corresponds to the average locaton of the center of the streamwse vortces near the wall. Note that for δ/r =0.00, S s maxmum n the vscous sublayer near y + =5; ths maxmum n S s lkely related to the generaton of the nternal layer, whch s known to be trggered at dscontnutes n the surface curvature (Baskaran et al., 1987, Webster et al., 1996). A closer nspecton of Fg. 1(b) dscloses that the large terms n the vscous sublayer for δ/r =0.00 are S and S, Fg.1 Profle of r.m.s. acoustc source at x =1.0: (a) Total source; (b) δ/r=0.00 due to the hgh turbulence ntenstes n the wall-normal and spanwse drectons, respectvely. In sum, the nternal layer that forms as a result of the dscontnuty n the surface curvature s responsble for the ncreased acoustc source generaton observed near the tralng edge of the bump. Fg.1 Contours of R ps : (a) at x + =0 n the y-z plane; (b) at y + =0 n the x-z plane; (c) at z + =0 n the x-y plane

8 Fg.14 Contours of acoustc densty fluctuatons at M=0.01 for δ/r=0.0: (a) Quadrupole; (b) Dpole; (c) Total A quanttatve statstcal descrpton of the relatonshp between wall pressure fluctuatons p w and acoustc source S s obtaned from the correlaton R ps ( x, y, z) and ts correspondng coeffcent R ps ( x, y, z), whch are defned as (,, ) = (, 0,, ) ( +,, +, ) R x y z ps p x z t S x x y z z t and 0 0 (,, ) (,, ) R x y z, (1) ps ps =, () prmssrms R x y z respectvely. The brackets denote an average over the spanwse drecton and tme. Usng the DNS database for a flat plate turbulent boundary layer (Km et al., 00), the correlaton and ts correspondng coeffcent are obtaned at the center of the computatonal doman, x 0 =100θ 0. The contour lnes of R ps ( x, y, z) n the y-z, x-z, and x-y planes are shown n Fg. 1. The contour levels span from -0. to 0. n ncrements of 0.0, and negatve correlatons are ndcated by dashed contours. In the contour map n the y-z plane (Fg. 1(a)), the maxmum correlaton coeffcent occurs drectly above the locaton of maxmum wall pressure fluctuatons. Note that no spanwse dsplacement s observed at y + =0 (Fg. 1(b)), ndcatng that the strongest correlaton coeffcents n the vcnty of the wall are observed drectly above the locatons of the maxmum wall pressure fluctuatons. Fgure 1(c) dscloses that the locaton of the maxmum Fg.15 Contours of acoustc densty fluctuatons at M=0.01 for δ/r=0.015: (a) Quadrupole; (b) Dpole; (c) Total correlaton coeffcent moves away from the wall wth ncreasng x +. Ths confrms the exstence of a tlted structure n the vcnty of the wall. It s seen that the y + locaton of the maxmum correlaton coeffcent ncreases from y + =15 at x + =-50 to y + =0 at x + =-10, whch gves a tlt angle of about 7.1. Fgures show contours of the far-feld acoustc densty fluctuatons for δ/r=0.0, 0.015, 0.00 and 0.045, respectvely. The contour levels range from to n ncrements of , and contours of negatve values are dashed. It s worth pontng out that the extremely small contour levels result from the M 4 and M factors n Eq.(10). In the present calculaton, based on the acoustc analogy, suffcent arthmetc precson s mantaned snce the far-feld and near-feld acoustc denstes are evaluated separately. The volume quadrupole contrbuton to the total acoustc densty propagaton s O(M 4 ), whereas the surface dpole contrbuton s O(M ). Thus, the effect of the quadrupole source on the far-feld acoustc densty fluctuatons s smaller than that of the dpole source at low Mach numbers. For δ/r=0.0, t s clearly shown n Fg. 14 that the total acoustc feld has dpole-lke characterstcs. The contrbuton of the volume quadrupoles to the total acoustc feld gradually ncreases wth ncreasng δ/r. The role of the volume quadrupole for δ/r =0.00 (Fg. 16) s more mportant than that for δ/r =0.015 (Fg. 15). As can be clearly seen n Fg. 17, the contrbuton of the volume quadrupole for δ/r =0.045 s much larger than the contrbutons for other cases. One can conclude from the acoustc feld results that the total acoustc feld shows dpolar characterstcs for δ/r=0.0, 0.015, and 0.00 (Fgs ), whereas quadrupolar characterstcs appear n the total acoustc feld for δ/r =0.045 (Fg. 17).

9 Fg.16 Contours of acoustc densty fluctuatons at M=0.01 for δ/r=0.00: (a) Quadrupole; (b) Dpole; (c) Total Therefore, the contrbuton of the quadrupole source ncreases wth ncreasng δ/r. 5 CONCLUSIONS A detaled numercal analyss of flow past a surface bump has been performed to scrutnze the effects of longtudnal surface curvature on wall pressure fluctuatons and flow nduced nose. Statstcal descrptons of the wall pressure fluctuatons were obtaned by performng DNSs of a turbulent boundary layer over a bump. The skn frcton coeffcent, C f, decreases when the boundary layer s subected to an adverse pressure gradent, but ncreases under favorable pressure gradents. The detachment behavor s nfluenced by streaky structures from upstream of the bump apex, whch vary markedly over tme. As a result, the locaton of detachment shows larger varatons than that of reattachment. Wall pressure fluctuatons are sgnfcantly enhanced near the tralng edge of the bump, where the boundary layer s subected to a strong adverse pressure gradent. The dynamc pressure s no longer an mportant parameter n boundary layers subected to a pressure gradent. The spanwse ntegral length scale of wall pressure fluctuatons ncreases near the tralng edge of the bump. The acoustc source S s small near the leadng edge of the bump and bump apex. Large acoustc sources are located near the tralng edge of the bump, where the streamwse pressure gradent s strongly adverse. The strength of the acoustc source ncreases wth ncreasng the radus of surface curvature δ/r. The nternal layer trggered by the dscontnuty n surface curvature near the tralng edge of the bump s responsble for ncreased acoustc source generaton n ths regon. The strongest correlatons between p w and S n the vcnty of the wall are observed drectly above the locatons of maxmum wall pressure fluctuatons. The effect of the quadrupole source on the far-feld acoustc densty fluctuatons s smaller than that Fg.17 Contours of acoustc densty fluctuatons at M=0.01 for δ/r=0.045: (a) Quadrupole; (b) Dpole; (c) Total of the dpole source. The contrbuton of the volume quadrupoles to the total acoustc feld gradually ncreases wth ncreasng δ/r. REFERENCES Baskaran, V., Smts, A.J. and Joubert, P.N. 1987, A turbulent flow over a curved hll. Part 1. Growth of an nternal boundary layer, J. Flud Mech., Vol. 18, pp Webster, D.R., DeGraaff, D.B. and Eaton, J.K. 1996, Turbulence characterstcs of a boundary layer over a two-dmensonal bump, J. Flud Mech., Vol. 0, pp Wu, X. and Squres, K.D. 1998, Numercal nvestgaton of the turbulent boundary layer over a bump, J. Flud Mech., Vol. 6, pp Km, J. 1989, On the structure of pressure fluctuatons n smulated turbulent channel flow, J. Flud Mech., Vol. 05, pp Cho, H. and Mon, P. 1990, On the space-tme characterstcs of wall-pressure fluctuatons, Phys. Fluds, Vol. A, pp Km, J., Cho, J.-I. and Sung, H.J. 00, Relatonshp between wall pressure fluctuatons and streamwse vortces n a turbulent boundary layer, Phys. Fluds, Vol. 14, pp Neves, J.C. and Mon, P. 1994, Effects of convex transverse curvature on wall-bounded turbulence. Part. The pressure fluctuatons, J. Flud Mech., Vol. 7, pp Na, Y. and Mon, P. 1998, The structure of wall-pressure fluctuatons n turbulent boundary layers wth adverse pressure gradent and separaton, J. Flud Mech., Vol. 77, pp

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