Flow equations To simulate the flow, the Navier-Stokes system that includes continuity and momentum equations is solved

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1 Smulaton of nose generaton and propagaton caused by the turbulent flow around bluff bodes Zamotn Krll e-mal: cq: Summary Accurate predctons of nose generaton and spread n turbulent separatng flows requre detaled smulatons of such flows to be performed. In ths report CFD smulatons of the unsteady turbulent flow feld behnd two outsde sde-vew vehcle mrror models are presented and analyzed. The obectve of ths study are: to nvestgate numercally the transent flow structure around the mrror models n a range of ncdent flow veloctes, to nterpret the expermental observatons avalable, to valdate the CFD tools Ansys CFX 1. and Ansys Fluent 6.3, and to dentfy computatonal requrements and predctve capabltes for a range of turbulence modellng approaches. In the smulatons, qualtatvely dfferent flow behavor and vortcal structure has been demonstrated for two mrror models consdered n agreement wth the suggestons earler made based on the measured data. Surface pressure dstrbutons and velocty felds as well as ther fluctuaton spectra are compared to the wnd tunnel measurements provded by GM. Despte a dscrepancy observed n the downwnd regon near the reflectng surface, a reasonable agreement was found. The performed smulatons wll provde a bass for subsequent aero-acoustc computatons. Statement of the problem Two dfferent full-scale mrror models were consdered. The computatonal doman s shown on Fgure 1. In the nozzle outlet, a unform mean velocty profle was assumed wth turbulence ntensty of, and ntegral length scale of lt =.7 m. The nlet turbulence s not felt to play an mportant role snce the nlet s located rather far upstream of the mrror. Ths allows the correct turbulent structures to establsh themselves before the ncomng flow reaches the mrror model. Quanttatve estmates of turbulent flow characterstcs are summarzed n the Table 1 for a range of nlet flow veloctes from 1 m/s to 5 m/s. In partcular, ntegral scales (length, tme, and velocty) are estmated aganst the Kolmogorov ones thereby demonstratng characterstc values for the fluctuaton spectrum of fully developed turbulence. Two mean flow veloctes, 3 m/s and 4 m/s, have been consdered at ths phase of the work. Flow equatons To smulate the flow, the Naver-Stokes system that ncludes contnuty and momentum equatons s solved

2 Both classcal Reynolds averaged Naver-Stokes (RANS) approach and ts combnaton wth Large-Eddy Smulaton (LES) that s referred as Detached Eddy Smulaton (DES) methodology have been exploted n ths study. When equatons (1) and () are treated as averaged (RANS) or fltered (LES), the stress tensor s equal to that for a Newtonan flud complemented by turbulent stresses, Where s the stran-rate tensor. Thus n equatons mean or fltered values are mpled nstead of nstantaneous quanttes (overbars are omtted over the averaged or fltered varables). Wthn the eddy vscosty concept, turbulent stresses are expressed through the turbulent vscosty,, and the mean flow stran rate: where, the turbulent knetc energy (TKE), and,, the turbulent vscosty, are defned by a turbulence model. Flow s assumed to be ncompressble ( ) wthn the velocty range consdered. Computatonal grds Mult-block computatonal grds used n the smulatons were constructed wth Ansys ICEM CFD 1.. The computatonal doman was dvded onto several sub-domans, each covered by a separate curvlnear structured grd. To assess grd dependency of the results, two grds wth total number of control volumes of.9m (referred below as coarse grd) and 7.15M (fne grd) have been used. Table. Characterstc szes of grd elements n near wake downstream the mrror models Characterstc grd szes are compared to turbulent length scales n Table 1. Flow Results Table surface pressure dstrbutons Adequate predcton of the surface pressure dstrbutons s necessary for accurate smulatons of nose orgnated from the dpole surface sources. Comparson of the pressure dstrbutons obtaned wth two grds s shown n Fgure for 3 m/s flow around MODEL A mrror model.

3 Mrror surface pressure dstrbutons Рronounced low-pressure regon develops at the top of the model, whch can nspre early flow separaton. We have found ndcatons of transent and mult-pont separaton and reattachment n ths regon. Ths can be seen n Fgure 3 where several local and unsteady separaton zones appear. Intensve fluctuatons of pressure n ths regon have also been predcted. Оn that ths regon produces consderable amount of nose. Worth notng, appearance of multple and unsteady separaton zones has yet to be further nvestgated to ensure ts physcal, not numercal mechansm. It could be done by mesh refnement n the streamwse drecton. The sound propagaton equaton The acoustc analogy s derved by rearrangement of the contnuty and momentum equatons, where s the stress tensor ncludng pressure, p, and vscous stresses, It s assumed that n a confned regon, Ω, flud flow feld s known from ths equatons. Whlst the flud (havng pressure p and densty ρ ) s at rest beyond Ω. The followng equaton can be derved Subtractng c t T ~ = from both sdes of last equaton yelds (1) where s the Lghthll s tensor, and s the speed of sound n the quescent flud. Equaton (1) s smlar to that of acoustc wave propagaton, the rght hand sde appears to be the source of the acoustc perturbatons. The Lghthll s analogy s therefore formulated as follows: densty fluctuatons n an arbtrarly movng real compressble flud are the same as those n a unform quescent flud subect to the mposed stresses as expressed by the rght hand sde of Eq. (1). Thus, n the Lghthll s acoustc analogy, the actual flow s replaced by the T dstrbuton produced by the actual flow. Then, Eq. (1) governs the sound propagaton n a unform acoustc medum at rest, subect to the T forcng. Equaton for Lghthll s tensor can be further rearranged neglectng molecular vscosty and thermal conductvty. In the deal sentropc flud µ = and p = c p therefore the Lghthll s tensor s reduced to T ~ = ρu u where densty s non-unform due to the acoustc perturbatons. However, n low-mach number flows, t can be reasonably assumed that ρ = ρ, and Eq. (1) smplfes to c t uu = ρ ()

4 The latter can be treated as the equaton of sound (.e. densty fluctuatons) propagaton n a quescent meda, and the rght hand sde of Eq. () s the expresson for source of sound produced by the flow n the regon Ω. Equaton () s also vald for perturbed densty, ~ ρ = ρ ρ Solvng Eq. () s possble after the flow feld s determned n Ω. That allows the sound pressure, ~ p = c ( ρ ) to be obtaned. ρ Solutons to the sound propagaton equaton The soluton of Eq. () s the convoluton of the rght hand sde wth the Green functon, ~ 1 dy p ( x, t) = [ T ] 4πc r Ω where s the dstance between the source pont y and the observer poston x, and the square brackets ndcate that the bracketed quantty s taken at the retarded tme,. If an observer at x s n the far feld then terms of order of and can be neglected and Eq. (3) s rewrtten as ~ 1 ( x y )( x y ) [ T ] ρ ( x, t) = dy (4) 4 3 4πc r t Ω For a numercal mplementaton of Lghthll s analogy, the temporal formulaton (4) s more approprate than that n Eq. (3) whch s based on the spatal dervatves of T. Note, nether Eq. (3) no Eq. (4) consder the effects of sold walls. To overcome the latter lmtaton, the soluton to Eq. () was extended by Curle [8] to nclude the effects of sold walls at rest. He solved Eq. () thereby obtanng the Krchhoff formula, (3) the equvalent and more convenent equaton can be derved: Where T s the stress tensor, and T ~ s the Lghthll s tensor. Equaton results n an mportant concluson. Indeed, t shows that sound generaton s determned by three contrbutng sources. The frst ntegral represents monopole sources dstrbuted over the surface S. The second one corresponds to dpole sources also dstrbuted over the surface S. Ths term ncludes sound refracton and dffracton at the surface. Fnally, the thrd ntegral represents the quadrupole sound emsson n the regon of Ω. Note, the surface S can be ether permeable or sold; f S s a sold mmovable wall then u =, and equaton smplfes: u The above formulated Lghthll s analogy can now be extended: densty fluctuatons n an arbtrarly real compressble flud movng nsde the regon Ω that contans sold walls are the same as those n a unform quescent flud affected by the dpoles dstrbuted over the wall surface S and the quadrupoles dstrbuted n the volume of Ω. Last equaton can be smplfed by neglectng effects of vscosty:

5 Acoustc pressure and sound pressure level n ar sensor locatons Acoustc pressures have been calculated and sound pressure level have been predcted for some cases: 3 m/s and 4m/s wnd speed, coarse and fne grd. The analyss of results ndcates that reasonable agreement between predcted and measured SPL has been obtaned although the predcted ones are underestmated, partcularly for frequences greater 1 khz. Acoustc pressure and sound pressure level n surface mcrophone locatons Measured SPL frequency dstrbutons are qualtatvely dfferent for the mcrophones located nsde and outsde the recrculatng zone behnd the mrror model. Sgnals recorded nsde the recrculatng zone are determned by pressure fluctuatons at the table surface, whlst those recorded outsde the recrculatng zone are manly due to the external nose. Fgure 1. The computatonal doman (a) Coarse grd (b) Fne grd Fgure. Surface pressure resoluton wth two grds. Fgure 3. Early separaton zone at the top of mrror. Velocty vectors colored by velocty magntude