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1 Journal of Computatonal Physcs 8 (9) Contents lsts avalable at ScenceDrect Journal of Computatonal Physcs journal homepage: A shock-capturng methodology based on adaptatve spatal flterng for hgh-order non-lnear computatons Chrstophe Bogey *,, Ncolas de Cacqueray, Chrstophe Bally 3 Laboratore de Mécanque des Fludes et d Acoustque, UMR CNRS 559, Ecole Centrale de Lyon, 6934 Ecully Cede, France artcle nfo abstract Artcle hstory: Receved June 8 Receved n revsed form 7 October 8 Accepted 3 October 8 Avalable onlne 7 November 8 Keywords: Shock-capturng Fnte dfferences Spatal flterng Hgh-order A shock-capturng methodology s developed for non-lnear computatons usng low-dsspaton schemes and centered fnte dfferences. It conssts n applyng an adaptatve second-order flterng to handle dscontnutes n combnaton wth a background selectve flterng to remove grd-to-grd oscllatons. The shock-capturng flterng s wrtten n ts conservatve form, and ts magntude s determned dynamcally from the flow solutons. A shock-detecton procedure based on a Jameson-lke shock sensor s derved so as to apply the shock-capturng flterng only around shocks. A second-order flter wth reduced errors n the Fourer space wth respect to the standard second-order flter s also desgned. Lnear and non-lnear D and D problems are solved to show that the methodology s capable of capturng shocks wthout provdng dsspaton outsde shocks. The shock detecton allows n partcular to dstngush shocks from lnear waves, and from vortces when t s performed from dlataton rather than from pressure. Fnally the methodology s smple to mplement and reasonable n terms of computatonal cost. Ó 8 Elsever Inc. All rghts reserved.. Introducton Issues [] specfc to computatonal aeroacoustcs (CAA) have led over the last 5 years to the desgn of approprate methods, reported for nstance n the revew of Colonus and Lele [], whch are less dspersve and less dsspatve than standard methods of computatonal fluds dynamcs (CFD). Centered dfferentaton schemes have n partcular been consdered to mnmze numercal dampng. They are however naccurate for the hgher wavenumbers dscretzed, and mght generate numercal nstabltes, specally for grd-to-grd oscllatons, and therefore are usually mplemented n combnaton wth flterng of the hgh-frequency waves nvolvng selectve flters [3 7] affectng the low-frequency waves n a neglgble manner. These methods have been appled successfully to compressble unsteady Naver Stokes computatons for predctng the nose generated by turbulent flows [,8,9], and have moreover been shown to be well suted to perform accurate large-eddy smulatons [,]. They can also be used for strongly non-lnear problems, such as the generaton of screech nose n supersonc jets as n Berland et al. [], but t s generally recognzed that they encounter serous problems for flows contanng dscontnutes such as shocks. Near shocks, the mplementaton of low-dsspaton schemes mght ndeed result n spurous Gbbs oscllatons due to spectral truncaton n the wavenumber space. * Correspondng author. Fa: E-mal addresses: chrstophe.bogey@ec-lyon.fr (C. Bogey), ncolas.cacqueray@ec-lyon.fr (N. de Cacqueray), chrstophe.bally@ec-lyon.fr (C. Bally). Research Scentst, CNRS. Ph.D. student. 3 Professor at Ecole Centrale de Lyon and Insttut Unverstare de France. -999/$ - see front matter Ó 8 Elsever Inc. All rghts reserved. do:.6/j.jcp.8..4

2 448 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) In order to prevent the appearance of Gbbs oscllatons n smulatons of shocked flows, the usual approach s based on shock-capturng upwnd-based schemes. Such schemes have been formulated snce the early eghtes by many researchers ncludng, among others, Harten et al. [3 8]. To handle shocks, these authors developed famous CFD algorthms such as the TVD (total varaton dmnshng) schemes makng use of flu or slope lmters, and the ENO (essentally non-oscllatory) and WENO (weghted ENO) schemes n whch an adaptatve stencl that adjusts to the smoothness of the solutons s appled, refer for nstance to the two revews made by Shu [9] and Prozzol [] for more detals. These schemes ensure hgh stablty, but they are n general of low accuracy, especally for tme-dependent problems. They mght provde unsatsfactory results for shock turbulence nteracton problems [], as well as ecessve numercal dampng on turbulent scales n large-eddy smulatons []. Attempts have therefore been made to mprove ther performance by modfyng ther desgn [3] or by ncreasng ther formal order [4]. In ths case, n order to assess the qualty of the solutons, n partcular n aeroacoustc studes, there s an urgent need of analysng the spectral propertes of the shock-capturng schemes n the Fourer space [5], and of checkng ther accuracy by solvng standard CAA test cases [6,7]. Another nterestng approach suggested by Adams and Sharff [8] s to couple compact/low-dsspaton schemes wth a shock-capturng scheme that s turned on around dscontnutes [8 3]. The adaptaton of the spatal scheme then requres the detecton of the strong non-lnear features wthn the computatonal doman. Ths has been done for nstance by Vsbal and Gatonde [3] who appled a shock detector to swtch between compact and shock-capturng schemes, but furthermore eplored a dfferent methodology where numercal flterng s adapted to the flow features. Wth the am of usng centered dfferentaton schemes to keep good resoluton characterstcs, one possblty s ndeed to make use of an adaptatve artfcal dsspaton model, correspondng also to a flterng of the solutons, whch s effectve near the dscontnutes but tends to have neglgble nfluence everywhere else. Jameson et al. [3] thus ntroduced addtonal terms n the Euler equatons that consst of a blend of second-order and fourth-order dsspatons wth non-lnear swtchng coeffcents. Ther method was appled by Pullam [33] and Swanson and Turkel [34] for steady nvscd flows around arfols, but t was found to be too dsspatve for unsteady problems. For aeroacoustcs purposes, Lockard and Morrs [35] and Km and Lee [36] proposed hgher-order versons of Jameson s model, n whch the selectve dsspaton of Tam et al. [4,5] s mplemented rather than the orgnal fourth-order dsspaton. Smlarly Vsbal and Gatonde [3], Hon et al. [37] and Emmert et al. [38] recently developed shock-capturng dsspaton models combnng second-order and hgh-order dffuson operators. One crucal pont n the methodology s the defnton of the shock detector, whch has to dstngush between shocks and gradents of any other knd n order to lmt the range of the shock-capturng dsspaton specfcally to the regons contanng shocks [39]. Detectors estmated from smple gradents [8,3], from second dervatves of pressure or densty [3,36 38] such as the Jameson detector, and from WENO-type smoothness crtera [3,4] have n partcular been used. Ducros et al. [4] also proposed a modfed verson of the Jameson detector takng nto account the local property of compressblty, whch s capable of dscrmnatng between turbulent fluctuatons and shocks [4,4]. Fnally, once the shock-detecton sensor s evaluated, the shock regon s dealt wth by means of a swtch whch has to specfy the type and amount of dsspaton to be specfed at each grd pont. In the present study, a shock-capturng methodology based on an adaptve spatal flterng s derved for hgh-accuracy non-lnear computatons ncludng low-dsspaton tme ntegraton and centered space dfferencng. Followng the works presented above, t conssts n applyng a background selectve flterng at each mesh pont to remove grd-to-grd oscllatons, n combnaton wth a shock-capturng flterng around dscontnutes. To smooth possble shocks n a proper manner, the shock-capturng flterng s wrtten n a conservatve form and s of second-order, but ts magntude has to be adjusted dynamcally from the flow solutons to be nl n regons of lnear propagaton, for well-resolved gradents and for turbulent fluctuatons, so that the approach should be approprate for unsteady CAA and CFD problems. To meet ths requrement, a second-order flter reducng phase errors wth respect to the standard second-order flter when appled wth a non-unform strength s frst desgned. A Jameson-lke shock sensor evaluated from the magntude of the hgher wavenumbers of the flow varables s then proposed. It can be estmated ether from pressure as classcally done, or from dlataton n order to gve weght to the local feature of compressblty n the procedure of shock detecton. The magntude of the shock-capturng flterng s fnally determned from the shock sensor n a smple way, usng a gven threshold parameter. The effcency of the shock-capturng methodology s assessed by solvng standard lnear and non-lnear problems [43,44,49,5] wth a low-dsspaton Runge Kutta algorthm and centered fnte dfferences, bult up n Bogey and Bally [7] to be well suted to CAA needs. Problems of acoustc wave and shock propagaton, vorte convecton, shock sound nteracton, shock tubes and shock vorte nteracton are specally consdered. The nfluence of some methodology parameters such as the flter shape, the use of pressure or dlataton for detectng the shock, and the threshold parameter provdng the flterng strength s thus dscussed, n order to draw recommendatons. The applcaton of the selectve flterng to the flues [5] rather than to the flow varables s also brefly dscussed for a non-lnear problem. The present paper s organzed as follows: the equatons governng the test problems and the background numercal algorthm, ncludng a sth-order selectve flter, are reported n Secton. The development of the shock-capturng flterng procedure s detaled n Secton 3, wth a focus on the mpact of the conservatve form of the flterng n the Fourer space, the defnton of the shock detector and the determnaton of the flterng magntude. The results obtaned for the test cases usng the adaptatve shock-capturng flterng are then shown n Secton 4. Concludng remarks are fnally provded n Secton 5.

3 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) Equatons and numercal algorthm.. Governng equatons To quantfy the effects of the shock-capturng flterng on the behavour and nteractons of dfferent knds of dsturbances, solutons of test cases wll be calculated n Secton 4 by solvng problems of acoustc and shock propagaton, vorte convecton, shock sound nteracton n a transonc nozzle, shock tubes and shock-vorte nteracton. They wll be computed from the one-dmensonal, the quas-one-dmensonal or the two-dmensonal Euler equatons wrtten n a conservatve form usng Cartesan coordnates. Dmensonless varables defned by ntroducng reference scales for densty, length and velocty (sound speed) are used.... One-dmensonal equatons The one-dmensonal Euler equatons are epressed n @ ¼ ; ðþ where the vector U of conservatve varables and the flu vector E are defned as U ¼½q; qu; qeš T and E ¼½qu; qu þ p; uðqe þ pþš T, and q, u, p denote densty, velocty and pressure. The total energy s gven by qe ¼ p=ðc Þþqu = wth c ¼ :4.... Quas-one-dmensonal equatons The quas-one-dmensonal Euler equatons are wrtten n @ þ Q ¼ ; ðþ where the varable vector U, the flu vector E and the source vector Q are respectvely gven by U ¼½q; qu; qeš T, E ¼½qu; qu þ p; uðqe þ pþš T and Q ¼ð=AÞðdA=dÞ½qu; qu ; uðqe þ pþš T, and A ¼ AðÞ s the cross-sectonal area...3. Two-dmensonal equatons The two-dmensonal Euler equatons are fnally ¼ ; where the varable vector U and the flu vectors E and F are provded by U ¼½q; qu; qv; qeš, E ¼½qu; qu þ p; quv; uðqe þ pþš and F ¼½qv; quv; qv þ p; vðqe þ pþš. The total energy s now gven by qe ¼ p=ðc Þþqðu þ v Þ=, where u and v are the two velocty components. ð3þ.. Numercal algorthm When Eqs. () (3) are solved, the spatal dervatves are appromated wth -pont fourth-order centered fnte dfferences, whch have been desgned [7] so as to generate neglgble phase errors down to waves dscretzed by four ponts per wavelength. Tme ntegraton s performed usng a s-stage second-order low-storage Runge Kutta algorthm dsplayng low-dsperson and low-dsspaton n the Fourer space [7]. These methods have been successfully mplemented n prevous studes to perform drect nose computatons for confguratons such as subsonc and supersonc jets [,5,53], and flows around an arfol [54]. Durng the computatons, a background numercal dsspaton s appled n the followng way: after each tme step, the conservatve varables U are fltered eplctly usng an -pont selectve flter at a magntude r sf, to provde at node U sf ¼ U r sf D sf ; where 6 r sf 6, and the flterng operator s gven by D sf ¼ X5 j¼ 5 d j U þj ð4þ ð5þ and d j are the flter coeffcents. The flterng procedure s conservatve as long as a constant magntude r sf s used. In what follows, ths magntude s moreover set to r sf ¼, mplyng that grd-to-grd oscllatons are completely removed after each tme teraton. The selectve flter nvolved n the present study s an -pont sth-order flter, whose coeffcents d j, reported n Table, have been determned so that ts dampng functon D k ðkdþ ¼d þ P 5 j¼ d j cosðjkdþ s lower than 5 over a large wavenumber range.

4 45 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) Table Coeffcents of the -pont sth-order selectve flter, wth d j ¼ d j. d ¼ : d ¼ : d ¼ : d 3 ¼ : d 4 ¼ : d 5 ¼ : π/6 π/8 π/4 π/ π kδ Fg.. Dampng functon, n logarthmc scales, as a functon of the wavenumber kd: 3 optmzed -pont flter of order 6, standard th-order flter, standard second-order flter. The dampng functon thus obtaned s shown n Fg.. Compared to the standard second-order flter,.e. to a typcal shock-capturng flter, the selectve flter dffers bascally by provdng apprecable dsspaton only for waves roughly over p= 6 kd 6 p, dscretzed by fewer than four grd ponts, whereas the second-order flter mght affect small wavenumbers. To evaluate accuracy lmts of the selectve flter, the two crtera D k ðkdþ 6 :5 3 and D k ðkdþ 6 :5 4 are used as prevously n Bogey and Bally [7], yeldng k=d ¼ 4:8 and k=d ¼ 5:74 n terms of number of ponts per wavelength. These lmts are lower for nstance than the lmts k=d ¼ 5:4 and k=d ¼ 6:96 found for the standard th-order flter also represented n Fg.. Fnally, boundary condtons based on characterstcs [56] are mplemented n the quas-one-dmensonal problems, whereas the non-radaton boundary condtons derved by Tam and Dong [55] are appled n the two-dmensonal cases. 3. Shock-capturng flterng The am s to develop a shock-capturng flterng procedure approprate for unsteady hgh-order smulatons. Therefore the flterng wll be of second-order, and wrtten n a conservatve form to accurately descrbe the propagaton of shocks, and ts magntude wll have to be adjusted dynamcally from the flow varables so that t s neglgble everywhere ecept around dscontnutes. 3.. Conservatve form of the flterng The shock-capturng flterng s appled at each tme step just after the background selectve flterng removng grd-togrd oscllatons. Snce ts magntude dependng on the shock detecton s epected to vary, the flterng operaton s wrtten n a conservatve form as the dfference between two dampng flues taken at the nterface of two adjacent cells as recommended by Km and Lee [36]. At pont on an unform grd, the conservatve varables U are thus fltered eplctly to yeld U sc ¼ U r sc D sc þ þ r sc D sc ; ð6þ where the flterng strength 6 r sc 6 s not constant, and the dampng functons D sc þ and D sc are estmated from the varables U usng the followng nterpolatons: D sc þ ¼ Xn c j U þj and D sc ¼ Xn c j U þj j¼ n j¼ n ð7þ To determne the coeffcents c j of the n-pont nterpolaton defnng the dampng functons, one consders the non-conservatve form of the flterng

5 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) U sc ¼ U r sc X n j¼ n d j U þj ð8þ and notes that Eqs. (6) and (8) must be equvalent when the flterng magntude s unform. For a gven n, the coeffcents c j are then drectly obtaned from the coeffcents d j of the correspondng non-conservatve centered flter. The values found for the standard second-order flter, here referred to as Fo, are collected n Table. The coeffcents c j calculated for the standard fourth-order flter Fo4 are also gven n the table, despte ths flterng s not dsspatve enough to handle strong dscontnutes n a proper manner and wll not be used to capture shocks subsequently. 3.. Characterstcs of the flterng n the Fourer space The effects of the shock-capturng flterng are nvestgated n the Fourer space by consderng the applcaton of the conservatve form (6) of the flterng wth a non-unform magntude r sc ¼ r sc þ þ Dr sc and r sc ¼ r sc Dr sc (by constructon 6 r sc 6 and :5 6 Dr sc 6 :5), yeldng U sc ¼ U r sc D sc þ D sc Dr sc D sc þ þ D sc : ð9þ Introducng the dampng functons (7) nto epresson (9) provdes ( ) ( ) ¼ U r sc c n U þn þ Xn ðc j c jþ ÞU þj c n U n Dr sc c n U þn þ Xn ðc j þ c jþ ÞU þj þ c n U n U sc j¼ n j¼ n ðþ In order that the second term n the rght-hand sde of Eq. () should be equvalent to a flterng at the magntude r sc provdng no dsperson, that s a flterng based on symmetrcal coeffcents, one has to set c n ¼ c n and c j c jþ ¼ c j c j. The coeffcents of the dampng functons are then antsymmetrc wth c j ¼ c j, and the flterng procedure () becomes ( ) U sc ¼ U r sc c U þ Xn ðc j c jþ ÞðU þj þ U j Þþc n ðu þn þ U n Þ j¼ ( ) Dr sc X n j¼ ðc j þ c jþ ÞðU þj U j Þþc n ðu þn U n Þ Applyng spatal Fourer transform to Eq. () allows us to wrte d U sc ¼ c U r sc D real ðkdþþdr sc D mag ðkdþ ; ðþ where D real ðkdþ s the transfer functon of the equvalent flter obtaned wth a unform flterng magntude, and D mag ðkdþ s the transfer functon of the phase errors generated by the varatons of the flterng strength. They are defned by ðþ D real ðkdþ ¼ c þ Xn ðc j c jþ Þ cosðjkdþþc n cosðnkdþ; j¼ and D mag ðkdþ ¼ Xn ðc j þ c jþ Þ snðjkdþ c n snðnkdþ: j¼ ð3þ ð4þ The transfer functons for the standard second- and fourth-order flters Fo and Fo4 are presented n Fg.. The profles for D real ðkdþ n Fg. (a) correspond to the dampng functons classcally observed as a functon of the wavenumber, wth the decrease of dsspaton as the order ncreases. Those for D mag ðkdþ n Fg. (b) suggest that the phase errors mght be mportant for the second-order flter. An attempt s now made to develop a specfc 4-pont conservatve flter for shock-capturng, referred to as Fopt, that dsplays dsspaton features smlar to those of the standard second-order flter Fo, but also generates reduced errors. The flter s then desgned so that ts dampng functon D real ðkdþ appromates the dampng functon D Fo realðkdþ of flter Fo, whle lowerng ts related phase errors gven by D mag ðkdþ for a gven range of wavenumbers. In practce the coeffcents c j of the flter Fopt are chosen so that the ntegral error Table Coeffcents c j for conservatve shock-capturng flterng: standard second-order flter (Fo), standard fourth-order flter (Fo4), and optmzed second-order flter (Fopt), wth c j ¼ c j. Fo Fo4 Fopt c /4 3/6.383 c /6.3967

6 45 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) (a) (b).8.8 D real (kd).6.4 D mag (kd) p/4 p/ 3p/4 p kd p/4 p/ 3p/4 p kd Fg.. Transfer functons, as a functon of the wavenumber, of the real and magnary parts, (a) D real ðkdþ and (b) D mag ðkdþ, for conservatve shockcapturng flterng: 3 standard second-order flter (Fo), standard fourth-order flter (Fo4), and -- optmzed second-order flter (Fopt). Z p h Z p D real ðkdþ D Fo real ðkdþ 5 dðkdþþ D mag ðkdþ dðkdþ ð5þ s mnmzed. The optmzaton s carred out by mposng 6 D r 6 for 6 kd 6 p. Two regularzaton constrants must also be satsfed: D r ðkd ¼ Þ ¼, whch s naturally ensured by epresson (3), and D r ðkd ¼ pþ ¼ yeldng c ¼ 4 þ Xn ð Þ j c j : j¼ ð6þ Therefore, for a 4-pont flter defned by antsymmetrc coeffcents, there s only one coeffcent to adjust. The coeffcents of the flter Fopt are reported n Table, and the correspondng transfer functons are presented n Fg.. Compared to the flter Fo, the optmzed flter shows phase errors decreased by a factor of about for small wavenumbers, whle beng sgnfcantly more dsspatve than the flter Fo4. Its shock-capturng capabltes wll be dscussed n Secton 4 devoted to the test cases Adaptatve flterng magntude The flterng strength r sc s to be estmated from the flow varables, so that t should be sgnfcant around dscontnutes but neglgble everywhere else. A procedure of shock detecton s therefore frst proposed. More precsely, n order to ndcate the presence of shocks wthn the computatonal doman, a shock detector, roughly smlar to that formulated by Jameson et al. [3] makng use of the second dervatve of pressure, s evaluated from the magntude of the hgh-wavenumber components of a varable that can be ether pressure or dlataton. Based on pressure, the present shock sensor s determned followng three steps. The pressure hgh-wavenumber components are frst etracted from varable p usng the second-order flter Fo, yeldng, at node Dp ¼ð p þ þ p p Þ=4: ð7þ The magntude of the hgh-pass fltered pressure s then calculated as Dp magn ¼ h Dp Dp þ þ ðdp Dp Þ ð8þ and the shock sensor s defned as the rato r epressed as r ¼ Dpmagn þ ; p ð9þ where ¼ 6 s ntroduced to avod numercal dvergence later n epresson (3). In some cases the use of pressure to detect shocks mght not be approprate for dstngushng between turbulent fluctuatons and shocks n an unambguous manner. To deal wth ths defcency, as also suggested by Ducros et al. [4], a possblty s to take nto account the local property of compressblty. Ths led us to perform the shock detecton from dlataton H ¼ r u rather than from pressure. The hgh-pass fltered dlataton s computed at node as DH ¼ ð H þ þ H H Þ=4 ðþ and ts ampltude as

7 ¼ h ð DH DH þ Þ þ ðdh DH Þ : ðþ DH magn The shock sensor based on dlataton s then calculated as C. Bogey et al. / Journal of Computatonal Physcs 8 (9) r ¼ DHmagn þ ; c ðþ =D where c ¼ cp =q s the square of the local sound speed. Once the value of the shock detector r s known, from pressure or dlataton, the strength of the flterng has to be gven. In the present approach, followng Vsbal and Gatonde [3] for nstance, a threshold parameter r th s used to specfy the regons where the shock-capturng flterng s employed. The flterng magntude s evaluated by the functon r sc ¼ r th þ r th ð3þ r r whch s represented n Fg. 3. For r 6 r th, the flterng magntude s r sc the hgh-wavenumber components of pressure or dlataton are apprecable, one gets < r sc ¼ as requred. For r > r th, that s when the level of <, and n partcular r sc! for r!þ. In ths way, the second-order flter s only swtched on when the gradents of pressure or dlataton are strong enough. The threshold parameter r th s typcally to be set between 6 and 4, a lower value correspondng to an applcaton of the shock-capturng flterng on a wder regon, leadng to smoother solutons. A threshold value of r th ¼ 5 wll be n addton shown later to provde approprate results for the dfferent test problems solved, and could be recommended as a reference parameter. For completeness, for the applcaton of the conservatve form (6) of the shock-capturng flterng, the values of r sc between the nodes are smply appromated by r sc þ ¼ rsc þ þ rsc and r sc þ r sc : ð4þ ¼ rsc 4. Test cases Two knds of test problems are solved to study the relevance and the effcency of the shock-capturng methodology. Frst lnear problems (acoustc propagaton and vorte convecton) are consdered n order to verfy that the shock-capturng flterng does not apply n these cases. Then non-lnear problems nvolvng shocks (shock propagaton, shock acoustc nteractons, shock tubes and shock vorte nteractons) are smulated to demonstrate the capablty of the methodology to take nto account dscontnutes n a proper manner. As prevously mentoned, the numercal algorthm used for the dscretzaton of the test cases combnes -pont low-dsperson centered fnte dfferences wth a 6-stage optmzed Runge Kutta schemes desgned n Bogey and Bally [7], and a background selectve flterng of the conservatve varables s mplemented after each tme step usng the -pont flter of sth-order presented n Secton., wth a magntude r sf ¼. 4.. Lnear problems Two test cases are frst computed n order to check whether the shock-capturng flterng s turned on n lnear problems nvolvng acoustcal or vortcal dsturbances..8 σ sc r/r th Fg. 3. Varatons of the shock-capturng flterng magntude r sc as a functon of the shock sensor r=r th.

8 454 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) Acoustc propagaton In order to compute the propagaton of an acoustc wave from the one-dmensonal Euler equatons (), wth or wthout makng use of the shock-capturng procedure, a pressure pulse s specfed by mposng at tme t ¼ the followng condtons: u ¼ ; p ¼ c þ : ep lnðþ and q ¼ ; b where b s the Gaussan half-wdth of the pulse. The pressure ampltude of the rght-gong travellng wave generated by the ntal pulse s 4 tmes smaller than the ambent pressure =c, so that non-lnear effects are neglgble durng the propagaton. The problem s solved on a unform grd wth a mesh spacng D ¼, wth a tme step Dt ¼ :8. To eplore the adaptablty of the shock-capturng method to lnear wave propagaton, two pulses, one well-resolved by the grd and another slghtly under-resolved, defned respectvely by half-wdths b ¼ and b ¼ 3 are dealt wth. The pressure profles thus obtaned at t ¼ wthout shock-capturng are presented n Fg. 4(a). There s no vsble dsperson nor dsspaton of the broader pulse wth b ¼ 3, whereas the shape of the pulse wth b ¼ has been modfed durng the propagaton wth a ampltude that has been n partcular notceably decreased by the selectve flterng. The pressure profles determned when the shock-capturng procedure s swtched on are not represented here, but they collapse perfectly wth the prevous profles for both pulse confguratons. The shock detectors r calculated at t ¼ from pressure or from dlataton are ndeed shown n Fg. 4(b) and to be lower than. These values are well below the threshold parameter whch s typcally between r th ¼ 6 and r th ¼ 4. Consequently the magntude of the shock-capturng flterng s set to r sc ¼, and the shock-capturng flterng s not employed n the present lnear problems Vorte convecton The convecton of a round vorte by a unform flow s now consdered by solvng the two-dmensonal Euler equatons (3). Att ¼, the followng ntal condtons are then mposed: u ¼ :5 y b ep lnðþ þ y ; v ¼ b ep lnðþ þ y b b ; p ¼ c ; and q ¼ n order to ntroduce a dvergence-free vorte at ¼ y ¼, smlarly to what was done n the frst CAA Workshop [43], whch wll be convected n the aal drecton at the dmensonless velocty.5 that s half the speed of sound. The computaton s performed on a grd contanng 8 ponts wth mesh spacngs D ¼ Dy ¼, wth a tme step Dt ¼ :5. When the shock-capturng procedure s appled, the standard second-order flter Fo s used n ts conservatve form, and a threshold parameter r th ¼ 4 s specfed for the shock detector. In addton, as prevously for the acoustc pulses, two geometrcal confguratons are studed: a well-resolved vorte defned by a Gaussan half-wdth b ¼ 5, and a vorte wth b ¼ 3 characterzed by a narrower core. As llustratons of the problem solutons, the vortcty and the pressure felds calculated at t ¼ 5 for the vorte wth b ¼ 5 wthout shock-capturng are represented n Fg. 5. As epected, the vorte has been convected by the aal flow so as to be located at ¼ 5 and y ¼ n Fg. 5(a). A regon wth negatve pressure nduced by the vorte s also observed at ths place n Fg. 5(b), whle a transtory crcular sound wave s notced all around the vortcal structure. To gve evdence of possble effects of the shock-capturng method on the present vortces, the pressure profles obtaned along y ¼ at tme t ¼ 5 wth or wthout shock-capturng are shown n Fg. 6. They dsplay pressure fluctuatons of aerodynamc nature centered on the vorte core at ¼ 5, and acoustc pressure waves at ¼ 3 and ¼ 8. Wth respect to the solutons computed wthout shock-capturng n Fg. 6(a), the shock-capturng procedure usng pressure as varable for the shock detecton appears to damp the solutons n Fg. 6(b), especally for the narrower vorte wth b ¼ 3. Smlar alteratons due to shock-capturng are however not observed n Fg. 6, when dlataton rather than pressure s used to evaluate the shock sensor. (a) (b).5 r/ r/ (g Fg. 4. Lnear acoustc propagaton. Solutons at t ¼ : (a) pressure computed wthout usng the shock-capturng procedure and, shock sensor r evaluated (b) from pressure and from dlataton usng the shock-capturng procedure. Gaussan pulse half-wdth: 3 b ¼ 3, b ¼.

9 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) (a) 6 (b) y/dy y/dy Fg. 5. Vorte convecton. Solutons at t ¼ 5 for the vorte wth Gaussan half-wdth b ¼ 5: (a) vortcty and (b) pressure, computed wthout usng the shock-capturng procedure. Representaton of contours: ¼½ :3; :5; :5Š for vortcty and cp ¼ þ½ :5; :; :3Š for pressure. (a) (b) g g g Fg. 6. Vorte convecton. Profles of pressure computed at t ¼ 5 along y ¼ : (a) wthout usng the shock-capturng procedure, and usng the shockcapturng procedure wth a shock detector evaluated from (b) pressure and dlataton. Gaussan half-wdth: 3 b ¼ 5, b ¼ 3. The dscrepances between the solutons obtaned usng the shock-capturng method n Fg. 6(b) and result from the choce of varable nvolved n the estmaton of the shock sensor. In the frst case, pressure s used, yeldng at t ¼ 5 for the aal shock sensor r the profles plotted n Fg. 7(a). Because of the mportant gradents of aerodynamc pressure around the vortces, the shock sensor s of sgnfcant magntude at 5. In ths way t mght ndcate the presence of a shock and lead to the applcaton of the shock-capturng flterng n the vorte regon. In the present problems, the second-order flter has been n partcular turned on n the begnnng of the calculatons, but s no more actve at t ¼ 5 because the threshold parameter s r th ¼ 4. A dfferent behavour s found when dlataton s used to evaluate the shock sensor r. In ths case, the shock sensor takes very small values around the vortces, as demonstrated by the profles of Fg. 7(b), and the shock-capturng flterng does not apply. The use of dlataton n the procedure of shock detecton therefore appears more approprate n vortcal flows because, contrary to pressure, t allows to dstngush a vorte from a shock. (a) (b) r/.5 r/ Fg. 7. Vorte convecton. Profles of the aal shock sensor r computed at t ¼ 5 along y ¼ : (a) from pressure, (b) from dlataton, usng the shockcapturng procedure. Gaussan half-wdth: 3 b ¼ 5, b ¼ 3.

10 456 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) Non-lnear problems Test cases are now smulated to look nto the capablty of the methodology to properly capture shocks wthout apprecably affectng the accuracy of the solutons on both sdes of the shocks, especally when the shocks nteract wth acoustc or aerodynamc perturbatons Shock propagaton The frst non-lnear problem studed s concerned wth shock propagaton. The test case s taken from the frst CAA Workshop [43]. It s problem from category, that s defned by the followng ntal perturbatons at tme t ¼ : u ¼ :5 ep lnðþ ; p ¼ 5 c þ c c u c and q ¼ þ c u c : The problem s solved from the one-dmensonal Euler equatons (), usng a mesh grd of spacng D ¼ and a tme step Dt ¼ :8, to provde pressure dstrbutons at t ¼. Solutons are frst computed wthout shock-capturng, by only applyng selectve flterng to the varables or to the flues. They are presented respectvely n Fg. 8(a) and (b). In the frst case, the ntal Gaussan pulse has become trangular n shape due to non-lnear effects. A shock s vsble at 48D, surrounded by hgh-frequency Gbbs oscllatons ndcatve of the spectral truncaton of pressure components. In the second case, the pressure pulse has been dspersed, and does not dsplay a satsfactory shape. The numercal approach consstng n flterng the flues, prevously shown to generate phase errors for lnear equatons [5], mght therefore be not sutable for strongly non-lnear problems. In what follows, the background selectve flterng s then appled to the conservatve varables as descrbed earler n the paper, n combnaton wth the shock-capturng method. Results obtaned by varyng the shock-capturng parameters are reported to assess the performance of the methodology. The pressure profle determned usng the non-conservatve form of the shock-capturng flterng, wth a shock detector evaluated from pressure, a threshold value r th ¼ 5 and the flter Fo, s presented n Fg. 8. The spurous Gbbs oscllatons occurrng wthout shock-capturng have been removed. However the shock s now located at 5D, farther downstream n the aal drecton. Ths llustrates that the conservatve form of the shock-capturng flterng s requred to properly calculate the speed of the shock propagaton. The solutons shown subsequently are therefore all computed usng conservatve flterng. (a).5 (b) g g g Fg. 8. Shock propagaton. Pressure computed at t ¼ usng: (a) selectve flterng of the varables wthout shock-capturng, (b) selectve flterng of the flues wthout shock-capturng, selectve flterng of the varables and shock-capturng based on non-conservatve flterng. (a).5 (b).4 g s sc s sc Fg. 9. Shock propagaton. Influence of the shock-detecton varable. Solutons computed at t ¼ : (a) pressure, (b) and magntude r sc of the shockcapturng flterng, usng flter Fo, a threshold parameter r th ¼ 5 and a shock sensor evaluated from: 3 pressure, dlataton (+, o grd ponts).

11 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) Solutons calculated wth a threshold value r th ¼ 5 and the standard flter Fo, usng pressure or dlataton to detect the shock, are represented n Fg. 9. The two pressure profles thus obtaned do not ehbt oscllaton n Fg. 9(a), and are even farly supermposed. Estmatng the shock detector from pressure or from dlataton s therefore nearly equvalent for the present problem of shock propagaton. In order to fnd nevertheless small dfferences, the magntudes of the shock-capturng flterng are plotted n Fg. 9(b) and. In the two cases the second-order flterng s seen to be swtched on over a lmted zone around the shock, contanng eght ponts usng pressure as shock-detecton varable, but only four ponts usng dlataton. Trackng the shock from dlataton rather than from pressure allows here to apply the flterng to fewer grd ponts. The nfluence of the threshold parameter r th s now nvestgated by dsplayng n Fg. solutons computed usng flter Fo and a shock detecton based on dlataton, wth r th ¼ 6 and r th ¼ 4.InFg. (a), the pressure profle predcted for r th ¼ 6 s smoother, whereas the profle for r th ¼ 4 shows remanng, albet of very low ampltude, Gbbs oscllatons near the shock. Ths suggests that the shock-capturng method s more dsspatve when the value of r th s decreased. More precsely, the shock-capturng flterng appears to be appled over a wder regon around the dscontnuty, as ndcated by the profles of the flterng magntude r sc n Fg.. Ths magntude s nl everywhere ecept for four grd ponts when the threshold parameter s r th ¼ 4, whereas ponts are affected by the second-order flterng when r th ¼ 6. Fnally the problem s solved usng the followng shock-capturng parameters: a threshold value r th ¼ 5, a shock sensor based on dlataton, and the standard flter Fo or the optmzed flter Fopt. The pressure dstrbutons obtaned n Fg. (a) are very smlar. The optmzed flter Fopt s therefore capable of properly capturng the shock. Furthermore one can note n Fg. that the magntude of the shock-capturng flterng s hgher when flter Fopt s used rather than flter Fo. Because flter Fopt s less dsspatve than flter Fo, t may have to be appled wth a hgher strength to handle the shock, ths strength beng determned dynamcally from the solutons. Nevertheless ths does seem to lead to a smoother soluton. On the contrary usng flter Fopt provdes a sharper shock than flter Fo n Fg.. Implementng the optmzed flter n the shock-capturng procedure may then be nterestng to reduce spurous dampng Shock acoustc nteracton The second non-lnear problem consdered s the category problem formulated n the thrd CAA Workshop [44] to smulate shock-sound nteracton n a transonc nozzle. To model ths problem, the quas-one-dmensonal Euler equatons () are solved over the computatonal doman 6 6, wth the area of the nozzle gven by (a).5 (b) g s sc s sc Fg.. Shock propagaton. Influence of the threshold parameter r th. Solutons computed at t ¼ : (a) pressure, (b) and magntude r sc of the shockcapturng flterng, usng flter Fo, a shock sensor evaluated from dlataton, and: 3 r th ¼ 4, r th ¼ 6 (+, o grd ponts). (a).5 (b) g s sc s sc Fg.. Shock propagaton. Influence of the shock-capturng flter. Solutons computed at t ¼ : (a) pressure, (b) and magntude r sc of the shockcapturng flterng, usng a shock sensor evaluated from dlataton, a threshold parameter r th ¼ 5 and 3 flter Fo, flter Fopt (+, o grd ponts).

12 458 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) γ /Δ Fg.. Shock propagaton. Influence of the shock-capturng flter. Pressure computed at t ¼, wth the same parameters as n Fg.. ( AðÞ ¼ :53657 :9886 epð lnðþð=:6þ Þ for P ; : :6654 epð lnðþð=:6þ Þ for < : At the nflow boundary, the mean pressure, velocty and pressure are specfed as q ¼, u ¼ :6533 and p ¼ =c. The mean pressure at the outflow boundary s set to p ¼ :6775 to create a shock. Once steady state s acheved for the mean solutons, low-ampltude acoustc oscllatory dsturbances are mposed at the nflow boundary for densty, velocty and pressure. Ther ampltudes are 5 tmes the mean nlet values, and ther pulsaton s ¼ :6p. Regardng the numercal parameters, the mesh grd contans 4 ponts and s characterzed by a constant spacng. The smulaton s carred out wth a CFL number of.8, provdng a tme step Dt ¼ :8D. At the boundary condtons, non-lnear boundary condtons based on characterstcs [56] are used as n a prevous reference [57]. Small correcton terms have been also added n order to prevent the drft of mean nflow and outflow values. Fnally, when the shock-capturng methodology s mplemented, the flterng s appled n ts conservatve form, and the shock detector s evaluated from dlataton. Solutons computed wthout shock-capturng are presented n Fg. 3. The profles of mean densty and pressure plotted n Fg. 3(a) dsplay the presence of a shock slghtly downstream of the nozzle narrowng, whose poston and ampltude are found n Fg. 3(b) to be n good agreement wth the analytcal soluton. The shock s thn and dscretzed by only three ponts, but t generates small oscllatons, whch are unfortunately of hgh ampltude wth respect to the acoustc dsturbances ntroduced at the nflow. The dstrbuton of pressure perturbatons obtaned when steady state solutons are reached for the oscllatory problem thus ehbts n Fg. 3 strong peaks at the shock poston, whereas the solutons n the upstream regon contanng the superposton of the ncdent waves and the waves reflected back at the throat, and n the downstream regon where the transmtted waves are travellng compare successfully wth the analytcal soluton [44]. The problem s then now solved usng the shock-capturng methodology wth a threshold value r th ¼ 5, a shock detecton from dlataton, and the standard flter Fo. Mean and fluctuatng solutons are represented n Fg. 4 as prevously. Ecept at the shock poston, they collapse well wth the solutons determned wthout shock-capturng, both for the mean profles n Fg. 4(a) and for the pressure waves n Fg. 4. The transmtted sound waves have n partcular been affected n a neglgble way by the shock-capturng method, and are n good agreement wth the analytcal soluton [44]. Ths lkely results from the features of the shock-detecton procedure, because, as n the frst non-lnear problem n Secton 4.., the shock-capturng flterng s only appled to fve ponts around the shock as t s ndcated by the values of the flterng magntude n Fg. 5. Usng shock-capturng, the shock has moreover been smoothed so as to be dscretzed by 4 or 5 mesh ponts, but stll agrees well wth the analytcal soluton n Fg. 4(b). The strong oscllatons observed n the dstrbuton of the fluctuatng pressure at the shock locaton wthout shock-capturng have also dsappeared n Fg. 4. They have been damped by the second-order adaptatve flterng. (a). (b).7 <p>,<r> <p> p / 5 Fg. 3. Shock acoustc nteracton. Solutons computed wthout shock-capturng: (a) 3 mean pressure and mean densty, (b) 3 mean pressure (+ grd ponts) and correspondng analytcal soluton, 3 fluctuatng pressure and þ correspondng analytcal soluton [44].

13 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) (a). (b).7 <p>,<r> <p> p / 5 Fg. 4. Shock acoustc nteracton. Solutons computed usng r th ¼ 5, a shock sensor evaluated from dlataton, and flter Fo: (a) 3 mean pressure and mean densty, (b) 3 mean pressure (+ grd ponts) and correspondng analytcal soluton, 3 fluctuatng pressure and + correspondng analytcal soluton [44]. σ sc.5.5 Fg. 5. Shock acoustc nteracton. Magntude r sc of the shock-capturng flterng, usng the standard second-order flter Fo, a shock sensor evaluated from dlataton, and a threshold parameter r th ¼ 5 (þ grd ponts). The shock acoustc nteracton s fnally smulated by mplementng flter Fopt rather than flter Fo n the shock-capturng procedure. The results determned for the mean pressure and densty, and for the fluctuatng pressure are shown n Fg. 6. They collapse those obtaned wth flter Fo n Fg. 4, ecept at the shock locaton. Usng the optmzed flter Fopt, the shock s ndeed well captured but seems sharper, the pressure gradent beng dscretzed by 5 grd ponts n Fg. 6(b), whch s one pont less than n Fg. 4(b). The varatons of fluctuatng pressure around the shock also appear less attenuated n Fg. 6 than n Fg Shock-tube problems Two standard shock-tube problems, namely Sod s and La s problems [8,45], are now consdered. They are solved from the one-dmensonal Euler equatons (), and ther ntal condtons are ðq; u; pþ ¼ð; ; Þ for < ; and ðq; u; pþ ¼ð:5; ; :Þ otherwse for the Sod test case, and (a). (b) <p>,<r> <p> p / 5 Fg. 6. Shock acoustc nteracton. Solutons computed usng r th ¼ 5, a shock sensor evaluated from dlataton, and flter Fopt: (a) 3 mean pressure and mean densty, (b) 3 mean pressure (þ grd ponts) and correspondng analytcal soluton, 3 fluctuatng pressure and + correspondng analytcal soluton [44].

14 46 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) ðq; u; pþ ¼ð:445; :698; 3:58Þ for < ; and ðq; u; pþ ¼ð:5; ; :57Þ otherwse for the La test case. For the two problems, the smulatons are made usng a mesh grd dscretzng a computatonal doman over 6 6, contanng ponts unformly spaced. The tme steps are Dt ¼ :5D for the Sod problem, and Dt ¼ :6D for the La problem. The shock-capturng methodology s employed wth a shock detector evaluated from the dlataton, and a threshold parameter r th ¼ 5, and apples the optmzed flter Fopt n ts conservatve form. The solutons obtaned for densty, velocty and pressure at tme t ¼ :4 for the Sod problem and at t ¼ :8 for the La problem are presented n Fgs. 7 and 8, respectvely. In both cases, the ntal condtons result n a shock wave propagatng to the rght, and a rarefacton wave propagatng to the left from the orgn. A central contact dscontnuty, vsble n the densty dstrbuton, s also generated. The numercal solutons show neglgble Gbbs oscllatons, and compare successfully wth the analytcal solutons derved from references [46,47]. The contact dscontnutes are n partcular preserved, but they appear to be slghtly damped by the shock-capturng procedure. Ths dampng occurs at the begnnng of the smulatons, when the shock waves and the contact dscontnutes are very close to one another. Later, however, when the two fronts are well separate, the shock-capturng flterng does not apply around the lnear contact dscontnutes [48], because of the shock detecton performed from the dlataton Shock vorte nteractons The fourth non-lnear problems eamned are the nteractons of a planar shock wave wth a sngle vorte. To assess the numercal methodology, the flow condtons, namely the shock Mach number and the vorte geometry and Mach number, are frst those of cases C and B, respectvely computed by Inoue and Hattor [49], hereafter referred to as I&H [49], and by Inoue [5]. These authors smulated the test problem at very low Reynolds numbers usng drect numercal smulaton (DNS). The effects of the Reynolds number on the physcal phenomena takng place durng the shock vorte nteracton were found to be very small, whch s also supported by good comparsons [49] between results obtaned from DNS and from the Euler smulatons performed by Ellzey et al. [58]. for slghtly dfferent flow parameters. Ths led us to compute the shock vorte problem from the D Euler equatons (3) n the present work. The shock wave s defned by an upstream Mach number M s ¼ u =c ¼ :, where the subscrpt denotes a quantty upstream of the shock and c s the sound velocty. The sngle vorte s assumed to be characterzed by the velocty dstrbutons of a Taylor vorte. The ntal tangental and radal veloctes of the vorte are epressed by u h ðrþ ¼M v r ep½ð r Þ=Š and u r ðrþ ¼; where the dstance from the vorte core r s non-dmensonalzed by the vorte radus R, and the Mach number of the vorte s M v ¼ u hma =c ¼ :5 where u hma denotes the mamum tangental velocty. The densty and pressure dstrbutons are gven by qðrþ ¼ c M v epð c r Þ and pðrþ ¼ð=cÞ½qðrÞŠ c : To study the nfluence of the grd resoluton, the problem s solved on two mesh grds wth constant spacngs D ¼ Dy: a coarse grd wth D ¼ :R and a fne grd wth D ¼ :5R. These grds contan and 6 8 ponts so as to both dscretze a computatonal doman etendng over 3R 6 6 R and 45R 6 y 6 45R. As a comparson, the grd spacng near the planar shock wave n the DNS of I&H [49] was D ¼ :5R. The shocks n the present calculatons are therefore sgnfcantly thcker. Intally the sngle vorte s located at ¼ R and y ¼, and the planar shock wave s specfed at ¼ by mposng densty, velocty and pressure varables correspondng to the left and rght states of a steady shock. Tme t s normalzed by R=c, and s adjusted so that the aal poston of the vorte s ¼ R at t ¼. The smulatons are carred out usng a CFL number of.6, provdng a tme step Dt ¼ :6D=c. The shock-capturng procedure s based on a (a). (b) r u p Fg. 7. Sod s shock-tube problem. 3 solutons computed at t ¼ :4 usng r th ¼ 5, a shock sensor evaluated from dlataton, and flter Fopt, and analytcal solutons [46,47], for: (a) densty, (b) velocty and pressure.

15 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) (a).4 (b) r u p Fg. 8. La s shock-tube problem. 3 solutons computed at t ¼ :8 usng r th ¼ 5, a shock sensor evaluated from dlataton, and flter Fopt, and analytcal solutons [46,47], for: (a) densty, (b) velocty and pressure. shock detector calculated from dlataton, as recommended n Secton 4.., to avod swtchng on the shock-capturng flterng n the vorte core, and on a threshold parameter set to r th ¼ 5. The results shown hereafter are fnally determned usng the optmzed flter Fopt, wrtten n ts conservatve form. Smlar results have been obtaned usng the standard flter Fo. The shock s however larger by one mesh sze, and grd-to-grd oscllatons are enhanced n ths case. Snapshots of the pressure feld Dp defned by Dp ¼ðp p Þ=p upstream of the shock, and Dp ¼ðp p s Þ=p s downstream, where p and p s are the upstream and downstream pressure values, are represented at three consecutve tmes n Fg. 9(a) for the coarse grd, and n Fg. 9(d) (f) for the fne grd. No sgnfcant Gbbs oscllaton s observed around the shocks, whch ndcates that the shocks have been correctly captured for both grds. The varaton of the shock thckness wth the grd resoluton can also be noted at t ¼ n Fg. 9(a) and. The shock beng dscretzed by about fve ponts n both cases, the shock thckness s more precsely around R for the coarse grd, and :R for the fne grd. Despte ths dfference, the pressure felds dsplayed n the top and n the bottom fgures look very much alke, and also agree well wth the correspondng (a) (b) /R 5 5 /R 5 5 /R (d) (e) (f) /R 5 5 /R 5 5 /R Fg. 9. Shock vorte nteracton, for shock and vorte Mach numbers M s ¼ : and M v ¼ :5. Representaton of the contours of the pressure feld Dp ¼½ :; :; :5; :; :; :5; :Š, obtaned wth the coarse grd (top), and wth the fne grd (bottom), at tme: (a) and (d) t ¼, (b) and (e) t ¼ :4, and (f) t ¼ 6. Computatons performed usng a shock sensor evaluated from dlataton, a threshold parameter r th ¼ 5, and flter Fopt.

16 46 C. Bogey et al. / Journal of Computatonal Physcs 8 (9) fgures of I&H [49]. The nose generaton mechansms here are consequently nearly ndependent of the Reynolds number as well as of the shock sze. As the vorte nteracts wth the shock, the shock wave deforms, and the nteractons generate sound waves n good concordance wth the observatons by I&H [49]. A precursor wave of quadrupolar nature s frst emtted, wth the clear appearance of four lobes n Fg. 9(e). The precursor wave s followed by a second sound wave, also of quadrupolar nature but of opposte sgn, whch can be seen n Fg. 9 and (f). Ths second sound wave results from the creaton of two reflected shock waves from the ncdent shock, whch wll be shown later. The spatal dstrbuton of the sound pressure s nvestgated more quanttatvely, and compared wth DNS data [49,5] n Fg.. Profles of pressure Dp are frst plotted n Fg. (a) aganst the dstance r for an angle h ¼ 45 wth respect to the downstream drecton, at tme t ¼ ; and 3. The profles obtaned wth the coarse and the fne grds are roughly supermposed, and they agree well wth the DNS curves of Inoue [5]. In partcular, as the waves propagate, a pressure peak characterzed by Dp > correspondng to a thrd sound wave appears after the precursor wave and the second sound wave. The crcumferental varatons of pressure Dp, assocated wth the precursor wave and wth the second sound wave, are then represented n Fg. (b). The quadrpolar features of the two radatons are observed. In addton the numercal results are weakly affected by the grd resoluton, and compare successfully wth the DNS solutons of I&H [49]. The only notable dscrepances are notced for angles around 8 for the second sound wave. In ths case, the results mght be nfluenced by the shock waves, dependng on the shock thckness. The way n whch the shock wave deforms as the nteractons wth the vorte are developng s emphaszed by three shadow graphs obtaned from the Laplacan of the densty n Fg. usng the fne grd. As the ntally planar shock wave passes through the vorte at tme t ¼ nfg. (a), ts shape s dstorted nto an S-shape. For shock and vorte Mach numbers M s ¼ : and M v ¼ :5, the nteractons are weak and a Mach reflecton s then observed [59] wth the formaton of two reflected waves whch elongate crcumferentally and swrl around the vorte n Fg. (b), as well as two slp-lnes emanatng from trple ponts n Fg.. (a) 6 (b) Dp/ Dp 3 4 r 9 8 q (deg.) Fg.. Shock vorte nteracton. (a) Radal profles of pressure Dp for h ¼ 45 from the vorte centre, at t ¼ : 3 fne grd, coarse grd, + Inoue [5],at t ¼ : fne grd, coarse grd, o Inoue [5], and at t ¼ 3: - - fne grd, coarse grd, Inoue [5]; (b) crcumferental dstrbutons of Dp at t ¼ 6, at r ¼ 6: (precursor): 3 fne grd, coarse grd, þ I&H [49], and at r ¼ 3:7 (second sound): fne grd, coarse grd, I&H [49]. Same computatons as n Fg. 9. Fg.. Shock vorte nteracton. Representaton of the shadowgraphs obtaned for r q wth the fne grd, at tme: (a) t ¼, (b) t ¼ 6, t ¼. Same computaton as n Fg. 9.