Linear interaction of a cylindrical entropy spot with a shock

Transcription

1 PHYSICS OF FLUIDS VOLUME 13, NUMBER 8 AUGUST 21 Linear interation of a ylindrial entropy spot with a shok David Fabre a) and Laurent Jaquin ONERA, 29, av. de la Division Leler, BP 72, F Châtillon, Frane Jörn Sesterhenn Fahgebiet Strömungsmehanik, Tehnishe Universität Münhen, Boltzmannstr. 15, Garhing, Germany Reeived 16 Marh 21; aepted 15 May 21 The interation of a ylindrial element of hot or old gas the entropy spot with a shok wave is onsidered. An exat solution in the limit of weak spot amplitudes is elaborated, using the linear interation analysis theory and the proedure of deomposition proposed by Ribner Tehnial Report No. 1164, NACA, The method is applied to an entropy spot with a Gaussian profile. Results are presented for a wide range of shok Mah numbers, with a speial interest at M 1 2. The resulting vortiity field onsists of a pair of primary ounter-rotating vorties, as well as a pair of seondary vorties of opposite sign and weaker amplitude. An expression for the irulation in half a plane is derived and ompared to existing results. The pressure field onsists of a ylindrial aousti wave whih propagates away from the transmitted spot and an evanesent nonpropagative field onfined behind the shok. For a hot spot, the ylindrial wave is a rarefation wave on its forward front and a ompression wave on its upstream propagating parts, and the nonpropagative field orresponds to a pressure defiit. The struture of the transmitted spot and the shok deformation are also disussed. The linear solution is ompared with numerial simulation results for M 1 2 and M 1 4. The omparison shows qualitative and quantitative agreement when linear as well as nonlinear spot amplitudes are onsidered. Finally, the method is applied to the ase of a onstant spot with a tophat profile, and the results are ompared to the ase of a Gaussian spot. This paper also ontains in the Appendix a general formulation of the linear interation problem for the three kinds of plane waves entropy, vortiity, and pressure waves impinging upon a shok. 21 Amerian Institute of Physis. DOI: 1.163/ I. INTRODUCTION It is well known that the interation of any of the three modes of flutuation identified by Kovasznay 1 the vortiity mode, the entropy mode, and the pressure mode with a shok wave gives rise to the three modes downstream of the shok and to a perturbation of the shok surfae. A theoretial basis of the problem has been given by the linear interation analysis LIA theory initiated by the pioneering studies of Ribner 2 and Moore 3 and followed by other authors. 4 7 This theory desribes the interation of a plane wave of any of the three modes with a shok and allows alulation of the amplitudes of the three generated and transmitted waves. Sine plane waves an be seen as the Fourier omponents of any flow field, the theory an, in priniple, be applied to the interation of an arbitrary weakly perturbed flow field with a shok. The approah has been applied by Ribner 8 to homogenous turbulene without entropy flutuations. The turbulene amplifiation rates and aousti generation levels predited by the theory have been ompared to diret numerial simulations, 9 11 and to rapid distortion theory. 12 Ribner also a Telephone: ; fax: ; eletroni mail: David.Fabre@onera.fr used the LIA method to haraterize the ylindrial sound wave formed by the interation of an isolated vortex with a shok wave. 13,14 In these referenes, the theoretial result has been suessfully ompared to a shok tube experiment. 15 The role of the entropy mode in shok-turbulene interation has been outlined in a series of analytial and omputational studies by Mahesh et al. 16,17 These authors have shown that for strong shoks, the shok-turbulene interation is strongly influened by the entropy flutuations. This motivates new interest in the ase of flows involving entropy flutuations. Apart from its aademi interest, the problem of shokentropy interation is relevant to aerospae appliations. For example, in supersoni wakes and jets, the entropy variations are found to be very large and the interation with oblique shoks is thought to be an important soure of noise. On the other hand, the mehanism of shok-indued vortiity generation desribed below ould be helpful to inrease the effiieny of high speed ombustion systems. 18,19 The entropy spot onsidered in this paper onstitutes one of the most simple forms of a loalized entropy perturbation; it onsists of a ylindrial element of fluid haraterized by a higher or lower temperature than the surrounding fluid. The interation of this entropy spot with a shok wave leads to a variety of physial phenomena that an be /21/13(8)/243/2/$ Amerian Institute of Physis Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

2 244 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn FIG. 1. Qualitative desription of the interation. desribed as follows, for the ase of a hot spot Fig. 1a: 1 The initially irular spot is ompressed and aquires an ellipti shape. 2 In the enter of the spot, the Mah number is lower than in the surrounding fluid. As a onsequene, the pressure jump is weaker, and a low pressure region forms behind the shok. This low pressure region leads to a deformation of the shok surfae. 3 Additionally, this low pressure region propagates away and forms a ylindrial sound wave. 4 Finally, the ombination of the density gradient and the pressure gradient leads to a barolini torque, whih reates a symmetrial vortiity struture. Thus the flow field an be thought of as an ellipti spot, a pair of ounter-rotating vorties, and a ylindrial rarefation wave. The same phenomenology applies to the ase of a old spot, if the phenomena are onsidered with the opposite sign. The later evolution of the spot is skethed in Fig. 1b. Under the influene of its own veloity, the transmitted spot rolls up and the high temperature regions onentrate into the vortial regions. In addition, if it is strong enough, the sound wave may steepen or flatten depending on its ompression or rarefation nature. The main features of the interation desribed above have been observed in a variety of shok-tube experiments, suh as a shok wave interating with a olumn of heated fluid resulting from a wire explosion, 2,21 spherial flames, 22,23 ylindrial and spherial bubbles of lighter and heavier gas, 22,24 and a ylindrial jet of light gas. 25 Note that some of these experiments involve additional effets suh as differenes in the gas properties, interfae effets, hemial reations, or three-dimensional effets. However, all the experiments seem to be basially governed by the shokentropy interation mehanism presented above. The problem has also reeived an important omputational effort, and the general features of the interation whih have been observed are in qualitative aordane with the experiments. However, differenes an be observed between the omputations whih used a smooth entropy distribution as an initial ondition e.g., a Gaussian spot 3, and those whih used a sharp one e.g., a onstant spot 27. A number of analyti developments for the problem have been undertaken. Haas and Sturtevant 24 developed a model based on geometrial aoustis to determine the geometry of the pressure waves. The method an only provide qualitative results, but whih are in aordane with experiments. It an apply to a strong entropy amplitude in the spot, but it is limited to weak shok waves. Very reently, Grasso and Pirozzoli 31 have formulated an aousti analogy of the problem that leads to an expression for the pressure field in the aousti wave. Their model applies for either weak shoks or weak spot amplitudes. Efforts have also been made to determine the irulation of the vortex pair, and several models have been elaborated, in partiular by Rudinger and Somers, 22 Pione et al., 26,27 and Yang et al. 29 However, a omplete analytial solution of the problem, desribing the vortiity and pressure generation as well as the shok deformation and the ompression of the spot, is still missing. The goal of this paper is to provide a omplete solution of this interation using the LIA method and the proedure of deomposition used by Ribner for the shok-vortex interation. 13,14 The paper is organized as follows: In Se. II the proedure of deomposition is desribed qualitatively; then in Se. III it is applied to the ase of a Gaussian entropy spot. In Se. IV detailed results are given for a Mah 2 shok, and the harateristi features of the solution are presented for a wide range of Mah numbers. In Se. V the linear solution is ompared with numerial simulation results. Finally, the appliation of the method to the ase of a onstant spot is presented in Se. VI and the differenes with the Gaussian spot are disussed. II. QUALITATIVE DESCRIPTION OF THE PROCEDURE In this setion the proedure used to treat the interation is desribed qualitatively. As stated in the Introdution, the interation of a plane entropy wave with a shok gives rise to a triad of waves behind the shok a detailed solution to this problem is given in Appendix A. Depending upon the angle of inidene of the impinging wave, two different regimes may be obtained. In the propagative regime see Fig. 2, the pressure wave propagates away as an aousti wave, and the three generated waves are in phase with the inident wave. In the nonpropagative regime Fig. 3, the amplitude of the pressure wave deays exponentially away from the shok, and there is a phase shift between the generated waves and the inident wave. Using one-dimensional Fourier synthesis, plane entropy waves in the same diretion an be arranged together to represent parallel entropy distributions. Suh a parallel entropy distribution, referred hereafter as an entropy slie, is represented in Fig. 4a. Positioning elementary entropy slies radially in all the angular diretions, just as the spokes of a wheel, one an onstrut a ylindrial entropy distribution, i.e., an entropy spot. This onstrution proedure of an iso- Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

3 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 245 FIG. 4. Illustration of the slie deomposition of the spot. FIG. 2. Interation of an entropy wave with a shok wave propagative regime. lated entropy spot as a sum of elementary entropy slies is skethed in Fig. 4. It is idential to the proedure used by Ribner to onstrut an isolated vortex as a sum of elementary shear flows. 13,14 When the spot is onveted through a shok, the interation of the elementary slies with the shok follows the same rules as the onstituent elementary waves. For eah elementary slie, the interation leads to a transmitted entropy slie, a generated vortiity slie, as well as a pressure perturbation and a shok deformation. This is represented in Fig. 5 for two partiular slies, whih have been hosen suh that the interation of the first slie angle is of the propagative type, and the interation of the seond slie angle is of the nonpropagative type. The generated entropy slies not represented in Fig. 5 will then fous on the enter of the transmitted spot. The resulting entropy spot an thus be obtained by rearranging the different transmitted entropy slies. Similarly, the ombination of the generated vortiity slies results in the vortial strutures. Note that the elementary vortiity slie generated for initial angle in Fig. 5 has an asymmetri profile. This is due to the phase shift on the onstituent plane waves. The interation of the slie with angle gives rise to an generated pressure slie with the same profile. This elementary pressure perturbation propagates away at the speed of sound C 2. At a time t after the instant when the enter of the spot has rossed the shok, the elementary pressure profiles orresponding to angles in the propagative range will all be tangent to a irle of radius RC 2 t entered on the transmitted spot. All these elementary slies will therefore ontribute to a ylindrial sound wave. This onstrution is illustrated in Fig. 6. The pressure perturbation resulting from the interation of the seond slie with angle in Fig. 5 has a different FIG. 3. Interation of an entropy wave with a shok wave nonpropagative regime. FIG. 5. Interation of two partiular elementary slies with the shok. Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

4 246 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn FIG. 6. Illustration of the formation of the ylindrial aousti wave. FIG. 7. Details about the geometrial proedure. Illustration of the hange of origin from the virtual enter O 1 of the inident spot, to the point O on the shok front, and then to the enter O of the transmitted spot. behavior. All the onstituent elementary pressure waves deay away from the shok independently from eah other, so the orresponding pressure field does not maintain an identifiable shape. Therefore, it has not been represented in Fig. 5. III. DETAILED ANALYSIS FOR A GAUSSIAN SPOT The initial flow field we will onsider hereafter onsists of an entropy spot with a Gaussian profile of radius a and amplitude entered on the point O 1, arried by a mean flow of temperature T 1, density 1 and veloity U 1, i.e., s e r2 /a 2, 1 C p where r is the radial oordinate relative to O 1, and where the entropy flutuation is made nondimensional with the speifi heat C p of the gas the ratio of speifi heats is C p /C v 1.4). In the limit of small amplitude, the orresponding temperature and density fields are given by T 1 (1 (s/c p )) and 1 (1(s/C p )). In order to desribe this spot as a superposition of plane entropy waves, we perform a two-dimensional Fourier transform of Eq. 1. Using polar oordinates (r,) in the physial plane and (k,) in the spetral plane, this results in s 1 C p 2 e ka 2 /4 e ikr os ka dka d, 2 where the real part must be retained. Reassembling the ontributions of all wave numbers of the same angle gives s 1 C p K a rˆ d, 3 where rˆr os() is the variable desribing planes of onstant phase in the angular diretion, and where Kz k 2 ek2 /4 oskzdk12zfz. 4 Funtion F(z) is the Dawson integral see Ref. 32, p. 298 defined as Fze z2 z e t 2 dt. Equation 3 is the mathematial expression of the slie deomposition introdued in Se. II and desribed in Fig. 4. The funtion K is the equivalent of the funtion g in Ribner s analysis of an isolated vortex Eq. 8 of Ref. 14. At a time t after the instant when the enter of the spot has rossed the shok, the virtual enter O 1 of the initial spot has been onveted a distane of U 1 t from the shok, and eah elementary slie rosses the shok at a point O orresponding to oordinates (U 1 t,u 1 t ot ) in the frame entered on O 1 see Fig. 7. The interation of eah elementary entropy wave e ikrˆ appearing in expression 2 gives rise to an entropy wave Z ss e iks ˆ, a vortiity wave Z sv e iks ˆ, and a pressure wave Z sv e ikp ˆe kp x. Here ˆ and ˆ are the variables desribing planes of onstant phase in the diretions s and p the origin is situated at the point O, and x is the distane from the shok front. The relative amplitudes Z ss, Z sv, Z sp, Z sx, and the geometrial harateristis k s, s, k p, p, of these waves are given in Appendix A. Making use of the geometrial relation ot s m ot with mu 1 /U 2 Eq. A8 of Appendix A, it an be seen that all the entropy and vortiity waves generated downstream of the shok fous on the point O with oordinates ((U 2 U 1 )t,) in the frame entered on O 1. Point O orresponds to the enter of the transmitted spot see Fig. 7. Using a frame of referene entered on point O leads to a simplifiation of the different omponents of the flow field. The oordinates of point O in that frame are x U 2 t, y U 2 t ot s. We introdue the variables rˆ sr os( s ) and rˆ p r os( p ), where (r,) are now polar oordinates entered on O. Using Eqs. A8 and A12 one may then verify that the hange of origin from O to O amounts to rˆ sˆ, 6 rˆ pˆ R, 7 where RC 2 t is the radius of the ylindrial sound wave at the instant onsidered the seond relation is only valid in the propagative regime. All these geometrial details are repro- 5 Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

5 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 247 dued in Fig. 7. Note that similar geometrial manipulations were used by Ribner 13,14 to study the shok/vortex interation. The expressions for the entropy and vortiity fields an then be simplified further by introduing the length suh that for eah plane wave, k s ka, and the Jaobian J s / s. Similarly, in order to simplify the evaluation of the pressure field in the propagative regime, we will introdue the length b suh that k p bka, and the Jaobian J p / p. Note that the length an be interpreted as the radius of the transmitted spot, when observed under the angle s. This is illustrated in Fig. 5. Similarly, the length b orresponds to the width of the elementary pressure slie at angular loation p. The expressions for /ak/k s, b/a k/k p, J s, and J p are given in Appendix A. Using all these expressions, the entropy, solenoidal veloity and vortiity fields take a onise form, similar to Eq. 3, s C p u U 2 v U 2 a IZ sv L U 2 RZ ss K RZ sv K RZ sv K rˆ s IZ ss L rˆ s IZ sv L rˆ s rˆ s os s J s d s, RZ sv K a J s d s. rˆ s IZ sv L rˆ s J s d s, 8 rˆ s sin s J s d s, 9 rˆ s 1 11 In these expressions, R and I denote the real and imaginary parts of the omplex transfer funtions, and L(z) is a new profile funtion, defined as Lz k 2 ek2 /4 sinkzdkze z2. 12 K and L are the derivatives of funtions K and L, with funtional form, Kz2z24z 2 Fz, 13 Lz12z 2 e z2. Applying the same proedure to the pressure field gives p p rˆ pr p K b Z sp J p d p Zsp d 2 e ka 2 /4 e kp iˆ xx kadka, 14 where the two terms on the right-hand side aount for the ontribution of the propagative and nonpropagative plane waves, respetively. Sine the plane pressure waves of the FIG. 8. Entropy field in the transmitted spot for M 1 2. Levels orrespond to.3,.1,.1,.3,.5,.1,.2,.4,.6, and.8. nonpropagative type deay exponentially independently from eah other, the seond term of this expression annot be simplified further. Finally the shok displaement takes the form, X a I RZ sx K yy sin s IZ sx L I yy sin s J s d s, 15 with funtions K I, L I defined as K I z 1 2 ek2 /4 oskzdk 2 ez2, 16 L I z 1 2 ek2 /4 sinkzdkfz. 17 Note that in these expressions, the entropy and solenoidal veloity fields do not depend upon time and are onveted like frozen patterns, whereas the pressure field and the shok deformation are expliitly time-dependent through R and y. Note also that Eqs. 9 and 1 only orrespond to the vortial part of the veloity field. The potential part of the veloity field, whih is related to the pressure field, is not detailed here. IV. RESULTS FOR A GAUSSIAN SPOT A. Entropy field The entropy field given by Eq. 8 is represented in Fig. 8 for M 1 2. As expeted, after ompression by the shok, the spot has aquired a roughly ellipti shape. One further noties the existene of a seondary spot of opposite sign in the wake of the main transmitted spot. Also, the entropy ontours in the periphery of the main spot are distorted bakward. It an be noted that the diretion of this distortion orresponds to the separating angle s between the propagative and nonpropagative regimes: for M 1 2, this ritial Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

6 248 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn FIG. 9. Maximum entropy flutuation in the transmitted spot, and minimum of entropy in the seondary spot multiplied by 1 as funtions of the shok Mah number. FIG. 11. Maximum vortiity in the primary and seondary vorties as funtions of the shok Mah number. angle is s 35 Eq. A1 of Appendix A. This effet an be attributed to the peak of the transfer funtion Z ss at this ritial angle. The maximum value of the entropy flutuation in the enter of the transmitted spot is represented in Fig. 9, for M 1 varying ontinuously from 1 to 4. This maximum value is found to derease as the shok strength inreases. It must be kept in mind that the base flow the unperturbed shok is not isentropi. Absolute values of the minimum in the seondary spot have also been reported in Fig. 9 for a few values of M 1. The amplitude of this seondary spot grows with the Mah number, but remains at least 1 times lower than the amplitude of the main spot. B. Vortiity field The vortiity field given by Eq. 11 is shown in Fig. 1 for M 1 2. As expeted, the flow field mainly onsists of a pair of ounter-rotating vorties. In addition, a seondary pair of weaker vorties of opposite sign is observed in the FIG. 1. Vortiity field from linear theory (M 1 2). Levels orrespond to.1,.3,.5, 1, 1.5, 2 and the dashed line orresponds to negative values. wake of the primary pair. As for the entropy field, the vorties are strethed in the diretion of the ritial angle s. This is also due to the peak in the transfer funtion at this partiular angle. Figure 11 displays the maximum values of the vortiity in the primary vorties for M 1 ranging ontinuously from 1 to 4 and in the seondary vorties for a few values of M 1. The expression of the veloity field given by Eqs. 9 and 1 an be manipulated to obtain an expression for the irulation in the upper or lower half plane. Calulations are reported in Appendix B. The nondimensional irulation takes the very simple form, 2m1 f, 18 au 2 m where mu 1 /U 2 is the ompression rate of the shok and f /2 for a Gaussian spot. Note that this value orresponds to the irulation in half a plane, and ontains the ontribution of the primary and seondary vorties. The irulation of the primary vorties, onsidered alone, may be slightly higher. We now ompare this expression with previous expressions proposed in the literature. The most elaborate model is due to Pione et al. 26 These authors omputed the vortiity by integrating the barolini prodution term during the interation, and integrated it over a half plane in order to evaluate the irulation. The hypothesis was made that the shok remains straight during the interation. Their result is V 2 1 V 2 2U 1 af ln 1 /, 19 where is the value of the density at the enter of the initial spot, V 2 U 1 U 2, and the fator f is also /2 for a Gaussian spot. Considering linear spot amplitudes (ln( 1 / )), and using our notation, Eq. 19 an be written P m2 1 f. 2 au 2 m Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

7 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 249 In the limit of weak shoks (m 1), Eqs. 18 and 2 are equivalent. For stronger shoks, Eq. 2 leads to an overpredition; for an M 1 2 shok, the ratio P / is 1.83, and it an be as muh as 3.5 in the limit of very strong shoks. In a more reent paper, Yang et al. 29 heked all the existing models inluding the one from Pione et al. against numerial results. They notied that all the models overprediated the omputed irulation for strong shoks. They proposed an expression based on dimensional analysis with orrelation whih fits most of their numerial results, Y 4aC 1 M P / 1 1 / Considering again linear spot amplitudes, and writing the pressure jump and the Mah number M 1 as funtions of the ompression rate m leads to Y 2m1 au 2 m. 22 So for weak spot amplitudes, the expression from Yang et al. 29 is equivalent to ours exept for the presene of the form fator f. However, Yang et al. did not onsider a Gaussian profile, but a sharper one. It will be shown in Se. VI that their result is retrieved when onsidering a onstant entropy spot. Finally we address the validity of the vortiity field predited by the linear solution during the later stages of the interation. If we exlude the possibility of threedimensional instabilities, the evolution after the interation is governed by the two-dimensional Navier Stokes equation. For the u-omponent of the veloity, u u u p u v t x y x 2 u x 2 2 u y Dimensional analysis shows that two time sales appear in this equation; a nonlinear time sale T nl a/u, where u is a harateristi value of the veloity field in the vorties, and a visous time sale T v a 2 /. The veloity field predited by the linear solution Eqs may be onsidered as valid for tt nl and tt v. Integrating both sides of Eq. 23 for x varying from to, and supposing that the veloity field remains symmetri during the roll-up, leads to d dt 2 u y 2 dx. 24 Equation 24 shows that the irulation in half a plane only evolves on the visous time sale T v. So the validity of the irulation predited by Eq. 18 only requires tt v. These results show that if the Reynolds number Reua/ T v /T nl of the vorties is high, then the later evolution will onsist of two stages: first, an invisid roll-up with onstant irulation on the time sale T nl, and then a slow visous diffusion on the time sale T v. FIG. 12. Pressure field p/ and shok deformation X /a from linear theory (M 1 2), at three different instants orresponding to R/a, 2, and 4. The step between two levels is.5 for R/a and.2 for R/a2 and 4. C. Pressure field The pressure field given by Eq. 14 is plotted in Fig. 12 for M 1 2 and for three different instants of the interation orresponding to R/a, R/a2, and R/a4. The first plot orresponds to the instant when the enter of the spot rosses the unperturbed shok. As expeted, the pressure field is then haraterized by a low pressure region into the spot. Later, the low pressure region propagates downstream and forms a ylindrial rarefation wave. In addition, we note that in the viinity of the shok front, the generated pressure wave is a ompression wave. Finally, we remark that a region of low pressure remains downstream of the shok. This near field orresponds to the ontribution of nonpropagative pressure waves. In the later stages of the interation, the pressure field onsists of a ylindrial aousti wave exept in the viinity of the shok. An asymptoti expression for this ylindrial wave is developed in Appendix C for R/a1. The result is p 2a R F;M 1K rr b, 25 with funtions F and K defined as F;M 1 J p Z sp a b, 26 p 1 K z z Kzzdz. 27 Expression 25 has the same form as the one given by Ribner for the shok/vortex interation Eq. 23 of Ref. 13. It onsists of the produt of the following terms: a b The first term 2a/R desribes the temporal evolution of the irular wave. Remember that the radius R C 2 t is a measure of time, so the temporal evolution is porportional to t 1/2, as expeted for a twodimensional aousti wave. The seond term F(;M 1 ) is the angular desription of the wave. This funtion is represented for several val- Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

8 241 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn FIG. 15. Angular amplitude of the aousti wave predited by Grasso and Pirozzoli s aousti analogy GP and by the linear theory LT, form and M 1 2. FIG. 13. Angular amplitude F(;M 1 ) of the ylindrial pressure wave. ues of M 1 in Fig. 13. It an be seen that for moderate Mah numbers, this funtion hanges its sign around the angle 9. The aousti wave therefore has an alternating struture. For a hot spot, it is a rarefation wave on its downstream propagating part 9, and a ompression wave on its upstream propagating part (9 p ). This formally orresponds to the emission of a dipolar aousti soure. For large Mah numbers, this alternating struture disappears, and the emission is similar to that resulting from a monopolar aousti soure. The last term K ((rr)/b ) is the radial desription of the wave. This funtion is represented in Fig. 14. It possesses a maximum K (.41)1.3666, and a seondary minimum K (1.22).635. So in its widest part, the sound wave resulting from a hot spot is a rarefation wave followed by a reompression of lower amplitude. The validity of the asymptoti expression 25 has been heked against the integral expression 14. The onvergene is rapid. For (R/a)4 and M 1 2, whih is the third instant plotted in Fig. 12, the asymptoti value is already reahed on the downstream front of the wave. This differs from the ase of shok vortex interation. 14 In this later ase, it was observed that the ylindrial sound wave was preeded by a preursor wave, resulting from the potential FIG. 14. Pressure profiles in the ylindrial aousti wave: K (z) for a Gaussian spot, and K C (z) for a onstant spot to be desribed in Se. VI. flow around the vortex ore. The amplitude of this wave was found to deay as t 1 rather than t 1/2. This is not observed in the ase of an entropy spot. Finally, one must remind that the asymptoti expression 25 an never be reahed in the viinity of the shok, beause at these loations, the near field an not be negleted. This near field was not inluded in Ribner s alulations for the ase of a shok/vortex interation. In a reent paper, Grasso and Pirozzoli 31 GP studied, both analytially and numerially, the sound generation resulting from a shok/spot interation. They onduted two analyti developments of this problem. The first one is a semianalytial model, whih applies in the limit of weak shoks. The seond is an aousti analogy that applies in the limit of either weak shoks or weak spot amplitudes. This seond model may be ompared to our linear solution. In Appendix D we reprodue GP s aousti analogy for the ase of a Gaussian spot with a weak amplitude. It is shown that in the limit R/a1, their result an be set to a form similar to our expression 25, p 2a R F GP;M 1 K rr b, 28 where the funtion K is the same than ours and where the length b is given by Eq. D17. The funtion F GP (;M 1 ) desribing the angular amplitude of the wave predited by GP s aousti analogy is given by Eq. D15. This funtion is plotted in Fig. 15 for M and M 1 2 and ompared to the funtion F(;M 1 ) obtained from the linear solution. It an be seen that on the downstream propagating parts of the wave 9, GP s aousti analogy and our linear solution are omparable, and that agreement is better for M than for M 1 2. On the other hand, on the upstream propagating parts of the wave (9 p ), for both ases M and M 1 2, the aousti levels predited by GP s aousti analogy are muh greater than those predited by our linear solution. We expet our linear solution to be more aurate beause it inludes all the details of the interation, whereas GP s aousti analogy makes the hypothesis that the shok remains straight during the interation. Also, GP s aousti analogy predits an aousti emission in an unbounded medium, and it needs to be orreted to aount for the aousti Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

9 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 2411 TABLE I. Convergene of numerial simulation results: Effet of mesh size M 1 2, R/a4. x.16a, y.32a x.8a, y.16a x.4a, y.8a Theory Min. pressure % 1% 1% Max. vortiity % 1% 1% FIG. 16. Maximum shok deformation as a funtion of the shok Mah number. waves refleted by the shok. For this later point, GP have shown that on the symmetry axis this orretion is at most 4% for M 1 2, but it may be greater on the upstream propagating parts of the wave. D. Shok displaement The shok displaement given by Eq. 15 is represented in Fig. 12 at three different instants, for M 1 2. The maximum value of the shok displaement is reahed approximately for the instant R/a1, when the transmitted spot has ompletely emerged from the shok. Later, the shok front deformation widens, and its maximum amplitude seems to derease as t 1. It has also been verified that for R/a, orresponding to instants before the interation, the shok displaement predited by Eq. 15 quikly tends to zero. This means that our expressions respet the ausality of the interation. The maximum displaement of the shok front during the interation has been plotted as a funtion of the Mah number in Fig. 16. The shok displaement due to the interation appears to be larger for weak shoks. V. COMPARISON WITH A NUMERICAL SIMULATION The linear solution developed herein is partiularly suited for a omparison with numerial simulation results. Here we use a ode based on a harateristi formulation of the Euler equations, as well as a shok-fitting tehnique to resolve the shok front. 33 We first give an outline of the numerial method, and then present omparisons of numerial results with the linear solution for M 1 2 and 4. This allows to demonstrate the validity of the linear theory when onsidering linear spot amplitudes, and to investigate the nature of the interation when onsidering nonlinear spot amplitudes. For this latter point, the results presented herein omplements those of Hussaini and Erlebaher. 3 A. Desription of the numerial method The idea of the ode is based on the -sheme by Moretti, 34 and is refurbished and put to work with a high order ompat upwind sheme and a Runge Kutta time integration. Here, we just want to give a rough idea of the ode and present the basis for a one-dimensional model problem. For a detailed desription, please see Ref. 33. The Euler equations in one spae dimension an be regarded as three transport equations for two Riemann variables, R u2/(1) and the entropy s. They take the form R 29 s t t s R t u R x s, R x u s x. 3 These equations an be rearranged to express the time derivatives of the primitive variables, 33 p t 2 X X p C v u t 1 2 X X, s t Xs, s t Xs, where the following notation has been introdued: X u R x s R x u 1 p x u x, X s u s x. 35 Eah of the spae derivatives X, X s is uniquely related to a propagation diretion whih is either u or u. We an onstrut a stable numerial sheme for the set of equations by hoosing an upwind differentiation with respet to the sign of the propagation diretion for eah of these terms. In our ase, we have used a fifth order ompat upwind sheme 35 for the spae derivatives, and a low storage third order Runge Kutta sheme 36 for time advanement. Please note that the ontinuity equation 31 might be simplified by Eq. 33, but this is only possible for isentropi flows. In the omputation of visous flows, one has to keep it as it stands. Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

10 2412 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn FIG. 17. Vortiity and pressure fields for M 1 2. Upper plots: superposition of linear theory and numerial simulation with 1 2. Middle plots: numerial simulation with 1 1. Lower plots: numerial simulation with 1. For eah ase, from left to right: pressure field for three different instants orresponding to R/a, 2, and 4, and vortiity field at instant orresponding to R/a4. Pressure levels are as in Fig. 12, and vortiity levels are as in Fig. 1. For two spae dimensions, there exists no equivalent formulation in terms of a finite number of Riemann variables beause the two aousti harateristis orresponding to X are degenerated from an infinite number of harateristis. But it is straightforward to deompose the equations formally in the same way for the y diretion and also for timedependent arbitrary grids. A set of equations generalizing the system an be obtained, see Ref. 33, Eq. 42. The resulting sheme is not onservative. Thus we annot apture shoks. In pratie, the sheme proves to be Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

11 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 2413 TABLE II. Comparison between linear theory and numerial simulation for M 1 2, R/a. Theory Min. pressure % 1% 1% 18% Max. vortiity % 1% 1% 6% Max. entropy % 1% 1% 2% Max. deformation % 1% 1% 23% TABLE IV. Comparison between linear theory and numerial simulation for M 1 4, R/a. Theory Min. pressure % 1% 2% 22% Max. vortiity % 1% 1% 6% Max. entropy % 1% 1% 4% Max. deformation % 1% 2% 15% robust enough to handle moderate shoks anyhow. 33 Strong shoks need to be fitted. We follow the idea of Moretti, 37 see, e.g., Erlebaher et al. 38 The grid is time dependent and the shok is taken to be the left, moving boundary. The other boundaries are taken to be nonrefleting. Aording to harateristi theory, four harateristis in two spae dimensions enter the shok from the left and one from the omputational domain. Given this information, it is possible though tedious to ompute X, X s, and X v u(v/x), as well as the shok aeleration. 38 This information atually forms the numerial boundary ondition whih the sheme sees. The shok loation and veloity are advaned in time with the same Runge Kutta sheme as the flow field, and the grid is deformed to follow the shok deformation. B. Results At first, we have validated the ode s onvergene by performing simulations with different mesh sizes. The dimensions of the omputational domain, initially retangular, were 1ay1a and x1a. We used a oarse grid x.16a, y.32a, an intermediate one x.8a, y.16a, and a refined one x.4a, y.8a. The Mah number was M 1 2 and the spot s amplitude was set to 1 4, in order to remain within the limits of validity of the linear solution. The extremum values of the pressure and the vortiity, dedued from the numerial simulations at a nondimensional time R/a4, are reported in Table I and ompared to the linear solution. As it an be seen, when onsidering the pressure field, the oarse grid is suffiient to obtain a good onvergene of the numerial simulation toward the linear solution. However, the vortiity field, a quantity whih is more dependent upon the disretization, needs a refined grid to onverge toward the linear solution. We performed a series of omputations to study the validity of the linear solution with regards to the amplitude of the inident spot. We used the refined grid speified above. The amplitude was set to 1 4,1 2,1 1, and 1. We onsidered two ases with Mah numbers of M 1 2 and 4. Figure 17 shows omputation results for M 1 2. We have represented the pressure levels for three instants orresponding to R/a, 2, and 4, and the vortiity field orresponding to the last of these instants. The upper plots are a superposition of the linear solution and of the numerial results with 1 2. As it an be seen, for the vortiity field, both plots are not distinguishable from eah other. For the pressure levels, very small differenes are found, in partiular for the level orresponding to zero, whih is not orretly resolved, and for the pressure in the near field the parts of the field lose to the shok front. However, onerning the near field, the differene does not orrespond to a bad aordane between the linear solution and the numerial simulation, but rather to a diffiulty in evaluating the double integral in Eq. 14. TABLE III. Comparison between linear theory and numerial simulation for M 1 2, R/a4. Theory Min. pressure % 1% 1% 17% Max. pressure % 1% 4% 33% Max. vortiity % 1% 2% 3% Max. entropy % 1% 1% 1% Max. deformation % 1% 1% 24% TABLE V. Comparison between linear theory and numerial simulation for M 1 4, R/a4. Theory Min. pressure % 1% 3% 28% Max. pressure % 1% 2% 65% Max. vortiity % 1% 2% 57% Max. entropy % 1% 1% 4% Max. deformation % 1% 1% 16% Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

12 2414 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn FIG. 18. Vortiity and pressure fields for M 1 4. Upper plots: superposition of linear theory and numerial simulation with 1 2. Middle plots: numerial simulation with 1 1. Lower plots: numerial simulation with 1. For eah ase, from left to right: pressure field for three different instants orresponding to R/a, 2, and 4, and vortiity field at instant orresponding to R/a4. Pressure levels are as in Fig. 12, and vortiity levels orrespond to.5, 1, 1.5, 2, 3, 5, 8. The middle and lower plots represent the same data from the numerial simulations with 1 1 and 1. For 1 1, very little differenes are found. For 1, nonlinear effets are learly visible. Considering the vortiity field, it is observed that the vorties are highly distorted, due to their self-indued roll-up. Considering the pressure field, one observes a low pressure region in the enter of the aousti wave. This orresponds to the nonlinear pressure defiit as- Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

13 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 2415 FIG. 19. Shok deformation x /a for M 1 2, at three different instants orresponding to R/a, 2, and 4. Linear theory full lines, 1 1 dashed lines, 1 dotted lines. FIG. 2. Shok deformation x /a for M 1 4, at three different instants orresponding to R/a, 2, and 4. Linear theory full lines, 1 1 dashed lines, 1 dotted lines. soiated with the vorties. One also observes that the pressure levels in the front of the aousti wave are tightened. This is due to the steepening of the aousti wave. However the shape of the aousti wave remains qualitatively the same. Tables II and III display quantitative omparisons between the numerial omputations and the linear solution. We have reported the extremum values of the pressure, the vortiity, the entropy and the shok displaement for the nondimensional times R/a and 4. As an be seen, for the simulations with 1 4 and 1 2, the differene with the linear solution is smaller than 1%. For the simulation with 1 1, the aordane with the linear solution is still exellent, differenes being of the order of 1%. The greater differene is found for the maximum pressure at the instant orresponding to R/a4. This is not surprising beause this region orresponds to the ontribution of the pressure omponents at the ritial angles, where it is known that the LIA theory is less aurate. 16 Considering the simulation with 1, i.e., a true nonlinear ase, it is striking to see that the reported values are still in good aordane with the theory. At the instant when the spot rosses the shok (R/a), one only observes a 6% differene for the maximum vortiity, and a 2% differene for the maximum entropy. Other reported quantities differ of about 2%, whih an still be regarded as a good aordane with the theory, onsidering that we have a 1% initial amplitude in the spot. At the instant orresponding to R/a4, as the vorties roll up, the maximum vortiity strays from the theory, but the other quantities still remain lose to the theory. In partiular, it is found that the maximum of entropy perturbation relaxes to the theoretial predition within 1%. Figure 18 and Tables IV and V display the same results obtained from the M 1 4 omputations. As for the M 1 2 ase, the omputations with 1 4,1 2, and 1 1 are in exellent agreement with the linear solution. For the 1 ase, the nonlinear effets are more intense. In partiular, on observes that the steepening of the aousti wave has lead to the apparition of a seondary shok wave. However, as reported by Table IV, at the instant R/a orresponding to the initial stages of the interation, the numerial simulation still keeps a orret aordane with the linear theory. This aordane deteriorates in the latter stages of the interation see Table V as the nonlinear effets beome predominant. Finally, Figs. 19 and 2 ompare the shok deformation from the linear theory and from the numerial simulations, respetively for M 1 2 and M 1 4. Again, it is observed that the simulation ompares very favorably with the linear theory up to 1 1, and that for the 1 ase, the agreement deteriorates as time is inreased. The results presented in this setion suggest that, even for large spot amplitudes, the shok spot interation is dominated by a linear mehanism, and that nonlinearities rather affet the pressure and vortiity fields in the postinteration stages. These onlusions omplement those of Hussaini and Erlebaher 3 who performed numerial simulations for M 1 2 and 1 and for spot amplitudes varying from.25.25, and onentrated on the nonlinear aspets of the interation. VI. THE CASE OF A CONSTANT ENTROPY SPOT The method used in Se. III for a Gaussian spot an be applied to spots with different profiles, provided that a Fourier deomposition of the profile is known. It is of interest to know how the atual spot profile influenes the interation. In this setion, we will apply the LIA method to a tophat spot with a onstant entropy value within its radius a. This onstitutes a good model for a ylindrial bubble of a different gas arried through a shok. A. Detailed alulation The initial entropy perturbation is taken as s 1 ra, ra. C p 36 Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

14 2416 Phys. Fluids, Vol. 13, No. 8, August 21 Fabre, Jaquin, and Sesterhenn The orresponding Fourier transform and the slie deomposition are, respetively, given by s 1 C p J1 kae ikrˆdkad, 37 s 1 C p K a rˆ d. 38 Here, J 1 is the Bessel funtion of the first kind and of order 1, and the funtion K C is given by a Weber Shafheitlin disontinuous integral see Ref. 32, p. 487, K C z1 z1, 1 z 2 1zz 2 1 z1. 39 The analysis an be arried out in a manner similar to that developed in Se. III. The entropy, veloity and vortiity fields are given by Eqs. 8 11, where the funtions K and L are replaed by the orresponding funtions K C and L C. The funtion L C is z z z1, L C 1z 2 4 z1. The pressure field is given by p p rˆ pr K C b Z sp J p d p p Zsp d J1 kae kp iˆ xx dka, 41 and the shok deformation is obtained from Eq. 15, with the new profile funtions K z CI 1z2 z1, 42 z1, z1, L zz CI 1 zsgnzz 2 1 B. Results z1. 43 Some diffiulties arise when evaluating the different omponents of the flow field beause the funtions entering the analysis are disontinuous with infinite branhes and integral expressions suh as Eq. 41 do not onverge numerially. As a onsequene, this setion primarily presents qualitative results. 1. Vortiity field FIG. 21. Pressure field resulting from the interation of a onstant spot with a M 1 2 shok (R/a4). The vortiity field onsists of two parts. The first part is singular, and orresponds to a vortex sheet loated on the interfae of the transmitted spot. The seond part is regular, and is loated within the transmitted spot. An important onsequene is that the vortiity levels are far more important than for a Gaussian spot. They are theoretially infinite andonsequently, the spot will roll up muh more rapidly. This an be observed in the numerial simulations of Pione and Boris, 27 whih showed a rapid roll-up after the interation. The expression 18 for the irulation of the vorties is still valid, but the value of the form fator f has to be set to 1 instead of /2 see Appendix B. Thus the expression proposed by Yang et al. and presented in Se. IV B is exatly retrieved. 2. Pressure field The pressure field is plotted in Fig. 21 for a M 1 2 shok, and for the instant orresponding to R/a4. Due to the problems of onvergene evoked above, Fig. 21 is not very aurate. However, it allows a qualitative desription of the pressure field. As for the Gaussian spot, the pressure field is omposed of a double ylindrial wave with a dipolar struture, and a low pressure near field above the shok. But the pressure variations are sharper than for the Gaussian spot. One an still use the asymptoti expression given by Eq. 25, but the funtion K (z) must be replaed by K z C 1 z KC zzdz. This funtion is plotted in Fig. 14. It ontains a jump at z 1, and a logarithmi singularity at z1. So, as for the vortiity, the pressure field theoretially reahes infinite values, whatever the initial amplitude of the spot. As a onsequene, nonlinear effets are likely to be important, even for weak initial spot amplitudes. VII. SUMMARY In this paper, we have used Ribner s linear interation analysis LIA theory to solve the problem of a ylindrial entropy spot of weak amplitude onveted through a shok wave. The interation leads to a transmitted spot, a pair of ounter-rotating vorties, a ylindrial sound wave, and a deformation of the shok front. Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see

15 Phys. Fluids, Vol. 13, No. 8, August 21 Linear interation of a ylindrial entropy 2417 The method was first applied to a Gaussian entropy spot. An exhaustive desription of the solution has been given for a Mah 2 shok, and the flow field has been haraterized for a wide range of Mah numbers. An expression for the irulation of the vorties has been given as a funtion of the shok strength. This expression was shown to be in aordane with the numerial omputations and qualitative preditions of Yang et al. 29 An asymptoti expression for the pressure in the ylindrial aousti wave has also been developed and ompared to a reent model given by Grasso and Pirozzoli. 31 The linear solution has been ompared to a numerial simulation result. Exellent qualitative and quantitative agreement was obtained, even for large spot amplitudes. This indiates that the features of the interation are essentially linear. Finally, the analytial method has been applied to a onstant entropy spot, whih is a model for a bubble of light gas arried by the flow. The interation onsists of the same phenomena, but the vortiity and pressure fields ontain singularities. As a result, nonlinear effets are expeted to be more important than for a Gaussian spot. APPENDIX A: LINEAR INTERACTION ANALYSIS LIA FOR PLANE WAVES In this appendix, we onsider the LIA problem for plane elementary waves onveted through a shok. As said in the Introdution, several solutions of this problem have been given with different methods. Ribner 2 treated the ase of inident vortiity waves. Moore 3 solved the problem for inident aousti waves. The solution of the problem for inident entropy waves was then given by Chang. 4 Here we propose a general form for the transfer funtions whih applies for the three kinds of inident waves. 39 These transfer funtions may be used to study the interation of perturbation fields ombining the three modes of flutuations with shoks. An appliation to different ases of homogeneous perturbation fields is given in Ref Formulation for an entropy wave onveted through a shok We onsider the interation of a plane elementary entropy wave with a shok, as represented in Figs. 2 and 3. The origin of the frame is loated on the unperturbed shok front, and we define the oordinates in the diretions, s, and p as rˆx os y sin, ˆ x os s y sin s, ˆ x os p y sin p. All the angles onsidered belong to,. We suppose that the upstream perturbation field onsists of an elementary entropy wave of wavelength 2/k, angle, and amplitude. This takes the form for x s 1 e ikrˆit, A1 C p where the real part is to be retained. This plane wave is superposed to a main flow field with pressure P 1, density 1, and veloity U 1. The ondition that the entropy wave is onveted with the mean flow simply reads ku 1 os. A2 The downstream (x) flow field is searhed as a mean field of veloity U 2, density 2, and pressure plus perturbations. The mean field is determined by the lassial Rankine Hugoniot jump relations. In partiular, the downstream Mah number M 2, the ompression rate m 2 / 1 U 1 /U 2, and the pressure jump /P 1 an be expressed as funtions of the upstream Mah number M 1, m 1M 1 2 2, A3 21M 1 M 2 21M 1 2 2M 1 2 1, A4 /P M A5 The ratio of speifi heats of the gas is 1.4 throughout this work. As represented in Figs. 2 and 3, the entropy and solenoidal veloity perturbations generated downstream are onsidered as plane waves of angle s and wavelength 2/k s. The pressure field is onsidered as a generalized pressure wave with amplitude proportional to e ikp ˆe kpx. This represents a perturbation with a onstant phase in the diretion p, but with an amplitude that may deay away from the shok. The parameter haraterizes the deay rate; one wavelength downstream of the shok, the amplitude of the perturbation is damped by a fator e 2. Aording to the angle of the inident entropy wave, this generalized wave may have a propagative nature, see Fig. 2 or a nonpropagative nature, see Fig. 3. We denote by Z ss, Z sv, and Z sp the relative amplitudes of the entropy, vortiity, and pressure waves. We also suppose that the shok front deforms with an amplitude Z sx. These amplitudes may be omplex. In that ase, the real part represents the omponent of the wave in phase with the inident wave, and the imaginary part represents the omponent in phase quadrature. So the perturbation field is searhed as for x) s C p Z ss e iks ˆ it, p Z sp e kp iˆ xit, u sin s Z U sv e iks ˆ it os p i p, 2 M 2 p v sin p os s Z U sv e iks ˆ it, 2 M 2 A6 Downloaded 18 Jul 21 to Redistribution subjet to AIP liense or opyright, see