论文李新亮的共享数据Li- Direct numerical simulation of compressible turbulent flows_Chinese phys Lett

Acta Mech Sin (2010) 26:795806 DOI 10.1007/s10409-010-0394-8 REVIEW Direct numerical simulation of compressible turbulent fl ows Xin-Liang Li De-Xun Fu Yan-Wen Ma Xian Liang Received: 15 September 2010 / Revised: 14 October 2010 / Accepted: 15 October 2010 / Published online: 22 December 2010 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010 AbstractThis paper reviews the authors recent studies on compressible turbulence by using direct numerical sim- ulation (DNS), including DNS of isotropic (decaying) tur- bulence, turbulent mixing-layer, turbulent boundary-layer and shock/boundary-layer interaction. Turbulence statistics, compressibility effects, turbulent kinetic energy budget and coherent structures are studied based on the DNS data. The mechanism of sound source in turbulent fl ows is also ana- lyzed. It shows that DNS is a powerful tool for the mecha- nistic study of compressible turbulence. KeywordsDirect numerical simulation Compressible turbulence Coherent structures Turbulent boundary-layer fl ow 1 Introduction With increasing demand of aeronautic and astronautic engi- neering, the study of compressible turbulent fl ow becomes a hot topic in fl uid mechanics. Direct numerical simulation (DNS), which directly solves the NavierStokes equations without any turbulence model, is a recently developed tool tostudyturbulentfl ow.Becausethereisnoturbulencemodel The project was supported by the National Basic Research Program of China (2009CB724100) and the National Natural Science Foundation of China (10632050, 10872205, 11072248). X.-L. Li (B) X. Liang LHD, Institute of Mechanics, Chinese Academy of Sciences, 100190 Beijing, China e-mail: lixl D.-X. Fu Y.-W. Ma LNM, Institute of Mechanics, Chinese Academy of Sciences, 100190 Beijing, China andallfl owscalesareresolved,DNScanprovidereliableand high-resolution fl ow data. However, due to its high compu- tational cost, DNS studies of turbulent fl ow with complex geometries and high Reynolds numbers remain a big chal- lenge.AccordingtoMoinandMahes1,DNSisnotabrute- force solution to engineering problems, but a research tool. Despite its limit (mainly the high cost), DNS is playing an increasingly important role in the study of turbulence mech- anism, turbulence control and turbulence modeling. Compared with DNS of incompressible fl ow, DNS of compressible fl ow is more challenging, especially at high Mach number. For instance, the shockwave poses great dif- fi culties to DNS studies. To resolve small scales of turbulent fl uctuations, numerical methods need to be low-dissipa- tive; however, most low-dissipation methods are not robust enough in the region of shockwaves, and numerical oscilla- tions often lead to failures of the simulation. In recent years, high-order WENO methods 2 are often used in DNS of compressible turbulent fl ows. To minimize the numerical dissipation of WENO schemes, many optimized or hybrid WENOmethods3havebeendeveloped.Samtaneyetal.4 performed DNS of decaying compressible turbulence by using asymmetrical fi nite difference scheme. Limited by the numerical schemes robustness, their turbulent Mach num- ber (Mt) can not be higher than 0.5. Li et al. 5,6 performed DNS of the decaying turbulence by using high-order upwind fi nitedifferenceschemesandagroupvelocitycontrol(GVC) scheme 7, and reached a higher turbulent Mach number of 0.7 5 and 0.95 7, respectively. Compared with DNS of isotropic turbulence, DNS of wall turbulence is more diffi cult. At the beginning, DNS case studies included temporally evolving boundary layers 8,9, and then spatially evolving ones. Rai et al. 10 per- formed DNS for spatially evolving fl at-plate boundary at free-streamMachnumber2.25byusingafi fth-orderupwind 123 796X.-L. Li et al. biased fi nite difference scheme. Pirozzoli and Grasso 11 performed the same DNS study later by using a seventh- order WENO scheme, and they investigated the effects of numerical schemes resolutiononthemeanvelocity profi les. Compared with the temporally evolving one, the spatially evolving DNS does not need “slow growth” or “extended temporal”assumptionsandthusminimizestheuncertaintyor artifi cial assumption 10,11. Li et al. 12 performed direct numerical simulations of spatially evolving boundary lay- ers over a blunt wedge with Ma=6 and boundary layers over fl at plates 1316 with the free-stream Mach number Ma=0.7, 2.25 and 6. The compressibility effects on the coherent structure were studied in these reports. This paper reviews the DNS studies of compressible tur- bulent fl ows by the authors, including DNS of compressible isotropic fl ow, mixing-layer fl ow, fl at-plate and blunt cone boundary layer fl ow and shock-boundary interaction fl ow. In these DNS studies, high-order fi nite difference Navier Stokes solver named “Hoam-OpenCFD” (now renamed as “OpenCFD-SC”) developed by the authors is used, and fl uid mechanisms, including compressibility effects, coher- entstructures,turbulentkineticenergybudgetsandthesound generation mechanisms are studied. Details can be found in the authors papers 57,1223 and book 24. 2 DNS of compressible isotropic turbulence and the sound source analysis Compressibleisotropic(decaying)turbulenceisanimportant model to study compressible free turbulence. We performed DNS cases of compressible isotropic (decaying) turbulence 5,7 and the passive scalar turbulence 6 in compressible turbulent fl ows. The Reynolds number based on the Taylors scale was Re= 72153, and the turbulent Mach number (based on the r.m.s. velocity and mean sound speed) was Mt= 0.20.95. Compressibility effects on the statistics of turbulent fl ow as well as the mechanics of shocklets in com- pressible turbulence were also studied, and the development of turbulent kinetic energy shows that high Mach number leadstomoredissipation.Scalinglawsincompressibleturbu- lencewerealsoanalyzed.Evidencewasobtainedthatscaling lawsandextendedselfsimilarity(ESS)holdinthecompress- ible turbulent fl ow in spite of the presence of shocklets, and compressibility has less effect on scaling exponents. Numer- ical results also show that the (extended) scaling exponents agree with the SL scaling law 25. The k1scaling range was found in passive scalars energy spectra, and the passive scalars spectra decay faster with the increase of turbulent Machnumber.Theextendedself-similarity(ESS)wasfound in the passive scalar of compressible turbulence. Details can be found in Refs. 57. Recently researchers have paid more and more attention to sound generation by turbulent fl ow. It is very important Table 1 Statistics of the turbulent fl ows DataReMtSu?Fu?Mesh D795.2132563 D2745.1532563 D381.50.530.4414.6522563 D4995.1073843 D5139.20.490.4915.0133843 Fig. 1 Distribution of axes of coherent eddies in fi eld D1 to study the mechanism of the sound source for sound con- trol. Tube-like eddies are thought to be the most important structures in isotropic turbulence. She et al. 26 found tube- like coherent eddies in homogeneous isotropic turbulence; Tanahashi et al. 27,28 gave the shapes and scales of the coherent eddies, and found that the eddys core is elliptic. On the basis of the DNS 29, we studied the sound radiation from the turbulent fl ows. The results show that the rotating elliptic eddies are important sound source. The sound fre- quency spectra were computed from the probability density function (PDF) of the coherent eddies rotation frequencies. Instantaneous fl ow fi elds of the compressible homoge- neous turbulence from the DNS 29 were used to study the mechanismofsoundsource.Statisticsoftheseinstantaneous fl ow fi elds are shown in Table 1, where Reis the Reynolds number based on Taylor scale and RMS velocity, Mtis the turbulent Mach number based on RMS velocity and mean sound speed, Su?is the skewness factor of u/x, and Fu?is the fl atness factor of u/x. The coherent eddy identifi cation scheme proposed by Tanahashietal.27isusedtoidentifythetube-likefi nestruc- turesinthecomputationsD1D5.Figure1showsdistribution 123 Direct numerical simulation of compressible turbulent fl ows797 Fig. 2 Mean fl uctuation density ?by using phaseaverage technique. a D1, b D3 Fig. 3 PDF of coherent eddies rotation frequency of the axes of the coherent eddies for fl ow fi eld D1. The thickness of the axis represents the amplitude of the second invariant of the velocity gradient tensor (Q) on the axis. The thicker axis denotes a stronger eddy. The statistical technique named as “phase averaged scheme” developed by Tanahashi et al. 27 is used to study the fl ow fi eld in each coherent eddies. According to this scheme, the fl ow should be rotated to the same orientation before being averaged. Figure 2 shows the phase-averaged densityfl uctuationaroundthecoherenteddy.Thedistribution of density fl uctuations is closely related to the elliptic fea- tureofthecoherentfi ne-scaleeddy.Densitytendstoincrease alongthemajoraxisoftheeddyanddecreasealongtheminor axis, which shows quadrupole characteristicstwo maxima and two minima in the circumference around the center. On the cross section, there is an in-fl ow in the direction of com- pression,andanout-fl owinthedirectionofstretching.Inthe direction of compression, dilatation becomes positive and in the direction of stretching dilatation becomes negative. This is the reason why the quadrupole is formed. Figure 3 shows the probability density function (PDF) of coherent eddies rotation frequency normalized by the mostexpectedfrequency.Lilleysfrequencyspectrummodel 30 (P() 4(1 + 22)3) and Poudmans model 31 (P() 7/2 ) are also shown in this fi gure. It shows that the PDF agrees very well with Lilleys model in the low fre- quency region and Poudmans model in the high frequency region, which indicates that the coherent eddies play a key role in the sound generation of isotropic turbulence. 3 DNS of compressible turbulent mixing-layer fl ow The turbulent mixing-layer is a basic model of free-shear turbulence, and it is also an important model for the study of fuel-air mixing in scramjet. We performed DNS of com- pressible mixing-layer fl ows for the study of development of coherent structures and compressibility effects. Figure 4 shows the schematic of the mixing-layer fl ow, and it is initialized by mean velocity profi le of hyper- bolic tangent with linear stability theory (LST) perturba- tions. Compressible NavierStokes equations with passive scale equation were solved directly by using our Fifth- order upwind compact scheme 22. The fl ow parameters Mc=0.8(Mc is convective Mach number) and Re=200. Details can be found in Refs. 20,21. Figure 5 shows the iso-surface of the instantaneous pres- sure at time t =36.63. Because the vortexs center has low pressure, the iso-surface of low pressure represents the shape of coherent eddies. This fi gure shows the formation of Lambda (?) vortex. According to linear stability analy- sis, the formation of Lambda vortex is the result of primary Fig. 4 Schematic of mixing-layer fl ow 123 798X.-L. Li et al. Fig. 5 Iso-surface of the instantaneous pressure at time t =36.63. a 3D view, b front view, c top view. Fig. 6 Three-dimensional contours of the passive scale at a t =9.6, b t =23.8, c t =36.6 Table 2 Flow and mesh parameters for the DNS of fl at-plate boundary layers CaseMaRe/inchTwMesh (Nx Ny Nz)?x+ ?y+ w ?z+ 10.705.00 1041.0981,000 100 32020.30 1.00 10.10 22.256.35 1051.9002,193 72 25614.10 1.10 6.60 36.002.00 1066.9804,000 90 2568.07 0.97 3.78 insatiabilityofcompressibleboundarylayer;however,intur- bulent boundary-layer fl ow, the formation of Lambda vortex is the result of the secondary instability. Figure 6ac show the three-dimensional instantaneous contours of the passive scalaratdifferenttimes,andthemixingprocessofthetwofl u- ids.Thisfi gureshowsthedevelopmentofthemushroom-like structures, which are typical structures in the mixing-layers. Details can be found in Refs. 20,21. 4 DNS of compressible turbulent fl at-plate boundary layer fl ow To study the mechanism of compressible wall turbulence, DNS of compressible channel fl ows 23 and fl at-plate boundary layer 1316 were performed by using high-order fi nite difference method. Table 2 shows the fl ow parame- ters of the fl at-plate boundary-layer fl ows. The free-stream Mach numbers were 0.7, 2.25 and 6. The schematic of com- putational confi guration is shown in Fig. 7. In case 1, LST perturbations(TSwaves)wereusedtotriggerthetransition, Fig. 7 Schematic of computational confi guration while blow and suction perturbations were used in the other two cases. Details of the computation can be found in Refs. 1216. Figure8showsthedistributionofskinfractioncoeffi cient CfinCase2,andthefastincreaseofCfdenotestheoccurring oftransition.ThetheoreticalvaluepredicatedbyBlasisustur- bulenceequationbasedonthemomentousthicknessandVan Dirst II transform 11 is also given in this fi gure. Figure 8 shows that our DNS result agrees very well with the theo- retical value. The mean Van Dirst velocity profi les in Fig. 9 are normalized by the wall-shear velocity at x =8.8, where the turbulence is full-developed. This fi gure shows that the 123 Direct numerical simulation of compressible turbulent fl ows799 Fig. 8 Distribution of skin fraction coeffi cient Fig. 9 Mean velocity profi le normalized by wall shear velocity (x =8.8) current result agree very well with that of Pirozzoli et al.s result 11, which validates the DNS. Comparisons of the normalized turbulence kinetic energy budget for Ma =6 (Case 3) and Ma =2.25 (Case 2) are shown in Fig. 10, where the solid lines are the normalized kinetic energy for Ma =6 and the dashed lines are that for Ma =2.25. Figure 10a shows that the difference of the tur- bulent transport term (T) between two Mach number cases is remarkable. Figure 10b shows that the term associated with the density fl uctuations (M term) for Ma =6 is much higherthanthatfor Ma =2.25,resultingfromthedifference between Reynolds average and Favre average at high Mach numbers, and the non-negligible term u? i . This fi gure also shows that the pressure-dilatation term ?dfor Ma =6 is much higher than that for Ma =2.25, but it remains small in contrast with the total dissipation. Figure 11 shows the nor- malized dilatation dissipation for Ma =2.25 and Ma =6, where d=(4/3) (u? i/xi)2 and it is the measurement of the compressibility effects. It shows that the dfor Ma =6 is much larger than that for Ma =2.25. However, the value of dis still very small even for Ma =6, and in general is less than 0.4% of the total dissipation. Details of the study of turbulent kinetic energy budget can be found in Refs. 13, 14,16. Figure 12 shows the typical coherent structures by using iso-surface of the second invariant of velocity gradient (Q), which shows the development of a hairpin-like vortex. Details of the mechanism of the hairpin vortex and hairpin vortexclustercanbefoundinRefs.15,24.Figure13shows thevisualizationofcoherentstructuresinthethreeDNScases by using the instantaneous iso-surface of Q. These fi gures show the compressible effects on the coherent structures. With the increasing of Mach number, the hairpin-like eddies are replaced by the semi-streamwise vortexes. The contours of the spanwise vorticity in the spanwise middle plane at different times are shown in Fig. 14. It Fig. 11 Normalized dilatation dissipation Fig. 10 Normalized turbulence kinetic energy budget. Solid line Ma = 6, dashed line Ma = 2.25 (P production, T turbulent transport, D turbulent diffusion, viscous dissipation, ?tpressure transport, ?d pressure-dilatation, M term associated with density fl uctuation) 123 800X.-L. Li et al. Fig. 12 Typical coherent structures in fl at-plate boundary layer (Case 1: Ma = 0.7) Fig. 13 Visualization of coherent structures by using iso-surface of Q. a Mach 0.7, b Mach 2.25 (full visualization and local enlarged plot), c Mach 6 shows that high-shear layers appear at t =3.0 in the region 5.5x 6.7. The spanwise vorticity in the high-shear layer ismuchhigher(about35times)thanthatintheenvironment fl ow. Then the high-shear layers go downstream and become unstable.Theinstabilityofthishigh-shearlayerleadstolam- inarbreakdown.Thehigh-shearlayerhasstrongcollocations with the quadrupole sound source. Details can be found in Ref. 16. 5 DNS of the blunt cone boundary layer transition Blunt cones are typical leading shapes of hypersonic vehi- cles, and correct prediction of transition locations of blunt cone boundary layers is of great importance for the aerodynamic design, thermal protection, and fl ying control for hypersonic vehicles. Good understanding of the transi- tion mechanism of hypersonic boundary layers is the foun- dationoftransitionpredictionandcontrol.DNSisapowerful tool to study transition or turbulence; however, the DNS of hypersonic blunt cone boundary layer turbulence is very rare becauseboththeMachnumberandtheReynoldsnumberare very high and the DNS is very diffi cult. We performed DNS cases of turbulent fl ow over a fi ve-degree half-cone-angle blunt cone. The free-stream Mach number is 6 and the angle of attack is 1 . The fl ow parameters are shown in Table 3. The seventh-order WENO scheme was used in the DNS, and very fi ne meshes (totally 320 million meshes) were used in 123 Direct numerical simulation of compressible turbulent fl ows801 Fig. 14 Contours of the spanwise vorticity on the spanwise middle plane at different time (Case 2: Mach 2.25) Table 3 Flow parameters RenMaAoA/()Tw/KT/Kh/() 10,00061294795 the boundary layer to resolve transition and turbulent fl ow. The schematic of the mesh and setup of the boundary condi- tions was shown in Fig. 15. Isothermal non-slide boundary conditionwasusedonthewall;non