Shock-Turbulence Interaction

Transcription

1 Shock-Turbulece Iteractio A.Sakurai ad M.Tsukamoto Tokyo Deki Uiversity, Nishikicho -, Kada, Chiyoda-ku, Tokyo, Japa Abstract. For the geeral purpose of ivestigatig pheomeo of shock-turbulece iteractio, we cosider here the problem of a plae shock wave propagatig i a turbulet flow field. We compute this by the kietic model approach with use of the Boltzma equatio i the BGK approimatio. We produce a plae shock wave by a shock tube flow type computatio ad ru this ito a prefabricated turbulet flow. We observe the shock thickess wideig ad compare it with the oe by a miture legth theory as well as data of a eperimetal study i [4]. Keywords: shock wave, shock iteractio, turbulece PACS: Ab, 5.. +y INTRODUCTION For the geeral purpose of ivestigatig pheomeo of shock-turbulece iteractio [], we cosider here the problem of a plae shock wave propagatig i a turbulet flow field. We compute this by the molecular kietic model theory approach of the Boltzma equatio with the BGK approimatio []. I practice, we prepare first a steady plae shock wave i uiform gas by a shock tube flow type computatio, ad the ru this ito a prefabricated two-dimesioal isotropic turbulet field []. Amog results, particular attetio is paid to the wideig of the shock frot thickess due to the iteractio with the turbulet flow. It is compared also with the shock thickess give by a miig legth theory [3] ad data by a correspodig eperimet [4]. BOLTZMANN-BGK EQUATION IN INTEGRAL FORM We cosider two-dimesioal flow field ad use the Boltzma BGK equatio i itegral form [] as f ( ξ, + ξ t, t+ t) f ( ξ,, t) =ν ( f f ) (), where f = f ( ξ,, t) is the molecular distributio fuctio, ξ= ( ξ, ξ, ξ ) is the molecular velocity, = (, y) is the y z spatial coordiate. ν = ν (, t) is the collisio frequecy ad f = f ( ξ,, t) is the local Mawellia: f = N T C T, C= ξ u () 3 ( π ) ep{ / } with the umber desity N, the temperature T ad flow velocity u which are epressed i o-dimesioal form based o a basic legth L ad the referecig umber desity N ad temperature T. Further we assume the Mawell molecular model to have ν = k / K. N, K= l / L, k = 8/ 5 π, where K, l represet Kudse umber ad mea free path respectively. For the coveiece of umerical computatio of two-dimesioal flow, we use the reduced distributio fuctios

2 g, h by = fdξ ad their local Mawellias g, h by ( g, h) (, ξz ) z g = ( N / πt )ep{ ( C + C ) / T}, h = Tg / (3) y So that we ca reduce the idepedet variables to ( ξ, ξ y,, y, t). PLANE SHOCK WAVE IN SHOCK TUBE TYPE COMPUTATION We produce first a steady plae shock wave i uiform gas by a shock tube flow type computatio. For this, we postulate a shock tube like cofiguratio as show i Fig, where two sectios separated by the membrae filled with gases of differet pressures p, p. Computatios are performed with use of the kietic model equatio above for the sudde removed of the separatig membrae. Oe of the results is show i Fig i pressure distributios at differet times where p p = P T N membrae P T N P R E S S U R E 4 3 t= FIGURE. Shock tube type cofiguratio. FIGURE. Plae shock wave profile propagatig i shock tube type computatio. TWO-DIMENSIONAL ISOTROPIC TURBULENT FLOW FIELD We produce a two-dimesioal isotropic turbulet flow[], that is space periodic i a square regio. A radom iitial coditio is set at time t=, as a local Mawellia f is set with their flow velocity u, umber desity N = ad temperature T = for = (, y) i a uit squire regio f N ( πt) 3 ep{ C / T} =, C= ξ u, (4) where u obeys the isotropic eergy spectrum E( k ) : 3 E( k) = π u( k) u( k ) = K k ep{ ( k / k) } (5) where u ( k) is the Fourier trasform of u defied as u ( k) = u ( ) ep( ik ) d. (6) where K is a adjustable costat for the ormalizatio, ad k (the wave umber magitude of eergy cotaiig eddies at the iitial state) is chose as k =. The idividual mode is give radomly as follows: Let

3 ad set with u k = π, = (, ), =,,, for i=, = ( u, v ) π i u = a (, )si { + y+ ε ( )}; a = E k, (7) ad similar epressios for v, where ε ( ) are radom umbers i [,]. Computatio is performed for K =.,time step t=., divisios for ad ξ = ξ =.5 for 5 < ξ, ξ < 5 i. The iitial Reyolds umber is R = 67 with M = 4.5 of the maimum value of the iitial velocity U = 6Kξ ad K =.6. Oe eample of results obtaied is show i Fig.3 for the desity cotours. e FIGURE 3. Desity cotours of the two-dimesioal turbulet flow field produced umerically by a radom iitial coditio havig a proper eergy spectrum. SHOCK WAVE IN TURBULENT FLOW FIELD We put the shock wave produced by the shock tube type computatio as above ito the turbulet flow as prefabricated as show above i Fig.3. I practice, we performed i the two ways (ⅰ),(ⅱ): (ⅰ) The shock wave is pushed ito the field from its left ed. (ⅱ) Place the wave stadig i the flow field Eamples of pressure distributio results are show i Fig.4 ad Fig.5 respectively for the cases (ⅰ) ad (ⅱ). A particular attetio is paid to the wideig of the shock frot thickess (TH) due to the iteractio with the turbulet flow, which is compared with the oe see i Fig.. The shock thickess TH see i Fig.5 is about.3 to the preset dimesioless scale. The wideig of the shock thickess due to turbulece is epected to be caused by the Reyolds stress ad it is aturally related to the scale of the miig legth i the turbulet flow field. The thickess give by a miig legth theory [3] is give as k M du TH =. log 9 k= u (8) e C M d

4 ,where C, k,m, u represet the velocity of soud, eddy kiematic viscosity, Mach umber of shock wave ad time average of the velocity, respectively. The TH value by eq (8) to the above eample case is comparable size of.7 to.3 above.. TH FIGURE 4. Plae shock wave pushed ito the turbulet flow field from its left ed FIGURE 5 Plae shock wave statioary i turbulet flow showig its thickess. The result is compared also with data by a correspodig eperimet [4] i which the iteractio betwee a shock wave produced by a shock tube with atmospheric grid turbulece i a low speed wid tuel is eamied. Its pressure data at various poits i the tuel is reproduced i Fig.6(a) ad it is compared with the temporal pressure data at =.5,y=.5 of the preset study i Fig.6(b). Notice the same kid of pressure value fluctuatios appearig i both cases. d e l t a P.5 (a) (b) FIGURE 6. (a) Reproductio of pressure data at various poits i wid tuel eperimet i [4]. (b) Preset umerical temporal pressure data at =.5, y=.5 SUMMARY Cosidered the problem of a plae shock wave propagatig i a turbulet flow field time Computed the flow field by the molecular kietic model approach of the Boltzma equatio with the BGK approimatio theory.

5 Ra a plae shock wave produced i a shock tube flow type computatio ito a prefabricated two dimesioal turbulet flow. Amog results, a particular attetio was paid to the wideig of shock frot thickess due to its iteractio with the turbulet flow, which is compared with the oe i the origial shock wave. Compared also with the shock thickess give by a miig legth theory[3], ad data by a correspodig eperimet[4]. REFERENCES. A. Hadjadi, High-fidelity umerical simulatio of shock/turbulece ad shock/vorte iteractios, ISSW7, Roue, 8, 5.. M.Tsukamoto ad A.Sakurai, Decayig isotropic turbulece i gas flow, ISCFD/CFD Joural, 6, 7, pp.4-47; A.Sakurai ad F.Takayama, Molecular kietic approach to the problem of compressible turbulece, Phys.Fluids, 5, 3, pp A.Sakurai, O the thickess of plae shock waves i a gas i turbulet motio, J.Phys.Soc.Japa, 5,95, pp D. Takagi, Nagoya Uivesity, Nagoya, Japa; A. Matsuda, A. Sasoh, S. Ito, K. Nagata ad Y. Sakai, "Pressure Modulatio of Weak Shock Waves Through Turbulet Flows," AIAA paper, AIAA--447,.