DNS of transition in hypersonic boundary-layer flows including high-temperature gas effects

Transcription

1 Center for Turbulence Research Annual Research Brefs DNS of transton n hypersonc boundary-layer flows ncludng hgh-temperature gas effects By C. Stemmer AND N. N. Mansour 1. Motvaton and Objectve Wnd-tunnel experments at hypersonc Mach numbers above 10 are extremely dffcult to undertake and facltes are lmted. Addtonally,the stagnaton condtons for free flght under atmospherc condtons can not be reproduced. Ths results n a lmted portablty of the wnd-tunnel results to atmospherc condtons. Therefore,numercal nvestgatons of hypersonc transton can be extremely valuable n developng an understandng of the transton process at hypersonc speeds. The objectve of ths effort s to develop an understandng of effects of nonequlbrum chemstry on transton. Our approach s to compare hypersonc transton on a flat plate under nonequlbrum chemcal and thermal condtons to hypersonc transton under equlbrum condtons. In the 1950 s and 60 s,a seres of hypersonc experments was conducted n free flght. The transton locaton could be found but no detals on the transtonal structures could be recorded n these experments see Schneder,1999,for a comprehensve revew of supersonc and hypersonc experments. Schneder also notes that the angles of attack of the test vehcles are uncertan. An ongong experment on transton at Ma = 21 n Novosbrsk,Russa Mronov & Maslov 2000,promses expermental verfcaton of the numercal fndngs to some extent. Further detaled experments on transton at hypersonc speeds cannot be expected n the near future. 2. Governng Equatons In order not to confuse the ndex notatons,the ndex refers to the speces 1-5 and no summaton s mpled on ths ndex,whereas the ndces j, k and l refer to the Cartesan drectons x, y and z and summaton from 1-3 s mpled. The contnuty equaton for chemcally-reactng compressble flows becomes ρ t + ρ u j + u D x,j = W, 2.1 j where W represents the speces producton terms see Eq and u D the dffuson veloctes see Eq Rewrtng ths equaton wth the speces concentratons rather than the denstes,t becomes ρ Dc Dt + ρ u D x,j =W, 2.2 j where the speces concentratons are gven by c = ρ ρ. 2.3

2 144 C. Stemmer & N. N. Mansour Note that snce c =1, 2.4 only 1 equatons have to be solved. The total mass s conserved ρ t + ρu j = The total momentum equatons are ρ Du j Dt = p + τ jk 2.6 x k wth uj τ jk = µ x k + u k The bulk vscosty s denoted by λ. The energy equaton for the total energy becomes ρ De + u ju j /2 Dt = q j + q vb j,j pu j,j + u k τ jk + + δ jk λu l,l. 2.7 ρ h u D,j,j 2.8 where e descrbes the nternal energy. The energy equaton for the vbratonal energy e vb n the case of vbratonal nonequlbrum s as follows e vb t + e vb u j + u D j = q vb j + Q T V + Q chem. 2.9 For the equlbrum case,the vbratonal temperature T vb s equal to the translatonal temperature T and eq s used wth T replacng T vb. The nternal energy for the complete system s a sum of the speces nternal energes takng nto account ther concentratons, e = c e The equlbrum nternal energy for one speces conssts of the translatonal,rotatonal and vbratonal energy and the heat of formaton. Note that atoms N and O delver no vbratonal and rotatonal contrbuton to the nternal energy e = e trans T +e rot T +e vb T vb + h f The nternal energy contrbutons from translaton,rotaton and vbraton are assembled through the specfc heats at constant volume as e = c trans v, T + c rot v, T + c vb v, T vb + h f The enthalpy s expressed as h = c trans p, T + c rot p, T + c vb p, + h f The nternal energy and enthalpy are connected through h = e + p ρ. 2.14

3 DNS of hypersonc transton 145 Mass Fracton c N O O N NO Temperature [K] Fgure 1. Composton of equlbrum ar at 1 atm. The flud s treated as an deal gas,where the followng equaton holds p = p = R ρ T M For the dffuson veloctes u D,Fck s law of dffuson s employed ρ u D j = ρd c, 2.16 where the dffuson coeffcent s ndependent of the speces. The translatonal and the vbratonal heat conducton s descrbed through Fourer s law q j = κ T, q vb j vb Tvb = κ Chemcal Modelng A fve speces N 2, O 2, N, O, NO model for ar wll be appled. The equlbrum composton for ar at constant pressure over temperature s shown n Fg. 1. The reacton rates k f and k b are modeled n an Arrhenus manner accordng to Park The model proposed by Park takes nto account the translatonal as well as the vbratonal temperature T vb for each speces. The vbratonal temperature descrbes the vbratonal relaxaton,whereas a translatonal temperature ncludes the rotatonal relaxaton,whch s assumed to take place nstantly. It only takes 9-12 molecule collsons for the rotatonal relaxaton to complete,whereas the vbratonal relaxaton takes 10 5 molecule collsons to reach a steady state the same order of magntude as for the chemcal relaxaton. The seventeen chemcal reactons thought to be suffcent for the modelng of ar under the condtons of nterest are as follows: The reacton partner M represents any of the

4 146 C. Stemmer & N. N. Mansour fve speces consdered; see Park,1989. N 2 +M N+N+M reac. 1 O 2 +M O+O+M reac. 2 NO+M N+O+M reac N 2 +O NO+N reac. 4 NO+O N+O 2 reac. 5 wth the producton terms M N2,M O2,M NO,M N andm O represent the speces masses : W N2 = M N2 R 1 + R 4 W O2 = M O2 R 2 R 5 W NO = M NO R 3 R 4 + R W N = M N 2R 1 R 3 R 4 R 5 W O = M O 2R 2 R 3 + R 4 + R 5 where R 1 = k f,1 ρn2 R 5 = k f,5 ρno M NO M N2 ρ M ρo2 ρ k f,2 M O2 M O + + k b,1 ρn M N 2 ρ M 2 ρo ρ k b,2 M O M R 2 = M R 3 = ρno ρ k f,3 + ρn k b,3 M NO M ρn2 ρo ρno R 4 = k f,4 + k b,4 M N2 M O ρo ρo2 + k b,5 M O2 M N ρn M NO M N ρn M N ρo ρ, M O and the forward reacton rates k f for the fve reactons consdered are M 2.20 k f,1 = TT vb 3/2 exp 59, 500/ TT vb for M = molecule k f,1 = TT vb 3/2 exp 59, 500/ TT vb for M = atom k f,2 = TT vb 8/5 exp 113, 200/ TT vb for M = molecule k f,2 = TT vb 8/5 exp 113, 200/ TT vb for M = atom k f,3 = exp 75, 500/ TT vb for M= N 2,O k f,3 = exp 75, 500/ TT vb for M= N,O,NO k f,4 = TT vb 1 exp 38, 370/ TT vb k f,5 = exp 19, 450/ TT vb. The backward reacton rates k b are calculated from the equlbrum rates through k b, = k f, /K eq, 2.22

5 DNS of hypersonc transton 147 The equlbrum rates are defned as K eq,1 =exp TT vb /10, log 10 10, 000/ TT vb , 000/ TT vb /TT vb K eq,2 =exp TT vb /10, log 10 10, 000/ TT vb , 000/ TT vb /TT vb K eq,3 =exp TT vb /10, log 10 10, 000/ TT vb , 000/ TT vb /TT vb 2.23 K eq,4 =exp TT vb /10, log 10 10, 000/ TT vb , 000/ TT vb /TT vb K eq,5 =exp TT vb /10, log 10 10, 000/ TT vb , 000/ TT vb /TT vb 2.2. Modelng of physcal and transport propertes The followng relatons are for a mxture of chemcally-reactng gases Specfc heat at constant volume The specfc heat at constant volume c v for atoms s descrbed through: c v, = c trans v, = 3 2 R The partal dervatves of the speces concentratons wth respect to the temperature are the contrbutons due to chemcal reactons. The specfc heat at constant volume c v for molecules Vncent & Kruger 1982 s made up as follows, where Θ vb c v, = c trans v, + c rot v, + c vb v, = 3 2 R + R + Θvb /T vb 2 e Θvb /T vb e Θvb /T vb 1 2 R, 2.25 s the characterstc temperature of vbraton of the molecular speces Specfc heat at constant pressure The specfc heat at constant pressure c p s descrbed by: c p, = c v, + R T Vscosty Blottner s formula wll be employed for the modelng of the vscosty Blottner,Johnson & Ells Ths approxmate formula s vald up to 10,000 K, far exceedng the temperature range of the flows nvestgated here. The coeffcents A µ,b µ and C µ are gven by Blottner et al. µ =0.1 exp [C µ + ln T B µ +lnt A µ ]. 2.27

6 148 C. Stemmer & N. N. Mansour Thermal conductvty The speces thermal conductvtes are descrbed employng Eucken s correcton,gven as Hrschfelder,Curtss & Brd 1964: κ = µ 5 2 ctrans v, + c rot v,, κ vb = µ c vb v, Mxng rules for vscosty and thermal conductvty The mxng rule n a mxture of gases,accordng to Wlke 1950,s n x µ µ mx n j=1 x 2.29 jφ j wth =1 [ 1+µ /µ j 1/2 M j /M 1/4] 2 Φ j = 8 + 8M /M j 1/2 and x = c /M n j=1 c j/m j. The same formula apples for the thermal conductvtes,replacng the vscosty µ by the thermal conductvty k. Further detals of the physcal modelng can be found,for example,n Sarma Dffuson coeffcent A constant Schmdt number Sc = 0.5 s assumed Hudson 1996 whch yelds for the dffuson coeffcent: D = µ ρsc = 2 µ 2.30 ρ Translatonal-vbratonal energy exchange Vbratonal energy s present only n the molecular speces N 2,O 2 and NO,whch are all modeled as harmonc oscllators. Therefore the followng equatons are vald. In case of the ncorporaton of anharmonc oscllatory molecules lke CO 2,dfferent relaxaton and energy expressons have to be appled Vncent & Kruger The translatonal-vbratonal energy exchange s descrbed through a Landau-Teller relaxaton model Vncent & Kruger 1982 as, Q T V = e vb,eq T e vb T vb c, 2.31 τ where the relaxaton tmes are determned for each speces as τ = 1 p C 1 expc 2 /T 1/3, 2.32 and the nonequlbrum vbratonal energy depends on the vbratonal temperature as e vb = Θvb /T vb e Θvb The equlbrum value for the vbratonal energy e vb,eq T replacng T vb. /T vb 1 R T vb follows the same expresson,wth

7 DNS of hypersonc transton 149 β arctan1/ma chem. reac. effects shock Ma>>1 b.l. δ/x Ma 2 / Re x Fgure 2. Schematc of shock locaton and boundary-layer edge for hypersonc boundary layers on a flat plate, showng dependence on Mach number The chemcal source term n Eq. 2.9 s expressed as the sum over the vbratonal nternal energy multpled wth the producton terms: Q chem = c e vb W Future Work A spatal fnte-dfference DNS code wll be appled on a Cartesan three-dmensonal grd on a flat plate. The code wll ncorporate a shock-capturng technque,snce the shock provoked by the flat-plate leadng edge s the major source of nonequlbrum. For the hgh Mach numbers,the locaton of the shock and the boundary-layer edge,whch s the area of lnear nstablty for hypersonc flows,merge,and the chemcal and thermal nonequlbrum n ths regon s expected to nfluence transton to a large extent Fg. 2; see also Anderson,1989. For the flght condtons nvestgated,the data n Fg. 3 are relevant. At a speed of V =5.9 Km/s,dssocaton of ntrogen and oxygen can be expected. For an alttude of h = 25 Km,chemcal and thermal equlbrum wll persst at a Mach number Ma = 20. At an alttude of about h = 100 Km Ma=20.8,full nonequlbrum condtons are present. Condtons are chosen such that onzaton wll not take place. Ths choce s consstent wth the return path of the shuttle as t enters the athmosphere.

8 150 C. Stemmer & N. N. Mansour Fgure 3. Flow regmes and thermochemcal phenomena n the stagnaton regon of a 30.5 cm radus sphere flyng n ar Gupta et al REFERENCES Anderson, J. D Hypersonc and and Hgh Temperature Gas Dynamcs. AIAA publcaton. Blottner, F. G., Johnson, M. & Ells, M Chemcally reactng vscous flow program for mult-component gas mxtures. Sanda Natl. Laboratores,SC-RR Gupta, R. N., Yos, M. J., Thompson, R. A. & Lee, K.-P A revew of reacton rates and thermodynamc and transport ropertes for an 11-speces ar Model for chemcal and thermal nonequlbrum calculatons to 30,000K. NASA RP Hrschfelder, J. O., Curtss, C. F. & Brd, R. A Molecular Theory of Gases and Lquds. Wley & Sons,New York. Hudson, M. J Lnear Stablty of Hypersonc Flows n Thermal and Chemcal Nonequlbrum. Ph.D. Thess,North Carolna State Unversty,Ralegh,NC. Mronov, S. G. & Maslov, A. A.,Expermental study of secondary stablty n a hypersonc shock layer on a flat plate. J. Flud Mech. 412, Park, C A revew of reacton rates n hgh temperature ar. AIAA Paper Sarma,G.S.R.2000 Physco-chemcal modelng n hypersonc flow smulaton. Prog. Aerospace Sc. 36, Schneder, S. P Flght data for boundary-layer transton at hypersonc and supersonc speeds. J. Spacecraft and Rockets 36,8-20. Vncent, W. G. & Kruger. C. H Introducton to Physcal Gas Dynamcs. Kreger,Malabar,FL. Wlke, S. P A Vscosty Equaton for Gas Mxtures. J. Comp. Phys. 18,