Modeling of Non-stationary Gas Flow in Annular Nozzle

Transcription

1 25 th ICDERS August Leeds UK Modeling of Non-stationay Gas Flow in Annula Nozzle Vladimi A.Levin 1 Nataliya E.Afonina 1 Valeiy G.Gomov 1 Ivan S.Manuylovich 1 Gennadiy D.Smekhov 1 Aleksande N.Khmelevsky 1 Vladimi V.Makov 2 1 Institute of mechanics of the Moscow State Univesity Moscow Russia 2 V.A.Steklov Mathematical Institute RAS Moscow Russia 1 Intoduction Seach fo new nozzle devices that can compete in jet engines with taditional Laval nozzles is one of the pomising diections on a way of impovement of thei oveall-mass and specific chaacteistics. In this espect the cetain inteest ae epesent annula and dual-slotted nozzles with deflecto in the fom of a spheical segment. As have shown expeiments [1 2] thee ae vaious modes of gas flow in such nozzle devices. In steady-state opeation they belong to the class of nozzles with the cental body [3 4]. In non-stationay peiodic pulsing opeating modes specified nozzles epesent high-fequency pulsed outlet devices. Such devices ae consideed to be pomising fo ealization of pulsed detonation egime of fuels combustion [5]. In this pape based on the Navie-Stokes equations it is caied numeical paametic study of the influence of vaious factos on the development ove time of the initial petubations caused by the device stat blown by ai unde a lamina flow model. In the calculations paametes of the numeical scheme and deteminative conditions of poblem wee vaied. As a esult in the calculation fist discoveed the flow egimes in which the stating petubation accompanying the stat of the annula nozzle do not decay and pass into the quasi-peiodic egime. Detemined the Fouie fequency spectum of the pessue fluctuations in the cente of the deflecto thust wall and the value of nozzle thust. Showed typical pulsed pessue signals obtained in the computational model and egisteed in the expeiments that ae pefomed in a pulsed aeodynamic setup using as a woking gas ai and combustion poducts of stoichiometic ai-acetylene mixtue. 2 Gas phase model Basic assumptions. The ai is consideed as an ideal one-tempeatue mixtue of molecula oxygen and nitogen with constant values of mola concentation of the mixtue components X O 2 =.21 and X N 2 =.79. Fo the desciption of molecula tanspot the appoximation Navie-Stokes is used. The equation of state. The themal equation of a condition has the fom p = ρrut / m whee R - a univesal gas constant m -aveage molecula ai weight. u Coespondence to: gomov@imec.msu.u 1

2 Non-stationay Gas Flow in Annula Nozzle Themodynamic model. Rotational and vibation enegy modes of ai components descibed by equilibium model "igid otato-hamonic oscillato" with chaacteistic vibation tempeatues T vo2= 2228 K and T vn 2= 3336 K. In this case the intenal enegy pe unit mass of gas e and heat capacity c p detemined by the expessions 2 5 R T X u Ru vk k 7 R R ( Tvk / T) exp( Tvk / T) Xk e= T + ; c u u p ; 2 m m k exp( Tvk / T) 1 = T 2 m + m k [exp( Tvk / T) 1] 2 Model of molecula tanspot. Molecula flow impulse tenso ˆ τ associated with the stain ate tenso ˆε as atio ˆ τ = µ ˆ ε whee the tenso components ˆε ae of the fom u u i j 2 u ε k ij = + δij xj xi 3 xk and dependence of facto of molecula viscosity on tempeatue is appoximated by a powe-law.683 fomula µ = aµ T. Molecula heat flux q is defined by expession T q = λ whee the coefficient of themal conductivity λ is detemined by µ numbe P =.7 : µc p λ = P 3 Basic equations and a numeical method c p and molecula Pandtl Basic equations. The Navie-Stokes equations in integal fom descibing unsteady axisymmetic gas flow in a cylindical coodinate system ( x y ϕ) have the fom UxdS + n xdl ds S F = δ S Ω S t whee S the fixed contol egion in a meidian plane (x y) δs bode aea n = ( nx ny) - unit outwad nomal to δs U a set of consevative vaiables pe unit volume F = F inv + F vis - the sum of inviscid and viscid flows U though a unit aea bounday aea Ω consists of the souce tems in the unit volume. Fo the consideed gas-phase model thee ae ρu ρuu pnn + x τ x F = ρuv + pnn + y τ y ρuh qh + uτx + vτ y 25 th ICDERS August Leeds 2

3 Non-stationay Gas Flow in Annula Nozzle U = { ρρ u ρv ρe T } ; T Ω = { b x }. Hee uv- velocity vecto components u!!! e = e +.5( u u) - total enegy pe unit of gas mass h = e + p / ρ - total enthalpy τx = ( τxx τxy) τy = ( τyx τyy) b = p + 2 u / x 2/3divu. x Numeical method. Unsteady gas motion equations ae solved numeically using an implicit diffeence scheme constucted by the finite volume method on a single block stuctued gid. In this appoach the system of diffeence equations consists of numeical analogs of the consevation equations fo tetahedal cells coveing a ated aea and diffeence appoximations of bounday conditions. The equations ae witten with espect to the values of input vaiables Z = { puvt } in the cells centes and in the side cells centes lying on the body suface. Gid cells ae fomed by the intesection of two inv discete families of cuves. Inviscid fluxes F G acoss cell boundaies ae computed by solving the L R exact Riemann poblem about the decomposition of an abitay discontinuity ZG =R( ZG Z G) whee R - is the task solution opeato. In the scheme of the fist ode of accuacy with espect to the L R spatial vaiables left Z G and ight Z G bounday values of input vaiables ae set equal to thei values at the cente of the coesponding cell. In the scheme of the second ode of accuacy L Z G and R Z G ae detemined using one-dimensional intepolation (extapolation) of values Z at the centes of nea-by cells on the given bounday between cells using a limite minmod. Viscous flows though the cell faces ae computed using the cental o one-sided diffeence fomulas of second-ode accuacy. Time deivatives ae appoximated by one-sided two-point fomulas of fist-ode accuacy o onesided thee-point fomulas of second-ode accuacy. At each time step diffeence equations ae solved using a two-laye iteative scheme. 4 Results of calculations and expeiments The calculations ae pefomed fo the flow domain including: - the deflecto (thust unit) with thust wall in fom of a spheical segment with adius of 36 mm and a height of 22 mm (base diamete d = 66.4 mm); - input annula nozzle diamete d with a height of the citical section h = 4.4 mm; - exhaust cone nozzle with half-angle 45 and a length of 15 mm; - sufficiently lage fly away egion. The calculations wee pefomed on gids with the numbe of points of 1x316 2x376 and 3x472. Gid nodes wee concentating nea the suface of the thust wall so that thee wee no less than 1 nodes in the bounday laye egion. The boundaies of the computational domain and the distibution of the nodes of computational mesh 1x316 fo ealy pesented geometic nozzle paametes ae shown in Figue 1. It was assumed that the blowing ai though the annula nozzle occus at the sound speed at given constant stagnation pessue and tempeatue P and T. The ai expiation fom the device though the exhaust conical nozzle occus in the gas medium with a pessue.1 atm and tempeatue = 3 K. Fo the poblem closue at the oute bounday ae used soft bounday conditions of extapolation type. The suface of the thust wall is assumed cooled to the tempeatue = 3 K o themally isolated. By hypothesis the stating of the device initially filled with stagnant ai occus instantly leading to the geneation of intense unsteady gas dynamic pocesses and significant incease of pessue. 25 th ICDERS August Leeds 3

4 Non-stationay Gas Flow in Annula Nozzle Figue 1. Configuation of ated domain and computational gid 1х316. As the esults of the calculations the futhe development of the stat petubation depends significantly on both the numeical scheme and the detemining paametes of the poblem. The most adical influence on the solution has the ode of the diffeence scheme accuacy in the spatial coodinates. Fo all consideed diffeence gids using fist-ode schemes the initial petubations ae damped. The settling time of a stationay solution inceases with inceasing "density" of the gid. In the calculations due the scheme of the second ode of accuacy the initial petubations in most cases conveted to quasi peiodic egime. The solution in the calculations of this scheme was established only in the case in which the gid step nea the suface of the thust wall exceeds a cetain citical value. Less influence on the solution has ode accuacy of scheme on time as well as the choice of values paametes minmod limite. As an example Figue 2 and Figue 3 show the time pessue dependence in the cental point of the thust wall and a foce of the thust D in the case of blowing cold ai (P = 19.8 atm T = 3K) Pa atm DN tms Figue 2. Dependence on time of pessue in the cental point of a thust wall at T =3К t ms Figue 3. Dependence on time the thust foce at T =3К. The oscillation amplitude is significantly inceased at blowing heated ai imitating by combustion poducts of stoichiometic ai-acetylene mixtue. On Figue 4 and 5 (simila to Figue 2 and 3) data fo T =3К ae pesented. 25 th ICDERS August Leeds 4

5 Levin V.A. Non-stationay Gas Flow in Annula Nozzle Pa atm D N t ms Figue 4. Dependence on time of pessue in the cental point of a thust wall at T =3К..5 1 t ms Figue 5. Dependence on time the thust foce at T =3К. Fouie analysis of time seies fo a numbe of paametes of the numeical solution shows that the spectum of the fundamental fequency at T = 3 K is in the ange of 2-3 khz. That coesponds fistly to the expeiments esults in measuing the thust and pessue which ae made in the aeodynamic pulse setup [1] and pesented in Figue 4 and 5 and secondly estimations based on the sound velocity and the adius of the annula nozzle. Figue 6. Dependence on time of pessue in the cental point of a thust wall (bottom beam) T=3К P=7.1 atm. Sweep velocity 5 ms/div. Nozzle was blown by ai. Figue 7. Dependence on time of pessue in the cental point of a thust wall at T=3К P=2.2 atm. Sweep velocity 2 ms/div. Nozzle was blown by C2H2 combustion poducts. These fequencies ae two odes of magnitude lowe than the oscillation fequency of the scheme oigin. When T = 3 K this ange is extended to 1 khz. Paametes of diffeence gid and numeical scheme as well as the ode of appoximation of the time deivatives ae affected on the spectal chaacteistics of the solutions appeciably. 25th ICDERS August Leeds 5

6 Non-stationay Gas Flow in Annula Nozzle Vaiation of the numeical scheme paametes consideably smalle influence on aveage values of the device powe chaacteistics. Fo compaison the steady-state value fo the fist ode scheme fo the spatial coodinates is equal of 1.28 atm fo the second ode scheme on a gid 2x376 with a lage step nea the thust wall suface atm the mean value on the same gid with minimal step atm. The coesponding values foces of the thust ae and 2471 N. Conclusions As a esult of a numeical study of the flow in the annula nozzle discoveed undamped quasipeiodic flow egimes with significant pessue pulsations on the suface thust wall simila to those obtained in the expeiments. Oscillation amplitude inceased significantly with inceasing the stagnation tempeatue of the blown gas. These modes ae obtained in calculations using the schemes of second ode accuacy in the spatial vaiables on the gid with sufficiently detailed definition of the bounday laye. Time scans pessue values depend stongly on the paametes of the numeical scheme. Significantly less affected by the vaiation of these paametes on the mean powe chaacteistics of the annula nozzle. The computational and expeimental eseaches pesented in the wok ae suppoted by RSF poject and accodingly. Refeences [1] Levin VA Peezhogin VN Khmelevsky AN. (1995). Featue of combustion poducts flow stuctue in spheical semi-open cavity. Combustion Explosion and Shock Waves. 31: 3. [2] Levin VA et al. (212). Gas dynamic and thust in the exhaust system of jet engine with an annula nozzle Combustion Explosion and Shock Waves. 48: 46. [3] Levin VA et al. (213). Expeimental and Numeical Simulation of the Flow in a Diving Module with an Annula and Linea Double-Slot Nozzle. High Tempeatue. 51: 681. [4] Levin VA et al. (213). Investigation an annula nozzle on combustion poducts of hydocabon fuels. Themophysics and Aeomechanics. 2: 265. [5] Levin VA Nechaev JN Taasov AI. (21) A new appoach to oganizing opeation cycles in pulsed detonation engines. In: High-Speed Deflagation and Detonation: Fundamentals and Contol. Eds. G.D. Roy S.M. Folov R.W. Netze and A.A. Boisov. Moscow 21 ELEX- KM Publishe p (ISBN ). 25 th ICDERS August Leeds 6