COMPUTATIONAL STUDY OF SUPERSONIC FLOW THROUGH A CONVERGING DIVERGING NOZZLE

Transcription

1 Engneerng e-transacton (ISSN ) Vol. 6, No. 1, June 011, pp 37-4 Onlne at Receved 5 October, 010; Accepted 30 December, 010 COMPUTATIONAL STUDY OF SUPERSONIC FLOW THROUGH A CONVERGING DIVERGING NOZZLE M.S.U. Chowdhury 1, J.U. Ahamed, P.M.O. Faruque 1 and M.M.K. Bhuya 1, 1 Department of Mechancal Engneerng Chttagong Unversty of Engneerng and Technology (CUET) Chttagong , Bangladesh Department of Mechancal Engneerng Unversty of Malaya, Kuala Lumpur, Malaysa Emal: amal93@yahoo.com ABSTRACT Computatonal soluton has been obtaned for Supersonc Flow through a Convergng Dvergng Nozzle. Varous characterstcs of compressble flud flow through nozzle s analyzed and determned. The nozzle geometry s assumed as crcular and axsymmetrc and flow as two dmensonal flow. Dscredted equaton s formed by dvdng the geometry nto 0 50 meshes. Ideal gas s assumed as the workng flud & Iteraton s done untl convergence. The varatons n statc pressure are decreasng gradually and consequently the velocty of the flow s ncreased. The varaton of velocty, Pressure, Temperature s determned along the length of the nozzle and plotted ts contour. Two dmensonal double precson (-DPP) s used for the analyss of the geometry. package very easly. Convergng dvergng nozzle s one of the most mportant devces used n all supersonc vehcles (Chma, 010). Moreover we are very much nterested to ncrease the speed more and more so t s mportant to carry out research on convergng dvergng nozzle. CFD made the research easer. The control volume, nozzle materal, ntal velocty of flud, geometry s to be selected frst for the analyss of back pressure, velocty vector, propertes, Mach number, pressure dstrbuton of supersonc flow through nozzle. Computatonal study for only U and V components are calculated here. The convergng dvergng geometry s shown n Fg.1. Key words: Convergng dvergng nozzle; Control volume; Back pressure; Mach number; Velocty vector, Contour. 1. INTRODUCTION Computatonal Flud Dynamcs (or CFD) s the analyss of systems nvolvng flud flow, heat transfer and assocated phenomena such as chemcal reactons by means of computer-based smulaton (Ferzger and Perl, 00; Srram, 009; Cherrared et al., 008). It s becomng very popular to solve flow problems wthout dong any experment. So t s very economcal and tme savng. By the development of hgh speed computer, there has been phenomenal growth n use of computer for the smulaton of the flow system. We can now dspense wth expermental methods n many cases even n aerospace desgn. Before, we have to analyze the flow system by manual dscretzaton and solved t by codng n computer language. But now t s easer to smulate the flow problem by software Convergng Fg.1 Convergng-dvergng nozzle geometry. CO-ORDINATE SYSTEM Dvergng Cartesan co-ordnate system s used for drawng the geometry of the nozzle. Moreover the geometry s assumed as two dmensonal and x- drecton y- drecton velocty u and v respectvely (Rahman et al., 010). To draw the geometry of the nozzle some parameter s to be assumed. For analyss, the whole structure s to be dvded nto small fragment called cell. The smaller the cell sze the fner the analyss. Each cell s composed of sx faces n case of three dmensonal analyses or one face n case of n case of 37

2 two dmensonal analyses. Agan each face s composed of four edges and each edge s created by onng pont of two nodal ponts as shown n Fgure (Anderson, 1995, 1990). D U (1) Dt x Momentum Equaton: U t U P U g () x x x I II III IV V Where, U x U x 3 U x k k I. Local change wth tme II. Momentum convecton III. Surface force IV. Molecular - dependent momentum exchange V. Mass- force Energy Equaton Fg. Process of meshng the geometry 3. GOVERNING EQUATIONS CFD s based on the fundamental governng equatons of flud dynamcs. The Equatons are contnuty, momentum, and energy equatons. To fnd out the propertes of the flow system the governng equatons are appled n every node (Ahmed et al., 010; Zafar, 003). Usng governng equaton the whole control volume s dscretzed and then the governng equatons are solved usng boundary condtons. Fnte volume method s used to solve the dscretzed equaton then ths flow system s solved on the bass of densty, because the densty-based formulaton may gve t an accuracy (.e. shock resoluton) advantage over the pressure-based solver for hgh-speed compressble flows (Labworth, 00). Applyng the mass, momentum and energy conservaton, assumng two-dmensonal, steady, compressble flow of varyng densty the governng equatons are: Contnuty Equaton: T c t Where, T U cu p x x x I. Local energy change wth tme II. Convectve term III. Pressure work IV. Heat flux (dffuson) T U ---- (3) x V. Irreversble transfer of mechancal energy nto heat These above governng equatons are appled over the control volume and then solved by proper selecton of nput condtons: flow system, solver selecton, boundary condtons, grd sze, edge dvson etc (Labworth, 00). 4. METHODOLOGY Snce the nozzle has a crcular cross-secton, t s reasonable to assume that the flow s ax-symmetrc and the geometry created to be two-dmensonal (Moner et al., 00; Cusdn and Muller, 005). ; Where, rx Now A r cross-secton at a dstance x s the radus of the 38

3 And A 0.1 x s assumed for generatng the nozzle geometry, then for the gven nozzle geometry, we get r 0.1 x 0.5 x ; 0.5 x (4) Ths s the equaton of the curved wall. GAMBIT s used for generatng the geometry. The ponts of the curve are connected by NURBS arc (Gambt, 004). The Nozzle geometry s drawn by usng the followng data shown n Table 1. Meshed edges, control volume and curved edges of nozzle are shown n Fgures 3, 4 and 5. Table 1 Data for Nozzle geometry x Fg. 5 Mesh control volume Boundary types for each of the edges are specfed n the Table. The nput condtons have to be defned one by one as: defnng solver, defnng the vscosty effect, defnng Energy Equaton, defnng Flud Propertes, (let deal gas), defnng Operatng Condtons, defnng Boundary condtons, defnng Equaton Type, Intalzng nlet propertes (Pressure = Pa; Axal velocty = ; Temperature = ), defnng convergence crtera(let the soluton wll be conversed at 10e -6 ), defnng number of teraton( let teraton number s 500) etc. s to be done. Table Boundary types of edges Edge Poston Name Type Left nlet PRESSURE_INLET Rght outlet PRESSURE_OUTLET Top wall WALL Bottom centerlne AXIS Fg. 3 Curved edge of the nozzle. The resduals of the teraton are prnted out as well as plotted n the graphcs wndow as they are calculated. Snce the teraton converges wthn 10-6 value shown n Fg. 6. So our crtera for solvng the problem are correct. 5. RESULTS AND DISCUSSIONS 5.1 Centerlne Velocty The varaton of the axal velocty s plotted along the centerlne as shown n the Fg. 7. Fg. 7 shows that the velocty of the centerlne s ncreasng gradually and t s maxmum at the ext of the nozzle.e. the velocty turnng from subsonc to supersonc gradually. Fg. 4 Meshed edges 39

4 5. Centerlne Pressure The varaton of the axal pressure s plotted along the centerlne and presented n the Fg. 8. The fgure shows that the pressure of the centerlne s decreasng gradually and t s the mnmum at the ext of the Fg. 6 Convergence dagram wth teratons Fg. 7 Centerlne velocty change wth respect to poston. 40

5 Fg. 8 Statc pressure changes wth respect to poston. 5.3 Vector Dsplay Fg. 9 shows the change of velocty along the flow drecton and red color represents the supersonc flow. The scale on the left of the Fg. 9 represents the value of temperature, pressure, velocty n the respectve fgure along left to rght of the nozzle. Fg. 9 Velocty vectors vs. poston colored by velocty. 6. CONCLUSIONS CFD reduces tme as well as cost of producton of flud dynamcs related products. It abates our expermental cost. Any flud flow and can be analyzed very easly. All the arcrafts (whose velocty s more than the velocty of sound) requres ths type of smulaton else the cost of producton of supersonc arcrafts wll be hgh and very much complcated. Based on the smulaton of supersonc flow through nozzle the followng concluson may be drawn: 1. The range of horzontal axs s taken from to +0.5 because of smplfyng the drawng as well as smulaton. 41

6 . Soluton s conversed at around 140 teraton, resduals s gnored after 10-6 value. 3. The velocty profle for the flow s sketched n X-Y plane. The profle gves us nformaton about the ncrement of the velocty n the rght sde of the nozzle. 4. The pressure profle shows that t reduces along the rght sde and the back pressure s around 1500 Pascal. Whch means that the pressure at ext of the nozzle must be 1500 else t wll not act as a nozzle.e. the flow wll not be supersonc. 5. If Mach number exceeds the value of 5, t creates hgh temperature whch causes the chemcal change of the flud. The flow through a convergng-dvergng nozzle s the mportant problems used for modelng the compressble flow for computatonal flud dynamcs. Occurrence of shock n the flow feld shows one of the most promnent effects of compressblty over flud flow REFERENCES Ahamed, J.U., Bhuya, M.M.K, Sadur, R., Masuk, H.H., Sarkar, M.A.R., Sayem, A.S.M., Islam, M Forced convecton heat transfer performance of porous twsted tape nsert. Engneerng e-transacton (ISSN ), 5(), Anderson, J Modern Compressble Flow: Wth Hstorcal Perspectve", nd Edton, Mc Graw Hll, New York, Anderson, J.D Computatonal Flud Dynamcs, McGraw-Hll Scence/Engneerng/Math; 1st edton, p. 574, ISBN / Cherrared, D., Saad, B., Glmar, M. and Rabah, D D modelsaton of streamwse necton n nteracton wth compressble transverse flow by two turbulence models. J. Appled Sc., 8, Chma, R.V., 010. Coupled analyss of an nlet and fan for a quet supersonc et, AIAA Paper: Also NASA/TM Cusdn, P. and Müller, J.D On the Performance of dscrete adont CFD codes usng automatc dfferentaton, Int. J. Num. Meth. Fluds, 47, Gambt, 004. A Tutoral Gude for creatng and meshng basc geometry, Fluent Incorporated. Ferzger, J.H. and Perl, M. 00. Computatonal methods for flud dynamcs, Sprnger, 3 rd revsed edton, p.18. Lapworth, B.L. 00. Challenges and Methodologes n the Desgn of Axal Flow Fans for Hgh Bypass Rato Gas Turbne Engnes usng Steady and Unsteady CFD: Advances of CFD n Flud Machnery Desgn, Elder Edton., Tourldaks and Yates: Professonal Engneerng Publshng. Moner, P., Muller, J.D. and Gles, M.B. 00. Edgebased multgrd and precondtonng for hybrd grds. AIAA Journal, 40(10), Rahman, M.., Nasrn, R., Bllah, M.M Effect of prandtl number on hydromagnetc mxed convecton n a double-ld drven cavty wth a heat-generatng obstacle, Engneerng e- Transacton, (ISSN ), 5(), Srram, M.A., Raan, N.K.S. and Kulkarn, P.S Computatonal Analyss of Flow through a Multple Nozzle Drven Laser Cavty and Dffuser, Computatonal Flud Dynamcs, 9, Zafar, M.A Numercal soluton of the flow feld n a channel wth porous meda. MSc Thess, Department of Mechancal Engneerng, Chttagong Unversty Engneerng and Technology (CUET), Bangladesh, Bangladesh, p