Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 UDK - UDC 62-225.5:532.525 Kratki znanstveni prispevek - Short scientific paper (1.03) Vzporedni vplivi pospeška in površinskega ogrevanja na stisljiv tok: simulacija vesoljske pogonske šobe s srednje veliko površinsko obrabo Parallel Effects of Acceleration and Surface Heating on Compressible Flow: Simulation of an Aerospace Propulsion Nozzle with a Medium Amount of Surface Wear A. Alper Ozalp (Uludag University, Turkey) Numerične simulacije vesoljskih pogonskih šob so, zaradi nujnosti sočasnega obravnavanja pospeška toka, stopenj prenosa toplote, hrapavosti površine, temperaturno odvisnih lastnosti zraka in sprememb gostote tokovnic zaradi stisljivosti toka, zelo zapletene. Da bi zagotovili pregled za večstransko obravnavo toka pogonskih šob, smo razvili nov računski model, ki vključuje osnosimetrično zveznost, vztrajnostne in energijske enačbe. Izvedli smo računske preizkuse z različnimi geometrijskimi oblikami šob in vstopnimi robnimi pogoji ter s skupno obravnavo površinskega toplotnega toka in hrapavosti. Izračuni so pokazali, da se vstopna moč pogonske šobe in njene izgube povečujejo z večanjem notranjega statičnega tlaka ter zmanjšujejo z zoževalnim kotom šobe in površinskim toplotnim tokom. Ugotovili smo, da je razmerje izgub glede na vstopno moč neodvisno od toplotnega toka, vendar pa se linearno zmanjšuje s povečanjem zoževalnih kotov. © 2007 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: pogonske šobe, stisljivi tok, pretočni koeficient, izgube moči) Numerical simulations of aerospace propulsion nozzles are very complex due to the necessity to simultaneously handle flow acceleration, momentum heat-transfer rates, surface roughness, temperature-dependent air properties and streamwise density variations due to the compressible character of the flow. To provide an overview for a multitask consideration of the propulsion-nozzle flows, a new computational model that integrates the axi-symmetrical continuity, the momentum and the energy equations has been developed. Numerical experiments were performed with various nozzle geometries, inlet-boundary conditions, with the combined handling of the surface heat flux and roughness conditions. The computations indicated that the input and loss power values of the propulsion nozzle increase with higher inlet stagnation pressures and decrease with higher nozzle convergence half angles and surface heat flux. The ratio oj the loss to the input power was found to be independent of the heat flux; however, it decreases linearly with an increase in the convergence half angles. © 2007 Journal of Mechanical Engineering. All rights reserved. (Keywords: propulsion nozzle, compressible flow, discharge coefficient, power losses) 0 INTRODUCTION Compressible flows are encountered in a wide variety of engineering applications, e.g., the flow accelerator of environmental control systems in commercial aircraft [1] which supplies fresh air to the passenger cabins of aircraft and the exhaust system of nuclear propulsion engines [2], which generate energy and thrust. Most aerospace applications are equipped with nozzles such that the overall system performance is significantly influenced by the flow acceleration, the surface heating, the inlet conditions and the wear-based friction. Recent studies have pointed out the considerable influence of high pressures and temperatures on the frictional behaviours of nozzle flows. In spite of the objective, an accurate : 3 Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 prediction in the design-oriented calculations of compressible flows is still a challenging task that is becoming increasingly important. The main design considerations for compressible-flow applications with nozzles are the flow geometries, the inlet-boundary conditions and the flow heat-transfer characteristics, where the performance predictions are reported by several experimental and numerical investigations. The effect of nozzle-exit over-pressure on vortex formation, with its contribution to nozzle thrust, was experimentally examined by Krueger and Gharib [3]. Significant losses in efficiency, due to heat transfer, especially when the ratio of the inlet stagnation to the back pressure converges to unity was determined by Lear et al. [4], who modelled the dissipative effects of heat transfer on the exit kinetic energy and the nozzle efficiency. Orieux et al. [5] illustrated the steady and transient performance of micro-nozzles for various nozzle geometries, ambient conditions and surface cooling, where the thrust values decreased both with cooling and with a narrower nozzle exit. The heat transfer and gas dynamics structure in a choked nozzle with cooling was experimentally investigated by Back et al. [6]. Instabilities in the propulsion of rockets, due to pressure and temperature fluctuations at the upstream of the rocket nozzle and due to the flow geometry, were numerically considered by Assovskii and Rashkovskii [7]. Bartz [8] handled the heat-transfer phenomena in compressible nozzle flows and considered the Nusselt number as a function of the inlet stagnation pressure and the convergence half angle, and Ahmad [9] correlated the variation of the nozzle discharge coefficients and surface heat-transfer values for various nozzle geometries. A 10o convergence half-angle nozzle with different working fluids and with a wide range of inlet stagnation pressures was experimentally considered by Massier et al. [10], who recorded lower discharge coefficients with a decrease in the inlet stagnation pressure. Paik et al. [11] studied the influence of flow geometry and Reynolds number on the variation of the discharge coefficients for sonic nozzles that are applied to gas flow-rate measurements, and reported higher discharge coefficients with an increase in the mass flow rate. Kim et al. [12] considered the effects of several kinds of gases and turbulence models with a wide range of Reynolds numbers on different sonic nozzle geometries. The combined effects of Reynolds number, area ratio and flow velocity on the critical pressure ratio of sonic nozzles were investigated by Park et al. [13]. Sato et al. [14] presented recent data on a real-time air-cooled propulsion ramjet engine. Ribault and Friedrich [15] investigated compressible flow behaviour along adiabatic and cooled walls by implementing the turbulent momentum and heat-transport analogies in a code. Although the available literature is highly concentrated on heating/cooling applications, inlet/ exit conditions and the geometrical structures of the nozzles, surface roughness (e) is becoming of major interest for compressible/incompressible nozzle flows. Gas-solid particle flows in the nozzles together with the high pressures and temperatures within the flow volume are the main sources of augmentations in the surface roughness. Kumar et al. [16] performed an experimental study of nozzle wear due to gassolid particle flow and determined an increase in the relative roughness (e/Din) values from 0.006 to 0.052. Bussiere and Mora [17] presented the real-time data of an Ariane 5 rocket-booster nozzle, where the relative roughness increased from a perfect surface finish to 0.012 during a flight that initiates with a launch and ends with the rocket in orbit. Although the surface roughness and the surface heat flux act simultaneously in real-time systems, the available literature deals with them separately. The combined effects have not yet been considered. To perform a comprehensive computational study, a new mathematical model, capable of implementing both the surface roughness and the surface heat flux (Q) conditions for aerospace propulsion nozzles was developed. Choked and un-choked cases were investigated for various convergence half angles (a), ratia of inlet stagnation to back pressure (b=(Po)in/Pb) and Q cases, and the proposed method was validated with the previous experimental and numerical reports. 1 MODELLING AND COMPUTATION The overall aim here is to build a predictive model for propulsion-nozzle flows in the presence of surface roughness and constant heat-flux conditions. The model should permit the determination of the necessary design parameters, such as the nozzle geometry and the inlet-boundary conditions for any given performance requirement depending on the applications. Thus, the main requirements are adaptability, simplicity and a short calculation time. The calculations rely on the principles of mass and en- 4 Ozalp A. A. Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 Stagnant Air at (To). (PJ m s v 'in Fig. 1. Schematic outline of the aerospace propulsion nozzle ergy conservation and on the momentum and state equations applied to the control volume, given in Fig. 1. It is assumed that the stagnation conditions of pressure and temperature in the storage tank, upstream of the nozzle, are homogeneous and, as in many numerical studies [2] to [13], the air velocity, pressure and temperature are considered to be uniform across any section normal to the flow axis. Since air properties, like the specific heat at constant pressure (C ), the kinematic viscosity ( n) and the Prandtl number (Pr), are substantially dependent on temperature (T) [18], they are characterized by 6th-order polynomials with an uncertainty of less than 0.02%, and the temperature dependency is indicated by the superscript T throughout the formulation. As the study is focused on flows with friction and heat transfer, the stagnation pressure (P ) o and stagnation temperature (T ) values will also vary in the flow direction with the variation in the Mach number (M). Thus, the conventional equations (Eqs. 1 and 2) for compressible, isentropic and one-dimensional flows are applicable only with the simultaneous handling of the momentum and energy equations. ,P\ , 1+ Mi2 2 g -1 2 1+ Mi 2 m = riUiAi Ui=MigRTi P ri ( ReDn RTi UD (1) (2) (3) (4) (5) (6) The nodal values (subscript i) of the mass . flow rate ( m ) the air velocity (U) and the density (r) can be calculated using Eqs. 3 to 5, where the mass flow rate, the most significant consideration from a numerical point of view, is kept constant in the flow direction. On the other hand, the diameter (D), based the Reynolds number (Eq. 6), is assigned to each differential cell with the mean cellular values of U, D and n. The friction coefficient (f) is a function of both ReD and e (Eq. 7), and the cell-based (subscript n) shear stress (t) and friction force (Ff ) can be expressed with Eqs. 8 to 9. 1 -3.6 log 6.9 eD (ReD)n { 3.7 r (^ (7) (8) (9) (Ff)n =tnp DnDxn The one-dimensional momentum (Eq. 10) and energy equations (Eq. 11) are applied to each differential cell in the nozzle, where the nodal properties, such as P, U and Cp, are interrelated with the contributions of cellular variants like Ff and the impulse (I). Eq. 11 represents the conservation of mechanical and thermal energy by the implementation of a cell-based surface flux and the frictional loss term. (10) Pi Ai +m. Ui = Pi+1 Ai+1 +m. Ui+1 +(Ff )n + In ()i U22 Q(As ) ()i +1 U22 (Ff )U n CPT Ti+ i + . n =CPT Ti+1+ i+1+ n m (11) Vargas and Bejan [1] evaluated heat-transfer data in their mathematical model for a compressible nozzle flow, where the Mach number was in the range Vzporedni vplivi pospeška in površinskega ogrevanja - Parallel Effects of Acceleration and Surface Heating 5 Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 tion data (P , T ), were calculated at the nodes of these cells, which are numbered from i=1 to n+1, whereas r, I, ReD and NuD were evaluated on a cell basis using the mean values of nodal inlet and exit data of each cell. As shown in Fig. 2, by disregarding the surface roughness and heat transfer, the flow of the solution logic first handles the problem as an isentropic type, which is manipulated as described by Laney [20]. The M value of the isentropic approach is the initial guess of the iterative solution procedure of the non-isentropic nozzle flow. The non-isentropic approach governs the complete equation set described above; however, if the solution scheme encounters singularities, like M>1, M =1 i ex & P

0.3 m (a=9o) is due to the higher acceleration rates of the flowing air, which can also be seen in Fig. 3. The streamwise variations of the fluid (Tf ) and surface (Ts) temperatures are presented in Fig. 5. As a consequence of the applied surface heat flux, the Ts values are above Tf for the complete a and b ranges and also throughout the flow volume. On the other hand, as the Ts values decrease in the streamwise direction, the opposite is true for Tf in all cases. The decrease rates of Ts become more significant towards the nozzle exit, especially for x>0.45 m, where the highest flow acceleration is determined for all the nozzles. Lower b and higher a indicates lower mass flow rates, which constitutes higher Ts and Tf, especially at the nozzle inlet. The durability of the nozzle material is directly related to the Ts, and the computations suggest the upstream nozzle sections should be carefully considered. On the other hand, surface wear is connected with Tf, and the vital regions appear towards the downstream regions of the nozzle, specifically at the exit plane. Fig. 6 presents the streamwise variations of surface heat-transfer rates for various convergence-half-angle and pressure-ratio cases, with the application of constant surface heat-flux values of 800 and 1600 kW/m2. The Nusselt numbers (NuD) were observed to increase in the flow direction for the complete set of investigated systems; however, a and Q appeared to cause the NuD to decrease in both the choked (b=2.00) and un-choked (b<2.00) cases. This outcome is highly dependent on the fact that narrower nozzles and higher heat-flux values contributed to lower mass flow rate values (Fig. 3), and thus M and U, which also decrease the amount of heat swept from the nozzle wall. On the other hand, the ratio of the exit to inlet Nusselt numbers l=(NuD)ex/(NuD)in increases with b and a, whereas it decreases with Q. Bartz [8] also reported increased l ratia with higher a and b; moreover, the typical report of Ahmad [9] for a nozzle with a=45o is l=3.9. The most significant ratio, evaluated in the present Fig. 4. Streamwise variation of non-dimensional pressure with various a, b and Q conditions 8 Ozalp A. A. Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 i kW/ii]2 — •:!;-. *:TS ).(Q = 1600 kW/m2 Fig. 5. Streamwise variation of flow and surface temperature with various a, ß and Q conditions kW/m2. Fig. 6. Streamwise variation of Nusselt number with various a, ß and Q conditions paper for the choking case of c=9o, is /1=2.70, and this is higher than the corresponding report of Back et al. [6] (/1=1.61) for the nozzle with c=15o. Fig. 6 further suggests that the application of different heat-flux values causes NuD to vary both at the inlet and exit planes of the nozzle. The most significant variations are recorded for the y?case of 1.01 for the complete a set, where these intervals are ±9.4% (inlet) and ±19.8% (exit) for o=3o and ±11.8% (inlet) and ±28.1% (exit) for the a=9o case. These values additionally imply that the effect of the wall’s heat flux on surface heat-transfer rates becomes more significant in higher a cases, and thus in tasks with lower mass flow rates. The combined effects of a, ReD and Q conditions on the discharge coefficient is given in Fig. 7a for the choked case of ß=2.00. It can be seen from Fig. 7a that the isentropic values are not only in agreement with the ISO 9300 (Paik et al. [11]) standardized correlation of C=f((ReD )J for choked nozzles but also with the experimental reports of Massier et al. [10] and Kim et al. [12]. A higher «produced a lower exit ReD, where the lower ReD and higher Q are accompanied by reduced values of Cd and point to lower mass flow rates, which show parallelism with the Mach-number variations of Fig. 3 and are similar to the reports of Paik et al. [11]. C = 0.2636+ 4.6279-10"2 ln ( Re„)„ (14) (15) Q=r1.0106 + 0.0074(lna)2 The adiabatic (g=0) correlations (Eqs. 14 and 15) of Ahmad [9] produce higher Cd values for the present a and ReD intervals; the gap is the outcome of the applied heat flux values; however, the gap decreases with lower a and with higher (ReJ^ values. The numerical results show, particularly for the ß=1600 kW/m2 case, that the nozzles of o=3o, 6o, 9o result in Rvalues of 0.935, 0.933 and 0.93 respectively, where Kim et al. [12] also reported a lower Cd with higher convergence half angles. The input power ( f) necessary to form the compressible flow within the nozzle and the amount of power loss ( flo) are the main considerations from the point of view" of the energy requirements to run the propulsion nozzles. Fig. 7(b) demonstrates that both fand ^ increase with higher y?and lower a, which indicates that the amount of air directed towards the nozzle is the characteristic design parameter. Increasing the surface heat flux from 800 to 1600 kW/m2 caused a decrease in >Fby 2.7% for all a Vzporedni vplivi pospeška in površinskega ogrevanja - Parallel Effects of Acceleration and Surface Heating 9 Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 kW/m2 kW/in2, Q,=16Ô0 kW/m rlsentropic Cases) (a) (b) Fig. 7. Variation of (a) discharge coefficient for the choked flow and (b) input and loss power values with various a, b and Q conditions values with b=2.00, where this influence is slightly lower than the Cd gap among the identical cases shown in Fig. 7a. However, the effect of Q on Yloss definitely varies with a; as the decrease rate for a=3o is 2.9% the corresponding values for a=6o and a=9o are 1.4% and 0.8%, respectively. Since the main source of Yloss is Ff (Eq. 9), Fig. 7b further suggests that Q also has an opposite effect on Ff, which is in agreement with the reports of Ribault and Friedrich [15], who determined higher friction-coefficient values (Eq. 7) with surface cooling. Lastly, the Yloss/Y ratio is dependent on both a and b and appears to be independent of Q: for b=1.01 the ratio is equal to 0.03% for the complete a set, whereas for b=2.00 the ratio exhibits a linear style and has values of 1.3%, 1.1% and 0.9% for a of 3o, 6o and 9o, respectively. 3 CONCLUSION A computational method for an investigation of compressible flow and heat-transfer characteristics in aerospace propulsion nozzles was developed. The model is capable of handling various flow geometries and inlet-boundary conditions, together with the simultaneous application of surface heating and roughness conditions. The main conclusions from the numerical experiments can be summarized as follows: Surface heating produces lower inlet Mach numbers but higher inlet non-dimensional pressure values; moreover, the effect of Q on both of the values is more apparent in flows with lower acceleration. The heat flux, especially in cases with high acceleration, directly increases the stagnation pressure and causes augmented static pressure values, which is remarkable from the point of view of surface wear and system damage, specifically for the nozzle section of 0.25 to 0.40 m. Moreover, the point of maximum shifts downstream in the flows with lower b and a reduced mass flow rate. The Nusselt numbers decrease with lower inlet stagnation pressures and with higher convergence half angles and heat-flux conditions; moreover, the effect of surface heat flux on the Nusselt numbers is more apparent in un-choked flows. The maximum values of Ts are recorded in the downstream nozzle sections, which is the very important for the nozzle material; and the decrease rates of Ts become significant towards the nozzle exit, especially for x>0.45 m, where the highest flow acceleration is determined for all the nozzle geometries. The augmentation of Tf in the flow direction is accompanied by the energy-transfer mechanism and also with the contribution of a lower Ts, and labels the exit-plane neighbourhood as vital from the point of view of wear. 4 NOMENCLATURE A cross-sectional area, m2 Ff frictional force, N Cd discharge coefficient I thrust, N Cp specific heat at constant pressure, J/kgK L nozzle length, m D nozzle diameter, mm m mass flow rate, kg/s f skin friction factor M Mach number 10 Ozalp A. A. Strojniški vestnik - Journal of Mechanical Engineering 53(2007)1, 3-12 Nud Nusselt number n kinematic viscosity, m2/s P pressure, Pa r density, kg/m3 Pr Prandtl number t shear stress, Pa Q surface heat flux, W/m2 Y power, kW R gas constant, J/kgK ReD Reynolds number Subscripts T temperature, K b, o back, stagnation U air velocity, m/s D diameter X streamwise direction, m ex, in exit, inlet i, n node, cell number Greek Letters loss loss power a convergence half angle, deg s heat-transfer surface b ratio of the inlet stagnation to the back pressure Superscripts e surface roughness, mm T temperature dependency g specific heat ratio __ cellular average 5 REFERENCES [I] Vargas, J.V.C., A. Bejan (2001) Thermodynamic optimization of finned crossflow heat exchangers for aircraft environmental control systems. Int J Heat Fluid Flow 22(2001), pp. 657-665. [2] Kammash, T., T. Godfroy (1997) An open cycle gas core fusion rocket for space exploration. Acta Astronautica 41(1997), pp. 229-237. 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Canale (1990) Numerical methods for engineers. McGraw Hill, Singapore. [20] Laney, C.B. (1998) Computational gasdynamics. Cambridge University Press, Cambridge. [21] Wu , J.S., K.C. Tseng (2001) Analysis of micro-scale gas flows with pressure boundaries using direct simulation Monte Carlo method. Computers and Fluids 30(2001), pp. 711-735. Author’s Address: Prof. Dr. A. Alper Ozalp Uludag University Department of Mechanical Eng. 16059 Gorukle, Bursa, Turkey aozalp@uludag.edu.tr Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 27.4.2005 25.10.2006 Received: Accepted: Open for discussion: 1 year 12 Ozalp A. A.