GT ANALYTIC ASSESSMENT OF AN EMBEDDED AIRCRAFT PROPULSION

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1 Proceedings of ASME Turbo Expo 6: Turbomachinery Technical Conference and Exposition GT6 June 3-7, 6, Seoul, South Korea GT6-575 ANALYTIC ASSESSMENT OF AN EMBEDDED AIRCRAFT PROPULSION Peter F. Pelz Ferdinand-J. Cloos Chair of Fluid Systems Technische Uniersität Darmstadt Darmstadt, Hesse, 69 Germany Jörg Sieber MTU Aero Engines AG München, Baaria, 995 Germany ABSTRACT This paper inestigates analytically the adantage of the embedded propulsion compared to a state of the art propulsion of an aircraft. Hereby, we are applying the integral method of boundary layer theory and potential theory to analyse the boundary layer thickness and the impact of the flow acceleration due to the embedded propulsion. The aircraft body is treated as a flat plate. The engine is treated as a momentum disc but there is a trade off, since the engine efficiency is effected by the boundary layer. The outcome of the energetic assessment is the following: the propulsion efficiency is increased by the embedded propulsion and the drag of the aircraft body is reduced. The optimized aircraft engine size depending on Reynolds number is gien. NOMENCLATURE a Induction factor. A Area. c Absolute elocity. c f Friction coefficient. c f Local friction coefficient. C Constant. e x Unit ector in x-direction. f + Friction coefficient. H Stream tube height. L Aircraft body length. Address all correspondence to this author. n Exponent of the elocity power law. p Pressure. P s Shaft power. R Gas constant. Re Reynolds number. S Thrust. t Stress ector. T Temperature. u Flow elocity within the boundary layer. u Friction elocity. U Flow elocity outside the boundary layer. Flying speed. W Body drag force. W f Friction force. x Axial coordinate. y Wall coordinate. α p Shaft power ratio. α w Friction force ratio. γ Isentropic exponent. δ Boundary layer thickness. δ Displacement boundary layer thickness. δ Momentum boundary layer thickness. ζ Dimensionless constant. η Efficiency. η Fr Propulsion efficiency Froud. κ Slenderness ratio. Dimensionless constant. κ Copyright c 6 by ASME

2 κ λ ν ρ τ w φ Dimensionless constant. Dimensionless constant. Kinematic iscosity. Air density. Wall shear stress. Flow potential. flow of the present theory is possible. Thus, the principle approach does not change for a compressible flow. By doing so, we yield two configurations; see Fig.. For the conentional propulsion representing the reference case, a pylon connects the aircraft engine with the aircraft body. For the embedded propulsion the aircraft engine is embedded in the aircraft body. INTRODUCTION NOSE TAIL AIRCRAFT BODY a CONVENTIONAL PROPULSION NOSE AIRCRAFT BODY TAIL W fo + W U = U = + c x L PYLON S o AIRCRAFT ENGINE b EMBEDDED PROPULSION AIRCRAFT ENGINE U = H y hx δ x Ux ux, y S U = + c H H U = + c x W f x = L W FIGURE. SCHEMATICAL DESCRIPTION OF THE BOUND- ARY LAYER AND STREAM TUBE FOR a THE CONVENTIONAL AND b THE EMBEDDED PROPULSION. FIGURE. EMBEDDED PROPULSION OF AN AIRCRAFT. Figure shows an aircraft engine embedded in the aircraft body. This set-up is one possibility among others of an embedded propulsion. For this topology there are two contradicting influences: first, the friction resistance W f of the hull will be reduced due to the boundary layer acceleration. Second, the engine efficiency will be negatie effected due to the non-uniform elocity profile at the engine inlet. Both effects are contradicting and can be traced back to the boundary layer thickness δ. For the first effect, the ratio of δ to hull length L will be important. For the second effect, the ratio of δ to the engine size H is releant. Hence, we expect an optimal slenderness ratio κ := H/L as a function of the Reynolds number for the minimal power consumption. This paper discusses the aerodynamic adantage of the embedded propulsion compared to an conentional propulsion by propulsion efficiency analysis. The influence of the propulsion interacting with the boundary layer is inestigated by Tillman, Hardin et al. [, ]. For different aircraft configurations the power balance method for performance estimation is applied by Sato [3] and Drela []. The aircraft body is treated as a flat plate. Following the idea of Rankine 65 [5] and Betz [6] the aircraft engine is modelled as a disc actuator. This approach is only alid for propeller engines. Thus, this analysis neglects the fuel mass flow and the flow is assumed to be incompressible. An extension to a compressible This paper is organized as follows: first, we discuss the axial momentum balance and we introduce the axial induction factors a := c/ and a := c/. Second, the outer flow will be discussed by potential theory. By the third step, we calculate the boundary layer thickness, e.g. the friction drag, with the results from the preious sections and compare the embedded propulsion with the conentional propulsion. At last, the first law of thermodynamics is applied to define the efficiency. By doing so, the optimal engine size is gien by the shaft power ratio of the two topologies. In the closure of this paper, we summarize our inestigation by three major findings. AXIAL MOMENTUM BALANCE For the embedded propulsion the axial momentum balance reads ρu dy ρu UH = W f p H, with the pressure p in front of the disc, the elocities u and U, the height H and the air density ρ. The friction force is W f = e x A k t da, Copyright c 6 by ASME

3 with A k the wetted area of the whole aircraft body. Using the displacement thickness δ, defined as usual as δ := and the momentum thickness δ := the left side of Eqn. yields u dy 3 U u u U U dy, Following the nomenclature of Glauert [7], the ratio of the total jet elocity and the elocity of transport, e.g. flying speed, is a := c = U, the so called axial induction factor of the disc. Close to the disc, the induction factor is definded as a := c = U. With the induction factors, the friction coefficients ρu dy ρu UH = ρu H δ δ ρu UH, 5 c f := W f ρ L, 3 ealuated in front of the disc at x = L. The continuity equation reads f + := c f κ, UH u dy = UH H δ U = 6 and the slenderness ratio κ := H L, 5 at the disc and with the Bernoulli equation p = ρ U U 7 we obtain a + λ a = f for the pressure in front of the disc, the axial momentum balance yields with U λ U + = W f ρ λ := H Hδ + δ H δ = δ +. 9 δ + is approximately the dimensionless momentum thickness neglecting terms higher order, thus δ + δ H. The axial momentum balance for the stream tube from the beginning of the aircraft body behind the disc for the induction factors a and a, reads ρhu U U = S p H, 7 with the pressure p behind the disc. The thrust S equals the sum of the friction force W f and body drag W at constant flying speed The body drag is gien by S = W f +W. W = ρhu U U. 9 3 Copyright c 6 by ASME

4 Using Bernoulli s equation behind the disc to the free jet stream, the pressure behind the disc is p = ρ U U. Following the nomenclature of Glauert, Eqn. 7 yields thus, the induction factor a is gien by a a = f +, FRICTION COEFFICIENT ff a = f + + a. With Eqn. 6 and the induction factors aδ +, f + and aδ +, f + are gien. A reference case without a boundary layer, e.g. δ + = and λ =, the well known Betz solution a = /a is included. For the general case, the induction factors aδ +, f + and aδ +, f + are gien by Fig. 3 and, respectiely..5.. DIMENSIONLESS BOUNDARY LAYER THICKNESS δδ + FIGURE. INDUCTION FACTOR a DEPENDENT ON BOUND- ARY LAYER THICKNESS AND FRICTION COEFFICIENT..3. FRICTION COEFFICIENT ff DIMENSIONLESS BOUNDARY LAYER THICKNESS δδ + FIGURE 3. INDUCTION FACTOR a DEPENDENT ON BOUND- ARY LAYER THICKNESS AND FRICTION COEFFICIENT. FRICTION COEFFICIENT ff + δδ + = DIMENSIONLESS BOUNDARY LAYER THICKNESS δδ δδ + = ff + FIGURE 5. INDUCTION FACTORS RATIO a/a DEPENDENT ON BOUNDARY LAYER THICKNESS AND FRICTION COEFFICIENT. THICK SOLID LINE INDICATES δ + =, DASHED LINE δ + = f +. Figure 5 shows the ratio a/a of the induction factors. The solid line indicates the Betz solution for δ + = and the dashed line indicates δ + = f + ; the solution for a constant pressure boundary layer as we will show in the following sections. For a compressible flow Eqn. has to be replaced by u/ + γ/γ p/ρ = const.. As usual the equation of state p = ρrt is as well needed as well as the energy equation in integral form [, 9]. OUTER FLOW Outside the boundary layer, e.g. the outer flow, the flow is irrotational. There, the elocity field is gien by U = φ. The flow upstream of the disc is described by a superposition of the following three sections: first, the potential of undisturbed approaching flow U x. Second, the line sink at x = L, H < y < H with the sink strength ch. Third, the combination of sinks and sources to consider the boundary layer at the Copyright c 6 by ASME

5 flat plate. Thus, we obtain φ =U x c π + L H U x π ln x L + y y dy +... dδ x dx ln x x + y dx. 3 With the integral method of the boundary layer theory and Eqn. 3 and, the axial momentum balance is dδ dx + du U dx δ + δ = τ w ρu = u U = c f, 9 the so called an Kármán momemtum equation and is alid for laminar and turbulent flow. The elocity of the outer flow is gien by Ux = φ x. y=δx The displacement of the boundary layer influences the elocity less than the propulsion does. Thus, the second term on the ride side of Eqn. 3 for the elocity profile within the boundary layer is negligible. The elocity of the outer flow Ux U a π H depends on the induction factor. x L x L + y dy 5 BOUNDARY LAYER AND DRAG Figure illustrates the boundary layer at the flat plate, e.g. the aircraft body. The drag force is the sum of friction force and body drag; see Eqn.. The friction force is the integral of the wall shear stress L W f = τ w dx. 6 The wall shear stress is τ w := ρu with the friction elocity u and yielding the local friction coefficient Hence, the friction force is c f := τ w ρu = u U. 7 c f := W f ρu L = L L U U c f dx. Reference Case The reference case, e.g. the conentional propulsion Fig., has a constant pressure boundary layer, thus, U = U = const.. Applying Eqn. 9 to the reference case, yields dδ /dx = c f. With an ansatz function for the axial elocity profile within the boundary layer, for example Prandtl s power law the friction coefficient is y n ux,y = Ux, 3 δx U L /5 c f =.7 =.7Re /5. 3 ν Equation 3 is the well known Blasius power law []. The solution of the integral method for boundary layer theory is ery robust against assumed ansatz functions [ ], in our case Eqn 3. Hence, to erify the ansatz function is not necessary. Boundary Layer of the Embedded Propulsion For the boundary layer of the embedded propulsion, we consider the outer flow solution Ux, see Eqn. 5, to sole the on Kármán momentum equation Eqn. 9. Using Eqn. 3 for the elocity profile within the boundary layer, we obtain δ δ := for the displacement thickness and δ δ := u U u y d = n U δ n +, 3 u y d = U δ n n + n +, 33 for the momentum thickness. Thus, δ = n + δ = hδ with h := n +. The choosen power law has to be calibrated to the 5 Copyright c 6 by ASME

6 iscous sublayer [] u yu n = C, 3 u ν with the constant C. For n = /7, the empirical constant C is.7 []. Hence, the wall shear stress is With these deiations and Uν n n+ τ w = ρ. 35 Cδ n δ := ζ δ, 36 with the mean elocity U and the aerodynamic, e.g. isotropic, efficiency η of the embedded propulsion and η of the reference case. Propulsion Efficiency of the Reference Case For a constant flying speed = U, the shaft power for the reference case is P s = S U /η and the Froud efficiency is gien by U η Fr, := S U U = η P s U = η U + c. 3 This is the well known definition of the propulsion efficiency [3]. ζ := n + n +, 37 n Propulsion Efficiency of the Embedded Propulsion For the embedded propulsion we calculate the propulsion efficiency to and κ := n n κ := n +, 39 we obtain the on Kármán momentum equation for the embedded propulsion δ κ dδ dx + [ du U dx + hδ κ + = CU ν n ] κ. ζ Hence, the momentum thickness for the embedded propulsion is δ x =.36 ν U 5 x U dx /5. FRIST LAW OF THERMODYNAMICS AND FROUD PROPULSION EFFICIENCY The shaft power is P s = UH p, η with the thrust η Fr := SU P s = η SU UH p, S = H ρ the pressure difference p = p p = ρ U UU +U, 5 and the dimensionless displacement thickness Thus, the propulsion efficiency is U U + δ + U 6 δ + := δ δ /H H δ /H δ H. 7 η Fr η = U U UU +U U U U + δ +U. Figure 6 shows the propulsion efficiency. Comparing both cases, one has to noe that the friction force as well as the shaft power differ but the body drag W is assumed 6 Copyright c 6 by ASME

7 PROPULSION EFFICIENCY ηη FFFF ηη SLENDERNESS RATIO κ Re= 9 Re= Re= 7 Re= 6 FIGURE 6. PROPULSION EFFICIENCY DEPENDENT ON SLENDERNESS RATIO κ FOR VARIOUS REYNOLDS NUMBER Re = U L/ν. FRICTION COEFFICIENT RATIO αα ww = c f cc fff.9..7 Re= 9 Re= Re= 7 Re= SLENDERNESS RATIO κ FIGURE 7. FRICTION COEFFICIENTS RATIO DEPENDENT ON SLENDERNESS RATIO FOR VARIOUS REYNOLDS NUMBER Re = U L/ν. to be approximately constant for both cases. The friction force ratio is α w := W f W f = c f c f 9 and is illustrated in Fig. 7. The flying speed is constant for both cases, thus the shaft power ratio is to analyse and not the propulsion efficiency. The shaft power ratio is α p := P s = η U P s η U ph 5 W +W f and is shown in Fig.. Applying Eqn. 9, Eqn. 5 yields α p = η η U U + c U U + δ +U U U U +U c f H L. 5 Conclusion This paper analyses the energetic assessment of two different aircraft propulsion topologies. For the first case, e.g. the reference case, the propulsion system is connect by a pylon to the aircraft body. For the second case, the propulsion system is embedded in the aircraft body. We derie the axial momentum balance for both cases including a solution for the boundary layer, e.g. the friction drag, and the outer flow. Hereby, we applied the integral method of boundary layer theory and the potential theory, respectiely. The outcome of this assessment are the following three major findings: SHAFT POWER RATIO αα PP = P s P s.9..7 κκ opt RRRR ηη ηη = Re= 9 Re= Re= 7 Re= SLENDERNESS RATIO κ FIGURE. SHAFT POWER RATIO DEPENDENT ON SLENDER- NESS RATIO FOR VARIOUS REYNOLDS NUMBER Re = U L/ν.. For increasing aircraft engine and increasing Reynolds number Re, the propulsion efficiency increases as well; see Fig. 6.. The optimized aircraft engine is gien by Fig. 7. The friction force is decreased depending on Reynolds number and aircraft engine size κ := H/L. 3. The energetic improement of the embedded propulsion is in relation to the reference case about %; see Fig.. References [] Tillman, T., Hardin, L., Moffitt, B., Sharma, O., Lord, W., Berton, J., and Arend, D.,. System-leel benefits of boundary layer ingesting propulsion. In Inited paper, presented at the 9th AIAA Aerospace Sciences Meeting, Orlando, FL. [] Hardin, L., Tillman, T., Sharma, O. P., Berton, J., 7 Copyright c 6 by ASME

8 and Arend, D. J.,. Aircraft system study of boundary layer ingesting propulsion. In th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. [3] Sato, S., et al.,. The power balance method for aerodynamic performance assessment. PhD thesis, Massachusetts Institute of Technology. [] Drela, M., 9. Power balance in aerodynamic flows. AIAA journal, 77, pp [5] Rankine, W., 65. On the mechanical principles of the action of propellers. Transactions of the Institution of Naal Architects, 6, pp [6] Betz, A., 96. Windenergie und ihre Ausnutzung durch Windmühlen. Vandenhoeck. [7] Glauert, H., 99. Grundlagen der Tragflügel- und Luftschraubentheorie aus dem Englischen übersetzt on H. Holl. Julius Springer-Verlag, Berlin. [] Becker, E., 966. Gasdynamik, Vol. 6. Teubner. [9] Saul, S., STONJEK, S., and Pelz, P. F., 5. Influence of compressibility on incidence losses of turbomachinery at subsonic operation. In International Conference on Fan Noise, Technology and Numerical Mehtods. [] Schlichting, H., 97. Boundary-Layer Theory. McGraw Hill. [] Pohlhausen, K., 9. Zur näherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht. ZAMM,, pp []. Kármán, T., 9. Über laminare und turbulente Reibung. ZAMM,, pp [3] Prandtl, L., 9. Führer durch die Strömungslehre. F. Vieweg & Sohn, Braunschweig. Copyright c 6 by ASME

DAY 19: Boundary Layer

DAY 19: Boundary Layer flat plate : let us neglect the shape of the leading edge for now flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence