ANALYTIC ASSESSMENT OF AN EMBEDDED AIRCRAFT PROPULSION

This paper investigates analytically the advantage 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 given.

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 negative effected due to the non-uniform velocity 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 relevant. 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 advantage of the embedded propulsion compared to an conventional propulsion by propulsion efficiency analysis. The influence of the propulsion interacting with the boundary layer is investigated by Tillman, Hardin et al. [1,2]. For different aircraft configurations the power balance method for performance estimation is applied by Sato [3] and Drela [4].
The aircraft body is treated as a flat plate. Following the idea of Rankine 1865 [5] and Betz [6] the aircraft engine is modelled as a disc actuator. This approach is only valid for propeller engines. Thus, this analysis neglects the fuel mass flow and the flow is assumed to be incompressible. An extension to a compressible 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. 2. For the conventional 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. This paper is organized as follows: first, we discuss the axial momentum balance and we introduce the axial induction factors a := c/v and a := c/v. 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 previous sections and compare the embedded propulsion with the conventional propulsion. At last, the first law of thermodynamics is applied to define the efficiency. By doing so, the optimal engine size is given by the shaft power ratio of the two topologies. In the closure of this paper, we summarize our investigation by three major findings.

AXIAL MOMENTUM BALANCE
For the embedded propulsion the axial momentum balance reads with the pressure p 1 in front of the disc, the velocities u and U, the height H and the air density ρ. The friction force is with A k the wetted area of the whole aircraft body. Using the displacement thickness δ 1 , defined as usual as and the momentum thickness the left side of Eqn. 1 yields The continuity equation reads at the disc and with the Bernoulli equation for the pressure in front of the disc, the axial momentum balance yields δ 2+ is approximately the dimensionless momentum thickness neglecting terms higher order, thus Following the nomenclature of Glauert [7], the ratio of the total jet velocity and the velocity of transport, e.g. flying speed, is the so called axial induction factor of the disc. Close to the disc, the induction factor is definded as With the induction factors, the friction coefficients and the slenderness ratio we obtain 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 with the pressure p 2 behind the disc. The thrust S equals the sum of the friction force W f and body drag W 0 at constant flying speed v The body drag is given by Using Bernoulli's equation behind the disc to the free jet stream, the pressure behind the disc is Following the nomenclature of Glauert, Eqn. 17 yields thus, the induction factor a is given by With Eqn. 16 and 22 the induction factors a(δ + , f + ) and a(δ + , f + ) are given. A reference case without a boundary layer, e.g. δ + = 0 and λ = 1, the well known Betz solution a = 1/2a is included. For the general case, the induction factors a(δ + , f + ) and a(δ + , f + ) are given by Fig. 3 and 4, respectively.   For a compressible flow Eqn. 20 has to be replaced by u/2 + γ/(γ − 1) p/ρ = const.. As usual the equation of state p = ρRT is as well needed as well as the energy equation in integral form [8,9].

OUTER FLOW
Outside the boundary layer, e.g. the outer flow, the flow is irrotational. There, the velocity field is given 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 1 x. Second, the line sink at x = L, −H < y < H with the sink strength −4cH. Third, the combination of sinks and sources to consider the boundary layer at the flat plate. Thus, we obtain ln (x − L) 2 + (y − y ) 2 dy + . . .
The velocity of the outer flow is given by The displacement of the boundary layer influences the velocity less than the propulsion does. Thus, the second term on the ride side of Eqn. 23 for the velocity profile within the boundary layer is negligible. The velocity of the outer flow depends on the induction factor. Figure 2 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. 18. The friction force is the integral of the wall shear stress

BOUNDARY LAYER AND DRAG
The wall shear stress is τ w := ρu 2 * with the friction velocity u * and yielding the local friction coefficient Hence, the friction force is With the integral method of the boundary layer theory and Eqn. 3 and 4, the axial momentum balance is the so called van Kármán momemtum equation and is valid for laminar and turbulent flow.

Reference Case
The reference case, e.g. the conventional propulsion Fig. 2, has a constant pressure boundary layer, thus, U = U 1 = const.. Applying Eqn. 29 to the reference case, yields 2dδ 2 /dx = c f . With an ansatz function for the axial velocity profile within the boundary layer, for example Prandtl's power law the friction coefficient is Equation 31 is the well known Blasius power law [10]. The solution of the integral method for boundary layer theory is very robust against assumed ansatz functions [10][11][12], in our case Eqn 30. Hence, to verify 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 U(x), see Eqn. 25, to solve the von Kármán momentum equation (Eqn. 29). Using Eqn. 30 for the velocity profile within the boundary layer, we obtain for the displacement thickness and for the momentum thickness. Thus, δ 1 = (2n + 1)δ 2 = hδ 2 with h := 2n + 1. The choosen power law has to be calibrated to the viscous sublayer [10] with the constant C. For n = 1/7, the empirical constant C is 8.74 [10]. Hence, the wall shear stress is With these deviations and and we obtain the von Kármán momentum equation for the embedded propulsion Hence, the momentum thickness for the embedded propulsion is

FRIST LAW OF THERMODYNAMICS AND FROUD PROPULSION EFFICIENCY
The shaft power is with the mean velocity U and the aerodynamic, e.g. isotropic, efficiency η of the embedded propulsion and η 0 of the reference case.

Propulsion Efficiency of the Reference Case
For a constant flying speed v = U, the shaft power for the reference case is P s0 = S 0 U 0 /η 0 and the Froud efficiency is given by (43) This is the well known definition of the propulsion efficiency [13].

Propulsion Efficiency of the Embedded Propulsion
For the embedded propulsion we calculate the propulsion efficiency to with the thrust the pressure difference and the dimensionless displacement thickness Thus, the propulsion efficiency is (48) 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 0 is assumed to be approximately constant for both cases. The friction force ratio is 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 and is shown in Fig. 8. Applying Eqn. 19, Eqn. 50 yields

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 derive 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, respectively. The outcome of this assessment are the following three major findings: