Based on the Aircraft Design Lecture Notes and Master Thesis of Roberto Segovia García. Hamburg University of Applied Sciences.
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1 Dieter Scholz Drag Estimation Based on the Aircraft Design Lecture Notes and Master Thesis of Roberto Segovia García. Hamburg University of Applied Sciences In cruise we calculate the drag coefficient from : Zero lift drag (Chapter 1) : Wave drag (Chapter 2) For cruise, but for take-off (with initial climb) and landing (with approach) the zero lift drag coefficient has further components, because high-lift devices may be deployed and/or the landing gear may be extended. Especially the landing gear adds a considerable amount of drag. For cruise the Oswald Factor e is calculated from the presentation "Estimating the Oswald Factor from Basic Aircraft Geometrical Parameters" (see under "Method only"). Without any further calculation e = e CR = 0.85 can be taken as a standard parameter (for moderate cruise Mach numbers). For take-off (with initial climb) and landing (with approach) the Oswald Factor e = e T/O = 0.7 can be taken as a standard parameter together with e CR = However, any other values of e CR : e T/O = e CR. 0.7/ Zero Lift Drag "clean" means: neither flaps, slats, nor the landing gear are extended.
2 V / Vs = 1.2 for take-off and initial climb V / Vs = 1.3 for approach and landing Estimating C D0 "clean" from E max : C D,0 A e 2 4 Em ax (for E max see Chapter 3) Estimating C D0 "clean" from wetted area See: Aircraft Design Lecture Notes, Section 13 for wetted area calculation. Estimating C D0 in more detail from a drag built up: See: Aircraft Design Lecture Notes, Section 13 for all required parameters.
3 2 Wave Drag 2.1 Introduction The aim of this Section is the analysis of the wave drag of the aircraft. The initial calculation of the wave drag in PrOPerA was based on the Boeing and Airbus philosophy (Scholz 1999), so the tool considers that the cruise Mach number was equal to drag divergence Mach number and the wave drag coefficient was a constant. For turbofans, this concept can be correct, but for the turboprops it is unknown if the aircraft flies at the drag divergence Mach number. For this reason, a method for the calculation of the wave drag is needed. With this implementation, the previous assumption is not needed and the wave drag is calculated for each design. 2.2 Calculation of the Drag Divergence Mach Number Equation obtained from Shevell Shevell 1980 describes an analytical method for the calculation of the crest critical Mach number,. Figure 2.1 Wave drag curve obtained from Shevell 1980
4 Δc D,wave /cos(φ 25 ) From the Figure 2.1, the relation between is represented with, which is the critical Mach number, and (2. 1) The mathematical approximation for the Shevell s curve that has been realized is the Equation 2.2. The curve is represented in the Figure 2.2. (2. 2) Where and Shevell curve M/M CC Figure 2.2 Representation of the approximation of the Shevell s curve The definition that has been taken for the calculation of is given by Boeing and is defined as the Mach number for which the wave drag coefficient has an increment of, or 20 drag counts [CTS] (2.3). The Equation 2.4 represents the calculation of. (2. 3) (2. 4)
5 2.3 Calculation of the Wave Drag Curve Analysis of the Real Wave Drag Curves The calculation has been based on experimental data from some aircraft. The aircraft that have been studied are: A , B , B , C-130H and BAe The wave drag curves of B , C-130H and BAe have been obtained from Roskam II 1985, Figure 2.3. Figure 2.3 Wave drag curves of B , C-130H and BAe , obtained from Roskam II With this empirical information, and have been measured. The values of and for the five airplanes are represented in the Table 2.1. Table 2.1 Values of and for the aircraft that have been analyzed A B B C-130H BAe The wave drag curves have been approximated with equations that have the same structure as 2.5. (2. 5) Due to the conditions of the problem, has to be zero at, so and. The resultant equation is 2.6.
6 (2. 6) The values of the constants and are given in the Table 2.2 for each aircraft. The curves for the A , B , B , C-130H and BAe have been represented in the Figure 2.4. The representation is based on the Equation 2.6 with the constants of the Table 2.2. A : Wave Drag B : Wave Drag ΔcD,wave ΔcD,wave B : Wave Drag C-130H: Wave Drag ΔcD,wave ΔcD,wave BAe : Wave Drag ΔcD,wave Figure 2.4 Wave drag curves for A , B , B , C-130H and BAe
7 Table 2.2 Values of and for the approximation of the wave drag curves of the aircraft that have been analyzed A B B C-130H BAe Tangent Equation with the Effect of the Sweep Angle for the Calculation of the Wave Drag Curve This equation has been developed based on the approximation of the Shevell s curve (2.2). The equation is represented by 2.7. (2. 7) The constants and are represented in the Table 2.3 and the SEE is represented in the Table 2.4, in [CTS]. The curve defined by 2.7 and the real curves (approximated with the Equation 2.6) are shown in the Figure 2.5. Table 2.3 Values of and for only one "optimized" Tangent Equation for all aircraft listed in Table 2.2 represented by Equation 2.7 Result Table 2.4 SEE obtained for each aircraft, using the Tangent Equation based on Shevell s curve (Shevell 1980), Equation 2.7 A B B C-130H BAe SEE [CTS]
8 Δc D,wave /cos 3 (φ 25 ) [CTS] 60 Mach number Drag rise 50 A B B C-130H 20 Bae Tangent equation with sweep Figure 2.5 Representation of the Tangent Equation based on Shevell s curve (Shevell 1980), Equation 2.7, and the real curves (approximated with the Equation 2.6) 2.4 Calculation of the Critical Mach Number, Calculation of obtained from the Tangent Equation with the Effect of Sweep Angle The critical Mach number is obtained from the Equation 2.7. The resultant equation for the calculation of is (2. 8) (2. 9) (2. 10) The constants and are given in the Table 2.3.
9 3 Estimating Maximum Glide Ratio, E max for typical passenger jets (see Fig. 1.1). Fig. 1.1 Aircraft plan forms and their relative wetted area S wet / S W
10 k E calculated: Oswald Factor e in cruise (see above). k E given as: k E k E 14.9 when calculated for standard parameters, which are considered to be: e = 0.85, see Table according to data in Raymer s book from our own statistics: k E short range aircraft k E medium range aircraft long range aircraft k E 4 Calculating the Glide Ratio, E For any lift coefficient (which may not be the optimum one):
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