4 Performance Calculation
4.1 Liftoff Distance
4.1.7 Aerodynamic Lift on Ground
The Lift force on ground plays an important role when determining the drag force due to roll-ing or brakroll-ing resistance, as the wheel load which is needed for the determination of the fric-tion force is dependant from the lift.
The Lift in general can be obtained from equation 4.49:
(4.49)
With
Lift coefficient on ground
Aircraft speed (CAS/EAS for the takeoff phase) Wing reference area
The Lift Coefficient on ground CL,G describes the lift coefficient for the complete aircraft. It needs to be broken down into the contributing components of the aircraft, notably the wings, horizontal tailplane and fuselage.
For the overall aircraft lift coefficient, equation 4.50 applies:
(4.50) With
Overall aircraft lift coefficient on ground
Zero lift coefficient
Wing lift curve slope
Wing Angle of Attack (AOA), on runway equals wing incidence angle
Lift contribution of the trimmed horizontal stabilizer Lift increment due to flap extension
Lift increment due to fuselage
Zero Lift and Wing Lift Coefficients
The wing lift coefficient consists of two parts, the zero lift coefficient of the wing and the lift due to an incidence angle of the airfoil.
(4.51)
With
Wing Lift Coefficient
The wing lift coefficient estimation is made according to the approaches outlined in DATCOM 1978. The estimation methods in this source are approximations derived from numerous tests. The estimations do not consider boundary layer disturbances through surface roughness, curvatures, heat transfer and pressure gradients, but will be of adequate precision for means of this report.
The Wing Lift Curve Slope is determined from equation 4.52
√ ( )
(4.52)
With
√ (4.53)
And
A Aspect ratio of the wing
Reciprocal value of the Mach number correction Correction factor of the airfoil section lift curve slope
Sweep angle at 50% cord
It is possible to reduce the correction factor of the airfoil section to 1, also the Mach number correction factor may be set to 1 due to the fact that the takeoff phase as outlined in section 4.1.5 is situated in a low Mach number region.
Hence, Eq. 4.53 simplifies to
√ ( )
(4.54)
In order to correct the sweep angle, a conversion according to Scholz 1999 is used.
(
) (4.55)
With
Chord location at which sweep angle is desired Chord location at which sweep angle is known
Taper ratio of the wing
For the Zero Lift Coefficient, the Eq. 4.56 applies, using the Wing Lift curve Slope from Eq.
4.54:
(4.56)
According to DATCOM1978, where the zero-lift angle of attack of the wing can be de-termined from Eq. 4.57. A linear spanwise twist distribution is assumed.
(4.57)
With
Profile zero angle of attack (between -2° and -4°, Anderson 2007) For NACA 64A airfoils, values are provided in Fig. 4.8
Wing twist angle tip to root in degrees, negative for washout
Mach Number correction acc. to Fig. 4.6
Change in zero lift angle of attack due to wing twist acc. to Fig. 4.7
Fig. 4.6 Determination of the Mach Correction Factor for Zero Lift Angle of Attack (DATCOM 1978)
Fig. 4.7 Determination of the Wing Twist to Zero Lift Angle of Attack Ratio (DATCOM 1978)
As the Wing Lift Curve Slope is dependent from the Mach number, the lift coefficient varies slightly with Mach number. If a time-step wise calculation method can be applied, the lift coefficient should therefore be calculated for the appropriate velocity at that time. For simplification, as the AOA does not change while the aircraft is on the runway, it can also be assumed constant.
Fig. 4.8 Section Angle of Zero Lift for NACA 64A profiles (NACA Report 903)
Lift Coefficient Increment due to Flaps
The flaps on the wing are acting as a lift augmentation device; therefore the lift coefficient of the aircraft wing is increased as shown in Eq. 4.50. The lift coefficient increment is cal-culated from the section lift coefficient with flap influence as shown in Eq. 4.58
(4.58)
With
Empirical correction factor for flap effectiveness acc. to Fig. 4.9
Section lift coefficient with flap influence acc. to Eq. 4.59
Wing area
“Flapped” area along the wing chord acc. to Fig. 4.10
Fig. 4.9 Empirical correction Factor for Flap Effectiveness (DATCOM 1978)
Fig. 4.10 Flapped Area of the Wing along the Chord Line(Roskam VI)
The section lift coefficient with flap influence as applicable for a single slotted flap is given by Eq. 4.59 in accordance to DATCOM 1978.
(4.59)
With
Flap Lift effectiveness parameter acc. to Fig. 4.11 Flap deflection angle in radians
Profile lift coefficient
Fig. 4.11 Lift Effectiveness Parameter by Flap Deflection Angle (DATCOM 1978)
The flap extension does not only have an effect in terms of lift coefficient increment, the flap extension also reduces the zero lift angle of attack due to the increased camber of the airfoil section. According to the empirical equation provided by DATCOM 1965, the zero lift angle increment in degrees can be estimated from the flap deflection angle.
(4.60)
With
Zero lift angle of attack change due to flap deflection Flap segment chord length
Wing chord length
Lift Coefficient Increment due to Fuselage Carryover
The lift coefficient of the aircraft as shown in Eq. 4.50 accounts for the lift contribution of the horizontal stabilizer. By the presence of the fuselage, an interaction between the lift created by the wing alone is also created. This effect shall be examined first, as it is related to the lift increment due to flap extension.
According to Torenbeek 1982, it has to be considered that the lift distribution across the wing is disturbed by the fuselage and reduced.
The easiest assumption would be to assume that the fuselage does not create any lift during takeoff, which may be true for large flap deflections when a large gap between the flaps and the fuselage exists. In this case it is valid that
(4.61)
With
Exposed wing area
Reference wing area
Fuselage area intersecting with the wing reference area
If this case shall be applied, in the lift force equation(see equation 4.49) the reference wing area should be set as the net wing area .
If a more elaborated approach is chosen, Torenbeek 1982 suggests the lift carry-over by the fuselage to be estimated from the lift generated by the wing center section if it was assumed to be extended to the aircraft centerline.
Thus, the lift coefficient increment due to fuselage interference would be
As a lift increment to the overall aircraft lift coefficient is added, even though the overall wing lift is disturbed by the fuselage, it is obvious that also in this case, the wing reference area in Eq. 4.50 needs to be substituted by the exposed wing area as shown for Eq. 4.61. In this case however, by using the increment, the lift over the fuselage is not assumed to be zero.
Other lift increments due to the fuselage, such as the lift created by the cylindrical body, are assumed to be zero, as the aircraft accelerating on the runway does not incur a fuselage angle of attack.
Lift force on the Horizontal Tailplane
For the horizontal tailplane, two major effects need to be considered for the takeoff run. These are the downwash created by the main wing, and the trim setting of the horizontal stabilizer for the takeoff.
In case of a variable incidence horizontal stabilizer that pivots around its rear spar, as is the case for the Learjet 35A/36A, the angle of attack of the complete horizontal stabilizer is al-tered for trim setting. For takeoff, the trim setting should be set such that the elevator forces necessary for the rotation are acceptable and the aircraft is trimmed out for the initial climb.
The more aft the Center of Gravity (CG) of the aircraft is located, the less nose-up force needs to be created. For the engine-out analysis, the most aft CG is the most conservative assump-tion that should be taken into account for the takeoff performance calculaassump-tion. Hence, the nec-essary trim force is reduced or may even be resulting in a slight nose-down trim setting as for the Learjet 35A/36A.
The downwash effect on the horizontal stabilizer refers to the fact that the free stream velocity vector on the horizontal tail plane is changed by the flow over the main wing. This leads to the air flow acting on the horizontal stabilizer to come from a slight upward angle, reducing the effective incidence angle of the horizontal stabilizer. This effect would therefore induce a slight nose-up pitch moment. However, if the aircraft is equipped with a T-tail such as the Learjet 35A/36A, according to Torenbeek 1982 the downwash effect on the horizontal stabi-lizer is very small and can be neglected.
For means of this performance calculation, both effects are expected to be acting in opposite directions and be of small magnitude, and have subsequently been considered negligible.