## Control Surfaces on the Tail Unit

In this section, a brief discussion will be given of the aerodynamic forces generated by the control-surface deflection of the tail unit and their effect on the force and

3 Figure 8-39 Position of the flap neutral point for flap designs of Fig. 8-38.

moment equilibrium of the whole airplane. For the case of zero control-surface deflection, the contributions of the horizontal tail and the vertical tail, respectively, to the aerodynamic forces of the whole airplane have been given in Secs. 7-2-1 and 7-3-1.

Elevator For the contribution of the horizontal tail with deflected elevator to the pitching moment of the whole airplane, Eqs. (7-3a) and (7-3b) yield

Here, from Fig. 7-5, rH is the distance of the lift force of the horizontal tail from the moment reference axis of the airplane.

The change in the moment caused by the elevator deflection at constant angle of attack is thus obtained as

(8-29)

Here, the quantity r’H of the previous equation has been replaced by the lever arm г"#, which is the distance of the flap neutral point from the moment reference axis of the horizontal tail.

For the two-dimensional flap wing in incompressible flow, the position of the flap neutral point is given in Fig. 8-15. The change in the pitching moment caused by the elevator deflection at constant lift coefficient (zero-moment coefficient) is obtained in analogy to Eq. (7-15) by substituting —(данІдг)н)^н f°r eH as

(8-30)

Here г#ДГ is the distance of the neutral point of the elevator from the neutral point of the whole airplane (see Fig. 7-6b).

Rudder The contribution of the vertical tail with a deflected rudder to the yawing moment of the whole airplane becomes, from Eqs. (7-49a) and (7-49b),

d°iv in dav Qv Av r’y

CMzV~~dav Pv~d^VvJ <7. XT

Here, r’y from Fig. 7-36 is the distance of the side force of the vertical tail from the moment reference axis of the airplane.

The change in the yawing moment caused by the rudder deflection is thus given as

дсщгУ _ d°iv dav Чу A v r’y (8-31)

Ьтіу dcxy Э7}y q^, A s

Here the quantity r’y of the previous equation has been replaced by the lever arm r’y, which is the distance of the flap neutral point from the moment reference axis of the vertical tail.

Rudder moments Information on the rudder moments of the airfoil of infinite span for incompressible flow is found in Sec. 8-2. The control-surface moments of the elevator and rudder and also of the ailerons cannot, in general, be computed with sufficient accuracy, because for the control-surface moments the transformation, from the airfoil of infinite span (plane problem) to the wing of finite span is not possible in a reliable way. The control-surface moments for control surfaces with balance provisions of Fig. 8-2 (inner balance, outer balance, balance tabs) are particularly difficult to determine because they are greatly affected by the boundary layer as well as by’ inviscid flow problems. The control-surface moments must therefore be determined largely through wind tunnel and flight tests (see Stiess [18]). Some wind tunnel measurements on the control-surface moments of tail surfaces with inner and outer balances were reported by Schlichting and Ulrich [39].

[1]The temperature gradient dT/dH determines the stability of the stratification in the stationary atmosphere. The stratification is more stable when the temperature decrease with increasing height becomes smaller. For dT/dH = 0 when n = 1, Eq. (1-13), the atmosphere is isothermal and has a very stable stratification. For n = у — 1.405, the stratification is adiabatic (isentropic) with dT/dH — —0.98 К per 100 m. This stratification is indifferent, because an air volume moving upward for a certain distance cools off through expansion at just the same rate as the temperature drops with height. The air volume maintains the temperature of the ambient air and is, therefore, in an indifferent equilibrium at every altitude. Negative temperature gradients of a larger magnitude than 0.98 K/100 m result in unstable stratification.

**[2]** Translator’s note: According to the definition given by NASA, the angle of sideslip is the angle between the direction of the incident flow and the symmetry plane of the airplane. The angle of yaw is referred to a chosen direction, which may sometimes be the direction of the airflow past the body, making the angle of yaw equal to the angle of sideslip. Under some conditions, however, as in turning, a different reference direction may be used.

[3]The angle 0 has been designated here as the angle of yaw. For the difference between angle of yaw and angle of sideslip see the footnote on page 13.

[4]These quantities may be called in the text simply “thickness” and “camber” when a misunderstanding is impossible.

[5]The influence of friction on lift will be considered in Sec. 2-6.

[6]The Joukowsky mapping function, Eq. (2-21), can be given in more general form in various ways, leading to additional profile shapes that are obtained from mapping circles. For example, when in Fig. 2-14a the mapping circle does not pass through the point +a on the real axis but rather through a point located somewhat farther outside, the sharp trailing edge of the normal Joukowsky profile is replaced by a rounded edge.

[7]It is necessary to take the Cauchy principal value

X-S

[8]Note that S/c = X — j | cos and 77 = zO.

[9]Note that, according to Jaeckel [30], the folio-wing relation applies:

[10]Translator’s note: Remembex that the term “nose iadius” does not necessarily imply a circular nose. The definition of nose radius is of the kind found in Figs. 243 and 244. The curvature can, therefore, be relatively large locally on the nose, even if the radius in the above sense is not small.

[11]In addition to the geometric twist, there is an aerodynamic twist, characterized by a twist angle measured against the profile zero-lift direction instead of the profile chord.

[12]To distinguish between the coefficients of the total forces and moments, the indices of which are always expressed in capital letters, lowercase letters will be used for the indices of the coefficients of local forces and moments.

[13]If the profile coefficient c^ is known over the span, it may be replaced by Cca — C]_,ao 27Г.

form and the constant induced downwash velocity over the wing span.

[14]Note that, according to [23],

[15]Here, the prime (‘) on the summation sign indicates that the term n — v is to be omitted in the summation.

[16]2a/ is the induced downwash angle fax behind the wing, f -» °°.

[17]Kraemer [79] conducted a more detailed study into the application of the momentum law to the computation of the induced drag; see Sears [79].

[18]The suction force is considered positive when acting upstream.

[19]In what follows, the index e of forces and moments and their coefficients, which indicates the axis system used, will be omitted.

[20]Foi flight mechanical computations, the axis of rotation coincides with the lateral axis through the airplane’s center of gravity.

position; x0 = axis of rotation for vanishing lift due to pitch.

[21]The value of the Tolling moment due to sideslip of the total airplane depends on the vertical position of the wing relative to the fuselage in addition to the dihedral.

[22]The profile drag is not taken into account.

[23]The trigonometric functions will be given as sinh’3 rather than arcsinh.

[24]For Ma = 0, Eq. (4-5) reduces to the well-known relationship of incompressible flow. It is not valid for hypersonic flow; see Sec. 4-3-5.

[25]rhe integral of the second equation is obtained through integration by parts.

[26]The opposite trend is found in transonic flow, in which the changes of the flow quantities are small in the lateral and strong in the longitudinal direction.

TThis formula and its comparison with measurements will be discussed in more detail in Sec. 5-3-3.

[27]In this section, the drag of the fuselage as obtained in inviscid flow (wave drag) is designated as Dp. Because of viscosity effects (friction), a contribution Dp must be added to this drag [see Eq. (5-17)].

[28]It can easily be seen that the flow pattern of the two counter-rotating vortices at у and yp and at (y + dy) and (yp + dyp), respectively, contains, as a streamline, the circle of radius R about the origin.

[29]For a blunt fuselage nose and tail, Eq. (6-6) gives finite values for dLp/dx, contrary to the exact values dLpjdx = 0.

[30]For the fuselage, a different transformation formula of the pressure coefficient was given by Eq. (5-53), where the angle of attack was transformed according to Eq. (5-52d). However, within the framework of the linear lift theory, the Eq. (6-32) for the fuselage is equivalent to Eqs. (5-52d) and (5-53).

[31] = – –00——- UK (6-31 a)

l/Malo — 1

w=—ot00lJoz (6-31 b)

The solid curves signify the mean values of the induced velocities over the circumference. They are essential for the computation of the lift distribution of the

[32]The stripe method is a procedure whereby the local lift coefficient is set proportional to the local angle of attack based on the lift slope of the plane problem, which from Eq. (4-46) is given for supersonic velocities by {dcijda)«, = 4/VMzL — 1.

[33]For simplicity it has been assumed that the ratio of the dynamic pressures qfjlq « is independent of the angle of attack a.

[34]The integral j must be evaluated after the Hadamard method of finite parts of divergent integrals.

[35]The index 0 has been omitted.

[36]The assistance of K. O. Arnold in preparing this section is gratefully acknowledged.