For high-fidelity flight simulations nonlinear aerody- namic coefficients are input to the simulation by means of extensive tables or function generators covering all the anticipated variations in the applicable parameters. How- ever, the coefficients are often approximately linear in the
regions of most interest, and simulations are often based on linearity assumptions, which permit acceptable fidelity to be achieved using greatly simplified data as input. The input tables can then be reduced to much smaller tables contain- ing only the values of the slopes of the linearized coefficient curves. These slopes, or derivatives, of the curves have been used extensively in aerodynamic stability analyses and are therefore called stability derivatives.
When stability derivatives are used, instantaneous aero- dynamic coefficients are calculated in the simulation by multiplying the appropriate derivative by the instantaneous value of the applicable variable (subpar. 5-2.1.5). If it is necessary to include nonlinearities, they can be approxi- mated by adding terms involving powers of the parameters.
Coefficient and derivative data for aerodynamic moments are based on specified locations of the center of mass of the missile. When the instantaneous center of mass differs from the one on which the input data are based, the moment coef- ficients must be corrected Eqs. 5-12).
To apply test data or empirical methods in a flight simulat- ion, it is necessary that the factors affecting aerodynamic forces and moments be presented in their proper relation- ship. This relationship has been determined by the method of dimensional analysis (Refs. 1 and 4) and confirmed by one-dimensional fluid mechanics theory, by potential theory (Ref. 1), and by wind tunnel and flight testing. For aerody- namic forces this relationship is given by the familiar form of the aerodynamic force equation employed extensively in aerodynamics:
(5-1) where
CF = F = S=
V=
p =
general aerodynamic force coefficient, dimen- sionlss
general force (aerodynamic), N aerodynamic reference area, m2
speed of a body, speed of air relative to a body, magnitude of velocity vector V, m/s
atmospheric density, kg/m3.
In practice F and CF are always expressed in terms of spe- cific components, e.g., drag force D and coefficient of drag CD (subpar, 5-2.1.4).
5-2.1.1 Dynamic Pressure Parameter
The term 0.5pV2 is a very important quantity, known as the dynamic pressure parameter, which was discussed in subpar. 3-3.2.3. The dynamic pressure parameter is equal to the kinetic energy per unit volume of air. Two equivalent forms of the dynamic pressure parameter were presented in Chapter 3 as
(3-5)
where
MN = Mach number, dimensionless Pa = ambient atmospheric pressure, Pa
Q = dynamic pressure parameter, Pa.
Whenever a fluid passes around an object there is a point at which the flow divides—part goes one way and part the other. This point of division is called the stagnation point because, theoretically, the molecules of fluid at this point are brought to rest relative to the object. At the stagnation point the rise in pressure, caused by the loss of all kinetic energy of the fluid is called the dynamic pressure, which in incompressible flow is equal to the dynamic pressure parameter Q. Thus in incompressible flow the force on the body per unit area at the stagnation point is equal to the dynamic pressure parameter.
Although the dynamic pressure actually acts only at the stagnation point, Eq. 5-1 shows that the total aerodynamic force F on the entire body is proportional to the dynamic pressure parameter.
In compressible flow the actual dynamic pressure is larger than the dynamic pressure parameter (Ref. 3); never- theless, accepted practice is to use the dynamic pressure parameter to characterize both incompressible and com- pressible flows. The difference is taken into account by the dependency of the force coefficient on Mach number.
5-2.102 Force Coefficient
The force coefficient CF is a dimensionless coefficient that accounts for all the factors that affect aerodynamic force except those included explicitly in Eq. 5-1. Force coefficients usually are based on tests and are defined as the ratio of the measured force to the product of the dynamic pressure parameter and the reference area, as shown by solving for the force coefficient in Eq. 5-1. The force coeffi- cient can be viewed as a proportionality between the actual aerodynamic force on a body and a reference force. The ref- erence force is the force that would result from applying a pressure equal to the dynamic pressure parameter to the ref- erence area.
The value of the aerodynamic force coefficient for a given body configuration is affected primarily by the shape of the body (including any control-surface deflections), the orientation of the body within the flow (angle of attack), and the flow conditions. For the types of flow generally encoun- tered in surface-to-air flight simulations, the flow conditions can be specified by two parameters: the Mach number and the Reynolds number.
5-2.1.2.1 Effect of Mach Number
As previously defined, the Mach number is the ratio of the missile speed, i.e., the relative speed of fluid flow, to the speed of sound in the ambient air. As the missile speed approaches and exceeds the speed of sound, the compress-
ibility characteristics of the air have a pronounced effect on the aerodynamic forces and moments. Shock waves are formed that affect the pressures and the distribution of pres- sures on the surface of the vehicle. These compressibility effects are taken into account in aerodynamic force and moment equations by including them in the aerodynamic coefficients. The compressibility effects are so important that aerodynamic force and moment coefficients for missiles are always given as functions of Mach number.
5-2.122 Effect of Reynolds Number
Another fluid-flow property that affects the values of the aerodynamic force and moment coefficients is characterized by the Reynolds number. The Reynolds number is a mea- sure of the ratio of the inertial properties of the fluid flow to the viscous properties. Reynolds number is given by
Reynolds number = where
aerodynamic reference length of body, m speed of a body, speed of air relative to a body, magnitude of velocity vector V, m/s
atmospheric dynamic viscosity, kg/(m-s) atmospheric density, kg/m3.
The reference length d is a scale factor that accounts for the effect of the size of the missile on the flow characteristics.
The missile diameter is often selected as the reference length, but the length of the missile body is also commonly used. Force coefficients are functions of Reynolds number.
When a force coefficient is given, the Reynolds number upon which it is based must also be given; in addition, the missile dimension used as a reference length for the Rey- nolds number must be specified.
Although Reynolds number varies as the missile changes speed and altitude, the effect on the aerodynamic force coef- ficients is generally small over the major portion of the flight of a typical surface-to-air missile. For this reason the variation of the force coefficients with Reynolds number is often neglected in missile flight simulation, except to ensure that the force coefficient data correspond to the general range of Reynolds numbers typical of full-scale missile flight. The variation of force coefficients with Reynolds number becomes important for missiles that fly to very high altitudes, greater than about 15 km, and in the interpretation of wind tunnel data based on small-scale models.
5-2.1.3 Reference Area
Aerodynamic pressures and tangential forces per unit area must be integrated over the surface area of the body to yield the force on the body. This dependence on area is represented in Eq. 5- I by the reference area S which is actu-
ally a scaling factor, rather than a specific area acted on by a specific pressure. The reference area accounts for the rela- tive size of the body in aerodynamic force calculations.
Under similar flow conditions, the aerodynamic forces on geometrically similar bodies are proportional to the respec- tive reference areas of the bodies. For geometrically similar bodies, the proportionality is constant, regardless of which area of the bodies is selected as the reference area. For a given body and flow, the product of the force coefficient and the reference area CFS must not vary with different choices for the reference area; therefore, the value of the force coef- ficient depends on which area of the body is used as the ref- erence area.
Although the selection of the particular area of the body to be employed as the reference area S is completely arbi- trary, common practice is to use the wing planform area for airplanes and the body cross-sectional area for missiles.
Other areas are sometimes used for specific applications.
For example, the body wetted area, i.e.,total surface exposed to the air, is used for calculations of viscous fric- tion forces; however, when these viscous components are combined with the pressure forces, a single reference area is needed. A coefficient based on wetted area is converted to the body cross-sectional area by
where
, dimensionless (5-3)
general aerodynamic coefficient based on body cross-sectional area, dimensionless
general aerodynamic coefficient based on wet- ted area dimensionless
cross-sectional area of body, m2 wetted area of body, m2.
When coefficients based on other reference areas are encountered, they can be converted to any desired reference area by equations of the form of Eq. 5-3 based on the equiv- alence of the product CFS.
An aerodynamic coefficient determined for a subscale model of a missile is directly applicable to the full-scale missile under similar flow conditions when used in Eq. 5-1, in which the corresponding reference area of the full-scale missile is used.
5-2.1.4 Components of Forces and Moments
The resultant aerodynamic force FA is always considered in terms of its components, either lift and drag or normal force and axial force, as discussed in subpar. 4-4.1. There- fore, values for a total aerodynamic force coefficient CF of a body are never used. The relationships among drag, axial force, lift, and normal force were given by Eqs. 4-13 and 4-
14. These equations show that under conditions of zero lift, i.e., angle of attack = O, drag and axial force are identical and lift and normal force are identical. In the general case in which the angle of attack ≠ O, the resultant aerodynamic force can be resolved into either lift/drag components or normal-force/axial-force components, and given aerody- namic data tables may be encountered in either or both sets of coordinates. For proposes of flight simulation, working with lift/drag components has the advantage that these force components produce accelerations perpendicular to and along the path of the vehicle. Therefore, discussions in the following paragraphs emphasize the use of lift and drag components.
The drag coefficient CD and the lift coefficient CL are typically used in Eq. 5-1 in place of CF to define the respec- tive components of the resultant aerodynamic force FA. The expressions for D and L are
(5-4) (5-5) where
aerodynamic drag coefficient dimensionless aerodynamic lift coefficient, dimensionless
magnitude of aerodynamic drag force vector D, N
magnitude of aerodynamic lift force vector L, N
aerodynamic reference area, m2
speed of a body, speed of air relative to a body, magnitude of velocity vector V, m/s
atmospheric density, kq/m3.
In calculations and discussions these components are han- dled. as separate but dated forces.
Theory and experiment show that the equation to calcu- late aerodynamic moments is analogous to that for aerody- namic forces but with the addition of a characteristic length term d as indicated previously by Eq. 3-4. For missiles this reference length is usually taken as the body diameter. As with aerodynamic forces, aerodynamic moments are always considered in terms of their components. The aerodynamic rolling mement i.e., the moment about the x-axis in the body reference frame, is calculated by substituting the roll- ing moment coefficient Cl into Eq. 3-4 in place of the gen- eral term CM. In a like manner, the aerodynamic moments about the other two axes are given by similar equations using the pitching moment coefficient CM and the yawing moment coefficient Cn. Expressions for the aerodynamic moment components about the coordinate axes are
(5-6)
where
Cl = aerodynamic roll moment coefficient about center of mass, dimensionless
Cm = aerodynamic pitch moment coefficient about center of mass, dimensionless
Cn = aerodynamic yaw moment coefficient about center of mass, dimensionless
d = reference length of body, m
LA,MA,NA = components of aerodynamic moment vector MA expressed in body coordinate system (roll, pitch, and yaw, respectively), N-m S = aerodynamic reference area, m2
V = speed of a body, speed of air relative to a body, magnitude of velocity vector V, m/s p = atmospheric density, kg/m.3
For missiles in a mature state of development, very detailed force and moment coefficient data maybe available as a function of Mach number, altitude, angle of attack, and sometimes control-surface deflection. In this case the most accurate use of these data in a simulation is through direct tabular data inputs with interpolation among the tabular val- ues rather than by use of the stability derivatives, which are discussed in subpar. 5-2.1.5 and par. 5-3.
5-2.1.5 Linearity Assumption
Aerodynamic coefficients are typically plotted as func- tions of parameters such as angle of attack control-surface deflection angle, and missile roll position. In general, these are not linear over all values of the parameters. The most applicable portion of the curves, however, are often approximately linear, and advantage has been taken of this fact in classical aerodynamic analyses. The assumption of linearity allows the behavior of a given aerodynamic coef- ficient relative to a given parameter to be described by specifying only the slope, or derivative, of the curve. The value of the aerodynamic coefficient is calculated easily by multiplying the value of the slope by the value of the parameter. For example, if the slope of the curve relating the coefficient of lift to the angle of attack is given as C , of attack by CL = CLa a. When the parameter is an angular rate, the derivative is made nondimensional by taking it with respect to the nondimensional velocity parameter wd/
(2V)* in which ω represents an angular rate. For example, the pitch damping derivative is given by
where Cm = Cmq =
aerodynamic pitch moment coefficient about center of mass, dimensionless
pitch damping derivative relative to pitch rate q, rad-l (deg-])
d = aerodynamic reference length of body, m q = pitch component of angular rate vector w ex-
pressed in body coordinate system, rad/s (deg/s)
V = speed of a body, speed of air relative to a body, magnitude of velocity vector V, m/s.
See Refs, 4 and 5 for justification of this formulation.