Shape and Flow Form Drag Skin Friction 0% 100% ≈10% ≈90% ≈90% ≈10% 100% 0% Figure 3.1 Figure 3.2

Drag refers to the forces acting opposite to the relative motion of any object moving with respect to a surrounding fluid. A spacecraft is subjected to drag forces when moving through a planet's atmosphere. This drag is greatest during launch and reentry, however, even a space vehicle in low orbit experiences some drag as it moves through the planet's thin upper atmosphere.

Types of drag are generally divided into the following categories:

• parasitic drag, consisting of
  • form drag
  • skin friction drag
  • interference drag
• induced drag
• wave drag

Form drag, or pressure drag, is caused by the air that is flowing over the body. The separation of air creates turbulence and results in pockets of low and high pressure that leaves a wake behind the body (thus the name pressure drag). This opposes forward motion and is a component of total drag. Since this drag is due to the shape, or form of the body, it is called form drag. Streamlining an aircraft, rocket or reentry vehicle will reduce form drag, and parts of a vehicle that do not lend themselves to streamlining are enclosed in covers called fairings that have a streamlined shape.

Skin friction drag is caused by the actual contact of the air particles against the surface of the body. This is the same as the friction between any two objects or substances. Because skin friction drag is an interaction between a solid and a gas, the magnitude of skin friction drag depends on the properties of both the solid and the gas. For the solid body, skin friction drag can be reduced by keeping a vehicle's surface highly polished and clean. For the gas, the magnitude of drag depends on the viscosity of the air. Along the solid surface of the body, a boundary layer of low energy flow is generated. The magnitude of the skin friction depends on the state of this flow.

Interference drag results when airflow around one part of a body (such as a fuselage) must occupy the same space as the airflow around another part (such as a wing). The two competing airflows must speed up in order to pass through the restricted area. This speeding up process requires extra energy and creates turbulence, resulting in a measurable increase in the form drag. This velocity increase is present at all airspeeds, but become even more important in the transonic range when the resulting velocity become sonic, producing shock waves.

Parasitic drag is simply the mathematical sum of form drag, skin friction, and interference drag. The phrase parasitic drag is mainly used when discussing lifting wings, since drag is generally small in comparison to lift. For flow around bluff bodies, drag usually dominates, thus the qualifier "parasitic" becomes meaningless. Form drag, skin friction drag, and interference drag on buff bodies are not considered elements of "parasitic drag", but directly as elements of drag.

A fluid flowing past the surface of a body exerts a force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction.

Lift is most commonly associated with the wing of a fixed-wing aircraft, although lift is also generated by lifting bodies. A lifting body is a fuselage with little or no conventional wing, in which the body itself produces lift. Even a simple space capsule, such as the Apollo command module, produces a significant and useful amount of lift. This lift is used to control the trajectory of the capsule, allowing reduced g-forces on the crew, as well as reducing the peak heat transfer into the capsule. While the common meaning of the word "lift" assumes that lift opposes weight, lift in the technical sense can be in any direction with respect to gravity, since it is defined with respect to the direction of flow rather than to the direction of gravity.

Lift depends on the density of the air, the velocity between the object and the air, the air's viscosity and compressibility, the surface area over which the air flows, the shape of the body, and the body's inclination to the flow. The dependence on body shape, inclination, air viscosity, and compressibility is very complex. These complex dependencies are dealt with by the introduction of a single variable, called the lift coefficient, designated C_L. This allows us to collect all the effects into a single equation. The lift equation states that lift force, F_L, is equal to a lift coefficient, C_L, times the density of the air, , times half of the square of the velocity, v, times the reference area, A.

For given air conditions, shape, and inclination of the object, we have to determine a value for C_L to determine the lift. For some simple flow conditions and geometries and low inclinations, aerodynamicists can determine the value of C_L mathematically. But, in general, this parameter is determined experimentally using a wind tunnel.

Lift-to-Drag Ratio

In aerodynamics, the lift-to-drag ratio, or L/D ratio, is the amount of lift generated by a wing or vehicle divided by the aerodynamic drag it creates by moving through the air. The term is calculated for any particular airspeed by measuring the lift generated, then dividing by the drag at that speed. These vary with speed, so the results are typically plotted on a graph. Because lift and drag are both aerodynamic forces, L/D ratio is an indication of the aerodynamic efficiency of the aircraft.

When the lift and drag coefficients are computed using the same reference area, the L/D ratio is,

The speed of sound, or acoustic velocity, is the distance traveled per unit time by a sound wave propagating through an elastic medium. In fluid dynamics, the speed of sound in a fluid medium (gas or liquid) is used as a relative measure for the speed of an object moving through the medium. The speed of an object divided by the speed of sound in the fluid is called the Mach number. The speed of sound, c, is given by the following equation.

where k is the specific heat ratio, R is the specific gas constant, and T is the air temperature in degrees Kelvin. For air on Earth, k is equal to 1.40 and R is equal to 287 N-m/kg-K. (In U.S. customary units, R = 1,715 ft-lb/slug-R, and T is in degrees Rankine).

Mach number, M is given by the equation,

As we have seen, the value of the drag and lift coefficients are strongly dependent on Mach number.

While the terms "subsonic" and "supersonic," in the purest sense, refer to speeds below and above the local speed of sound respectively, aerodynamicists often use the same terms to talk about particular ranges of Mach values. Flight can be roughly classified into the following seven categories:

Dynamic pressure, q, is the kinetic energy per unit volume of a fluid particle. It is defined by the equation,

An important aspect of dynamic pressure is that the aerodynamic stress, i.e. the stress within a structure subject to aerodynamic forces, experienced by an aircraft or missile is proportional to q. Therefore, by looking at the variation of q during flight, it is possible to determine how the stress will vary and, in particular, when it will reach its maximum value. The point of maximum aerodynamic load is often referred to as max Q and it is a critical parameter in many applications, such as during spacecraft launch.

It can be seen that dynamic pressure is part of the drag and lift equations. Substituting equation (3.6) into equations (3.1) and (3.2), we can rewrite the drag and lift equation in terms of q.

The ballistic coefficient, BC, of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the negative acceleration, that is, a high BC indicates a low negative acceleration. Ballistic coefficient is a function of mass, drag coefficient, and reference area (as used in the definition of C_D).

From Newton's second law we see that dividing drag by mass gives the negative acceleration produced by the restraining drag force. From equations (3.1), (3.6) and (3.9), and dividing by m, we obtain

Ballistic coefficient has a significant effect on the flight behavior of satellites and reentry vehicles. We will examine this in more detail later.

Angle of attack, denoted by , is the term used to describe the angle between the chord line of an airfoil and the vector representing the relative motion between the airfoil and the oncoming air. In the case of a rocket, angle of attack is the angle between the longitudinal axis of the rocket and the direction of the air flowing past it. Angle of attack changes during flight as the vehicle changes direction and/or attitude.

The drag and lift forces act through the body's center of pressure. The amount of force produced depends on the angle of attack. If the angle of attack is small, the drag and lift are comparatively small. Increasing the angle of attack increases lift up to a point; however, too high an angle of attack results in a loss of lift and can cause an aircraft to stall. The critical angle of attack is the angle of attack that produces maximum lift.

The variation in drag and lift forces that result from changing angle of attack are accounted for in the value of the drag and lift coefficients. Since C_D and C_L vary with angle of attack, their values are typically plotted on a graph.

Angle of attack generally refers the angle measured in the nose up/nose down direction (pitch). Angle of attack is usually defined to be positive when the relative wind is coming from below the nose of the aircraft. Sideslip angle, , is the angle between the body's reference line and the oncoming flow measured in the side-to-side direction (yaw). Sideslip angle is usually positive when the relative wind is coming from the right of the nose.

The center of pressure, CP, is the point on a body where the total sum of the aerodynamic pressure field acts. Mathematically it can be said that the net pressure force on the body acts through this point. In the same way that the weight of all the rocket components acts through the center of gravity, the aerodynamic forces act through the center of pressure. Typically the center of pressure is referenced to the nose of the vehicle.

There are several important problems to consider when determining the center of pressure for an object. As we change angle of attack, the pressure at every point on the object changes. And, therefore, the location of the center of pressure changes as well. In general, the pressure variation around the object also imparts a torque, or "twisting force", to the rocket. If an object is not restrained in some way it will rotate as it moves through the air.

To resolve some of these design problems, engineers prefer to characterize the forces on an object by the aerodynamic force coupled with an aerodynamic moment to account for the torque. It was found both experimentally and analytically that, if the aerodynamic force is applied at a location 1/4 of the length back from the leading edge on most low speed objects, the magnitude of the aerodynamic moment remains nearly constant with angle of attack. Engineers call the location where the aerodynamic moment remains constant the aerodynamic center. (For supersonic airfoils, the aerodynamic center is nearer the 1/2 chord location.) Using the aerodynamic center as the location where the aerodynamic force is applied eliminates the problem of the movement of the center of pressure with angle of attack in aerodynamic analysis.

When computing the stability of a rocket, we usually apply the aerodynamic forces at the aerodynamic center of airfoils and compute the center of pressure of the vehicle as an area-weighted average of the centers of the components.

For a rocket to be aerodynamically stable, the center of pressure must be located aft of the center of gravity. This ensures that any increased forces resulting from an increased angle of attack results in an increased restoring moment that drives the rocket back to the trimmed position. For small rockets using aerodynamics as the sole means of stabilization, the CP is typically one to two rocket diameters aft of the CG. Large launch vehicles, which operate part of the time outside the atmosphere, include additional means of stabilization, such as engine gimbaling and/or attitude control thrusters.

The center of pressure can be calculated but, in general, this is a complicated procedure requiring the use of calculus. The aerodynamic forces are the result of pressure variations around the surface of the rocket. In general, we must determine the integral of the pressure times the unit normal, times the area, times the distance from a reference line. Then divide by the integral of the pressure times the unit normal, times the area. This procedure involves much work and is too complex to demonstrate here. Simpler methods for locating the center of pressure have been developed that are quite satisfactory for small and model rockets.

Simple Method

A simple version of the calculation procedure can be used to compute the center of pressure of a model rocket. It is assumed that we already know the projected area and location, relative to some reference location, of each of the major parts of the rocket: nose, body tube, and fins. The approximate location of the rocket's center of pressure from the reference line is given by the following equation.

where A_N, A_B and A_F are, respectively, the projected areas of the nose, body tube, and fins; and X_N, X_B and X_F are the distances of each area's centroid from the reference line.

Barrowman Method

Another method for determining the center of pressure is that documented in 1967 by James S. Barrowman of NASA's Sounding Rocket Branch at Goddard Space Flight Center. This simple algebraic method is able to compute to high accuracy the center of pressure of a rocket flying subsonically at small angles of attack.

LN = length of nose d = diameter at base of nose dF = diameter at front of transition dR = diameter at rear of transition LT = length of transition XP = distance from tip of nose to front of transition CR = fin root chord CT = fin tip chord S = fin semispan LF = length of fin mid-chord line R = radius of body at aft end XR = distance between fin root leading edge and fin tip leading edge parallel to body XB = distance from nose tip to fin root chord leading edge N = number of fins

Figure 3.3

Nose

Conical Transition

Use for both increasing and decreasing diameters. For multiple transitions, calculate each separately using different X_P.

Fins

For multiple sets of fins, calculate each set separately using different X_B.

Location of CP

The total rocket force coefficient is the sum of the force coefficients of each part calculated,

The distance of the rocket CP from the nose tip is the sum of the products of the force coefficient and CP of each part divided by the total rocket C_N,

Figure 3.4 Figure 3.5

Figure 3.4 shows the forces acting on a launch vehicle. Note that the weight, W of the vehicle is concentrated at its center of gravity, CG, and the aerodynamic forces—lift, F_L, and drag, F_D—are concentrated at its center of pressure, CP. This configuration, with the center of gravity ahead of the center of pressure, is stable since the lift and drag forces tend to provide a restoring torque about the vehicle's center of gravity. Ideally, thrust, F_T, acts through the centerline of the vehicle, but we can develop a control torque by gimballing the engine nozzle. The flight path angle, , is measured from the local horizon to the velocity vector, and the angle of attack, , is measured from the velocity vector to the centerline of the vehicle.

Aerodynamic forces are used differently on a rocket than on an airplane. On an airplane, lift is used to overcome the weight of the aircraft, but on a rocket, thrust is used in opposition to weight. Because the center of pressure is not normally located at the center of gravity of the rocket, aerodynamic forces can cause the rocket to rotate in flight. The lift of a rocket is a side force used to stabilize and control the direction of flight. While most aircraft have a high lift to drag ratio, the drag of a rocket is usually much greater than the lift.

All forms of drag affect a rocket in flight, with each contributing to the drag coefficient. Skin friction drag is caused by friction between air molecules moving at different speeds in the boundary layer that surrounds the rocket. This effect is coupled to the overall surface area of the rocket. At subsonic speeds, skin friction is normally the largest contributor to overall drag, and even at supersonic speeds, it is a main contributor. Form drag, or pressure drag, is the drag caused by a rocket pushing aside the air in front of it. At subsonic speeds, pressure drag normally contributes only slightly to the overall drag. At supersonic speeds, pressure drag (then commonly called wave drag) can be significant, especially if the forward facing parts of a rocket are blunt. The base area of a rocket produces form drag caused by the wake trailing behind the rocket. During powered flight, base drag is significantly reduced at high altitude as the wake tends to get filled by the expanding exhaust from the engine. Interference drag is generated where two surfaces are intersected at an angle, causing the flow patterns to interfere with each other. For a rocket, this drag is generally caused by the interference of flow over the fins and that over the rocket body. Parasitic drag is caused by objects that are attached to the outside surface of the rocket. Lift-induced drag occurs when the rocket nose, body and fins are inclined to the flight direction.

Optimal design in terms of minimum drag depends heavily on the velocity in question. For rockets spending most of their flight at subsonic speeds, the nosecones should be rounded and they should have boattails. Similarly, the leading edges of the fins should be rounded and have sharp trailing edges. Maximum fin thickness should be closer to the leading edge than to the trailing edge. For rockets spending most of their time at supersonic speeds, it is essential that all forward facing surfaces, i.e. nosecone and leading edges, are sharp. Boattailing and sharp trailing edges are also desirable. The fins should be thin with maximum thickness midway between the leading and trailing edges. In general, the total surface area of any rocket should be kept as small as possible. Also, the cross-sectional area should be kept at its minimum practical value. When drag considerations contradict stability considerations, then stability conditions are of greater importance.

Lift occurs when a flow of gas is turned by a solid object. The flow is turned in one direction, and the lift is generated in the opposite direction, according to Newton's third law of action and reaction. For a rocket, the nose cone, body, and fins can turn the flow and become a source of lift if the rocket is inclined to the flight direction.

As shown in Figure 3.4, the drag and lift forces are defined in relation to the direction of the oncoming air. In some cases it would be convenient to consider the aerodynamic forces relative to the longitudinal axis of the rocket. For this purpose we define the forces F_A and F_N as shown in Figure 3.5, where F_A is the axial force and F_N is the normal force. The geometrical relation between these two sets of components are,

When the rocket is flying at zero angle of attack, drag force and axial force coincide, i.e. F_D=F_A, and F_L=F_N=0.

The axial force and the normal force can be defined using dimensionless coefficients C_A and C_N, where

At very small angles of attack we can approximate C_N to be linear with , where C_N,α is the normal force coefficient gradient, dC_N/d, per degree.

Note that the normal force produces a turning moment, m, around the rocket's center of gravity. The magnitude of this moment is given by the equation,

where X_CG and X_CP are the distances of the CG and CP from a reference point, typically taken as the tip of the rocket's nose.

It is common to normalize moments by specifying a characteristic length, d, where the choice of d is commonly the diameter of the rocket. We define m, called the pitch moment, as follows

where C_m is the pitch moment coefficient. Note that, C_m/C_N = (X_CG-X_CP)/d.

Force Coefficients

Mathematical methods exist for the computation of a rocket's drag coefficient. These methods typically involve computing the contribution to the drag coefficient from all parts of the rocket and from all types of drag, and then taking a mathematical sum of the individual contributions to obtain the total drag coefficient. Generally speaking, these methods are lengthier than we wish to present in the web page. However, if the reader wishes to study these techniques from the original source documents, we refer you to the following: