28 Bluff Body Flows

Introduction

Aerospace engineers are more often than not interested in the aerodynamics of smooth, slender, streamlined shapes that gradually taper to a sharp point at their trailing edges, e.g., airfoils and wings. However, non-streamlined or unstreamlined shapes with blunt front and/or rear faces, called bluff bodies (sometimes referred to as blunt bodies), are also encountered in many engineering applications. While the drag force on a body shape comprises two primary contributors, skin-friction drag and pressure drag, the total drag on bluff bodies is typically dominated by pressure drag. This outcome results from the effects of the large low-pressure zone formed in the body’s wake, as illustrated in Figure 1.

The differences in the flows between a streamlined body and a bluff body. The bluff body induces flow separation and a large, broad wake, characterized by low pressure and high drag.

Understanding bluff-body aerodynamics is essential because of its wide-ranging applications across various engineering fields. Studying how non-streamlined objects interact with airflow helps in designing more efficient structures and vehicles. For instance, understanding unsteady wind loads on bluff bodies, such as buildings and bridges, is essential for ensuring aeroelastic stability and preventing failures. Optimizing vehicle shapes reduces drag, improves fuel efficiency, and leads to better performance. The aerodynamics of bluff bodies also play a role in many sports, including golf, tennis, and baseball. For example, the dimples on a golf ball create turbulent flow, reducing drag and allowing the ball to travel farther. In tennis, the spin imparted on the ball affects its trajectory and speed. Similarly, in baseball and cricket, the stitching on the ball and the manner in which it is thrown, including spin, produce aerodynamic effects that vary with the players’ skill.

Learning Objectives

  • Distinguish the fundamental differences between the flows of streamlined versus bluff bodies.
  • Understand the use of reference areas in defining bluff body drag coefficients and the concept of equivalent drag area.
  • Appreciate why the drag of bluff bodies with smooth surfaces, such as circular cylinders and spheres, shows sensitivities to Reynolds number variations.
  • Become familiar with the Magnus effect on spinning cylinders and spheres.
  • Know about the process of streamlining of aircraft and terrestrial vehicles.
  • Be able to calculate the drag from a parachute and other bluff bodies.

Drag Coefficients

For bluff bodies, the interest is usually in the drag on that body, mainly because experiments have found that drag is the dominant force. This observation, however, does not imply that bluff bodies cannot produce lift because many do. Nevertheless, initially examining only the drag characteristics of such bodies is convenient. Furthermore, bluff bodies may produce pitching moments, which are sometimes necessary for specific engineering applications, such as determining torsional loads.

Recall that the two-dimensional drag coefficient is

(1)   \begin{equation*} C_d = \frac{D'}{\frac{1}{2} \, \varrho_{\infty} \, V_{\infty}^2 \, l} \end{equation*}

where {D'} is the drag per unit span, and {l} is a characteristic length, e.g., for a circular cylinder, l = d, where {d} is its diameter. The lowercase drag coefficient C_d refers to a two-dimensional (sectional) value based on a characteristic length, whereas the uppercase C_D denotes a three-dimensional drag coefficient defined using a reference area.

For a prismatic body of finite span or length b, such as a long circular cylinder, the total drag is D = D' \, b. Therefore, the corresponding drag coefficient may be written as

(2)   \begin{equation*} C_D = \frac{D}{\frac{1}{2} \, \varrho_{\infty} \, V_{\infty}^2 \, l \, b} \end{equation*}

where the product l\,b is the reference area. For an ideal two-dimensional body, the sectional coefficient C_d is normally used; for a finite-length body, C_D is based on the selected reference area. In general, for a three-dimensional object, the drag coefficient is defined as

(3)   \begin{equation*} C_D = \frac{D}{\frac{1}{2} \, \varrho_{\infty} \, V_{\infty}^2 \, A_{\rm ref}} \end{equation*}

where A_{\rm ref} denotes the reference area, typically the projected frontal area. Using a sphere of diameter {d} as an example, its projected area is {A_{\rm ref} = \pi d^2/4}. Likewise, a cube with side length {l} will have a reference area of A_{\rm ref} = l^2.

However, caution is warranted when defining drag coefficients for three-dimensional bodies, as the drag coefficient depends on the choice of the body’s reference area, which may not be unique. For example, on the one hand, for specific bodies of revolution, the volumetric drag coefficient is used by convention, with the reference area given by the square of the cube root of the volume. On the other hand, in hydrodynamics, the analysis of submerged streamlined bodies uses the wetted surface area as the reference area. Therefore, to consistently use and compare drag coefficients, it is imperative to adopt the convention used in the particular field of application.

To avoid ambiguity in the definitions of drag coefficients across different body shapes, the term “drag coefficient” per se is not always used. Instead, the equivalent drag area is used, often denoted by {\scriptstyle{f}} or A_{\mathrm{eq}}. The equivalent drag area is defined as

(4)   \begin{equation*} f \equiv A_{\mathrm{eq}} = \frac{D}{\frac{1}{2} \, \varrho_{\infty} \, V_{\infty}^2} \end{equation*}

where {\scriptstyle{f}} or A_{\mathrm{eq}} would be measured in units of area.

The advantage of using equivalent drag areas is that they eliminate ambiguity in defining a reference area. This approach is often used to estimate the drag of complex shapes composed of many simpler shapes, such as an entire airplane. In such a case, the equivalent drag area of the complex shape is obtained by a drag synthesis of the equivalent drag areas of N simple shapes, i.e.,

(5)   \begin{equation*} f = f_1 + f_2 + f_3 + \cdots + f_N \end{equation*}

However, it would be incorrect to write

(6)   \begin{equation*} { C_D = C_{D_{1}} + C_{D_{2}} + C_{D_{3}} + \cdots + C_{D_{N}} } \end{equation*}

because the reference areas on which each value of C_D is based could be different, and generally will be. To take into account component interference, i.e., to make some allowance for flow interference effects between the different components, then

(7)   \begin{equation*} f = K \left( f_1 + f_2 + f_3 + \cdots + f_N \right) \end{equation*}

where K > 1 and often K = 1.2 will be used for aircraft drag synthesis without any other information, i.e., a 20% drag penalty from component interference effects. A general rule is not to add drag coefficients unless the reference area is the same for all coefficients.

Drag Coefficients of Simple Shapes

There are many bluff body shapes for which the drag (or drag coefficient) must be known for engineering purposes. However, in practice, far fewer such shapes are encountered, most of which have been well-studied, and the drag coefficients have been measured in wind tunnels. Furthermore, flows around bluff bodies are often unsteady and can produce various types of vortex shedding, as shown in the animation in Figure 2. Therefore, the measured or computed drag is usually reported as a time-averaged value taken over many cycles of the unsteady wake motion.

CFD simulation of unsteady flow and periodic vortex shedding produced by a bluff-body shape.

Drag coefficients for standard two- and three-dimensional bluff-body shapes are widely published and serve as a valuable resource for engineers. These shapes can be further categorized into two primary subclasses of bluff bodies: 1. Those with sharp, angular shapes tend to have fixed separation locations and drag coefficients that are relatively insensitive to the Reynolds number. 2. Those with smooth contours exhibit variable flow separation points and may be sensitive to the Reynolds number. These effects are summarized in the schematic of Figure 3.

The aerodynamics of smooth bluff bodies depend on the Reynolds number, but angular bodies with sharp corners tend to have fixed flow separation points.

Of course, there is always the possibility that some bluff body shapes combine smooth contours and sharp edges. In this case, determining the drag coefficient and the effects of the Reynolds number may require specialized wind-tunnel measurements. Values for some common bluff-body shapes are shown in the tables in Figures 4 and 5.

 

Measured drag coefficients of some simple two-dimensional bluff bodies.

For example, a sphere or a cylinder is smooth, whereas a cube is angular. The flows about smooth bluff bodies or those with rounded corners generally exhibit some sensitivity to the Reynolds number, and their drag coefficients vary as a consequence, as shown in the tables. In contrast, angular shapes exhibit little to no variation with changes in the Reynolds number, at least after reaching a minimum threshold.

 

Measured drag coefficients of some simple three-dimensional bluff bodies.

Check Your Understanding #1 – Calculating the drag on a bluff body

What will be the drag on a solid hemisphere with a frontal area of 0.26 m2 and a flow speed of 100 m/s? Hint: The drag will depend on which side of the hemisphere points into the wind.
Show solution/hide solution.

The drag force on the hemisphere is given by

    \[ D = \frac{1}{2} \, \varrho \, V ^2 \, C_D \, A_{\rm ref} \]

where A_{\rm ref} = 0.26 m2 and V = 100 m/s, and if MSL ISA conditions are assumed then \varrho = 1.225 kg/m3. Looking up the value of a solid hemisphere’s drag coefficient in the table in Figure 5, it can be seen that the C_D value depends on whether the smooth side or the flat side faces the relative wind. In the former case, C_D = 0.42, and in the latter case, C_D = 1.17. Inserting the values for the smooth side facing the flow gives

    \[ D = \frac{1}{2} \, \varrho \, V ^2 \, C_D \, A_{\rm ref} = 0.5 \times 1.225 \times 100.0^2 \times 0.42 \times 0.26 = 668.9 \mbox{ N} \]

And for the flat side facing the flow, the drag is

    \[ D = \frac{1}{2} \, \varrho \, V ^2 \, C_D \, A_{\rm ref} = 0.5 \times 1.225 \times 100.0^2 \times 1.17 \times 0.26 = 1,863.2 \mbox{ N} \]

Flow Patterns Around a Circular Cylinder

A classic example of a well-studied two-dimensional bluff body flow is that of a circular cylinder, for which the flow readily separates near its maximum thickness or just downstream; representative examples are shown in Figure 6. A wake is produced downstream of the cylinder, which usually contains turbulence and various sizes of vortical eddies. As might be expected for a smooth body, the flow about the cylinder and the resulting wake structure and drag are sensitive to variations in Reynolds number, which in this case is defined based on the diameter of the cylinder, {d}.

Various flow states can be produced around a circular cylinder, depending primarily on the Reynolds number.

At very low Reynolds numbers, near unity, known as “creeping flow” or “Stokes flow,” the behavior is significantly influenced by viscosity rather than inertia, and the flow remains steady and attached. At slightly higher Reynolds numbers, beginning at about Re \sim 5, a stable pair of symmetric recirculating vortices forms in the wake of the cylinder. The flow remains steady, but a high drag coefficient is observed.

As the Reynolds number is increased, the steady symmetric wake becomes unstable and develops alternate vortex shedding, producing two rows of counter-rotating vortices known as a von Kármán vortex street. This unsteady flow creates fluctuating aerodynamic forces on the cylinder, especially an alternating lift force transverse to the freestream; the drag also fluctuates, usually at twice the shedding frequency. However, no time-averaged lift is produced because the mean flow remains symmetric with respect to the freestream direction, even though the instantaneous vortex shedding is alternating and unsteady. This periodic flow also creates a fluctuating pressure field that manifests as an Aeolian tone (after Aeolus, the Greek god of the wind). This effect is the source of the buzzing or “singing” sound sometimes heard when the wind blows over high-tension power wires.

As the Reynolds number increases through the subcritical range, the wake becomes increasingly turbulent and aperiodic, although alternate vortex shedding remains present. The boundary layer generally remains laminar at separation, while transition and turbulence develop in the separated shear layers and wake. A major downstream movement of the separation points does not occur until the critical Reynolds-number regime, where boundary-layer transition occurs before separation and produces the drag crisis discussed below.

In the ideal inviscid limit, the flow would remain attached and symmetric about the body, resulting in zero drag. This result is known as d’Alembert’s paradox, after the mathematician Jean le Rond d’Alembert. The paradox is that ideal inviscid, incompressible, steady flow theory predicts zero drag on a body, whereas real bodies moving through fluids experience drag because viscosity produces boundary layers, flow separation, wakes, and drag.

Drag of Circular Cylinders & Spheres

The corresponding drag coefficient on a circular cylinder as a function of the Reynolds number is shown in Figure 7. Note that the results are for a smooth cylinder; surface roughness will alter the quantitative relationships. The drag coefficient for a sphere is also shown for reference. The drag coefficient decreases quickly with increasing Reynolds number Re (note the logarithmic scales) until Re reaches approximately 10^3, after which the drag remains nominally constant. This outcome arises because, after the initial low-Reynolds-number regime, the separated wake structure becomes relatively established. Hence, the drag coefficient changes more slowly as the Reynolds number increases until the critical regime is approached.

The drag coefficient of a circular cylinder and a sphere as a function of Reynolds number based on diameter. Notice the logarithmic scales.

With further increases in Re, it will be apparent that there is a critical value of Re where the drag coefficient suddenly decreases by a substantial factor. This condition, called the critical Reynolds number, is often termed the drag crisis (after Hoerner) and corresponds to boundary-layer transition occurring before separation. The resulting turbulent boundary layer has greater momentum near the wall, so it can remain attached farther around the rear of the cylinder before separating.

The critical Reynolds number for a cylinder is about Re = 2 \times 10^5. The narrower wake subsequently reduces pressure drag and overall drag. However, with further increases in Re, the drag coefficient rises again as the boundary-layer transition and separation locations continue to change, causing the wake to broaden. Pressure drag remains the dominant contribution, although turbulent skin friction also increases.

The drag coefficient of smooth and rough spheres as a function of Reynolds number. The results for a circular disk are also shown as a reference.

The aerodynamic behavior of a sphere is qualitatively similar to that of a circular cylinder, as shown in Figure 8. Again, there is a sudden drop in drag as the Reynolds number reaches a critical value because the boundary layer transitions to turbulence before separating, allowing separation to move farther downstream. However, in this case, the surface roughness effects are of some interest, which can cause the drag reduction on the sphere to occur at a lower value of the critical Reynolds number.

Check Your Understanding #2 – Drag coefficient of a circular cylinder

Consider nominally two-dimensional flow past a long circular cylinder of diameter D = 0.4~\mathrm{m}. The freestream velocity is V_{\infty} = 50~\mathrm{m/s}. The properties of air at standard conditions are \varrho_{\infty} = 1.225~\mathrm{kg/m^3} and \mu_{\infty} = 1.7894 \times 10^{-5}~\mathrm{kg\,m^{-1}\,s^{-1}}.

  1. Determine the Reynolds number for this flow.
  2. Identify the flow regime around the cylinder.
  3. Estimate the sectional drag coefficient, C_d, for the cylinder.
Show solution/hide solution.
  1. The Reynolds number is

    (8)   \begin{equation*} Re_D = \frac{\varrho_{\infty} \, V_{\infty} \, D}{\mu_{\infty}} \end{equation*}

    Substituting values gives

    (9)   \begin{equation*} Re_D = \frac{1.225 \times 50.0 \times 0.4}{1.7894 \times 10^{-5}} = 1.37 \times 10^6 \end{equation*}

  2. This Reynolds number lies above the nominal critical value associated with boundary-layer transition and the drag crisis on a smooth circular cylinder. The flow is in the supercritical or post-critical range, although the precise transition and separation state depends strongly on surface roughness, freestream turbulence, and the experimental conditions.
  3. From the smooth-cylinder drag curve in Figure 7, the sectional drag coefficient at this Reynolds number is approximately C_d \approx 0.5 to 0.7. The exact value depends on surface roughness, freestream turbulence, aspect ratio, and the details of the post-critical drag behavior.

Roughness Effects on Cylinders & Spheres

Surface roughness modifies the flow around cylinders and spheres, primarily affecting their drag by promoting earlier transition of the boundary layer to turbulence. When the transition occurs before laminar boundary-layer separation, the turbulent boundary layer can resist adverse pressure gradients better, flow separation is delayed, the downstream wake narrows, base pressure rises, and the total drag coefficient C_D drops sharply. This behavior first occurs at or near a critical Reynolds number, Re_{\text{crit}}. The underlying flow physics is similar for both circular cylinders and spheres, although the details of the wake structure and the numerical values of C_D differ.

This behavior is illustrated in Figure 9, using flow visualization of a sphere, where the roughness causes a noticeable reduction in the width and extent of the downstream wake. It explains the reduction in drag, i.e., the roughness does not change the actual Reynolds number, but it promotes boundary-layer transition and makes the separation behavior resemble that of a smooth sphere at a higher Reynolds number.

Flow visualization about a sphere with a smooth surface (left) at a Reynolds number of about Re = 3 \times 10^5, and the same sphere at the same Reynolds number with surface roughness (right).

The explanation for this behavior is that surface roughness causes the laminar boundary layer on the upstream side of the sphere to transition earlier into a turbulent boundary layer, which can remain attached to the sphere’s surface much longer than a laminar boundary layer. The consequence of this behavior is a narrower low-pressure wake, leading to lower drag on the sphere. In effect, adding roughness produces the same outcome that would naturally occur at higher Reynolds numbers, i.e., a turbulent boundary layer and delayed separation. This behavior is often interpreted as an increase in an effective Reynolds number, in the sense that the flow undergoes transition and separation characteristics similar to those at higher actual Reynolds numbers. This is why roughness is often described as increasing the effective Reynolds number, as it shifts the flow state into the post-critical regime at a lower actual Reynolds number.

The drag coefficient measurements for a cylinder in Figure 10 indicate that increasing roughness generally reduces the effective critical Reynolds number, thereby causing the drag crisis to occur at a lower Reynolds number. The influence of roughness on the external flow over cylinders and spheres can be described using the nondimensional parameter \dfrac{\epsilon}{D}, where \epsilon is a characteristic surface roughness height and D is the cylinder or sphere diameter. This ratio provides a convenient way to compare the effects of roughness across different body sizes. However, roughness does not simply reduce drag at all Reynolds numbers. Well below the drag crisis, roughness may increase skin-friction drag or move separation unfavorably; near the critical regime, it can promote earlier boundary-layer transition, delay separation, narrow the wake, and sharply reduce pressure drag.

The drag coefficient of a circular cylinder as a function of Reynolds number and surface roughness. Increasing roughness shifts the drag crisis to lower Reynolds numbers.

The underlying flow physics is similar for spheres, as shown in Figure 11, but the drag coefficients are considerably lower. Once the transition occurs before separation, the turbulent boundary layer remains attached farther around the rear of the sphere, narrowing the downstream wake and thereby reducing pressure drag. The details of the transition process and wake structure differ between cylinders and spheres, but the overall sequence of changes in the flow regimes is analogous. A smooth sphere has a subcritical drag coefficient of about C_D \approx 0.45 to 0.50, followed by a sharp decrease through the drag crisis. Surface roughness, including golf-ball dimples, can shift the crisis to lower Reynolds numbers, allowing much lower drag coefficients over part of the golf-ball flight range. However, the resulting value of C_D depends strongly on the Reynolds number and the roughness height, so no single rough-sphere value applies universally.

The drag coefficient of smooth and rough spheres as a function of Reynolds number. Surface roughness shifts the drag crisis to lower Reynolds numbers. Data adapted from Achenbach.

This overall behavior is summarized in the table below.[1]

Representative drag coefficients for smooth versus rough or tripped cylinders and spheres. Exact values depend on Reynolds number and {\epsilon/D}.
Body Regime (Re) Smooth C_D Rough/Tripped C_D Notes
Cylinder Subcritical ≈ 1.1–1.5 ≈ 1.1–1.5 Roughness may increase drag before crisis
Cylinder Critical Drops near Re \sim 2\times10^5 Drops at lower Re Roughness shifts the drag crisis left
Cylinder Post-critical ≈ 0.5–0.9 ≈ 0.9–1.2 Rough cylinders may have higher post-critical drag
Sphere Subcritical ≈ 0.45–0.50 ≈ 0.45–0.50 Laminar separation; broad wake
Sphere Critical Drops at higher Re Can drop to ≈ 0.1–0.3 Roughness/dimples can trigger earlier crisis
Sphere Post-critical ≈ 0.1–0.2 ≈ 0.1–0.4 Values depend strongly on roughness height

Why Do Golf Balls Have Dimples?

Ever wonder why a golf ball has dimples? The clue is in Figure 11. Did you know that in the earliest days of golf, in the windy, cold links of Scotland, kilted golfers[2] made a remarkable discovery? They found that older, worn, and rougher balls traveled farther than new ones. This discovery was not merely an observation; it significantly affected the game’s outcomes. Golfers[3] using these rough balls gained a considerable competitive edge, requiring many fewer strokes from the tee to the hole.

Scottish golfers found that well-worn balls often flew farther than new smooth ones, an observation that eventually led to the modern dimpled golf ball.

Unsurprisingly, the unique behavior of these rough spheres caught the attention of scientists, who began studying their aerodynamics. The sphere drag data discussed previously in Figure 11 show that surface roughness can shift the drag crisis to lower Reynolds numbers, allowing a rough sphere to have much less drag than a smooth sphere over part of the golf-ball flight range. The Reynolds number for a golf ball’s flight is approximately 8 \times 10^4 to 2.1 \times 10^5. This discovery also led to the standardization of golf-ball dimensions and aerodynamic characteristics so that the outcome of the game would depend primarily on the golfer’s skill rather than on the particular ball being used. Roughness reduces the ball’s aerodynamic drag in a way that depends on Reynolds number, so controlling the ball’s dimensions and surface construction helps maintain consistent performance standards.

The reduction in drag of the dimpled ball at the Reynolds number of its flight causes it to travel much farther than a smooth ball of the same diameter.

A modern golf ball is a machine-dimpled sphere, and the dimples provide a controlled form of surface roughness to deliberately alter its aerodynamics. The dimples cause the boundary layer on the upstream side of the ball to transition from laminar to turbulent flow earlier than it would on a smooth sphere, delaying flow separation and thereby reducing pressure drag. At Reynolds numbers corresponding to golf-ball flight, which are typically of order Re = 10^5, the flow on a smooth ball would usually remain laminar over much of the forward surface and separate relatively early, producing a broad wake and higher pressure drag, as shown in Figure 13.

However, a turbulent boundary layer, created using surface roughness, will remain attached to the ball’s surface much longer than a laminar boundary layer before it separates. This behavior results in a narrower low-pressure wake and lower pressure drag. Interestingly, creating turbulence reduces drag in this case, i.e., the dimples serve as a form of flow control. However, do not expect dimples to work on streamlined bodies, such as wings. For streamlined bodies, the objective is usually to keep the flow attached and to minimize the total drag. Laminar flow can reduce skin-friction drag, but surface roughness or dimples generally increase skin friction and can also disturb the carefully designed pressure distribution. Therefore, adding dimples to a streamlined body will usually increase drag rather than reduce it.

The consequence of reduced drag is that a golfer can hit the ball farther. The trajectory of a golf ball with quadratic drag cannot generally be represented by a simple closed-form range equation because the horizontal and vertical velocity components change continuously during flight. However, a useful first approximation can be obtained by neglecting lift, assuming a constant drag coefficient, and numerically integrating the two-dimensional equations of motion. If x is horizontal distance, z is height, and V = \sqrt{\overbigdot{x}^2+\overbigdot{z}^2}, then

(10)   \begin{equation*} m \, \overbigddot{x} = -\frac{1}{2} \, \varrho \, A \, C_D \, V \, \overbigdot{x} \end{equation*}

and

(11)   \begin{equation*} m \, \overbigddot{z} = -m \, g -\frac{1}{2} \, \varrho \, A \, C_D \, V \, \overbigdot{z} \end{equation*}

These equations are integrated from the initial conditions when the ball is struck at \overbigdot{x}(0)=V_0\cos\theta, \overbigdot{z}(0)=V_0\sin\theta, x(0)=0, and z(0)=0 until the ball returns to z=0. This model still neglects backspin, which produces lift and can significantly affect the range, as well as variations of C_D with Reynolds number and spin. Nevertheless, it provides a physically consistent way to show how reducing the drag coefficient increases the range.

Check Your Understanding #3 – Reynolds number and drag coefficient of a golf ball

What is a golf ball’s flight Reynolds number? What are representative drag coefficients for a smooth sphere and for a rough or dimpled golf ball under these flight conditions? Why is this important regarding the flight distance of a golf ball? To answer this question, the issues to address are: What is the diameter of a golf ball? How fast does a golf ball travel? How does surface roughness affect this distance?

Show solution/hide solution.

A little research will reveal that, according to golf rules, a golf ball’s diameter must be at least 1.68 inches (42.67 mm). Additionally, an amateur golfer typically hits a ball at an average speed of 125 mph (201.2 kph) with a driver club. However, those speeds can vary significantly with a golfer’s skill level, ranging from about 60 mph (96.6 kph) to 160 mph (257.5 kph). Therefore, the Reynolds number of a golf ball can only be established within certain bounds.

The Reynolds number of the golf ball based on its diameter is

    \[ Re_D = \frac{\varrho \, V D}{\mu} \]

where \varrho and \mu can be assumed to take MSL ISA values, i.e., in SI units, \varrho = 1.225 kg/m3 and \mu = 1.7894 \times 10^{-5} kg m-1 s-1. Inserting the values using 257.5 kph or 71.53 m/s gives

    \[ Re_D = \frac{1.225 \times 71.53 \times 0.04267}{1.7894 \times 10^{-5}} = 2.09 \times 10^5 \]

and for 201.2 kph or 55.88 m/s gives

    \[ Re_D = \frac{1.225 \times 55.88 \times 0.04267}{1.7894 \times 10^{-5}} = 1.63 \times 10^5 \]

and for 96.6 kph or 26.82 m/s gives

    \[ Re_D = \frac{1.225 \times 26.82 \times 0.04267}{1.7894 \times 10^{-5}} = 7.8 \times 10^4 \]

Therefore, the Reynolds number for a golf ball’s flight is approximately 8 \times 10^4 to 2.1 \times 10^5.

This Reynolds-number range is important because it lies near the range where surface roughness can strongly affect boundary-layer transition and separation on a sphere. According to Figure 11, a smooth sphere in this range typically has a drag coefficient of about C_D \approx 0.45 to 0.50, depending on Reynolds number and surface condition. A rough or dimpled sphere can have a much lower drag coefficient if the roughness shifts the drag crisis into this Reynolds-number range. Representative values for a dimpled golf ball are often of the order of C_D \approx 0.25 to 0.35, but the exact value depends on Reynolds number, spin rate, dimple geometry, and surface condition. Therefore, these values should be interpreted as representative engineering estimates.

This outcome is important because the dimples reduce the drag of the ball in a manner that depends on the Reynolds number and, therefore, on how hard it is struck. Standardizing golf-ball dimensions and aerodynamic characteristics helps ensure that all conforming balls satisfy the same equipment rules.

This reduction in drag is very important regarding the flight distance of a golf ball. To illustrate the effect, consider a launch speed of 135 mph, or V_0 = 60.39~\mathrm{m/s}, a launch angle of \theta = 15^\circ, a golf-ball mass of m = 0.0459~\mathrm{kg}, and a diameter of d = 0.04267~\mathrm{m}. The projected frontal area is

    \[ A = \frac{\pi \, d^2}{4} = 1.43 \times 10^{-3}~\mathrm{m^2} \]

Neglecting lift and assuming a constant drag coefficient, the trajectory is obtained by numerically integrating

    \[ m \, \ddot{x} = -\frac{1}{2} \, \varrho \, A \, C_D \, V \, \overbigdot{x} \]

and

    \[ m \, \ddot{z} = -m \, g -\frac{1}{2} \, \varrho \, A \, C_D \, V \, \overbigdot{z} \]

where

    \[ V = \sqrt{\overbigdot{x}^2+\overbigdot{z}^2} \]

Using \varrho = 1.225~\mathrm{kg/m^3} and g = 9.81~\mathrm{m/s^2}, a representative smooth-sphere value of C_D = 0.50 gives a range of approximately

    \[ R_{\mathrm{smooth}} \approx 93~\mathrm{m} \]

or about 101 yards. Repeating the numerical integration using a representative dimpled-ball value of C_D = 0.30 gives

    \[ R_{\mathrm{dimpled}} \approx 114~\mathrm{m} \]

or about 124 yards. In this drag-only comparison, the lower drag coefficient increases the range by about 21 m, or approximately 23%. These values are not intended to predict the full range of an actual golf shot because the model neglects backspin-induced lift, wind, launch height, and variations in C_D during flight. However, the calculation provides a physically consistent demonstration of why dimples can substantially increase the flight distance.

Aerodynamics of Spinning Cylinders & Spheres

Another effect experienced by golf balls and other balls used in various games and sports is a lift force produced by their spinning motion, such as backspin or topspin. The ball’s spin spoils the flow’s horizontal symmetry, creating differential pressure and producing a lift force. This phenomenon is known as the Magnus effect, named after Heinrich Magnus, the German scientist who first studied it in the 19th century. The effect is illustrated in Figure 14. In some cases, Magnus observed a tendency of a cannonball to curve or veer significantly away from its expected flight path. He concluded that this behavior was a result of the residual spinning motion of the cannonball as it left the barrel. Isaac Newton also observed this same behavior while watching people play tennis.

The spin of a ball (sphere), in this case, a topspin or a backspin, will cause a type of lift called a Magnus lift.

The dependence of the Magnus lift coefficient, C_l, on the translational and rotational speeds of a smooth cylinder of diameter D has been investigated and measured in wind tunnels, a summary of some results being reproduced in Figure 15, where {V_\infty} is its translational or freestream velocity and \Omega is the rotational angular velocity. The results can be generalized in terms of a non-dimensional spin rate, i.e.

(12)   \begin{equation*} S = \frac{\Omega \, D}{2\, V_\infty} = \frac{\Omega \, R}{V_\infty} \end{equation*}

where R = D/2. Notice the additional dependency on the Reynolds number, as given by

(13)   \begin{equation*} Re_D = \frac{\varrho_\infty \, V_\infty D}{\mu_\infty} \end{equation*}

Indeed, the relationship between the Magnus lift coefficient and the Reynolds number is highly nonlinear. These effects are related to a “moving wall” effect that alters the boundary-layer characteristics over the top and bottom halves of the cylinder, resulting in premature laminar separation on the upper half and delayed separation on the lower half. This latter behavior is also characteristic of spheres and is central to the physics of all ball games. Under some combinations of Reynolds number, spin ratio, and surface condition, a “reverse Magnus effect” can occur, in which the lift acts opposite to the usual Magnus-force direction. This behavior is again tied to boundary-layer transition and flow separation.

Results showing the Magnus lift coefficient, C_l, on the translational and rotational speeds of a smooth cylinder.

For an idealized inviscid rotating cylinder, the lift per unit span may be written as

(14)   \begin{equation*} L' = 2\pi \, \varrho_\infty \, V_\infty \, R^2 \, \Omega \end{equation*}

The corresponding lift coefficient, normalized by the dynamic pressure and cylinder diameter D = 2R, is

(15)   \begin{equation*} C_l = \frac{L'} {\tfrac{1}{2} \, \varrho_\infty \, V_\infty^2 \, D} = 2\pi \left( \frac{\Omega \, R}{V_\infty} \right) = 2\pi \, S \end{equation*}

This linear relation provides an idealized reference for the Magnus lift at small spin ratios. For a real cylinder, the measured lift also depends on the Reynolds number, surface roughness, and the locations of boundary-layer transition and separation. Consequently, the relationship between C_l and S becomes nonlinear, as shown in Figure 15, and the lift may eventually saturate at higher spin ratios. Under some combinations of Reynolds number, spin ratio, and surface condition, the reverse Magnus effect may also occur.

Surface roughness is known to promote earlier transition to turbulence and to alter the separation pattern, thereby changing the resulting Magnus force.[4] This effect may be represented empirically by allowing a lift-scaling factor and a saturation parameter to depend on both Re_D and the roughness parameter {\epsilon/D}, i.e.,

(16)   \begin{equation*} F_S = F_S(Re_D,\epsilon/D) \qquad \beta = \beta(Re_D,\epsilon/D) \end{equation*}

where \epsilon is a characteristic roughness height. This notation emphasizes that roughness alters boundary-layer transition and separation behavior rather than the actual Reynolds number.

For a spinning sphere of radius R, measurements show that the Magnus lift still increases with the spin ratio S, but with a smaller initial slope than the 2\pi value obtained for an idealized rotating cylinder. A generic empirical representation is

(17)   \begin{equation*} C_{L,\text{sph}}(S,Re_D) = F_{S,\text{sph}}(Re_D) \frac{\alpha \, S} {1+\beta_{\text{sph}}(Re_D)\,S} \end{equation*}

where \alpha is an empirical constant that sets the nominal small-spin slope, {F_{S,\text{sph}}(Re_D)} is a Reynolds-number-dependent correction factor that accounts for changes in boundary-layer transition and separation, and \beta_{\text{sph}}(Re_D) is an empirical saturation parameter that controls how rapidly the lift departs from a linear dependence on S. The values of these quantities must be obtained from experimental data for the particular sphere and surface condition being considered.

At small spin ratios, for which \beta_{\text{sph}}S \ll 1, the relation reduces to

(18)   \begin{equation*} C_{L,\text{sph}} \simeq \alpha \, F_{S,\text{sph}}(Re_D)\,S \end{equation*}

so that \alpha F_{S,\text{sph}} represents the effective initial slope of the C_L versus S curve. At larger spin ratios, the denominator limits the continued growth of the lift coefficient, giving the asymptotic value

(19)   \begin{equation*} C_{L,\max,\text{sph}}(Re_D) \simeq \frac{\alpha}{\beta_{\text{sph}}(Re_D)} F_{S,\text{sph}}(Re_D) \end{equation*}

This form is intended as a convenient empirical correlation rather than a universal law because the fitted functions depend on Reynolds number, surface roughness, seams or dimples, and other details of the sphere.

Topspin increases the relative flow speed over the lower side of the ball, reducing the pressure there and producing a downward aerodynamic force that causes the ball to dip more quickly. Backspin produces the opposite effect, generating an upward force that allows the ball to remain aloft longer and travel farther. Sidespin produces a lateral force, giving rise to curved trajectories such as a baseball curveball or a soccer banana kick.

The magnitude and direction of these forces depend on the spin rate, Reynolds number, and surface condition of the ball, including seams, dimples, or surface roughness. At sufficiently high spin ratios, the Magnus lift may begin to saturate, so further increases in spin produce progressively smaller changes in aerodynamic force.

Complex Three-Dimensional Bluff Bodies

The flows produced by more general three-dimensional bluff body shapes are considerably more complex. Other types of flow phenomena are shown in Figure 16, which depicts the behavior of the so-called “airwake” over a ship’s surface. Scarf or horseshoe vortices tend to form and wrap around the base (or foot) of bluff bodies that protrude from a surface (wall), in this case, the funnel. These vortices are, again, another source of drag. In addition, complex flow separation and reattachment regions on the body often make the a priori estimation of drag on such bodies very difficult.

The flow around a complex bluff body, especially one that protrudes from a surface, is highly complex and three-dimensional. In this case, the flow concerns the rear of a ship with a flight deck for helicopter operations.

Vortex-Induced Vibration

One concern with three-dimensional bluff-body flows is that the aerodynamic forces generated by vortex shedding can be sufficiently strong to excite the structure itself. One of the most important effects in this regard is vortex-induced vibration (VIV), where the unsteady aerodynamic excitation produced by vortex shedding causes the structure to vibrate. VIV problems can arise in many engineering applications, not just in aerospace systems, wherever periodic vortex shedding occurs. Long or flexible structures such as wires, chimneys, towers, cables, and bridge decks can be particularly susceptible.

The Strouhal number is commonly used to characterize vortex shedding. It relates the shedding frequency to the flow velocity and a characteristic dimension of the body (the diameter d for a cylinder). The Strouhal number St is defined as

(20)   \begin{equation*} St = \frac{f_{\rm st} \, d}{V_{\infty}} \end{equation*}

where f_{\rm st} is the vortex shedding frequency and {V_{\infty}} is the freestream velocity. For a circular cylinder, the Strouhal number is approximately St \approx 0.2 over a wide range of Reynolds numbers.

The lock-in phenomenon occurs when the vortex-shedding frequency approaches a system or structure’s natural vibration frequency, leading to severe oscillation. If the aerodynamic excitation is strong enough and the structural damping is insufficient, the resulting cyclic stresses can quickly cause fatigue damage or failure. In aeroelastic problems, related self-excited instabilities such as flutter can also occur, and the oscillation amplitude may grow until nonlinear effects limit the motion or structural failure occurs.

Wakes Shed from Bridges

In civil engineering, unsteady aerodynamic loading on suspension bridges, which are bluff bodies in cross-section, is a significant concern and can lead to bridge failures. When the Tacoma Narrows Bridge over Puget Sound in Washington state collapsed in 1940 because of torsional flutter in strong winds, it was captured on film. The lessons learned have become a textbook example of aeroelastic instability in bluff bridge decks. Vortex-induced vibration and torsional flutter are related concerns for flexible bridge structures, but they are not the same phenomenon.

Figure 17 shows the essence of the problem. The bluff “H”-shaped cross-section of the bridge causes separated flow and unsteady aerodynamic loading on the bridge deck. These loads can produce vertical and torsional oscillations, and if the aerodynamic moments feed energy into the structural motion faster than structural damping can dissipate it, the oscillations can grow rapidly. Depending on the bridge’s characteristics, such as weight, torsional stiffness, damping, and natural frequencies, the deformations can be substantial and may quickly exceed 20 degrees.

In cross-section, a bridge deck can cause vortex shedding and oscillatory aerodynamic forces that can lead the bridge to twist and oscillate in the wind.

Twisting of the bridge deck changes the local angle of attack and modifies the separated flow around the deck, which can increase the aerodynamic moments that drive the motion. While the deck structure resists lifting and twisting because of its inherent stiffness and damping, the motion can become unstable if the aerodynamic forces feed energy into the structure faster than damping can dissipate it. In some cases, it may be possible to modify the bridge’s cross-sectional shape to reduce excitation from separated flow, improve aerodynamic damping, and increase the critical wind speed for instability, thereby mitigating torsional flutter.

Streamlining

Streamlining a bluff or otherwise unstreamlined body shape is an effective technique for reducing drag. For example, the basic idea is shown in Figure 18. Adding a tail and/or nose fairing can significantly delay flow separation, reducing drag. However, a consideration is that the fairing adds extra weight to the aircraft.

Streamlining a body can reduce drag, although it is not always a viable design option.

While streamlining may not be a viable engineering option in some cases because of practical or cost constraints, it is always worthwhile to reduce an aircraft’s overall drag and improve its performance, even at the expense of a slight increase in structural weight. For example, carefully blending a wing to the fuselage using a fairing can weaken the horseshoe vortex system and reduce interference drag through improved streamlining, as shown in the photograph in Figure 19 for a general aviation airplane, with an almost negligible increase in airframe weight. For a commercial aircraft, the wing-root fairing is designed to reduce interference drag and may also help smooth the aircraft’s cross-sectional area distribution for transonic flight. The area rule states that wave drag is reduced when the total cross-sectional area varies smoothly and continuously along the aircraft’s length. In practical terms, abrupt changes in the slope or curvature of this area distribution should be avoided, especially where wings, nacelles, pylons, or tail surfaces add cross-sectional area.

A wing root fairing can help reduce interference drag between the wing and the fuselage, thereby decreasing the total aircraft drag.

An airplane with a fixed landing gear can benefit from adding fairings and spats around the landing gear and wheels. Of course, retracting the wheels entirely is the only way to almost eliminate the drag. However, the additional weight and higher cost of a mechanical system for retractable landing gear are usually not viable for smaller aircraft, such as those in the general aviation class.

Other aircraft that can benefit from streamlining include helicopters, which have relatively unstreamlined airframes for their utility. Using fairings at strategic locations on a helicopter’s airframe can reduce drag by 10% to 30%, allowing it to fly faster and/or achieve better range or endurance.

Streamlining of Terrestrial Vehicles

The aerodynamic drag of terrestrial vehicles, such as automobiles, trucks, and trains, increases significantly at higher driving speeds. For a given drag coefficient, C_D, the drag, D, on the vehicle will increase with the square of its speed V, i.e.,

(21)   \begin{equation*} D = \frac{1}{2} \, \varrho \, V^2 \, C_D \, A_{\rm ref} \end{equation*}

So the power required to overcome the aerodynamic drag, P_{\rm req}, will increase with the cube of the driving speed, i.e.,

(22)   \begin{equation*} P_{\rm req} = D V = \frac{1}{2} \, \varrho \, V^3 \, C_D \, A_{\rm ref} \end{equation*}

Therefore, the aerodynamic power required increases rapidly with speed. For a fixed distance traveled, however, the aerodynamic energy required is proportional to the drag and therefore increases with the square of the speed. Rolling resistance, or friction, also affects the net resistance on the vehicle and the fuel or energy needed for propulsion; however, aerodynamic effects dominate at higher speeds.

As with flight vehicles, terrestrial vehicles can reduce energy consumption by lowering drag, i.e., by streamlining to minimize C_D. Aerospace engineers possess the training and expertise to understand the steps required to achieve profitable drag reductions. Much of the testing of terrestrial vehicles is conducted in wind tunnels, although CFD solutions can also be beneficial.

Automobiles & Cars

The automotive industry has become proficient at streamlining vehicle designs to reduce fuel consumption and internal noise, a process that typically requires extensive wind tunnel testing. Turbulence about the vehicle, such as from rear-view mirrors, can be a significant source of internal noise. The photograph in Figure 20 below shows an example of flow visualization about an automobile. The judicious use of smoke filaments and appropriate lighting can provide insight into flow patterns, indicating where flow is attached or separated, thereby informing decisions about where streamlining may be needed.

Example of smoke flow visualization about an automobile in a wind tunnel.

As shown in Figure 21, the careful streamlining of automobile shapes can reduce drag. The result is a significant decrease in fuel consumption and a notable reduction in internal noise, as a result of corresponding reductions in turbulence. However, streamlining can only be taken so far, beyond which compromises in the vehicle’s utility may become problematic. Exposed wheels are a significant source of drag, so there has been a trend toward more careful streamlining of the wheel well and the vehicle’s underside.

Streamlining an automobile by rounding off sharp corners can cut drag by half and commensurately improve fuel consumption.

Trucks (Lorries) & Buses

As shown in Figure 22, trucks (lorries) and buses are aerodynamically inefficient because of their “boxy” shapes, which lead to high base-pressure drag. For most trucks, their average fuel consumption is 50% to 200% higher than that of automobiles. For example, an 18-wheeler truck used to transport most goods to supermarkets and similar establishments might achieve 5 to 8 miles per gallon (2.1 to 3.4 km per liter). Still, these values can vary significantly depending on engine size, driving speed, and load (weight). The large, rectangular, box-like shapes often seen being towed by tractor-trailers are poor aerodynamic shapes, characterized by high drag. These boxes are typically referred to as intermodal shipping containers or ISO containers (ISO 6346); the older term CONEX is sometimes used informally, especially for containerized cargo units derived from military usage.

Tractor-trailers are bluff bodies and suffer from high drag and poor fuel economy.

Engineers have four significant areas of interest in obtaining meaningful aerodynamic drag reduction on trucks and lorries. These are:

  1. The frontal shape of the tractor part, which is somewhat unstreamlined, is partly because of utility requirements (engine access) and cooling needs.
  2. The gap between the tractor’s end and the trailer’s leading edge (or the front of the CONEX container) can cause flow separation and turbulence.
  3. The shapes of the sides and underbody of the trailer, which shed turbulence and create drag.
  4. The very back of the trailer is a significant source of pressure drag.

Continuous improvements to the aerodynamics of trucks have reduced their drag, resulting in increased fuel efficiency. Minimizing the gap between the trailer and the tractor is essential, so fairings on the top of the tractor are often used, as shown in Figure 23. In addition, rubber skirts can be installed beneath the trailer and along its sides to redirect high-speed airflow away from the trailer’s underside, thereby reducing drag from the wheels, which also act as bluff bodies.

Reductions in drag can be achieved by minimizing the gap between the tractor and trailer, modifying the trailer’s sides and underbody, and adding a fairing at the rear.

Trailer tails are increasingly used on trucks, which are pop-out afterbody fairings, as also shown in Figure 23. Although relatively crude in shape, they help reduce drag from flow separation and turbulence at the trailer’s rear end. Figure 24 shows that even a short trailer tail reduces base-pressure drag. The corresponding reduction in fuel consumption makes their use worthwhile when considering the overall fuel costs of the truck fleet. Indeed, a slight decrease in drag can significantly reduce fuel costs when the number of vehicles and the total miles traveled are considered cumulatively.

Trailer tails (in this case, the effects of a 2-ft. long trailer tail versus a 4-ft. long tail) can significantly reduce a truck’s drag and, thus, improve its fuel efficiency.

Streamlining the aerodynamics of buses and coaches is also essential in improving fuel efficiency. Manufacturers have begun introducing increasingly more streamlined shapes with minimized frontal areas and curved or tapered roofs. Spoilers, underbody skirts, and wheel covers further improve airflow management, while the careful design of doors, mirrors, and other external components helps reduce turbulence. While employing wind tunnel testing and computational fluid dynamics simulations during the design phase of a bus or coach is a significant investment in time and cost, the resulting improved aerodynamic performance can yield substantial fuel savings and a lower environmental impact.

Trains

Traveling by train has long been a vital mode of transportation. High-speed railway networks continue to grow, particularly in Europe, China, and Japan. As with all vehicles, the aerodynamic drag of a train increases with the square of its speed, so the power and energy required to propel it increase significantly at higher speeds. Related issues for trains include supplying the necessary traction from the wheels to the rails and the significant pressure changes (felt by passengers) when trains pass each other at high speeds or travel through tunnels.

Because the length-to-width ratio of a train is much larger than that of any other ground vehicle, the aerodynamic characteristics of trains tend to be more complex than those of cars, trucks, or airplanes. Nevertheless, significant drag reductions can be achieved through streamlining, as shown in the photograph below. The basic principles are similar to those for airplanes, including a streamlined nose and rounded corners along the train’s entire length. Again, much of the work on the most effective drag-reduction measures for high-speed trains using various forms of streamlining is conducted in wind tunnels.

A high-speed train with a highly streamlined nose and rounded corners on the carriages can significantly reduce drag.

Creating Drag – Parachutes

A parachute is an aerodynamic decelerator that reduces the speed of a descending or moving object by producing aerodynamic force and removing mechanical energy from the system. Parachutes operate over a wide range of flight regimes and include several distinctly different configurations. Personnel and cargo parachutes generally operate at subsonic speeds, aircraft drag parachutes provide deceleration during landing or rejected takeoff, and spacecraft parachutes may deploy in low-density atmospheres at high subsonic or supersonic Mach numbers.

Parachutes may be divided broadly into drag parachutes and ram-air parachutes, as illustrated in Figure 26. Drag parachutes include round, conical, annular, ribbon, ringsail, and disk-gap-band canopies. Their aerodynamic force acts predominantly opposite to the direction of motion, although they may also produce lateral forces and oscillatory moments. These configurations are used for emergency descent, cargo delivery, aircraft braking, store recovery, and spacecraft entry and descent. Ram-air parachutes, often called parafoils, are inflated fabric wings formed from interconnected cells. They produce both lift and drag, travel forward while descending, and provide substantially greater maneuverability and landing control. Ram-air canopies are widely used for sport parachuting and many personnel applications.

Force and velocity relationships for drag and ram-air parachutes.

Drag Parachutes

For a drag parachute, the aerodynamic force acts predominantly opposite to the direction of motion. The drag is written as

(23)   \begin{equation*} D = \frac{1}{2} \, \varrho \, V^2 \, C_D \, A_{\rm ref} \end{equation*}

where C_D is the drag coefficient and A_{\rm ref} is the specified reference area of the canopy. Depending on the parachute configuration and the source of the aerodynamic data, the reference area may be the projected frontal area, the nominal canopy area, or another defined canopy area. A quoted drag coefficient must be used with the reference-area definition on which it is based.

For a drag parachute descending nearly vertically at a steady speed, the airspeed relative to the surrounding atmosphere is approximately equal to the rate of descent V_d. The total supported weight W includes the payload, harness, suspension lines, and canopy. At terminal descent,

(24)   \begin{equation*} W - D = 0 \end{equation*}

Therefore,

(25)   \begin{equation*} W = \frac{1}{2} \, \varrho \, V_d^2 \, C_D \, A_{\rm ref} \end{equation*}

and the steady rate of descent is

(26)   \begin{equation*} V_d = \sqrt{ \frac{2 \, W} {\varrho \, C_D \, A_{\rm ref}} } \end{equation*}

The descent speed increases with supported weight and decreases with increasing canopy area or drag coefficient.

A hemispherical shell provides a useful idealized model of a drag parachute. For an open hemispherical shell of diameter d, the projected frontal area is

(27)   \begin{equation*} A_{\rm ref} = \frac{\pi \, d^2}{4} \end{equation*}

A representative drag coefficient based on this projected area is approximately C_D = 1.42, although the value for an actual parachute also depends on canopy depth, porosity, venting, suspension geometry, and degree of inflation. Substitution of the projected area gives

(28)   \begin{equation*} V_d = \sqrt{ \frac{8 \, W} {\varrho \, C_D \, \pi \, d^2} } \end{equation*}

Check Your Understanding #4 – Rate of descent of a conventional parachute

Consider an open hemispherical parachute supporting a total weight of W = 980~\mathrm{N} in SI units or W = 220~\mathrm{lb} in USC units. The canopy diameter is d = 7.32~\mathrm{m} or d = 24~\mathrm{ft}, and its drag coefficient based on projected frontal area is C_D = 1.42. At MSL ISA conditions, determine the steady rate of descent in both SI and USC units. For this estimate, neglect the drag of the suspension lines, harness, and payload relative to the canopy drag.

Show solution/hide solution.

For steady vertical descent,

(29)   \begin{equation*} V_d = \sqrt{ \frac{2 \, W} {\varrho \, C_D \, A_{\rm ref}} } \end{equation*}

In SI units, the projected area is

(30)   \begin{equation*} A_{\rm ref} = \frac{\pi \times 7.32^2}{4} = 42.1~\mathrm{m^2} \end{equation*}

At MSL ISA conditions, \varrho = 1.225~\mathrm{kg/m^3}, so

(31)   \begin{equation*} V_d = \sqrt{ \frac{2 \times 980} {1.225 \times 1.42 \times 42.1} } = 5.17~\mathrm{m/s} \end{equation*}

In USC units, the projected area is

(32)   \begin{equation*} A_{\rm ref} = \frac{\pi \times 24^2}{4} = 452.4~\mathrm{ft^2} \end{equation*}

At MSL ISA conditions, \varrho = 0.002377~\mathrm{slug/ft^3}, so

(33)   \begin{equation*} V_d = \sqrt{ \frac{2 \times 220} {0.002377 \times 1.42 \times 452.4} } = 17.0~\mathrm{ft/s} \end{equation*}

The two results are equivalent apart from rounding.

Round and related drag parachutes are rarely exact hemispheres. Their canopies may be flat-circular, conical, extended-skirt, annular, cruciform, ribbon, ringsail, or disk-gap-band configurations. An apex vent or distributed geometric porosity allows air to pass through the canopy, which can reduce oscillation and improve stability. Slots, gaps, ribbons, and rings may also control inflation loads and improve structural survivability. These features alter both the drag coefficient and the unsteady aerodynamic behavior.

Ram-Air Parachutes

A ram-air parachute operates as a flexible wing rather than primarily as a drag-producing device. Air enters openings along the leading edge and pressurizes interconnected cells, producing an airfoil-shaped canopy. The canopy travels forward while descending along an inclined flight path and produces both lift and drag. Unlike a rigid wing, its aerodynamic shape depends on internal pressure, fabric permeability, suspension-line loading, brake settings, and aerodynamic forces. The canopy geometry and its aerodynamic characteristics are therefore closely coupled.

The aerodynamic forces are written as

(34)   \begin{equation*} L = \frac{1}{2} \, \varrho \, V^2 \, C_L \, A_{\rm ref} \qquad \text{and} \qquad D = \frac{1}{2} \, \varrho \, V^2 \, C_D \, A_{\rm ref} \end{equation*}

where C_L and C_D are the lift and drag coefficients, V is the airspeed along the flight path, and A_{\rm ref} is normally the specified canopy planform area. The lift acts perpendicular to the flight path, while the drag acts opposite to the direction of motion. The coefficients must be used with the reference-area convention associated with the measured data.

Consider a steady gliding descent at a flight-path angle \gamma below the horizontal. The forward speed and rate of descent are V_{\rm forward} = V \, \cos\gamma and V_d = V \, \sin\gamma. For steady, unaccelerated flight, D = W \, \sin\gamma and L = W \, \cos\gamma, so

(35)   \begin{equation*} \tan\gamma = \frac{D}{L} = \frac{C_D}{C_L} \end{equation*}

The ratio of forward speed to rate of descent is

(36)   \begin{equation*} \frac{V_{\rm forward}}{V_d} = \frac{1}{\tan\gamma} = \frac{L}{D} = \frac{C_L}{C_D} \end{equation*}

Therefore, the steady glide ratio of a ram-air parachute is equal to its aerodynamic lift-to-drag ratio. A higher value of L/D allows the parachute to travel farther forward for a given loss of altitude.

The resultant aerodynamic force balances the supported weight, so

(37)   \begin{equation*} W = \frac{1}{2} \, \varrho \, V^2 \, A_{\rm ref} \sqrt{C_L^2 + C_D^2} \end{equation*}

and hence

(38)   \begin{equation*} V = \sqrt{ \frac{2 \, W} {\varrho \, A_{\rm ref} \, \sqrt{C_L^2 + C_D^2}} } \end{equation*}

The flight-path angle is \gamma = \tan^{-1}(C_D/C_L).

Figure 27 shows measured values of L/D versus angle of attack for two semi-rigid MC-4/5 ram-air parachutes having the same geometry but different fabric permeability. One parachute used porous fabric, while the other used nonporous fabric. Both had an aspect ratio of 2 and were tested at a Reynolds number of approximately 8.2 \times 10^5. The measurements represent the canopies alone and do not include the drag of the suspension lines, slider, harness, or payload.

Measured lift-to-drag ratio versus angle of attack for porous and nonporous semi-rigid ram-air parachute canopies.

As shown in Figure 27, L/D initially increases with angle of attack, reaches a maximum, and then decreases for both parachutes. The porous parachute reaches a maximum lift-to-drag ratio of approximately (L/D)_{\max} = 5.1 near \alpha = 8^\circ. The nonporous parachute reaches a slightly higher maximum value of approximately (L/D)_{\max} = 5.3 near \alpha = 9^\circ. The nonporous parachute has a maximum L/D approximately 4% higher than that of the porous parachute in these tests. The shallowest steady glide angle occurs at the maximum value of L/D, with

(39)   \begin{equation*} \gamma_{\min} = \tan^{-1} \left( \frac{1}{(L/D)_{\max}} \right) \end{equation*}

For the porous parachute, \gamma_{\min} = \tan^{-1}(1/5.1) = 11.1^\circ, while for the nonporous parachute, \gamma_{\min} = \tan^{-1}(1/5.3) = 10.7^\circ. The difference in minimum glide angle is approximately 0.4^\circ for the isolated canopies.

These glide angles apply only to the measured canopies. The suspension lines, slider, harness, payload, and parachutist produce additional drag but essentially no lift. The total drag of the complete parachute system may be expressed as

(40)   \begin{equation*} D_{\rm total} = D_{\rm canopy} + D_{\rm lines} + D_{\rm slider} + D_{\rm harness} + D_{\rm payload} \end{equation*}

The corresponding system lift-to-drag ratio is

(41)   \begin{equation*} \left( \frac{L}{D} \right)_{\rm system} = \frac{L_{\rm canopy}} {D_{\rm total}} \end{equation*}

Consequently, the complete-system glide ratio is lower than the canopy-only values shown in Figure 27.

If all drag coefficients are defined using the same dynamic pressure and the canopy planform area as the common reference area, then the aggregate drag coefficient of the suspension lines, slider, harness, and payload may be written as

(42)   \begin{equation*} C_{D,\rm additional} = \frac{ D_{\rm lines} + D_{\rm slider} + D_{\rm harness} + D_{\rm payload} }{ \tfrac{1}{2} \, \varrho \, V^2 \, A_{\rm ref} } \end{equation*}

The complete-system drag coefficient is then

(43)   \begin{equation*} C_{D,\rm system} = C_{D,\rm canopy} + C_{D,\rm additional} \end{equation*}

The drag coefficients can be added in this form only because they are based on the same dynamic pressure and reference area. The complete-system glide relations become

(44)   \begin{equation*} \tan\gamma = \frac{C_{D,\rm system}}{C_L} \end{equation*}

and

(45)   \begin{equation*} V = \sqrt{ \frac{2 \, W} {\varrho \, A_{\rm ref} \sqrt{ C_L^2 + C_{D,\rm system}^2 }} } \end{equation*}

Check Your Understanding #5 – Performance of a ram-air parachute

Consider a porous ram-air parachute supporting a total weight of W = 980~\mathrm{N} in SI units or W = 220~\mathrm{lb} in USC units. The canopy reference area is A_{\rm ref} = 26.0~\mathrm{m^2} or A_{\rm ref} = 280~\mathrm{ft^2}. Near \alpha = 8^\circ, the measured canopy coefficients are approximately C_L = 0.56 and C_{D,\rm canopy} = 0.11, based on the canopy planform area. The complete parachute system has a measured glide ratio of (L/D)_{\rm system} = 3.0. Assume that this system glide condition corresponds approximately to the same canopy operating point and that the canopy produces essentially all of the lift.

Determine:

  1. The effective complete-system drag coefficient.
  2. The aggregate drag coefficient of the suspension lines, slider, harness, and payload.
  3. The steady flight-path angle.
  4. The airspeed, forward speed, and rate of descent in both SI and USC units.
Show solution/hide solution.

Because all coefficients are referenced to the same canopy planform area, the effective complete-system drag coefficient is

(46)   \begin{equation*} C_{D,\rm system} = \frac{C_L}{(L/D)_{\rm system}} = \frac{0.56}{3.0} = 0.187 \end{equation*}

The aggregate drag coefficient of the suspension lines, slider, harness, and payload is

(47)   \begin{equation*} C_{D,\rm additional} = C_{D,\rm system} - C_{D,\rm canopy} = 0.187 - 0.11 = 0.077 \end{equation*}

This subtraction is valid because both drag coefficients are based on the same dynamic pressure and canopy planform area.

The flight-path angle is

(48)   \begin{equation*} \gamma = \tan^{-1} \left( \frac{C_{D,\rm system}}{C_L} \right) = \tan^{-1} \left( \frac{0.187}{0.56} \right) = 18.4^\circ \end{equation*}

In SI units,

(49)   \begin{equation*} V = \sqrt{ \frac{2 \times 980} {1.225 \times 26.0 \times \sqrt{0.56^2 + 0.187^2}} } = 10.21~\mathrm{m/s} \end{equation*}

The forward speed is

(50)   \begin{equation*} V_{\rm forward} = V \, \cos\gamma = 10.21 \, \cos 18.4^\circ = 9.69~\mathrm{m/s} \end{equation*}

The rate of descent is

(51)   \begin{equation*} V_d = V \, \sin\gamma = 10.21 \, \sin 18.4^\circ = 3.23~\mathrm{m/s} \end{equation*}

In USC units,

(52)   \begin{equation*} V = \sqrt{ \frac{2 \times 220} {0.002377 \times 280 \times \sqrt{0.56^2 + 0.187^2}} } = 33.5~\mathrm{ft/s} \end{equation*}

The forward speed is

(53)   \begin{equation*} V_{\rm forward} = V \, \cos\gamma = 33.5 \, \cos 18.4^\circ = 31.8~\mathrm{ft/s} \end{equation*}

The rate of descent is

(54)   \begin{equation*} V_d = V \, \sin\gamma = 33.5 \, \sin 18.4^\circ = 10.6~\mathrm{ft/s} \end{equation*}

The measured complete-system glide ratio determines only the aggregate additional drag. It does not identify the separate drag contributions of the suspension lines, slider, harness, and payload.

Beyond the steady glide condition, control inputs deform the ram-air canopy or change its trailing-edge shape, thereby altering its lift coefficient, drag coefficient, flight-path angle, and airspeed. These changes allow the pilot to turn, control the glide path, and flare before landing. During a flare, the canopy operates transiently, so the steady-flight equations do not by themselves describe the complete maneuver.

For both drag and ram-air parachutes, deployment is a highly unsteady inflation process in which the aerodynamic force can rise rapidly. The resulting opening load depends on deployment speed, atmospheric density, canopy geometry, fabric porosity, payload mass, suspension-system properties, and inflation time. Reefing may be used to restrict the canopy during the initial stage of deployment and then allow it to expand progressively, thereby reducing peak opening loads. The canopy and suspended payload also form a coupled aerodynamic and structural system, and canopy oscillation, breathing, collapse, reinflation, wake interaction, and pendular motion of the payload can affect stability and loads. Consequently, the applicable aerodynamic coefficients must correspond to the particular canopy geometry, reference-area convention, inflation state, Reynolds number, porosity, control setting, and operating condition.

Landing Drag Parachutes

A drag parachute may also be used to decelerate an aircraft after touchdown. In this application, the parachute does not support the vehicle’s weight but produces an additional horizontal retarding force that supplements the aerodynamic drag and the braking forces at the wheels. The longitudinal equation of motion during the landing roll may be written as

(55)   \begin{equation*} m \, \frac{dV}{dt} = - D_{\rm airframe} - D_{\rm chute} - F_{\rm brakes} - F_{\rm rolling} \end{equation*}

where D_{\rm chute} is the drag produced by the parachute. If the chute is fully inflated, then

(56)   \begin{equation*} D_{\rm chute} = \frac{1}{2} \, \varrho \, V^2 \, C_{D,\rm chute} \, A_{\rm ref} \end{equation*}

Because the drag varies approximately with V^2, it is most effective immediately after touchdown, when the speed is highest. As the vehicle slows, the parachute force decreases rapidly, and the wheel brakes become increasingly important.

The Space Shuttle Orbiter provides a useful example. A drag-parachute system was added beginning with STS-49 in 1992 (Figure 28) to improve landing deceleration and reduce the demand on the wheel brakes.[5] The system used a 40-ft nominal-diameter conical ribbon main parachute extracted by a 9-ft pilot parachute. The main canopy had to deploy well behind the Orbiter because its broad aft body, vertical tail, engine nozzles, and orbital-maneuvering-system pods produced a large turbulent wake.

The Space Shuttle used a 40-ft nominal-diameter conical ribbon main parachute extracted by a 9-ft pilot parachute.

Deployment timing was important. The chute was deployed after main-gear touchdown while the nose was being lowered toward the runway. If the full parachute force had developed immediately, the high attachment point on the vertical tail would have produced a substantial nose-up pitching moment between the drag-parachute attachment point and the aircraft center of gravity. This moment could interfere with the derotation maneuver and, under some conditions, increase the tendency of the Orbiter to become airborne again.

To control the opening force and pitching moment, the main parachute was initially reefed. Reefing restricted the inflated canopy area during the first stage of deployment and limited the initial drag. After the Orbiter had slowed and the nose gear was approaching the runway, the reefing restraint was released and the canopy inflated more fully. The drag parachute was retained during the high-speed portion of the ground roll and then jettisoned before the aircraft slowed to taxiing speed.

Bluff Bodies at Supersonic Speeds

The behavior of bluff bodies at high speeds is also of interest to aerospace engineers, particularly for spacecraft re-entering Earth’s atmosphere. The drag of supersonic projectiles has been studied in wind tunnels. The results show that their drag is comprised not only of a high value of pressure drag from flow separation on the aft part of the body but also wave drag from the formation of shock waves, as illustrated in Figure 29 for a sphere (ball) flying at supersonic speeds. Notice the strong compression bow shock that stands off from the leading edge of the sphere, as well as the turbulence in the wake behind the sphere.

Shadowgraph flow visualization of a sphere flying at a supersonic Mach number of 1.53.

The effects of Reynolds number on the drag of a sphere have been previously discussed, but what about the effects of Mach number? Classic high-Reynolds-number data for spheres flying at Mach numbers between 0.6 and 2.0 date back to the 19th century, when Francis Bashforth conducted drag measurements on spherical projectiles moving at high subsonic and supersonic speeds between 1870 and 1880, as shown in Figure 30. Interestingly, after the rapid rise in drag through the transonic range, the drag coefficient tends to plateau at higher Mach numbers because the flow is then dominated by a detached bow shock, high forebody pressure, and a separated wake whose overall pressure distribution changes more slowly with further increases in Mach number.

Francis Bashforth measured the drag coefficient of spherical projectiles, showing that the drag coefficient varies through the transonic range but becomes much less sensitive to Mach number at higher supersonic speeds.

H. Julian “Harvey” Allen and Alfred J. Eggers Jr., aeronautical engineers at NACA, developed the blunt-body concept for atmospheric re-entry vehicles. Their approach recognized that bluff or “blunt” bodies moving at very high supersonic and hypersonic Mach numbers generate strong shock waves, known as bow shocks, that form ahead of the body, as shown in Figure 31. Notice that the bow shock is positioned away from the body surface, a distance referred to as the standoff distance. Across this detached shock, much of the kinetic energy of the incoming flow is converted into internal energy in the shock layer, and the large standoff distance keeps much of the hottest gas away from the surface. Therefore, compared with a sharp-nosed body, a blunt body can reduce the rate of heat transfer into the vehicle.

A bluff body at a high supersonic Mach number represents a space capsule during re-entry into the atmosphere.

For otherwise similar hypersonic flow conditions, the stagnation-point convective heating rate decreases approximately with the square root of the nose radius, i.e.,

(57)   \begin{equation*} \overbigdot{q}_{\rm stag} \, \propto \, \frac{1}{\sqrt{R_{\rm le}}} \end{equation*}

where {R_{\rm le}} is the local nose radius of curvature. Therefore, increasing the nose radius reduces the stagnation-region heating rate, although the complete heating relation also depends on velocity, density, gas properties, chemistry, and wall conditions. Also, note the significant flow separation and turbulence in the body’s wake in this image. The bluff-body shape also generates significant pressure drag, helping slow the spacecraft as it re-enters the atmosphere until it is sufficiently decelerated for a parachute to be deployed, illustrating how drag can be beneficial.

Towed Banners

A towed banner provides a useful counterexample to the simplest definition of a bluff body. Banners towed behind general aviation airplanes (Figure 32) are used for aerial advertising and consist of a flexible fabric suspended on a towline that deforms and billows in the airstream, producing a large effective drag area dominated by pressure drag and an unsteady wake. A rigid bluff body usually has a fixed geometry that enforces large-scale flow separation and a well-defined wake. A banner does not impose separation in this deterministic way; the coupled interaction between aerodynamic loading and fabric deformation governs its instantaneous shape. Consequently, there are no fixed separation points or uniquely defined wake structures, and the flow remains inherently unsteady with continuously evolving regions of separation and vortex shedding.

Banners towed behind general aviation airplanes are used for advertising.

The key point is that the banner undergoes continuous, unsteady motion, including billowing, fluttering, and oscillation, producing a broad downstream wake and a drag comparable to that of a bluff body. The drag behavior of a banner is best expressed in terms of drag coefficient, C_D, defined by

(58)   \begin{equation*} C_D = \frac{D}{\tfrac{1}{2}\,\varrho_{\infty} \, V_{\infty}^2 \, A} \end{equation*}

where D is the measured drag of the banner of physical area A.

The drag coefficient measurements in Figure 33 for four banners in a wind tunnel show that, for a given banner, the drag coefficient is approximately constant over a wide range of wind speeds. This behavior confirms that the time-averaged drag scales approximately with the dynamic pressure, even though the banner’s instantaneous motion may be highly unsteady. Notice that C_D depends strongly on the aspect ratio of the banner, A\!R. For a banner of length L and width W, the planform area is A = L \, W and the aspect ratio is

(59)   \begin{equation*} A\!R = \frac{L}{W} \end{equation*}

For banners of the same fabric area, lower-aspect-ratio shapes produce a larger drag coefficient than longer, more slender banners.

Drag coefficient of fabric banners as a function of wind speed for different aspect ratios.

Banners are known to produce a turbulent downstream wake because of their unsteady deformations, so the aspect-ratio dependence may be interpreted using a wake-scaling argument. Although the separation is not fixed deterministically, the time-averaged wake still governs the drag. Therefore,

(60)   \begin{equation*} D \sim \varrho_{\infty} \, V_{\infty}^2 \, A_{\mathrm{wake}} \end{equation*}

where A_{\mathrm{wake}} is an effective cross-sectional area of the disturbed flow. If the characteristic lateral extent of the wake is denoted by \delta, then A_{\mathrm{wake}} \sim L\,\delta. As a purely heuristic scaling model, suppose that the transverse excursion of the banner scales with its width but decreases with increasing aspect ratio, for example, as

(61)   \begin{equation*} \delta \sim \frac{W}{\sqrt{A\!R}} \end{equation*}

Therefore,

(62)   \begin{equation*} A_{\mathrm{wake}} \sim \frac{L \, W}{\sqrt{A\!R}} = \frac{A}{\sqrt{A\!R}} \end{equation*}

It follows that

(63)   \begin{equation*} C_D = \frac{A_{\mathrm{eq}}}{A} \sim \frac{A_{\mathrm{wake}}}{A} \sim A\!R^{-1/2} \end{equation*}

which is qualitatively consistent with the measured trend in Figure 33 that the equivalent drag area decreases with increasing aspect ratio. The exponent -1/2 should not be interpreted as a universal law.

A physical interpretation of this drag behavior is that lower-aspect-ratio banners undergo greater transverse excursions and produce broader wakes (see Figure 34). In contrast, higher-aspect-ratio banners or streamers tend to align more closely with the flow, undergo smaller deformations, and generate narrower downstream wakes, resulting in lower drag. The oscillatory motion itself is generally multimodal in both the lateral and longitudinal directions, and the production of any appreciable pressure difference between the sides of the banner may cause it to curl up along its longitudinal edges. At lower wind speeds, large-amplitude, low-frequency motions tend to dominate, whereas at higher wind speeds the motion shifts toward smaller-amplitude, higher-frequency oscillations. Transitions between these modes can produce noticeable changes in drag, as verified by the wind tunnel tests. The contribution of viscous shear stresses (skin-friction drag) is also present, but it is not the dominant source of drag on a towed banner.

Testing suggests that long, slender banners tend to produce less drag than short, wide banners.

Another interpretation of banner drag can be framed in terms of energy. As the freestream flows past the banner, energy is continuously transferred from the mean flow into the turbulent wake and into the unsteady deformation of the flexible fabric, so that

(64)   \begin{equation*} D\,V_{\infty} \sim \overbigdot{E}_{\mathrm{wake}} + \overbigdot{E}_{\mathrm{diss}} \end{equation*}

where \overbigdot{E}_{\mathrm{wake}} is the rate of kinetic energy dissipation in the wake and \overbigdot{E}_{\mathrm{diss}} represents additional dissipation associated with fabric deformation and oscillatory motion, the balance depending in part on the fabric properties. For a towed system, this energy must be supplied by the towing aircraft, so the additional power required is

(65)   \begin{equation*} \Delta P_{\mathrm{tow}} = D \, V_{\infty} \end{equation*}

which shows that the drag arises from a continual drain of energy from the flow into wake losses and structural motion; consequently, banner towing demands a significant increase in power for sustained flight.

Check Your Understanding #6 – How draggy is a banner compared to a parachute?

Figure 33 shows wind tunnel measurements of the drag coefficient, C_D, for flexible fabric banners of different aspect ratios, A\!R, for a reference area of 0.075 m2. The goal is to consider a banner with A\!R=10 and compare its drag with that of a hemispherical parachute. For the hemispherical canopy, the drag coefficient is approximately C_{D,\mathrm{hem}} = 1.42, based on its projected frontal area.

  1. Show that the equivalent drag area of the hemispherical parachute is A_{\mathrm{eq,hem}} = 0.71 A_f.
  2. From the plotted data in Figure 33, estimate C_D for the AR=10 banner and explain its significance.
  3. Compare the drag of the banner with that of the hemispherical parachute for the same fabric area.
  4. Consider a banner 2 m high and 20 m long. Estimate its equivalent drag area, and show that for this case, its drag is approximately the same as that of a hemispherical parachute whose diameter is about equal to the banner height.
Show solution/hide solution.

1. For a hemispherical parachute canopy of radius R, the fabric area is A_f = 2\pi R^2 and the projected frontal area is

    \[ A_p = \pi R^2 = \frac{A_f}{2} \]

Therefore, because the drag coefficient based on projected frontal area is C_{D,\mathrm{hem}} = 1.42, the equivalent drag area of the hemispherical parachute is

    \[ A_{\mathrm{eq,hem}} = C_{D,\mathrm{hem}} \, A_p = 1.42\left(\frac{A_f}{2}\right) = 0.71 \, A_f \]

2. From Figure 33, for A\!R=10, the drag coefficient of the banner is C_D \approx 0.13. For the test banner, the fabric area is A_f = 0.075\ \mathrm{m}^2, so

    \[ A_{\mathrm{eq,ban}} = C_D \, A_f \approx 0.13 \times 0.075 \approx 0.01\ \mathrm{m}^2 \]

and its significance is that this drag coefficient directly gives the equivalent drag area when multiplied by the fabric area.

3. For the same fabric area, then

    \[ A_{\mathrm{eq,ban}} \approx 0.13 \, A_f \qquad \text{and} \qquad A_{\mathrm{eq,hem}} = 0.71 \, A_f \]

Therefore,

    \[ \frac{A_{\mathrm{eq,ban}}}{A_{\mathrm{eq,hem}}} = \frac{0.13}{0.71} \approx 0.18 \]

so the A\!R=10 banner produces only about 18% of the drag of a hemispherical parachute made from the same amount of fabric. Note that this comparison cannot be performed using the drag coefficient because the reference areas differ.

4. Scaling to a banner 2 m high and 20 m long, then A_f = 40\ \mathrm{m}^2 and A\!R = 10. Its equivalent drag area is A_{\mathrm{eq,ban}} \approx 0.13 \times 40 = 5.2\ \mathrm{m}^2. For a hemispherical parachute of diameter d, then

    \[ A_{\mathrm{eq,hem}} = 1.42 \frac{\pi \, d^2}{4} \]

Equating this to the banner drag area gives

    \[ 1.42 \frac{\pi \, d^2}{4} = 5.2 \]

so d \approx 2.1\ \mathrm{m}. Therefore, in this case, the drag of the 2 m by 20 m banner is approximately equivalent to that of a hemispherical parachute whose diameter is about equal to the banner height. As a final step, one can show that this equivalence is not a general result but depends on the banner’s aspect ratio.

Summary & Closure

While aerospace engineers are more often concerned with minimizing drag through streamlining, in other cases, creating drag can also be beneficial. The study of bluff-body shapes, with their almost infinite possibilities, is not merely theoretical; it also has practical applications. Many standard shapes have been studied in wind tunnels, and their drag characteristics are well known and published as functions of the Reynolds number and, in some cases, the Mach number. Examples include projectiles and re-entry vehicles, which are crucial in aerospace engineering.

The flow features and drag characteristics of angular bluff body shapes are primarily independent of variations in Reynolds number, whereas smooth bluff bodies show more sensitivity to Reynolds number. Most smooth bluff bodies (e.g., circular cylinders and spheres) exhibit a critical Reynolds number, below and above which the flow state changes significantly and the resulting drag behavior differs. Drag reduction for terrestrial vehicles remains a substantial challenge for reducing fuel consumption. While conventional streamlining practices are highly effective, they are approaching diminishing returns. However, the future may see more innovative concepts, such as active flow-control devices, to further reduce drag.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Make a list of the parts of an airplane that are potentially bluff-body-producing drag elements. Besides removing them, what steps could be taken to reduce the drag of these elements?
  • A rectangular sensor package is desired to be mounted on the external fuselage of a high-speed aircraft. What are the potential engineering concerns?
  • If dimples can reduce drag on golf balls, then why are dimples not used on the surfaces of airplanes or automobiles?
  • For the nose cone of a rocket, is it best to use a sharply pointed nose or a blunt, rounded nose, and why?

Other Useful Online Resources

For more information on bluff bodies, check out some of these online resources:

  1. Data ranges adapted from Achenbach (1971), Schlichting & Gersten (Boundary-Layer Theory), Hoerner (Fluid-Dynamic Drag), and modern golf-ball aerodynamics studies. The numerical values are representative only; exact values depend on Reynolds number, surface roughness, freestream turbulence, and the definition of the roughness parameter. ↵
  2. They were probably not wearing kilts but trews (trousers), although it still makes a good story. ↵
  3. In Scots, the term is gowfers. ↵
  4. P. W. Bearman and J. K. Harvey, “Golf ball aerodynamics,” Aeronautical Quarterly, Vol. 27, 1976, pp. 112–122, as well as R. D. Mehta, “Aerodynamics of Sports Balls,” Annual Review of Fluid Mechanics, Vol. 17, 1985, pp. 151–189. ↵
  5. R. E. Meyerson, “Space Shuttle Orbiter Drag Parachute Design,” AIAA Aerodynamic Decelerator Systems Conference, 2001; and C. H. Lowry, “Space Shuttle Orbiter Drag Chute Summary,” 22nd AIAA Aerodynamic Decelerator Systems Technology Conference, 2013. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n25

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