# 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 are called *bluff bodies *(or sometimes they are called *blunt bodies*) are also encountered in many engineering applications. While the drag force on a body shape comprises the effects of the two primary contributors, skin friction drag and pressure drag, the total drag on bluff bodies is typically dominated by the pressure drag component. This outcome arises because of the effects of the large low-pressure zone produced in the wake at the rear of the body, as illustrated in the diagram below.

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 show sensitivities to Reynolds number variations.
- Become familiar with the Magnus effect on spinning cylinders and spheres.

# 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, examining just the drag characteristics of such bodies is convenient in the first instance. Furthermore, bluff bodies may also produce pitching moments, which sometimes need to be known for certain types of engineering work, e.g., to determine torsional loads.

Recall that the two-dimensional drag coefficient is

(1)

where is the drag per unit span, and is a characteristic length, e.g., for a circular cylinder, , where is its diameter.

In most cases, the drag coefficients for two-dimensional bluff bodies are presented in terms of *per length*, , so the drag coefficient is defined as

(2)

where the product is equivalent to an area. In general, for a three-dimensional object, the drag coefficient is defined as

(3)

where is defined as the reference area, usually the *projected frontal area*. Using a sphere of diameter as an example, its projected area is . Likewise, a cube with side length will have a reference area of

However, caution must be used when defining drag coefficients for three-dimensional bodies because the drag coefficient always depends on the definition 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, in which the reference area is 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, adopting the convention used in the particular field of application is imperative.

To avoid ambiguity in the definitions of drag coefficients for different body shapes, drag coefficients *per se* are not always used. Instead, the *equivalent drag area* is used, given the symbol . The equivalent drag area is defined as

(4)

where would be measured in units of area.

The advantage of using equivalent drag areas is that there is no ambiguity in defining a reference area. This approach is often used when estimating the drag of complex shapes made up of many simpler shapes, e.g., the drag of 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 simple shapes, i.e.,

(5)

However, it would be incorrect to write

(6)

because the reference areas on which each value of 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)

where and often will be used for aircraft drag synthesis without any other information, i.e., a 20% drag penalty from component interference effects.

# Drag Coefficients of Simple Shapes

There are many bluff body shapes for which the drag (or drag coefficient) may need to 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 measured in wind tunnels. Furthermore, flows about bluff bodies are often unsteady and can produce various types of vortex shedding, as shown in the animation below. Therefore, the measurement or computation of drag is usually made from a relatively long period, known as an *ensemble time average*.

Drag coefficients for standard two-dimensional and three-dimensional bluff body shapes are published in many sources and are a valuable resource for engineers. These shapes can be further categorized into two primary sub-classes of bluff bodies:

- Those with sharp angular shapes tend to have fixed points of flow separation and drag coefficients that are insensitive to the Reynolds number.
- Those with smooth contours will have variable points of flow separation and can exhibit some sensitivity to the Reynolds number.

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 specific measurements in the wind tunnel to establish the drag coefficients. Values for some common bluff-body shapes are shown in the tables below.

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 show little or no variations with changes in the Reynolds number, at least after reaching a minimum threshold.

Example #1 – Calculating the drag on a bluff body

What will be the drag on a solid hemisphere of frontal area 0.26 m^{2} in a flow speed of 100 m/s? Hint: Notice that the drag will depend on what side of the hemisphere points into the wind.

The drag force on the hemisphere is given by

where m^{2} and m/s, and if MSL ISA conditions are assumed then kg/m^{3}. Looking up the value of a solid hemisphere’s drag coefficient in the table above, it can be seen that the value depends on whether the *smooth side* or the *flat side* faces the relative wind. In the former case, then , and in the latter case, then . Inserting the values for the smooth side facing the flow gives

And for the flat side facing, the flow gives

# Flow Patterns About a Circular Cylinder

A classic example of a well-studied two-dimensional bluff body flow is that about a circular cylinder, for which the flow readily separates near its maximum thickness or just downstream, representative examples being shown below. 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, .

At very low Reynolds numbers near unity, called a “creeping flow” or Stokes flow, the behavior of the flow is influenced much more by viscosity effects than the effects of inertia. At slightly higher Reynolds numbers but below 5, a stable pair of symmetric vortices are formed in the wake of the cylinder; the flow is steady mainly, but a high drag coefficient is produced.

As the Reynolds number is further increased, the drag coefficient decreases to a point. The wake vortices move further downstream of the body in the form of alternative vortex shedding, producing two rows of counter-spinning vortices called a Von Kármán vortex street. This is an unsteady flow, creating an unsteady drag on the cylinder. However, no time-averaged unsteady lift is produced because of the strong asymmetry in the flow with respect to the flow direction. Interestingly, this latter type of periodically alternating flow also produces a periodic pressure field that manifests as a sound or a noise called 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, the periodic flow obtained about the cylinder at the lower Reynolds numbers becomes highly aperiodic, but still with the alternate shedding of vortices. Here, the flow in the wake is more turbulent. However, the boundary layer, which is of the turbulent kind, separates further down the surface, producing a narrower wake. Because the downstream wake is narrower, there is a smaller region of lower pressure on the downstream side, thereby giving lower drag.

As the Reynolds number increases, inertia effects begin to dominate over viscous effects. As a result, the flow becomes attached everywhere with a much smaller wake downstream of the cylinder, i.e., it begins to approach an inviscid flow. Because the flow is symmetric both upstream and downstream, the cylinder has no lift or drag force. This outcome is called d’Alembert’s Paradox, after the mathematician Jean le Rond d’Alembert, who first solved this problem by concluding that his theory for the flow did not agree with experiments made on cylinders in the wind tunnel. The reason was that d’Alembert did not consider the action of viscosity in his solution, i.e., he considered only an *inviscid* flow. However, as shown in the images above, this flow type does not occur naturally because a real fluid is always viscous to some degree, and the Reynolds number is finite.

# Drag of Circular Cylinders & Spheres

The corresponding drag coefficient on a circular cylinder as a function of the Reynolds number is shown in the figure below. Notice the results are for a “smooth” cylinder, the effects of surface roughness changing the quantitative relationships. The drag coefficient for a sphere is also shown for reference. The drag reduces quickly with increasing Reynolds number (notice the logarithmic scales) until reaches about 10^{3}, after which the drag stays nominally constant. This outcome arises because the point of flow separation (laminar boundary layer separation) is delayed to a greater downstream distance, eventually separating close to the points of maximum thickness.

With further increases in , it will be apparent that there is a critical value of where the drag suddenly decreases by almost an order of magnitude. This condition, called the *critical Reynolds number*, corresponds to the reattachment of the previously separated laminar boundary layer as a turbulent boundary layer, which finally separates at a further downstream distance on the rear part of the cylinder. The critical Reynolds number for a cylinder is about 2 x 10^{5}. The narrower wake subsequently produced leads to a reduction in pressure drag and overall drag. However, notice that with yet further increases in , the drag increases again because of the increasingly higher surface shear stresses (i.e., skin friction) produced by the turbulent boundary layer as the flow speed and Reynolds number increase.

The aerodynamic behavior of a sphere is qualitatively similar to that of a circular cylinder, as shown below. Again, there is a sudden drop in drag as the Reynolds number reaches a critical value because of the change from laminar flow separation to turbulent separation. 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.

This latter behavior is shown below using flow visualization, the roughness causing a significant reduction in the width and extent of the downstream wake. It explains the reduction in drag, i.e., the roughness manifests as an increase in the effective value of the Reynolds number. The surface roughness causes the laminar boundary layer on the upstream side of the sphere to transition immediately into a turbulent boundary layer, which can remain attached to the sphere’s surface much longer than for a laminar boundary layer. The consequence of this behavior is the creation of a narrower low-pressure wake, so there is a lower drag on the sphere.

Worked Example #2 – Drag coefficient of a circular cylinder

Consider the airflow past a circular cylinder with a diameter, , of 0.4 meters. The free-stream velocity, , is 50 m/s. The properties of air at standard conditions are as follows: = 1.225 kg/m and = 1.7894 x 10 kg m s.

- Determine the Reynolds number for this flow.
- Classify the flow regime around the cylinder (laminar or turbulent).
- Determine the drag coefficient, , for the cylinder.

- Reynolds number is given by
Inserting the values gives

- The flow regime can be determined based on the Reynolds number. The critical Reynolds number for the transition from laminar to turbulent flow typically occurs at about 2 x 10 for flow around a circular cylinder. In this case, the Reynolds number is well above this threshold, and the flow regime around the cylinder is turbulent.
- The drag coefficient, , for a circular cylinder in turbulent flow can be estimated from the chart, which is approximately 0.14.

# Why do Golf Balls Have Dimples?

Ever wonder why a golf ball has dimples? In the early days, golfers found the most curious outcome that old, worn, and rough balls traveled further than new balls. Golfers using the rough balls required fewer strokes from the tee to the hole, so gained a competitive advantage. It did not take long before the aerodynamics of spheres were studied in wind tunnels, finding that rough spheres can, under certain conditions, have less aerodynamic drag than smooth spheres. It soon became clear that some standardization of golf balls was required in the interest of making the sport fair to all participants.

A modern golf ball is a machine-dimpled sphere, and the dimples act as surface roughness to deliberately change its aerodynamics. The dimples, which are a standardized form of roughness, cause the boundary layer on the upstream side of the ball to transition from laminar to turbulent, delaying flow separation and thereby reducing its drag. At the Reynolds numbers corresponding to the ball’s flight, which is about , the flow on a smooth ball would usually be laminar, and the flow would easily separate near the equator and cause higher pressure drag, as shown in the figure below. However, a turbulent boundary layer, which can be created using surface roughness, will remain attached to the ball’s surface much longer than a laminar boundary before it separates. This behavior creates a narrower low-pressure wake and less pressure drag. It is an interesting outcome that creating turbulence reduces drag!

Example #3 – Reynolds number of a golf ball

What is the flight Reynolds number of a golf ball? What is the drag coefficient on the ball at the conditions of flight? Why is this important in regard to 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?

A little research will show that according to golf rules, a golf ball’s diameter must be at least 1.68 inches or 42.67 mm. Also, an amateur golfer typically hits a ball at an average speed of 135 mph or 217.4 kph with a driver club. However, those speeds can vary significantly with a golfer’s skill level and may range from a low of 110 mph (177 kph) to as high as 160 mph (257.5 kph) for a professional golfer.

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

where and can be assumed to take MSL ISA values, i.e., in SI units, then = 1.225 kg/m^{3} and = 1.48 x 10^{-5} kg m^{-1} s^{-1}. Inserting the values using 257.5 kph or 71.53 m/s gives

and for 217.4 kph or 60.39 m/s gives

Therefore, the Reynolds number for a golf ball’s flight is approximately 2.2 x 10^{5} to as much as 3 x 10^{5}.

Examining the versus Reynolds number for this range, as shown in the graph below, it can be seen that based on measurements then, the drag coefficient of a rough ball in this range will be less than half of a smooth ball, i.e., the dimples act to increase in the effective Reynolds number. Of course, this outcome is supported by experience and other empirical data from golfing.

# Aerodynamics of Spinning Balls

Another effect experienced by golf balls and other balls used in other games and sports is a lift force produced by their spinning motion, e.g., a backspin or topspin. The spin of a ball spoils the horizontal symmetry of the flow, creating differential pressure and producing a lift force. This behavior is called the Magnus effect and is named after Heinrich Magnus, the German scientist who first investigated it in the 19th century, as illustrated in the figure below. In some cases, Magnus observed a tendency of a cannonball to curve or veer significantly away from their expected flight path. He concluded that this behavior was because of their residual spinning motion as the cannonball left the barrel. Isaac Newton also noticed this same behavior while observing people playing tennis.

The dependence of the Magnus lift coefficient, , on the translational and rotational speeds of a smooth cylinder of diameter has been investigated and measured in the wind tunnel, some results being reproduced in the figure below, where is its translational speed and is the rotational angular velocity. The results are presented in terms of a non-dimensional spin rate, . Notice the additional dependency on the Reynolds number, as given by . A negative lift or a so-called “reverse” Magnus effect occurs at or about the critical Reynolds number. This behavior ties to a “moving wall” effect that affects the boundary layer characteristics over the top and bottom halves of the cylinder, in effect causing premature laminar separation on the upper half and delayed separation on the lower half. This latter behavior is also characteristic of spheres and balls.

Therefore, the explanation for the curving behavior of a ball lies in aerodynamics. In the case of a topspin, there is a higher flow velocity (hence a lower pressure) over the ball’s upper surface, thereby producing a positive (upward) lift force. Therefore, a topspin will cause a ball to fly higher but not always further. A backspin will cause a ball to dive more quickly toward the ground. Backspin and topspin effects are used in many other ball games, such as golf, baseball, soccer, etc., which adds to the player’s skill and the game’s unpredictability (and fun). The effects of Reynolds number and surface roughness mean that the curving characteristics depend on the size and surface finish of the ball, e.g., a smooth ping pong ball versus the seams found on a baseball or soccer ball.

# Complex Three-Dimensional Bluff Bodies

The flows produced about more general three-dimensional bluff body shapes are considerably more complex, and some of the other types of flow phenomena involved are shown in the figure below, which describes the behavior of the so-called “airwake” produced over the surface of a ship. *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.

# Vortex Induced Vibration

One of the concerns with three-dimensional bluff body flows is that the aerodynamics can produce such high forces to cause excitations of the structure it is made of. One of the most concerning effects in this regard is vortex-induced vibration (VIV), where the aerodynamic excitation produced by vortex shedding causes the structure to vibrate or even enter into some flutter. VIV problems can occur in many engineering applications, not just limited to the aerospace field, where flow-induced vortex shedding is possible, particularly with long, slender cantilevered structures such as wires, chimneys, and suspension bridges.

The Strouhal number is usually used to quantify VIV effects, which relates the frequency of shedding to the velocity of the flow and a characteristic dimension of the body (diameter in the case of a cylinder). The Strouhal number is defined as

(8)

where is the vortex shedding frequency (or the frequency in radians per second), and is the free-stream flow velocity. The Strouhal number of a cylinder is 0.2 and is constant over a wide range of Reynolds numbers.

The lock-in phenomenon occurs when the vortex shedding frequency becomes close to the natural vibration frequency of a system or structure, resulting in a severe oscillation or even flutter. Although the onset of flutter will usually always reach a limit cycle in some form, the associated high cyclic stresses produced on a metallic structure can quickly cause fatigue failure.

# Bridges

In civil engineering, vortex shedding from suspension bridges, which are bluff bodies in cross-section, is a significant cause of concern and can lead to bridge failures. When the Tacoma Narrows Bridge over Puget Sound in Washington state collapsed in 1940 because of a torsional flutter produced in a stiff wind, it was captured on film. The lessons learned have become a textbook example of vortex-induced vibration. However, this bridge is not the only one to have suffered from the effects of VIV and flutter.

The figure below shows the essence of the problem. The bluff “H” shape of the bridge’s cross-section causes vortex shedding, leading to a periodic unsteady aerodynamic force. The upshot of this behavior is that the bridge deck begins to twist in the wind, as shown in the figure below. Depending on the weight and dynamic characteristics of the bridge, the deformations can be substantial and may quickly exceed 20 degrees.

The twisting of the bridge deck causes the flow separation to increase, forming stronger vortices and higher aerodynamic loads that further twist the deck. While the deck structure resists this lifting and twisting because of its inherent torsional stiffness, the behavior can lead to a “lock-in” event and the development of “torsional flutter.” In some cases, it may be possible to modify the cross-sectional shape of the bridge to reduce the vortex shedding and increase the wind speed for torsional flutter to occur.

# Streamlining

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

While streamlining may not be a viable engineering option in some cases because of practicality or cost, it is always a worthy effort on an airplane to reduce its overall drag and increase its performance, even at the expense of some increase in overall structural weight. For example, blending a wing carefully to the fuselage using a fairing can eliminate the horseshoe vortex and reduce interference drag from the resulting streamlining achieved, as shown in the photograph below for a general aviation airplane, for an almost negligible increase in airframe weight. For a commercial aircraft, the wing root fairing is designed to reduce interference drag and becomes part of the transonic drag reduction process using Whitcomb’s *area rule*. The area rule states that its cross-sectional area should change smoothly and continuously rather than abruptly to minimize drag on an aircraft.

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 of almost eliminating the drag. However, the extra weight and higher cost of a mechanical system to retract the landing gear are not usually viable for a smaller aircraft, e.g., one in the general aviation class.

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

# Streamlining of Terrestrial Vehicles

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

(9)

so the power required to propel the vehicle will increase with the cube of its speed over the ground, i.e.,

(10)

Therefore, it is also apparent that the fuel or energy required will increase with the cube of the speed. Rolling resistance or friction also affects the net resistance on the vehicle and the fuel or energy needed for propulsion, but the aerodynamic effects dominate at higher speeds.

As with flight vehicles, much can be done to reduce energy consumption on terrestrial vehicles by reducing drag, i.e., by using streamlining to reduce the value of as much as practical. To this end, aerospace engineers have the training and expertise to understand the needed steps to make profitable drag reductions. Much testing of terrestrial vehicles is done in wind tunnels, although CFD solutions can be helpful too.

## Automobiles & Cars

The automotive industry has become proficient in streamlining vehicles to reduce fuel consumption and internal noise, which usually requires much wind tunnel testing. Turbulence about the vehicle, such as from rear-view mirrors, can be a significant source of internal noise. An example of flow visualization is shown in the photograph below. The judicious use of smoke filaments and suitable lighting can provide insight into the flow patterns and where the flow is attached or separated, helping to decide where streamlining may be needed.

As shown in the figure below, careful streamlining of automobile shapes can reduce drag. The result is significant increases in fuel economy and notable reductions in internal noise from corresponding reductions in turbulence. However, streamlining can only be done to a point where compromises in the vehicle’s utility may become problematic. Exposed wheels are a significant drag source, so there has been a trend toward more carefully streamlining the wheel well and the vehicle’s underside.

## Trucks & Lorries

Trucks and lorries are very aerodynamically inefficient because of their bluff-body shapes and resulting high base pressure drag, with most trucks averaging a fuel consumption between 50% and 200% higher than automobiles. The large CONEX rectangular box-like shapes often seen being towed by tractor-trailers are poor aerodynamic shapes with high drag. These boxes are often called *Intermodal Shipping Containers* or ISO boxes (ISO 6346). The fuel required to transport the load also depends upon the freight’s weight, i.e., from the effects of rolling friction.

Continuous improvements to the aerodynamics of trucks have reduced their drag and saved more fuel. There are four significant areas of interest to engineers in terms of drag reduction:

- The frontal shape of the tractor, which is very unstreamlined.
- The gap between the tractor’s end and the trailer’s beginning or the front of the CONEX box.
- The shapes of the sides and underbody of the trailer.
- The very back of the trailer is a significant source of pressure drag, as shown in the figure below.

Minimizing the gap between the trailer and the tractor is essential, so fairings on the top of the tractor are often used. In addition, skirts can be placed underneath and along the trailer’s sides, redirecting the higher-speed airflow away from the trailer’s underside and reducing the drag from the wheels, which act as bluff bodies.

Today, *trailer tails* are increasingly used as pop-out afterbody fairings, as shown in the figure’s lower image above. Although relatively crude in shape, they help reduce the drag caused by flow separation and turbulence at the trailer’s rear end. The corresponding reduction in fuel consumption makes them worthwhile when overall fleet costs are concerned. A slight decrease in drag can significantly reduce fuel costs when the number of vehicles and the miles traveled are cumulatively considered.

## Trains

Traveling by train has always been an essential 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 the speed, so drag and energy requirements to propel the train increase significantly at higher speeds. Related issues for trains include the ability to supply the needed traction from the wheels to the rails and the significant pressure changes (which are felt by the 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 for any other ground vehicle, the aerodynamic characteristics of trains tend to be more complex than those of cars, trucks, or airplanes. Nevertheless, significant reductions in drag can be achieved by streamlining, an example being shown in the photograph below. The basic principles needed are not unlike those for airplanes, including the need for a streamlined nose shape and rounded corners along the entire length of the train. Again, much of the work on the most effective steps for drag reduction on high-speed trains using types of streamlining is done in wind tunnels.

# Creating Drag – Parachutes

A parachute is used to slow the motion of an object through the air by creating drag. This situation is one example where the creation of high drag has a good outcome! Parachutes come in many shapes and sizes, but the classic design of a parachute canopy is a hemispherical shell, as shown in the figure below.

The drag coefficient of a hemispherical shell based on the projected frontal area is about 1.42. Notice that for an open hemispherical shell of diameter , its projected frontal area (i.e., the reference area on which the drag coefficient is based) is just the area of a circle of this diameter. The drag coefficient for most parachutes is close to 1.42, depending on their construction.

If the weight suspended from the parachute is , and if the weight of the parachute itself is neglected, then steady force equilibrium gives

(11)

where is the drag of the parachute. The drag is given by

(12)

where is the projected frontal area of the parachute, is the density of the air, and for an open hemispherical shell. The projected frontal area is , where is the diameter. Solving for (the rate of descent) gives

(13)

Notice that that rate of descent increases for a higher weight and/or for a smaller area parachute.

For an average person, parachutes are designed to give a decent speed between 10 to 20 mph (16 to 32 kph). Of course, a more detailed analysis of this problem would consider the weight of the parachute and the backpack, although it would likely only increase the actual rate of descent by a small amount. Holes or vents can be introduced into the canopy to improve the parachute’s stability and give it some forward motion, which aids in the parachutist’s control of the descent; the vents also change the drag coefficient somewhat.

Example #4 – Rate of descent with a parachute

A general aviation aircraft is fitted with a ballistic parachute recovery system for improved safety in the case of a severe emergency. Consider a scenario where the aircraft’s pilot has to deploy this parachute system. The aircraft has a net mass of 1,200 kg, a projected (horizontal planform) cross-sectional area of 25 m^{2}, and a corresponding drag coefficient of 1.8 based on this reference area. The parachute has an area of 150 m^{2} and a drag coefficient of 1.5 based on this area. Determine the vertical airspeed at which the aircraft will parachute to a landing. Assume an average air density of 1.01 kg m^{-3} during the descent. Ignore the parachute’s weight and assume that the aircraft descends in a flat horizontal wings-level attitude.

The aircraft’s vertical drag coefficient is two orders of magnitude greater than expected during normal flight. The airplane is streamlined in normal flight, with minimal pressure drag and boundary layer separation. However, during descent with a parachute, the airplane behaves as a bluff body with large amounts of flow separation and corresponding pressure drag.

Therefore, the net vertical drag will equal the airplane’s weight during a steady descent. The net drag will be the sum of that produced by the parachute plus that from the airplane itself, i.e.,

where is the rate of descent, is the projected cross-sectional area of the airplane, is the drag coefficient of the airplane based on this reference, is the projected cross-sectional area of the parachute, is the drag coefficient of the parachute based on this reference area. Simplifying gives

and rearranging to solve for gives

Substituting in the known numerical values gives

which seems reasonable, but it will still be a reasonably firm landing!

# Bluff Bodies at Supersonic Speeds

The behavior of bluff bodies at high speeds is also fascinating to aerospace engineers, especially 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 of not only 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, an example being shown in the figure below for spheres (balls) 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.

The effects of Reynolds number on the drag of a sphere have been previously discussed, but what about the effects of Mach number? The most accurate 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 did drag measurements on supersonically spherical projectiles, which are shown below. Interestingly, it can be seen that the drag coefficient plateaus in value for Mach numbers greater than 1.4.

H. Julian “Harvey” Allen, a key aeronautical engineer at NASA, designed a spacecraft returning to re-enter the Earth’s atmosphere. His approach recognized that bluff or “blunt” bodies moving at very high supersonic and hypersonic Mach numbers generate strong shock waves that form in front of the body, known as a *bow shock*, as shown in the image below. Notice that the bow shock is away from the body’s surface and helps dissipate heat into the surrounding air rather than into the body itself. Also notice the significant flow separation and turbulence in the body’s wake.

Because there are significant changes in the temperature of the flows across strong shock waves, the shock waves that stand off from the surface allow much more heat to be dissipated into the air rather than into the body itself. The bluff body shape also creates a lot of pressure drag, helping to slow down the spacecraft as it re-enters the atmosphere until it is slow enough such that a parachute can be deployed, which is another example of where the creation of drag on a body can be beneficial.

# Summary & Closure

While aerospace engineers are more often than not concerned with minimizing drag using, in part, the principles of streamlining, in other cases, the creation of drag can be helpful too. There are almost an infinite number of possible bluff body shapes. However, many standard shapes have been studied in the wind tunnel, and their drag characteristics are known and published as functions of Reynolds number and, in some cases, with Mach number too, e.g., projectiles and re-entry vehicles.

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) have a critical Reynolds number, below and above which there is a significant change in the flow state and resulting drag behavior. Drag reduction of terrestrial vehicles continues to be a substantial issue in reducing fuel consumption, with conventional streamlining practices being very effective but are now reaching the point of diminishing returns. However, the future may see more innovative concepts, possibly in the form of active flow control devices, to help 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:

- Fantastic educational film on The Secret of Flight 7: Problem of Drag.

For interesting images of vortex shedding from cylinders, check out Professor Matthew Juniper’s page, which includes more readings and videos. - Fluid mechanics video lecture on bluff bodies.
- Fluid mechanics video lecture on the drag forces produced on bluff bodies.
- Inflatable trucktails for tractor-trailers!
- A CFD simulation of the supersonic flow over a bluff body.
- YouTube video on the aerodynamics of golf balls.
- Video on the intricate aerodynamics of the dimples on golf balls.
- You can find Lord Raleigh’s original paper on Aeolian tones here.
- A video of galloping power lines in a strong wind caused by vortex shedding.