26 Aerodynamics of Bluff Bodies

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 (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 is because of the effects of the large low-pressure zone produced in the wake at the rear of the body, as illustrated in the figure below.

A bluff body generally produces large scale flow separation, originating at the extremities or corners of the body, and a  broad wake containing low pressures, thereby producing high overall drag.

Objectives of this Lesson

  • Distinguish the basic differences between the flows about 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.

Drag Coefficients

For bluff bodies, the interest is usually in the drag on that body, mainly because it is the dominant force. This observation, however, does not imply that bluff bodies cannot produce lift. Nevertheless, in the first instance, examining just the drag characteristics of such bodies is convenient. Furthermore, bluff bodies may also produce pitching moments, which in some cases may need to be known for certain types of engineering work.

Recall that the two-dimensional drag coefficient is

(1)   \begin{equation*} C_d = \frac{D'}{\frac{1}{2} \rho_{\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 then l = d where d is the diameter.

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

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

where the product l b is equivalent to an area.

In general, for a three-dimensional object then the drag coefficient is defined as

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

where A_{\rm ref} is a reference area and this is usually the projected frontal area. Using a sphere of diameter d as an example, then its projected area is A_{\rm ref} = \pi d^2/4.

However, caution must be used when defining drag coefficients for three-dimensional bodies because the drag coefficient depends on the definition of the body’s reference area, which may not be unique. For example, for specific bodies of revolution, then the volumetric drag coefficient is used by convention, in which the reference area is the square of the cube root of the volume. In hydrodynamics, the analysis of submerged streamlined bodies uses the wetted surface area as the reference area. Therefore, for consistency in using and comparing 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 for different body shapes, drag coefficients per se are not always used but instead the equivalent drag area is used, which is given the symbol f. The equivalent drag area is defined as

(4)   \begin{equation*} f = \frac{D}{\frac{1}{2} \rho_{\infty} V_{\infty}^2} \end{equation*}

where f 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 is the approach 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 so it can be written that

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

However, it would be incorrect to write

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

where K > 1 and often K = 1.2 will be used for aircraft drag synthesis in the absence of any other information, i.e., a 20% drag penalty from component interference effects.

Drag Coefficients of Simple Shapes

There is an infinite number of bluff body shapes for which the drag (or drag coefficient) may need to be known for engineering purposes. However, in practice, far fewer 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 as an ensemble time average.

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

Drag coefficients for standard two-dimensional and three-dimensional bluff body shapes are published in various sources and are a valuable resource for engineers. Values for some common bluff-body shapes are shown in the tables below. Notice also that these shapes can be further categorized into two primary sub-classes of bluff bodies, namely: 1. Those with sharp angular shapes; 2. Those with smooth shapes. Of course, there is always the possibility that some bluff body shapes have a combination of both smooth contours and sharp edges.

The measured drag coefficients of some simple two-dimensional bluff bodies.

For example, a sphere or a cylinder is smooth, whereas an I-beam or 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 result, as shown in the tables. In contrast, angular shapes show little or no variations with the Reynolds number.

The measured drag coefficients of some simple three-dimensional bluff bodies.

Example #1 – Calculating the Drag on a Bluff Body

What will be the drag on a solid hemisphere of frontal area 0.26 m2 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

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

where A_{\rm ref} = 0.26m2 and V = 100 m/s, and if MSL ISA conditions are assumed then \rho = 1.225 kg/m3. Looking up the value of the drag coefficient of a solid hemisphere in the table given above, 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, then C_D = 0.42, and in the latter case then C_D = 1.17.

Inserting the values gives for the smooth side facing the flow gives

    \[ D = \frac{1}{2} \rho  V ^2 \, C_D \, A_{\rm ref}} = 0.5 \times 1.225 \times 100.0^2 \times 0.42 \times 0.26 = 667 \mbox{ N} \]

and for the flat side facing the flow gives

    \[ D = \frac{1}{2} \rho  V ^2 \, C_D \, A_{\rm ref}} = 0.5 \times 1.225 \times 100.0^2 \times 1.17 \times 0.26 = 1,863 \mbox{ N} \]

Flow Patterns About a Circular Cylinder

A classic example of a well-studied two-dimensional bluff body flow is about a circular cylinder, for which the flow readily separates near its maximum thickness, 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 is sensitive to variations in Reynolds number, which in this case is defined based on the diameter of the cylinder, d.

Images showing the variety of different flow states that can be produced by a circular cylinder, which will depend primarily on the Reynolds number.

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, and so it produces 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 separates further down the surface, producing a narrower wake. Because the wake is narrower, this operating condition also has lower drag.

As the Reynolds number increases, inertia effects 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, there is no lift or drag force on the cylinder. This outcome is called d’Alembert’s Paradox, after the mathematician Jean le Rond d’Alembert, who first solved this problem with the conclusion that his theory for the flow did not agree with experiments that had been 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 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. Also shown for reference is the drag coefficient for a sphere. The drag reduces quickly with increasing Reynolds number Re (notice the logarithmic scales) until Re reaches about 10^3, after which point 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.

The drag coefficient of a circular cylinder as a function of Reynolds number based on its diameter. Notice the logarithmic scale.

With further increases in Re, it will be apparent that there is a critical value of Re where the drag suddenly decreases by almost an order of magnitude. This condition, which is called the critical Reynolds number, corresponds to the reattachment of the previously separated laminar boundary layer as a turbulent boundary layer, which then finally separates at a further downstream distance on the rear part of the cylinder. The narrower wake subsequently produced leads to a reduction in pressure drag and so overall drag. However, notice that with yet further increases in Re, 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, of some interest, in this case, is the surface roughness effects, which can cause the drag reduction on the sphere to occur at a lower value of the critical Reynolds number.

The drag coefficient of a smooth and rough sphere as a function of Reynolds number based on diameter. The results for a circular disk is also shown as a reference.

This latter behavior is shown below using flow visualization, the roughness causing a large reduction in the width and extent of the downstream wake, and so 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 surface of the sphere much longer than for a laminar boundary. The consequence of this behavior is the creation of a narrower low-pressure wake, so there is a lower drag on the sphere.

Flow visualization of the flow about a sphere, with and without the application of surface roughness. (a) Smooth sphere, (b) Rough sphere.

Why Golf Balls Have Dimples

Ever wonder why a golf ball has dimples? In the early days, golfers found a curious outcome that old, worn, and rough balls traveled further than new balls. 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.

A golf ball is a dimpled sphere, and the dimples act as a type of surface roughness to deliberately change its aerodynamics. The dimples 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 flight of the ball, which is about 3 \times 10^5, then, normally, the flow on a smooth ball would be laminar, and the flow would easily separate and cause high pressure drag, as shown in the figure below. However, a turbulent boundary layer will remain attached to the surface of the ball much longer than a laminar boundary before it separates, and so this creates a narrower low-pressure wake and hence less pressure drag.

Sketches of the flow about smooth and rough golf balls. The corresponding reduction in drag with the dimpled ball at the Reynolds number of its flight causes it to travel much farther, in fact about twice as far as if the ball were to be smooth.

Example #2 – Calculation of Reynolds Number on 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.

To answer this question, the first issues to address are:

  1. What is the diameter of a golf ball?
  2. How fast does a golf ball travel?

A little research will show that according to the rules of golf, then the diameter of a golf ball must not be less than 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 or 177 kph to as high as 160 mph or 257.5 kph.

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

    \[ Re = \frac{\rho V d}{\mu} \]

where \rho and \mu can be assumed to take MSL ISA values, i.e., in SI units then \rho = 1.225 kg/m3 and \mu = 1.48 x 10-5 kg m-1 s-1. Inserting the values using 257.5 kph or 71.53 m/s gives

    \[ Re = \frac{\rho V d}{\mu} = \frac{1.225 \times 71.53 \times 0.04267}{1.48 \times 10^{-5}} = 2.13 \times 10^5 \]

and for 217.4 kph or 60.39 m/s gives

    \[ Re = \frac{\rho V d}{\mu} = \frac{1.225 \times 60.39 \times 0.04267}{1.48 \times 10^{-5}} = 2.53 \times 10^5 \]

Therefore, the Reynolds number for the flight of a golf ball is approximately in the range of 2.2 x 105 to perhaps as much as 3 x105.

Examining the C_D 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 the world of golfing.

Aerodynamics of Spinning Balls

Another effect experienced by golf balls, as well as other balls used for numerous games and sports, is a lift force produced by its spinning motion, e.g., a backspin or topspin. The spin of a ball spoils the horizontal symmetry of the flow about it, creating differential pressure and producing a lift force. This behavior is called the Magnus effect and named after Heinrich Magnus, the German scientist who first investigated it, as illustrated in the figure below.

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

In some cases, Magnus observed cannonballs’ tendency 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 cannon ball left the barrel. Isaac Newton also noticed this same behavior while observing people playing tennis.

The explanation for the curving behavior of the ball lies in aerodynamics. In the case of a backspin, 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 backspin will cause a golf ball to fly higher and often further. A topspin 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).

Complex Three-Dimensional Bluff Bodies

The flows that are 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, there are often complex flow separation and reattachment regions on the body itself, making the a priori estimation of drag on such bodies very difficult.

The flow around a complex bluff body, especially those protruding from a surface, contains significant complexity and three-dimensionality. In this case, the flow is about the rear of a ship that has a flight deck for helicopter operations.

One of the concerns with three-dimensional bluff body flows is that the aerodynamics can produce such high aerodynamic 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 type of 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 d in the case of a cylinder). The Strouhal number St is defined as

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

where f_{\rm st} is the vortex shedding frequency (or the frequency in radians per second), and V_{\infty} 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. In civil engineering, vortex shedding from suspension bridges is a significant cause of concern and can lead to bridge failures.

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 delay or even eliminate flow separation, streamlining the resulting flow and reducing the drag.

While streamlining may not be a viable engineering option in some cases, 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.

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 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 to minimize drag on an aircraft, then its cross-sectional area should change smoothly and continuously rather than abruptly.

A wing root fairing can help reduce the interference drag between the wing and the fuselage, and so decrease the total aircraft drag by about 10% for an almost negligible increase in airframe weight.

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 completely 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.

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 its drag by 10% to 30%.

Streamlining of Terrestrial Vehicles

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

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

so the power required to propel the vehicle P_{\rm req} will increase with the cube of its speed over the ground, i.e.,

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

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 required for propulsion, but at higher speeds the aerodynamic effects dominate.

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 C_D. 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

The automotive industry has become proficient in streamlining vehicles to reduce fuel consumption and internal noise, which usually requires much wind tunnel testing. An example of flow visualization is shown in the photograph below. The judicious use of smoke filaments and suitable lighting can provide much insight into the flow patterns and where the flow is attached or separated, therefore helping to make decisions about streamlining.

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

As shown in the figure below, the improvements that can be obtained with careful streamlining of automobile shapes can result in significant fuel economy increases and notable reductions in internal noise from corresponding reductions in turbulence. However, streamlining can only be done to a point, after which compromises in the utility of the vehicle 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 underside of the vehicle.

The use of streamlining on an automobile by rounding off sharp corners alone can cut drag by half and improve the fuel consumption by over 100%.

Trucks

Trucks 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 that have high drag. These boxes are often called Intermodal Shipping Containers or an ISO box (ISO 6346). The fuel required also depends upon the weight of the freight being carried, i.e., from the effects of rolling friction.

Continuous improvements to the aerodynamics of trucks have reduced drag and saved more fuel. There are four significant areas of interest to engineers: 1. The frontal shape of the tractor; 2. The gap between the tractor’s end and the trailer’s beginning or the front of the CONEX box; 3. The shapes of the sides and underbody of the trailer; 4. The very back of the trailer, which is a significant source of pressure drag, as shown in the figure below.

Flow separation from a tractor-trailer is a major source of drag. However, ven relatively crude pop-out afterbody fairings or trailer tails can help significantly to reduce drag and improve the fuel consumption commercial trucking fleets.

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 also shown in the lower image of the figure above. Although relatively crude in shape, they help reduce the drag caused by flow separation and turbulence at the rear end of the trailer. The corresponding reduction in fuel consumption makes them worthwhile when overall fleet costs are concerned. A slight reduction 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.

A high-speed train showing the highly streamlined nose and the rounded corners on the carriages, which are necessary to reduce drag.

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.

Parachutes are used for a variety of purposes, the classic canopy shape being hemispherical.

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 d, 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. For most parachutes, the drag coefficient is close to 1.42, depending on its exact construction.

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

(11)   \begin{equation*} W_p - D = 0 \end{equation*}

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

(12)   \begin{equation*} D = \frac{1}{2} \rho A C_D V^2 \end{equation*}

where A is the projected frontal area of the parachute, \rho is the density of the air, and C_D = 1.42 for an open hemispherical shell. The projected frontal area is A = \pi d^2/4 where d is the diameter. Solving for V (the rate of descent) gives

(13)   \begin{equation*} V = \sqrt{\frac{2 W_p}{\rho A C_D}} \end{equation*}

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 itself 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 changing the drag coefficient somewhat.

Example #3 – 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 m2, and a corresponding drag coefficient of 1.8 based on this reference area. The parachute has an area of 150 m2 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 what would be expected during normal flight. In normal flight, the airplane is streamlined, with minimal pressure drag and boundary layer separation. However, during decent with a parachute, the airplane behaves as a bluff body with large amounts of flow separation and corresponding pressure drag.

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

    \[ W = M \, g = \frac{1}{2} \rho V_d^2 S_a C_{D_{a}} + \frac{1}{2} \rho V_d^2 S_p C_{D_{p}} \]

where V_d is the rate of descent, S_a is the projected cross-sectional area of the airplane, C_{D{a}} is the drag coefficient of the airplane based on this reference,  S_p is the projected cross-sectional area of the parachute, C_{D{p}} is the drag coefficient of the parachute based on this reference area. Simplifying gives

    \[ M \, g = \frac{1}{2} \rho V_d^2 \left(  S_a C_{D_{a}} + S_p C_{D_{p}} \right) \]

and rearranging to solve for V_d gives

    \[ V_d = \sqrt{ \frac{2 M \, g}{\rho \left(S_a C_{D_{a}} + S_p C_{D_{p}} \right)}} \]

Substituting in the known numerical values gives

    \[ V_d = \sqrt{ \frac{2  \times 1,200.0 \times 9.81 }{1.01 \times \left(25.0 \times 1.8 + 150.0 \times 1.5 \right)} } = 9.29 \mbox{ m/s} \]

which seems reasonable.

Bluff Bodies at Supersonic Speeds

The behavior of bluff bodies at high speeds is also of much interest to aerospace engineers, especially for spacecraft that re-enter the 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.

Schlieren flow visualization image 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? 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.

The drag coefficient of supersonic spheres, as measured by Bashforth.

H. Julian “Harvey” Allen of NASA developed the design of 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, which are known as bow shocks, as shown in the image below.

A bluff body at supersonic speeds, in this case representing a space capsule during reentry into the Earth’s atmosphere. The bow shock is away from the body’s surface and so helps dissipate heat into the surrounding air rather than into the body itself. Notice also the significant flow separation and turbulence in the wake of the body.

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 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 significant 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 the use of more innovative concepts, if not active flow concepts, 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?
  • It is desired to mount a rectangular sensor package 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: