Drag, a Nightmare and a Challenge

Abstract

Motion through a fluid is accompanied by a resistive force. Overcoming that is the price one has to pay to move. Drag is associated with the pressure forces on a structure as well as the shear forces experienced by virtue of the finite air viscosity and the tendency of air to be dragged along by the moving surface. Both kinds of forces are always present to varying degrees and they influence one another. Fortunately, the shear forces are small, though not negligibly small, compared to the dynamic forces associated with changing speed and they involved complicated physics in a small portion of the flow field known as a boundary layer.

Lift is, of course, not the only force of interest to the airplane builder because a resistive force must be overcome with propulsion. That force is called drag and the subject of much of the design work around an airplane. While lift can be determined from a look at inviscid air flow, drag cannot. That makes for very much more complicated analysis and even a display of the physics.

A classical aerodynamics textbook author will prove the assertion of no drag in potential flow by developing the field equations about a cylinder with circulation. Such a model is built by the addition of a source and a sink, adjacent to each other, and a uniform flow. Such a source-sink combination is called a doublet and the flow picture is as shown in the left side of Fig. 4.1. Without circulation, the results show two stagnation points (one where the flow encounters the cylinder at 9 o’clock and another where it departs from the cylinder at 3 o’clock). The addition of a vortex at the center will cause the stagnation points to move from these positions closer to 6 o’clock depending on the strength of the vortex. The fact that the stagnation points relocate near the bottom means the pressure is higher at the bottom than the top, hence there is lift. Because the flow is symmetrical about the vertical axis, there is no drag.

Fig. 4.1

Streamline pattern for potential flow (no friction). Left image is of a source-sink doublet in uniform flow without circulation. The stagnation points are at the 9 and 3 o’clock positions. The image at right is identical with the addition of clockwise circulation by a vortex at the center so that stagnation points move to 5 and 7 o’clock positions

So, what is drag? It has a number of components that are best illustrated by reference to geometries simpler than those of an airfoil.

4.1 The Air Near a Surface

The simplest example to illustrate friction flow kinematics is a flat plate aligned with the flow direction. In a fluid such as air, molecules immediately adjacent to the surface are in mechanical equilibrium with it and must therefore be at rest. This is the no-slip condition we invoked in connection with the Kutta-Joukowski condition for lift. Air molecules further from the surface share dynamic equilibrium with their neighbors: slow ones near the surface and faster moving ones further out. The net result is a flow velocity that varies between zero at the surface and full freestream velocity far out. The region of varying velocity is called a boundary layer¹ (Fig. 4.2).

Fig. 4.2

A flat plate with velocity distributions created by initially uniform flow from the left

4.1.1 The Boundary Layer Comes in Various Thicknesses

The boundary layer is characterized by its thickness. One can choose to describe the boundary layer thickness by the location where the flow is 95% of freestream speed or one can choose 99%. Such a thickness is arbitrarily dependent on the definition. There are more precise ways to describe it. For example, one can speak of the air volume flow rate that is not carried by the region next to the wall and define a boundary layer width representing the volume flow deficit. Such a thickness is called a displacement thickness. This region is not carrying the flow that it would carry, absent the boundary layer. As a result, the thin plate is seen by the oncoming flow as a thickening wedge-shaped object. The unavailable volume causes the freestream to be displaced. Thus, the boundary layer alters the effective shape of the surface on which the layer grows. So much for thinking that we prescribe the flow direction by specifying an airfoil shape!

Another example of thickness definition is associated with the momentum that has been removed from the flow by the shearing viscous forces. The boundary layer thickness based on this idea is called a momentum thickness. It is numerically smaller than the displacement thickness and is a precise characterization of the boundary layer as far as its resistive force exerted on the moving fluid. Figure 4.3 illustrates these thicknesses and the displacement of the freestream flow away from the plate.

Fig. 4.3

Thicknesses of the boundary layer. Its arbitrary edge is labeled (E). The momentum thickness (M) characterizes the momentum removed by friction forces. The thickness at any point is the drag experienced by the plate to this point. The thicker displacement thickness (D) is a measure of the volume no longer available to the freestream, hence the diversion of the streamline of the flow approaching the plate

Behind the flat plate, the momentum deficit is a wake that can be used to measure the viscous drag experienced by the flat plate by examining the velocity distribution (in the flow direction) in that wake. Figure 4.4 illustrates the wake’s velocity distribution on and behind the airfoil. The white area behind the velocity distributions is the volume deficit. There is also an associated momentum deficit. Further back, the wake grows in width and the velocities more closely approach the incoming velocity while preserving the same momentum deficit as a measure of the drag the flow experienced.

Fig. 4.4

The trailing edge of the plate shown in Fig. 4.3 or of an airfoil. Flow is from the left. The points marked M are the edges of the momentum thickness of the boundary layer. To the right is a sketch of the velocity distribution in the wake behind the plate. The momentum deficit associated with the broad unshaded area in the wake is the same as on the trailing edge

If the drag force resulting from such a flow situation were the only one, life would be relatively simple for the determination of drag. Real aerodynamic surface are not flat plates nor are they necessarily oriented with the freestream direction. Because of this misalignment of surfaces with the freestream flow direction, both shear and pressure forces impact the drag experienced.

The matter is further complicated in that the pressure distribution on an airfoil affects the shear experienced and hence affects the shape and progression of the boundary layer. The pressure history of the boundary layer consequently has an impact on the viscous drag and the effective shape of the hardware. It is not just a matter of the freestream flowing along a surface. The shape may even be altered sufficiently that the freestream flow fails to follow the surface altogether. This is flow separation from the surface.

An extreme example of flow separation is encountered when a flat plate is faced normal to the flow, as if to stop the flow (Fig. 4.5). There will still be a boundary layer growing from the impact stagnation point outward along the surface. Here the boundary layer flowing outward will not have sufficient energy to turn the corner to re-assemble the downstream flow to resemble what is seen upstream. The flow simply separates from the edges. In a way, one can think of this as an application of the Kutta condition. The net result is a high pressure on the upstream side of the plate and the lower static pressure of the environment downstream of the plate. The pressure difference leads to pressure drag. This is the drag felt by a hand that is held out the moving car window.

Fig. 4.5

Flow approaching a plate normal to the flow. The stagnation point is at A where the total pressure of the flow is experienced. A boundary layer grows from A to B. The recirculating region behind the plate is essentially at flow static pressure

4.2 How Important is Fluid Friction?

Air, like any fluid,² be it water, oil, or honey, displays a resistance to being sheared. A fluid, by nature, yields to any shearing forces. By contrast, solids do not. The shear stress within a fluid is a force (per unit area, like pressure) on an area subject to antiparallel motion. It is this kind of motion that leads to the rotational nature of fluid affected by friction. The spatial velocity gradient applied to a fluid (picture two flat plates with a fluid between them) and the stress are proportional to one another. The proportionality constant is called the viscosity.³ Such a constant is a valid descriptor of what are called Newtonian fluids. Most of the examples cited above are fairly close to Newtonian, especially air. Examples of non-Newtonian fluids are, for example, blood, the honey cited above and many others, for which a descriptive relation between shear and stresses are more complicated because the effective viscosities that are stress-dependent. That is indeed a more complicated world because the relationship between stress and deformation is not simple. It is non-linear. For relatively simple air, life is mercifully easier for modelling its behavior. The magnitudes of the viscosities for the fluid examples above vary substantially and it is quite small for air. How small? you may ask.

4.3 Mr. Reynolds and His Number

For flow issues generally, and especially aerodynamic situations, one would want to have some measure of the relative importance of the dynamic forces being exploited to provide forces on wings and the resistive forces associated with friction. Thus, we might form a (non-dimensional) fraction with dynamic pressure in the numerator and viscous shear in the denominator. Naturally, the viscosity will appear in such a number. With such a formulation for air (and its numerically small viscosity), we should expect that under normal circumstances where friction effects are small, such a numerical parameter would be very large. In fact, for an inviscid fluid, it would be infinite.

Such a number is called the Reynolds number⁴ and symbolized by Re. An interesting aspect of this number is that it includes a scale length that originated from the measure of the velocity gradient associated with the shear. For an internal flow, such as through a pipe, the diameter is an obvious choice for that length scale. For an external flow like that around an airfoil, the choice for a length scale to describe a viscous flow is a bit of a challenge for a variety of length scales could be used. By far the easiest is to choose some easily identifiable physical dimension of the flow. For an airfoil, the chord is a pretty good choice. Typically, for a whole airplane the wing chord is also commonly used for it is that dimension that best characterizes the flow along the airplane’s wing surface. For the localized development of a boundary layer, the path length along which the flow proceeds, a good dimension is the length to any point from the start of the development. In general, the length scale in the Reynolds number is included in the symbol as a subscript, such a “c” for chord. Other length scales may be used. An important one is “x” for path length to characterize the “age” of the boundary layer. This length scale is the relevant one to describe the transition from laminar to turbulent flow to be described further on.

Before going on, it would be good to state the definition of the Reynolds number:

$$ {{Re}} \,{\text{or}}\,{{Re}}_c = \frac{\rho Vc}{\mu }\;\;{\text{or}}\;\;[{\text{density}}\,x\,{\text{speed}}\,x\,{\text{length}}\,{\text{scale}}/{\text{viscosity}}] $$

Here ρ is the air density; V the velocity; c the chord, and μ is the air viscosity. Note the kinematic viscosity is ν = μ/ρ.

In any system of units, the viscosity of air (μ) is numerically small and, for most aerodynamic purposes, only slightly dependent on the temperature. That smallness will make the Reynolds number in the aviation environment large. Typically, we might expect numbers in the hundreds of thousands up to several million.

4.4 Scale Model Testing

When airplanes did not fly high or fast, similar values of density and speed would apply to a wind tunnel model or to a real airplane. Thus, the model size is the determinant of Reynolds number and the values (characterizing the importance of friction) will not be identical in a wind tunnel model and on a full-sized, geometrically similar article. There is, as a consequence, some risk that wind tunnel data cannot be used to predict full-size airplane performance to a degree that might be desired because the flow conditions are not identical. Under such circumstances, experience or mathematical adjustments to model data may or may not be effective in predicting full size airplane performance from data obtained from a wind tunnel test.

For example, a small 4-seat general aviation airplane might have a Reynolds number of four million and a one twentieth scale model would like sport a number closer to 200,000 which lies well under the number of half a million where one might expect the flow to be predominantly laminar rather than turbulent at the higher number. That wrinkle is taken up in the next section and involves the observation that the boundary layer along a flat plate changes its character when one compares laminar and turbulent flows. We know this because transition to turbulent flow occurs at a Reynolds number based on length of about 500,000. Performance predictions are indeed difficult!

An interesting situation is presented by wind tunnel modeling for a modern airliner. An example might be a 1/20 model of a Boeing 747 run in a transonic wind tunnel. A transonic tunnel operates at higher speed and is specifically designed to operate at Mach number close to one. The effective Reynolds number would be around three million. The number calculated for the real airplane operating in a much lower density and with a larger chord will give a Reynolds number close to 2.5 million. In this case, the compensating scale and density bring the two numbers fairly close. Wind tunnel test data extrapolation is quite modest and carries a correspondingly modest risk to being accurate.

There is some flexibility in choosing wind tunnel parameters that might give better matches between Reynolds number for the wind tunnel test and the real article. The available options are, however, expensive, or technically unrealistic because they violate other similarity rules, specifically those related to compressibility, or flight Mach number. Increasing the size of a wind tunnel (so the model can be larger) may be an option. Another is to operate the tunnel at elevated pressure to increase air density. Finally, gases other than air might be employed. All of these options are costly and not commonly used. The options are limited.

Fortunately, numerical modeling executed on a computer is a great help in allowing airplanes to be designed with reasonably good performance predictions. An observer might even wonder whether the configuration similarities of modern airliners, especially the twin-engine ones, might just have led engineers to identify the best way to build an efficient airplane.

4.5 Turbulent Flow

Back to basics. What about this laminar and turbulent flow issue? It turns out that to make things really interesting, nature added yet another level of complexity to viscous fluid motion. Specifically, the motion of molecules sharing dynamic information to result in the velocity profile of a boundary layer works only to a point. Under conditions where it does, the flow is rather well behaved and steady. This is so-called laminar flow. Laminar flow is smooth and quiet like a placid river. In many aeronautical circumstances, this lovely state of affairs cannot be maintained as the flow proceeds along the solid surface because the thermal motion of air molecules is not up to the task of maintaining the dynamic equilibrium we might wish for. There comes a point where the steady motion becomes chaotic involving unsteady eddies that are large at the edge of the layer and smaller closer to the body. The size variation is imposed by the presence of the boundary. These eddies are very much more effective than the random thermal motion of individual molecules in transferring the low momentum of air molecules near the wall to the freestream air passing by further out. The presence of eddies is associated with a significant increase in skin friction drag. This is turbulent flow. The NASA image (Fig. 4.6) shows the nature of the boundary layer for the two regimes wherein it can be found. There are numerous examples of laminar to turbulent flow transitions in nature. The flow from the kitchen water faucet can undergo such a change as can the smoke rising from a burning stationary cigarette.

Fig. 4.6

Velocity distributions in laminar and turbulent boundary layer (NASA Glenn Research Center). Figure at right is of a smoke visualization of transition on a flat plate boundary layer viewed from above (From Alfredsson & Matsubara, 2000)

In a simple flat plate experiment like Fig. 4.2, a Reynolds number based on flow length from the leading edge of about half a million (500,000) will result in turbulent flow. This number may vary a bit depending on the initial turbulence present or the roughness of the surface. Most modern airplanes operate at Reynolds numbers well over half a million (500,000) so that turbulence on aircraft surfaces is normal and unavoidable. The difference in noise levels between the front interior (First Class) of a commercial airliner and that at the rear is a reflection of the more intense turbulence of an “older” boundary layer, i.e., further downstream from its start.

To give some sense of scale for a flat plate experiment in sea level air, the (99%) laminar boundary layer thickness at the point of transition one foot (30 cm) from the leading is approximately 0.1 in. or 3 mm. This experiment would require the air move at about 75 feet/s (25 m/sec), a low speed for an airplane (~80 km/hr or 50 mph in more conventional units of measure). Raising that speed to 750 feet/s (at sea level) would move the transition point toward to a position about one inch from the leading edge and the boundary layer thickness at transition would be very thin ~0.01 in.! One can get a realistic appreciation of the boundary thickness on a real aircraft depicted in this book by examination of the inlet of the Concorde supersonic airliner shown in Fig. 13.6. The varying width space between the wing underside and the inlet upper side is but a few centimeters (about an inch) wide. It varies because the boundary layer has developed further on the inboard side of the inlet structure.

The no-slip condition also applies to turbulent flow albeit in a very thin laminar sublayer where interactions prevail on a molecular scale. The thickness of the turbulent boundary is larger than that of the laminar layer. On an airplane, the boundary layer is very thin near leading edges growing to several inches near the aft end of a well-designed body on which it grew. See, for example, the discussion of diverters in connection with jet engine inlets (Chap. 13) and the vortex generators described in the next section.

Description of the behavior of boundary layers on complex body shapes has long been a challenge in mathematical modeling and in wind tunnel testing as the scale dependence of Reynolds number suggests. The descriptive equations are hard to solve because the mechanism of transmitting shear forces changes character between laminar and turbulent flow. Viscosity is no longer the only descriptor of the fluid’s frictional properties. The transition also involves a number of factors, among them, how long has the flow experienced viscous stresses, the pressure history, turbulence in the incoming air, surface roughness, etc. The nature of interactions between three-dimensional eddies of vorticity continues to pose challenges for those interested in mathematical modelling.

Having characterized the importance of fluid friction, we can look at more realistic flows to illustrate the nature of pressure drag cited earlier in connection with the general topic of drag. Even though flow around a cylinder or sphere is not of great importance to the performance of a modern airplane, it is interesting and describes the basic idea of the physics around this component of drag.

Under flow conditions described by low Reynolds number (Re, based on diameter), the flow around a model cylinder in a wind tunnel may be as shown in Fig. 4.7. As Re is increased, the flow becomes unsteady because it is unstable (~100). This is the range wherein the so-called Karmann vortex street is formed. It becomes steady again in the region of interest to aviation (Re ~ 10,000) and, in this regime, the flow is generally laminar.

Fig. 4.7

Steady flow visualization around a cylinder at low Reynolds number (Re ~ 26) (Photograph by Sadatoshi Taneda published in “An Album of Fluid Motion, by Milton Van Dyke)

The point where the flow separates from the body of the cylinder is important in determining the drag force experienced because it determines the area on which the low (static) pressure (in the wake) acts on the rear side of the cylinder. In this regime, the oncoming flow proceeds around the front side of the cylinder much like flow without friction, although a little viscous drag is experienced there. The pressure is close to stagnation pressure on the front side of the cylinder and is lowest near the 12 o’clock position because of the locally high velocity. The flow separation will occur near there.

At higher Re, (>300,000) the flow becomes turbulent and tends to separate at closer to the 1 o’clock position. The picture of the flow field will resemble Fig. 4.7 with important differences. These include a very turbulent wake reaching far downstream and the absence of any large-scale organized fluid motion.

With a low-pressure wake zone at the rear and a high pressure in the front, the net force resisting the flow motion is the pressure drag that dominates the drag performance of objects like cylinders and spheres. The serious student may wish to examine graphically the laboratory result of the reduced wake area by noting that a plot of drag coefficient for flow across a cylinder exhibits a significant drop in turbulent flow (~300,000). This variation in cylinder drag coefficient is attributable to the shifting flow separation point from near 11 o’clock at low Re to 1 o’clock at fully turbulent flow Reynolds number.

On an airfoil or wing, both viscous and pressure drag components are in play. A good aerodynamicist minimizes drag by providing an opportunity for flow to remain attached to the body to recover as much of the higher pressure on the aft end of the body. This is done by increasing the chord of an airfoil-like shape. Such an increase will, however, present a greater surface area which, in turn, increases viscous drag. This is streamlining to obtain a configuration with minimum drag. Beyond streamlining is tailoring the surfaces of the airfoil to optimize the pressure distribution for minimal deleterious effects on drag while maximizing lift. Control of the boundary layer evolution in such a way as to remain laminar has proved beneficial. A notable example is the so-called laminar flow wing of the P-51 Mustang during WW II.

The shape of a nice airfoil section, such as in the photo of Fig. 3.4 illustrates streamlining. For some thickness to length (or chord) ratio, there is a minimum in total drag. For example, streamlining reduces the drag from a round wire by about a factor of 10 for an airfoil shape about 10 times longer than wide and with the same thickness as the wire diameter. Biplane builders who necessarily used lots of wires for shear strength in their wings were eager users of streamlined wires when their advantage became clear.

An interesting aspect of the transition from laminar to turbulent flow is that the turbulent layer is more energetic so that it tends to be able to flow further into a region of higher pressure without separating from the body surface. Golf players found that if a ball is dimpled to promote turbulent flow, its lower drag would allow it to travel further!

In wing design, engineers have struggled with means to mitigate the deleterious effect of the boundary layer on a wing. These include energizing the flow near the surface with air from another source (such as an engine) or removing the boundary layer altogether by sucking it away through a porous surface. Such steps are employed, primarily when the need to do so is imperative, because implementation requires power from an engine. An example is on the surfaces of the inlet of an airplane capable of supersonic speeds. There, the interactions between possible shock waves and the boundary layer can be problematic. More commonly, clever use of available air may also be of help. Such methods are incorporated in flap systems used during takeoff and landing, where the flow deflection angles are large. Multiple element flaps are examples of such designs. The use of vortex generators often accomplishes drag reduction on airplane surfaces. Let’s examine their functionality.

4.6 Vortex Generators

Since the low energy boundary layer air is the culprit that leads to flow separation from the wing and the associated drag as well as stall, it seems logical that increasing the energy in boundary layer air might allow for better performance on both these counts. Indeed, it does. The tool available is the vortex generator. I’ll abbreviate this set of words as VG. That is what the pros call it! It is a very simple vane, usually quite small, at most on the order of inches in size and protrudes into the freestream air past the boundary layer edge. It is mounted on the surface of the area where flow separation is a problem. That may be on the rearmost quarter of a wing’s upper surface, near a wing’s leading edge for stall control, or on the converging rear section of an airplane fuselage. The shape can vary: a rectangular fin, a triangular fin or a pair of these elements at opposing angles of attack to the airflow.

The function of the VG is to create a trailing vortex from its tip so that exchange between the low energy boundary layer air and the higher energy freestream air is forced. By such means, the deleterious effects of flow separation from the surface are pushed to more severe operating conditions, such as higher angles of attack (Figs. 4.8 and 4.9).

Fig. 4.8

VG pair array on the upper surface on the Boeing 737 prototype at the Museum of Flight. Such VGs may be located closer to the leading edge on a general aviation aircraft because the latter usually don’t have leading edge flaps. The sketch illustrates the function of a single VG. Note the velocity deficit approaching the VG (red arrows) is shared with a larger part of the flow behind the VG (green arrows). The no-slip condition applies always

Fig. 4.9

An example of two arrays of VGs on the wing of a Douglas A4D US Navy jet fighter. Museum of Flight display

If you thought that the drag picture for an airplane is only about laminar and turbulent flow, or only about pressure and viscous drag, you would be mistaken. The vortices we need for lift also have something to say about drag. Their contribution to drag will also be tackled in Chap. 6.

For now, we have to step back to understand how pressures on airplane surfaces arise from the motion of air over them.