What is a viscous flow? Answer: A flow where the effects of viscosity, thermal conduction, and mass diffusion are important. The phenomenon of mass diffusion is important in a gas with gradients in its chemical species, for example, the flow of air over a surface through which helium is being injected or the chemically reacting flow through a jet engine or over a high-speed reentry body. In this book, we are not concerned with the effects of diffusion, and therefore we treat a viscous flow as one where only viscosity and thermal conduction are important.
First, consider the influence of viscosity. Imagine two solid surfaces slipping over each other, such as this book being pushed across a table. Clearly, there will be a frictional force between these objects which will retard their relative motion. The same is true for the flow of a fluid over a solid surface; the influence of friction between the surface and the fluid adjacent to the surface acts to create a frictional force which retards the relative motion. This has an effect on both the surface and the fluid. The surface feels a “tugging” force in the direction of the flow, tangential to the surface. This tangential force per unit area is defined as the shear stress r, first introduced in Section 1.5 and illustrated in Figure 15.2. As an equal and opposite reaction, the fluid adjacent to the surface feels a retarding force which decreases its local flow velocity, as shown in insert a of Figure 15.2. Indeed, the influence of friction is to create V = 0 right at the body surface—this is called the по-slip condition which dominates viscous flow. In any real continuum fluid flow over a solid surface, the flow velocity is zero at the surface. Just above the surface, the flow velocity is finite, but retarded, as shown in insert a. If и represents the coordinate normal to the surface, then in
Figure 1 5.2 Effect of viscosity on a body in a moving fluid: shear stress and separated flow.
the region near the surface, V = V(n), where V = 0 at n = 0, and V increases as n increases. The plot of V versus n as shown in insert a is called a velocity profile. Clearly, the region of flow near the surface has velocity gradients, 9 V/дп, which are due to the frictional force between the surface and the fliud.
In addition to the generation of shear stress, friction also plays another (but related) role in dictating the flow over the body in Figure 15.2. Consider a fluid element moving in the viscous flow near a surface, as sketched in Figure 15.3. Assume that the flow is in its earliest moments of being started. At the station si, the velocity of the fluid element is Vi. Assume that the flow over the surface produces an increasing pressure distribution in the flow direction (i. e., assume p3 > рг > Pi). Such a region of increasing pressure is called an adverse pressure gradient. Now follow the fluid element as it moves downstream. The motion of the element is already retarded by the effect of friction; in addition, it must work its way along the flow against an increasing pressure, which tends to further reduce its velocity. Consequently, at station 2 along the surface, its velocity V2 is less than Vi. As the fluid element continues to move downstream, it may completely “run out of steam,” come to a stop, and then, under the action of the adverse pressure gradient, actually reverse its direction and start moving back upstream. This “reversed flow” is illustrated at station S3 in Figure 15.3, where the fluid element is now moving upstream at the velocity V3. The picture shown in Figure 15.3 is meant to show the flow details very near the surface at the very initiation of the flow. In the bigger picture of this flow at later times shown in Figure 15.2, the consequence of such reversed-flow phenomena is to cause the flow to separate from
Figure 1 5.3 Separated flow induced by an adverse pressure gradient. This picture corresponds to the early evolution of the flow; once the flow separates from the surface between points 2 and 3, the fluid element shown at S3 is in reality different from that shown at S] and S2 because the primary flow moves away from the surface, as shown in Figure 15.2.
the surface and create a large wake of recirculating flow downstream of the surface. The point of separation on the surface in Figure 15.2 occurs where dV/dn = 0 at the surface, as sketched in insert b of Figure 15.2. Beyond this point, reversed flow occurs. Therefore, in addition to the generation of shear stress, the influence of friction can cause the flow over a body to separate from the surface. When such separated flow occurs, the pressure distribution over the surface is greatly altered. The primary flow over the body in Figure 15.2 no longer sees the complete body shape; rather, it sees the body shape upstream of the separation point, but downstream of the separation point it sees a greatly deformed “effective body” due to the large separated region. The net effect is to create a pressure distribution over the actual body surface which results in an integrated force in the flow direction, that is, a drag. To see this more clearly, consider the pressure distribution over the upper surface of the body as sketched in Figure 15.4. If the flow were attached, the pressure over the downstream portion of the body would be given by the dashed curve. Flowever, for separated flow, the pressure over the downstream portion of the body is smaller, given by the solid curve in Figure 15.4. Now return to Figure 15.2. Note that the pressure over the upper rearward surface contributes a force in the negative drag direction; that is, p acting over the element of surface ds shown in Figure 15.2 has a horizontal component in the upstream direction. If the flow were inviscid, subsonic, and attached and the body were two-dimensional, the forward-acting components of the pressure distribution shown in Figure 15.2 would exactly cancel the rearward-acting components due to the pressure distribution over other parts of the body such that the net, integrated pressure distribution would give zero drag. This would be d’Alembert’s paradox discussed in Chapter 3. Flowever, for the viscous, separated flow, we see that p is reduced in the separated region; hence, it can no longer fully cancel the pressure distribution over the remainder of the body. The net result is the production of drag; this is called the pressure drag due to flow separation and is denoted by Dp.
Figure 1 5.4 Schematic of the pressure
distributions for attached and separated flow over the upper surface of the body illustrated in Figure 15.2.
In summary, we see that the effects of viscosity are to produce two types of drag as follows:
Df is the skin friction drag, that is, the component in the drag direction of the integral of the shear stress r over the body.
Dp is the pressure drag due to separation, that is, the component in the drag direction of the integral of the pressure distribution over the body.
Dp is sometimes called form drag. The sum Df + Dp is called the profile drag of a two-dimensional body. For a three-dimensional body such as a complete airplane, the sum Df + Dp is frequently called parasite drag. (See Reference 2 for a more extensive discussion of the classification of different drag contributions.)
The occurrence of separated flow over an aerodynamic body not only increases the drag but also results in a substantial loss of lift. Such separated flow is the cause of airfoil stall as discussed in Section 4.3. For these reasons, the study, understanding, and prediction of separated flow is an important aspect of viscous flow.
Let us turn our attention to the influence of thermal conduction—another overall physical characteristic of viscous flow in addition to friction. Again, let us draw an analogy from two solid bodies slipping over each other, such as the motion of this book over the top of a table. If we would press hard on the book, and vigorously rub it back and forth over the table, the cover of the book as well as the table top would soon become warm. Some of the energy we expend in pushing the book over the table will be dissipated by friction, and this shows up as a form of heating of the bodies. The same phenomenon occurs in the flow of a fluid over a body. The moving fluid has a certain amount of kinetic energy; in the process of flowing over a surface, the flow velocity is decreased by the influence of friction, as discussed earlier, and hence the kinetic energy is decreased. This lost kinetic energy reappears in the form of internal energy of the fluid, hence causing the temperature to rise. This phenomenon is called viscous dissipation within the fluid. In turn, when the fluid temperature increases, there is an overall temperature difference between the warmer fluid and the cooler body. We know from experience that heat is transferred from a warmer body to a cooler body; therefore, heat will be transferred from the warmer fluid to the cooler surface. This is the mechanism of aerodynamic heating of a body. Aerodynamic heating becomes more severe as the flow velocity increases, because more kinetic energy is dissipated by friction, and hence the overall temperature difference between the warm fluid and the cool surface increases. As discussed in Chapter 14, at hypersonic speeds, aerodynamic heating becomes a dominant aspect of the flow.
All the aspects discussed above—shear stress, flow separation, aerodynamic heating, etc.—are dominated by a single major question in viscous flow, namely, Is the flow laminar or turbulent? Consider the viscous flow over a surface as sketched in Figure 15.5. If the path lines of various fluid elements are smooth and regular, as shown in Figure 15.5a, the flow is called laminar flow. In contrast, if the motion of a fluid element is very irregular and tortuous, as shown in Figure 15.5b, the flow is called turbulent flow. Because of the agitated motion in a turbulent flow, the higher-energy fluid elements from the outer regions of the flow are pumped close to the surface. Hence, the average flow velocity near a solid surface is larger for a turbulent flow
(b) Turbulent flow
Figure 1 5.5 Path lines for laminar and turbulent flows.
in comparison with laminar flow. This comparison is shown in Figure 15.6, which gives velocity profiles for laminar and turbulent flow. Note that immediately above the surface, the turbulent flow velocities are much larger than the laminar values. If (3 V/3n)„=0 denotes the velocity gradient at the surface, we have
Because of this difference, the frictional effects are more severe for a turbulent flow; both the shear stress and aerodynamic heating are larger for the turbulent flow in comparison with laminar flow. However, turbulent flow has a major redeeming value; because the energy of the fluid elements close to the surface is larger in a turbulent flow, a turbulent flow does not separate from the surface as readily as a laminar flow. If the flow over a body is turbulent, it is less likely to separate from the body surface, and if flow separation does occur, the separated region will be smaller. As a result, the pressure drag due to flow separation Dp will be smaller for turbulent flow.
This discussion points out one of the great compromises in aerodynamics. For the flow over a body, is laminar or turbulent flow preferable? There is no pat answer; it depends on the shape of the body. In general, if the body is slender, as sketched in Figure 15.7a, the friction drag Df is much greater than Dp. For this case, because Df is smaller for laminar than for turbulent flow, laminar flow is desirable for slender bodies. In contrast, if the body is blunt, as sketched in Figure 15.7b, Dp is much greater than Df. For this case, because Dp is smaller for turbulent than for laminar flow, turbulent flow is desirable for blunt bodies. The above comments are not allinclusive; they simply state general trends, and for any given body, the aerodynamic virtues of laminar versus turbulent flow must always be assessed.
Although, from the above discussion, laminar flow is preferable for some cases, and turbulent flow for other cases, in reality we have little control over what actually happens. Nature makes the ultimate decision as to whether a flow will be laminar or turbulent. There is a general principle in nature that a system, when left to itself, will always move toward its state of maximum disorder. To bring order to the system, we generally have to exert some work on the system or expend energy in some manner. (This analogy can be carried over to daily life; a room will soon become cluttered and disordered unless we exert some effort to keep it clean.) Since turbulent flow is much more “disordered” than laminar flow, nature will always favor the occurrence of turbulent flow. Indeed, in the vast majority of practical aerodynamic problems, turbulent flow is usually present.
Let us examine this phenomenon in more detail. Consider the viscous flow over a flat plate, as sketched in Figure 15.8. The flow immediately upstream of the leading edge is uniform at the freestream velocity. However, downstream of the leading edge, the influence of friction will begin to retard the flow adjacent to the surface, and the extent of this retarded flow will grow higher above the plate as we move downstream, as shown in Figure 15.8. To begin with, the flow just downstream of the leading edge will be laminar. However, after a certain distance, instabilities will appear in the laminar flow; these instabilities rapidly grow, causing transition to turbulent flow. The transition from laminar to turbulent flow takes place over a finite region, as sketched in Figure 15.8. However, for purposes of analysis, we frequently model the
Figure 1 5.7 Drag on slender and blunt bodies.
Figure 15.8 Transition from laminar to turbulent flow.
transition region as a single point, called the transition point, upstream of which the flow is laminar and downstream of which the flow is turbulent. The distance from the leading edge to the transition point is denoted by xCT. The value of xcr depends on a whole host of phenomena. For example, some characteristics which encourage transition from laminar to turbulent flow, and hence reduce xa, are:
1. Increased surface roughness. Indeed, to promote turbulent flow over a body, rough grit can be placed on the surface near the leading edge to “trip” the laminar flow into turbulent flow. This is a frequently used technique in wind-tunnel testing. Also, the dimples on the surface of a golf ball are designed to encourage turbulent flow, thus reducing Dp. In contrast, in situations where we desire large regions of laminar flow, such as the flow over the NACA six-series laminar-flow airfoils, the surface should be as smooth as possible. The main reason why such airfoils do not produce in actual flight the large regions of laminar flow observed in the laboratory is that manufacturing irregularities and bug spots (believe it or not) roughen the surface and promote early transition to turbulent flow.
2. Increased turbulence in the free stream. This is particularly a problem in wind – tunnel testing; if two wind tunnels have different levels of freestream turbulence, then data generated in one tunnel are not repeatable in the other.
3. Adverse pressure gradients. In addition to causing flow-field separation as discussed earlier, an adverse pressure gradient strongly favors transition to turbulent flow. In contrast, strong favorable pressure gradients (where p decreases in the downstream direction) tend to preserve initially laminar flow.
4. Heating of the fluid by the surface. If the surface temperature is warmer than the adjacent fluid, such that heat is transferred to the fluid from the surface, the instabilities in the laminar flow will be amplified, thus favoring early transition. In contrast, a cold wall will tend to encourage laminar flow.
There are many other parameters which influence transition; see Reference 42 for a more extensive discussion. Among these are the similarity parameters of the flow, principally Mach number and Reynolds number. High values of Мж and low values of Re tend to encourage laminar flow; hence, for high-altitude hypersonic flight, laminar flow can be quite extensive. The Reynolds number itself is a dominant factor
in transition to turbulent flow. Referring to Figure 15.8, we define a critical Reynolds number, Recr, as
D ______ Poo F-x-l’cT
The value of Recr for a given body under specified conditions is difficult to predict; indeed, the analysis of transition is still a very active area of modem aerodynamic research. As a rule of thumb in practical applications, we frequently take Recr ~ 500,000; if the flow at a given x station is such that Re = рж V^x/poo is considerably below 500,000, then the flow at that station is ihost likely laminar, and if the value of Re is much larger than 500,000, then the flow is most likely turbulent.
To obtain a better feeling for Recr, let us imagine that the flat plate in Figure 15.8 is a wind-tunnel model. Assume that we carry out an experiment under standard sea level conditions [рж — 1.23 kg/m3 and = 1.79 x 10-5 kg/(m ■ s)] and measure xcr for a certain freestream velocity; for example, say that xCI = 0.05 m when Voo = 120 m/s. In turn, this measured value of xcr determines the measured Recr as
Hence, for the given flow conditions and the surface characteristics of the flat plate, transition will occur whenever the local Re exceeds 412,000. For example, if we double Voo, that is, = 240 m/s, then we will observe transition to occur at xcr =
0. 05/2 = 0.025 m, such that Recr remains the same value of 412,000.
This brings to an end our introductory qualitative discussion of viscous flow. The physical principles and trends discussed in this section are very important, and you should study them carefully and feel comfortable with them before progressing further.