The Physics of Energy, page 119
Example 29.3 Variations in Pressure, Temperature, and Density of Air in Motion
Consider airflow around the tip of a rotating wind turbine blade. Relative to the blade, the ambient air in thelocal wind is moving at a speed of m/s. How do the pressure, temperature, and density of the air vary between different locations in the flow?
According to Bernoulli’s equation (29.16), the (static) pressure of a fluid increases as the velocity of the fluid decreases. The change in static pressure is given by the (negative of the) dynamic pressure . To get an upper bound on the magnitude of the fluctuations, we consider the pressure difference between the flow at m/s and m/s.
The fractional variation in its pressure is roughly
Assuming that the air is a compressible ideal gas with temperature and pressure that vary adiabatically with fluctuations in volume, is a constant for a given mass of air, with . It follows that
From , we have
The proportional changes in all these quantities are therefore relatively small, with the smallest fractional change in the temperature.
29.2.3 Circulation and Vorticity
Two closely related but distinct features of fluid flow that arise in several aspects of the study of wind power are vorticity and circulation. Vortices and eddies are phenomena familiar from bathtubs and rivers. Whirlpools are large-scale vortices. Vortices can appear in steady flow, as when water drains from a bathtub, or in unsteady flow, where vortices break loose from a steady flow and wander downstream in quasi-regular way (Figure 29.5). A vortex signals a net circulation of fluid. Circulation can be quantified by constructing the line integral of the velocity around a closed curve
(29.17)
is known as the circulation around the curve . The sign of depends on the direction chosen for the integration around the curve .
Figure 29.5 (a) A vortex formed in the steady flow of water down a drain. (Credit: Jessica Rose) (b) A vortex street moving downstream from an obstruction. (Credit: Jürgen Wagner reproduced under CC-BY-SA 4.0 license via Wikimedia Commons)
A particularly simple example of circulation is a symmetric flow of an incompressible, ideal fluid around a circular cylinder (see Figure 29.6),
(29.18)
where we have chosen the radial dependence so that the circulation is constant – and equal to Γ – on all curves that surround the cylinder (Problem 29.10). The figure shows a cross section through the cylinder and the fluid circulating around it. It is not hard to show that this velocity field satisfies eq. (29.10) as an incompressible fluid must (Problem 29.10).
Figure 29.6 (a) A vortex circulating around a cylindrical obstruction in a fluid. is nonzero but the vorticity vanishes everywhere in the fluid. The polar coordinates are defined in the figure, as are the unit vectors and r.
Suppose that, in contrast to the previous example, a surface S can be found that spans the curve in eq. (29.17) and lies entirely within the fluid. In that case, the integral that defines can be converted to an integral over the surface S by use of Stokes theorem (B.27),
(29.19)
Under these circumstances the circulation around a curve in a fluid can be regarded as a surface integral of a local quantity
(29.20)
which is known as the vorticity. measures the magnitude of local circulation of the fluid and points along the axis around which the fluid is circulating. Two examples of flows with vorticity are shown in Figure 29.7. The flow on the left is obviously a vortex; the flow on the right does not appear to obviously circulate, but increases in velocity from left to right. When υ is integrated around the circle , the result is nonzero because the (positive) contributions on the right are larger than the (negative) contributions on the left.
Figure 29.7 Two flows with vorticity, illustrated by velocity vectors of the flow. In (a) the fluid is clearly circulating. In (b) the flow is linear, but the velocity increases to the right, making the integral around a closed curve nonzero.
If a fluid flow has nonzero vorticity at a given point x, then the circulation around any sufficiently small closed curve encircling that point and the axis of vorticity will be nonzero. Thus, vorticity implies circulation. The converse, however, is not true. The symmetric flow around a cylinder shown in Figure 29.6 provides a counter-example: the velocity defined in eq. (29.18) satisfies everywhere outside the cylinder (Problem 29.10), so it has zero vorticity even though its circulation is not zero. This is possible because the fluid velocity flow vector υ is not defined as a continuous field extending into the interior of the cylinder. A fluid flow in which the vorticity is zero is called irrotational. The fact that an irrotational flow can nevertheless have nonzero circulation around an obstructing object figures centrally in the explanation of lift in §29.4.
For a steady flow, vorticity, like all other properties of the fluid, is time-independent. The time dependence of vorticity in a non-steady flow can be more complex, as illustrated in Figure 29.5(b) for example. The British physicist William Thomson (Lord Kelvin, for whom the absolute temperature scale is named) proved that any flow of an ideal (inviscid) barotropic3 fluid that is irrotational will remain irrotational. Furthermore, Kelvin’s circulation theorem states that the vorticity of a packet of fluid is constant as the packet moves through the flow. Hence vorticity, once created, is persistent. Viscous effects eventually dissipate vorticity in real fluids, as when, for example, a vortex stirred up in a glass of water eventually comes to rest.
Circulation and Vorticity
The circulation
measures the extent to which a fluid circulates around a closed curve . If the vorticity is non-vanishing at a point within the fluid, then there are curves surrounding that point about which the circulation is nonzero.
A fluid in which is termed irrotational. When a curve surrounds an obstruction in the fluid such as a cylinder, the circulation can be nonzero even if everywhere in the fluid. Irrotational flows can have non-vanishing circulation in the presence of obstructions.
29.3Viscosity
Internal friction plays an essential role in the motion of fluids. The term viscosity refers both to the phenomenon of friction in fluid flow and to the parameter that measures the tendency of a fluid to experience internal frictional forces. A fluid’s viscosity is an intrinsic property like its density or heat capacity. In everyday experience we associate viscosity with the thickness of a fluid as well as with its resistance to flow. A relatively low-viscosity liquid such as water splashes when it is poured and stays in motion for a relatively long time when it is stirred. In contrast, a high-viscosity liquid such as honey pours smoothly and comes quickly to rest when stirred.
Viscosity can never be entirely ignored when considering the interaction of objects with a flowing fluid. Molecules of the fluid interact strongly with the molecules of the object at the interface, leading to the no-slip condition: the fluid must come to rest relative to the object at the object’s surface. A commonly encountered, though counter-intuitive, example of this phenomenon is the build-up of dust on the surface of a fan blade (Figure 29.8). Even though the blade may move quickly through the air when turning, dust that has accumulated on the blade is not swept away because the air is at rest at the blade’s surface. The layer directly above an object’s surface through which a fluid recovers its free-stream velocity is known as the boundary layer; one example being the planetary boundary layer that was analyzed in the previous chapter. Viscosity cannot be ignored in the boundary layer because it is responsible for bringing the fluid to rest. If a fluid has low viscosity (or more precisely, a high Reynolds number, §29.3.2), however, the boundary layer is thin and viscosity is not important in the bulk flow.
Figure 29.8 Dust accumulates on a ceiling fan despite its rapid rotation because the air comes to rest in the boundary layer at the blade’s surface. (Credit: Noelle Talley, Homemakerchic.com)
29.3.1 Defining Viscosity
Viscosity is associated with shear stress. We have already encountered this concept under static conditions, but viscous shear characterizes a fluid in motion rather than at rest. Suppose a fluid is flowing steadily in the x-direction above the plane , where there is a solid surface, as shown in Figure 29.9. This could illustrate water flowing in a pipe, a wind stream blowing over Earth’s surface as discussed in §28.1.5, or the air passing over the blade of a wind turbine. Far above the plane the fluid is flowing at a speed v, but at the surface the fluid is at rest as required by the no-slip condition. The boundary layer is conventionally defined as the domain in z over which the fluid recovers 99% of its bulk velocity . A flow such as the one shown in Figure 29.9, where layers of fluid slip past one another, is known as laminar flow, because the fluid moves in quasi-parallel layers with (ideally) no disruption. At a small enough velocity , all fluids flow in this way. Without viscosity, however, laminar flow is unstable since the slightest perturbation would never damp out, but would instead propagate indefinitely downstream of its origin.
Figure 29.9 A viscous flow past a plate in the xy-plane. High above the plate the fluid flows with a bulk velocity , while at the plate’s surface the fluid is at rest. The fluid recovers its bulk velocity over a distance D. Note that this laminar flow has nonzero vorticity, as indicated in the figure.
In a boundary layer, such as the one shown in Figure 29.9, the fluid at each value of z exerts a drag force on the layers of fluid directly above and below, which are moving at different speeds. The force on the layer above acts in the -direction, opposing the motion of the fluid, much like the frictional force familiar from particle mechanics. The frictional force per unit area acts in the xy-plane, so it is a shear stress. The viscosity η of the fluid is defined as the coefficient of proportionality between the frictional force and the velocity gradient that gives rise to the force.
(29.21)
(A partial derivative is necessary because can, in principle, depend on the other coordinates x and y.)
The units of viscosity are momentum per unit area, or kg/m s, and the SI unit is the pascal-second (1 Pa s = 1 kg/m s). A cgs unit, the poise (abbreviated P) is commonly used (1 P = 1 gm/cm s = 0.1 Pa s). Common liquids have viscosities on the scale of poise = 1 centipoise. For example, at 20 , the viscosities of water, di-ethyl ether, and olive oil are 1.002, 0.233, and 84 centipoise, respectively. Gases typically have viscosities on the scale of millipoise. For example, air at STP has poise. The viscosities of gases depend only weakly on density and pressure.
Viscosities can be determined by studying laminar flow in a straight pipe of circular cross section. In this configuration the total mass flowing through the pipe per unit time is related to the pressure drop from one end of the pipe to the other by Poiseuille’s law,
(29.22)
where R and l are the radius and length of the pipe and ρ and η are the density and viscosity of the fluid. Poiseuille’s law, which is derived in Example 29.4, summarizes many commonly observed aspects of viscous flow. For example, the fact that with all other variables fixed expresses the phenomenon that pressure drops along the length of a pipe through which water is flowing. The fact that agrees with the observation that fire hoses, which must maintain high pressure, are very thick. Poiseuille’s law provides a simple method to estimate the viscosity of different fluids, by measuring the flow rate as a function of pressure in a pipe of fixed length and radius.
Example 29.4 Poiseuille’s Law
Poiseuille’s law relates pressure, viscosity, and flow rate in a cylindrical pipe. It illustrates how viscosity affects flow in a practical situation and also shows how it can be measured. Consider the uniform flow of an incompressible fluid through a straight pipe with circular cross section as shown in the figure at right. We assume that the flow is steady and laminar, that the pipe is very long, and that we can ignore boundary effects at the ends of the cylinder. From the translational and rotational symmetries along and around the pipe’s central axis, it follows that the fluid velocity only has an component, and the speed depends only on the distance r from the center of the pipe. Thus, . The no-slip condition requires the fluid to be at rest at the walls, . Consider a (mathematical) cylinder of radius r and length l in the fluid. From the definition of viscosity, the shear stress on this cylinder is . The total force on the cylinder’s horizontal surface (with area ) is then . The minus sign indicates that the viscosity opposes the motion. Since the flow is steady, this force must be cancelled by the force generated by the pressure difference between the ends of the cylinder, which is ,
This simple differential equation can be integrated, remembering that , to get an equation for ,
υ vanishes at and increases proportionally to the pressure that drives the flow; it decreases with the viscosity (increased friction) and with the length of the cylinder (also increased friction).
The total mass of fluid flowing through the pipe per unit time, , can be obtained by integrating the mass flux in the -direction, , over the cross section of the pipe,
which is Poiseuille’s law.
29.3.2 Reynold’s Number and the Variation of Viscous Effects with Length Scale
When an object is placed in a fluid flow with a homogeneous velocity v far from the object, viscous shear stress gives rise to a drag force on the object. The viscous drag on the object is proportional to both the viscosity and the gradient of the velocity in the vicinity of the object (eq. (29.21)), , where K is some relevant characteristic dimension of the object. The object also experiences a drag force proportional to the dynamic pressure of the fluid . (See the discussion of drag forces in §2.3.) The dimensionless ratio of these two sources of drag,
(29.23)
defines the Reynolds number (Re) named after the Anglo-Irish engineer Osborn Reynolds.
The character of fluid flow in the vicinity of an object is determined by the Reynolds number. At large Reynolds number, viscous drag is negligible. Minute fluctuations in fluid motion do not damp out, and lead to turbulence. At small Reynolds number, viscous damping dominates and fluid flow is laminar. For a fluid flowing in a long cylindrical pipe, for example, where K is taken to be the diameter and v is the average speed of flow, the flow is always laminar if and always turbulent if . In between is a transition region, where both laminar and turbulent flow are possible depending on details such as the roughness of the pipe’s inner surface. Examples of flows at low and high Reynolds numbers are shown in Figure 29.10. The same fluid behaves very differently depending on the speed v and the scale K of the flow. A person ( m) moving through water at m/s, for example, experiences a turbulent environment since under these conditions. In contrast, the flow of water at cm/s past a small creature ( cm) is predominantly laminar since under these conditions. At human scales ( m/s, m) water would have to be replaced by honey () to have ! Note that at low enough velocities and/or small enough scales any fluid flow is laminar and dominated by viscous forces.
Reynolds Number and Viscosity
The Reynolds number of a particular fluid flow is given by the ratio of inertial forces to viscous forces within the fluid, and is a characteristic of a flow pattern. For fluid flow relative to a surface,
where v is the maximum velocity of the fluid relative to the surface, K is a length scale characteristic of the surface, and η is the viscosity of the fluid, given by the ratio of shear stress to velocity gradient. Flow at low Reynolds number is typically smooth and steady laminar flow, while flow at high Reynolds number is turbulent and chaotic. For air at STP,
Figure 29.10 Flows past a cylinder (in cross section) at (a) low and (b) high Reynolds number. At low Re the flow is everywhere laminar. At high Re there is a turbulent wake behind the cylinder, but the flow is laminar outside the wake.
The length scale K that enters the definition of Reynolds number is somewhat arbitrary. For a pipe of circular cross section, for example, either the radius or the diameter could be used. (It is conventional to use the diameter.) For a pipe with some other cross section, K is chosen so that the transition from laminar to turbulent flow occurs close to the same values of Re as for a circular pipe. This leads to some ambiguity in the definition of Reynolds number in any specific isolated situation. This ambiguity cancels out, however, when Reynold’s number is used to compare the behaviors of two geometrically similar objects of different overall size. For example, the flow of air with speed v around a sphere of radius R will have the same character as the flow of air at speed around a sphere of radius since the Reynolds numbers of the two flows are the same.
The similarity of flows at the same Reynolds number makes it possible to use the results of tests at a small scale to predict the behavior of full-scale systems. Consider, for example, the aerodynamics of a large wind turbine blade. Early in the design process, it is impractical to test a variety of potential designs at full scale. Reynolds number scaling implies that useful information can be obtained by testing a scaled down version of the blade, provided that the density and velocity of the airstream are adjusted to give a Reynolds number corresponding to actual operating conditions. Thus, for example, the aerodynamic characteristics of a wind turbine blade of length 50 meters, can be studied on a 1/10th scale model if the product of the density and velocity of the airstream is increased by a factor of 10, so Re is unchanged.
