At present there are no constitutive equations that are valid over the entire range of densities and velocities encountered in the storage and handling of granular materials. Most of the available equations fall into one of two regimes: (i) slow flow and (ii) rapid flow. In the slow flow regime, the solids fraction ν is high and forces are exerted across interparticle contacts which last for a long time compared to the contact time in the rapid flow regime. The contacts occur during the sliding and rolling of particles relative to each other. In the rapid flow regime, the solids fraction is low, and momentum is transferred mainly by collisions between particles and by free flight of particles between collisions. Consider the flow of a granular material between two parallel plates. If V is the relative velocity of the plates and H is the gap between them, the stresses are found to be approximately independent of the nominal shear rate γ ≡ V/H in the slow flow regime (small γ and high ν) and to increase strongly with γ in the rapid flow regime (large γ and low ν).

In devices such as hoppers and chutes, both the regimes can occur in different spatial regions, and there can also be transition regions where the nature of the flow changes from one regime to the other. Given a device and a set of operating conditions, it is difficult to determine a priori the type of flow regime that is likely to prevail. However, the following criterion can be used as a very rough guideline (Savage and Hutter, 1989).

In this chapter, we apply the hydrodynamic equations derived in Chapter 7 to some simple flow problems, and compare the results with experimental data, where available. In each of the problems, we shall first employ the heuristic description of Haff (1983) (see §7.2), followed by the kinetic theory model that was described in Chapter 7. The simplifications made in the heuristic high-density theory allow relatively easy solution of the equations of motion, and give physical insight into the behavior. The results of the kinetic theory, when compared with those of the heuristic high-density theory, gives us an understanding of the effects of the compressibility of the granular medium. We also see how the results of the high-density theory appear as a certain limit of the kinetic theory.

All the problems that we consider, indeed all problems of practical interest, require the specification of boundary conditions at solid walls. In some cases, such as flow in inclined chutes, we also require boundary conditions at a free surface, i.e., the interface between the granular medium and the atmosphere. For the flow of fluids, the no-slip boundary condition is usually imposed at solidwalls, thereby specifying the velocity of the fluid at thewalls. The temperature of the fluid at the walls is set by specifying the temperature of the walls, or by specifying the flux of energy at the walls. In addition, the pressure at some of the boundaries must be specified. Boundary conditions for granular materials are significantly different: they usually display considerable slip at solid boundaries, and the grain temperature clearly has little to do with the thermodynamic temperature of the walls.

PRELIMINARIES AND SCALING

A characteristic feature of slow, quasi-static flow of granular materials, considered in the previous chapters, is the rate independence of the stress. As discussed earlier, in the slow flow regime, grains are in abiding contact and friction is the dominant mechanism for generating shear forces. In this chapter we consider the contrasting regime of rapid flow, in which grains are in continuous fluctuational motion, and come into contact only during very brief collisions.We shall see that the stress in this regime of flow is rate dependent; indeed, the stress varies as the square of the shear rate for shear flow with a spatially uniform shear rate.

Our physical picture of rapid flow is that of grains in a state of constant agitation, with interactions between them occurring only through instantaneous collisions, as shown in Fig. 7.1. To simplify our analysis, we shall assume uniformity in size and shape, and consider the granular material to be composed of smooth spheres. By this, we mean that there is no tangential force exerted by one sphere on the other at the point of contact. This picture is identical to that of molecules in a gas, which is why granular materials in this state are often referred to as “granular gases” in the literature. However, there is a fundamental and crucial difference between a granular material and a gas: collisions or interactions between molecules are elastic, i.e., the net energy of a colliding pair is conserved, but collisions between grains are inelastic.

In Chapter 7, we considered the kinetic theory of a granular gas composed of smooth inelastic spheres. However, the particles in most granular materials one normally encounters are rough, and it is therefore desirable to extend the theory to rough spheres. It is clear that particle roughness will result in the transmission of a tangential impulse during collision. We shall see that the tangential impulse is partly determined by the angular velocities of the colliding pair. As a consequence, the hydrodynamic balance of the angular momentum field, which was implicitly satisfied for smooth particles, must be enforced here.

The dimensional analysis of §7.1 does not depend on whether the particles are smooth or rough. Hence the scaling of the stress with the shear rate, particle size, and density for a granular gas composed of rough particles will be the same as that for smooth particles. However, we shall see that the form and symmetry of the stress tensor is different and that additional hydrodynamic variables are required to describe the flow of a rough granular gas.

Pidduck (1922) was the first to develop a kinetic theory for a gas of rough spherical molecules. He considered “perfectly rough” spheres, which conserve the total kinetic energy of a particle pair during collision. He determined the equilibrium distribution function and applied the Chapman–Enskog analysis to determine the first correction from equilibrium for a dilute gas; his analysis is presented (with a few corrections) in Chapman and Cowling (1964, chap. 11). McCoy et al. (1966) extended the analysis to dense gases, and Lun (1991) further extended it to a dense gas of nearly elastic, nearly perfectly rough spheres.

A bunker is a combination of a bin and a hopper (Fig. 1.5). The abrupt change in geometry at the bin–hopper transition results in a rich variety of flow patterns, and oscillatory wall stresses. These features pose formidable difficulties for modeling bunker flow. At present, there are no satisfactory models which can capture all the observed features.

Some of the experimental observations are summarized below. This is followed by a discussion of models for the bin section, the transition region, and the hopper section.

EXPERIMENTAL OBSERVATIONS

Flow Regimes

The patterns observed by Nguyen et al. (1980) for the flow of sand through a bunker are shown in Fig. 4.1. For θw < 40°, mass flow (type A) occurs regardless of the value of H/W (Fig. 4.2). Here W and H are the half-width of the bunker and the height of the free surface of the material relative to the exit slot, respectively. (For small values of H, the free surface is not flat. In this case, H is the elevation of the point of intersection of the free surface with the bunker wall.) For θw > 70°, funnel flow of type B occurs when H/W exceeds a critical value (≍3) and funnel flow of type C occurs for a smaller value of H/W. As indicated by the hatched regions in Fig. 4.2, the boundaries between various regimes are not sharply defined. For θw = 60°, there is an interesting transition from mass flow to funnel flow, and back again to mass flow as H/W decreases from high values.

The flow of granular materials such as sand, snow, coal, and catalyst particles is a common occurrence in natural and industrial settings. Unfortunately, the mechanics of these materials is not well understood. Experiments reveal complex and, at times, unexpected behavior, whereas existing theories are often tentative and do not represent the entire range of observed behavior. Nevertheless, significant advances have been made in the understanding of the mechanics of granular flows, and the time is ripe for an account of experimental observations and theoretical models pertaining to flow in relatively simple geometries.

The importance of understanding granular flows need not be overstated – a large fraction of the materials handled and processed in the chemical, metallurgical, pharmaceutical, and food-processing industries are granular in nature. The flow and transportation of these materials are often critical operations in these processes. In most cases, the design of processes and equipment is based largely on experience and empirical rules. An appreciation of the underlying principles may be helpful in developing better design and operating procedures.

Some of the early investigations of granular flow were motivated by the need to understand the deformation of soils subjected to external loads, such as large structures. The deformation rates in these processes are usually very small. Theoretical models for these slow flows have increased in sophistication and complexity over the years, borrowing concepts from metal plasticity and soil mechanics. A contrasting picture of granular flow has emerged over the last three decades. This is believed to be applicable to rapid flows, where the deformation rates are large.

General Background of Flexible-Wing Flyers

In the development of MAVs, there are three main approaches, which are based on flapping-wings, rotating wings, and fixed wings for generating lift. We focus on the fixed, flexible-wing aerodynamics in this chapter. It is well known that flying animals typically have flexible wings to adapt to the flow environment. Birds have different layers of feathers, all flexible and often connected to each other. Hence, they can adjust the wing planform for a particular flight mode. The flapping modes of bats are more complicated than those of birds. Bats have more than two dozen independently controlled joints in the wing (Swartz, 1997) and highly deforming bones (Swartz et al., 1992) that enable them to fly at either a positive or a negative AoA, to dynamically change wing camber, and to create a complex 3D wing topology to achieve extraordinary flight performance. Bats have compliant thin-membrane surfaces, and their flight is characterized by highly unsteady and 3D wing motions (Figure 3.1). Measurements by Tian et al. (2006) have shown that bats exhibit highly articulated motion, in complete contrast to the relatively simple flapping motion of birds and insects. They have shown that bats can execute a 180° turn in a compact and fast manner: flying in and turning back in the space of less than one half of its wingspan and accomplishing the turn within three wing beats with turn rates exceeding 200°/s.

Low Reynolds number aerodynamics is important to a number of natural and manmade flyers. Birds, bats, and insects have been of interest to biologists for years, and active study in the aerospace engineering community has been increasing rapidly. Part of the reason is the advent of micro air vehicles (MAVs). With a maximal dimension of 15 cm and nominal flight speeds of around 10 m/s, MAVs are capable of performing missions such as environmental monitoring, survelliance, and assessment in hostile environments. In contrast to civilian transport and many military flight vehicles, these small flyers operate in the low Reynolds number regime of 105 or lower. It is well established that the aerodynamic characteristics, such as the lift-to-drag ratio of a flight vehicle, change considerably between the low and high Reynolds number regimes. In particular, flow separation and laminar–turbulent transition can result in substantial change in effective airfoil shape and reduce aerodynamic performance. Because these flyers are lightweight and operate at low speeds, they are sensitive to wind gusts. Furthermore, their wing structures are flexible and tend to deform during flight. Consequently, the aero/fluid and structural dynamics of these flyers are closely linked to each other, making the entire flight vehicle difficult to analyze.

The primary focus of this book is on the aerodynamics associated with fixed and flapping wings. Chapter 1 offers a general introduction to low Reynolds number flight vehicles, considering both biological flyers and MAVs, followed by a summary of the scaling laws, which relate the aerodynamics and flight characteristics to a flyer's size on the basis of simple geometric and dynamics analyses.

Bird, bat, and insect flight has fascinated humans for many centuries. As enthusiastically observed by Dial (1994), most species of animals fly. There are nearly a million species of flying insects, and of the living 13,000 warm-blooded vertebrate species (i.e., birds and mammals), 10,000 (9000 birds and 1000 bats) have taken to the skies. With respect to maneuvering a body efficiently through space, birds represent one of nature's finest locomotion experiments. Although aeronautical technology has advanced rapidly over the past 100 years, nature's flying machines, which have evolved over 150 million years, are still impressive. Considering that humans move at top speeds of 3–4 body lengths per second, a race horse runs approximately 7 body lengths per second, a cheetah accomplishes 18 body lengths per second (Norberg, 1990), a supersonic aircraft such as the SR-71, “Blackbird,” traveling near Mach 3 (~2000 mph) covers about 32 body lengths per second, it is amazing that a common pigeon (Columba livia) frequently attains speeds of 50 mph, which converts to 75 body lengths per second. A European starling (Sturnus vulgaris) is capable of flying at 120 body lengths per second, and various species of swifts are even more impressive, over 140 body lengths per second. The roll rate of highly aerobatic aircraft (e.g., the A-4 Skyhawk) is approximately 720°/s, and a Barn Swallow (Hirundo rustics) has a roll rate in excess of 5000°/s. The maximum positive G-forces permitted in most general aviation aircraft is 4–5 G and select military aircraft withstand 8–10 G.