We now present exact solutions to some flow problems obtained by solving the governing differential equations. Though exact solutions are known to only a small number of problems due to the complexity of the governing equations, these solutions are very valuable not only for the insight that they offer, but also as a means of testing the accuracy of numerical schemes such as the one presented in [49]. All the solutions that are presented in this chapter have been obtained by guessing (fully, or in terms of some function to be determined) the velocity field. If one guesses the stress field, then of course, one would have to check the compatibility conditions presented in Vol. I; the same compatibility conditions derived for linear elasticity hold, with the rate of deformation tensor and velocity playing the roles of the linearized strain tensor and displacements, respectively.

In contrast to the control volume approach, the solution for each unknown field, such as the velocity or pressure field, is obtained at each point of the flow domain as a function of the position vector and time, from which other quantities of interest (e.g., the drag force) can be derived. Needless to say, this detailed information is obtained only at the expense of the increased complexity of having to solve the governing differential equations. Since the fluid is assumed to be incompressible, the pressure field is not constitutively related to the density and pressure, and is either prescribed, or has to be determined from the governing equations and boundary conditions. Often, we shall be interested in finding the temperature field. Since, in this chapter, we assume the fluid to be incompressible, and the viscosity of the fluid to be constant, the equations of momentum and energy are decoupled. Hence, we can first find the velocity and pressure fields using the continuity and momentum equations, before proceeding to find the temperature using the energy equation.

We consider only laminar, fully developed flows in this chapter.

The subject of compressible flow has wide applications in high speed flows such as those occurring in rocket or gas turbine engines, or the flow around supersonic airplanes; we now turn our attention to the analysis of such flows. With the governing equations discussed in Chapter 1, and a knowledge of the appropriate constitutive laws, it should be possible to carry out the analysis carried out in this chapter for any gas. For reasons of simplicity, however, we shall restrict ourselves to a perfect gas. For a numerical implementation of the equations presented in this chapter, see [50].

As mentioned in Section 1.3.9, under the assumption of incompressibility, the equations of continuity and momentum are sufficient to solve for the velocity and pressure fields. However, for compressible flows, since the density is not constant, the equations of continuity, momentum and energy conservation have to be considered simultaneously in order to obtain a solution to a flow problem.

In reality, every fluid is compressible. However, for liquid flows and for flow of gases with low Mach numbers, the density changes are so small that the assumption of incompressibility can be made with reasonable accuracy. We have pointed out in Section 1.6, following our discussion of the Eckert number, that the assumption of incompressibility can be made when the Mach number M defined as the ratio of the speed of flow to the speed of sound is less than 0.3.

Compressible flows can be classified based on the Mach number as follows:

1. Subsonic flow: The Mach number number is less than one everywhere in the flow. If, further, the Mach number everywhere is less than 0.3, the incompressibility assumption can be made.

2. Transonic flow: The Mach number in the flow lies in the range 0.8 to 1.0. Shock waves appear, and the flow is characterized by mixed regions of subsonic and supersonic flow.

3. […]

The significance of the vorticity field has been stressed at a number of points in the discussion thus far. For example, the Helmholtz decomposition showed that the vorticity can be used with the dilatation to re-create the velocity field. Furthermore, vorticity does often concentrate into small regions, as in a tornado, and so by following the vorticity dynamics, one has an economical and insightful means of capturing the essence of the flow field. Another significant example is in the case of turbulent flows, where in many ways the dynamics of the vorticity field offers the most direct means of understanding such key processes in the flow as momentum transport and the passage of energy between scales.

An equation that governs the physics of the vorticity field in a moving fluid can be obtained by taking a curl of the Navier-Stokes equation (16.2). Under many circumstances, as in the case of incompressible flow, the pressure is removed by the action of taking the curl, so that the resulting vorticity equation offers a route toward determining the flow without directly taking the pressure into account. This may represent an advantage in the analysis of some flows by providing a more straightforward approach toward understanding the physics of the flow than can occur with the coupled velocity and pressure. In particular, the pressure continuously adapts to the evolution of the velocity field so that it may be more difficult to establish the origin and cause of flow behaviors in this case than through examining the vorticity field and its evolution. In flows for which vortical structures dominate the physics, as in turbulent flow, there very well may be a definite advantage to analyzing the flow through numerical schemes that model the vorticity equation. In this case, the terms in the vorticity equation represent the physical processes that are most relevant for understanding the dynamics of the flow field.

What Is a Fluid?

What distinguishes fluids from solids? Because we all have an intuitive understanding of the difference, the question is really about precisely identifying the formal differences between them. In this regard, it is useful to consider the different ways that fluids and solids react to applied forces. For example, in the case of a solid, the material offers resistance if we press down on it. If the applied force is not so large as to shatter the solid, then it is clear that the solid is quite capable of resisting the force so as to reach a state of equilibrium. The solid arrives at a state where it ceases to move or deform.

Consider now specifically the case of a gas, as shown in Fig. 1.1. In the figure, a piston is pushing down on the gas, and although initially the gas might compress because of the applied force, it is also able to eventually reach a point for any given applied force where it does not compress further. In other words, the gas is capable of resisting the downward force in the same way as the solid. We may conclude that resistance to a normal force is not a good candidate for framing the distinguishing properties of fluids and solids.

The situation for fluids and solids is different if we consider an applied shear force, as in the experiment indicated in Fig. 1.2. Following the application of a shear force to the top surface of the solid, as shown in Fig. 1.2(a), an equilibrium is reached in which the body has deformed a fixed amount. Alternatively, if the container holds a fluid, as in Fig. 1.2(b), and a shear force is applied to the top lid, the fluid cannot prevent the lid from sliding to the side. This is true no matter how small the applied force may be. This is not to say that the fluid does not offer resistance – it does – but the resistance it offers cannot be enough to create a stationary equilibrium.