Fluid dynamic principles that are fundamental to understanding the motion of fluids in radial compressors are highlighted. These include the continuity and the momentum equations in various forms. These equations are then used to delineate the effect of the fluid motion on pressure gradients on the flow. The simple radial equilibrium equation for a circumferentially averaged flow is introduced. Special features of the flow in radial compressors due to the radial motion are considered, such as the effects of the Coriolis and centrifugal forces. The relative eddy, which gives rise to the slip factor of a radial impeller, is explained. A short overview of boundary layer flows of relevance to radial compressors is provided. The flow in radial compressor impellers is strongly affected by secondary flows and tip clearance flows, and an outline is provided of the current understanding of the physics related to these. The phenomenon of jet-wake flow in compressors is described.

Introduction

Any material that flows in response to an applied stress is a fluid. Although solids acquire a finite deformation or strain upon being stressed, fluids deform continuously under the action of applied forces. In solids, stresses are related to strains; in fluids, stresses are related to rates of strain. Strains in solids are a consequence of spatial variations or gradients in the displacements of elements from their equilibrium positions. Strain rates in fluids are a result of gradients in the velocities or rates of displacement of fluid elements. Velocity gradients are equivalent to strain rates, so stresses in fluids are related to velocity gradients. The equation connecting stresses with velocity gradients in a fluid is known as the rheological law for the fluid. The simplest fluid, and as a consequence the one most often studied, is the Newtonian or linear fluid, in which the rate of strain or velocity gradient is directly proportional to the applied stress; the constant of proportionality is known as the viscosity. We deal only with Newtonian viscous fluids throughout this chapter. Non-Newtonian fluid behavior is discussed in Chapter 7. Fluid mechanics is the science of fluid motion. It uses the basic principles of mass, momentum, and energy conservation together with the rheological or constitutive law for the fluid to describe how the fluid moves under an applied force.

Many problems involving fluid mechanics arise in geodynamics. Obvious examples involve flows of groundwater and magma.

In this chapter, basic concepts in fluid mechanics are introduced. Firstly, the definition of a fluid is discussed in depth with the conclusion that a fluid is such a substance that cannot generate internal shear stresses by static deformation alone. Secondly, some important properties of fluids are discussed, which includes viscosity of fluids, surface tension of liquids, equation of state for gases, compressibility of gases, and thermal conductivity of gases. Lastly, some important concepts in fluid mechanics are discussed, which includes the concept of continuum and forces in a fluid. Within these discussions, fluid is compared to solid in both microscopic and macroscopic to reveal the mechanism of its mechanical property. Viscosity of fluid is compared to friction and elasticity of solid to give readers a better idea how it works microscopically. Forces is classified as body force and surface force for further analysis. Finally, continuum hypothesis is introduced to deem the fluid as continuously separable, which tells the reader that fluid mechanics is a kind of macroscopic mechanics that conforms Newtonian mechanics and thermodynamics.

In this Chapter

We introduce the fundamental concepts of fluid mechanics. Our focus will be on mantle convection, but we will consider a variety of other geodynamic applications. These applications utilize Newtonian fluids in which the stress is proportional to the spatial gradient of velocity. The constant of proportionality is the viscosity. Solutions for isothermal problems require an equation for conservation of mass and a force balance equation. In our applications the force balance includes the pressure forces, viscous terms, and the gravitational body force. Temperature variations require addition of a buoyancy force and an energy equation. In addition to the terms included in the heat equation in Chapter 4, terms are required to account for the advection of heat (energy). Unlike the heat equation, the equations for fluid flow are usually nonlinear, for example, the product of velocity and temperature gradient in the energy equation. This nonlinearity greatly increases the difficulty of obtaining analytical solutions.

One of the important problems we will consider in this chapter is postglacial rebound. Under the load of ice during the last ice age, the continental crust was depressed in order to achieve isostatic compensation. The surface of Greenland is currently depressed below sea level due to the load of the Greenland Ice Cap. At the end of the last ice age, about 8000 years ago, large quantities of ice melted. The removal of this ice load results in a “rebound” of the Earth’s surface in order to re-establish the isostatic balance. This rebound demonstrated beyond doubt the fluid behavior of the Earth’s mantle. The rate of rebound quantified the viscosity of the mantle.

Fluid mechanics is one of the fields that is contributing significantly to the current resurgence of interest in variational methods. Historically, calculus of variations has not been taught or utilized as a core mathematical tool in the arsenal of the fluid mechanics practitioner; it is a rare fluid mechanics textbook that even mentions variational calculus. However, recent research developments are revolutionizing the field by reframing certain fluid mechanics phenomena within a variational framework. This is particularly the case in the areas of flow control and hydrodynamic stability.

Traditionally, flow control has been implemented in an ad hoc manner involving a significant amount of trial and error within an empirical (experimental or numerical) framework. Beginning in the 1990s, flow control is increasingly being framed within the context of optimal control theory. As will be seen in Chapter 10, the solution to optimal control formulations, particularly for large problems involving many degrees of freedom, is very computationally intensive. It is only recently that the computational resources have become available that are capable of solving realistic fluid mechanics scenarios within such an optimal control framework.

Similarly, significant advances are being made in our understanding of stability of fluid flows through the application of transient growth analysis that seeks the “optimal,” or most unstable, initial perturbation (disturbance) that results in the greatest growth of the instability.