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Essentials of Fluid Mechanics

The basic fundamentals of fluid mechanics are essential for understanding the fluid dynamics of pumping machinery. This chapter aims to provide a quick revision of the definitions and basic laws of fluid dynamics that are important for a thorough understanding of the material presented in this book. Of particular interest are the kinematics of fluid flow; the three conservation principles of mass, momentum, and energy; relevant dimensionless parameters; laminar and turbulent flows; and friction losses in piping systems. Some applications of relevance to pumping machinery are also considered.

1.1 Kinematics of Fluid Flow

To fully describe the fluid motion in a flow field it is necessary to know the flow velocity and acceleration of fluid particles at every point in the field. This may be a simple task in laminar flows but may be difficult in turbulent flows. If we use the Eulerian method and utilize Cartesian coordinates, the velocity vector at any point in a flow field can be expressed as

where is the velocity vector; u, v, and w are the velocity components in the x, y, and z directions; and , and are unit vectors in the respective directions. In general, each of the velocity components can be a function of position and time, and accordingly we can write

The components of acceleration in the three directions can be expressed as

The acceleration vector becomes

This vector can be split into two components, the local component, , and the convective component, , that can be expressed as

1.1.1 Types of Flows

The flow field can be described as steady or unsteady, uniform or non-uniform, compressible or incompressible, rotational or irrotational, one-, two-, or three-dimensional, and can also be described as laminar or turbulent. The flow is said to be steady if the velocity vector at any point in the flow field does not change with time.

Accordingly, the local component of acceleration () vanishes if the flow is steady. The flow can also be described as uniform if the velocity vector does not change in the streamwise direction. For example, the pipe flow shown in Figure 1.1 is uniform since the velocity vector does not change downstream, but the flow in the bend shown in Figure 1.2 is non-uniform.

Figure 1.1 Laminar flow in a pipe as an example of uniform flow

Figure 1.2 Flow in a 90° bend as an example of non-uniform flow

The flow is described as incompressible if the density change within the flow field does not exceed 5%. Accordingly, most of the flows in engineering applications are incompressible as, for example, flow of different liquids in pipelines and flow of air over a building. However, compressible flows occur in various applications such as flow in the nozzles of gas and steam turbines and in high speed flow in centrifugal and axial compressors. In general, the flow becomes compressible if the flow velocity is comparable to the local speed of sound. For example, the flow of air in any flow field can be assumed incompressible up to a Mach number of 0.3.

The flow is called one-dimensional (1-D) if the flow parameters are the same throughout any cross-section. These parameters (such as the velocity) may change from one section to another. As an approximation, we may call pipe or nozzle flows 1-D if we are interested in describing the average velocity and its variation along the flow passage. Figure 1.3 shows an example of 1-D flow in a pipe with constriction. On the other hand, the flow is called 2-D if it is not 1-D and is identical in parallel planes. For example, the viscous flow between the two diverging plates shown in Figure 1.4 is two-dimensional. In this case, two coordinates are needed to describe the velocity field.

Figure 1.3 One-dimensional flow in a pipe with constriction

Figure 1.4 Two-dimensional flow between two diverging plates

If the flow is not 1-D or 2-D, it is then three-dimensional. For example, flow of exhaust gases out of a smoke stack is three-dimensional. Also, air flow over a car or over an airplane is three-dimensional.

1.1.2 Fluid Rotation and Vorticity

The rate of rotation of a fluid element represents the time rate of the angular displacement with respect to a given axis. The relationship between the velocity components and the rate of rotation can be expressed as

where ?x, ?y, ?z represent the rate of rotation around the x, y, and z axes.

The vorticity ? is defined as twice the rate of rotation. Accordingly, the vorticity vector can be expressed as

The flow is called irrotational when the rate of rotation around the three axes is zero. In this case, we must have ?x = ?y = ?z = 0 for irrotational flow. The components of the vorticity vector in cylindrical coordinates can be written as

1.2 Conservation Principles

1.2.1 Conservation of Mass

Considering the general case of a compressible flow through the control volume (c.v.) shown in Figure 1.5 and assuming that n is a unit vector normal to the elementary surface area dA and v is the flow velocity through this area, then the conservation of mass equation takes the form

Figure 1.5 A schematic of an arbitrary control volume showing the flow velocity through a small elementary surface area

where ? is the fluid density, is the fluid velocity, dV is an elementary volume, and t is the time.

When the control volume tends to a point, the equation tends to the differential form,

where u, v, and w are the velocity components in the x, y, and z directions. If the flow is incompressible, the above equation can be reduced to

In the special case of 1-D steady flow in a control volume with one inlet and one exit (Figure 1.6), the conservation of mass equation takes the simple form,

Figure 1.6 One-dimensional flow in a diverging flow passage

where is the mass flow rate, V is the flow velocity, and A is the cross-sectional area.

1.2.2 Conservation of Momentum

1.2.2.1 Conservation of Linear Momentum

In the general case of unsteady flow of a compressible fluid, the linear momentum conservation equation (deduced from the Reynolds transport equation) can be expressed as

where the term represents the vectorial summation of all forces acting on the fluid body and is its linear momentum.

In case of steady flow, the first term on the right-hand side of Eq. (1.12) vanishes and the equation is reduced to

The right-hand side of the above equation represents the net rate of outflow of linear momentum through the control surface. In the special case of steady one-dimensional flow, the equation can be written in the form

When the control volume is very small (tends to a point), the momentum equation tends to the following differential form (known as the Navier-Stokes equation):

If the flow is frictionless (µ = 0), the diffusion term, , vanishes and the equation becomes

The above equation is well-known as Euler's equation. The equation can be applied along a streamline to yield the following 1-D Euler equation

where as is the acceleration in the streamwise direction. If the above equation is further simplified for the case of steady, incompressible, frictionless flow, it results in Bernoulli's equation, which can be written as

The application of the momentum equation normal to the streamline results in an equation similar to (1.16) and can be written as

where n is a coordinate normal to the streamline.

1.2.2.2 Conservation of Angular Momentum

In the general case of unsteady flow of a compressible fluid, the angular momentum conservation equation (deduced from the Reynolds transport equation) can be expressed as:

where the term represents the vectorial summation of all moment acting on the fluid body within the control volume, is the velocity vector and is the...