Chapter 3 FLUID-FLOW CONCEPTS AND BASIC EQUATIONS

The statics of fluids: almost an exact science Nature of flow of a real fluid is very complex By an analysis based on mechanics, thermodynamics, and orderly experimentation, large hydraulic structures and efficient fluid machines have been produced. This chapter: the concepts needed for analysis of fluid motion the basic equations that enable us to predict fluid behavior (motion, continuity, and momentum and the first and second laws of thermodynamics) the control-volume approach is utilized in the derivation of the continuity, energy, and momentum equations in general, one-dimensional-flow theory is developed in this chapter

3.1 Flow Characteristics; Definitions Flow: turbulent, laminar; real, ideal; reversible, irreversible; steady, unsteady; uniform, nonuniform; rotational, irrotational Turbulent flow: most prevalent in engineering practice the fluid particles (small molar masses) move in very irregular paths causing an exchange of momentum from one portion of the fluid to another the fluid particles can range in size from very small (say a few thousand molecules) to very large (thousands of cubic meters in a large swirl in a river or in an atmospheric gust) the turbulence sets up greater shear stresses throughout the fluid and causes more irreversibilities or losses the losses vary about as the 1.7 to 2 power of the velocity; in laminar flow, they vary as the first power of the velocity.

Laminar flow: fluid particles move along smooth paths in laminas, or layers, with one layer gliding smoothly over an adjacent layer governed by Newton's law of viscosity [Eq. (1.1.1) or extensions of it to three-dimensional flow], which relates shear stress to rate of angular deformation the action of viscosity damps out turbulent tendencies is not stable in situations involving combinations of low viscosity, high velocity, or large flow passages and breaks down into turbulent flow An equation similar in form to Newton's law of viscosity may be written for turbulent flow: η: not a fluid property alone; depends upon the fluid motion and the density - the eddy viscosity. In many practical flow situations, both viscosity and turbulence contribute to the shear stress:

An ideal fluid: frictionless and incompressible and should not be confused with a perfect gas The assumption of an ideal fluid is helpful in analyzing flow situations involving large expanses of fluids, as in the motion of an airplane or a submarine A frictionless fluid is nonviscous, and its flow processes are reversible. The layer of fluid in the immediate neighborhood of an actual flow boundary that has had its velocity relative to the boundary affected by viscous shear is called the boundary layer may be laminar or turbulent, depending generally upon its length, the viscosity, the velocity of the flow near them, and the boundary roughness Adiabatic flow : no heat is transferred to or from the fluid Reversible adiabatic (frictionless adiabatic flow): isentropic flow

Steady flow occurs when conditions at any point in the fluid do not change with the time : ∂v/∂t = 0, space is held constant no change in density ρ, pressure p, or temperature T with time at any point Turbulent flow (due to the erratic motion of the fluid particles): small fluctuations occurring at any point; the definition for steady flow must be generalized somewhat to provide for these fluctuations: Fig.3.1 (a plot of velocity against time, at some point in turbulent flow) When the temporal mean velocity does not change with the time, the flow is said to be steady. The same generalization applies to density, pressure, temperature The flow is unsteady when conditions at any point change with the time, ∂v/∂t ≠ 0 Steady: water being pumped through a fixed system at a constant rate Unsteady: water being pumped through a fixed system at an increasing rate

Figure 3.1 Velocity at a point in steady turbulent flow

Uniform flow: at every point the velocity vector is identically the same (in magnitude and direction) for any given instant ∂v/∂s = 0, time is held constant and δs is a displacement in any direction : no change in the velocity vector in any direction throughout the fluid at any one instant says nothing about the change in velocity at a point with time. In flow of a real fluid in an open or closed conduit: even though the velocity vector at the boundary is always zero when all parallel cross sections through the conduct are identical (i.e., when the conduit is prismatic) and the average velocity at each cross section is the same at any given instant, the flow is said to be uniform Flow such that the velocity vector varies from place to place at any instant ∂v/∂s ≠ 0: nonuniform flow A liquid being pumped through a long straight pipe has uniform flow A liquid flowing through a reducing section or through a curved pipe has nonuniform flow.

Examples of steady and unsteady flow and of uniform and nonuniform flow liquid flow through a long pipe at a constant rate is steady uniform flow liquid flow through a long pipe at a decreasing rate is unsteady uniform flow flow through an expanding tube at a constant rate is steady nonuniform flow flow through an expanding tube at an increasing rate is unsteady nonuniform flow

Rotation of a fluid particle about a given axis, say the z axis: the average angular velocity of two infinitesimal line elements in the particle that are at right angles to each other and to the given axis If the fluid particles within a region have rotation about any axis, the flow is called rotational flow, or vortex flow If the fluid within a region has no rotation, the flow is called irrotational flow If a fluid is at rest and is frictionless, any later motion of this fluid will be irrotational.

One-dimensional flow neglects variations of changes in velocity, pressure, etc., transverse to the main flow direction Conditions at a cross section are expressed in terms of average values of velocity, density, and other properties Flow through a pipe Two-dimensional flow: all particles are assumed to flow in parallel planes along identical paths in each of these planes  no changes in flow normal to these planes Three-dimensional flow is the most general flow in which the velocity components u, v, w in mutually perpendicular directions are functions of space coordinates and time x, y, z, and t Methods of analysis are generally complex mathematically, and only simple geometrical flow boundaries can be handled

Streamline: continuous line drawn through the fluid so that it has the direction of the velocity vector at every point There can be no flow across a streamline Since a particle moves in the direction of the streamline at any instant, its displacement δs, having components δx, δy, δz, has the direction of the velocity vector q with components u, v, w in the x, y, z directions, respectively  : the differential equations of a streamline; any continuous line that satisfies them is a streamline

Steady flow (no change in direction of the velocity vector at any point): the streamline has a fixed inclination at every point - fixed in space A particle always moves tangent to the streamline  in steady flow the path of a particle is a streamline In unsteady flow (the direction of the velocity vector at any point may change with time): a streamline may shift in space from instant to instant A particle then follows one streamline one instant, another one the next instant, and so on, so that the path of the particle may have no resemblance to any given instantaneous streamline. A dye or smoke is frequently injected into a fluid in order to trace its subsequent motion. The resulting dye or smoke trails are called streak lines. In steady flow a streak line is a streamline and the path of a particle.

Streamlines in two-dimensional flow can be obtained by inserting fine, bright particles (aluminum dust) into the fluid, brilliantly lighting one plane, and taking a photograph of the streaks made in a short time interval. Tracing on the picture continuous lines that have the direction of the streaks at every point portrays the streamlines for either steady or unsteady flow. Fig. 3.2: illustration of an incompressible two-dimensional flow; the streamlines are drawn so that, per unit time, the volume flowing between adjacent streamlines is the same if unit depth is considered normal to the plane of the figure  when the streamlines are closer together, the velocity must be greater, and vice versa. If u is the average velocity between two adjacent stream-lines at some position where they are h apart, the flow rate Δq is A stream tube : made by all the streamlines passing through a small, closed curve. In steady flow it is fixed in space and can have no flow through its walls because the velocity vector has no component normal to the tube surface

Figure 3.2 Streamlines for steady flow around a cylinder between parallel walls

Example 3.1 In two-dimensional, incompressible steady flow around an airfoil the streamlines are drawn so that they are 10 mm apart at a great distance from the airfoil, where the velocity is 40 m/s. What is the velocity near the airfoil, where the streamlines are 75 mm apart? and

3.2 The Concepts of System and Control Volume The free-body diagram (Chap. 2): a convenient way to show forces exerted on some arbitrary fixed mass – a special case of a system. A system refers to a definite mass of material and distinguishes it from all other matter, called its surroundings. The boundaries of a system form a closed surface; it may vary with time, so that it contains the same mass during changes in its condition (for example, a kilogram of gas may be confined in a cylinder and be compressed by motion of a piston; the system boundary coinciding with the end of the piston then moves with the piston) The system may contain an infinitesimal mass or a large finite mass of fluids and solids at the will of the investigator.

The law of conservation of mass: the mass within a system remains constant with time (disregarding relativity effects): m is the total mass. Newton's second law of motion is usually expressed for a system as m is the constant mass of the system; ∑F refers to the resultant of all external forces acting on the system, including body forces such as gravity; v is the velocity of the center of mass of the system.

A control volume refers to a region in space and is useful in the analysis of situations where flow occurs into and out of the space The boundary of a control volume is its control surface The size and shape of the control volume are entirely arbitrary, but frequently they are made to coincide with solid boundaries in parts; in other parts they are drawn normal to the flow directions as a matter of simplification By superposition of a uniform velocity on a system and its surroundings a convenient situation for application of the control volume may sometimes be found, e.g., determination of sound-wave velocity in a medium The control-volume concept is used in the derivation of continuity, momentum, and energy equations. as well as in the solution of many types of problems The control volume is also referred to as an open system.

The continuity relation, i.e., the law of conservation of mass Regardless of the nature of  the flow, all flow situation are subject to the following relations, which may be expressed in analytic form: Newton's laws of motion, which must hold for every particle at every instant The continuity relation, i.e., the law of conservation of mass The first and second laws of thermodynamics Boundary conditions; analytical statements that a real fluid has zero velocity relative to a boundary at a boundary or that frictionless fluids cannot penetrate a boundary

Figure 3.3 System with identical control volume at time t in a velocity field

Fig.3.3: some general new situation, in which the velocity of a fluid is given relative to an xyz coordinate system (to formulate the relation between equations applied to a system and those applied to a control volume) At time t consider a certain mass of fluid that is contained within a system, having the dotted-line boundaries indicated. Also consider a control volume, fixed relative to the xyz axes, that exactly coincides with the system at time t. At time t + δt the system has moved somewhat, since each mass particle moves at the velocity associated with its location. N - the total amount of some property (mass, energy, momentum) within the system at time t; η - the amount of this property, per unit mass, throughout the fluid.The time rate of increase of N for the system is now formulated in terms of the control volume.

At t + δt, Fig. 3.3b, the system comprises volumes II and III, while at time t it occupies volume II, Fig. 3.3a. The increase in property N in the system in time δt is given by (*) The term on the left is the average time rate of increase of N within the system during time δt. In the limit as δt approaches zero, it becomes dN/dt. If the limit is taken as δt approaches zero for the first term on the right-hand side of the equation, the first two integrals are the amount of N in the control volume at t+δt and the third integral is the amount of N in the control volume at time t. The limit is

The next term, which is the time rate of flow of N out of the control volume, in the limit, may be written : dA, Fig 3.3c, is the vector representing an area element of the outflow area Similarly, the last term of Eq. (*), which is the rate of flow of N into the control volume, is, in the limit, The minus sign is needed as v · dA (or cos α) is negative for inflow, Fig. 3.3d Collecting the reorganized terms of Eq. (*) gives : time rate of increase of N within a system is just equal to the time rate of increase of the property N within the control volume (fixed relative to xyz) plus the net rate of efflux of N across the control-volume boundary

3.3 Application of The Control Volume    to Continuity, Energy, and Momentum The continuity equations are developed from the general principle of conservation of mass, Eq. (3.2.1), which states that the mass within a system remains constant with time, i.e. In Eq. (3.2.6) let N be the mass of the system m. Then η is the mass per unit mass, : the continuity equation for a control volume : the time rate of increase of mass within a control volume is just equal to the net rate of mass inflow to the control volume.

Energy Equation The first law of thermodynamics for a system: that the heat QH added to a system minus the work W done by the system depends only upon the initial and final states of the system - the internal energy E or by the above Eq.: The work done by the system on its surroundings: the work Wpr done by pressure forces on the moving boundaries the work Ws done by shear forces such as the torque exerted on a rotating shaft. The work done by pressure forces in time δt is

By use of the definitions of the work terms In the absence of nuclear, electrical, magnetic, and surface-tension effects, the internal energy e of a pure substance is the sum of potential, kinetic, and  "intrinsic" energies. The intrinsic energy u per unit mass is due to molecular spacing and forces (dependent upon p, ρ, or T):

Linear-Momentum Equation Newton's second law for a system, Eq. (3.2.2), is used as the basis for finding the linear-momentum equation for a control volume by use of Eq. (3.2.6) Let N be the linear momentum mv of the system, and let η be the linear momentum per unit mass ρv/ρ. Then by use of Eqs. (3.2.2) and (3.2.6) : the resultant force acting on a control volume is equal to the time rate of increase of linear momentum within the control volume plus the net efflux of linear momentum from the control volume. Equations (3.3.1). (3.3.6), and (3.3.8) provide the relations for analysis of many of the problems of fluid mechanics - a bridge from the solid-dynamics relations of the system to the convenient control-volume relations of fluid flow