# The most common methods to identify velocity of flow are pathlines, streaklines and streamlines.

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1 4 FLUID FLOW 4.1 Introduction Many civil engineering problems in fluid mechanics are concerned with fluids in motion. The distribution of potable water, the collection of domestic sewage and storm water, and the wave actions on offshore structures are common examples. The viscosity of water is small and therefore in most hydraulic problems associated with civil engineering it is reasonable to ignore the effect of shear forces. 4.2 Velocity of Flow The definition of velocity in a fluid is much more complicated than in the case of a rigid solid. Basically this follows from the fact that the individual particles of a solid are bound together whereas the particles of a fluid can move independently of each other. The most common methods to identify velocity of flow are pathlines, streaklines and streamlines. Pathline - trace the position of a particle at successive intervals of time starting from a given point. Streakline -trace of all particles that have previously passed through a common point. Streamline -an imaginary curve that is tangential to the velocity vectors of a connected series of fluid particles. In unsteady flow, streamlines, pathlines and streaklines are all different, but in steady flow, streamlines, pathlines and streaklines are identical. The streamline is thus a line representing the direction of flow of the series of particles at a given instant. Because the streamline is always tangential to the flow, it follows that there is no flow across a streamline. P.4-1

2 4.3 Types of Flow There are several types of flow that occur in practice: uniform and non-uniform flow, steady and unsteady flow, laminar and turbulent flow. Some of them can be explained by means of streamlines Uniform and Non-uniform Flow In uniform flow the velocity (including its magnitude and direction) does not change from one point to another along any of the streamlines in the flow field. Uniform flow Non-uniform flow This means that both the area and velocity of the flow must be the same at every cross-section, and the streamlines must be straight and parallel. If the streamlines are not straight, there will be a change in the direction of the flow. If the streamlines are not parallel, there will be change in the magnitude of the flow. In non-uniform flow the velocity changes from point to point along streamlines. Mathematically, uniform and non-uniform flow can be defined as: dv = 0 ds (uniform flow) dv 0 ds (non-uniform flow) where V = velocity of flow s = position measuring along a streamline P.4-2

3 4.3.2 Steady and Unsteady Flow Steady flow means the velocity at any point in the flow field does not change with respect to time. If the velocity at a point changes over time, then the flow is unsteady. Mathematically, steady flow and unsteady flow can be presented as follows: dv = 0 dt (steady flow) dv 0 dt (unsteady flow) where V = velocity of flow t = time of study Discharge at a constant rate through a pipe is a common example at steady flow. If the pipe is of a constant diameter, the flow is uniform and steady. In fact, most of the civil engineering hydraulic problems are concerned with steady flow. An example of non-uniform and unsteady flow occurring together is the case of the flow from a nozzle when for some reason there is a change in the discharge rate. There are, therefore, four possible types of flow. Steady uniform flow. Conditions do not change with position or time. The velocity and cross-sectional area of the stream of fluid are the same at each cross-section; e.g. flow of a liquid through a pipe of uniform bore running completely full at constant velocity. Steady non-uniform flow. Conditions change from point to point but not with time. The velocity and cross-sectional area of the stream may vary from crosssection to cross-section, but, for each cross-section, they will not vary with time; e.g. flow of a liquid at a constant rate through a tapering pipe running completely full. Unsteady uniform flow. At a given instant of time the velocity at every point is the same, but this velocity will change with time; e.g. accelerating flow of a P.4-3

4 liquid through a pipe of uniform bore running full, such as would occur when a pump is started up. Unsteady non-uniform flow. The cross-sectional area and velocity vary from point to point and also change with time; e.g. a wave travelling along a channel Real and Ideal Flow When a real fluid flows past a boundary, the fluid immediately in contact with the boundary will have the same velocity as the boundary. The velocity of successive layers of fluid will increase as moving away from the boundary. Boundary layer V = 0.99Vf Ideal Fluid Free velocity =Vf V = 0.99Vf Boundary layer Real Fluid The part of the flow adjoining the boundary in which this change of velocity occurs is known as the boundary layer. In this region, shear stresses are developed between layers of fluid moving with different velocities as a result of viscosity. The thickness of the boundary layer is defined as the distance from the boundary at which the velocity becomes equal to 99% of the free stream velocity. Outside this boundary layer, the effect of the shear stresses due to the boundary can be ignored and the fluid can be treated as if it were an ideal fluid. If the fluid velocity is high and its viscosity low, the boundary layer is comparatively thin, and the assumption that a real fluid can be treated as an ideal fluid greatly simplifies the analysis of the flow and still leads to useful results One, Two and Three Dimensional Flow P.4-4

5 When there is no obstruction or channelling, fluid flow can be thought of as three-dimensional flow. In situations where the velocity of flow in one co-ordinate direction has no changes, the flow can be described as two-dimensional. The flow between two parallel plates is an example. If the velocity of flow is constant across each section but changes in only one direction, it can be described as one-dimensional. The average flow in a duct can be considered as one-dimensional. 4.4 Flow Rate and Mean Velocity The quantity of fluid flowing per unit time across any section is called the flow rate or the discharge. It may be expressed in terms of volume flow rate, m 3 /s; weight flow rate, kn/s or mass flow rate, kg/s. In dealing with incompressible fluids, volume flow rate is commonly used, whereas weight flow rate or mass flow rate is more convenient with compressible fluids. In a real fluid, the velocity adjacent to a solid boundary will be zero. For a pipe, the velocity profile would be as shown in fig (a) below for laminar flow and fig (b) for turbulent flow. u r r dr R u (a) Laminar flow (b) Turbulent flow Flow rate is the volume rate of flow passing a given section of the flow stream. It is also called discharged. Mathematically, flow rate can be defined as follows: Q = A V. da where Q = flow rate, m 3 /s P.4-5

6 If v is constant, Q = V.A V = velocity of flow, m/s da = area normal to the direction of velocity, m 2 In many practical problems, such as the flow of water through a pipe, the diameter of the pipe, and the discharge are given, and the velocity of the flow is the determined from, Q V = A The velocity so obtained is called the average velocity or mean velocity. By definition, it is simply the discharge divided by the cross-sectional area. P.4-6

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