MEE 309 Engineering Fluid Mechanics UJ PDF

MEE 309 ENGINEERING FLUID MECHANICS (2 UNITS) LECTURE NOTE  MEE 309: ENGINEERING FLUID MECHANICS INTRODUCTION Fluid mechanics is that discipline within the broad field of applied mechanics concerned with the behavior of liquids and gases at rest or in moti...

MEE 309 ENGINEERING FLUID MECHANICS (2 UNITS) LECTURE NOTE  MEE 309: ENGINEERING FLUID MECHANICS INTRODUCTION Fluid mechanics is that discipline within the broad field of applied mechanics concerned with the behavior of liquids and gases at rest or in motion. This field of mechanics obviously encompasses a vast array of problems that may vary from the study of blood flow in the capillaries (which are only a few microns in diameter) to the flow of crude oil through a 4-ft-diameter pipe. Fluid mechanics principles are needed to explain why airplanes are made streamlined with smooth surfaces for the most efficient flight, whereas golf balls are made with rough surfaces (dimpled) to increase their efficiency. For example:  How can a rocket generate thrust without having any air to push against in outer space?  Why can’t you hear a supersonic airplane until it has gone past you?  How can a river flow downstream with a significant velocity even though the slope of the surface is so small that it could not be detected with an ordinary level?  How can information obtained from model airplanes be used to design the real thing?  How much greater gas mileage can be obtained by improved aerodynamic design of cars and trucks? The list of applications and questions goes on and on—but the point here is that fluid mechanics is a very important, practical subject. FLUID KINEMATICS Fluid kinematics is a branch of Fluid Mechanics which involves the position, velocity and acceleration of the fluid, and the description and visualization of its motion without being concerned with the specific forces necessary to produce the motion (the dynamics of the motion). In general, fluids flow. That is, there is a net motion of molecules from one point in space to another point as a function of time. A typical portion of fluid contains so many molecules that it becomes totally unrealistic (except in special cases) to attempt to account for the motion of individual molecules. Rather, the continuum hypothesis is employed and fluids are being considered to be made up of fluid particles that interact with each other and with their surroundings. Thus, the flow of a fluid can be described in terms of the motion of fluid particles rather than individual molecules. This motion can be described in terms of the velocity and acceleration of the fluid particles. Department of Mechanical Engineering, UJ Page 1  MEE 309: ENGINEERING FLUID MECHANICS Fig. 1: Particle location in terms of its position vector. At a given instant in time, a description of any fluid property (such as density, pressure, velocity, and acceleration) may be given as a function of the fluid’s location. This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of the flow. The specific field representation may be different at different times, so that to describe a fluid flow we must determine the various parameters not only as a function of the spatial coordinates (x, y, z, for example) but also as a function of time, t. FLOW DESCRIPTIONS There are two general approaches in analyzing fluid mechanics problems. The first method, called the Eulerian method, uses the field concept. In this case, the fluid motion is given by completely prescribing the necessary properties (pressure, density, velocity, etc.) as functions of space and time. From this method we obtain information about the flow in terms of what happens at fixed points in space as the fluid flows past those points. The second method, called the Lagrangian method, involves following individual fluid particles as they move about and determining how the fluid properties associated with these particles change as a function of time. That is, the fluid particles are “tagged” or identified, and their properties determined as they move. The difference between the two methods of analyzing fluid flow problems can be seen in the example of smoke discharging from a chimney, as shown in Fig. 2 below. Department of Mechanical Engineering, UJ Page 2  MEE 309: ENGINEERING FLUID MECHANICS Fig. 2: Eulerian and Lagrangian descriptions of temperature of flowing fluid. In the Eulerian method, temperature-measuring device would be attached to the top of the chimney (point 0) to record the temperature at that point as a function of time. At different times there are different fluid particles passing by the stationary device. Thus, one would obtain the temperature, T, for that location (𝑥 = 𝑥0, 𝑦 = 𝑦0 and 𝑧 = 𝑧0 ) as a function of time. That is, 𝑇 = 𝑇(𝑥0 , 𝑦0 , 𝑧0 , 𝑡). The use of numerous temperature-measuring devices fixed at various locations would provide the temperature field, 𝑇 = 𝑇(𝑥, 𝑦, 𝑧, 𝑡). The temperature of a particle as a function of time would not be known unless the location of the particle is known as a function of time. In the Lagrangian method, one would attach the temperature-measuring device to a particular fluid particle (particle A) and record that particle’s temperature as it moves about. Thus, one would obtain that particle’s temperature as a function of time, T = TA (t). The use of many such measuring devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time. The temperature would not be known as a function of position unless the location of each particle is known as a function of time. If enough information in Eulerian form is available, Lagrangian information can be derived from the Eulerian data—and vice versa. In fluid mechanics it is usually easier to use the Eulerian method to describe a flow—in either experimental or analytical investigations. There are, however, certain instances in which the Lagrangian method is more convenient. For example, some numerical fluid mechanics calculations are based on determining the motion of individual fluid particles (based on the appropriate interactions among the particles), thereby describing the motion in Lagrangian terms. Department of Mechanical Engineering, UJ Page 3  MEE 309: ENGINEERING FLUID MECHANICS VELOCITY FIELD One of the most important fluid variables is the velocity field, ………………… (1) where u, v and w are the x, y, and z components of the velocity vector. By definition, the velocity of a particle is the time rate of change of the position vector for that particle. From Fig. 1, the position of particle A relative to the coordinate system is given by its position vector, rA, which (if the particle is moving) is a function of time. Velocity of the particle = drA/dt =VA Velocity vector V = v(x, y, z, t) Since the velocity is a vector, it has both a direction and a magnitude. The magnitude of V = |V| = (u2+v2+w2)1/2 = V = speed of the fluid By determining V and 𝜃 for other locations in the 𝑥 − 𝑦 plane, the velocity field can be sketched as shown in the Fig. 3 above. For example, on the line 𝑦 = 𝑥 the velocity is at −450 angle relative −𝑦⁄ to the 𝑥 axis(𝑡𝑎𝑛𝜃 = 𝑣⁄𝑢 = 𝑥 = −1). At the origin 𝑥 = 𝑦 = 0 so that V = 0. This point is a stagnation point. The farther from the origin the fluid is, the faster it is flowing (as seen from eqn. i). ACCELERATION FIELD As discussed above fluid motion can be described by either (1) following individual particles (Lagrangian description) or (2) remaining fixed in space and observing different particles as they pass by (Eulerian description). In either case, to apply Newton’s second law (F = ma) we must be able to describe the particle acceleration in an appropriate form. For the infrequently used Lagrangian method, we describe the fluid acceleration just – a = a (t) as is done in solid body dynamics— for each particle. For the Eulerian description we describe the acceleration field as a function of position and time without actually following any particular particle. This is analogous to describing the flow in terms of the velocity field, V = V (x, y, z, t) rather than the velocity for particular particles. Consider a fluid particle moving along its pathline as is shown in Fig. 4. In general, the particle’s velocity, denoted VA for particle A, is a function of its location and the time. That is, VA = VA (rA, t) = VA [xA(t), yA(t), zA(t), t] ………………………. (2) where xA = xA(t), yA = yA(t) and zA = zA(t) define the location of the moving particle. By definition, the acceleration of a particle is the time rate of change of its velocity. Since the velocity may be a Department of Mechanical Engineering, UJ Page 4  MEE 309: ENGINEERING FLUID MECHANICS function of both position and time, its value may change because of the change in time as well as a change in the particle’s position. Thus, we use the chain rule of differentiation to obtain the acceleration of particle A, denoted aA as ………………. (3) Fig. 4: Velocity and position of particle A at time t. Using the fact that the particle velocity components are given by and Eqn. 3 becomes ……………………… (4) Since Eqn. 4 is valid for any particle, we can drop the reference to particle A and obtain the acceleration field from the velocity field as: ……………………………... (5) This is a resultant vector whose scalar components can be written as: and …………………………… (6) Department of Mechanical Engineering, UJ Page 5  MEE 309: ENGINEERING FLUID MECHANICS The above result is often written shorthand notation as: ……… (7) is termed the material derivative or substantial derivative. Now, resultant velocity: 𝑉 = √𝑢2 + 𝑣 2 + 𝑤 2 ……………………… (8) Resultant acceleration, 𝑎 = √𝑎𝑥2 + 𝑎𝑦2 + 𝑎𝑧2 ……………………… (9) In vector notation, Velocity vector: 𝑉 = 𝑢𝑖 + 𝑣𝑗 + 𝑤𝑘 ………………………. (10) 𝑑𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝜕𝑉 Acceleration vector: 𝑎 = = (𝑢 𝜕𝑥 + 𝑣 𝜕𝑦 + 𝑤 𝜕𝑧 ) + 𝜕𝑡 …................ (11) 𝑑𝑡 𝑎 = 𝑎𝑥 𝑖 + 𝑎𝑦 𝑗 + 𝑎 𝑧 𝑘 and |𝑉| = √𝑢2 + 𝑣 2 + 𝑤 2 |𝑎| = √𝑎𝑥2 + 𝑎𝑦2 + 𝑎𝑧2 Vectorially, 𝜕𝑉 𝑎 = (𝑉. ∇)𝑉 + 𝜕𝑡 ……………………….. (12) The velocity, in general is a function of space (s) and time (t) i.e. 𝑉 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡) or 𝑉 = 𝑓(𝑠, 𝑡) and the acceleration, 𝑑𝑉 𝜕𝑉 𝑑𝑠 𝜕𝑉 𝑎= =. + 𝜕𝑡 𝑑𝑡 𝜕𝑠 𝑑𝑡 𝜕𝑉 𝜕𝑉 Therefore 𝑎 = 𝑉 𝜕𝑠 + 𝜕𝑡 ………………………….. (13) Thus the acceleration consists of two parts: 𝜕𝑉 𝑉 𝜕𝑠 : This part is due to change in position or movement and is called convective acceleration. Department of Mechanical Engineering, UJ Page 6  MEE 309: ENGINEERING FLUID MECHANICS 𝜕𝑉 : This part is with respect to time at a given location and is called local (or temporal) 𝜕𝑡 acceleration. Therefore 𝜕𝑉 Local acceleration = …………………………… (14) 𝜕𝑡 𝜕𝑢 𝜕𝑣 𝜕𝑤 = , , 𝜕𝑡 in Eqn. 6 above. 𝜕𝑡 𝜕𝑡 Tangential and Normal Acceleration: When the motion is curvilinear Eqn. 13 gives the tangential acceleration. A particle moving in a 𝑉2 curved path (Fig. 5) will always have a normal acceleration 𝑎𝑛 = towards the centre of the 𝑟 curved path (r being the radius of the path), though its tangential acceleration (as) may be zero as in the case of uniform circular motion. Fig. 5: Tangential and normal acceleration. Generally, for motion along a curved path a = as + an 𝜕𝑉 𝜕𝑉 𝑉2 = (𝑉 𝜕𝑠 + 𝜕𝑡 + ) ………………………. (15) 𝑟 Department of Mechanical Engineering, UJ Page 7  MEE 309: ENGINEERING FLUID MECHANICS ONE, TWO AND THREE DIMENSIONAL FLOWS Generally, a fluid flow is a rather complex three-dimensional, time-dependent phenomenon—𝑽 = 𝑽(𝑥, 𝑦, 𝑧, 𝑡) = 𝑢𝒊 + 𝑣𝒋 + 𝑤𝒌. In many situations, however, it is possible to make simplifying assumptions that allow a much easier understanding of the problem without sacrificing needed accuracy. One of these simplifications involves approximating a real flow as a simpler one- or two-dimensional flow. In almost any flow situation, the velocity field actually contains all three velocity components (u, v and w, for example). In many situations the three-dimensional flow characteristics are important in terms of the physical effects they produce. One-dimensional flow: The flow in which two of the velocity components are assumed to be negligible, leaving the velocity field to be approximated as one-dimensional flow field. Mathematically V = ui, v = 0, w = 0 Example: Flow in a pipe where average flow parameters are considered for analysis. Fig. 6: One dimensional flow Although there are very few, if any, flows that are truly one-dimensional, there are many flow fields for which the one-dimensional flow assumption provides a reasonable approximation. There are also many flow situations for which use of a one-dimensional flow field assumption will give completely erroneous results. Two-dimensional flow: In many situations one of the velocity components may be small relative to the two other components. In situations of this kind it may be reasonable to neglect the smaller component and assume two-dimensional flow. This is flow in which the velocity is a function of time and two rectangular space coordinates. That is, V = ui + vj, where u, v and w are functions of x and y (and possibly time, t). Examples: (i) Flow between parallel plates of infinite extent. (ii) Flow in the main stream of a wide river. Department of Mechanical Engineering, UJ Page 8  MEE 309: ENGINEERING FLUID MECHANICS Fig. 7: Two- dimensional flow. Three-dimensional flow: It is the type of flow in which the velocity is a function of time and three mutually perpendicular directions. That is, 𝑽 = 𝑽(𝑥, 𝑦, 𝑧, 𝑡) = 𝑢𝒊 + 𝑣𝒋 + 𝑤𝒌. Example: Flow in a converging or diverging pipe of channel. ASSIGNMENT 1: In a fluid the velocity field is given by: 𝑉 = (6 + 2𝑥𝑦 + 𝑡 2 )𝒊 − (𝑥𝑦 2 + 10𝑡)𝒋 + 25𝒌 Determine: (i) The velocity components u, v, w at any point in the flow field. (ii) The speed at point (1, 2, 3). (iii) The speed at time 𝑡 = 1s at point (3, 0, 2). (iv) What is the acceleration of a particle at (0, 0, 2) at time t = 2s? STEADY AND UNSTEADY FLOW: Steady Flow: This a type of flow in which the fluid characteristics (velocity, pressure, density, etc.) at a point in space do not vary with time. Mathematically, 𝜕𝑢 𝜕𝑣 𝜕𝑤 ( 𝜕𝑡 ) = 0; ( 𝜕𝑡 ) = 0; ( 𝜕𝑡 ) =0 𝑥0 ,𝑦0 ,𝑧0 𝑥0 ,𝑦0 ,𝑧0 𝑥0 ,𝑦0 ,𝑧0 𝜕𝑝 𝜕𝜌 ( 𝜕𝑡 ) = 0; ( 𝜕𝑡 ) = 0; and so on 𝑥0 ,𝑦0 ,𝑧0 𝑥0 ,𝑦0 ,𝑧0 where (𝑥0 , 𝑦0 , 𝑧0 , 𝑡) is a fixed point in a fluid field. Example: Flow through a prismatic or non-prismatic conduit at a constant flow rate Q m3/s is steady. (A prismatic conduit has a constant size shape and has a velocity equation in the form 𝑢 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 which is independent of t). Unsteady Flow: It is a flow type in which the velocity, pressure or density of fluid at a point in space change w.r.t. time. Mathematically, 𝜕𝑢 𝜕𝑣 𝜕𝑤 ( 𝜕𝑡 ) ≠ 0; ( 𝜕𝑡 ) ≠ 0; ( 𝜕𝑡 ) ≠0 𝑥0 ,𝑦0 ,𝑧0 𝑥0 ,𝑦0 ,𝑧0 𝑥0 ,𝑦0 ,𝑧0 𝜕𝑝 𝜕𝜌 ( 𝜕𝑡 ) ≠ 0; ( 𝜕𝑡 ) ≠ 0; and so on 𝑥0 ,𝑦0 ,𝑧0 𝑥0 ,𝑦0 ,𝑧0 Velocity equation is in the form 𝑢 = 𝑎𝑥 2 + 𝑏𝑥𝑡 Department of Mechanical Engineering, UJ Page 9  MEE 309: ENGINEERING FLUID MECHANICS In reality, almost all flows are unsteady in some sense. That is, the velocity does vary with time. It is not difficult to believe that unsteady flows are usually more difficult to analyse (and to investigate experimentally) than are steady flows. Hence, considerable simplicity often results if one can make the assumption of steady flow without compromising the usefulness of the results. Among the various types of unsteady flows are non-periodic flow, periodic flow, and truly random flow. An example of a non-periodic, unsteady flow is that produced by turning off a faucet to stop the flow of water. Usually this unsteady flow process is quite mundane and the forces developed as a result of the unsteady effects need not be considered. However, if the water is turned off suddenly (as with an electrically operated valve in a dishwasher), the unsteady effects can become important [as in the “water hammer” effects made apparent by the loud banging of the pipes under such conditions]. In other flows the unsteady effects may be periodic, occurring time after time in basically the same manner. The periodic injection of the air-gasoline mixture into the cylinder of an automobile engine is such an example. The unsteady effects are quite regular and repeatable in a regular sequence. They are very important in the operation of the engine. In many situations the unsteady character of a flow is quite random. That is, there is no repeatable sequence or regular variation to the unsteadiness. This behaviour occurs in turbulent flow and is absent from laminar flow. Rate of Flow or Discharge Rate of flow (or discharge) is defined as the quantity of a fluid flowing per second through a section of a pipe of a channel. It is generally denoted by Q. Consider a fluid flowing through a pipe. Let, A = Area of cross-section of the pipe, and V = Average volume of the fluid. ∴ Discharge, Q = Area x average velocity Q = A.V m3/s ………………………………………………………. (16) Compressible and Incompressible Flows: i) Compressible flow is that type of flow in which the density (ρ) of the fluid changes from point to point (or in other word density is not constant for this flow). Mathematically, ρ ≠ constant. Example is the flow of gases through orifices, nozzle, gas turbines etc. ii) Incompressible flow is that type of flow in which density is constant for the fluid flow. Liquids are generally considered flowing incompressible. Mathematically, ρ = constant. Example is subsonic aerodynamics. Department of Mechanical Engineering, UJ Page 10  MEE 309: ENGINEERING FLUID MECHANICS CONTINUITY EQUATION The continuity equation is based on the principle of conservation of mass and states as follows: if no fluid is added or removed from the pipe in any length then the mass passing across different sections shall be same. Consider two cross-section of a pipe as shown in Fig. 8 below: 2 1 1 2 Fig. 8: Fluid flow through a pipe Let, A1 = Area of the pipe at section 1 – 1 V1 = Velocity of the fluid at section 1 – 1 𝜌1 = Density of the fluid at section 1 – 1 and, A2, V2, 𝜌2 , are corresponding values at sections 2 – 2 The total quantity of fluid passing through section 1 – 1 = 𝜌1 𝐴1 𝑉1 and the total quantity of fluid passing through section 2 – 2 = 𝜌2 𝐴2 𝑉2 From the law of conservation of matter (theorem of continuity), we have 𝜌1 𝐴1 𝑉1 = 𝜌2 𝐴2 𝑉2 …………………………….. (17) Eqn. 17 is applicable to both compressible and incompressible fluids and is called Continuity Equation. In the case of incompressible fluids, 𝜌1 = 𝜌2 and the continuity Eqn. 16 reduces to 𝐴1 𝑉1 = 𝐴2 𝑉2 ………………………………… (18) Department of Mechanical Engineering, UJ Page 11  MEE 309: ENGINEERING FLUID MECHANICS Continuity Equation in Cartesian Co-ordinates Consider a fluid element (control volume) – parallelopiped with sides dx, dy, and dx as shown in Fig. 9. Y X Z Fig. 9: Fluid element in three-dimensional flow. Let ρ = Mass density of the fluid at a particular instant; u, v, w = Components of velocity of flow entering the three faces of the parallelopiped. Rate of mass of fluid entering the face ABCD (i.e. fluid influx) = ρ × velocity in X-direction × area of ABCD = ρ udy dz …………………………………………………… (i) Rate of mass of fluid leaving the face EFGH (i.e. fluid efflux) 𝜕 = 𝜌𝑢𝑑𝑦𝑑𝑧 + 𝜕𝑥 (𝜌𝑢𝑑𝑦𝑑𝑧)𝑑𝑥 ………………………………. (ii) The gain in mass per unit time due to flow in the X-direction is given by the difference between the fluid influx and efflux. ∴ Mass accumulated per unit time due to flow in X-direction 𝜕 = 𝜌𝑢𝑑𝑦𝑑𝑧 − [𝜌𝑢 + 𝜕𝑥 (𝜌𝑢)𝑑𝑥] 𝑑𝑦𝑑𝑧 𝜕 = − 𝑑𝑥 (𝜌𝑢)𝑑𝑥𝑑𝑦𝑑𝑧 ……………………………………. (iii) Similarly, the gain in fluid mass per unit time in the parallelopiped due to flow in Y and Z- directions 𝜕 = − 𝑑𝑦 (𝜌𝑣)𝑑𝑥𝑑𝑦𝑑𝑧 (in Y-direction).…………………… (iv) 𝜕 = − 𝑑𝑧 (𝜌𝑤)𝑑𝑥𝑑𝑦𝑑𝑧 (in Z-direction).…………………… (v) The total (or net) gain in fluid mass per unit time for fluid along three co-ordinate axes 𝜕 𝜕 𝜕 = −[ (𝜌𝑢) + (𝜌𝑣) + (𝜌𝑤)] 𝑑𝑥𝑑𝑦𝑑𝑧 …………… (vi) 𝜕𝑥 𝜕𝑦 𝜕𝑧 Department of Mechanical Engineering, UJ Page 12  MEE 309: ENGINEERING FLUID MECHANICS Rate of change of mass of the parallelopiped (control volume) 𝜕 = 𝜕𝑡 (𝜌𝑑𝑥𝑑𝑦𝑑𝑧) …………………………………………. (vii) Equating eqns. (vi) and (vii), we get 𝜕 𝜕 𝜕 𝜕 − [𝜕𝑥 (𝜌𝑢) + 𝜕𝑦 (𝜌𝑣) + 𝜕𝑧 (𝜌𝑤)] 𝑑𝑥𝑑𝑦𝑑𝑧 = 𝜕𝑡 (𝜌𝑑𝑥𝑑𝑦𝑑𝑧) Simplification and rearrangement of terms would reduce the above expression to 𝜕 𝜕 𝜕 𝜕𝜌 (𝜌𝑢) + (𝜌𝑣) + (𝜌𝑤) + = 0 ………………………… (19) 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑡 Eqn. (19) is the general equation of continuity in three-dimensions and is applicable to any type of flow and for any fluid whether compressible or incompressible. 𝜕𝜌 For steady flow ( 𝑑𝑡 = 0) incompressible fluid (ρ = constant) the equation reduces to 𝜕𝑢 𝜕𝑣 𝜕𝑤 + 𝜕𝑦 + = 0 ……………………………………………...... (20) 𝜕𝑥 𝜕𝑧 For two-dimensional flow, Eqn. (20) reduces to 𝜕𝑢 𝜕𝑣 + 𝜕𝑦 = 0 (since w = 0) 𝜕𝑥 For one-dimensional flow, say in X-direction, Eqn. (20) takes the form 𝜕𝑢 =0 (since v = 0, w = 0) 𝜕𝑥 Integrating with respect to x, we get u = constant If the area of flow is a then the rate of flow is Q = a.u = constant for steady flow which is the same Eqn. (18) and states that if area of flow a is constant the velocity of flow u will also be constant. Types of Flow Lines Although fluid motion can be quite complicated, there are various concepts that can be used to help in the visualization and analysis of flow fields. To this end we discuss the use of streamlines, streaklines, and path lines in flow analysis. The streamline is often used in analytical work while the streak line and path line are often used in experimental work. Pathline: A pathline (Fig. 10) is the path followed by a fluid particle in motion. A path line shows the direction of particular particle as it moves ahead. In general, this is the curve in three-dimensional space. However, if the conditions are such that the flow is two-dimensional the curve becomes two-dimensional. Department of Mechanical Engineering, UJ Page 13  MEE 309: ENGINEERING FLUID MECHANICS 1 2 3 Fig. 10: Pathlines Streamline: A streamline (Fig. 11) is a line that is everywhere tangent to the velocity field. If the flow is steady, nothing at a fixed point (including the velocity direction) changes with time, so the streamlines are fixed lines in space. For unsteady flows the streamlines may change shape with time. Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity field. Fig. 11: Streamline Equation of a streamline in a three-dimensional flow is given as: 𝑑𝑥 𝑑𝑦 𝑑𝑧 = =........................................... (21) 𝑢 𝑣 𝑤 For two-dimensional flows the slope of the streamline, must be equal to the tangent of the angle that the velocity vector makes with the x axis or 𝑑𝑦 𝑣 = 𝑢 …………………………………. (22) 𝑑𝑥 If the velocity field is known as a function of x and y (and t if the flow is unsteady), this equation can be integrated to give the equation of the streamlines. The following points about streamlines are worth noting: 1. A streamline cannot intersect itself neither can two streamlines cross each other. 2. There cannot be any movement of the fluid mass across the streamlines. 3. Streamline spacing varies inversely as the velocity; converging of streamlines in any particular direction shows accelerated flow in that direction. 4. Whereas a pathline gives the path of one particular particle at successive instants of time, a streamline indicates the direction of a number of particles at the same instant. 5. The series of streamlines represent the flow pattern at an instant.  In steady flow, the pattern of streamlines remains invariant with time. The pathlines and streamlines will then be identical. Department of Mechanical Engineering, UJ Page 14  MEE 309: ENGINEERING FLUID MECHANICS  In unsteady flow, the pattern of streamlines may or may not remain the same at the next instant. Stream Tube: A stream tube (Fig. 12) is a fluid mass bounded by a group of streamlines. The contents of a stream tube are known as “current filament”. Examples of stream tube are pipes and nozzles. Following points about stream tube are worth noting: The stream tube has finite dimensions. As there is no flow perpendicular to streamlines, therefore, there is no flow across the surface (called stream surface) of the stream tube. The stream surface functions as if it were a solid wall. The shape of a stream tube changes from one instant to another because of change in the position of streamlines. Fig. 12: Stream Tube Stream Function: The stream function is defined as a function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to this direction. It is denoted by ψ (psi). In case of two-dimensional flow, the stream function may be defined mathematically as ψ = f(x, y, t) …………….for unsteady flow, and ψ = f(x, y) …………….for steady flow, such that 𝜕ψ 𝜕ψ 𝑢= and 𝑣 = − ……………………. (23) 𝜕𝑦 𝜕𝑥 The continuity equation for two-dimensional flow is 𝜕𝑢 𝜕𝑣 + 𝜕𝑦 = 0 𝑑𝑥 Substituting the value of u and v from Eqn. (23), we get 𝜕 𝜕ψ 𝜕 𝜕ψ ( 𝜕𝑦 ) + 𝜕𝑦 (− 𝜕𝑥 ) = 0 𝜕𝑥 𝜕2 ψ 𝜕2 ψ − 𝜕𝑥𝜕𝑦 = 0 𝜕𝑥𝜕𝑦 Hence existence of ψ means a possible case of fluid flow. The flow may be rotational or irrotational. The rotational component 𝜔𝑧 is given by Department of Mechanical Engineering, UJ Page 15  MEE 309: ENGINEERING FLUID MECHANICS 1 𝜕𝑣 𝜕𝑢 𝜔𝑧 = 2 (𝜕𝑥 − 𝜕𝑦) Substituting the value of u and v from Eqn. (23), we get 1 𝜕 𝜕ψ 𝜕 𝜕ψ 𝜔𝑧 = 2 [𝜕𝑥 (− 𝜕𝑥 ) − 𝜕𝑦 ( 𝜕𝑦 )] 1 𝜕2 ψ 𝜕2 ψ or 𝜔𝑧 = 2 ( 𝜕𝑥 2 + 𝜕𝑦 2 ) …………………………….. (24) This equation is known as Poisson’s equation. For irrotational flow since 𝜔𝑧 = 0, Eqn. (24) becomes 𝜕2 ψ 𝜕2 ψ + 𝜕𝑦 2 = 0 i.e. ∆2 ψ = 0 𝜕𝑥 2 which is the Laplace equation in ψ. Streakline: A streakline is a curve which gives an instantaneous picture of the location of the fluid particles, which have passed through a given point. Example of streakline is the path taken by smoke coming out of chimney (Fig. 13). Fig. 13: Streaklines at t = t1. VORTEX: A vortex represents a flow in which the streamlines are concentric circles. A vortex motion is characterized by a flow pattern wherein the streamlines are curved. When fluid flows between curved streamlines, centrifugal forces are set up and these are counter-balanced by the pressure force acting in the radial direction. Let and ψ = −K ln r where K is a constant. In this case the streamlines are concentric circles as illustrated in the Fig. 14, with 𝑣𝑟 = 0 and ……………………….. (25) Department of Mechanical Engineering, UJ Page 16  MEE 309: ENGINEERING FLUID MECHANICS Fig. 14: Streamline path of a vortex. From Eqn. 25 it can be seen that the tangential velocity varies inversely with the distance from the origin, with a singularity occurring at r = 0 (where the velocity becomes infinite). This type of vortex flow is called irrotational or free vortex flow and is defined as a flow in which the fluid mass rotates without any external impressed contact force. The whole fluid mass rotates either due to fluid pressure itself or the gravity or due to rotation previously imparted. Example of free vortex flow is the swirling motion of the water as it drains from a bathtub. If the vortex flow is such that the fluid mass is made to rotate by means of some external agency – generally mechanical power which imparted a constant torque on the fluid mass – it is called rotational or forced vortex flow. Example of forced vortex flow is the motion of a liquid contained in a tank that is rotated about its axis with angular velocity 𝜔. CIRCULATION: A mathematical concept commonly associated with vortex motion is that of circulation. The circulation, Г, is defined as the line integral of the tangential component of the velocity taken around a closed curve in the flow field. In equation form, Г can be expressed as Г = ∳𝐶 𝑉𝑑𝑠 ………………………….……… (26) where the integral sign means that the integration is taken around a closed curve, C, in the counterclockwise direction, and ds is a differential length along the curve. Circulation around regular curves can be obtained by integration. Consider a fluid element ABCD in rotational motion. Let the velocity components along the sides of the element be as shown in Fig. 15. Starting from A and proceeding anticlockwise, we have 𝜕𝑣 𝜕𝑢 𝑑Γ = 𝑢d𝑥 + (𝑣 + 𝜕𝑥 d𝑥) d𝑦 − (𝑢 + 𝜕𝑦 d𝑦) d𝑥 − 𝑣d𝑦 𝜕𝑣 𝜕𝑢 = (𝜕𝑥 − 𝜕𝑦) d𝑥. d𝑦 Department of Mechanical Engineering, UJ Page 17  MEE 309: ENGINEERING FLUID MECHANICS Y 𝜕𝑢 𝑢 + 𝜕𝑦 d𝑦 C D Fluid element 𝜕𝑣 𝑣= dy d𝑥 𝜕𝑥 v B u Fig. 15: ACirculation. dx X 𝜕𝑣 𝜕𝑢 But (𝜕𝑥 − 𝜕𝑦) = ζ ……………………………… (27) for two-dimensional flow in x – y plane and, therefore, is the vorticity of element about the z axis. Vorticity is defined as a vector that is twice the rotation vector; that is, ζ = 2𝜔 If a flow possesses vorticity, it is rotational. The flow is irrotational if rotation,𝜔, is zero. For a three-dimensional flow the rotation is possible about three axes. The expression for rotation 𝜔𝑧 , 𝜔𝑥 and 𝜔𝑦 can be obtained as follows: 1 𝜕𝑣 𝜕𝑢 𝜔𝑧 = 2 (𝜕𝑥 − 𝜕𝑦) 1 𝜕𝑤 𝜕𝑣 𝜔𝑥 = 2 ( 𝜕𝑦 − 𝜕𝑧 ) ………………………… (28) 1 𝜕𝑢 𝜕𝑤 𝜔𝑦 = 2 ( 𝜕𝑧 − 𝜕𝑥 )} In the vector notation, the above Eqn. 28 can be rewritten as 1 𝜔 = 2 [𝜔𝑥 𝑖 + 𝜔𝑦 𝑗 + 𝜔𝑧 𝑘] 1 = 2 (∆ × 𝑉) The vector (∆ × 𝑉) is the curl of velocity vector. Department of Mechanical Engineering, UJ Page 18  MEE 309: ENGINEERING FLUID MECHANICS Velocity Potential: The velocity potential is defined as a scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. Mathematically, the velocity potential,𝜙, is defined as: 𝜙 = 𝑓(𝑥, 𝑦, 𝑧, 𝑡) for unsteady flow, and 𝜙 = (𝑥, 𝑦, 𝑧) for steady flow, such that 𝜕∅ 𝑢 = − 𝜕𝑥 𝜕∅ 𝑣 = − 𝜕𝑦 ………………………………. (29) 𝜕∅ 𝑤 = − 𝜕𝑧 } where u, v and w are the components of velocity in the x, y and z directions respectively. The negative sign signifies that ∅ decreases with an increase in the value of x, y and z. In other words it indicates that the flow is always in the direction of decreasing∅. For an incompressible steady flow the continuity equation is 𝜕𝑢 𝜕𝑣 𝜕𝑤 + 𝜕𝑦 + =0 𝜕𝑥 𝜕𝑧 Substituting the values of u, v and w in terms of ∅ in Eqn. 29, we get 𝜕 𝜕∅ 𝜕 𝜕∅ 𝜕 𝜕∅ (− 𝜕𝑥 ) + 𝜕𝑦 (− 𝜕𝑦) + 𝜕𝑧 (− 𝜕𝑧 ) = 0 𝜕𝑥 𝜕2 ∅ 𝜕2 ∅ 𝜕2 ∅ + 𝜕𝑦 2 + 𝜕𝑧 2 = 0 ………………….. (30) 𝜕𝑥 2 This equation is called Laplace’s equation. Thus, inviscid, incompressible, irrotational flow fields are governed by Laplace’s equation. This type of flow is commonly called a potential flow. To complete the mathematical formulation of a given problem, boundary conditions have to be specified. These are usually velocities specified on the boundaries of the flow field of interest. It follows that if the potential function can be determined, then the velocity at all points in the flow field can be determined from Eqn. 29. The velocity potential is a consequence of the irrotationality of the flow field, whereas the stream function is a consequence of conservation of mass. It is to be noted, however, that the velocity potential can be defined for a general three-dimensional flow, whereas the stream function is restricted to two-dimensional flows. Department of Mechanical Engineering, UJ Page 19  MEE 309: ENGINEERING FLUID MECHANICS SOME EXAMPLES OF POTENTIAL FLOW: Uniform Flow: The simplest potential flow is one for which the streamlines are all straight and parallel, and the magnitude of the velocity is constant. This type of flow is called a uniform flow. For example, consider a uniform flow in the positive x direction as is illustrated in Fig. 16a. In this instance, u = U and v = 0, and in terms of the velocity potential Integrating these two equations yield 𝜙 = 𝑈𝑥 + 𝐶 where C is an arbitrary constant, which can be set equal to zero. Thus, for a uniform flow in the positive x direction 𝜙 = 𝑈𝑥................................................ (31) Fig. 16: Uniform flow: (a) in the x direction; (b) in an arbitrary direction, α. The corresponding stream function can be obtained in a similar manner, since 𝜕𝜓 𝜕𝜓 =𝑈 =0 𝜕𝑦 𝜕𝑥 and, therefore, 𝜓 = 𝑈𝑦................................................. (32) These results can be generalized to provide the velocity potential and stream function for a uniform flow at an angle α with the x axis, as in Fig. 16b. For this case 𝜙 = 𝑈(𝑥 cos 𝛼 + 𝑦 sin 𝛼).................... (33) and 𝜓 = 𝑈(𝑦 cos 𝛼 − 𝑥 sin 𝛼).................... (34) Source and Sink: A source or sink represents a purely radial flow. Let us consider a fluid flowing radially outward from a line through the origin perpendicular to the x–y plane as is shown in Fig. 17. Let m be the Department of Mechanical Engineering, UJ Page 20  MEE 309: ENGINEERING FLUID MECHANICS volume rate of flow emanating from the line (per unit length), and therefore to satisfy conservation of mass or Fig. 17: The streamline pattern for a source. Also, since the flow is purely radial flow, 𝑣𝜃 = 0, the corresponding velocity potential can be obtained by integrating the equations below: It follows that........................................................... (35) If m is positive (i.e. m > 0), the flow is radially outward, and the flow is considered to be a source flow. If m is negative (i.e. m < 0), the flow is toward the origin, and the flow is considered to be a sink flow. The flow rate, m, is the strength of the source or sink. The stream function for the source can be obtained by integrating the relationships to yield............................................................... (36) We note that at the origin where r = 0 the velocity becomes infinite, which is of course physically impossible. Thus, sources and sinks do not really exist in real flow fields, and the line representing the source or sink is a mathematical singularity in the flow field. However, some real flows can be approximated at points away from the origin by using sources or sinks. Also, the velocity potential Department of Mechanical Engineering, UJ Page 21  MEE 309: ENGINEERING FLUID MECHANICS representing this hypothetical flow can be combined with other basic velocity potentials to describe approximately some real flow fields. Doublet: A doublet is another basic potential flow which is formed by appropriately combining a source and sink in a special way. Consider the equal strength, source-sink pair of Fig. 18. The combined stream function for the pair is which can be written as................. (37) Fig. 18: The combination of a source and sink of equal strength located along x axis. From Fig. 18 it follows that and Substituting these results in Eqn. 36 gives Department of Mechanical Engineering, UJ Page 22  MEE 309: ENGINEERING FLUID MECHANICS so that................................................ (38) For small value of a.................................. (39) since the tangent of an angle approaches the value of the angle for small angles. A Doublet is formed by letting a source and sink approach one another (a 0) while increasing the strength m (m ∞) so that the product 𝑚𝑎⁄𝜋 remain constant. 2 2 In this case, since 𝑟⁄(𝑟 − 𝑎 ) → 1⁄𝑟, Eqn. 39 reduces to.................................................................. (40) where K, a constant equal to 𝑚𝑎 ⁄𝜋, is called the strength of the doublet. The corresponding velocity potential for the doublet is........................................................................ (41) Plots of lines of constant 𝜓 reveal that the streamlines for a doublet are circles through the origin tangent to the x axis as shown in Fig. 19. Fig. 19: Streamlines for a doublet. Flow around a Circular Cylinder: A flow pattern equivalent to that of an ideal fluid passing a stationary circular cylinder, with its axis perpendicular to the direction of flow, is obtained by combining a doublet with uniform flow. This combination gives for stream function.............................................. (42) Department of Mechanical Engineering, UJ Page 23  MEE 309: ENGINEERING FLUID MECHANICS and for the velocity potential........................................... (43) In order for the stream function to represent flow around a circular cylinder it is necessary that 𝜓 = constant for r = a where a, is the radius of the cylinder. Since Eqn. 42 can be written as it follows that 𝜓 = 0 for r = a if which indicates that the doublet strength, K, must be equal to Ua2. Thus, the stream function for flow around a circular cylinder can be expressed as........................................... (44) and corresponding velocity potential is.......................................... (45) A sketch of the streamlines for this flow field is shown in Fig. 20. Fig. 20: The flow around a circular cylinder. The velocity components can be obtained from either Eqn. 44 or 45 as......................... (46).................... (47) Department of Mechanical Engineering, UJ Page 24  MEE 309: ENGINEERING FLUID MECHANICS FLOW THROUGH PIPES The importance of fluid (liquid or gas) transportation in our daily life in a closed conduit (commonly called a pipe if it is of round cross section or a duct if it is not round) cannot be over emphasized. A close look of the world around us will indicate that there is a wide variety of applications of pipe flow. Such applications range from the large, man-made pipeline that carries crude oil, to the more complex natural systems of “pipes” that carry blood throughout our body and air into and out of our lungs. Other examples include the water pipes in our homes and the distribution system that delivers the water from the city well to the house. Numerous hoses and pipes carry hydraulic fluid or other fluids to various components of vehicles and machines. The flow in a pipe is termed pipe flow only when the fluid completely fills the cross-section and there is no free surface of fluid. Some of the basic components of a typical pipe system are shown in Fig. 21. They include the pipes themselves, which can be of more than one diameter, the various fittings used to connect the individual pipes to form the desired system, the flow-rate control devices (valves), and the pumps or turbines that add energy to or remove energy from the fluid. Fig. 21: Typical pipe system components. LAMINAR FLOW: The flow of a fluid in a pipe may be laminar flow or it may be turbulent flow depending on the 𝜌𝑉𝑙 characteristic Reynolds number , where l is the characteristic length. Examples of laminar 𝜇 viscous flow include among other flow past tiny bodies, underground flow, movement of blood in the arteries of a human body, flow of oil in measuring instrument and rise of water in plants through their roots. Generally, a laminar flow has the following characteristic: ‘No slip’ at the boundary Department of Mechanical Engineering, UJ Page 25  MEE 309: ENGINEERING FLUID MECHANICS 𝑑𝑢 There is a shear between fluid layers which is due to viscosity and is given by 𝜏 = 𝜇. 𝑑𝑦 for flow in X-direction. The flow is rotational. There is continuous dissipation of energy due to viscous shear, therefore, to maintain the flow energy must be supplied externally. Loss of energy is proportional to first power of velocity and first power of viscosity. No mixing between different fluid layers (except by molecular motion, which is negligible). 𝜌𝑉𝑙 The flow remains laminar for as long as is less than critical value of Reynolds number. 𝜇 REYNOLDS EXPERIMENT: The flow of a fluid in a pipe may be laminar flow or it may be turbulent flow. This was demonstrated by Osborne Reynolds in 1883 through an experiment (using simple apparatus as shown in Fig. 22) in which water was discharged from a tank through a glass tube. The rate of flow could be controlled by a valve at the outlet, and a fine filament of dye injected at the entrance to the tube. Fig. 22: Reynolds Apparatus. At low velocities it was found out that the dye filament (a streakline) remained as a well-defined line as it flows along, with only slight blurring due to molecular diffusion of the dye into the surrounding water. For a somewhat larger “intermediate velocity” the dye filament fluctuates in time and space, and intermittent bursts of irregular behaviour appear along the streak. On the other hand, for “large enough velocities” the dye filament almost immediately becomes blurred and spreads across the entire pipe in a random fashion. These three characteristics, denoted as laminar, transitional, and turbulent flow, respectively, are illustrated in Fig. 23. Department of Mechanical Engineering, UJ Page 26  MEE 309: ENGINEERING FLUID MECHANICS Fig. 23: Time dependence of fluid velocity at a point. The Reynolds number ranges for which laminar, transitional, or turbulent pipe flows are obtained cannot be precisely given. The actual transition from laminar to turbulent flow may take place at various Reynolds numbers, depending on how much the flow is disturbed by vibrations of the pipe, roughness of the entrance region, and the like. For general engineering purposes (i.e., without undue precautions to eliminate such disturbances), the following values are appropriate: The flow in a round pipe is laminar if the Reynolds number is less than approximately 2100. The flow in a round pipe is turbulent if the Reynolds number is greater than approximately 4000. For Reynolds numbers between these two limits, the flow may switch between laminar and turbulent conditions in an apparently random fashion (transitional flow). On the basis of his experiment Reynolds discovered that: In case of laminar flow: The loss of pressure head ∝ velocity. In case of turbulent flow: the loss of head is approximately ∝ V2. {More exactly the loss of head ∝ Vn where n varies from 1.75 to 2.0}. FLOW OF VISCOUS FLUID IN CIRCULAR PIPES: Fully developed steady flow in a constant diameter pipe may be driven by gravity and/or pressure forces. For horizontal pipe flow, gravity has no effect except for a hydrostatic pressure variation across the pipe, 𝛾𝐷, that is usually negligible. It is the pressure difference, ∆𝑝 = 𝑝1 − 𝑝2 , between one section of the horizontal pipe and another which forces the fluid through the pipe. Viscous effects provide the restraining force that exactly balances the pressure force, thereby allowing the fluid to flow through the pipe with no acceleration. If viscous effects were absent in such flows, the pressure would be constant throughout the pipe, except for the hydrostatic variation. Hagen – Poiseuille Law Hagen-Poiseuille Law, valid only for laminar flow, is based on the assumptions that the fluid follows Newton’s law of viscosity and there is no slip of fluid particles at the boundary (i.e. the fluid particles adjacent to the pipe will have zero velocity). Fig. 24 shows a horizontal circular pipe of radius R, having laminar flow of fluid through it. Consider a small concentric cylinder (fluid element) of radius r and length dx as a free body. Department of Mechanical Engineering, UJ Page 27  MEE 309: ENGINEERING FLUID MECHANICS Fig. 24: Viscous/laminar flow through a circular pipe. If 𝜏 is the shear stress, the shear force F is given by 𝐹 = 𝜏 × 2𝜋 × 𝑑𝑥 Let p be the intensity of pressure at the left end and the intensity of pressure at the right end be 𝜕𝑝 (𝑝 + 𝜕𝑥. 𝑑𝑥) Thus the forces acting on the fluid element are: The shear force, 𝜏 × 2𝜋 × 𝑑𝑥 on the surface of fluid element. The pressure force, 𝑝 × 𝜋𝑟 2 on the left. 𝜕𝑝 The pressure force, (𝑝 + 𝜕𝑥. 𝑑𝑥) 𝜋𝑟 2 on the right end. For steady flow, the net force on the cylinder must be zero. 𝜕𝑝 [𝑝 × 𝜋𝑟 2 − (𝑝 + 𝜕𝑥. 𝑑𝑥) 𝜋𝑟 2 ] − 𝜏 × 2𝜋𝑟 × 𝑑𝑥 = 0 Simplifying gives 𝜕𝑝 𝑟 𝜏 = − 𝜕𝑥. 2 …………………………………. (48) Eqn. 47 shows that flow will occur only if pressure gradient exists in the direction of flow. The negative sign shows that pressure decreases in the direction of flow. Also Eqn. 48 indicates that the shear stress varies linearly across the section (see Fig. 25). Its value is zero at the centre of pipe (r = 0) and maximum at the pipe wall given by 𝜕𝑝 𝑅 𝜏0 = − 𝜕𝑥 ( 2 ) ……………………………… (49) Department of Mechanical Engineering, UJ Page 28  MEE 309: ENGINEERING FLUID MECHANICS Fig. 25: Shear stress and velocity distribution across a section. From Newton’s law of viscosity, 𝑑𝑢 𝜏 = 𝜇. 𝑑𝑦 ……………………………. (i) In this equation, the distance y is measured from the boundary. The radial distance r is related to distance y by the relation y = R-r or dy = - dr Eqn. (i) then becomes 𝑑𝑢 𝜏 = −𝜇 𝑑𝑟 …………………………………... (50) Comparing the two values of 𝜏 from Eqns. 48 and 50, we have 𝑑𝑢 𝜕𝑝 𝑟 −𝜇 𝑑𝑟 = − 𝜕𝑥. 2 1 𝜕𝑝 or, 𝑑𝑢 = 2𝜇 (𝜕𝑥 ) 𝑟. 𝑑𝑟 Integrating the above equation w.r.t. ‘r’, we get 1 𝜕𝑝 𝑢 = 4𝜇. 𝜕𝑥 𝑟 2 + 𝐶 ………………………..… (51) Where C is the constant of integration and its value is obtained from the boundary condition: At r = R, u = 0 1 𝜕𝑝 1 𝜕𝑝 ∴ 0 = 4𝜇. 𝜕𝑥 𝑅 2 + 𝐶 or 𝐶 = − 4𝜇. 𝜕𝑥 𝑅 2 Substituting this value of C in Eqn. 51, we get 1 𝜕𝑝 1 𝜕𝑝 𝑢 = 4𝜇. 𝜕𝑥 𝑟 2 − 4𝜇. 𝜕𝑥 𝑅 2 1 𝜕𝑝 𝑢 = − 4𝜇. 𝜕𝑥 (𝑅 2 − r 2 ) ………………….….. (52) Eqn. 52 shows that the velocity distribution curve (Fig. 25) is parabolic in the radial coordinate. The maximum velocity (umax) occurs at the pipe centerline, and the minimum velocity (zero) at the pipe wall. Maximum velocity is given by 1 𝜕𝑝 𝑢max = − 4𝜇. 𝜕𝑥 𝑅 2 …………………………. (53) Department of Mechanical Engineering, UJ Page 29  MEE 309: ENGINEERING FLUID MECHANICS From Eqns. 51 and 52, we have 𝑟 2 𝑢 = 𝑢max [1 − (𝑅) ] ……………………….. (54) Eqn. 54 is the most commonly used equation for the velocity profile for laminar flow through pipes. This equation can be used to calculate the discharge as follows: 𝑑𝑄 = 𝑢 × 2𝜋𝑟 × 𝑑𝑟 𝑟 2 = 𝑢max [1 − (𝑅) ] 2𝜋𝑟. 𝑑𝑟 Total discharge, 𝑄 = ∫ 𝑑𝑄 𝑅 𝑟 2 = ∫0 𝑢max [1 − (𝑅) ] 2𝜋𝑟. 𝑑𝑟 𝑅 𝑟3 = 2𝜋𝑢max ∫0 (𝑟 − 𝑅2 ) 𝑅 𝑟2 𝑟4 𝑅2 𝑅4 = 2𝜋𝑢max [ 2 − 4𝑅2 ] = 2𝜋𝑢max [ 2 − 4𝑅2 ] 0 𝜋 = 2 𝑢max 𝑅 2 𝜋 𝑄 𝑢 𝑅2 𝑢max 2 max Average velocity of flow, ū = 𝐴 = = ………………… (55) 𝜋𝑅 2 2 Eqn. 55 shows that the average velocity is half the maximum velocity. Substituting Eqn. 54 in Eqn. 55, we have 1 𝜕𝑝 ū = − 8𝜇. 𝜕𝑥. 𝑅 2 8𝜇ū or −𝜕𝑝 =. 𝜕𝑥 𝑅2 The pressure difference between two sections 1 and 2 at distance x1 and x2 (see Fig. 24) is given by 𝑝 8𝜇ū 𝑥 − ∫𝑝 2 𝜕𝑝 = 2 ∫ 𝜕𝑥 1 𝑅 2 𝑥1 8𝜇ū 8𝜇ū𝐿 or (𝑝1 − 𝑝2 ) = (𝑥2 − 𝑥1 ) = 𝑅2 𝑅2 32𝜇ū𝐿 or (𝑝1 − 𝑝2 ) = …………………………. (56) 𝐷2 where D is the diameter of the pipe, and L is the length. Eqn. 56 is known as the Hagen-Poiseuille equation. The Hagen-Poiseuille theory is restricted to laminar flow (those with Reynolds numbers less than approximately 2100) in a horizontal pipe. The adjustment necessary to account for non-horizontal pipes, as shown in Fig. 26, can be easily included by replacing the pressure drop, ∆𝑝, by the combined effect of pressure and gravity, ∆𝑝 − 𝛾ℓsin𝜃, where 𝜃 is the angle between the pipe and the horizontal. (Note that 𝜃 > 0 if the flow is uphill, while 𝜃 < 0 if the flow is downhill.) This can be seen from the force balance in the x direction (along the pipe axis) on the cylinder of fluid shown in Fig. 26b. The net force in the x Department of Mechanical Engineering, UJ Page 30  MEE 309: ENGINEERING FLUID MECHANICS direction is a combination of the pressure force in that direction, ∆𝑝𝜋𝑟 2, and the component of weight in that direction, −𝛾𝜋𝑟 2 ℓsin𝜃. The result is given by Fig. 26: Free-body diagram of a fluid cylinder for flow in a non-horizontal pipe. Thus, all of the results for the horizontal pipe are valid provided the pressure gradient is adjusted for the elevation term, that is, ∆𝑝 is replaced by ∆𝑝 − 𝛾ℓsin𝜃 so that …………………… (57) and ………………… (58) It is seen that the driving force for pipe flow can be either a pressure drop in the flow direction, ∆𝑝, or the component of weight in the flow direction, −𝛾ℓsin𝜃. If the flow is downhill, gravity helps the flow (a smaller pressure drop is required; sin𝜃 < 0). If the flow is uphill, gravity works against the flow (a larger pressure drop is required; sin𝜃 > 0). Note that 𝛾ℓsin𝜃 = 𝛾∆𝑧 (where ∆𝑧 is the change in elevation) is a hydrostatic type pressure term. If there is no flow, V = 0 and∆𝑝 = 𝛾ℓsin𝜃 = 𝛾∆𝑧, as expected for fluid statics. Assignment 2: An oil of 8 poise and specific gravity 0.9 is flowing through a horizontal pipe of 50mm diameter. If the pressure drop in 100 m length of the pipe 2000kN/m2, determine (i) Rate of flow of oil; (ii) Centre-line velocity; (iii) Total frictional drag 100 m length of pipe; Department of Mechanical Engineering, UJ Page 31  MEE 309: ENGINEERING FLUID MECHANICS (iv) Power required to maintain the flow; (v) Velocity gradient at the pipe wall; (vi) Velocity and shear stress at 10mm from the wall. FRICTION FACTOR FOR LAMINAR FLOW: In a pipe of diameter D in which a viscous fluid of viscosity 𝜇 is flowing with a velocity ū, the loss of pressure head, hf, is given by Eqn. 56 as 32𝜇ū𝐿 ℎ𝑓 = ………………………… (i) 𝑤𝐷 2 where, L = Length of the pipe, and w = Weight density of the fluid. The loss of head due to friction is given by 4𝑓𝐿𝑉 2 4𝑓𝐿ū2 ℎ𝑓 = = ………………… (ii) 𝐷×2g 𝐷×2g where, f is the co-efficient of friction between pipe and fluid, and V = ū. From eqn. (i) and (ii), we have 32𝜇ū𝐿 4𝑓𝐿ū2 = 𝑤𝐷 2 𝐷×2g 32𝜇ū𝐿×𝐷×2g 16𝜇 or, 𝑓= = ū𝜌𝐷 (since 𝑤 = 𝜌. g) 4𝐿.ū2.𝜌.𝑔.𝐷 2 16𝜇 1 = 𝜌𝑉𝐷 = 16 × 𝑅𝑒 (Since ū = V) 𝜌𝑉𝐷 where, 𝑅𝑒 (= ) is the Reynolds number 𝜇 16 i.e. 𝑓 = 𝑅𝑒 ………………………………………. (59) TURBULENT FLOW IN PIPES: In a pipe, a laminar flow occurs when Reynolds number (Re) is < 2000 and a turbulent flow occurs when Re > 4000. The fluid motion in a turbulent flow is irregular and chaotic leading to a complete mixing of fluid due to collision of fluid masses with one another. The fluid masses are interchanged between adjacent layers. As the fluid masses in adjacent layers have different velocities, interchange of fluid masses between adjacent layers is accompanied by a transfer of momentum which causes additional shear stresses of high magnitude between adjacent layers. The shear in turbulent flow is mainly due to momentum transfer. Generally, turbulent flow conditions are far more likely in most engineering situations. Fig. 27, shows the velocity distribution curves for laminar and turbulent flows in a pipe. Department of Mechanical Engineering, UJ Page 32  MEE 309: ENGINEERING FLUID MECHANICS Fig. 27: Velocity distribution curves for laminar and turbulent flows in a pipe. In turbulent flow the flatness of velocity distribution curve in the core region away from the wall is because of the mixing of fluid layers and exchange of momentum between them. The velocity distribution which is parabolic in laminar flow, tends to follow power law and logarithmic law in turbulent flow. FRICTION FACTOR FOR TURBULENT FLOW: Consider a small element of fluid within a conduit, as shown in Fig. 28. The flow is assumed to be uniform and steady so that the fluid acceleration in the direction of flow is zero. Fig. 28: Forces on a control volume in a pipe flow. Let, 𝑝1= Intensity of pressure at section 1, 𝑝2 = Intensity of pressure at section 2, D = Diameter of the pipe, ′ 𝑓 = Non-dimensional factor (whose value depends upon the material and nature of the surface), and ℎ𝑓 = Loss of head due to friction. Propelling force on the flowing fluid between the two sections is = (𝑝1 − 𝑝2 )𝐴 (where, A = area of cross-section of the pipe) Frictional resistance force = 𝑓 ′ 𝑃𝐿𝑉 2 where, P = Wetted perimeter, and V = Average flow velocity. Under equilibrium condition, Department of Mechanical Engineering, UJ Page 33  MEE 309: ENGINEERING FLUID MECHANICS Propelling force = Frictional resistance force i.e. (𝑝1 − 𝑝2 )𝐴 = 𝑓 ′ 𝑃𝐿𝑉 2 Dividing both sides by weight density w, we have 𝑝1 −𝑝2 𝑓′ ( )𝐴 = 𝑃𝐿𝑉 2 𝑤 𝑤 𝑓′ or, ℎ𝑓 = 𝑃𝐿𝑉 2 𝑤 2g𝑓 ′ 𝑃 𝐿𝑉 2 2g𝑓 ′ 𝐿 𝑉2 or, ℎ𝑓 = (𝐴) = × 𝑚 × 2g ………… (60) 𝑤 2g 𝑤 𝐴 The ratio 𝑃 is called the hydraulic mean depth or hydraulic radius, denoted by m (or R). 𝐿 𝑉2 2g𝑓 ′ The term (𝑚 × 2g ) has dimension of ℎ𝑓 and thus the term is a non-dimensional quantity and 𝑤 can be replaced with another constant f. 𝐿 𝑉2 ∴ ℎ𝑓 = 𝑓 × 𝑚 × 2g …………………………… (61) In the case of a circular pipe, 𝜋 𝐴 ×𝐷 2 𝐷 4 Hydraulic mean depth, 𝑚=𝑃= = 𝜋𝐷 4 Substituting this value in Eqn. 60, we get 𝐿 𝑉2 ℎ𝑓 = 𝑓 × 𝐷/4 × 2g 4𝑓𝐿𝑉 2 ℎ𝑓 = ………………………………… (62) 𝐷×2g The factor f is known as Darcy coefficient of friction. Eqn. 62 is known as Darcy-Weisbach equation and holds good for all types of flows provided a proper value of f is chosen. Sometimes Eqn. 62 is written as 𝑓1 𝐿𝑉 2 ℎ𝑓 = 𝐷×2g where 𝑓1 = 4f. It will be seen from Eqn. 62 that all the parameters, with the exception of the friction factor f, are measurable. Results of extensive experimentation in this area led to the establishment of the following proportional relationship: (i) ℎ𝑓 ∝ 𝑙; (ii) ℎ𝑓 ∝ 𝑉 2 ; (iii) ℎ𝑓 ∝ 1⁄𝐷 ; (iv) ℎ𝑓 depends on the surface roughness of the pipe; (v) ℎ𝑓 depends on the fluid density and viscosity; (vi) ℎ𝑓 is independent of pressure. Department of Mechanical Engineering, UJ Page 34  MEE 309: ENGINEERING FLUID MECHANICS Expression for co-efficient of friction in terms of shear stress: (𝑝1 − 𝑝2 )𝐴 = Force due to shear stress, 𝜏0 where 𝜏0 = shear stress at the pipe wall = Shear stress (𝜏0 ) × surface area = 𝜏0 × 𝜋𝐷𝐿 𝜋 or (𝑝1 − 𝑝2 ) 𝐷2 = 𝜏0 𝜋𝐷𝐿 4 4𝜏0 ×𝐿 ∴ (𝑝1 − 𝑝2 ) = ………………………………. (63) 𝐷 Eqn. 63 gives the pressure drop, (𝑝1 − 𝑝2 ), in terms of wall shear stress, 𝜏0. Eqn. 62 can be written as 𝑝1 −𝑝2 4𝑓𝐿𝑉 2 ℎ𝑓 = = 𝑤 𝐷×2g 4𝑓𝐿𝑉 2 (𝑝1 − 𝑝2 ) = × 𝑤 …………………. (64) 𝐷×2g Equating Eqns. 63 and 64 and simplifying, we get 𝑓𝜌𝑉 2 𝜏0 = 2 2𝜏0 ∴ 𝑓 = 𝜌𝑉 2 ……………………….... (65) LOSS OF HEAD IN A PIPE SYSTEM: When fluid flows in a pipe, it experiences some resistance to its motion, due to which its velocity and ultimately the head of fluid available is reduced. This loss of energy (or head) is classified as either major energy losses – due to friction or minor energy losses – due to (i) sudden enlargement/contraction of pipe; (ii) an obstruction in pipe; and (iii) pipe fittings (such as valves, bends, tees, and the like). In most cases, the majority of the system loss is associated with the friction in the straight portions of the pipes, the major losses. However, in some other cases the minor losses are greater than the major losses. MAJOR LOSSES: These losses which are due to friction may be calculated using either Darcy-Weisbach formula or Chezy’s formula. Darcy-Weisbach Formula: Darcy-Weisbach formula (generally used for flow through pipes) is given by 4𝑓𝐿𝑉 2 ℎ𝑓 = ……………………………………. (66) 𝐷×2g where ℎ𝑓 = Loss of head due to friction f = Coefficient of friction (a function of Reynolds number, Re) 0.0791 𝑓 = (𝑅𝑒)1/4 for Re varying from 4000 to 106 Department of Mechanical Engineering, UJ Page 35  MEE 309: ENGINEERING FLUID MECHANICS 16 𝑓 = 𝑅𝑒 for Re < 2000 (i.e. laminar/viscous flow) L = Length of the pipe, V = Mean velocity of flow, and D = Diameter of the pipe. Chezy’s Formula: Reference to Fig. 28, equilibrium between the propelling force due to pressure difference and the frictional resistance gives: (𝑝1 − 𝑝2 )𝐴 = 𝑓 ′ 𝑃𝐿𝑉 2 (𝑝1 −𝑝2 ) 𝑓′ or, = 𝑃𝐿𝑉 2 𝑤 𝑤 𝑓′ 𝑃 or, ℎ𝑓 = 𝐿𝑉 2 𝑤𝐴 𝑤 𝐴 ℎ𝑓 ∴ Mean velocity, 𝑉 = √𝑓′ × √𝑃 × 𝐿 𝑤 where, the factor √𝑓′, is called the Chezy’s constant, C; 𝐴 area of flow the ratio (= wetted perimeter) is called the “hydraulic mean depth” or “hydraulic radius” and 𝑃 denoted m (or R); ℎ𝑓 the ratio prescribes the loss of head per unit length of pipe and is denoted by i or S (slope). 𝐿 ∴ Mean velocity, 𝑉 = 𝐶 √𝑚 𝑖 ………………………………… (67) Eqn. 67 is known as Chezy’s formula. The formula is useful in finding the head loss due to friction is the mean flow velocity through the pipe and the value of Chezy’s constant is known. Chezy’s formula is generally used for the flow through open channels. MINOR LOSSES: Whereas the major loss of head is due to friction, the minor loss of head, which often a result of the dissipation of kinetic energy, includes the following cases: 1. Loss of head due to sudden expansion/contraction, 2. Loss of head at the entrance/exit of a pipe, 3. Loss of head due to obstruction in the pipe, 4. Loss of head in various pipe fittings and bends. LOSS OF HEAD DUE TO SUDDEN EXPANSIONS: Generally, the losses due to pipe or duct fittings are determined experimentally. However, the case of a sudden expansion in a pipe or duct may be determined analytically. Fig. 29 shows a liquid flowing through a pipe which has sudden expansion. Due to sudden enlargement, the flow is decelerated abruptly and eddies are developed resulting in loss of head. Consider two sections 1 – 1 (before enlargement) and 2 – 2 (after enlargement) Department of Mechanical Engineering, UJ Page 36  MEE 309: ENGINEERING FLUID MECHANICS Fig. 29: Loss of head due to sudden expansion. Let A1 = Area of pipe at section 1 – 1. = 𝜋⁄4 𝐷12 (where D1 is the diameter of the pipe), p1 = Intensity of pressure at section 1 – 1, V1=Velocity of flow at section 1 – 1, 𝐴2 (𝜋⁄4 𝐷22 ), 𝑝2 and 𝑉2 = Corresponding values at section 2 – 2, 𝑝0 = Intensity of pressure of the liquid eddies on the area (A2 – A1), he = Loss of head due to sudden enlargement. Applying Bernoulli’s equation to sections 1 – 1 and 2 – 2, we have 𝑝1 𝑉2 𝑝2 𝑉2 + 2g1 + 𝑧1 = + 2g2 + 𝑧2 + Loss of head due to sudden expansion (ℎ𝑒 ) 𝑤 𝑤 But 𝑧1 = 𝑧2 (pipe being horizontal) 𝑝1 𝑉2 𝑝2 𝑉2 ∴ + 2g1 = + 2g2 + ℎ𝑒 𝑤 𝑤 𝑝1 𝑝2 𝑉2 𝑉2 ℎ𝑒 = ( 𝑤 − ) + ( 2g1 − 2g2 ) …………………….. (i) 𝑤 Now, the force acting on liquid in the control volume (between sections 1 – 1 and 2 – 2) in the flow direction is given by: 𝐹𝑥 = 𝑝1. 𝐴1 + 𝑝0 (𝐴2 − 𝐴1 ) − 𝑝2. 𝐴2 Assuming 𝑝0 = 𝑝1, we have 𝐹𝑥 = 𝑝1. 𝐴1 + 𝑝1 (𝐴2 − 𝐴1 ) − 𝑝2. 𝐴2 = 𝑝1. 𝐴2 − 𝑝2. 𝐴2 = (𝑝1 − 𝑝2 )𝐴2 ……………… (ii) Consider momentum of liquid at the sections 1 – 1 and 2 – 2; Momentum of liquid/sec at section 1 – 1 = mass x velocity = 𝜌𝐴1 𝑉1 × 𝑉1 = 𝜌𝐴1 𝑉12 Momentum of liquid/sec at section 2 – 2 = 𝜌𝐴2 𝑉2 × 𝑉2 = 𝜌𝐴2 𝑉22 ∴ Change in momentum of liquid/sec = 𝜌𝐴2 𝑉22 − 𝜌𝐴1 𝑉12 But from continuity equation, we have 𝐴1 𝑉1 = 𝐴2 𝑉2 Department of Mechanical Engineering, UJ Page 37  MEE 309: ENGINEERING FLUID MECHANICS 𝐴2 𝑉2 or, 𝐴1 = 𝑉1 ∴ Change of momentum/sec 𝐴2 𝑉2 = 𝜌𝐴2 𝑉22 − 𝜌. 𝑉12 𝑉1 = 𝜌𝐴2 𝑉22 − 𝜌𝐴2 𝑉1 𝑉2 = 𝜌𝐴2 (𝑉22 − 𝑉1 𝑉2 ) …………………….…… (iii) Now, net force = change in momentum ∴ (𝑝1 − 𝑝2 )𝐴2 = 𝜌𝐴2 (𝑉22 − 𝑉1 𝑉2 ) 𝑝1 −𝑝2 or, = 𝑉22 − 𝑉1 𝑉2 𝜌 Dividing both sides by g, we get 𝑝1 𝑝2 𝑉22 −𝑉1 𝑉2 − = (since 𝜌g = 𝑤) 𝑤 𝑤 g 𝑝 𝑝2 Substituting the value ( 𝑤1 − ) in eqn. (i), we get 𝑤 𝑉22 −𝑉1 𝑉2 𝑉2 𝑉2 ℎ𝑒 = + 2g1 − 2g2 g (𝑉1 −𝑉2 )2 ∴ ℎ𝑒 = …………………………………..… (68) 2g LOSS OF HEAD DUE TO SUDDEN CONTRACTION: Due to sudden contraction, the streamlines converge to a minimum cross-section called the vena contracta and then expand to fill the downstream pipe (Fig. 30) Fig. 30: Loss of head due to sudden contraction. Let, 𝐴𝑐 = Area of flow at section C – C, 𝑉𝑐 = Velocity of flow at section C – C, 𝐴2 = Area of flow at section 2 – 2, 𝑉2 = Velocity of flow at section 2 – 2, and ℎ𝑐 = Loss of head due to sudden contraction. Loss of head due to sudden contraction = Loss up to vena contracta + Loss due to sudden enlargement beyond vena contacta Department of Mechanical Engineering, UJ Page 38  MEE 309: ENGINEERING FLUID MECHANICS (𝑉𝑐 −𝑉2 )2 or, ℎ𝑐 = negligibly small + …………………… (i) 2g From continuity equation, we have 𝐴𝑐 𝑉𝑐 = 𝐴2 𝑉2 𝑉𝑐 𝐴 1 1 𝐴 = 𝐴2 = 𝐴 =𝐶 (since 𝐶𝑐 = 𝐴𝑐 ) 𝑉2 𝑐 ( 𝑐⁄𝐴 ) 𝑐 2 2 𝑉 𝑉𝑐 = 𝐶2 𝑐 Substituting the value of 𝑉𝑐 in eqn. (i), we get 𝑉 2 ( 2 −𝑉2 ) 𝑉22 1 2 𝐶2 ℎ𝑐 = = ( − 1) 2g 2g 𝐶 𝑐 𝑉22 1 2 i.e. ℎ𝑐 = ( − 1) …………………………………… (69) 2g 𝐶𝑐 𝑉2 In general, ℎ𝑐 = 𝑘 2g2 1 2 where, 𝑘 = (𝐶 − 1) 𝑐 k is known as the loss coefficient. Table 1 below shows some experimental values of 𝐶𝑐 and the corresponding values of k obtained with sharp edges. Table 1: Loss coefficient for sudden contraction. 𝐴2 0.1 0.3 0.5 0.7 1.0 ⁄𝐴 1 𝐶𝑐 0.61 0.632 0.673 0.73 1.0 k 0.41 0.34 0.24 0.14 0 LOSS OF HEAD DUE TO OBSTRUCTION IN PIPE: The loss of head due to an obstruction in pipe takes place on account of the reduction in the cross- sectional area of the pipe by the presence of obstruction which is followed by an abrupt enlargement of the stream beyond the obstruction (Fig. 31). Fig. 31: Loss of head due to obstruction in pipe. Department of Mechanical Engineering, UJ Page 39  MEE 309: ENGINEERING FLUID MECHANICS Head loss due to obstruction (ℎobs. ) is given by: 𝐴 2 𝑉2 ℎobs. = (𝐶 ) ……………………………… (70) 𝑐 (𝐴−𝑎) 2g where, A = Area of the pipe a = Maximum area of obstruction, and V = Velocity of liquid in pipe. LOSS OF HEAD AT THE ENTRANCE TO PIPE: A fluid may flow from a reservoir into a pipe through any number of different shaped entrance regions as are sketched in Fig. 32. Each geometry has an associated loss coefficient. Loss of head at the entrance to pipe (ℎ𝑖 ) is given by: 𝑉2 ℎ𝑖 = 𝑘 2g …………………………………………. (71) where, V = Velocity of liquid in pipe. k = Loss coefficient. Fig. 32: Entrance flow condition and loss coefficient. (a) Reentrant, k = 0.8 (a) sharp-edge, k = 0.5, (c) slightly rounded, k = 0.2, (d) well-rounded, k = 0.04 LOSS OF HEAD AT THE EXIT OF A PIPE: Loss of head at the exit of a pipe is denoted by ℎ0 and is given by: 𝑉2 ℎ0 = …………………………………………… (72) 2g where, V = Velocity of liquid at outlet of pipe. Department of Mechanical Engineering, UJ Page 40  MEE 309: ENGINEERING FLUID MECHANICS LOSS OF HEAD DUE TO BEND IN THE PIPE: The loss of head in bends (ℎ𝑏 ) provided in pipes may be expressed as: 𝑉2 ℎ𝑏 = 𝑘 2g ………………………………………... (73) where, V = Mean velocity of flow of fluid, and k = coefficient of bend; it depends on the angle of bend, radius of curvature of bend and diameter of pipe. LOSS OF HEAD IN VARIOUS PIPE FITTINGS: The loss of head in various pipe fittings (such as valves, couplings, etc.) may be represented by: 𝑉2 ℎ𝑓𝑖𝑡𝑡𝑖𝑛𝑔𝑠 = 𝑘 2g …………………………………… (74) where, V = Mean velocity of flow in pipe, and k = loss coefficient; it depends on the type of the pipe fitting. It is non-dimensional constant and its value is obtained experimentally for any fitting. Table 2 sets out some typical values. Table 2: Head loss coefficients for a range of pipe fittings. Fitting Loss coefficient, k Gate valve (open to 75 per cent shut) 0.25 25 Globe valve 10 Spherical plug valve (fully open) 0.1 Pump foot valve 1.5 Return bend 2.2 0 90 elbow 0.9 0 45 elbow 0.4 Large-radius 900 bend 0.6 Tee junction 1.8 Sharpe pipe entry 0.5 Radiused pipe entry 0.0 Sharp pipe exit 0.5 Department of Mechanical Engineering, UJ Page 41  MEE 309: ENGINEERING FLUID MECHANICS DIMENSIONAL ALAYSIS Although several practical engineering problems involving fluid mechanics can be solved by using the equations and analytical procedures, there remain a large number of problems that rely on experimentally obtained data for their solution. In fact, it is probably fair to say that very few problems involving real fluids can be solved by analysis alone. The solution to many problems is achieved through the use of a combination of analysis and experimental data. Thus, engineers working on fluid mechanics problems should be familiar with the experimental approach to these problems so that they can interpret and make use of data obtained by others, such as might appear in handbooks, or be able to plan and execute the necessary experiments in their own laboratories. An obvious goal of any experiment is to make the results as widely applicable as possible. To achieve this end, the concept of similitude is often used so that measurements made on one system (for example, in the laboratory) can be used to describe the behavior of other similar systems (outside the laboratory). The laboratory systems are usually thought of as models and are used to study the phenomenon of interest under carefully controlled conditions. From these model studies, empirical formulations can be developed, or specific predictions of one or more characteristics of some other similar system can be made. Dimensional analysis is a mathematical technique which makes use of the study of the dimension for solving several engineering problem. Each physical phenomenon can be expressed by an equation giving relationship between different quantities; such quantities are dimensional and non- dimensional. Dimensional analysis helps in determining a systematic arrangement of variables in the physical relationship, combining dimensional variables to form non-dimensional parameters. It is based on the principle of dimensional homogeneity and uses the dimensions of relevant variables affecting the phenomenon. Dimensional analysis has become an important tool for analyzing fluid flow problems and also useful in presenting experimental results in a concise form. Dimensional analysis may be used to: (i) test the dimensional homogeneity of any equation; (ii) derive rational formulae for a flow phenomenon; (iii) derive equations expressed in terms of non- dimensional parameters to show the relative significance of each parameter; and plan model tests and present experimental results in a systematic manner; thus making it possible to analyze the complex fluid flow phenomenon. Dimensional analysis has the following advantages: (a) It expresses the functional relationship between the variables in dimensionless terms. (b) In hydraulic model studies it reduces the number of variables involved in a physical phenomenon, generally by three. (c) By the proper selection of variables, the dimensionless parameters can be used to make certain logical deduction about the problem. Department of Mechanical Engineering, UJ Page 42  MEE 309: ENGINEERING FLUID MECHANICS (d) Design curves, by the use of dimensional analysis, can be developed from experimental data or direct solution of the problem. (e) It enables getting up a theoretical equation in a simplified dimensional form. (f) Dimensional analysis provides partial solutions to the problems that are too complex to be dealt with mathematically. (g) The conversion of units of quantities from one system to another is facilitated. Dimensions The various physical quantities used in fluid phenomenon can be expressed in terms of fundamental quantities otherwise known as primary quantities. The fundamental quantities are mass, length, time and temperature, designated by the letters M, L, T, and 𝜃 respectively. Temperature is specially useful in compressible flow. Other quantities such as velocity, area, acceleration etc. are expressed in terms of fundamental quantities and are referred to as derived or secondary quantities. The expression for a derived quantity in terms of primary quantities is called the dimension of the physical quantity. A quantity may either be expressed dimensionally in M-L-T or F-L-T system (some engineers prefer to use force instead of mass as fundamental quantity because the force is easy to measure). Table 3 gives the dimensions of various quantities used in both the system. Department of Mechanical Engineering, UJ Page 43  MEE 309: ENGINEERING FLUID MECHANICS Table 3: Quantities used in Fluid Mechanics and Heat Transfer and their Dimensions Department of Mechanical Engineering, UJ Page 44  MEE 309: ENGINEERING FLUID MECHANICS IMPULSE-MOMENTUM EQUATION: The impulse-momentum equation is one of the basic tools for the solution of flow problems. Its application leads to the solution of problems in fluid mechanics which cannot be solved by energy principles alone. Sometimes it is used in conjunction with energy equation to obtain complete solution of engineering problems. The momentum equation is based on the law of conservation of momentum or momentum principle which states as follows: the net force acting on a mass of fluid is equal to change in momentum of flow per unit time in that direction. From Newton’s second law, 𝐹 = 𝑚𝑎 Where m = Mass of fluid, F = Force acting on the fluid, and a = Acceleration (acting in the same direction as F) 𝑑𝑣 But acceleration, 𝑎= 𝑑𝑡 𝑑𝑣 𝑑(𝑚𝑣) ∴ 𝐹 = 𝑚. 𝑑𝑡 = ………………………... (74) 𝑑𝑡 (‘m’ is taken inside the differential, being constant) This equation is known as Impulse-momentum equation. And may be stated as follows: The impulse of a force F acting on a fluid mass ‘m’ in a short interval of time, dt, is equal to the change in momentum, d(mv), in the direction of force. The impulse-momentum equations are often called simply momentum equations. The impulse-momentum equations can the applied in solving the following types of problems: (1) To determine the resultant force acting on the boundary of flow passage by a stream of fluid as the stream changes its direction, magnitude or both. Problems of this type are: (i) Jet propulsion, (ii) Moving vanes, (iii) Pipes bends, (iv) Reducers, etc. (2) To determine the characteristic of flow when there is an abrupt change of flow section. Problems of this type are: (i) Sudden enlargement in a pipe (ii) Hydraulic jump in a channel, etc. ROTOR-DYNAMIC MACHINES: Rotor-dynamic machines are machines that have a rotating part called impeller, through which the fluid flow is continuous. Rotor-dynamic machines can be classified into three classes namely axial flow machines, centrifugal machines and mixed flow machines depending on the direction of fluid flow in relation to the plane of impeller rotation. In axial flow machines the flow is perpendicular to the impeller and, hence, along its axis of rotation, as shown in Fig. 33a and 34a. In centrifugal machines (sometimes called ‘radial flow’), although the fluid approaches the impeller axially, it turns at the machine’s inlet so that the flow through the impeller is in the plane of the impeller rotation (Fig. 33b and 34b). Mixed flow machines dirive their names from the fact that the flow through their impellers is partly axial and Department of Mechanical Engineering, UJ Page 45  MEE 309: ENGINEERING FLUID MECHANICS partly radial. Each type of machine has advantages and disadvantages for different applications and in terms of fluid-mechanical performance. Fig. 33: Axial flow and centrifugal impellers. Fig. 34: (a) A radial-flow machine, (b) An axial-flow machine. Department of Mechanical Engineering, UJ Page 46  MEE 309: ENGINEERING FLUID MECHANICS Fig. 35 shows a mixed flow fan impeller from the discharge side. It should be noted that the hub is conical; thus the direction of flow leaving the impeller is somewhere between the axial and radial. Fig. 35: A mixed flow fan impeller. Pumps and turbines (sometimes called fluid machines) occur in a wide variety of configurations. In general, pumps add energy to the fluid—they do work on the fluid; turbines extract energy from the fluid—the fluid does work on them. The term “pump” will be used to generically refer to all pumping machines, including pumps, fans, blowers, and compressors. Both pumps and turbines can be axial flow, mixed flow or radial flow. All impellers consist of a supporting disc or cylinder and blades attached to it. It is the motion of the blades which is related to the motion of the fluid, one doing the work on the other or vice versa. In any case, there are forces exerted on the blades and, since they rotate with the impeller, torque is transmitted because of the rate of change of angular momentum. One-dimensional theory: Although the real flow through an impeller is complex (