1. FLUID MECHANICS ME-10 MODULE - 2 KINEMATICS OF FLUID FLOW Presented by: Ayush Agrawal (Asst. Professor) Civil Engineering Department Jabalpur Engineering College

2. Fluid Kinematics 2 C Branch of fluid mechanics which deals with response of fluids in motion without considering forces and energies in them. C The study of kinematics is often referred to as the geometry of motion. CAR surface pressure contours and streamlines Flow around cylindrical object

3. Fluid Flow Type: 2. Mass flow rate 3.Weigh flow rate Rate of flow: Quantity of fluid passing through any section in a unit time. Rate of flow  Quantity of fluid time  volume of fluid time 3  weight of fluid 1.Volume flow rate:  mass of fluid time

4. Fluid Flow C 1.Volume flow rate: C 2. Mass flow rate C 3.Weigh flow rate V L A Longitudinal SectionCross Section Q  volume of fluid  AL timet M  mass of fluid    AL  timet t  g  AL    AL   G G  time weight of fluid C Let’s consider a pipe in which a fluid is flowing with mean velocity,V. C Let, in unit time, t, volume of fluid (AL) passes through section X-X, Units X 4 X

5. Types of Flow 5 C Depending upon fluid properties C Ideal and Real flow C Incompressible and compressible C Depending upon properties of flow C Laminar and turbulent flows C Steady and unsteady flow C Uniform and Non-uniform flow

6. Ideal and Real flow C Real fluid flows implies friction effects. Ideal fluid flow is hypothetical; it assumes no friction. Velocity distribution of pipe flow 6

7. Compressible and incompressible flows Incompressible fluid flows assumes the fluid have constant density with respect to temparature and pressure while in compressible fluid flows density is variable and becomes function of temperature and pressure. P1P1 P2P2 v1v1 v2 v2v2 P1P1 P2P2 v1v1 v2v2 Incompressible fluid 7 Compressible fluid

8. Laminar and turbulent flow C The flow in laminations (layers) is termed as laminar flow while the case when fluid flow layers intermix with each other is termed as turbulent flow. C Reynold’s number is used to differentiate between laminar and turbulent flows. 8 Transition of flow from Laminar to turbulent Laminar flow Turbulent flow

9. Steady and Unsteady flows 9 C Steady flow: It is the flow in which conditions/properties of flow remains constant w.r.t. time at a particular section but the condition may be different at different sections. C Flow conditions: velocity, pressure, density or cross-sectional area etc. C e.g., A constant discharge through a pipe. C Unsteady flow: It is the flow in which conditions of flow changes w.r.t. time at a particular section. C e.g., A variable discharge through a pipe V Longitudinal Section X X tt VV  0;  V  contt  0;  V  variable tt VV

10. Uniform and Non-uniform flow C Uniform flow: It is the flow in which conditions of flow remains constant from section to section at particular/any given time C e.g., Constant discharge though a constant diameter pipe C Non-uniform flow: It is the flow in which conditions of flow does not remain constant from section to section. C e.g., Constant discharge through variable diameter pipe 10 V X X xx Longitudinal Section  V  0;  V  contt  0;  V  variable xx VV V Longitudinal Section X

11. Describe flow condition C Constant discharge though non variable diameter pipe X V Longitudinal X Steady-non-uniform flow Longitudinal Section X Unsteady flow !! Non-uniform flow !! C Variable discharge though non variable diameter pipe X V unsteady-non-uniform flow 111 Section Steady flow !! Non-uniform flow !!

12. Flow Combinations 12 TypeType 1. Steady Uniform flow 2. Steady non-uniform flow 3. Unsteady Uniform flow 4. Unsteady non-uniform flow Example Flow at constant rate through a duct of uniform cross-section Flow at constant rate through a duct of non-uniform cross-section (tapering pipe) Flow at varying rates through a long straight pipe of uniform cross-section. Flow at varying rates through a duct of non-uniform cross-section.

13. One, Two and Three Dimensional Flows C Although in general all fluids flow three-dimensionally, with pressures and velocities and other flow properties varying in all directions, in many cases the greatest changes only occur in two directions or even only in one. In these cases changes in the other direction can be effectively ignored making analysis much more simple. C Flow is one dimensional if the flow parameters (such as velocity, pressure, depth etc.) at a given instant in time only vary in the direction of flow and not across the cross-section Mean velocity Water surface Longitudinal section of rectangular channel Cross-sectionVelocity profile 13

14. One, Two and Three Dimensional Flows C Flow is two-dimensional if it can be assumed that the flow parameters vary in the direction of flow and in one direction at right angles to this direction Two-dimensional flow over a weir C Flow is three-dimensional if the flow parameters vary in all three directions of flow Three-dimensional flow in stilling basin 14

15. Visualization of flow Pattern C The flow velocity is the basic description of how a fluid moves in time and space, but in order to visualize the flow pattern it is useful to define some other properties of the flow. These definitions correspond to various experimental methods of visualizing fluid flow. CAR surface pressure contours and streamlines Flow around cylindrical object 15

16. Visualization of flow Pattern Streamlines around a wing shaped body Flow around a skying athlete 16

17. Path line and stream line C Pathline: It is trace made by single particle over a period of time. C Streamline show the mean direction of a number of particles at the same instance of time. C Character of Streamline C 1. Streamlines can not cross each other. (otherwise, the cross point will have two tangential lines.) C 2. Streamline can't be a folding line, but a smooth curve. C 3. Streamline cluster density reflects the magnitude of velocity. (Dense streamlines mean large velocity; while sparse streamlines mean small velocity. ） Flow around cylindrical object 17

18. Stream line A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. It has the direction of the velocity vector at each point of flow across the streamline. The path line of a particle is same as Streamline in case of Steady Flow.

19. Mean Velocity and Discharge C V A Longitudinal SectionCross Section Q  AV Volume flow rate: Q  volume of fluid    tV  A time  t V ∆tXV ∆tX 19 X C Let’s consider a fluid flowing with mean velocity,V, in a pipe of uniform cross-section.Thus volume of fluid that passes through section XX in unit time, ∆ t, becomes; Volume of fluid    tV  A M   AV G   AV Similarly

20. Fluid System and Control Volume 20 C Fluid system refers to a specific mass of fluid within the boundaries defined by close surface. The shape of system and so the boundaries may change with time, as when fluid moves and deforms, so the system containing it also moves and deforms. C Control volume refers to a fixed region in space, which does not move or change shape. It is region in which fluid flow into and out.

21. Rotational & Irrotational Flow Rotational Flows :-  The flow in which fuid particle while flowing along stream lines rotate about their own axis is called as rotational flow.  Irrotational Flows:-  The flow in which the fluid particle while flowing along stream lines do not rotate about their axis is called as irrotational flow.

23. Streakline and streamtubes C A Streakline is the locus of fluid particles that have passed sequentially through a prescribed point in the flow. C It is an instantaneous picture of the position of all particles in flow that have passed through a given point. Easy to generate in experiments like dye in a water flow, or smoke in an airflow. C Streamtube is an imaginary tube whose boundary consists of streamlines. C The volume flow rate must be the same for all cross sections of the stream tube. 23

25. Continuity C Matter cannot be created or destroyed - (it is simply changed in to a different form of matter). C This principle is know as the conservation of mass and we use it in the analysis of flowing fluids. C The principle is applied to fixed volumes, known as control volumes shown in figure: An arbitrarily shaped control volume. 25 For any control volume the principle of conservation of mass says Mass entering per unit time -Mass leaving per unit time = Increase of mass in the control volume per unit time

26. Continuity Equation (One Dimensional) A stream tube C For steady flow there is no increase in the mass within the control volume, so Mass entering per unit time = Mass leaving per unit time C Derivation: C Lets consider a stream tube. C ρ 1, v 1 and A 1 are mass density, velocity and cross-sectional area at section 1. Similarly, ρ 2, v 2 and A 2 are mass density, velocity and cross- sectional area at section 2. C M 1   1 A 1 V 1 M 2   2 A 2 V 2 d Md M 26 dt st dt  d  M st  M MM M 21  1 A 1 V 1   2 A 2 V 2  According to mass conservation

27. Continuity Equation C For steady flow condition C Assuming incompressible fluid, Therefore, according to mass conservation for steady flow of incompressible fluids volume flow rate remains same from section to section. d  M st  / dt  0  1 A 1 V 1   2 A 2 V 2  0   1 A 1 V 1   2 A 2 V 2 M   1 A 1 V 1   2 A 2 V 2 C Hence, for stead flow condition,mass flow rate at section 1= mass flow rate at section 2. i.e., mass flow rate is constant. C Similarly G   1 gA 1 V 1   2 gA 2 V 2  1   2   A 1 V 1  A 2 V 2 Q 1  Q 2 Q 1  Q 2  Q 3  Q 4 27

34. Velocity Potential Function  It 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. It is denoted by ‘φ’ (phi).  The negative sign indicates that the flow takes place in the direction in which ‘φ’ decreases.

37. Hence,

38. Stream Function  It is defined as the scalar function of space and time, such that its partial derivative with respect to any direction gives the velocity component at right angles to that direction. It is denoted by 'Ѱ’ and defined for two dimensional flow. Laplace equation

40. i.If stream function exists, it is a possible case of fluid flow which may be rotational or irrotational. ii.If stream function satisfies the Laplace equation, it is a possible case of an irrotational flow.

44. Flow Net  A grid obtained by drawing a series of stream lines and equipotential lines is known as a flow net.  Flow net provides a simple graphical technique for studying two – dimensional irrotational flows, when the mathematical calculation is difficult.  The stream lines and equipotential lines are mutually perpendicular to each other.  A flow net analysis assist in the design of an efficient boundary shapes.  It is also used to calculate the flow at ground level.

75. Circulation and Vorticity  Circulation and vorticity are two primary measures of rotation in a fluid.  Circulation, which is a scalar integral quantity, is a macroscopic measure of rotation for a finite area of the fluid.  Vorticity however is a vector field that gives a microscopic measure of the rotation at any point in the fluid. VORTICITYCIRCULATION

76.  The circulation, C, about a closed contour in a fluid is defined as the line integral evaluated along the contour of the component of the velocity vector that is locally tangent to the contour. “Meaning” of Circulation :  Circulation can be considered as the amount of force that pushes along a closed boundary or path. Circulation is the total “push” you get when going along a path, such as a circle. Circulation

77. Vorticity  Vorticity is the tendency for elements of the fluid to spin.  Vorticity can be related to the amount of circulation or rotation (or more strictly the angular rate of rotation) in a fluid. Vertical Component of Vorticity :  In large-scale dynamic meteorology, we are in concerned only with the vertical components of absolute and relative vorticity.  The vertical component of vorticity is defined as the circulation about a closed contour in the horizontal plane divided by the area enclosed, in the limit where the area approaches zero.

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