Kinematics of Flow. Fluid Kinematics Fluid motion -Types of fluid - Velocity and acceleration -...

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Kinematics of Flow

Transcript of Kinematics of Flow. Fluid Kinematics Fluid motion -Types of fluid - Velocity and acceleration -...

Page 1: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Kinematics of Flow

Page 2: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Fluid Kinematics

Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation

Potential Flows -Velocity Potential - Stream function - Flow net (Cir and Vorticity) - Source, Sink and Doublet

Page 3: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Section I

Page 4: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Fluid Kinematics: It is defined as the branch of science which deals with motion of particles without considering the force causing the motion.

Types of fluid flow:1) Steady flow and unsteady flow2) Uniform and no-uniform flow3) Laminar and turbulent flow4) Compressible and in compressible flow5) Rotational and irrigational flows6) One, two and three dimensional flow

Rate of flow or Discharge Q: m3/s or liters/s

Q = A V

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Continuity equation

The equation based on the principle of conservation of mass is called continuity equation. Thus the quantity of fluid per second is constant.Consider two cross-section of a pipe

Let V1 = Average velocity Section 1-1 ς1 = Density at section 1-1 A1 = Area of pipe at section 1-1And V2, ς2, A2 are corresponding values at section 2-2

Rate of flow at section 1-1 = ς1 A1 V1

Rate of flow at section 2-2 = ς2 A2 V2

ς1 A1 V1 = ς2 A2 V2 Continuity equation

A1 V1 = A2 V2 Incompressible fluids

Page 6: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex The diameter of a pipe at the section 1 and 2 are 10 cm and 15 cm respectively. Find the discharge through the pipe if the velocity of water flowing through the pipe at section 1 is 5 m/s. Determine also the velocity at section 2.

D1 =10cm D2 =15cm

V1 =5 m/s

(1)(2)

Ans Q = 0.03927 m3/s, V2 = 2.22 m/s

Page 7: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex A 30 cm diameter pipe, conveying water, branches into two pipes of diameters 20 cm and 15 cm repectively. If the average velocity in the 30 cm diameter pipe is 2.5 m/s, find the discharge in this pipe. Also determine the velocity in 15 cm pipe if the average velocity in 20 cm diameter pipe is 2 m/s.

V1 = 2.5 m/sD1 = 30 cm

V3 = ?D3 = 15 cm

V2 = 2 m/sD2 = 20 cm

Ans Q3 = 0.1139 m3/s, V3 = 6.44 m/s

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Ex Water flows through a pipe AB 1.2 m diameter at 3 m/s and then passes through a pipe BC 1.5 m dia. At C, the pipe branches. Branch CD is 0.8 m in diameter and carries one-third of the flow in AB. The flow velocity in branch CE is 2.5 m/s. Find the i) Volume rate of flow in AB, ii) The velocity in BC, iii) The velocity in CD and iv) The diameter of CE.

Ans Q = 3.393 m3/s, VBC = 1.92 m/s, VCD = 2.25 m/s, DCE = 1.07 m

Page 9: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex A 25 cm diameter pipe carries oil of Sp.gr. 0.9 at a velocity of 3 m/s. At another section the diameter is 20 cm. Find the velocity at this section and also mass rate of flow of oil.

Ans: V3= 4.68 m/s, Mass rate of flow = 132.23 kg/s

Page 10: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Fluid element of length dx, dy and dz (In x, y, z direction)Velocity = u, v and w

Mass of fluid entering the face ABCD = ς x Velocity-x x Area ABCD = ς u dy dz

Mass of fluid leaving the face EFGH

Continuity equation in Three-Dimensions

Gain of mass in x-direction = Mass through EFGH- Mass ABCD

Page 11: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Similarly the net gain of mass in y and z direction

Net gain of mass

Since there is no accumulation of mass

Page 12: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Velocity function:

Acceleration function:

Page 13: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex The velocity vector in a fluid flow is given.

Find the velocity and acceleration of a fluid particle at (2, 1, 3) at time t=1.

Ans V , u=32, v=-40, w=2, R=51.26: a, ax=1536, ay=320, az=2, R=1568.9

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Section II

Page 15: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Velocity Potential Function: it is define 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 defined by ø

Page 16: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Stream Function: it is define as a scalar function of space and time such that its partial derivation with respect to any direction gives the velocity component at right angles to that direction.

It is defined by Ψ

Page 17: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Equipotential line: A line along which the velocity potential ø is constant, is called equipotential line.

ø = ConstantdΨ =0

dø = 0

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Line of constant Stream Function:

Ψ = constant dΨ =0

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Flow Net: A grid obtained by drawing a series of equipotential lines and stream lines is called a flow net.

Relation between stream function and velocity Potential function:

and

Page 20: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex The velocity potential function is given by

Calculate the velocity components at the point (4, 5).

Ans u = -40 units, v =50 units

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Ex A Stream function is given by

Calculate the velocity components and also magnitude and direction of the resultant velocity at any point.

Ans u = 6 units/sec, v =5 units/sec, R =7.81 unit/s, θ=39 48

Page 22: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Types of Motion:

1) Linear Translation or Pure Translation:It is define as the movement of a fluid element in such a way that it

moves bodily from one position to another position and the two axes ab and cd represented in new position by a’b’ and c’d’.

Page 23: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

2) Linear deformation:It is define as the deformation of a fluid element in linear direction

when the element moves.

3) Angular deformation or shear deformation:It is define as the average change in angle contained by two adjacent

sides.

Page 24: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

4) Rotation: it is defined as the movement of a fluid element in such a way that both of it’s axes (H & V) rotate in the same direction.

Page 25: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Vortex flow:Vortex flow is defined as the flow of a fluid along a curved path or the flow of a rotating mass of fluid is known a ‘Vortex flow’.

1) Force vortex flow: it is defined as that type of vortex flow, in which some external torque is required to rotate the fluid mass. The fluid mass in this type of flow, rotates at constant angular velocity, ω

The tangential velocity of any fluid particle is given by

v = ω x r

Page 26: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Examples of forced vortex:1. A Vertical cylinder containing liquid which is rotated about its

central axis with a constant angular velocity ω2. Flow of liquid inside the impeller of a centrifugal pump3. Flow of water through the runner of turbine.

Page 27: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

2) Free vortex flow: When no external torque is required to rotate the fluid mass, that type of flow is called free vortex flow.

Example1) Flow of liquid through a hole provided at the bottom of a

container2) Flow of liquid around a circular bend in a pipe3) A whirlpool in river4) Flow of fluid in a centrifugal pump casing

Angular momentum = Mass x Velocity = m x vMoment of momentum = Momentum x r = m v r

Time rate of change of angular momentum =

For free vortex

mvr = Constant

Page 28: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex A fluid is given by V = 8x3i-10x2yj.Find the shear strain rate and state whether the flow is rotational or irrotational.

Ans Shear strain= -10xy, Wz =-10xy

Page 29: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Ex The velocity components in two-dimensional flow are

u=y3/3 +2x-x2y v = xy2-2y-x3/3Show that these components represent a possible case of an irrotational flow.

Two-dimensional flow:

Page 30: Kinematics of Flow. Fluid Kinematics  Fluid motion -Types of fluid - Velocity and acceleration - Continuity equation  Potential Flows -Velocity Potential.

Prepared by, Dr Dhruvesh Patel

Prepared by, Dr Dhruvesh Patel