Last Updated on: 10-02-2020€¦ · DYNAMICS OF FLUID FLOW: Module-3 Fluid Kinematics and Dynamics...

Last Updated on:

Module-3

Fluid Kinematics and Dynamics

[email protected]

98795 10743

Fluid Mechanics & Hydraulics (3140611)

10-02-2020

Prof. Mehul Pujara

2Module-3 Fluid Kinematics and Dynamics Darshan Institute of Engineering & Technology, Rajkot

Fluid:

• The term fluid refers to both aliquid and a gas; it is generallydefined as a state of matter inwhich the material flows freelyunder the action of a shearstress.


A branch of fluid mechanics whichdeals with the study of velocity andacceleration of the particles offluids in motion and theirdistribution in space withoutconsidering any forces or energyinvolved.

It provides

(i) an idea about the rate offlow

(II) an idea about differenttypes of velocities of flow

Fluid Kinematics???

• Equation of Continuity is the first and fundamental equation of flow.


1. Mass flow rate ( 𝑚)

2. Volume flow rate( 𝑄)

Rate of Flow:


Volume Flow Rate:


Mass Flow Rate:


Control Volume Approach:

• A system is whatever the engineer selects for study.• Open System (Control Volume system)• Closed System (Control Mass system)

• The closed system involves selection and analysisof a specific collection of matter, the closedsystem is a Lagrangian concept.

• The control volume involves selection and analysisof a region in space, the CV is an Eulerian concept.


Observer concentrates on themovement of single particle.

Observer has to move with thefluid particle to observe itsmovement

The path and changes in velocity,acceleration, pressure and densityof a single particle are described.

Not commonly use.

Langrangian Concept:

P

PP

P

𝑢 =𝜕𝑥

𝜕𝑡𝑣 =

𝜕𝑦

𝜕𝑡𝑤 =

𝜕𝑧

𝜕𝑡

𝑎𝑥 =𝜕2𝑥

𝜕𝑡2𝑎𝑦 =

𝜕2𝑦

𝜕𝑡2𝑎𝑧 =

𝜕2𝑧

𝜕𝑡2


Observer concentrates on various fixed point particles.

Observer remains steady and observe the change in the fluidparameters at the fixed point only

Method describes the overall flow characteristics at various points as fluid particle pass.

Commonly used.

Eulerian Concept:

u = f1 (x,y,z,t), v = f2(x,y,z,t), W = f3(x,y,z,t)


1. Path line

2. Stream line

3. Stream tube

4. Streak line or filament line

5. Potential line

Lines of flow to describe the motion of the fluid:


The path traced by single fluid particle inmotion over a period of time is called itspath line.

The path line shows the direction ofvelocity of the particle as it moves ahead.

During steady flow, path line coincide withstream line.

However path line fluctuates betweendifferent stream lines during an unsteadyflow.

It is real line showing successive position ofone particle.

Particle may cross its path line.

Path line:


A stream line is imaginary line drawnthrough a flowing fluid such that the tangentat each point on the line indicates thedirection of the velocity of the fluid particleat that point.

It shows positions of various fluid particles.

Stream line can not intersect each other,they are always parallel.

No flow across stream line.

Stream line:


Stream Tube:

A circular space formed by the collection ofstream lines passing through the perimeterof a closed curve in a steady flow.

Stream tube behaves as a solid surfacetube.

It may be of regular or irregular shape.


Real line showing an instantaneouspicture of the position of all fluidparticles in the flow which havepassed from a given point.

The line of smoke from chimney isnothing but a streak line.

It may change instant to instant.

Flow across streak line is possible.

Streak line or Filament line:


The lines of equal velocity potential are called potential line.

Stream line and potential line are at right angle to each other.

Potential Line:


Steady or Unsteady

Uniform or Non-uniform

Laminar or Turbulent

Compressible or Incompressible

Rotational or Irrotational

One, Two or Three Dimensional

Types of fluid flow:


Fluid characteristics like velocity,acceleration, pressure and densitydo not change with time at anypoint in the fluid is called steadyflow.

𝜕𝑣

𝜕𝑡=0,

𝜕𝑃

𝜕𝑡=0,

𝜕𝜌

𝜕𝑡=0 ,

𝜕2𝑣

𝜕𝑡2=0

Flow of water with constantdischarge through pipeline.

If fluid properties changes with timeits called unsteady flow.

Steady vs. Unsteady Flow:


Velocity of the flow does not changealong its direction of flow at any point oftime is called Uniform flow.

Flow through a constant diameter pipeline is uniform flow.

𝜕𝑣

𝜕𝑠=0

Velocity of flow changes with its directionof flow any instant of time is called Non-uniform flow.

Uniform vs. Non-uniform flow:


Fluid particles move in layers witheach layer sliding over the other, theflow is called laminar Flow.

Velocity of flow is very small.

Reynolds number<2100

The flow in which adjacent layerscross each other and the layers donot move along the well definedpath is called turbulent Flow.

Reynolds number>4000

Laminar vs. Turbulent Flow:


The volume and hence density of thefluid does not remain constant duringthe flow is called compressible flow.

Gases are compressible.

The flow in which the changes involume and thereby density of the fluidare insignificant, the flow isincompressible.

Liquids are incompressible.

For gases in subsonic aerodynamiccondition air flow can be considerincompressible.

Compressible vs. Incompressible Flow:


Ex.

Fluid particles rotate about their own axiswhile flowing along stream lines, is calledrotational flow.

Motion of a liquid in a rotating cylinder is arotational flow.

The flow in which fluid particles do notrotate about their own axis while flowingalong stream lines is called Irrotationalflow.

Flow of water in emptying wash basing isIrrotational flow.

Rotational vs. Irrotational Flow:


One, Two and Three Dimensional Flows:

Flow parameter such as velocity is afunction of time and one space co-ordinate only, say x is called OneDimensional flow.

Ex. u=f(x), v=0 and w=0

Flow parameter such as velocity is afunction of time and two space co-ordinate only, say x and y is called TwoDimensional flow.

Ex. u=f1(x,y), v=f2(x,y) and w=0

Flow parameter such as velocity is afunction of time and three space co-ordinate only, say x,y and z is calledThree Dimensional flow.

Ex. u=f1(x,y,z), v=f2(x,y,z) and w=f3(x,y,z)

Ex.

Ex.

Ex.


Critical depth (yc) is defined as the depth of flow where energy is at aminimum for a particular discharge.

For a given value of specific energy, the critical depth gives the greatestdischarge, or conversely, for a given discharge, the specific energy is aminimum for the critical depth.

Critical Depth:


Subcritical flow occurs when the actual water depth is greater than criticaldepth.

Subcritical flow is dominated by gravitational forces and behaves in a slowor stable way.

It is defined as having a Froude number less than one.

Supercritical flow is dominated by inertia forces and behaves as rapid orunstable flow.

When the actual depth is less than critical depth it is classified assupercritical.

Supercritical flow has a Froude number greater than one.

Critical flow is the transition or control flow that possesses the minimumpossible energy for that flow rate. Critical flow has a Froude number equalto one.

Sub Critical, Critical and Super Critical flow:


CONTINUITY EQUATION:

It follows “Principle of conservationof mass”

“Mass neither be created nor bedestroyed, total mass of systemremains constant.”

Mass offluidenteringunit time

Mass offluidmovingout perunit time

Rate ofincreaseof massper unittime

- =



Considering flow though the stream tube.

Mass of fluid entering into stream tube perunit time is

= Density * Discharge

= Density * Area * velocity

= 𝜌 𝐴 𝑣 (1)

Mass of fluid coming out from stream tubeper unit time is

= 𝜌 𝐴 𝑣 +𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠ds (2)

Net mass of fluid reaming in stream tubeper unit time is

= -𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠ds (3)

𝜌 𝐴 𝑣

𝜌 𝐴 𝑣 +𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠ds


The mass of fluid in stream tube is

= density * volume of fluid in tube

= 𝜌 𝐴 ds (4)

Rate of increase of mass of fluid with timein stream tube is

=𝜕 (𝜌 𝐴𝑑𝑠 )

𝜕𝑡

=𝜕 (𝜌 𝐴 )

𝜕𝑡𝑑𝑠 (5)

Net mass remained in the stream tube per unittime is equal to rate of increase of mass withthe time.

Equation (3) = Equation (5)

-𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠𝑑𝑠 =

𝜕 (𝜌 𝐴 )

𝜕𝑡𝑑𝑠


-𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠= 𝜕 (𝜌 𝐴 )

𝜕𝑡

𝜕 (𝜌 𝐴 )

𝜕𝑡+ 𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠= 0 (6)

Equation (6) is known as continuityequation for one dimensional flow, it isapplicable to all cases of flow.


For Steady flow𝜕 (𝜌 𝐴 )

𝜕𝑡= 0

Therefore𝜕 (𝜌 𝐴 𝑣)

𝜕𝑠= 0

𝜌 𝐴 𝑣 = Constant

At section 1 and 2 we can write continuityequation

𝜌1 𝐴1 𝑣1 = 𝜌2 𝐴2 𝑣2

In case of incompressible flow

𝜌1 = 𝜌2

𝐴1 𝑣1 = 𝐴2 𝑣2


1

2


Hydrodynamics

Aerodynamics

Electromagnetism

Quantum mechanics

Flow of fluid through pipe, ducts or tubes

Rivers

Process plants

Power plants

Dairies

Logistics in general

CONTINUITY EQUATION APPLICATION:

1

2


Consider a fluid element of lengths dx, dy anddz in the direction of x, y and z.

Let inlet velocity components are u, v and w inthe direction of x, y and z respectively

Mass of fluid entering the face ABCD persecond is

= density * velocity in x-direction * area of ABCD

= ρ * u * (dy*dz)

Mass of fluid leaving the face EFGH per secondis

= ρu dydz +𝜕 (ρu dydz)

𝜕𝑥dx

CONTINUITY EQUATION IN THREE DIMENSTIONS:


Net gain of mass in x direction is

= ρu dydz - [ ρu dydz +𝜕 (ρu dydz)

𝜕𝑥dx]

= -𝜕 (ρu dydz)

𝜕𝑥dx

= -𝜕 (ρu)𝜕𝑥

dx dy dz

Similarly net gain of mass in y direction is

= -𝜕 (ρv)𝜕𝑦

dx dy dz

Net gain of mass in z direction is

= -𝜕 (ρw)

𝜕𝑧dx dy dz



Total Net gain of mass per unit time is

= - [𝜕 (ρu)𝜕𝑥

+𝜕 (ρv)𝜕𝑦

+𝜕 (ρw)

𝜕𝑧] dx dy dz

The net increase of mass per unit time in thefluid element must be equal to the rate ofincrease of mass of fluid in the element.

The mass of fluid is

= density * volume

= ρ * dx dy dz

The rate of increase of mass of fluid in theelement

=𝑑ρ

𝑑𝑡dx dy dz



Equation two expression

- [𝜕 (ρu)𝜕𝑥

+𝜕 (ρv)𝜕𝑦

+𝜕 (ρw)

𝜕𝑧] dx dy dz =

𝑑ρ

𝑑𝑡dx dy dz

𝜕𝜌

𝜕𝑡+𝜕 (ρu)𝜕𝑥

+𝜕 (ρv)𝜕𝑦

+𝜕 (ρw)

𝜕𝑧= 0

The above equation is continuity equation incartesian coordinates in general form.

It is applicable to Steady and unsteady flow

Uniform and non-uniform flow

Compressible and incompressible flow



General Continuity equation is

𝜕𝜌

𝜕𝑡+𝜕 (ρu)𝜕𝑥

+𝜕 (ρv)𝜕𝑦

+𝜕 (ρw)

𝜕𝑧= 0

For Steady flow, the equation is

𝜕 (ρu)𝜕𝑥

+𝜕 (ρv)𝜕𝑥

+𝜕 (ρw)

𝜕𝑥= 0

For incompressible flow, the equation is𝜕u𝜕𝑥

+𝜕v𝜕𝑦

+𝜕w𝜕𝑧

= 0

For One dimensional flow, the equation is

𝜕u𝜕𝑥

= 0



It is the study of fluid motion with the forces causing flow.

The dynamic bhaviour of the fluid flow is analyzed by the Newton’s second law of motion.

F = m a

In fluid flow, the following forces are present:

Fg, gravity force

Fp, pressure force

Fv, viscous force

Ft, turbulent force

Fc, force due to compressibility

F = Fg + Fp + Fv + Ft + Fc

If Fc is neglected

F = Fg + Fp + Fv + Ft

Equation is known as Reynold’s Equation of motion.

DYNAMICS OF FLUID FLOW:


If Ft is neglected

F = Fg + Fp + Fv

Equation is known as Navier-Stokes Equation.

If is Fv neglected

F = Fg + Fp

Equation is known as Euler’s equation of motion.

DYNAMICS OF FLUID FLOW:


The below equation which consider effect of gravitational and pressure force is known as Euler’s equation of motion.

F = Fg + Fp

Consider a stream-line in which flow is taking place.

Consider cylindrical element of cross section dA and length ds. The forces acting on it is: Pressure force pdA in direction of flow

pressure force (p + 𝜕𝑝

𝜕𝑠ds) dA opposite to

the direction of flow

Weight of element ρ g dA ds

EULER’S EQUATION OF MOTION:


Let θ is the angle between the direction of flow and the line of action of the weight of element.

pdA - (p + 𝝏𝒑

𝝏𝒔ds) dA - ρ g dA ds cos θ = m * as

pdA - (p + 𝜕𝑝

𝜕𝑠ds) dA - ρ g dA ds cos θ = ρ dA ds * as

Now,

as = 𝒅𝒗

𝒅𝒕, where v is a function of s and t.

= 𝜕𝑣

𝜕𝑠

𝑑𝑠

𝑑𝑡+

𝜕𝑣

𝜕𝑡

= v 𝜕𝑣

𝜕𝑠+ 𝜕𝑣

𝜕𝑡

If Flow is steady then 𝜕𝑣

𝜕𝑡= 0

as = v 𝝏𝒗

𝝏𝒔



Substituting value of as in equation,

-𝝏𝒑

𝝏𝒔ds dA - ρ g dA ds cos θ = ρ dA ds * v

𝝏𝒗

𝝏𝒔

Dividing by ρ ds dA above equation,

-𝝏𝒑

ρ𝝏𝒔- g cos θ = v

𝝏𝒗

𝝏𝒔

𝝏𝒑

ρ𝝏𝒔+ g cos θ + v

𝝏𝒗

𝝏𝒔= 0

Now,

cos θ =𝒅𝒛

𝒅𝒔

The above equation can be rewrite

𝟏

ρ𝝏𝒑

𝝏𝒔+ g

𝒅𝒛

𝒅𝒔+ v

𝝏𝒗

𝝏𝒔= 0


𝒅𝒑

ρ+ g dz + v dv= 0

The above equation is known as Euler’s Equation of Motion.


It States, “The sum of pressure energy(pressure head), the kinetic energy (velocityhead) and the potential energy (potentialhead) is constant along a stream line.”

Integrating Euler’s equation of motion

𝑑𝑝

𝜌+ 𝑔𝑑𝑧 + 𝑣𝑑𝑣 = 0

Or p/ 𝜌 + gZ + v2/2 = Constant

Or p/ 𝜌g + v2/2g + Z = Constant

Substituting 𝜌g = 𝛾

p/𝛾 + v2/2g + Z = Constant

The above equation known as Bernoulli’sequation.

BERNOULLI’S EQUATION:

p/𝛾 = Pressure Headv2/2g = Velocity HeadZ = Potential Head

p/𝛾 + v2/2g + Z = Total Head or Hydrodynamic Head


Assumption made in Bernoulli’s equation:

Flow is steady: No variation with respect to time

Fluid is ideal: No frictional effect due to viscosity

Flow is incompressible: density constant

One-dimensional flow

Velocity of flow is uniform.

Bernoulli’s equation at section 1&2

p1/𝛾 + v12/2g + Z1 = p2/𝛾 + v2

2/2g + Z2

BERNOULLI’S EQUATION:

1 2


It is defined as a scalar function of space and time such that its negativederivative with respect to any direction gives the fluid velocity in thatdirection.

Mathematically, the velocity potential is defined as ∅ = f (x , y, z) for steadyflow such that,

u = -𝜕∅

𝜕𝑥

v = -𝜕∅

𝜕𝑦

w = -𝜕∅

𝜕𝑧where u, v and w are the components of velocity in x, y and z directionsrespectively.

VELOCITY POTENTIAL FUNCTION:


The continuity equation for an incompressible steady flow is𝜕u𝜕𝑥

+𝜕v𝜕𝑦

+𝜕w𝜕𝑧

= 0

Substituting the values of u, v and w from previous equation, we get𝜕

𝜕𝑥(-𝜕∅

𝜕𝑥) +

𝜕

𝜕𝑦(-𝜕∅

𝜕𝑦) +

𝜕

𝜕𝑦(-𝜕∅

𝜕𝑧) = 0

𝜕2∅

𝜕𝑥2+𝜕2∅

𝜕𝑦2 +𝜕2∅

𝜕𝑧2= 0

The equation is known as Laplace equation.

The rotational component is

ωz =1

2(𝜕v𝜕𝑥

−𝜕u𝜕𝑦

), ωy =1

2(𝜕u𝜕𝑧

−𝜕w𝜕𝑥

), ωx =1

2(𝜕w𝜕𝑦

−𝜕v𝜕𝑧

)

or

ωz =1

2(−

𝜕2∅

𝜕𝑥𝜕𝑦+

𝜕2∅

𝜕𝑦𝜕𝑥), ωy =

1

2(−

𝜕2∅

𝜕𝑧𝜕𝑥+

𝜕2∅

𝜕𝑥𝜕𝑧), ωx =

1

2(−

𝜕2∅

𝜕𝑦𝜕𝑧+

𝜕2∅

𝜕𝑧𝜕𝑦),




ωz =1

2(−

𝜕2∅

𝜕𝑥𝜕𝑦+

𝜕2∅

𝜕𝑦𝜕𝑥), ωy =

1

2(−

𝜕2∅

𝜕𝑧𝜕𝑥+

𝜕2∅

𝜕𝑥𝜕𝑧), ωx =

1

2(−

𝜕2∅

𝜕𝑦𝜕𝑧+

𝜕2∅

𝜕𝑧𝜕𝑦)

If ∅ is continuous function then,

𝜕2∅

𝜕𝑥𝜕𝑦=

𝜕2∅

𝜕𝑦𝜕𝑥,𝜕2∅

𝜕𝑧𝜕𝑥=

𝜕2∅

𝜕𝑥𝜕𝑧,𝜕2∅

𝜕𝑦𝜕𝑧=

𝜕2∅

𝜕𝑧𝜕𝑦

ωz = ωy = ωx = 0

When rotational components are zero, the flow is called Irrotational.

The properties of potential function are:

1. If velocity potential (∅) is exist, the flow should be Irrotational.

2. If velocity potential (∅ ) is satisfies the Laplace equation, itrepresents steady incompressible Irrotational flow.



It is defined as the scalar function of space and time, such that its partialderivative with respect to any direction gives the velocity component at rightangles to that direction.

It is defined only for two dimension flow.

Mathematically, for steady flow it is defined as 𝜑= f (x, y) such that

𝜕𝜑

𝜕𝑥= v

𝜕𝜑

𝜕𝑦= -u

where u and v are the components of velocity in x and y directionsrespectively.

STREAM FUNCTION:


The continuity equation for two dimensional flow is,

𝜕u𝜕𝑥

+𝜕v𝜕𝑦

= 0

𝜕

𝜕𝑥−

𝜕𝜑

𝜕𝑦+

𝜕

𝜕𝑦(𝜕𝜑

𝜕𝑥) = 0

−𝜕2𝜑

𝜕𝑥𝜕𝑦+

𝜕2𝜑

𝜕𝑦𝜕𝑥= 0


ωz =1

2(𝜕v𝜕𝑥

−𝜕u𝜕𝑦

) =1

2(𝜕2𝜑

𝜕𝑥2+𝜕2𝜑

𝜕𝑦2)

For Irrotational flow ωz = 0 so,

𝜕2𝜑

𝜕𝑥2+𝜕2𝜑

𝜕𝑦2 = 0 , it is Laplace equation for 𝜑.

STREAM FUNCTION:


The velocity potential is,

u = -𝜕∅

𝜕𝑥

v = -𝜕∅

𝜕𝑦

The stream function is,

𝜕𝜑

𝜕𝑥= v

𝜕𝜑

𝜕𝑦= -u

So

u = -𝜕∅

𝜕𝑥= -

𝜕𝜑

𝜕𝑦& v = -

𝜕∅

𝜕𝑦=𝜕𝜑

𝜕𝑥

𝜕∅

𝜕𝑥=

𝜕𝜑

𝜕𝑦&

𝜕∅

𝜕𝑦= -

𝜕𝜑

𝜕𝑥

Relation between Stream Function and Velocity Potential Function:


A grid obtained by drawing a series of equipotential lines and stream linesis called a flow net.

The flow net is an important tool in analyzing two-dimensional irrotationalflow problems.

FLOW NET:


Example:


References:

1. Fluid Mechanics and Fluid Power Engineering by D.S. Kumar, S.K.Kataria & Sons

2. Fluid Mechanics and Hydraulic Machines by R.K. Bansal, LaxmiPublications

3. Fluid Mechanics and Hydraulic Machines by R.K. Rajput, S.Chand & Co

4. Fluid Mechanics; Fundamentals and Applications by John. M. CimbalaYunus A. Cengel, McGraw-Hill Publication

Last Updated on: 10-02-2020€¦ · DYNAMICS OF FLUID FLOW: Module-3 Fluid Kinematics and Dynamics...

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Transcript of Last Updated on: 10-02-2020€¦ · DYNAMICS OF FLUID FLOW: Module-3 Fluid Kinematics and Dynamics...