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All that begins . . .

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Faculty of Mechanical EngineeringDepartment of Thermo Fluids

SKMM 2313 Mechanics of Fluids 1Introduction to Fluids in Motion

« An excerpt (mostly) from Potter et al. (2012) »

Abu Hasan ‘ABDULLAH

November 2017

Outline

1 Introduction2 Description of Fluid Motion

I Lagrangian and Eulerian Descriptions of MotionI Pathlines, Streaklines, and StreamlinesI AccelerationI Angular Velocity and Vorticity

3 Classification of Fluid FlowsI One-, Two-, and Three-Dimensional FlowsI Viscous and Inviscid FlowsI Laminar and Turbulent FlowsI Incompressible and Compressible Flows

4 The Bernoulli Equation

abu.hasan.abdullah c b n a 2017 SKMM 2313 Mechanics of Fluids 1 Introduction to Fluids in Motion 1 / 35

Introduction

This chapter serves as an introduction to all the following chapters that deal withfluid motions.General equations of fluid motion are very difficult to solve; consequently,engineers tend to simplify by making certain assumptions especially those relatedto fluid properties:

I under certain conditions, the viscosity can affect the flow significantly; in others, viscouseffects can be neglected, greatly simplifying the equations without significantly alteringthe predictions.

I compressibility of a gas in motion is taken into account if velocities are very high. Butcompressibility effects are not taken into account to predict wind forces on buildings orto predict any other physical quantity that is a direct effect of wind because wind speedsare simply not high enough.

This chapter is divided into THREE sections1 general approaches used to describe fluid motion–Lagrangian vs. Eulerian descriptions2 overview of different types of flow: compressible vs. incompressible flows, viscous vs.

inviscid flows.3 introduction to an equation that establishes how pressures and velocities vary in a flow

field—the Bernoulli equation


Description of Fluid Motion

In analyzing fluid motion we need toI describe of physical quantities as a function of space and time coordinates,I introduce the different flow lines that are useful in describing a fluid flow,I present mathematical description of the motion.

Each fluid particle is a small mass of fluid. A large number of them occupies a smallvolume δV that moves with the flow. If fluid is

I incompressible: the volume does not change in magnitude but may deform.I compressible: volume changes its magnitude as the it deforms.

In both cases the particles are considered to move through a flow field as an entity.


Description of Fluid MotionLagrangian Description of Motion

In particle mechanics attention is focused on individual particles. Motion of eachparticle is observed as a function of space and time through these physicalquantities:

position: s(x0, y0, z0, t)

velocity: V(x0, y0, z0, t)

acceleration: a(x0, y0, z0, t)

The point (x0, y0, z0) locates the starting point of each particle.

This is the Lagrangian description, named after Joseph L. Lagrange (1736–1813).


Description of Fluid MotionEulerian Description of Motion

If we identify points in space and then observe the velocity of particles passing eachpoint; we can observe the rate of change of velocity as the particles pass each point(x, y, z), that is,

∂V/∂x, ∂V/∂y, ∂V/∂z

and we can observe if the velocity, at each particular point, is changing with time, t

∂V/∂t

The flow properties, such as velocity, are functions of both space and time. InCartesian coordinates the velocity is expressed as V = V(x, y, z, t). The region offlow that is being considered is called a flow field.

This is the Eulerian description of fluid motion, named after Leonhard Euler(1707–1783),


Description of Fluid MotionSteady Flow

If the quantities of interest do not depend on time, i.e.

V = V(x, y, z)

the flow is said to be a steady flow.

Most of the flows of interest in this introductory course are steady flows.

For a steady flow, all flow quantities at a particular point are independent of time,i.e.

∂V∂t

= 0∂p∂t

= 0∂ρ

∂t= 0 (1)

Note:I In general, properties of a fluid particle do vary with time.I In a steady flow, however, properties do not vary with time at a fixed point.


Description of Fluid MotionPathlines, Streaklines, and Streamlines

Pathline:I the locus of points traversed by a given particle as it travels in a field of flow;I provides us with a “history” of the particle’s locations.

Streakline:I an instantaneous line whose points are occupied by all particles originating from some

specified point in the flow field.I tells us where the particles are “right now.”

Streamlines:I a line in the flow possessing the following property: the velocity vector of each particle

occupying a point on the streamline is tangent to the streamline.I equation expressing the velocity vector is

V × dr = 0 (2)

cross product of two vectors in the same direction is zero!

Figure 1: Streamline in a flow field.


Description of Fluid MotionPathlines, Streaklines, and Streamlines

Streamtube:I a tube whose walls are streamlinesI since the velocity is tangent to a streamline, no fluid can cross the walls of a streamtubeI Examples:

F a pipe is a streamtube since its walls are streamlinesF an open channel is a streamtube since no fluid crosses the walls of the channel.


Description of Fluid MotionAcceleration

Figure 2: Velocity of a fluid particle.

A particular particle shown in Figure 2 had its velocity changes from V(t) at time tto V(t + dt) at time t + dt. The acceleration is, by definition,

a =dVdt

(3)

The velocity vector V is given in component form as

V = ui + vj + wk (4)

where (u, v,w) are the velocity components in the x-, y-, and z- directions,



With V = V(x, y, z, t), the chain rule from calculus gives

dV =∂V∂x

dx +∂V∂y

dy +∂V∂z

dz +∂V∂t

dt (5)

This gives acceleration as

a =dVdt

=∂V∂x

dxdt

+∂V∂y

dydt

+∂V∂z

dzdt

+∂V∂t

dtdt

(6)

since, for a particular particle,

dxdt

= udydt

= vdzdt

= w (7)

the acceleration is then

a = u∂V∂x

+ v∂V∂y

+ w∂V∂z

+∂V∂t

(8)



The scalar component equations of the vector Eq. 8 for Cartesian coordinates arewritten as the acceleration is then

ax =∂u∂t

+ u∂u∂x

+ v∂u∂y

+ w∂u∂z

ay =∂v∂t

+ u∂v∂x

+ v∂v∂y

+ w∂v∂z

az =∂w∂t

+ u∂w∂x

+ v∂w∂y

+ w∂w∂z

(9)

The time-derivative term on the right side of Eqs. 8 and 9 for the acceleration iscalled the local acceleration and the remaining terms on the right side in eachequation form the convective acceleration. Hence the acceleration of a fluid particleis the sum of the local acceleration and convective acceleration.



Eq. 3 allows us to simplify and write Eq. 8 as

a =DVDt

(10)

where

DDt

= u∂

∂x+ v

∂

∂y+ w

∂

∂z+∂

∂t(11)

The derivative in Eq. 11 is called the substantial derivative, or material derivative.Why?

The substantial derivative can be used with other dependent variables; forexample, DT/Dt would represent the rate of change of the temperature, T, of afluid particle as we followed the particle along.


Description of Fluid MotionAngular Velocity and Vorticity

As a particle travels along it may rotate or deform. There are certain flows, orregions of a flow, in which the fluid particles do not rotate and are referred to asirrotational flows.

Figure 3: Fluid particle occupying an infinitesimal parallelepiped at a particular instant.

A small fluid particle occupying an infinitesimal volume that has the xy-face asshown in Figure 3. The angular velocity Ωz about the z-axis is the average of theangular velocity of line segment AB and line segment CD.



The two angular velocities, counterclockwise being positive,

ΩAB =vB − vA

dx=

[v +

∂v∂x

dx2−(

v− ∂v∂x

dx2

)]/dx =

∂v∂x

(12)

ΩCD = −uD − uC

dy= −

[u +

∂u∂y

dy2−(

u− ∂u∂y

dy2

)]/dy = −∂u

∂y(13)

Hence, the angular velocity Ωz of the fluid particle is

Ωz =12

(ΩAB + ΩCD) =12

(∂v∂x− ∂u∂y

)(14)

Similarly, the angular velocity about the y-axis is

Ωy =12

(∂u∂z− ∂w∂x

)(15)

and x-axis

Ωx =12

(∂w∂y− ∂v∂z

)(16)



Vorticity ω is commonly defined to be twice the angular velocity;

ω = 2× Ω (17)

and its three components are

ωx =∂w∂y− ∂v∂z

ωy =∂u∂z− ∂w∂x

ωz =∂v∂x− ∂u∂y

(18)

An irrotational flow possesses no vorticity!

Figure 4: Irrotational and rotational flow.



The deformation of the particle of Figure 3 is the rate of change of the angle thatline segment AB makes with line segment CD. If AB is rotating with an angularvelocity different from that of CD, the particle is deforming.

The deformation is represented by the rate-of-strain tensor∗;its component εxy inthe xy-plane is given by

εxy =12

(ΩAB − ΩCD) =12

(∂v∂x

+∂u∂y

)(19)

For the xz-plane and the yz-plane we have

εxz =12

(∂w∂x

+∂u∂z

)εyz =

12

(∂w∂y

+∂v∂z

)(20)

Observe that the rate-of-strain tensor is symmetric:

εxy = εyx εxz = εzx εyz = εzy

* a mathematical object analogous to but more general than a vector, represented by an array of componentsthat are functions of the coordinates of a space (see https://en.wikipedia.org/wiki/Tensor).


Classification of Fluid FlowsOne-, Two-, and Three-Dimensional Flows

In Eulerian description of motion the velocity vector, in general, depends on threespace variables and time i.e. V = V(x, y, z, t) i.e. the flow is a three-dimensionalflow, because the velocity vector depends on three space coordinates.A two-dimensional flow is a flow in which the velocity vector depends on only twospace variables i.e. V = V(x, y).In axisymmetric flow the velocity vector would depend in r and θ i.e. V = V(r, θ);the flow in Figure 5 is considered two dimensional if described in a cylindricalcoordinate system. In Figure 5 the flow is normal to a plane surface; the fluiddecelerates and comes to rest at the stagnation point.

Figure 5: Two-dimensional, stagnation point flow.



A one-dimensional flow is a flow in which the velocity vector depends on only onespace variable i.e. V = V(x).Such flows occur away from geometry changes in long, straight pipes or betweenparallel plates, as shown in Figure 6.

I The velocity in the pipe varies only with r i.e. u = u(r).I The velocity between parallel plates varies only with the coordinate y i.e. u = u(y).

Figure 6: One-dimensional flow: (a) flow in a pipe; (b) flow between parallel plates.

When the velocity profile does not vary with respect to the space coordinate in thedirection of flow, the flows may be referred to as developed flows, as in Figure 6.



When the velocity, and other fluid properties, are constant over the area, the flow issaid to be a uniform flow, see Figure 7.

Figure 7: Uniform velocity profiles.


Classification of Fluid FlowsViscous and Inviscid Flows

An inviscid flow is one in which viscous effects do not significantly influence theflow and are thus neglected:

I we model an inviscid flow analytically by simply letting the viscosity be zero,I the primary class modeled as inviscid flows, is external flows i.e. a flow which exists

exterior to bodies,I of primary importance in flows around streamlined bodies, such as flow around an

airfoil or a hydrofoilI viscous effects are confined to a thin boundary layer attached to the boundary, see

Figure 8.

Figure 8: Flow around an airfoil.


Classification of Fluid FlowsViscous and Inviscid Flows

In a viscous flow the effects of viscosity are important and cannot be ignored:I include a broad class of internal flows, such as flows in pipes and conduits and in open

channels,I viscous effects cause substantial “losses” and account for the huge amounts of energy

that must be used to transport oil and gas in pipelines,I no-slip condition resulting in zero velocity at the wall and the resulting shear stresses

lead directly to these losses.


Classification of Fluid FlowsLaminar and Turbulent Flows

A viscous flow can be classified as either a laminar flow or a turbulent flow.In a laminar flow

I fluid flows with no significant mixing of neighboring fluid particles,I flow would not mix with the neighboring fluid except by molecular activity,I flow would retain its identity for a relatively long period of time,I viscous shear stresses always influence a laminar flow,I flow may be highly time dependent, due to the erratic motion of a piston as shown by

the output of a velocity probe in Figure 9(a) or it may be steady, see Figure 9(b).

Figure 9: Velocity as a function of time in a laminar flow: (a) unsteady flow; (b) steady flow.



In a turbulent flowI fluid motions vary irregularly so that quantities such as velocity and pressure show a

random variation with time and space coordinates,I physical quantities are often described by statistical averages and in this sense we can

define a “steady” turbulent flow as a flow in which the time-average physical quantitiesdo not change in time,

I instantaneous velocity measurements in an unsteady and a steady turbulent flow isshown in Figure 10.

Figure 10: Velocity as a function of time in a turbulent flow: (a) unsteady flow; (b) “steady”flow.



A flow is laminar or turbulent depending on three physical parameters describingthe flow conditions:

I length scale of the flow field, such as the thickness of a boundary layer or the diameterof a pipe, L,

I velocity scale such as a spatial average of the velocity, V; for a large enough velocity theflow may be turbulent,

I kinematic viscosity, ν; for a small enough viscosity the flow may be turbulent.

These three parameters are combined into a single parameter to predict the flowregime; it is called the Reynolds number, named after Osborne Reynolds(1842–1912), a dimensionless parameter, and defined as

Re =VLν

(21)

If Re is small, the flow is laminar; if Re is large, the flow is turbulent.This can be more precisely stated by defining a critical Reynolds number, Recrit, sothat

Re < Recrit : laminar flow



Recrit is different for every geometryI Recrit ≈ 2000 for a flow inside a rough-walled pipe,I Recrit ≈ 1500 for flow between parallel plates using the average velocity and the

distance between the plates.In a boundary layer on a flat plate, the length scale changes with distance from theupstream leading edge. A Reynolds number is calculated using the length x as thecharacteristic length. For a certain xT , Re becomes Recrit and the flow undergoestransition from laminar to turbulent.

I Recrit = 106 for a smooth rigid plate in a uniform flow with a low free-streamfluctuation level,

I Recrit = 3 × 105 for most engineering applications where a rough wall, or highfree-stream fluctuation level is assumed.

Figure 11: Boundary layer flow on a flat plate.



In an inviscid flow it is NOT appropriate to refer as laminar or as turbulent.

The inviscid flow is often called the free stream, which can be irrotational orrotational; most often it is irrotational.


Classification of Fluid FlowsIncompressible and Compressible Flows

An incompressible flow exists if the density of each fluid particle remains relativelyconstant as it moves through the flow field, i.e.

DρDt

= 0 (22)

examplesI atmospheric flow, in which ρ = ρ(z), where z is vertical,I flows that involve adjacent layers of fresh and salt water,I low-speed gas flows, such as the atmospheric flow referred to above.

Mach number, named after Ernst Mach (1838–1916), defined as

M =Vc

: where speed of sound c =√

kRT (23)

is used to decide whether a gas flow can be studied as an incompressible flow; forstandard air this corresponds to a velocity below about 100 m/s,

I If M < 0.3, density variations are at most 3% and the flow is assumed incompressible;I If M > 0.3, density variations influence the flow and compressibility effects should be

accounted for.


The Bernoulli Equation

In the derivation of Bernoulli equation it is assumed that viscous effects i.e. shearstresses

τ = µdudy

introduced by velocity gradients du/dy are neglected.The derivation starts with the application of Newton’s second law to a fluidparticle—summing up forces in the s-direction of motion:

p dA−(

p +∂p∂s

ds)

dA− ρ g ds dA cos θ = ρ ds dA as (24)

Figure 12: Particle moving along a streamline.



Acceleration of the particle in the s-direction, as, is given by

as = V∂V∂s

+∂V∂t

(25)

where ∂V/∂t = 0 because the flow is assumed steady. Also

dh = ds cos θ =∂h∂s

ds (26)

so that

cos θ =∂h∂s

(27)

Dividing by ds dA, and substituting for as and cos θ, Eq. 27 becomes

−∂p∂s− ρg

∂h∂s

= ρV∂V∂s

(28)



If density, ρ, is assumed constant and that

V∂V∂s

=∂( 1

2 V2)

∂s

then Eq. 28 can be re-written as

∂

∂s

(V2

2+

pρ

+ gh)

= 0 (29)

which is satisfied if, along the streamline,

V2

2+

pρ

+ gh = constant (30)

Note: The constant may have a different value on a different streamline.



Between two points on the same streamline,

V21

2+

p1

ρ+ gh1 =

V22

2+

p2

ρ+ gh2 (31)

This is the Bernoulli equation, named after Daniel Bernoulli (1700–1782).

Dividing Eq. 31 throughout by g leads to an alternative form:

V21

2g+

p1

γ+ h1 =

V22

2g+

p2

γ+ h2 (32)

Recap: Assumptions made in derivation of the Bernoulli equationI Inviscid flow (no shear stresses)I Steady flow (∂V/∂t = 0)I Along a streamline (as = V ∂V/∂s)I Constant density (∂ρ/∂s = 0)I Inertial reference frame



In the Bernoulli equation, the sums:

pγ

+ h : piezometric head

pγ

+V2

2g+ h : total head

and

p : static pressure

p + 12ρV2 = pT : total or stagnation pressure



A piezometer, Figure 13 (a), is used to measure static pressure, p, in a pipe.

Figure 13: Pressure probes: (a) piezometer; (b) pitot probe; (c) pitot-static probe.

A pitot probe, Figure 13 (b), is used to measure the total pressure, pT , in a fluidflow. Point 2 just inside the pitot tube is a stagnation point i.e. velocity there iszero.

A pitot-static probe is used to measure the difference between total and staticpressure, pT − p, with one probe, Figure 13 (c).



In Figure 13, the velocity at point 1 can be determined by applying the Bernoulliequation between points 1 and 2:

p1

γ+

V21

2g=

p2

γ(33)

where point 2 is a stagnation point so that V2 = 0. This gives

V1 =

√2ρ

(p2 − p1) (34)


Bibliography

1 MERLE C. POTTER, DAVID C. WIGGERT, BASSEM RAMADAN & TOM I-P. SHIH (2012):Mechanics of Fluids, 4ed, Cengage Learning (ISBN-13: 978-0-495-66773-5)

2 FRANK M. WHITE (2011): Fluid Mechanics, 7ed, McGraw-Hill (ISBN 0073529346)3 BRUCE R. MUNSON, THEODORE H. OKIISHI, WADE W. HUEBSCH & ALRIC P. ROTHMAYER

(2013): Fundamentals of Fluid Mechanics, 7ed, John Wiley & Sons (ISBN 9781118116135)4 YUNUS A. CENGEL & JOHN M. CIMBALA (2014): Fundamentals of Fluid Mechanics, 6ed,

McGraw-Hill (ISBN 9780073380322)5 JOHN F. DOUGLAS, ET AL. (2013): Fluid Mechanics, 5ed, Pearson (ISBN 9780131292932)6 DONALD F. ELGER, ET AL. (2013): Engineering Fluid Mechanics, 5ed, John Wiley & Sons

(ISBN 9781118164297)7 PHILIP J. PRITCHARD & JOHN C. LEYLEGIAN (2011): Fox and MacDonald’s Introduction to

Fluid Mechanics, 8ed, John Wiley & Sons (ISBN 9780470547557)


. . . must end

. . . and I end my presentation with two supplications

my Lord! increase me in knowledge(TAA-HAA (20):114)

O Allah! We ask You for knowledge that is of benefit(IBN MAJAH)

peace be upon youabuhasan/content/teaching/skmm2313/skmm... · 2017-11-01 · pathline: i the locus...

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