Lectures 1-2 Introduction Fluid Mechanics1

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  • 7/27/2019 Lectures 1-2 Introduction Fluid Mechanics1

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    Lectures 1 & 2

    INTRODUCTION: CONCEPTS AND DEFINITIONS(Fluid Mechanics; 4th edition; Frank M. White)

    Dr. Vinh Q. Tang

    [email protected]

    Office: ME2186

    MAAE 2300 - Fluid Mechanics

    Department of Mechanical and Aerospace Engineering

    Carleton University

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    Contents

    1. Basic Concepts & Definitions

    2. Dimensions and Units

    3. Properties of the Velocity Field4. Thermodynamic Properties

    5. Transport Properties

    6. Flow Patterns

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    1. Basic Concepts & Definitions

    Fluid Mechanics - Study of fluids at rest, in

    motion, and the effects of fluids on

    boundaries

    Fluid - A substance which moves and deforms

    continuously as a result of an applied shear

    stress

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    Continuum Concept

    Fluids consist of discrete particles, the molecules.But for engineering purpose, we only needaverage effects due to many molecules.

    We can assume that fluids (and solids) arecontinua, i.e., continuous distributions ofmatter

    This is satisfactory if the mean free path, , of theparticle is much less than the significant length of our

    problem. Example for hydrogen at 150C and 1 atm: =1.8x10-7 m, so

    for many practical problems continuum assumption is ok.

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    Pressure and Velocity

    Two important properties in the study of fluid mechanics:

    Pressure: The normal stress on any plane through a

    fluid element at rest The direction of pressure forces will always be

    perpendicular to the surface of interest.

    Velocity: The rate of change of position at a point in aflow field.

    It is used to specify flow field characteristics; flow rate;momentum; and viscous effects for a fluid in motion

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    2. Dimensions and Units

    Primary Dimensions in SI and BG Systems SI: International System of units

    BG: British gravitational units

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    Secondary Dimensions in

    Fluid Mechanics

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    Other Units

    S.I.: 1 Newton (N) = 1 kg m/s2

    British Gravitational System: 1 lbf = 1 slug ft/s2

    (1 slug = 32.174 lbm)

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    Unit Consistency

    Units in an equation must be consistent Example 1Mechanical Energy = Kinetic Energy + Potential Energy

    2

    2

    1mV mgzME

    )550)(81.9)(1.7()23)(1.7(21 2

    2 cmsmkg

    hrkmkgME

    Substituting given values of parameters

    As shown the units are not consistent.

    Must convert

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    )100

    1)(550)(81.9)(1.7()1000()sec3600

    ()23)(1.7(21

    2222

    cmmcm

    smkg

    kmmhr

    hrkmkgME

    )550)(81.9)(1.7()23)(1.7(2

    12

    2

    cms

    mkghr

    kmkgME

    Recall:

    Converting units:

    2222 /.383/.9.144 smkgsmkg

    )/.1)(/.9.527(2

    22

    smkg

    N

    smkg

    mN.9.527

    N.m1J1where9.527 J

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    Other Notes on Units

    Weight is a force (but mass is not) Force is defined by the acceleration that it

    produces on a standard mass:

    On earth 1kg weighs:

    amF .)1).(1(1

    2s

    mkgN

    )81.9)(1()).(1( 2s

    mkggkg earth

    N81.9

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    English system:

    On earth 1lbm weighs:

    To give 1 slug an acceleration of 1 ft/sec2 requires a force of 1 lbf

    )sec

    2.32).(1(12

    ftlblb

    mf

    )sec2.32).(1()).(1( 2ft

    lbglb mearthm

    flb1

    definitionby2.321m

    lbslug

    mlbslug 2.321

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    2

    sec

    .2.321ft

    lblbmf

    2sec.11ft

    sluglbf

    From previous slide:

    mlbslug 2.321 And :

    Thus :

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    Example 2

    Given: Following pump power requirements

    Determine: The power required in kW

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    Recall

    Substitute values and introduce conversions, we

    get:

    (Note: We used 1 lbf = 1 slug . ft/s2

    )

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    Tips

    At the start of the problem, convert all

    parameters with units to the base units being

    used in the problem, e.g.:

    For S.I. problems, convert all parameters to kg,

    m, and s

    For BG problems, convert all parameters to slug,

    ft, and s Then convert the final answer to the desired

    final units.

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    3. Properties of the velocity Field (2)

    Mass flow rate

    Note: Vn is measured relative to the

    moving boundary

    Vn : the component of the

    velocity normal to the

    surface area across which

    the fluid flows

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    4. Thermodynamic Properties

    The usual thermodynamic properties are also

    important in fluid mechanics

    P : Pressure (kPa, psi)

    T : Temperature (0C, 0F)

    : Density (kg/m3, slug/ft3)

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    State Relations for Gases

    All gases at high temperatures and low pressures

    (relative to their critical point) are in good agreement

    with theperfect-gas law

    Each gas has its own constant R, equal to a universal constant

    divided by the molecular weight

    where = 49,700 ft2/(s2.R) = 8314 m2/(s2.K).

    Most problems in this class are for air, with M = 28.97

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    Standard Air Density

    Recall:

    Substitute values for and M(for air), we get:

    Density can be determined from

    Where, standard atmospheric pressure is 2116 lbf/ft2, and

    standard temperature is 60F or 520R. Thus standard airdensity is

    P= RT

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    Specific heats

    As a first approximation in airflow analysis we

    commonly take cp, cv, and k to be constant The ratio of specific heats of a perfect gas : ; kair=1.4

    Actually, for all gases, cp and cvincrease gradually withtemperature, and k decreases gradually

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    State Relations for Liquids

    Liquids are nearly incompressible and have a

    single reasonably constant specific heat

    Thus an idealized state relation for a liquid is

    Water is normally taken to have a density of 1.94slugs/ft3 and a specific heat cp = 25,200 ft

    2/(s2..R).

    The steam tables may be used if more accuracyis

    required.

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    The density of a liquid usually decreases slightly with

    temperature and increases moderately with

    pressure.

    If we neglect the temperature effect, an empirical

    pressure-density relation for a liquid is:

    where B and n are dimensionless parameters which vary

    slightly with temperature and pa anda are standard

    atmospheric values. Water can be fitted approximately to

    the values B 3000 and n 7.

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    Alternative for Density

    : Specific Weight (weight per unit volume)(N/m3, lbf/ft3)

    For H2O: = 9790 N/m3 = 62.4 lbf/ft3

    For Air: = 11.8 N/m3 = 0.0752 lbf/ft3

    S.G. : Specific Gravity = / (ref)

    (ref)= (water at 1 atm, 20C) for liquids = 998 kg/m3

    = (air at 1 atm, 20C) for gases = 1.205 kg/m3

    = g

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    Example 3

    Determine the static pressure difference indicated by

    an 18 cm column of fluid (liquid) with a specific

    gravity of 0.85.

    P = g h= (SG) . . h

    = (0.85) (9790 N/m3 ) (0.18 m)

    = 1498 N/m2

    = 1.5 kPa

    Note:

    See note

    below

    g

    gSG

    liquid

    water

    liquid

    .)(SG).(

    .water

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    5. Transport Properties

    e.g.:

    coefficient of viscosity (dynamic viscosity)

    {M / L t }

    kinematic viscosity ( / ) { L2 / t }

    They relate to the diffusion of momentum due to

    shear stresses, as seen in following slides

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    Stress and Pressure

    AF

    AreaForceStress :

    Area

    ForceTangentialstress, Shear

    Area

    ForceNormalStress, Normal

    A

    ve)(compressiFStresseCompressiv:Pr essure

    Fluid cannot

    sustain tensile

    stress

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    A solid can resist a shear stress by a static

    deformation; a fluid cannot

    Any shear stress applied to a fluid, no matter

    how small, will result in motion of that fluid

    The fluid moves and deforms continuously as

    long as the shear stress is applied.

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    Newtonian fluid

    A fluid which has a linear relationship between shear

    stress and velocity gradient

    Note: The linearity coefficient in

    the equation is the coefficient

    of viscosity

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    Classic Problem: Viscous flow induced by

    relative motion between two parallel plates

    Note: No slip at either plate

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    No-Slip Condition

    At solid boundaries, the solid and fluid

    molecules interlock, and there is no relative

    velocity between the fluid and the solid (i.e.,

    no slip) Its based on empirical fact, no matter how

    smooth the solid surface is. Exceptions are for

    conditions at very low pressures, such as inouterspace.

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    Surface Tension

    Molecules deep within the liquid repel eachother because of their close packing

    Molecules at the surface are less dense and

    attract each other. Since half of theirneighbors are missing, the mechanical effect isthat the surface is in tension

    We can account adequately for surface effectsin fluid mechanics with the concept of surfacetension

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    Surface tension acts in the plane of the liquid

    surface, and its magnitude measured in N/m (see

    Example 1.9) The two most common interfaces are water-air and

    mercury-air

    For a clean surface at 20C 68F, the measuredsurface tension is:

    0.0050 lbf/ft = 0.073 N/m air-water

    0.033 lbf/ft = 0.48 N/m air-mercury

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    Pressure change across a curved interface due to

    surface tension

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    Flow regions

    Flows constrained by solid surfaces can typically bedivided into two regimes:

    A) Flow near a bounding surface with1. significant velocity gradients

    2. significant shear stresses This flow region is referred to as a "boundary layer

    B) Flows far from bounding surface with

    1. negligible velocity gradients2. negligible shear stresses

    3. significant inertia effects

    This flow region is referred to as "free stream" or "inviscidflow region"

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    Reynolds number (Re)

    An important parameter in identifying the characteristics ofthe flow regions

    This physically represents the ratio of inertia forces in theflow to viscous forces

    For most flows of engineering significance, both thecharacteristics of the flow and the important effects due tothe flow, e.g., drag, pressure drop, aerodynamic loads, etc.,are dependent on this parameter

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    Laminar Flow & Turbulent Flow

    Very low Re indicates viscous creeping motion

    where inertia effects are negligible

    Moderate Re implies a smoothly varying

    laminarflow

    High Re probably spells turbulent flow

    which is slowly varying in the time-mean but has

    superimposed strong random high-frequency

    fluctuations

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    Low Re : laminar flow

    High Re : turbulent flow

    Moderate Re : transition flow

    Flow issuing at constant speed from a pipe:

    (a) high viscosity, low-Reynolds-number, laminar flow;

    (b) low-viscosity, high-Reynolds-number, turbulent flow.

    (a) (b)

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    Formation of a turbulent puff in pipe flow:(a)and (b) nearthe entrance;

    (c) somewhat downstream;

    (d) far downstream

    (a)

    (b)

    (c)

    (d)

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    6. Flow Patterns

    Fluid flow is generally three-dimensional

    Pressures and velocities change in all directions

    But, one or two-dimensional flow analysis can

    be useful for many practical applications

    Where the changes are most significant in one or

    two directions only

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    One-Dimensional Flow

    Conditions vary only in the direction of flow notacross the cross-section

    In certain pipe flow application we can assume 1-D flow

    (Average or mean velocity over the cross section is used)

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    Streamlines & Streamtube

    The most common method of flow-pattern presentation:(a)Streamlines are everywhere tangent to the local velocity

    vector

    (b)Streamtube is formedby a closed collection of streamlines

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    Streamline over an aerofoil an example

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    Examples

    Example 1.1

    Example 1.3

    Example 1.7

    Example 1.9