1. FLUID MECHANICS BASICS: A BRIEF REVIEW...1 1. FLUID MECHANICS BASICS: A BRIEF REVIEW Fluid...

NGM_JF006_1: Computational Fluid Dynamics Széchenyi University Instructor: D. Feszty, T. Jakubík Audi Hungaria Dept. of Whole Vehicle Development

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1. FLUID MECHANICS BASICS: A BRIEF REVIEW

Fluid Mechanics is the science dealing with the action of forces on fluids.

Fluids are substances, which flow freely. This is because particles in the fluid can easily move

and change their relative position.

See the PowerPoint slideshow called “FluidsI_Introduction.pdf” for examples on

application of Fluid Mechanics in mechanical, aerospace, biomedical, renewable energy,

civil engineering.

1.1. DEFINITIONS

1.1.1. Distinction between solids, liquids, gases

It is the molecular structure which distinguishes solids, liquids and gases:

Tab. 1. Properties of solids. liquids, fluids

Solids Liquids Gases

Molecular structure Lattice formation None None

Molecular motion No motion Free motion Free motion

Molecular spacing Constant & small Constant & small Varying & large

Volume Constant: defined by molecular spacing

Constant: defined by molecular spacing

Any: gas completely fills space

Volume defined Volume defined Any volume


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1.1.2. Fluid as a continuum

The action of forces on fluids can be determined either by:

- counting the effect of each and every molecule

OR

- considering the average effect of the molecules in a given volume, i.e. treating the fluid

as a continuum

Continuum can be assumed only when the number of molecules in a small volume (which is

much smaller than the body immersed in the fluid) is sufficiently great so that average effects

within the volume will be varying smoothly with time.

Example: in a volume as large as a dust particle, there are:

106 molecules of air 1 molecule of air at H = 0 m altitude (sea-level) at H = 305 km altitude

air CAN be treated air CANNOT be treated as continuum as continuum

Fig.1. Even solids can behave like fluid: think about sand in a sand glass or sand flowing out of your hand.

The behaviour of a collection of sand particles (which are solid particles themselves) satisfies the behaviour

of a liquid described in Table 1, as well as that of a continuum, hence it behaves as a fluid in these particular

cases. All laws of Fluid Mechanics will be valid for them.


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1.1.3. Branches of Fluid Mechanics

Hydrodynamics ( = const.)

study of fluids with virtually no density change ( = const.)

Examples: all liquids, low-speed gases (with speed less

than Mach 0.3, where Mach 1 is the speed of sound)

topic of Fluid Mechanics I

Fluid Mechanics Gas dynamics (varying )

fluids with significant density change ( ≠ const.)

Examples: high-speed gases (with speeds more than Mach

0.3)

topic of Fluid Mechanics II at many universities

Aerodynamics

study of external or internal air flows

Examples: air past aircraft, rocket engineering, cars, trains..

topic of Aerodynamics and Heat Transfer at aerospace

universities

1.1.4. Streams of Fluid Mechanics

Fluid statics (v = const.)

study of fluids in rest (v = const.)

Fluid Mechanics Fluid kinematics (v > 0, no forces considered)

study of fluids in motion (v > 0), but

WITHOUT considering the effects of associated forces

Fluid dynamics (v > 0 and forces considered)

study of fluids in motion (v >0)

WITH considering the effects of associated forces


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1.2. FLUID PROPERTIES

The physical condition of fluids is described by the fluid properties. These are expressed in

terms of a limited number of basic dimensions, such as length, mass (or force), time and

temperature, which in turn are quantified by basic units.

1.2.1. Units: the English and SI system

English (traditional) units:

Are based on the basic units of:

Mass [slug] or [lbm] (pound mass)

Length [ft]

Time [s]

Temperature [°F] (Fahrenheit) or [°R] (Rankine)

And all other properties are derived from these basic units, e.g.

Density [slug/ft3]

Pressure [psi] (pound per square inch)

etc.

SI (Systéme International, or also called metric) units:

Are based on the basic units of:

Mass [kg]

Length [m]

Time [s]

Temperature [K] (Kelvin) or [°C] (Celsius)

And all other properties are derived from these basic units, e.g.

Density [kg/m3]

Pressure [Pa] (Pascal) = [N/m2]

etc.

Also note that in the SI system, prefixes are used to indicate multiplication of units by

powers of 10, i.e.:

G (giga) = 109

M (mega) = 106

k (kilo) = 103

c (centi) = 10-2

m (mili) = 10-3

(micro) = 10-6


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Conversion between SI and English units:

1 ft = 0.3048 m

1 lbm = 0.453594 kg (Note: lbm = pound mass)

1 slug = 14.593903 kg

K = 273 + °C

°F = 5/9 °C + 32

°R = °F + 460

Note:

0 K is the lowest absolute temperature, the absolute zero (-

273.15 °C)

0 °C is the water freezing temperature

100 °C is the water boiling temperature

1 K = 1 °C 1 °R = 1°F

Abs. zero at: 0 K -273 °C 0 °R -460 °F

1 lbf = 4.4482216 N

Note: “lbf” stands for “pound force”

1 atm = 101,325 Pa

1 bar = 100,000 Pa

1 psi = 6,894.75 Pa

Comment about consistency:

never mix English and SI units in the same formula, i.e. [slugs] with [kg], etc.

always convert consistent units to the same base, e.g.

0.1 GPa + 10,000 Pa = 0.1 x 106 Pa + 10,000 Pa = 110,000 Pa

1.2.2. Properties: definitions

system: - is a system of particles or a given quantity of matter in Fluid Mechanics

extensive properties: - properties related to the total mass of the system

- represented by UPPERCASE letters, e.g. M, W, T, etc.

intensive properties: - properties independent of the amount of fluid

- represented by lowercase letters, e.g. p, , etc.


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1.2.3. Basic properties of fluids

Property Notation SI unit English unit

Mass M kg slug, lbm

Weight W N lbf

Volume V m3 ft3

Density kg/m3 slugs/ft3

Specific weight w N/m3 lbf/ft3

Temperature T K, °C °F, °R

Pressure p Pa = N/m2 psi, atm, bar

Viscosity N.s/m2 lbf.s/ft2

1.2.4. Definition of stress and pressure

Pressure is basically a specific type of stress. The general definition of stress is:

Force Stress = ------------------------------------ N/m2 or Pa Area over which force acts Depending on the direction of the force, we can distinguish between: Shear stress:

𝜏 =𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒

𝑎𝑟𝑒𝑎=

𝐹𝑇

𝐴

Normal stress:

𝜎 =𝑛𝑜𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒

𝑎𝑟𝑒𝑎=

𝐹𝑁

𝐴


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SHEAR STRESS NORMAL STRESS However, since fluids cannot sustain tensible stress, only the compressive stress can be considered for them, which is in fact called “pressure”: pressure = compressive stress

𝑝 =𝑛𝑜𝑟𝑚𝑎𝑙 𝑓𝑜𝑟𝑐𝑒

𝑎𝑟𝑒𝑎=

𝐹𝑁

𝐴

Moreover, fluids cannot sustain shear stress without motion, i.e. whenever shear stress is applied to fluids, motion occurs. This is the basic difference between FLUIDS and SOLIDS.

SOLID FLUID


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Formal definition of FLUID:

A FLUID is a substance, which cannot sustain shear stress statically, i.e. without motion.

1.2.5. The no-slip condition Notice that in the figure above, for a liquid point “A” remained stationary on the surface above and it is only point “B”, which moves due to the application of shear stress. This is due to the fact, that at solid boundaries the fluid and solid molecules “interlock” and there is no relative motion between them. This is called the “no-slip condition” in Fluid Mechanics.

The no-slip condition always occurs, no matter how “smooth” the solid surface is (there will always be some finite surface roughness). In other words, the no-slip condition means that in a real fluid the velocity at the wall is always zero. Note that often we will consider idealized fluids, in which we will neglect the no-slip condition. 1.2.6. Viscosity In plain terms, viscosity is essentially the measure of how “sticky” (or viscous) a fluid is. For example, honey is more viscous than water and water is more viscous than air. The scientific explanation of viscosity is the following:

When shear stress () is applied to a fluid, there will be a “velocity profile” in the fluid.


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A: stationary due to no-slip condition

B: moves due to shear stress ()

The velocity profile is proportional (denoted by the ≈ sign) to the shear stress () as:

𝑣𝐵 − 𝑣𝐴

∆𝑦 ≈ 𝜏

𝑑𝑣

𝑑𝑦≈ 𝜏

And the constant of proportionality is nothing else than viscosity () pronounced as “mu”.

𝜇𝑑𝑣

𝑑𝑦≈ 𝜏

There are two types of viscosity defined in Fluid Mechanics:

Dynamic (or absolute) viscosity: units in [N.s/m2]

Kinematic viscosity: 𝜗 = 𝜇

𝜌 units in [m2/s]

Note: the upper definition is called “dynamic” viscosity, because a force term (the Newton) occurs in the units. Recall from Sec. 1.4. that when we call something “dynamic” that means that we consider the forces acting on the object. In real life, the velocity profile v = f(y) is not linear as shown in the sketches above, but usually non-linear:

and this is called the “boundary layer” of the fluid


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In real life, all object on the ground, let them be stationary (e.g. buildings, skyscrapers, stadiums, wind turbines, etc.) or moving (cars, aircraft, trains, etc.) lie in the boundary layer of the Earth, when wind is blowing. A moving car or aircraft creates its own boundary layer around the surface. The boundary layer thickness around a moving car or aircraft is in the order of 5-20 mm, whereas the boundary layer thickness of Earth (due to wind blowing) can be as much as 100 m high. This is one reason why wind turbines are so tall: to place the actual rotors as high as possible, where the wind velocity is higher than close to the ground. 1.3. FLUIDS IN MOTION

In Fluid Kinematics or Fluid Dynamics, the knowledge of the velocity distribution throughout the

flowfield is imperative for determining the forces acting on the body immersed in the fluid.

For example, to determine the lift generated by an airplane wing, we need to know the velocity

distribution around it. Or, to determine the wind loads acting on a skyscraper or a wind turbine,

we need to know the velocity distribution along the building or the turbine blades.

In Fluid Mechanics, there are two ways to express the equations for fluids in motion: in the

Lagrangian or in the Eulerian viewpoint.

1.3.1. Lagrangian viewpoint

In this approach, individual fluid particles are considered and their motion “tracked”:

Position of fluid particle: 𝑟(𝑡) = 𝑥𝑖 + 𝑦𝑗 + 𝑧�⃗⃗�

with: x, y, z – coordinates

𝑖,𝑗, �⃗⃗� – unit vectors

Velocity of fluid particle: �⃗�(𝑡) =𝑑𝑟

𝑑𝑡=

𝑑𝑥

𝑑𝑡𝑖 +

𝑑𝑦

𝑑𝑡𝑗 +

𝑑𝑧

𝑑𝑡�⃗⃗�

with: u, v, w – velocity components in x, y, z coordinates


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To describe the entire flowfield, each and every particle has to be considered simultaneously. Good for rarefied gas dynamics (i.e. edge of space), where the number of particles is relatively small.

We don’t distinguish between individual particles. Instead, we consider their total effect in an elementary small volume (such as point P in space). Good for fluids, where the number of particles is large.

The above expressions are for one (1) individual particle only. For the entire flowfield, the

motion of ALL fluid particles has to be considered simultaneously. This is usually a very

challenging task for large number of particles, such as air at sea level or water. Hence, we

usually prefer to use the second approach…

1.3.2. Eulerian viewpoint

In this approach, we are focussing on a certain point in space and consider the motion of fluid

particles passing that point as time goes by.

Position in space: P[x, y, z]

Velocity at point P: u = f1(x, y, z, t)

v = f2(x, y, z, t)

w = f3(x, y, z, t)

1.3.3. Graphical illustration

The difference between the Lagrangian and Eulerian viewpoints can be illustrated graphically

as:

LAGRANGIAN EULERIAN

particle 1

particle 2

elementary small volume

inflow

w

outflow

w

dx

dy

dz


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1.3.4. Flow pattern

In Fluid Mechanics, the pattern of the flow is usually traced (or visualized) via either pathlines,

streaklines or streamlines. Their definition is:

PATHLINE: a line defining the path of a given particle of fluid (for example: recording

the positions of a floating cork on the surface of a river)

STREAKLINE: traces made by a dye or smoke injected at a given point in the flowfield

(e.g. smoke tunnel visualization)

STREAMLINE: a line to which velocity vectors are tangent at all points along itself

The difference between these three definitions can be illustrated on the following example:

Consider a 4 m/s velocity vector acting at a fixed point A in the coordinate system. Between

time 0-4.99 s, the velocity vector is horizontal, however, at time 5 s it turns by 70 degrees

upwards and stays in this direction for the remaining time. Then, the pathline, streakline and

streamline visualization would be:

PATHLINE: STREAKLINE: STREAMLINE:

t = 0 s

t = 4.5 s

t = 5.5 s

t = 10 s

A

A A A

A A A

A A

A A A

A’

A’

A’

A’

A’

A’


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STREAMTUBE: In Fluid Mechanics, we frequently refer to the term “streamtube”, which

means a fictitious tube with walls composed not by solid material, but by

streamlines. This means – just like for solid tubes – no flow through the

walls since streamlines are curves to which velocity is always tangent.

1.3.5. Uniform and non-uniform flow (variation of flow in space)

UNIFORM FLOW: no change in the velocity from point to point along any of the streamlines.

This means that streamlines must be straight and parallel.

𝜕�⃗⃗�

𝜕𝑠= 0

Note: “s” is direction along flow between parallel plates open channel flow Streamline (no friction) (no friction)

streamline

streamtube A wind turbine’s wake can be considered to

be a streamtube.

water surface

Source: http://electrical-engineering-

portal.com/aerodynamics-of-horizontal-axis-wind-

turbines

http://electrical-engineering-portal.com/aerodynamics-of-horizontal-axis-wind-turbines




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NON-UNIFORM FLOW: change of velocity along the streamline

(typically means non-parallel or non-straight streamlines)

𝜕�⃗⃗�

𝜕𝑠≠ 0

converging flow vortex flow (change of velocity (change of velocity magnitude along direction along

streamline) streamline) 1.3.6. Steady vs. unsteady flow (variation of flow in time) STEADY FLOW: at any given point, the velocity does NOT vary with time in magnitude or

direction. Example: airfoil at low angle of attack.

𝜕�⃗�

𝜕𝑡= 0

Computational streaklines (left) and smoke traces in a wind tunnel (right) around an airfoil. Source: hu.wikipedia.org & NASA


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UNSTEADY FLOW: at a given point in the flowfield, the velocity does vary by time. Example: airfoil at high angle of attack.

𝜕�⃗�

𝜕𝑡≠ 0

Source: NASA


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1.3.7. Laminar vs. turbulent flow LAMINAR FLOW: a very “organized” flow, in which fluid “layers” shear on each other. TURBULENT FLOW: characterised by mixing action throughout the flowfield, caused by eddies

of varying size within the flow. Turbulent flow is essentially an unsteady flow but due to the small-scale

of eddies we usually look at average effects and so we consider it steady flow.

1.3.8. 1D, 2D and 3D flow The most general representation of a fluid particle’s velocity is by 3 components: [u, v, w]. 1D flow: flow conditions change along one dimension only and are constant over

any plane perpendicular to the streamlines.

Source: http://www.sailtheory.com/resistance.html

http://www.sailtheory.com/resistance.html


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2D flow: flow conditions change along 2 dimensions only (third component of velocity is zero) flow in this plane is essentially 2D infinitely long wing 3D flow: flow conditions change in all 3 dimensions. For example, flow around a

skyscraper or a racecar.

Source: http://www.cg.tuwien.ac.at/research/vis/Miscellaneous/ESIS/

Note: Due to viscous effects (no-slip condition at the wall, Sec. 2.5 – 2.6) real flows are not truly 1D, but it is often sufficient to assume they are and the use of mean values in each plane is sufficient.

Source: http://scmfastslow.blogspot.ca/2010/11/aerodynamics.html

Source: http://www.5dt.com/products/ifloviz04.html &

http://en.wikipedia.org/wiki/Lift_(force)

http://www.cg.tuwien.ac.at/research/vis/Miscellaneous/ESIS/

http://scmfastslow.blogspot.ca/2010/11/aerodynamics.html

http://www.5dt.com/products/ifloviz04.html

http://en.wikipedia.org/wiki/Lift_(force)


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1.4. CONSERVATION LAWS In Fluid Mechanics, the vast majority of theories and methods are based on the following 3 conservation laws:

- conservation of mass - conservation of momentum - conservation of energy

These 3 principles will form the fundamentals of Computational Fluid Dynamics (CFD) too. We will derive them specifically for CFD purposes in the next Chapter, however, for now let us recall their meaning and the form in which we have learned them in Fluid Mechanics I. 1.4.1. Control Volume Recall that the conservation laws were applied to a Control Volume in Fluid Mechanics. Control Volume is a specific, suitably chosen volume, in which we monitor or “control” the behaviour of fluid. (Note the analogy to the Transit Zone at an Airport – we control who can get in and who gets out and as such, this is a “controlled zone” of people. A “controlled zone” for fluid particles – what we call a Control Volume - is the same: we monitor in it that what goes in and what goes out.) It can be large or small, and of any shape. In CFD, we will use thousands or millions of very small hexahedral (cube) or tetrahedral (pyramid) shape Control Volumes.

Control Volume of hexahedral shape Control Volume of tetrahedral shape

Advantage: we only need to know the properties at the edges of the Control Volume, not inside. This allows us to get around the complexities of the objects or machines inside the Control Volume. Example: if we put a Control Volume (CV) around a jet engine and we know what is the inflow and outflow of the CV, we can calculate the thrust of the engine even without considering the complexities of the jet engine itself.


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Aircraft Jet engine in a Control Volume.

1.4.2. Conservation of Mass This law essentially says: mass outflow – mass inflow = decrease of mass in CV net rate of mass in CV For a Control Volume as the one shown below, the conservation of mass will take the form (see Fluid Mechanics I for full derivation):

Control Surface (CS)


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For steady flows (𝜕

𝜕𝑡= 0):

For uniform 1D flow (replace the integral with summation) :

… in other words (for steady, 1D flows), what goes in, must come out: fluid (or mass) cannot disappear!

1.4.3. Conservation of Momentum This law is based on Newton’s 2nd law:

�⃗� = 𝑚�⃗� = 𝑚𝑑�⃗⃗�

𝑑𝑡

and it allows to determine the force (F) which the fluid exerts on its surroundings. For the Control Volume shown earlier, it will take the form of (see Fluid Mechanics I for derivation):

fluid velocity relative to the CV surface, i.e. the inflow or outflow velocity from the Control Volume fluid velocity relative to an inertial reference frame. Makes a big difference for moving Control Volumes


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For steady (𝜕

𝜕𝑡= 0) and uniform flows (replace integral with summation) it becomes:

1.4.4. Conservation of energy Allows to determine the power generated by fluid as well as the effect of heating or cooling the fluid.

For the Control Volume shown earlier and by adding heat Q:

By splitting the power term (�̇�) to rate of flow work (𝑊�̇�) and rate of shaft work (𝑊𝑠̇ ):

and assuming steady (𝜕

𝜕𝑡= 0) one dimensional flow (replace integral with summa) we get:

Heat addition (negatiue if heat is

generated: 𝑑𝑄

𝑑𝑡= �̇�

Control Surface (CS)

p1, T1

p2, T2


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1.5. SOLUTION ALONG A STREAMLINE We can derive useful solutions for fluid flows along a streamline too, i.e. without using the Control Volume method. Such solution allows us to assess the fluid properties between two points along a streamline.

The most widely used method for a solution along a streamline is the Bernoulli equation, which

was derived for the following assumptions:

- incompressible fluid (constant density, i.e. = const)

- inviscid fluid (zero viscosity, i.e. = 0)

- no energy added or extracted between the two points

- losses due to friction are negligible


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static hydrostatic dynamic pressure pressure pressure

Also recall that:

static pressure (p) pressure, which one would feel when moving along with

the particles in the fluid

dynamic pressure pressure associated with the motion of the fluid

stagnation pressure (p0) pressure, which would occur when the fluid is brought to

rest. For incompressible fluid (note: this is NOT valid for

compressible fluid)

Bernoulli equation essentially says that for a fluid where elevation effects are negligible

(gz along a streamline is constant), the stagnation pressure (p0) along a streamline is

constant. Warning!

- It is a common mistake for students to use the Bernoulli equations for any case in Fluid Mechanics. This is wrong! Bernoulli equation can only be used for the conditions listed among the assumptions made above, i.e. incompressible, inviscid, 1D flow.

- It is a big mistake to apply it to compressible flows or to the flow across

machines, which generate or extract energy, such as wind turbines, propellers, pumps, jet engines, etc!!

With this, we have completed the review of the relevant fundamentals from Fluid Mechanics. When we apply the conservation of mass, momentum and energy to a small fluid element, then we obtain the governing equations of Computational Fluid Dynamics, the so-called Navier Stokes equations. These will be derived in the next Chapter.

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