Fluid Mechanics

Question 1

Mechanics -

Answer 1

Mechanics - physical science that deals with both stationary and moving bodies under the influence of forces.
The branch of mechanics that deals with bodies at rest is called statics, while the branch that deals with bodies in motion is called dynamics.

Question 2

Fluid mechanics -

Answer 2

Fluid mechanics is the science that deals with the behaviour of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries.

Question 3

Hydrodynamics -

Hydraulics -

Answer 3

The study of the motion of fluids that are practically incompressible (such as liquids, especially water, and gases at low speeds) is usually referred to as hydrodynamics.
A subcategory of hydrodynamics is hydraulics, which deals with liquid flows in pipes and open channels.

Question 4

Gas dynamics

Aerodynamics
Propulsion

Meterology
Oceonography
Hydrology

Answer 4

Gas dynamics deals with the flow of fluids that undergo significant density changes, such as the flow of gases through nozzles at high speeds.

The category aerodynamics deals with the flow of gases (especially air) over bodies such as aircraft, rockets, and automobiles at high or low speeds.
Propulsion (Internal Flows)

Meterology
Oceonography
Hydrology

Question 5

stress -
types

Fluid -

a fluid at rest is at a state of ______ shear stress

No-slip condition -

Boundary Layer -

Flow separation -

No temp. jump -

Brief History of Fluid Mechanics

Answer 5

stress - resistance developed as a result of applied force
tangential or shear stress
Normal stress

Fluid - Substance that continues to undergo deformation no matter how small the shear force is applied.

a fluid at rest is at a state of zero shear stress

No-slip condition - relative velocity of fluid in contact with solid is zero because of viscosity

Boundary Layer - the region near the surface where viscosity effects are significant

Flow separation - in regions of adverse pressure gradient flow separates from surface

No temp. jump - a fluid and a solid surface have the same temperature at the points of contact

Question 6

Fluid Flow Classification
Viscous Flow vs Inviscid Flow
Internal Flows vs External Flows -

Laminar Flow vs Turbulent Flow - 
Natural vs Forced Flow - 

Steady Flows vs Unsteady Flows
Uniform Vs Non-uniform

System, Boundary
    Closed system (Control Mass System), Open System (Control Volume)

Answer 6

Fluid Flow Classification
Viscous Flow vs Inviscid Flow
Internal Flows vs External Flows - flow is completely bounded vs flow is unbounded
Laminar Flow vs Turbulent Flow -
Natural vs Forced Flow - like flow due to buoyancy effects vs flow due to fan
Steady Flows vs Unsteady Flows
Uniform Vs Non-uniform

System, Boundary
   Closed system (Control Mass System), Open System (Control Volume)

Question 7

Experimental approach -

Analytical Approach -

Accuracy -
Precision -

Property -
Intensive -
Extensive -

State of a system -

Continuum -

Specific properties -
Specific weight -
Specific gravity -

EOS -

at ______ pressure and _____ temperature gas behaves as ideal gas

Answer 7

Experimental approach - expensive, time-consuming & often impractical but closest approximation to reality within limits of experimental error
Analytical Approach - fast, inexpensive but accuracy depends on assumptions & model used

Accuracy - nearness to true value
Precision - repeatability

Property - any characteristic of a system
Intensive - nature of system on mass/extent of system P, T
Extensive - mass of system not on nature m, V

State of a system - The number of properties required to fix the state of a system is given by the state postulate: The state of a simple compressible system is completely specified by two independent, intensive properties.

Continuum - a continuous, homogeneous matter with no spacing, that is, a continuum

Specific properties - per unit mass
Specific weight - rho-g
Specific gravity - ratio of density of fluid to that of water or air

EOS - equation that relates P, T & rho
like Ideal gas equation
at low pressure and high-temperature gas behaves as ideal gas

Question 8

Saturation T -

Saturation P -

Vapour Pressure -

Partial Pressure -

Cavitation -

Answer 8

One to one correspondence between P & T during phase change process
Saturation T - At a given pressure, the temperature at which a pure substance changes phase is called the saturation temperature T sat .
Saturation P - At a given temperature, the pressure at which a pure substance changes phase is called the saturation pressure P sat.

Vapour Pressure - the pressure exerted by a vapor in phase equilibrium with its liquid, at a given temperature.
Partial Pressure - Partial pressure is defined as the pressure of a gas or vapor in a mixture with other gases
The rate of evaporation from open water bodies such as lakes is controlled by the difference between the vapor pressure and the partial pressure.
Cavitation - when the pressure in liquid-flow systems drops below the vapor pressure at some locations (tip of impeller or suction sides of pumps) it results in vaporization and formation of bubbles, which are swept away from the low-pressure regions and on collapsing generates highly destructive high-pressure waves. It is a common cause for drop-in performance and even the erosion of impeller blades.

Question 9

Energy -

Microscopic energy -

internal energy of a system.

Macroscopic energy -

Mechanical energy

Heat -

Sensible energy -

Latent energy -

calorie (1 cal = 4.1868 J) -

Simple Compressible Systems -

Answer 9

Energy - exist in various forms such as thermal, mechanical, kinetic, potential, electrical, magnetic, chemical, and nuclear
Microscopic energy - energy related to the molecular structure and the degree of the molecular activity of a system.
Thermal (sensible, latent), chemical, Nuclear energy
The sum of all microscopic forms of energy is called the internal energy of a system.

Macroscopic energy - related to motion and influence of external effects such as gravity, magnetism, electricity.
K.E, P.E

Mechanical energy (flow+k.e+p.e) - can be converted to mechanical work
Heat - energy transfer due to temp. difference
Sensible energy - that portion of internal energy associated with the activeness of molecules & is proportional to temperature
Latent energy - that portion of internal energy associated with the molecular arrangement or phase change 

calorie (1 cal = 4.1868 J) - energy needed to raise the temperature of 1 g of water at 14.5°C by 1°C.

Simple Compressible Systems - absence of magnetic, electric and surface tension effects

Question 10

Bulk modulus of elasticity or coefficient of compressibility -

Isothermal compressibilty (alpha) (inverse of Coefficient of Compressibility) -

Water hammer -

Coefficient of Volume expansion (beta) -

Answer 10

Bulk modulus of elasticity or coefficient of compressibility - change in pressure corresponding to fractional change in volume at constant T. Infinite for incompressible fluid
Isothermal compressibilty (alpha) (inverse of Coefficient of Compressibility) - fractional change in volume corresponding to a unit change in pressure for constant T

Water hammer - when fluid flow encounters abrupt flow restriction (such as sudden closure of valve) it is locally compressed producing acoustic waves which resemble the sound produced when a pipe is hammered. It is very dangerous & can even damage the pipeline.

Coefficient of Volume expansion (beta) - fractional change in volume corresponding to unit change in T at const P

Question 11

viscosity -

Newtonian fluid -

The viscosity of liquids _______ and the viscosity of gases ______ with increase in temperature

kinetic theory of gases predicts the viscosity of gases to be proportional to the _____________

surface tension -

why liquid droplets take spherical shape.

capillary effect -

if cohesive forces _____ than adhesive forces - fall in capillary height, contact angle will be _____ 90 degree

Answer 11

viscosity - internal resistance of a fluid to motion. cohesive forces between molecules & molecular momentum transfer
Newtonian fluid - stress is proportional to strain rate
The viscosity of liquids decreases and the viscosity of gases increases with increase in temperature
kinetic theory of gases predicts the viscosity of gases to be proportional to the square root of temperature

surface tension - the surface of the liquid acts like a stretched elastic membrane under tension due to cohesive forces between the molecules.
Net attractive force acting on the molecule at the surface of the liquid tends to pull them towards the interior of the liquid thereby compressing them which causes the liquid to minimize its surface area. For a given volume min. surface area is that of a sphere so liquid droplets take spherical shape.

capillary effect - the rise or fall of a liquid in small-diameter tube inserted into the liquid
if cohesive forces greater than adhesive forces - fall in capillary height, contact angle will be >90
if cohesive forces less than adhesive forces - rise in capillary height, contact angle will be <90

Question 12

Pressure -
Pressure is the _______ force per unit area, & is a ______________
Absolute Pressure -

Gauge Pressure -

Vacuum pressure -

Pressure at a point -

Pascal’s Law -

Pressure measuring devices -

Barometer -
Atmospheric pressure values -

Answer 12

Pressure - normal force exerted by a fluid per unit area
Pressure is the compressive force per unit area, & is a scalar
Absolute Pressure - actual pressure at a given position is called the absolute pressure, and it is measured relative to absolute vacuum (i.e., absolute zero pressure)
Gauge Pressure - difference between absolute pressure and local atmospheric pressure
Vacuum pressure - pressure below atmospheric pressure & is the difference between atmospheric pressure & absolute pressure
Pressure at a point - the pressure at a point in a fluid has the same magnitude in all directions (from force balance)
Variation of pressure with depth - rhogh (this principle used in manometer)
Pascal’s Law - the pressure applied to a confined fluid increases the pressure throughout by the same amount
Hydraulic brakes & lifts are based on this principle

Pressure measuring devices - manometer, bourdon tube, pressure transducers
Barometer - measures atmospheric pressure
Atmospheric pressure values - 1.01325 bar, 101.325 kPa, 1 atm, 760mm of Hg, 760 torr are same. 1 bar = 100kPa

Question 13

Fluid statics -

Hydrostatic forces on submerged plane surfaces -

centre of pressure -

Hydrostatic forces on submerged curved surfaces

Buoyancy -

Archimedes principle -

Answer 13

Fluid statics - forces on fluid at rest (no shear force only pressure force)

Hydrostatic forces on submerged plane surfaces -
The magnitude of the resultant force acting on a plane surface of a completely submerged plate in a homogeneous (constant density) fluid is equal to the product of the pressure Pc at the centroid of the surface and the area A of the surface.
centre of pressure - intersection between line of action of resultant force and surface

Hydrostatic forces on submerged curved surfaces

1. The horizontal component of the hydrostatic force acting on a curved surface is equal (in both magnitude and the line of action) to the hydrostatic force acting on the vertical projection of the curved surface.
2. The vertical component of the hydrostatic force acting on a curved surface is equal to the hydrostatic force acting on the horizontal projection of the curved surface, plus (minus, if acting in the opposite direction) the weight of the fluid block.

Buoyancy - fluid exerts an upward force on a body immersed in it that tends to lift the body
The buoyancy force is caused by the increase of pressure in a fluid with depth
Weight of the fluid displaced by the fluid
It is independent of the density of solid and its distance from free surface.
Archimedes principle - The buoyant force acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body, and it acts upward through the centroid of the displaced volume.

Question 14

Stability
stable -
neutrally stable -
unstable -

stability of immersed body
inherently stable in the vertical direction
rotational stability of an immersed body depends on the relative locations of the center of gravity G of the body and the center of buoyancy B (i.e centroid of displaced volume)

stable
unstable
neutrally stable

What about a case where the center of gravity is not vertically aligned with the center of buoyancy ?

Answer 14

Stability
stable - any small disturbance generates a restoring force that returns it to its initial position.
neutrally stable - if someone moves the ball, it would stay at its new location. It has no tendency to move back to its original location, nor does it continue to move away
unstable - any disturbance, even an infinitesimal one, body does not return to its original position; rather it diverges from it.

stability of immersed body
For an immersed or floating body in static equilibrium, the weight and the buoyant force acting on the body balance each other, and such bodies are inherently stable in the vertical direction
rotational stability of an immersed body depends on the relative locations of the center of gravity G of the body and the center of buoyancy B (i.e centroid of displaced volume)

centre of G is directly below centre of Buoyancy - stable
a stable design for a submarine calls for the engines and the cabins for the crew to be located at the lower half in order to shift the weight to the bottom as much as possible. Hot-air or helium balloons (which can be viewed as being immersed in air) are also stable since the cage that carries the load is at the bottom

center of gravity G is directly above point B - unstable
A body for which G and B coincide - neutrally stable

What about a case where the center of gravity is not vertically aligned with the center of buoyancy (Fig. 3–45)? It is not really appropriate to discuss stability for this case since the body is not in a state of equilibrium. In other words, it cannot be at rest and would rotate toward its stable state even without any disturbance.

Question 15

stability for floating body
metacentric height
metacentre - 
stable - 
unstable - 

Fluids in rigid body motion

rotation of cylinder -

Answer 15

stability for floating body
measure of stability for floating bodies is the metacentric height GM
distance between the center of gravity G and the metacenter M
metacentre - the intersection point of the lines of action of the buoyant force through the body before and after rotation
stable - if point M is above point G i.e GM +ve
unstable - if point M is below point G i.e GM -ve
a boat can tilt to some maximum angle (20 degree) without capsizing, but beyond that angle, it overturns (and sinks)

Fluids in rigid body motion

• eg transportation in tankers or rotation of cylinder (no motion between fluid layers relative to each other)
• equation of motion for rigid body
• acceleration on a straight path
• free surface & const. pressure lines are parallel inclined surfaces

rotation of cylinder - forced vortex motion

• equations of motion
• free surface and constant pressure lines are paraboloids of revolution

Question 16

Fluid kinematics -

Lagrangian approach -
Eulerian approach -

Flow field - Pressure, Velocity & Acceleration fields

convective term - (V.del)
total or material derivative (D/Dt) - local (del/delt) + convective (V.del)

Flow visualization

streamlines -
streamtube -
Pathline -
streaklines -
For a steady flow streamline, pathline & streakline are identical but can be quite different for unsteady flow
Main difference is that a streamline _________, while a streakline and a pathline _____________

Timeline -

Answer 16

Fluid kinematics - deals with describing the motion of fluids without necessarily considering the forces and moments that cause the motion

Lagrangian approach - particle-based approach to study
Eulerian approach - region-based approach or control volume based approach

Flow field - Pressure, Velocity & Acceleration fields

convective term - (V.del)
total or material derivative (D/Dt) - local (del/delt) + convective (V.del)

Flow visualization

streamlines - A streamline is a curve tangent to which gives the local velocity vector at that instant
streamtube - bundle of streamlines
Streamlines are useful indicators of the direction of fluid motion throughout the flow field at an instant e.g, regions of recirculating flow
and separation of a fluid off of a solid wall are easily identified by the streamline pattern. Streamlines cannot be directly observed experimentally
except in steady flow fields, in which they are coincident with pathlines and streaklines

Pathline - actual path traversed by fluid particle
A modern experimental technique called particle image velocimetry (PIV) utilizes particle pathlines to measure the velocity field over an entire
plane in a flow. In PIV, tiny tracer particles are suspended in the fluid & flow field is inferred from them using photography & computers

streaklines - locus of all particles that have previously passed through same location
by introducing dye or smoke we get streaklines
smoke wire (thin wire coated with mineral oil which breaks into beads due to surface tension & on applying current results in streakline of smoke)

For a steady flow streamline, pathline & streakline are identical but can be quite different for unsteady flow
Main difference is that a streamline represents flow pattern at a given instant in time, while a streakline and a pathline are flow patterns that have a time history associated with them.

Timeline - set of adjacent fluid particles marked at the same instant in time
Timelines are particularly useful in situations where the uniformity of a flow (or lack thereof) is to be examined
Hydrogen bubble wire (experimentally in a water channel) - when electric current is passed through the wire, electrolysis of the water occurs and tiny hydrogen gas bubbles form at the wire

Question 17

Refractive Flow Visualization Techniques -

shadowgraph technique -
schlieren technique -

Interferometry -

Kármán vortex street -

Plots of fluid flow data -
profile plot -
vector plots -
contour plot -

Types of motion or Deformation of fluid elements

- strain rate tensor 
Vorticity and Rotationality
Rotational flow - 
- circular flows may be \_\_\_\_\_\_\_\_
fluid in boundary layer region is \_\_\_\_\_\_\_\_ & outside it is \_\_\_\_\_\_\_\_\_\_\_\_

Thus if a flow originates in an irrotational region, it remains irrotational until some ______ process alters it.

solid body rotation
line vortex

Reynolds Transport Theorem -

Answer 17

Refractive Flow Visualization Techniques - based on the refractive property i.e the light rays bend as they travel from one medium to another owing to a difference in index of refraction. used where density changes from one location in the flow to another, such as natural convection flows (temperature differences cause the density variations), mixing flows (fluid species cause the density variations), and supersonic flows (shock waves and expansion waves cause the density variations).

shadowgraph technique -
schlieren technique -

Interferometry - visualization technique that utilizes the phase change of light as it passes through air of varying densities

Kármán vortex street - unsteady vortices shed in an alternating pattern, in a flow past the cylinder

Plots of fluid flow data -
profile plot - indicates how the value of a scalar property varies along some desired direction in the flow field.
vector plots - an array of arrows indicating the magnitude and direction of a vector property at an instant in time.
contour plot - shows curves of constant values of a scalar property (or magnitude of a vector property) at an instant in time.

vector plot - gives both magnitude & direction unlike streamlines that give direction only
contour plots - quickly reveal regions of high (or low) values of the flow property being studied

Types of motion or Deformation of fluid elements

• translation, rotation, linear strain & shear strain
• as fluid elements may be in constant motion, motion & deformation is described using rates.
• rate of translation (velocity, V)
• rate of rotation (angular velocity, omega)
• linear strain rate (epsilonxx, yy, zz), volumetric strain rate (zero for incompressible flow)
• shear strain rate
• strain rate tensor (a combination of linear strain rate and shear strain rate into one symmetric second-order tensor)

Vorticity and Rotationality
vorticity - curl of V ( a measure of rotation of a fluid particle)
- it is twice the angular velocity

Rotational flow - if vorticity is non-zero
- circular flows may be rotational
fluid in boundary layer region is rotational & outside it is irrotational
The vorticity of a fluid element cannot change except through the action of viscosity, nonuniform heating (temperature gradients), or other
nonuniform phenomena. Thus if a flow originates in an irrotational region, it remains irrotational until some nonuniform process alters it.

solid body rotation (v=wr) - circular streamlines, flow is rotational (as vorticity is non-zero) [roundabout]
line vortex (v=k/r) - circular streamlines, flow is irrotaional (as vorticity is zero) [ferris wheel]

Reynolds Transport Theorem - RTT relates the rates of change of an extensive property between a closed system and for a control volume system.

Question 18

Ch.5 Mass, Bernoulli & Energy Equations

conservation of mass -
conservation of mass relation written for differential control volume is continuity equation

Newton’s second law of motion -
conservation of momentum -

conservation of energy -

del - used for path function that have____ differentials like heat, work & mass transfer
d - for point functions that have ____differentials

A simple rule in selecting a control volume is to make the control surface _______ toflow at all locations whereit crosses fluid flow, whenever possible.
some practical application like syringe involve deforming CV

mechanical energy -

mechanical energy of a fluid ( __+__+__ ) does not change during flow ifits ____, ____, ____, and _____ remain constant. In the absenceof any losses, the mechanical energy change represents the mechanicalwork supplied to the fluid or extracted from the fluid.

The transfer of mechanical energy is usually accomplished by a ________, and thus mechanical work is often referred to as _________.

A pump or a fan receives ___ (usually from an electric motor) and transfers itto the fluid as ____ energy (less frictional losses).
A turbine, on the other hand, converts the ___ energy of a fluid to ____ work.

Answer 18

conservation of mass - mass of system remains constant during a process
conservation of mass relation written for differential control volume is continuity equation

Newton’s second law of motion - rate of change of momentum of a body is equal to net force acting on the body
conservation of momentum - net force is zero, momentum is conserved.

conservation of energy - net energy transfer to or from a system during a process be equal to the change in the energy content of the system

del - used for path function that have inexact differentials like heat, work & mass transfer
d - for point functions that have exact differentials

A simple rule in selecting a control volume is to make the control surface normal to flow at all locations where it crosses fluid flow, whenever possible.
some practical application like syringe involve deforming CV

mechanical energy - the form of energy that can be converted to mechanical work completely and directly by an ideal mechanical device such as an ideal turbine. Kinetic and potential energies are the familiar forms of mechanical energy. Thermal energy is not mechanical energy, however, since it cannot be converted to work directly and completely (the second law of thermodynamics)

mechanical energy of a fluid (flow energy+k.e+p.e) does not change during flow if its pressure, density, velocity, and elevation remain constant. In the absence of any losses, the mechanical energy change represents the mechanical work supplied to the fluid or extracted from the fluid.

The transfer of mechanical energy is usually accomplished by a rotating shaft, and thus mechanical work is often referred to as shaft work. A pump
or a fan receives shaft work (usually from an electric motor) and transfers it to the fluid as mechanical energy (less frictional losses). A turbine, on the
other hand, converts the mechanical energy of a fluid to shaft work.

Question 19

Bernoulli equation - in mechanical energy terms

Bernoulli equation - in pressure head terms

Limitations of Bernoulli’s equation -

static pressure (P) - 
Dynamic pressure (rhoV^2/2) - 
stagnation pressure (Pstag)- 
Hydrostatic pressure (rhogz) - 

Piezometer tube -

Pitot static tube -

Answer 19

Bernoulli equation - The mechanical energy ( i.e sum of the kinetic, potential, and flow energies) of a fluid particle is constant along a streamline for a steady, incompressible and frictionless flow

Bernoulli equation - the sum of total pressure ( i.e static, dynamic & hydrostatic pressure) is constant along a streamline.

Limitations of Bernoulli’s equation - steady flow, frictionless flow, no shaft work & no heat transfer, incompressible flow & flow along a streamline

static pressure (P) - actual thermodynamic pressure, it doesn't incorporate any dynamics effects
Dynamic pressure (rhoV^2/2) - pressure rise when fluid brought to rest isentropically
stagnation pressure (Pstag)- sum of static plus dynamic pressure
Hydrostatic pressure (rhogz) - it accounts for elevation effects

If static and stagnation pressure of a flowing liquid are greater than atmospheric pressure, a vertical transparent tube called a piezometer tube (or simply a piezometer) can be attached to the pressure tap and to the Pitot tube.

Piezometer tube - measures static pressure for +ve gauge pressure of flowing liquid
Pitot static tube - directly measures dynamics pressure

Question 20

Bernoulli equation - in terms of Mechanical energy

Bernoulli equation - in terms of pressure energy

Limitations of Bernoulli’s equation -

static pressure -
Dynamic pressure -
stagnation pressure -

Hydrostatic pressure -

If static and stagnation pressure of a flowing liquid are greater than atmospheric pressure, a vertical transparent tube called a piezometer tube (or simply a piezometer) can beattached to the pressure tap and to the Pitot tube.

Piezometer tube -
Pitot static tube - directly measures dynamics pressure

Bernoulli Equation - in terms of head

Hydraulic Grade Line -
Energy Grade Line -

For stationary bodies such as reservoirs or lakes, the EGL and HGL_____ with the free surface of the liquid. The elevation of the freesurface z in such cases represents _____ the EGL and the HGL since thevelocity is ____ and the static pressure (gage) is ____

The EGL is always a distance _____ above the HGL. These two linesapproach each other as the velocity _____, and they diverge as thevelocity _____. The height of the HGL decreases as the velocity______, and vice versa.

In an idealized Bernoulli-type flow, EGL is _____ and its height(or not) remains _____. This would also be the case for HGL when the flowvelocity is ______

The mechanical energy loss due to frictional effects (conversion tothermal energy) causes the EGL and HGL to slope______ in thedirection of flow

For open-channel flow, the HGL coincides with the free surface of theliquid, and the EGL is a distance ____ above the free surface

At a pipe exit, the pressure head is zero (atmospheric pressure) and thusthe HGL ____ with the pipe outlet

A _____ occurs in EGL and HGL whenever mechanical energy isadded to the fluid (by a pump, for example). Likewise, a ____occurs in EGL and HGL whenever mechanical energy is removed fromthe fluid (by a turbine, for example)

The pressure (gage) of a fluid is _____ at locations where the HGLintersects the fluid. The pressure in a flow section that lies above the HGLis _____, and the pressure in a section that lies below the HGL is_____

first law of thermodynamics -

The energy content of a fixed quantity of mass (a closed system) can bechanged by ____ mechanisms:

Thermal Energy -

Heat -

work -

shaft work
work done by pressure forces

Answer 20

Bernoulli Equation - The sum of the pressure, velocity, and elevation heads along a streamline is constant for steady, incompressible & frictionless flows
why heads used - convenient to represent the level of mechanical energy graphically using heights to facilitate visualization of the various terms of the Bernoulli equation

static Pressure head - P/rhog
velocity head - v^2/2g
elevation head - z

Hydraulic Grade Line - The line that represents the sum of the static pressure and the elevation heads.
Energy Grade Line - The line that represents the total head of the fluid

For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid. The elevation of the free surface z in such cases represents both the EGL and the HGL since the velocity is zero and the static pressure (gage) is zero

The EGL is always a distance V^2/2g above the HGL. These two lines approach each other as the velocity decreases, and they diverge as the velocity increases. The height of the HGL decreases as the velocity increases, and vice versa.

In an idealized Bernoulli-type flow, EGL is horizontal and its height remains constant. This would also be the case for HGL when the flow velocity is constant

The mechanical energy loss due to frictional effects (conversion to thermal energy) causes the EGL and HGL to slope downward in the direction of flow

For open-channel flow, the HGL coincides with the free surface of the liquid, and the EGL is a distance V^2/2g above the free surface

At a pipe exit, the pressure head is zero (atmospheric pressure) and thus the HGL coincides with the pipe outlet

A steep jump occurs in EGL and HGL whenever mechanical energy is added to the fluid (by a pump, for example). Likewise, a steep drop (Horizontal) occurs in EGL and HGL whenever mechanical energy is removed from the fluid (by a turbine, for example)

The pressure (gage) of a fluid is zero at locations where the HGL intersects the fluid. The pressure in a flow section that lies above the HGL is negative, and the pressure in a section that lies below the HGL is positive

first law of thermodynamics (conservation of energy principle) - Energy can neither be created nor be destroyed during a process; it can only change forms.

The energy content of a fixed quantity of mass (a closed system) can be changed by two mechanisms: heat transfer Q and work transfer W

Thermal Energy - sensible & latent forms of internal energy

Heat - energy transfer by virtue of temperature difference. The direction of Heat transfer is always from higher temp to lower temp.

work - energy interaction associated with a force acting through a distance. A rising piston, a rotating shaft, and an electric wire crossing the
system boundary are all associated with work interactions.

shaft work
work done by pressure forces

Question 21

Ch.8 Flow in Pipes

Why fluids, especially liquids, are transported in circular pipes.?

Laminar & Turbulent Flow
Reynolds Number -
Critical Reynolds Number value -
Hydraulic Diameter -

Transition from Laminar to Turbulent flow also depends on -

In Most practical conditions Transition regime
In Pipe flow -
in Boundary layer -

laminar flow can be maintained at much higher Reynolds numbers (______) in very smooth pipes by avoidingflow disturbances and pipe vibrations.

Entrance Region -

Entrance Length Formula - for laminar & for turbulent

Laminar flow in Pipes -

Turbulent flow in Pipes -
terms such as -

Unlike laminar flow, the expressions for the velocity profile in a turbulentflow are based on both __________

the velocity profile is parabolic in laminarflow but is much fuller in turbulent flow, with a sharp drop near the pipewall. Turbulent flow along a wall can be considered to consist of fourregions ______, characterized by the distance from the wall.

Colebrook equation -
Moody chart -

Vena contracta -

Minor Losses -
Losses more in sudden contraction or sudden expansion?

Piping Network & Pump Selection -

Flow rate & Velocity Measurements -

Obstruction Flowmeters:
Positive Displacement Flow meters

Answer 21

Why fluids, especially liquids, are transported in circular pipes.?
Pipes with a circular cross-section can withstand large pressure differences between the inside and the outside without undergoing significant distortion. Noncircular pipes are usually used in applications such as the heating and cooling systems of buildings where the pressure difference is relatively small.

Laminar & Turbulent Flow
Reynolds Number - Ratio of Inertia force to Viscous force
Critical Reynolds Number value - depends on geometry & flow conditions
Hydraulic Diameter - Dh=4A/P

Transition from Laminar to Turbulent flow also depends on - surface roughness, pipe vibrations, and fluctuations in the flow

In Most practical conditions Transition regime
In Pipe flow - 2300 - 4000
in Boundary layer - 10^6

laminar flow can be maintained at much higher Reynolds numbers (upto 10^5) in very smooth pipes by avoiding flow disturbances and pipe vibrations.

Entrance Region - The region beyond the entrance region in which the velocity profile is fully developed and remains unchanged is called the hydrodynamically fully developed region. The flow is said to be fully developed when the normal-
ized temperature profile remains unchanged as well.

Entrance Length Formula - for laminar & for turbulent

Laminar flow in Pipes -

Turbulent flow in Pipes -
terms such as - rho_u’v’ & rho_u’2 called Reynolds shear stresses

Eddy motion and thus eddy diffusivities are much larger than their molecular counterparts in the core region of a turbulent boundary layer. The eddy motion loses its intensity close to the wall and diminishes at the wall because of the no-slip condition (uЈ and vЈ are identically zero at a stationary wall). Therefore, the velocity profile is very slowly changing in the core region of a turbulent boundary layer, but very steep in the thin layer adjacent to the wall, resulting in large velocity gradients at the wall surface. So it is no surprise that the wall shear stress is much larger in turbulent flow than it is in laminar flow

Unlike laminar flow, the expressions for the velocity profile in a turbulent flow are based on both analysis and measurements, and thus they are semi-empirical in nature with constants determined from experimental data.

the velocity profile is parabolic in laminar flow but is much fuller in turbulent flow, with a sharp drop near the pipe wall. Turbulent flow along a wall can be considered to consist of four regions (VBOT), characterized by the distance from the wall.
(V) Viscous sublayer - (0 - 5 wall units) law of the wall
very thin layer next to wall, viscous effects are dominant, velocity profile in this layer is very nearly linear, and flow is streamlined
(B) Buffer layer -
Next to the viscous sublayer, turbulent effects are becoming significant, but the flow is still dominated by viscous effects.
(O) Overlap layer - logarithmic law
Above the buffer layer, in which the turbulent effects are much more significant, but still not dominant.
(T) Turbulent layer -
Above the transition layer, turbulent effects dominate over viscous effects.

Colebrook equation - implicit relation obtained by curve fitting exercise on experimental data of friction factor dependence on Re & relative roughness
Moody chart - used to obtain friction factor values for relative roughness & Reynold values

Vena contracta - the point in a fluid stream where the diameter of the stream is the least, and fluid velocity is at its maximum
It is formed as fluid streamlines can’t change direction abruptly.
The streamlines are unable to closely follow the sharp angle in the pipe/tank wall. The converging streamlines follow a smooth path, which results in the narrowing of the jet.

Minor Losses -
Losses more in sudden contraction or sudden expansion?
sudden expansion because of flow separation.

Piping Network & Pump Selection -

Flow rate & Velocity Measurements -
Pitot and Pitot-Static Probes
Obstruction Flowmeters: Orifice, Venturi, and Nozzle Meters
Positive Displacement Flow meters

Question 22

Ch.9 Differential Analysis of Fluid Flow

Control Volume Technique -

Differential analysis -

Conservation of mass -

Stream function -

Conservation of Linear momentum -

Navier-Stokes Equation -

Differential Analysis of Fluid flow -

Some flow types -
Couette flow -
Poiseuille flow -

Creeping flow -
Potential flow -

Answer 22

Ch.9 Differential Analysis of Fluid Flow

Control Volume Technique - control volume technique is useful when we are interested in the overall features of a flow, such as mass flow rate into and out of the control volume ornet forces applied to bodies.
The interior of the control volume is in fact treated like a “black box” in control volume analysis—we cannot obtain detailed knowledge about flow properties such as velocity or pressure at points inside the control volume.

Differential analysis, on the other hand, involves application of differentialequations of fluid motion to any and every point in the flow field over a regioncalled the flow domain.When solved, these differentialequations yield details about the velocity, density, pressure, etc., at every pointthroughout the entire flow domain.

Conservation of mass -

Stream function -

Conservation of Linear momentum -

Navier-Stokes Equation -

Differential Analysis of Fluid flow -

Some flow types -
Couette flow - flow between two parallel plates
Poiseuille flow - flow inside a pipe or channel

Creeping flow - inertial terms are negligible
Potential flow - inviscid & irrotational flow

Question 23

Ch.10 Approximate Solutions of N-S equations

Answer 23

Ch.10 Approximate Solutions of N-S equations

Question 24

Ch.11 Flow over Bodies: Drag & Lift

Two Dimensional flow -

Axisymmetric flow -

Bluff body -
Streamlined shape -

DRAG AND LIFT

A stationary fluid exerts only _______ forces on the surface of abody immersed in it.
A moving fluid, however, also exerts __________
forces on the surface because of the _______

Drag force -
Lift force -

The fluid forces also may generate moments and cause the body to rotate.
Rolling moment -
Yawing moment -
Pitching moment -

Drag formula & Lift formula

Thedrag and lift coefficients are primarily functions of the shape of the body, but in some cases, they also depend on the Reynolds number and the surfaceroughness

terminal velocity &

Answer 24

Ch.11 Flow over Bodies: Drag & Lift

Two Dimensional flow -The flow over a body is said to be two-dimensionalwhen the body is very long and of constant cross-section and the flow isnormal to the body.
The two-dimensional idealization is appropriate when the body is sufficiently long so that the end effects are negligible and the approach flow is
uniform.

Axisymmetric flow - when the body possesses rotationalsymmetry about an axis in the flow direction. The flow in this case is alsotwo-dimensional and is said to be axisymmetric.

Bluff body - Body (such as a building) tends to blockthe flow and is said to be bluff or blunt.
Streamlined shape - shapes having smooth changes in shape

DRAG AND LIFT

The force a flowing fluid exerts on a bodyalong the flow directionis called drag.
A stationary fluid exerts only normal pressure forces on the surface of abody immersed in it. A moving fluid, however, also exerts tangential shear
forces on the surface because of the no-slip condition caused by viscouseffects.

Drag force - the drag force is due to the combined effects of pressure andwall shear forces in the flow direction
Lift force - The components of the pressure andwall shear forces in the direction normal to the flow is called lift.

For two-dimensional flows, the resultant of the pressure and shear forcescan be split into two components: one in the direction of flow, which is the
drag force, and another in the direction normal to flow, which is the lift.
For three-dimensional flows, there is also a side forcecomponent in the direction normal to the page

The fluid forces also may generate moments and cause the body to rotate.
Rolling moment - The moment about the flow direction is called the rolling moment
Yawing moment - themoment about the lift direction is called the yawing moment, and
Pitching moment - themoment about the side force direction is called the pitching moment.

Drag formula & Lift formula

it is found convenient to work with appropriate dimensionless numbers that represent the drag and lift characteristics of the body. These numbers are the drag coefficient C D , and the lift coefficient C L

Thedrag and lift coefficients are primarily functions of the shape of the body, but in some cases, they also depend on the Reynolds number and the surfaceroughness

the maximum velocity a falling body can attain and iscalled the terminal velocity & all the forcesbalance each other and the net force acting on the body (and thus its acceleration) is zero. The forces acting on a falling body are usually the drag force, the buoyant force, and the weight of the body

Question 25

FRICTION AND PRESSURE DRAG

Reducing Drag by Streamlining

Flow Separation

DRAG COEFFICIENTSOF COMMON GEOMETRIES

PARALLEL FLOW OVER FLAT PLATES

FLOW OVER CYLINDERS AND SPHERES

Answer 25

FRICTION AND PRESSURE DRAG

part of drag that is due directly to wall shear stress is called theskin friction drag (or just friction drag) since it is caused by frictional effects, and
the part that is due directly to pressure P is called thepressure drag (also called the form drag because of its strong dependenceon the form or shape of the body)

friction drag is the component of the wall shear force in the direction of flow, and thus it depends on the orientation of the body as well asthe magnitude of the wall shear stress.
Friction drag is a strong function of viscosity, and increases with increasingviscosity.
The frictiondrag is also proportional to the surface area.
The friction drag coefficient is independent of surface roughness in laminar flow, but is a strong function of surface roughness in turbulent flow due to surface roughness elements protruding further into the boundary layer.
The friction drag coefficient is analogous to the friction factor in pipe flow, and its value depends on the flow regime.

The pressure drag is proportional to the frontal area and to the differencebetween the pressures acting on the front and back of the immersed body.
The pressure drag becomes most significant when thevelocity of the fluid is too high for the fluid to be able to follow the curvature of the body, and thus the fluid separates from the body at some pointand creates a very low pressure region in the back.
The pressure drag in thiscase is due to the large pressure difference between the front and back sidesof the body.

Reducing Drag by Streamlining

streamlining has opposite effects on pressure andfriction drags.
It decreases pressure drag by delaying boundary layer separation and thus reducing the pressure difference between the front and back ofthe body and increases the friction drag by increasing the surface area. Theend result depends on which effect dominates. Therefore, any optimizationstudy to reduce the drag of a body must consider both effects and mustattempt to minimize the sum of the two.

Streamlining has the added benefit of reducing vibration and noise.
Streamlining should be considered only for blunt bodies that are subjectedto high-velocity fluid flow (and thus high Reynolds numbers) for which
flow separation is a real possibility. It is not necessary for bodies that typically involve low Reynolds number flows (e.g., creeping flows in which
Re Ͻ 1), since the drag in those cases is almostentirely due to friction drag, and streamlining will only increase the surfacearea and thus the total drag

Flow Separation
At sufficiently high velocities, the fluid streamdetaches itself from the surface of the body. This is called flow separation.

The region of flow trailing the body wherethe effects of the body on velocity are felt is called the wake.
The separated region comes to an end when the two flow streams reattach.Therefore, the separated region is an enclosed volume, whereas the wakekeeps growing behind the body until the fluid in the wake region regains itsvelocity and the velocity profile becomes nearly flat again. Viscous androtational effects are the most significant in the boundary layer, the separated region, and the wake.

Flow separation on the top surface of a wing reduces lift drasticallyand may cause the airplane to stall. drag and lift are strongly dependent on the shape of the body, and any effect that causes the shape to change has a profound effect on the drag and lift.
important consequence of flow separation is the formation and shedding of circulating fluid chunks, called vortices, in the wake region. Theperiodic generation of these vortices downstream is referred to as vortexshedding.

DRAG COEFFICIENTSOF COMMON GEOMETRIES

The drag coefficients for mostgeometries (but not all) remainessentially constant at Reynoldsnumbers above about 10^4 .
This is due to the flow at high Reynolds numbers becoming fully turbulent. However, this is not the case for rounded bodies such as circular cylinders and spheres.

at low Reynolds numbers, the shape of the body does not have a major influence on the drag coefficient

the drag coefficient at highReynolds numbers. First of all, the orientation of the body relative to thedirection of flow has a major influence on the drag coefficient.

Asa rule of thumb, the percentage of fuel savings due to reduced drag is abouthalf the percentage of drag reduction.

the ideal shapeof a vehicle is the basic teardrop, with a drag coefficient of about 0.1 for theturbulent flow case

PARALLEL FLOW OVER FLAT PLATES

FLOW OVER CYLINDERS AND SPHERES
A sudden drop in the drag coefficient somewhere in the range of10^5 < Re < 10^6. This large reduction in C D isdue to the flow in the boundary layer becoming turbulent, which movesthe separation point further on the rear of the body, reducing the size ofthe wake and thus the magnitude of the pressure drag. This is in contrastto streamlined bodies, which experience an increase in the dragcoefficient (mostly due to friction drag) when the boundary layerbecomes turbulent.
The delay of separation in turbulent flow is caused by the rapid fluctuations of the fluid in the transversedirection, which enables the turbulent boundary layer to travel farther alongthe surface before separation occurs, resulting in a narrower wake and asmaller pressure drag.
In the range of Reynolds numbers where the flow changes fromlaminar to turbulent, even the drag force F D decreases as the velocity (and
thus the Reynolds number) increases. This results in a sudden decrease indrag of a flying body (sometimes called the drag crisis) and instabilities in
flight.

surface roughness, in general, increases the dragcoefficient in turbulent flow. This is especially the case for streamlined bodies. For blunt bodies such as a circular cylinder or sphere, however, anincrease in the surface roughness may actually decrease the drag coefficient.
roughening the surface can be used togreat advantage in reducing drag, but it can also backfire on us if we are notcareful—specifically, if we do not operate in the right range of the Reynoldsnumber.
golf balls are intentionally roughened toinduce turbulence at a lower Reynolds number to take advantage of the sharpdrop in the drag coefficient at the onset of turbulence in the boundary layer.The occurrence of turbulent flow reduces the drag coefficient of a golf ball by about half.
Tennis balls travel short distances and never reach turbulent range.

Question 26

LIFT

Answer 26

LIFT

airfoils are specifically designed to generate lift while keeping thedrag at a minimum.
some devices such asthe spoilers and inverted airfoils on racing cars are designed for the opposite purpose of avoiding lift or even generating negative lift to improve traction and control.
lift inpractice can be taken to be due entirely to the pressure distribution on thesurfaces of the body, and thus the shape of the body has the primary influence on lift. Then the primary consideration in the design of airfoils is minimizing the average pressure at the upper surface while maximizing it at thelower surface.
lift is practically independent of the surface roughness since roughness affects the wall shear, not the pressure.
the contribution of viscous effects to lift is negligible, weshould be able to determine the lift acting on an airfoil by simply integrating the pressure distribution around the airfoil. The pressure changes in theflow direction along the surface, but it remains essentially constant throughthe boundary layer in a direction normal to the surface. Therefore, it seems reasonable to ignore the very thin boundary layer on the airfoil and calculate the pressure distribution around the airfoil from the relatively simple potential flow theory (zero vorticity, irrotational flow) forwhich net viscous forces are zero for flow past an airfoil.
At zero angle of attack, the lift produced by the symmetrical airfoil is zero, as expected because of symmetry, and the stagnationpoints are at the leading and trailing edges.
For the nonsymmetrical airfoil,which is at a small angle of attack, the front stagnation point has moveddown below the leading edge, and the rear stagnation point has moved up to the upper surface close to the trailing edge. To our surprise, the lift produced is calculated again to be zero—a clear contradiction of experimentalobservations and measurements. Obviously, the theory needs to be modifiedto bring it in line with the observed phenomenon.
the two flow streams from the top and the bottom sides of the airfoil meet at the trailing edge, yielding a smooth flow downstream parallel to the trailing edge. Lift is generated because the flow velocity at the top surface is higher, and thus the pressure on that surface is lower due to the Bernoulli effect.
after a sudden increase inangle of attack, a counterclockwisestarting vortex is shed from the airfoil,while clockwise circulation appearsaround the airfoil, causing lift tobe generated.
It is desirable for airfoils to generate the most lift while producing theleast drag.
The flaps are used to alter the shape of thewings during takeoff and landing to maximize lift and to enable the aircraftto land or take off at low speeds.
Tip vortices that interact with the free stream impose forces on the wingtips in all directions, including the flow direction. The component of the
force in the flow direction adds to drag and is called induced drag.
end effects can be minimized by attaching endplates or winglets atthe tips of the wings perpendicular to the top surface. The endplates function by blocking some of the leakage around the wing tips, which results ina considerable reduction in the strength of the tip vortices and the induceddrag.
large commercial planes (and soaring birds likealbatrosses) have long and narrow wings -bodies with large aspect ratios fly moreefficiently, but they are less maneuverable because of their larger moment ofinertia
fighter planes (and fighter birds like falcons) haveshort and wide wings - manoeuvre better since the wings are closer to the central part
The lift coefficient increases almost linearly with the angle of attack areaches a maximum at about angle16° and then starts to decrease sharply.
This decrease of lift with further increase in the angle of attack is calledstall, and it is caused by flow separation and the formation of a wide wake
region over the top surface of the airfoil. Stall is highly undesirable sinceit also increases drag.
At zero angle of attack (0°), the lift coefficient is zero forsymmetrical airfoils but nonzero for nonsymmetrical ones with greatercurvature at the top surface.

Question 27

Drag -

Skin friction -
Pressure Drag -
Interference drag -
Induced drag -

upwash -

Lift Generated by Spinning

Magnus effect.

Answer 27

Drag -
Skin friction - shear stresses (surface)
Pressure Drag - difference in front & back pressure (shape)
Interference drag - mixing of airflow streamlines between airframe components such as wing & fuselage
Induced drag - due to wingtip vortices (winglets/ endplates are used to prevent wingtip vortices)

upwash - birds & fighter planes fly in V formation to take advantage of upwash which helps in supporting weight.

Lift Generated by Spinning

The phenomenon of producing lift by the rotationof a solid body is called the Magnus effect.
The stagnation points shift down, and the flow is no longer symmetric aboutthe horizontal plane that passes through the center of the cylinder. The average pressure on the upper half is less than the average pressure at the lowerhalf because of the Bernoulli effect, and thus there is a net upward force(lift) acting on the cylinder.

Question 28

Ch. 12 Compressible Flows

STAGNATION PROPERTIES

Enthalpy -
Stagnation Enthalpy or Total Enthalpy -

stagnation enthalpy -
Isentropic stagnation enthalpy -

Dynamic Temperature -

SPEED OF SOUND AND MACH NUMBER
Speed of Sound -
Ma -
Subsonic , Supersonic , Hypersonic , Transonic

ONE-DIMENSIONAL ISENTROPIC FLOW

Supersonic flow continues accelerating even when the section is diverging because _______

ISENTROPIC FLOW THROUGH NOZZLES

Converging Nozzles
the effects of back pressure (i.e., the pressure applied at the nozzle discharge region) on the exit velocity, the mass flow rate, and the pressure distribution along the nozzle.

Converging–Diverging Nozzles

Answer 28

Ch. 12 Compressible Flows

STAGNATION PROPERTIES

Enthalpy - Internal Energy + Flow Energy
Stagnation Enthalpy or Total Enthalpy - Enthalpy + Kinetic Energy

when potential energy is negligible - total enthalpy represents the total energy of flowing fluid stream.
static enthalpy - ordinary enthalpy

stagnation enthalpy - enthalpy of a fluid when it is brought to rest adiabatically
Isentropic stagnation enthalpy - when the stagnation process is reversible & adiabatic.
Dynamic Temperature - V2/(2Cp)

SPEED OF SOUND AND MACH NUMBER
Speed of Sound - the speed at which an infinitesimally small pressure wave travels through a medium (c=sqrt(gammaRT))
Ma - ratio of velocity of object to the velocity of sound in that medium, Ma = V/c
Subsonic < 1 , Supersonic > 1 , Hypersonic&raquo_space; 1 , Transonic around 1

ONE-DIMENSIONAL ISENTROPIC FLOW

Supersonic flow continues accelerating even when the section is diverging because mass flow is constant large decrease in density makes acceleration in the diverging section possible.

ISENTROPIC FLOW THROUGH NOZZLES

Converging Nozzles
the effects of back pressure (i.e., the pressure applied at the nozzle discharge region) on the exit velocity, the mass flow rate, and the pressure distribution along the nozzle.

Converging–Diverging Nozzles

Question 29

SHOCK WAVES AND EXPANSION WAVES

Shock waves -
Normal shock waves -

Decrease - V, Ma, Po
Increase - P, T, To S, rho

Fanno line -
Rayleigh line -

Oblique Shocks -

Prandtl-Meyer Waves -

Answer 29

SHOCK WAVES AND EXPANSION WAVES

Shock waves - abrupt changes in fluid properties occuring in a very thin section of fluid flow.
Normal shock waves - shock waves that occur in a plane normal to the direction of flow are called normal shock waves.
Flow across the shock wave is highly irreversible & can’t be approximated as isentropic.

Decrease - V, Ma, Po
Increase - P, T, To S, rho

Fanno line - locus of states that have same stagnation enthalpy & mass flow rate on h-s diagram.
Rayleigh line - locus of states that have same momentum & mass on h-s diagram.

Oblique Shocks - produced in compressing flows
Prandtl-Meyer Expansion Waves - Expanding Flows produces Expansion waves (not shock waves).
Flow doesn’t turn suddenly but gradually & the flow region is isentropic in expanding region. Composed of infinite number of Mach waves called Prandtl-Meyer expansion waves.

Question 30

DUCT FLOW WITH HEAT TRANSFER AND NEGLIGIBLE FRICTION (RAYLEIGH FLOW)

Answer 30

DUCT FLOW WITH HEAT TRANSFER AND NEGLIGIBLE FRICTION (RAYLEIGH FLOW)

Plot of Rayleigh flow on a T-s diagram is called the Rayleigh line.
The states on the upper arm of the Rayleigh line above point a are subsonic, and the states on the lower arm below point a are supersonic. Therefore, a process proceeds to the right on the Rayleigh line with heat addition and to the left with heat rejection regardless of the initial value of the Mach number.
Heating increases the Mach number for subsonic flow, but decreases it for supersonic flow. The flow Mach number approaches Ma=1 in both cases (from 0 in subsonic flow and from infinity in supersonic flow) during heating.

Table that shows the effects of heating & cooling on the properties of Rayleigh flow for subsonic & supersonic case.

Question 31

ADIABATIC DUCT FLOW WITH FRICTION (FANNO FLOW)

Answer 31

ADIABATIC DUCT FLOW WITH FRICTION (FANNO FLOW)

Plot of Fanno flow on T-s diagram is called Fanno line.
Friction causes entropy to increase, and thus a process always proceeds to the right along the Fanno line. At the point of maximum entropy, the Mach number is Ma=1. All states on the upper part of the Fanno line are subsonic, and all states on the lower part are supersonic.

Friction increases the Mach number for subsonic Fanno flow, but decreases it for supersonic Fanno flow. The Mach number approaches unity (Ma=1) in both cases.

The energy balance requires that stagnation temperature T_o=T+V^2/2C_p remain constant during Fanno flow. But the actual temperature may change. Velocity increases and thus temperature decreases during subsonic flow, but the opposite occurs during supersonic flow.

Table that shows the effects of friction on properties of Fanno flow, for subsonic & supersonic.

Question 32

vortex flows

Solid-Body Rotation

Irrotational Vortex

Rankine Vortex

Answer 32

Flows in circular paths are called vortex flows, some basic forms of which are described
in what follows.

Solid-Body Rotation

circular cylinder rotating with constant angular velocity

Irrotational Vortex

ur = constant

Rankine Vortex

Real vortices, such as a bathtub vortex or an atmospheric cyclone, have a core
that rotates nearly like a solid body and an approximately irrotational far field
An idealization of such a behavior is called the Rankine vortex, in which the vorticity is assumed uniform
within a core of radius R and zero outside the core