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HIGHSPEED AERODYNAMICS

AER134 3 0 0 100

UNIT-1

1. ONE DIMENSIONAL COMPRESSIBLE FLOW 9Energy, momentum, continuity and state equations, Velocity of sound,

Adiabatic steady state flow equations, Flow through converging, diverging

passages, Performance under various back pressures.

Possible Classifications of Continuum Fluid Mechanics

Inviscid Viscous

(= 0)

Laminar

Turbulent

InternalExternal

Compressible Incompressible

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A fluid problem is called compressible if changes in the density of the fluid havesignificant effects on the solution. If the density changes have negligible effectson the solution, the fluid is called incompressible and the changes in density are

ignored.

In order to determine whether to use compressible or incompressible fluiddynamics, the Mach number of the problem is evaluated. As a rough guide,compressible effects can be ignored at Mach numbers below approximately 0.3.Nearly all problems involving liquids are in this regime and modeled as

incompressible.

The incompressible Navier-Stokes equations are simplifications of the Navier-Stokes equations in which the density has been assumed to be constant. These can

be used to solve incompressible problems.

Liquids such as water, by their very nature, are incompressible. It will require

enormous forces to squeeze a gallon of water to fit into a volume that is 0.9999gallons. While computing fluid dynamics of liquids, it is therefore customary toassume that the density of the liquid is constant, and the flow is "incompressible.

Other fluids such as air are clearly compressible. It is easy to change the densityof air inside a small volume by squeezing it with relatively small amounts of

force. Thus, flows involving air or gases are usually compressible. At small

velocities compared to the speed of sound, however, the density of the fluid does

not change from point to point, even when other properties such as pressure orvelocity are changing.

Newton was right in assuming that air is made of molecules, which collide witheach other, and obey particle physics.

The average distance air molecules (Nitrogen, Oxygen, etc.) can travel beforecollision with a neighbour molecule is called the mean free path. Normally thismean free path is very small (of the order of 10-8 m) compared to the dimensions

of the wing.

Thus, billions of collisions will occur by the time a molecule travels from theleading edge of a wing to the trailing edge.

When this many molecules and collisions are involved, it is reasonable to assumethat air is a continuous medium, not discrete particles.

Properties such as density, pressure and temperature become continuouslydefinable quantities, which are averages of molecule properties taken over

millions of molecules.

This assumption that the fluid is a continuous medium with continuously varyingproperties is called the concept of continuum.

The concept of continuum fails in the outer edges of rarefied atmosphere wheresatellites operate, and during the early phases of reentry.

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The following laws are frequently used in dealing with a variety of compressible

flow problems:

(i) Law of conservation of mass

(Continuity equation)

(ii) Newtons second law of motion(Momentum equation)

(iii) First law of thermodynamics

(Energy equation).

(iv) Second law of thermodynamics

(Entropy relation)

The aerodynamic forces and moments on, the body are due to only two basic

sources:

Pressure distribution over the body surface Shear stress distribution over the body surface

No matter how complex the body shape may be, the aerodynamic forces and

moments on the body are due entirely to the above two basic sources.The only mechanisms nature has for communicating a force to a body moving

through a fluid are pressure and shear stress distributions on the body surface.

Both pressure p and shear stress have dimensions of force per unit area (pounds per

square foot or Newton per square meter).

As sketched in Figure 1 p acts normal to the surface, and (toe) acts tangential to thesurface. Shear stress is due to the "tugging action" on the surface, which is caused by

friction between the body and the air. The net effect of thep and distributions integratedover the complete body surface is a resultant aerodynamic forceR and momentM on the

body, as sketched in Figure 2.

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Temperature Variation in the Atmosphere

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The analysis of compressible flow is based on three fundamental physical principles;

in turn, these principles are expressed in terms of the basic flow equations. They

are:

2. Principle:Time rate of change of momentum of a body equals the net force

exerted on it. (Newton's second law.)

3. Principle: Energy can be neither created nor destroyed, it can only change in

form.

To summarize and reinforce the physical significance of the force on a moving fluidelement, let us display Newton's second law in diagrammatic form as follows:

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The following equations are general they apply no matter what type of gas is being

considered. Also, in general they must be solved numerically for the propertiesbehind the shock wave.

For a calorically perfect gas, we can immediately add the thermodynamic relations suchas equation of state and enthalpy.

and

The above five equations with five unknowns, 2, u2, p2, h2, and T2 can be solved

algebraically.

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ONE-DIMENSIONAL FLOW EQUATIONS

Consider the flow through a one-dimensional region, as represented by the shaded area in

Fig 3.5. This region may be a normal shock wave, or it may be a region with heataddition, in either case, the flow properties change as a function of x as the gas flows

through the region. To the left of this region, the flow field velocity, pressure,temperature, density, and internal energy are u1, p1, T1, 1, and e1, respectively. To theright of this region, the properties have changed, and are given by u2, p2, T2, 2, and e2.

Assume that the left- and right-hand sides each have an area equal to A

perpendicular to the flow. Also, assume that the flow is steady, such that all

derivatives with respect to time are zero, and assume that body forces are not

present.

Fig. 3.5 Rectangular control volume for one-dimensional flow

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Flow through Converging-Diverging passages

Flow Through Nozzle

Assumptions for the analysis are:

the flow is one dimensional and steady

the fluid is an ideal gas with constant thermodynamics properties, and

the flow is isentropic (i.e., adiabatic and frictionless)

The Governing conservation relations are

Mass: u A = constant

Or

)1(0=++A

dA

u

dud

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Momentum:

dx

dP

dx

duu =

or

( )20=+ dPduu

Energy

2tan

2

22

cc uhtconsuh +==+

when uc~ 0

Energy conservation relation becomes,

)3(2

2

chu

h =+

Equation of state : p = R T (4)

Isentropic Relation:

)5(

1/

=

=

ccc T

T

P

P

,

which is valid for a single component, ideal gas.

Equations 1 to 5 constitute the basic set of equations from which all the downstreamparameters can be derived.

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Velocity variation with Area

The nozzle is specified by the area term.

+=

u

dud

A

dA

We have to relate to u to get the desired result. For this, we make use of the concept ofvelocity of sound, i.e., the speed of propagation of pressure waves in a fluid medium

under isentropic conditions.

tconss

Pa

tan

2

=

=

From Eq.(2)

dPduu =

and using the above equation we can rewrite the momentum equation as,

daduu2

=

u

duM

u

du

a

ud 22

2

==

Where M is the Mach number at the local station.

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We have,

+=

u

dud

A

dA

Subtitute the value of d/in the above continuity equation, we get,

)6(1

/2

=M

AdA

u

du

The task of the nozzle is to accelerate the flow, i.e., to make du positive. This can be

achieved by making dA negative (decreasing area, converging) when M < 1 as it happens

at chamber conditions, and by dA positive (increasing area, diverging) when M > 1 i.e.,supersonic conditions. Evidently, the transition from subsonic to supersonic occurs at theminimum cross sectional station ( M=1 and du to be finite require dA to be zero in Eq.6).

A convergent - divergent nozzle or a De-lavel nozzle is essential to accelerate a stagnant

or subsonic gaseous medium to supersonic velocities.

The driving pressure gradient for the flow comes from the difference between Pc andPa. But the downstream pressure Pa will be sensed by the flow only when M < 1 every

where in the nozzle. Once M = 1 is attained at the throat supersonic flow prevails in the

divergent part of the nozzle and the flow does not sense Pa, i.e., the mass flow rate

becomes independent of Pa. Suh a condition is termed as choked flow. However, all

upstream conditions get reflected at all stations, i.e., one can continue to increase themass flow rate by suitably changing the chamber conditions.

Critical Ratio for choking to occur

Applying the energy conservation equation to the throat.

ct

th

uh =+

2

2

Since no compositional changes are assumed to take place; i.e., by virtue of theassumption made, h can be set equal to the sensible enthalpy, i.e C pT

cpt

tp TCu

TC =+2

2

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SubstitutingTRau tt ==

As choking holds good.

cpt

tp TCTR

TC =+2

on simplification

)7(

2

1+=

t

c

T

T

and by the isentropic relation

)8(2

11/

+=

t

c

P

P

When = 1.2

7.1

1.1

=

=

t

c

t

c

P

P

andT

T

Note: If in any situation a pressure ratio of 1.7 is measured across an orifice or

nozzle (i.e at sea level condition if a Pc of about 1.7 atm or 25 psi is recorded) one

can be certain that choking will have been occurred.

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Discussion on Area-velocity relation

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