lecture hydro

8/12/2019 Lecture Hydro

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Fluid Flow

AE 1350


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Motivation An airplane of prescribed configuration is flying 3 km

above the ground at a speed of 112 m/s At a specific point along the wing, the pressure and shear

stress are dictated by the laws of nature From these quantities, the forces (lift and drag) and moments

on the vehicle can be determined and its flight quantified A rocket nozzle of prescribed shape and size is fed

given amounts of fuel and oxidizer. The flow velocity and pressure at the nozzle exit are dictated

by the laws of nature From these quantities, the thrust on the vehicle can be

determined and its flight quantified For these reasons and others, an understanding of fluid

dynamics is vital to our study of atmospheric flight


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Gas Properties Knowledge of P, , T and

Vat each point in the flow fully defines the flowfield,

With knowledge of these quantities, we can fully describe the forces and momentsexerted on a general body in a flowfield.

The aerodynamic force exerted by the flow on the surface of a vehicle stems fromtwo fundamental sources: pressure and shear stress

Each of these distributions can vary in magnitude and direction along the body(they are point properties)

Integrating the pressure and shear stress distributions across the body yields aresultant force (or forces) these are the basic sources of all aerodynamic forces.

z)y,(x,VV

z)y,T(x,T

z)y,(x,

z)y,P(x,P

rr ====

At the surface, the shear stress distribution acts tangential to the surface.

Pressure distribution acts normal to the aircraft surface.


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Governing Equations

The laws of nature or governing aerodynamic equationsare formulated by applying basic physical principles to aflowing gas. Mass can not be created or destroyed Continuity Eq. F =m a (Newtons 2 nd law) Momentum Eq.

Energy can not be created or destroyed Energy Eq.

In this lecture, we will derive these governing flow

equations and apply them in a useful manner


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Conservation of Mass: Continuity Equation Consider a stream tube, i.e. a collection of streamlines that form a

tube-like shape. The cross-sectional area of this tube can change size as one moves

along with the flow However, as long as the flow is steady (time-invariant), within this

tube, mass can not be created or destroyed.

The mass that enters the stream tube from the left (e.g. at the rateof 1 kg/sec) must leave on the right at the same rate (1 kg/sec).


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Conservation of Mass: Continuity Equation

In a similar manner, the mass flow rate across area A 2 can beshown to be:

Since mass can not be created or destroyed, and all the mass thatenters A 1 must exit A 2 (no mass can exit across a streamtube),

Mass flux is constant in steady fluid flow

2222

2 VAdt

dm m ==&

constantAV

VAVA

mm222111

21

===

&&


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Incompressible Flow In reality, air is a compressible fluid.

Its density WILL change if the temperature changes, or if some externalforce is applied.

Example: A child squeezing a balloon

A flow is said to be incompressible if there are no changes indensity caused by the velocity of the flow.

Theory and observational evidence suggest that most flows may

be treated as incompressible (i.e. constant density) until theMach number is sufficiently high (> 0.4 or so) The variability of density in aerodynamic flows is particularly important

at high subsonic speeds and for all supersonic vehicles and rocket engines

For an incompressible flow ( = constant), the continuityequation becomes:

constantAV

VAVA

mm2211

21

=

== &&


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Incompressible Flow This is why a common garden nozzle reduces its area with

distance:

In this case,

This same principle is used in the design of nozzles for subsonicwind tunnels or in the acceleration of subsonic flow towardssupersonic conditions in a rocket nozzle

1 12

12

21

VVAA

V

mm

>=

= &&

A1

A2

V1 V2


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Continuity

For incompressible flow,

AV = constant

If area between streamlinesis high, the velocity is low

If area between streamlinesis low, the velocity is high

High VelocityLow Velocity

What can we say about the flow velocity around this cylinder?


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High Velocity

Low Velocity

ContinuityFor incompressible flow,

AV = constant

If area between streamlinesis high, the velocity is low

If area between streamlinesis low, the velocity is high

In regions where thestreamlines squeeze together,the flow velocity is high


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Viscous Effects Streamlines describe the path the

fluid particles will take. Flow velocity is tangential to

the streamline. Viscosity alters the shape of streamlines

around blunt bodies. Scientists inject smoke particles intostreamlines to make them visible tothe naked eye.

Viscous effects are a function ofReynolds number

At low Reynolds numbers, the flow isapproximated as inviscid (frictionless)

Inviscid (ideal) flow

Viscous flow


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Streamlines over a Cylinder (Low Reynolds Number of 10)

dia.CylinderD

VD Number Reynolds

=

=

where


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Streamlines over a Cylinder (High Reynolds Number of 2000)

dia.CylinderD

VD Number Reynolds

=

=

where


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The Momentum Equation

Continuity equation is only part of the story. For example, it tells us nothing about the pressure distribution, yet

we know that pressure is an important flow quantity To get at the forces (pressure), we need to examine Newtons 2 nd

law as applied to fluid dynamics. In general, Newtons 2 nd law can be expressed as:

The force exerted is equal to time rate of change of momentum;from which, for a constant mass system, we have the relation,

)v(mdtd

F rr =

amF rr =


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Momentum Equation Consider a small slice of stream tube

Newtons 2 nd law states that the time rate of change of

momentum of the fluid particles within this stream tubeslice must equal to the forces acting on it To begin, lets examine the momentum rate across stream tube

entrance and exit planes Where momentum rate = mass flux * velocity


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Momentum EquationPV

A

P+dp+dV+dVA+dA

Momentum rate in is equal to mass flow rate times velocity: [ VA]V = V2A Momentum rate out is equal to mass flow rate times velocity:

[(+d)(V+dV)(A+dA)](V+dV) = [ VA](V+dV) (from the Continuity Equation)

Rate of change of momentum = ( Momentum rate out ) ( Momentum rate in )

VAdVAV-dV)VA(V(mv)dtd 2 =+=


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Momentum Equation

P

VA

P+dp+dV+dVA+dA

This leaves pressure differential as the only cause of a force. Since the pressure isalways normal to the surface and oriented inward, we have

Adp-F

dpdA-Adp-F

dA)dp)(A(PPAF

+=

++=

Now, lets consider the forces acting on this stream tube slice For the time being, we will ignore the presence of friction (inviscid flow) The change in gravitational attraction across a small element is also quite small


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Momentum Equation Rate of change of momentum = AVdV Forces acting on the stream tube = -Adp

Recall that we have neglected all forces other than pressure

(e.g., viscous, gravity, electrical and magnetic forces) Equating terms (and dividing by A), we get:

VdVdp

Adp-VAdV

So,Adp-Fand

VAdV(mv)dtd

where

(mv)dtd

F

=

=

=

==

This equation is generallytermed Eulers equation. Itis the momentum equationfor steady, inviscid flow


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Use of Eulers Equation Eulers equation was derived for the case of steady, inviscid flow

and can be used to relate pressure and velocity across the flowfield It can be applied to either compressible or incompressible flow

For incompressible flows ( constant), Eulers equation can beintegrated to relate two points on the same streamline:

This equation is termed Bernoullis Equation Note that if all the streamlines far upstream have the same value of

P, and V (as is generally the case), the constant in Bernoullisequation is the same for all streamlines and this equation can beapplied to two points along different streamlines.

constantPV21

0dpVdV

0dpVdV

2 =+=+

=+


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Summary on the Application of theEuler and Bernoulli Equations

These equations are a statement of Newtons 2 nd law for fluid dynamics

The Euler & Bernoulli equations relate pressure and velocity in a flowfield Eulers equation is derived assuming an inviscid (frictionless) flow Bernoullis equation is only valid for inviscid, incompressible flow As derived, Bernoullis equation relates point properties along a streamline, but in

cases with uniform upstream conditions, can be used to relate flow properties atany two points in the flow.

For compressible flow, Eulers equation may be used, but must be treated as avariable in the integration. Bernoullis equation should not be used for

compressible flow.

constant pV21 2

=+VdVdp =


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Pitot Tube: Application of Bernoullis

Equation to Determine Airspeed Measure local pressure of air:

pS , static pressure

Measure pressure after bringing flow to zero speedrelative to instrument (i.e., stagnate flow):

pT , total pressure From Bernoulli:

Determine velocity of flow with respect to instrument:

Really only need difference between pS & pT (not an absolutemeasurement). Determine from standard atmosphere or other means

TS

2

p pV2

1

=+

( ) p p2VST =

ps

pT


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Pitot Tube

(Wikopedia)

Measure pressure difference ( )

p p2V ST

=


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Pitot-static system

Total pressure probe(s) not behind propulsion sources,near front of aircraft usuallyon sides of fuselage near frontor front of a wing

Static pressure port(s) in

location with minimumdisturbance from aircraft itself,usually along sides of fuselage

Airspeed indicator showsmeasured airspeed to pilot

2 on each side of this Boeing 757


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Airspeed Indicator

Mechanically usesBernoullis equation to

show airspeed notcorrected for actual density Aircraft are typically

flown via this indicatedairspeed (not corrected fordensity, mechanical errors,etc.). Generally indicatedin Knots.

( )

standard

ST

p p2

V =


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How Does the Venturi Meter Work?

222

211

1

2

2

1

21

222111

V21

PV21

P:Bernoulli

AA

VV

Thus,:flowibleincompressFor

AVAV

+=+

=

==

= ( )

( )

( )

11

22

21

121

1

1222

212

1

122

1

222

1

1222

21

AVm

:rateflowmassCompute

AA

1PP2V

:VforSolve

PP2

AA

1V

PP2

V

V1V

PPV21

V21

=

=

=

=

=

&


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Compressible Flow

AE 1350


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So far For an incompressible flow, conservation of mass and

momentum gave us:

For compressible flow, we need another equation toallow us to solve for variability in density:

Const PV

Const AV

=+

=2

21

0=+=

dPVdV Const AV


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Compressible Flow As stated earlier, when the velocity exceeds about 100

m/s (Mach 0.4), the flow can no longer be assumed to beincompressible. This is the situation for a large

percentage of aerodynamic applications. In high speed flow analysis, the kinetic energy of the

flow particles must be accounted for and significantchanges in temperature are associated with significant

changes in energy (flow speed). In addition to the previous relations, we will need tomake use of the first law of thermodynamics.

Definitions: Specific internal energy: energy per unit mass stored in random

motion of molecules, e Specific enthalpy: h = e+p/ = e + RT


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First Law of Thermodynamics

Let e denote the specific internalenergy of the gas

The only means by which e can beincreased (or decreased) is by heataddition to (or removal from) thesystem, or work done on (or by) the

system. This law can not be derived, but has

been verified from observations.

wqde +=

Heat added per

unit mass

Work done on thesystem per unit massdue to body forcessuch as gravity, andpressure forces

Change ininternal energyper unit mass


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Neglecting gravitational, magnetic and electrical forces, the onlymeans for doing work on the fluid system is pressure

Work is defined as force times distance, so:

Where sdA is the change in volume of the unit mass of gascreated by the inward displacement of the boundary surface dueto the pressure force. Since pressure is pushing in, the volume isdecreasing (dv < 0) and the work done is positive, so:

The First Law can now be re-written as:

(P)sdAw

(PdA)sw==

Pdvw =

Pdvqde =


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=

=

1

Pdq de

Pdvq de


dp1

qdh

dp1

1

Pdde)P

d(edh

+=

+

+=+=

From the definition of enthalpy, the first law can also be expressed as:

Thermodynamic Process


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Thermodynamic Process The means by which changes in the fluid properties of a system take place

Examples: Constant volume process: gas inside a rigid volume (e.g., propellant tank). As heat

is added, P and T will change. By definition, dv=0

Constant pressure process: piston system in which pressure is maintained as heat isadded. As heat is added to this system, T and will change. By definition, dp=0

TChyields,0Tat0hlettingandretemperatuof functionanotisCthatAssuming

dTCqdh:lawFirst

pressureconstantatdTdq

CdefineWe

P

P

P

P

= ==

==

=

CvTe

yields,0Tat0elettingandretemperatuof functionanotisCthatAssuming

dTCqde :lawFirst

olumeconstant vatdTdq

CdefineWe

v

v

v

===

==

=

These equations are significant. They are derived from the First Law of Thermodynamics intowhich the definitions of specific heat have been applied. They directly relate e or h to T. While

not specifically proven here, they are applicable to any process in which a perfect gas is used.


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A li ti f Fi t L f Th d i


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0 p

2

v

v

00 p0

0

2

TC2

VP

TC

P

TCP

eh

re.temperatustagnationcalledisTwhereTCh

hConstant2

Vh

=++

+=+==

==+

Internal Energy

Pressure work Kinetic Energy

Generalization of Bernoullisequation for compressible flow.Either form of this equation istermed the CompressibleEnergy Eq. Note that Eulersequation (conservation of

momentum) is implicit to thisrelation

Application of First Law of Thermodynamics

for Adiabatic, Inviscid Compressible Flow


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Isentropic Flow

Adiabatic: no heat is added or taken away ( q=0) Reversible: no frictional or other dissipative effects

Isentropic: BOTH! (Adiabatic + Reversible)


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Isentropic FlowIn such a flow,

( ) 11V

V

V

CRlnC1

RlnlnTC

1

RdTdT

C

:soRT,Pand TCe

1d

TPde

T1

becomes 1

Pd-qde:lawFirst

0Tq

0q

+=+

=

=

==

=

=

=

=

Isentropic Flow


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Isentropic Flow From the previous slide:

For air,

And by definition,

So we get,

Use (2) in (1) above:

( ) (1) CRlnlnTC 1V +=

R CR R 25

R 27

C V p +=+==

VCCp

=

(2) 1

R C

R CC

CCp

V

VV

V

=

+=

=

( )

1-

1

3

2

1

TC

Clnln(T)1

1CRlnlnT1

R

=

+=

+=


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i l


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Isentropic Flow

=

1

2

1

2

p p

From ideal gas law: RT p =

=

2

1

1

2

1

2

RT RT

p p

p p

=

1

21

1

2

T T

p p

( )1/

1

2

1

2

=

T T

p p Given a temperature variation,

can find the correspondingpressure change


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Isentropic Flow Summary


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Isentropic Flow Summary Our isentropic, compressible flow relations can be expressed as a function of

Mach number as:

1

2

00

11

211

00

20

02

02

V p

0 p

2

p

M2

1-1

PP

M2

1-

1T

T

M2

1-1

TT

T

T

RT

V

2

1-1

1RT

2V

1RT

1

R CC

TC2

VTC

+=

=

+=

=

+=

=+

=+

==

=+

The quantities P 0, 0, and T 0 are calledstagnation pressure, stagnation densityand stagnation temperature. Theyrepresent the properties the flow wouldhave if brought to rest reversibly andadiabatically.

Note that:

1

000TT

PP

=

=

Summary


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Summary For steady incompressible flow ( = constant) of a frictionless (inviscid) fluid,

we need only concern ourselves with conservation of mass (Continuity) andmomentum (Bernoulli):

For steady, adiabatic and frictionless (isentropic) compressible flow, P, T, andV are all variables, and are related through conservation of mass (Continuity),momentum (Energy), the first law of thermodynamics (Isentropic flow relations)and the Equation of State

constantPV

2

1

constantAV

2 =+

=

RTP

TT

PP

constant2

Vh

constantAV

1

000

2

=

=

=

=+

=

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