icmm tgzielinski idealflow.slides

7/29/2019 ICMM TGZielinski IdealFlow.slides

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Introduction Ideal ow theory Basic aerodynamics

Fundamentals of Fluid Dynamics:Ideal Flow Theory

Introductory Course on Multiphysics Modelling

TOMASZ G. Z IELINSKI(after: D.J. A CHESON s Elementary Fluid Dynamics )

Institute of Fundamental Technological Research

Warsaw

Poland

http://www.ippt.gov.pl/~tzielins/
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.ippt.gov.pl/~tzielins/http://www.ippt.gov.pl/~tzielins/mailto:[email protected]


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Outline

1 IntroductionMathematical preliminariesPreliminary ideas and denitionsConvective derivative


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Outline


2 Ideal ow theory

Ideal uidIncompressibility conditionEulers equations of motionBernoulli theoremsVorticity


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Outline


2 Ideal ow theory


3 Basic aerodynamicsSteady ow past a xed wingFluid circulation round a wingKuttaJoukowski theorem and conditionConcluding remarks


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Outline


2 Ideal ow theory




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Mathematical preliminaries

Theorem (The divergence theorem)

Let the region V

be bounded by a simple surface S

with unit outward normal n . Then:

S

f n dS =V

f dV ; in particular S

f n dS =V

f dV .

Theorem (Stokes theorem)

Let C be a simple closed curve spanned by a surface S with unit normal n . n

S

C

Then:

C

f d x =

S

f n dS .

Greens theorem in the plane may be viewed as a special case ofStokes theorem (with f = u ( x, y), v( x, y), 0 ):

C

u d x+ v d y =S

v

x

u

yd xd y .



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Preliminary ideas and denitionsA usual way of describing a uid ow is by means of the owvelocity dened at any point x = ( x, y, z) and at any time t :

u = u ( x, t) = u ( x, t), v( x, t), w( x, t) .

Here, u , v, w are the velocity components in Cartesian coordinates.



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Preliminary ideas and denitions

u = u ( x, t) = u ( x, t), v( x, t), w( x, t) .

Denition (Steady ow)

A steady ow is one for which

u

t= 0 , that is, u = u ( x) = u ( x), v( x), w( x) .



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u = u ( x, t) = u ( x, t), v( x, t), w( x, t) .



u


Denition (Two-dimensional ow)

A two-dimensional ow is of the form

u = u ( x, t), v( x, t), 0 where x = ( x, y) .



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y y


u = u ( x, t) = u ( x, t), v( x, t), w( x, t) .



u


Denition (Two-dimensional ow)

A two-dimensional ow is of the form

u = u ( x, t), v( x, t), 0 where x = ( x, y) .

Denition (Two-dimensional steady ow)

A two-dimensional steady ow is of the form

u = u ( x), v( x), 0 where x = ( x, y) .



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Streamlines and particle paths (pathlines)

Denition (Streamline)

A streamline is a curve which, at any particular time t , has the samedirection as u ( x, t) at each point. A streamline x = x( s), y = y( s), z = z( s)( s is a parameter) is obtained by solving at a particular time t :

d xd s

u

=

d yd s

v

=d zd s

w

.



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d xd s

u

=

d yd s

v

=d zd s

w

.

For a steady ow the streamline pattern is the same at all times,and uid particles travel along them.In an unsteady ow, streamlines and particle paths are usually

quite different.



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d xd s

u

=

d yd s

v

=d zd s

w

.

Example

Consider a two-dimensional ow described as follows

u ( x, t) = u 0 , v( x, t) = at , w( x, t) = 0,

where u 0 and a are positive constants. Now, notice that:in this ow streamlines are (always) straight lines,

uid particles follow parabolic paths: x(t) = u0

t , y(t) = a t2

2 y = ax

2

2u 20.



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u ( x, t) = u 0 , v( x, t) = at , w( x, t) = 0,

x

y

t = 0

in this ow streamlines are (always) straight lines,

uid particles follow parabolic paths: x(t) = u 0 t , y(t) =a t 2

2 y =ax 2

2u 20 .



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u ( x, t) = u 0 , v( x, t) = at , w( x, t) = 0,

x

y

t = t



2 y =ax 2

2u 20 .



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u ( x, t) = u 0 , v( x, t) = at , w( x, t) = 0,

x

y

t = 2 t



2 y =ax 2

2u 20 .



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u ( x, t) = u 0 , v( x, t) = at , w( x, t) = 0,

x

y

t = 0

t = t

t = 2 t



2 y =ax 2

2u 20 .



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Convective derivative

Let f ( x, t) denote some quantity of interest in the uid motion.



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Note that f

tmeans the rate of change of f at a xed

position x in space.



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Note that f


position x in space.The rate of change of f following the uid is

D f D t

=dd t

f x(t), t = f x

d xd t

+ f t

where x(t) = x(t), y(t), z(t) is understood to change with time atthe local ow velocity u , so that

d xd t

=d xd t

,d yd t

,d zd t

= u , v, w .

Therefore, f x

d xd t

= f x

d xd t

+ f y

d yd t

+ f z

d zd t



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Note that f



D f D t

=dd t

f x(t), t = f x

d xd t

+ f t

where x(t) = x(t), y(t), z(t) is understood to change with time atthe local ow velocity u , so that

d xd t

=d xd t

,d yd t

,d zd t

= u , v, w .

Therefore, f x

d xd t

= f x

u + f y

v+ f z

w = (u ) f .


d


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Note that f



D f D t

=dd t

f x(t), t = f x

d xd t

+ f t

f x

d xd t

= f x

u + f y

v+ f z

w = (u ) f .

Denition (Convective derivative)

The convective derivative (a.k.a. the material , particle ,Lagrangian , or total-time derivative ) is a derivative taken withrespect to a coordinate system moving with velocity u (i.e., followingthe uid):

D f D t

= f t

+ (u ) f .


C i d i i


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Denition (Convective derivative)

The convective derivative (a.k.a. the material , particle ,Lagrangian , or total-time derivative ) is a derivative taken withrespect to a coordinate system moving with velocity u (i.e., followingthe uid):

D f D t

= f t

+ (u ) f .

By applying the convective derivative to the velocity components u , v,w in turn it follows that the acceleration of a uid particle is

Du

D t=

u

t+ (u )u .


C i d i i


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By applying the convective derivative to the velocity components u , v,

w in turn it follows that the acceleration of a uid particle isDuD t

=ut

+ (u )u .

Example

Consider uid in uniform rotation with angular velocity , so that:

u = y , v = x , w = 0.

The ow is steady so ut = 0 , but

DuD t

= y

x+ x

y

y, x, 0 = 2 x, y, 0 .

represents the centrifugal acceleration 2 x2 + y2 towards therotation axis.


O tli


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Outline

1 Introduction

Mathematical preliminariesPreliminary ideas and denitionsConvective derivative

2 Ideal ow theoryIdeal uidIncompressibility conditionEulers equations of motionBernoulli theoremsVorticity



Id l id


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Ideal uid

Denition ( Ideal uid )

Properties of an ideal uid :1 it is incompressible : no uid element can change in volume as

it moves;


Ideal uid


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Ideal uid



it moves;2 it has constant density : is the same for all uid elements and

for all time t (a consequence of incompressibility);


Ideal uid


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Ideal uid




for all time t (a consequence of incompressibility);3 it is inviscid , so that the force exerted across a geometrical

surface element n S within the uid is p n S ,

where p ( x, t) is a scalar function of pressure , independent on thenormal n .


Ideal uid


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Ideal uid




for all time t (a consequence of incompressibility);3 it is inviscid , so that the force exerted across a geometrical

surface element n S within the uid is p n S ,

where p ( x, t) is a scalar function of pressure , independent on thenormal n .

RemarksAll uids are to some extent compressible and viscous.Air , being highly compressible , can behave like anincompressible uid if the ow speed is much smaller thanthe speed of sound .


Incompressibility condition


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S

V

Sn

Let S be a xed closed surface drawnin the uid, with unit outwardnormal n , enclosing a region V .




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S

V

Sn

Let S be a xed closed surface drawnin the uid, with unit outwardnormal n , enclosing a region V .Fluid enters the enclosed region V atsome places on S , and leaves it atothers.




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S

V

Sn

uLet S be a xed closed surface drawnin the uid, with unit outwardnormal n , enclosing a region V .Fluid enters the enclosed region V atsome places on S , and leaves it atothers.

The velocity component along theoutward normal is u n , so thevolume of uid leaving through asmall surface element S in unit timeis u n S .




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S

V

Sn

uLet S be a xed closed surface drawnin the uid, with unit outwardnormal n , enclosing a region V .Fluid enters the enclosed region V atsome places on S , and leaves it atothers.

The velocity component along theoutward normal is u n , so thevolume of uid leaving through asmall surface element S in unit timeis u n S .The net volume rate at which uid isleaving V equals

S

u n dS

and must be zero for an

incompressible uid.




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S

V

Sn

uNow, using the divergence theorem andthe assumption of the smoothness ofow (i.e., continuous velocity gradient)gives


For all regions within an incompressibleuid

S

u n d S =V

u dV = 0

which (assuming that u is continuous)yields the following important constrainton the velocity eld:

u = 0

(everywhere in the uid).


Eulers equations of motion


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Euler s equations of motionThe net force exerted on an arbitrary uid element is

S

p n dS = V

p dV

(the negative sign arises because n points out of S ). Now, providedthat p is continuous it will be almost constant over a small element V . The net force on such a small element due to the

pressure of the surrounding uid will therefore be p V .




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u e s equat o s o ot oThe net force exerted on an arbitrary uid element is

S

p n dS = V

p dV

(the negative sign arises because n points out of S ). Now, providedthat p is continuous it will be almost constant over a small element V . The net force on such a small element due to the

pressure of the surrounding uid will therefore be p V .

The principle of linear momentum implies that the following forcesmust be equal for a uid element of volume V :

p + g V the total (external) net force acting on theelement in the presence of gravitational body force per unitmass g Nkg =

ms2 (the gravity acceleration),

V DuDt the inertial force , that is, the product of the elementsmass (which is conserved) and its acceleration.




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q

The principle of linear momentum implies that the following forcesmust be equal for a uid element of volume V :

p + g V the total (external) net force acting on theelement in the presence of gravitational body force per unitmass g N

kg = m

s2 (the gravity acceleration),

V DuDt the inertial force , that is, the product of the elementsmass (which is conserved) and its acceleration.

This results in the Eulers momentum equation for an ideal uid.Another equation of motion is the incompressibility constraint.

Eulers equations of motion for an ideal uid

DuD t

= 1 p + g , u = 0




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Boundary and interface-coupling conditions

Impermeable surface. The uid cannot ow through the boundary

which means that the normal velocity is constrained(no-penetration condition):

u n = u n .

Here, u n is the prescribed normal velocity of the impermeableboundary ( u

n= 0 for motionless, rigid boundary).




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u n = u n .



Free surface. Pressure condition (involving surface tension effects):

p = p0 2T mean .

Here: p0 is the ambient pressure,T is the surface tension,mean is the local mean curvature of the free surface.


Eulers equations of motiond d f l d


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u n = u n .



Free surface. Pressure condition (involving surface tension effects):

p = p0 2T mean .

Here: p0 is the ambient pressure,T is the surface tension,mean is the local mean curvature of the free surface.

Fluid interface. The continuity of normal velocity and pressure at theinterface between uids 1 and 2:

u (1) u (2) n = 0 , p (1) p(2) = 0 2 T (1) + T (2) mean .


Bernoulli theorems


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The gravitational force , being conservative , can be written as thegradient of a potential :

g = where = gz m2

s2 ,


Bernoulli theorems


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The gravitational force , being conservative , can be written as thegradient of a potential :

g = where = gz m2

s2 ,

and the momentum equationut

+ (u ) u

( u ) u + ( 12 u 2 )

= p

+

can be cast into the following form

ut

+ ( u ) u = H where H = p

+12

u 2 + .


Bernoulli theorems


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u

t+ ( u ) u = H where H =

p+

1

2u 2 + .

For steady ow ( ut = 0 ): ( u ) u = H u

(u ) H = 0 .

Theorem (Bernoulli streamline theorem)

If an ideal uid is in steady ow, then H is constant along a streamline.


Bernoulli theorems


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u


p+

1

2u 2 + .


(u ) H = 0 .



Denition ( Irrotational ow )

A ow is irrotational if u = 0 .


Bernoulli theorems


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u


p+

1

2u 2 + .


(u ) H = 0 .



Denition ( Irrotational ow )

A ow is irrotational if u = 0 .

For steady irrotational ow :

H = 0 .

Theorem (Bernoulli theorem for irrotational ow)

If an ideal uid is in steady irrotational ow, then H is constant throughout the whole uid.


Vorticity: rotational and irrotational ow


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Denition ( Vorticity )

Vorticity is a concept of central importance in uid dynamics; it isdened as

= u 1s .




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= u 1s .

For an irrotational ow : = 0 .Vorticity has nothing directly to do with any global rotation of the

uid.




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= u 1s .


uid.Interpretation of vorticity (in 2D)For two-dimensional ow (when u = u ( x, t), v( x, t), 0 ):

= 0, 0, where =v x

u y

.




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= u 1s .


uid.Interpretation of vorticity (in 2D)For two-dimensional ow (when u = u ( x, t), v( x, t), 0 ):

= 0, 0, where =v x

u y

.

Then, at any point of the ow eld 2 represents the average angularvelocity of two short uid line-elements that happen, at that instant,to be mutually perpendicular.




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Interpretation of vorticity (in 2D)For two-dimensional ow (when u = u ( x, t), v( x, t), 0 ):

= 0, 0, where = v x

u y

.

A B

C

x

y

u x

x

v x

x

u y

y

v y

y

average angular velocity:AB + AC

2=

12

v x

u y

=2




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Interpretation of vorticity (in 2D)For two-dimensional ow (when u = u ( x, t), v( x, t), 0 ):

= 0, 0, where = v x

u y

.

A B

C

x

y

u x

x

v x

x

u y

y

v y

y

average angular velocity:AB + AC

2=

12

v x

u y

=2

Thus, the vorticity acts as a measure of the local rotation , or spin,

of uid elements.


Line vortex ow and uniformly rotating ow


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Flow may be written in cylindrical polar coordinates (r , , z):

u = u r er + u e + u z e z .

Consider the following steady, two-dimensional ows (with u = u (r ),u r = u z = 0 , and being a constant):Line vortex ow: u = r e

Uniformly rotating ow: u = r e

u = u eu

1r

u r




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u = u r er + u e + u z e z .



u = u eu

1r

u r




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u = u r er + u e + u z e z .



u = u eu

1r

u r




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u = u r er + u e + u z e z .



u = u eu

1r

u r




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u = u r er + u e + u z e z .

Consider the following steady, two-dimensional ows (with u = u (r ),u r = u z = 0 , and being a constant):Line vortex ow: u = r e although the uid is clearly rotating in a

global sense the ow is in fact irrotational since = 0 (except atr = 0 , where neither u nor is dened);

Uniformly rotating ow: u = r e here, = 0, 0, 2 and avorticity meter is carried around as if embedded in a rigid body.

u = u eu

1r

u r


Rankine vortex


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Denition ( Rankine vortex )

Rankine vortex is a steady, two-dimensional ow described as

u = u e with u = r for r a (uniformly rotating ow), a 2

r for r > a (line vortex ow),

where and a are constants.


Rankine vortex


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Denition ( Rankine vortex )

Rankine vortex is a steady, two-dimensional ow described as

u = u e with u = r for r a (uniformly rotating ow), a 2

r for r > a (line vortex ow),

where and a are constants. Therefore,

= e z with =1r

(ru )r

1r

( r 2 )r = 2 for r a (vortex core ),

1r

( a 2 )r = 0 for r > a (irrotational).


Rankine vortex


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r

u

a

a

u =

r for r a a 2r for r > a

r

2

a

= 2 for r a0 for r > a

The Rankine vortex serves as a simple idealized model for areal vortex.Real vortices are typically characterized by fairly small vortexcores (a is small) in which, by denition, the vorticity isconcentrated .Outside the core the ow is irrotational .


Vorticity equation


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ut

+ u = H

t+ ( u ) = 0

t+ (u ) ( ) u + u

0 u

0= 0 .

Here, the fourth term vanishes because the uid is incompressiblewhile the fth term vanishes because = 0 .


Vorticity equation


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ut

+ u = H

t+ ( u ) = 0

t+ (u ) ( ) u + u

0 u

0= 0 .

Here, the fourth term vanishes because the uid is incompressiblewhile the fth term vanishes because = 0 .

Vorticity equation

t+ (u ) = ( ) u , or

D

D t= ( ) u .

The vorticity equation is extremely valuable: as a matter of fact, itinvolves only u since the pressure has been eliminated and = u .


Vorticity equation


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Vorticity equation

t + (u ) = ( ) u , orD

D t = ( ) u .


In the two-dimensional ow (u = u ( x, t), v( x, t), 0 , = 0, 0, ( x, t) )of an ideal uid subject to a conservative body force g the vorticity of each individual uid element is conserved :

DD t

= 0.


Vorticity equation


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Vorticity equation

t + (u ) = ( ) u , orD

D t = ( ) u .


In the two-dimensional ow (u = u ( x, t), v( x, t), 0 , = 0, 0, ( x, t) )of an ideal uid subject to a conservative body force g the vorticity of each individual uid element is conserved :

DD t

= 0.

In the steady, two-dimensional ow of an ideal uid subject to aconservative body force g the vorticity is constant along astreamline :

(u ) = 0.


Outline


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2 Ideal ow theoryIdeal uid

Incompressibility conditionEulers equations of motionBernoulli theoremsVorticity



Steady ow past a xed wing


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Steady ow past a wing at small angle of attack (incidence) istypically irrotational .




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There are no regions of closed streamlines in the ow; all thestreamlines can be traced back to .




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There are no regions of closed streamlines in the ow; all thestreamlines can be traced back to .The vorticity is constant along each streamline , and hence equal oneach one to whatever it is on that particular streamline at .




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There are no regions of closed streamlines in the ow; all thestreamlines can be traced back to .The vorticity is constant along each streamline , and hence equal oneach one to whatever it is on that particular streamline at .As the ow is uniform at , the vorticity is zero on all streamlinesthere; hence, it is zero throughout the ow eld around the wing .



Typical measured pressure distribution on a wing in steady ow:


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yp p g ythe pressures on the uppersurface are substantially lowerthan the free-stream value p ;the pressures on the lowersurface are a little higherthan p ;

in fact, the wing gets most of itslift from a suction effect on itsupper surface .



Typical measured pressure distribution on a wing in steady ow:


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yp p g ythe pressures on the uppersurface are substantially lowerthan the free-stream value p ;the pressures on the lowersurface are a little higherthan p ;

in fact, the wing gets most of itslift from a suction effect on itsupper surface .

Why the pressures above the wing are less than those below?

The ow is irrotational and the Bernoulli theorem states that

H = p +12 u 2 is constant throughout 2D irrotational ows .

Explaining the pressure differences, and hence the lift on thewing, thus reduces to explaining why the ow speeds above thewing are greater than those below .An explanation is in terms of the concept of circulation .


Fluid circulation round a wing


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Denition ( Circulation )

The circulation round some closed curve C lying in the uid regionis dened as =

C

u d x m2

s .

IfS

is the region enclosed by the curveC

then the Stokes theoremgives =

C

u d x =S

( u ) n dS ,

or in the two-dimensional context

=

C

u d x+ v d y =S

v x

u y

d xd y .

Notice that in the surface integrals vorticity terms appear.



= 0 if the closed curve C is spanned by a surface S which lies


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p ywholly in the region of irrotational ow, that is, = 0 for anyclosed curve not enclosing the wing .This cannot be stated for any closed curve that does enclosethe wing . What can be stated is that such circuits have thesame value of .


Fluid circulation round a wing = 0 if the closed curve C is spanned by a surface S which lies


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wholly in the region of irrotational ow, that is, = 0 for anyclosed curve not enclosing the wing .This cannot be stated for any closed curve that does enclosethe wing . What can be stated is that such circuits have thesame value of .

Proof.Consider two arbitrary closed curves, ABCA and DFED, enclosing a wing.

0 = ABCADEFDA= ABCA + AD + DEFD + DA

= ABCA + AD DFED AD ABCA = DFED



= 0 if the closed curve C is spanned by a surface S which lies


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p ywholly in the region of irrotational ow, that is, = 0 for anyclosed curve not enclosing the wing .This cannot be stated for any closed curve that does enclosethe wing . What can be stated is that such circuits have thesame value of .

Therefore:

circulation round a wing is permissible in a steady irrotational ow.

Yet, two questions still arise:1 Why there should be any circulation?2 Why it should be negative, corresponding to larger ow speeds

above the wing than below?

The answers are given by the KuttaJoukowski condition .


KuttaJoukowski theorem and condition

Consider a steady, irrotational ow of uid round a wing. According to


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Consider a steady, irrotational ow of uid round a wing. According toideal ow theory, the drag on the wing (the force per unit length of

wing parallel to the oncoming stream) is zero. What is the lift (theforce per unit length of wing perpendicular to the stream) is stated bythe following theorem.

Theorem (Kutta-Joukowski lift theorem )

Let be the uid density and U the ow speed at innity. Then, the lift of the wing is

F y = U Nm ,

where is the uid circulation around the wing.



Theorem (K tta Jo ko ski lift theorem )


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Theorem (Kutta-Joukowski lift theorem )

Let be the uid density and U the ow speed at innity. Then, the lift of the wing is

F y = U Nm ,

where is the uid circulation around the wing.

Obviously, of a great importance is here the fact that = 0 . In thecase of a wing with a sharp trailing edge this can be explained asfollows: one good reason for non-zero circulation is that there wouldotherwise be a singularity (innity) in the velocity eld. This is statedby the KuttaJoukowski condition .

= 0 = K < 0



Obviously, of a great importance is here the fact that = 0 . In the


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y, g pcase of a wing with a sharp trailing edge this can be explained as

follows: one good reason for non-zero circulation

is that there wouldotherwise be a singularity (innity) in the velocity eld.

= 0 = K < 0

KuttaJoukowski condition/hypothesis

The circulation is such that the ow leaves the trailing edge

smoothly , or, equivalently, that the ow speed at the trailingedge is nite .The ow speed is nite at the trailing edge only for one valueof the circulation around the wing: the critical value K. Thisparticular ow will correspond to the steady ow that is actuallyobserved.



KuttaJoukowski condition/hypothesis


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The circulation is such that the ow leaves the trailing edge

smoothly , or, equivalently, that the ow speed at the trailingedge is nite .The ow speed is nite at the trailing edge only for one valueof the circulation around the wing: the critical value K. Thisparticular ow will correspond to the steady ow that is actuallyobserved.

The critical value K depends on the ow speed at innity U , and onthe size, shape, and orientation of the wing.

Thin, symmetrical wings

For a thin and symmetrical wing of length L, making an angle with the oncoming stream, the critical value of circulation is

K U L sin .



Thin symmetrical wings


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Thin, symmetrical wings

For a thin and symmetrical wing of length L, making an angle with the oncoming stream, the critical value of circulation is

K U L sin .

Using this formula for the lifttheorem gives the following result

F y U 2 L sin ,

which is in excellent accord withexperiment provided that theangle of attack is small , that isonly a few degrees, depending onthe shape of the wing.

F y

c

Inviscid theory < c 6 12

Experiment


Concluding remarks

KuttaJoukowski hypothesis provides a rational explanation forh l d d h


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the circulation round a wing in steady ight.It says nothing about the dynamical process by which thatcirculation is generated when a wing starts from a state of rest.


Concluding remarks

KuttaJoukowski hypothesis provides a rational explanation forh i l i d i i d i h


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Starting vortex

The circulation is generated by the so-called starting vortex , whichis a concentration of vorticity which forms at the trailing edge of

a wing as it accelerates from rest in a uid. It leaves the wing (whichnow has an equal but opposite bound vortex round it), and rapidlydecays through the action of viscosity.


Concluding remarks

KuttaJoukowski hypothesis provides a rational explanation forth i l ti d i i t d i ht


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Starting vortex

The circulation is generated by the so-called starting vortex , whichis a concentration of vorticity which forms at the trailing edge of

a wing as it accelerates from rest in a uid.

Question : Is a starting vortex theoretically explicable?


Concluding remarks



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A legitimate conclusion on the basis of ideal ow theory:

If the wing and uid are initially at rest, the vorticity is initially zerofor each uid element.It remains zero since the vorticity is conserved for each uidelement.

Therefore there should be no starting vortex.


Concluding remarks



85/85

A legitimate conclusion on the basis of ideal ow theory:

If the wing and uid are initially at rest, the vorticity is initially zerofor each uid element.It remains zero since the vorticity is conserved for each uidelement.

Therefore there should be no starting vortex.

An explanation of the starting vortex:

Ideal ow theory accounts well for the steady ow past a wing.The explanation of how that ow became established involvesviscous effects in a crucial way.But air, in some sense, is hardly viscous at all! Yet, viscouseffects are sufciently subtle that shedding of the vortex , whichis an essentially viscous process, occurs no matter how smallthe viscosity of the uid happened to be.

icmm tgzielinski idealflow.slides

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