chapter 4 simplification to the conservation...

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Chapter 4 Simplification to the Conservation Equations

4.1 Governing Equations for Ideal Fluid Flows Isotropic Fluid: Isotropic, Newtonian are assumed to have linear relationship between stress and rate of strain. A fluid is said to be isotropic when the relation between the components of stress and those of rate of train is the same in all directions. It is said to be Newtonian when this relationship is linear, that is when the fluid obeys stokes law of friction. Ideal flow: Non-heat conducting, inviscid, incompressible, homogeneous fluid is defined as ideal fluid. Assumptions used are:

• Non-heat conductive • Homogeneous • Incompressible • Inviscid flow

Controlling equations:

Continuity equation: 0)( =•∇+∂∂ V

tr

ρρ

Momentum equation: 0~)()(=−•∇−∇+•∇+

∂∂ fPVV

tV rrrr

ρτρρ

Energy equation: ⇒ drop out since non-heat conductive assumption State equation: TRP ρ=

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Simplification of continuity equation:

0

0

00)(

=•∇⇒

=•∇+⇒

=•∇+∇•+∂∂

⇒=•∇+∂∂

V

VtD

D

VVt

Vt

r

r

rrr

ρρ

ρρρρρ

• This equation does not imply that the density is a constant through the flow field for all

times [Eulerian frame]

• It simply states that the density of an identifiable fluid element remains unchanged. In order words, if, at some internal instant, the fluid is inhomogeneous (i.e., whose fluid element are of different density), it will remain inhomogeneous for all other times. If the fluid is homogenous (i.e., once whose elements are all of the same density) at some initial instant, it will remain homogenous for all other time.

• For an incompressible, inhomogeneous fluid there will be both local and spatial variation

of density although the material rate of change of density is zero.

• For all incompressible homogeneous fluid, however, the local and spatial variation varnish independently, and the density is constant in time as well as in space. i,e., incompressible, homogenous fluid, =ρ constant.

Simplification of momentum equation:

fPtD

VD

fPVVt

VVt

V

fPVVVVt

Vt

V

fPVVtV

inviscidtodue

inviscidtodue

rr

r876rrr

rr

r876rrrrr

r

rrrr

ρρ

ρτρρρ

ρτρρρρ

ρτρρ

+−∇=⇒

=−•∇−∇+∇•+∂∂

+•∇+∂∂

⇒

=−•∇−∇+∇•+•∇+∂∂

+∂∂

⇒

=−•∇−∇+•∇+∂

∂

=

=

0~])([)]([

0~)()(

0~)()(

0

0

fPtD

VD rr

ρρ +−∇= is also called Euler equation

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For non-heat-conducting, inviscid, incompressible, homogeneous fluid, density =ρ constant, internal energy =e constant (i.e., =T constant) for the entire flow. The dependent variables become PV ,

r.

For ideal fluid, the governing equations are: 1). Continuity equation: 0=•∇ V

r

2). Euler equation fPtD

VD rr

ρρ +−∇= or fPtD

VD rr

+−∇= )(ρ

For the Euler equation, it can be re-written as:

fPVVt

V

fPtD

VD

rrrr

rr

+−∇=∇•+∂∂

⇒

+−∇=

)()(

)(

ρ

ρ

According to vector identity relation:

)()()()()( BAABABBABArrrrrrrrrr

×∇×+×∇×+∇•+∇•=•∇ Using VBA

rrr== , then

)]()[(2)()()()()( VVVVVVVVVVVVVVrrrrrrrrrrrrrr

×∇×+∇•=×∇×+×∇×+∇•+∇•=•∇

)()2

()( VVVVVVrr

rrrr

×∇×−•

∇=∇•⇒

Therefore, Euler equation can be expressed as:

fPVVVVt

V rrrrrr

+−∇=×∇×−•

∇+∂∂ )()()

2(

ρ

Rearrange the above equation, we get:

fVVPVVt

V rrrrrr

=×∇×−+•

∇+∂∂ )()

2(

ρ

Let us consider body forces that are conservative only.

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A necessary and sufficient condition for fr

to be conservation is that 0=×∇ fr

. That is fr

is irrotational through the region chosen. Since 0=×∇ f

r , we can represent the force field f

r as

the gradient of a scalar field U, i.e., Uf ∇=r

If fr

is a conservative force field, then the work done by fr

given by ∫ •=b

a

rdfW rr is

independent of path for any points between a and b.

Proof: )()( aUbUdUrdUrdfWb

a

b

a

b

a

−==•∇=•= ∫∫∫rrr

Thus, W is only a function of U at the two end points and not dependent on the path. Substitute for f

r into the Euler equation, it becomes:

UVVPVVt

V∇=×∇×−+

•∇+

∂∂ )()

2(

rrrrr

ρ

Rearrange the equation, it can express as:

0)()2

( =×∇×−−+•

∇+∂∂ VVUPVV

tV rr

rrr

ρ.

This is a differential equation which holds at every point in an ideal fluid (i.e. non-heat conducting, homogeneous, incompressible, inviscid flow). There are two special cases for which this equation can be integrated in closed form: 1). Irrotational flow 2). Steady flow along streamline

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4.2 Rotation, Vorticity and Circulation: Fluid rotation Definition: Fluid rotation at a point O (also called the angular velocity at the point) is the average angular velocity of two infinitesimal and mutually perpendicular fluid lines OA and OB instantaneously passing thorough point O. A fluid line is a line passing through a set of fluid particles of fixed identity. The average fluid rotation at a point is given by

)(21 V

rv ×∇=ϖ .

Vorticity Definition: Vorticity is defined to be V

rvr×∇==Ω ϖ2 .

In an irrotational velocity field as know as irrotational flow, i.e., 0=×∇ Vr

and 0=Ωr

. Circulation Definition: Circulation is defined as ∫−=Γ

C

ldVrr

, where C is a

closed path. Circulation is the value of the line integral of the velocity around a closed curve C. By Stokes’ theorem:

AdVldVC S

rrrr•×∇−=−=Γ ∫ ∫∫ )(

Or

AdldVC S

rrrr•Ω−=−=Γ ∫ ∫∫ where S is the area enclosed by curve C.

O

A B

X

Y

Z

X

Y Vr

ldr

O

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4.3 Potential Flow and Potential Function Potential Flow Definition: A non-heat conducting, homogeneous, inviscid, incompressible (i.e., ideal fluid), and irrotational flow is defined as potential flow. The dependent variables of ideal fluid are P and V

r . The governing equations of the Fluid flow

are: 1). Continuity equation: 0=•∇ V

r

2). Euler equation 0)

2(

0)()2

(

=−+•

∇+∂∂

⇒

=×∇×−−+•

∇+∂∂

UPVVt

V

VVUPVVt

V

ρ

ρrrr

rrrrr

Velocity Potential (φ ) Definition: Velocity potential is defined only for ideal irrotational flow for either steady or unsteady flows as:

φ∇=Vr

Since 3

332

221

11

332211

ˆˆˆ

ˆˆˆ

eqh

eqh

eqh

eVeVeVV

∂∂

+∂∂

+∂∂

=∇

++=φφφφ

r

Therefore: 33

322

211

11;1;1

qhV

qhV

qhV

∂∂

=∂∂

=∂∂

=φφφ

Note: Potential function φ is defined for either 2-D or 3-D, steady or unsteady flow. As long as

the flow is irrotational, the potential function φ exists. Since 0=•∇ V

r and φ∇=V

r ⇒ 02 =∇=∇•∇=•∇ φφV

r

02 =∇ φ ⇒ Laplace Equation.

Therefore, the potential function satisfies the Laplace Equation.

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4.4 Integral of Euler Equation in Irrotational Flows

The Euler equation fPVVVVt

V rrrrrr

+−∇=×∇×−•

∇+∂∂ )()()

2(

ρ

If fr

is conservative, then, fr

can be represented as the gradient of a scalar field U, i.e.,

Uf ∇=r

Therefore, the above equation can be expressed as

0)()2

( =×∇×−−+•

∇+∂∂ VVUPVV

tV rr

rrr

ρ.

If the flow is irrotational , then 0=×∇ V

r. The above equation can simplified as:

0)2

( =−+•

∇+∂∂ UPVV

tV

ρ

rrr

Since the velocity field is irrotational (i.e., potential flow), then φ∇=V

r

We can represent )(tt

V∂∂

∇=∂∂ φr

and above equation becomes

0)2

()( =−+•

∇+∂∂

∇ UPVVt ρφ

rr

⇒ 0)2

( =−+•

+∂∂

∇ UPVVt ρφ

rr

Taking a dot product with ld

r, an element length along any arbitrary path, we get:

0)2

( =•−+•

+∂∂

∇ ldUPVVt

rrr

ρφ

By definition, ()() dld =•∇

r, and therefore,

0)2

()2

( =−+•

+∂∂

=•−+•

+∂∂

∇ UPVVt

dldUPVVt ρ

φρ

φrr

rrr

Integrating the above equation leads to the equation known as unsteady Bernoulli equation.

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)(2

0)2

(

tFUPVVt

UPVVt

d

=−+•

+∂∂

⇒

=−+•

+∂∂

∫

ρφ

ρφ

rr

rr

At any instant of time, F(t) is a constant for all points of the fluid field. Because the integration is only with respect to space, the constant is independent of space coordinates, where as it may, in general, still depend on time. Body force most often encountered is that due to gravity.

Uf ∇=r

Or 33

322

211

1 ;;qh

Ufqh

Ufqh

Uf∂∂

=∂∂

=∂∂

=

If we choose a Cartesian coordinate system with the Z-axial to be positive when pointing upward and normal to the surface of the earth,. The force on a body of mass m is given by (0,0,-mg). Thus for f

r which is body force vector per unit mass. It becomes ),0,0(),,( gffff zyx −==

r.

0

00

ˆˆˆ

=

−∂∂

∂∂

∂∂

=×∇

gzyx

kji

fr

⇒ Gravity force field is conservative.

ZgUgZU

−=⇒−=∂∂ + Constant C

By choosing reference as 0=U at 0=Z leads to Constant C=0. Then, ZgU −= . The unsteady Bernoulli equation in the conservative, irritation body force (gravity) and irrotational velocity field becomes:

)(2

tFgZPVVt

=++•

+∂∂

ρφ

rr

If the flow is steady, the equation will become

CgZPVV=++

•ρ2

rr

C is a constant all over the flow field.

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4.5 Path Lines, Streak Line and Streamlines Pathlines This is the path traced out by any one particle of the fluid in motion. It is a history of the particle’s location of the same particle. Streaklines Streakline is defined as an instantaneous line whose points are occupied by all the particles originating from some specific points in the fluid field. Streamlines A streamline is defined as an imaginary line drawn in the fluid whose tangent at any point is in the direction of the velocity vector at that point.

• By definition there is no flow across it at any point. • Any streamline may be replaced by a solid boundary without modifying the flow. • Any solid boundary is itself a streamline of the flow around it.

Streakline, pathlines and streamlines are in general different in unsteady flows, while in steady flow both are identical. According to the definition of streamlines, no fluid can cross a stream surface or the walls of a stream tube. By definition, the equation of a streamline is 0=× SdV

vr,

where Sdv

is an elemental length along a streamline. Vr

is the flow velocity vector.

0=× SdVvr

0)(ˆ)(ˆ)(ˆ

ˆˆˆ

1122213

33111322233321

332211

321

321

=−+

−+−=

=×

dqhVdqhVedqhVdqhVedqhVdqhVe

dqhdqhdqhVVVeee

SdVvr

In differential form:

z

x y

P(xi, yi, zi) Sdv

Vr

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000

112221

331113

223332

=−=−=−

dqhVdqhVdqhVdqhVdqhVdqhV

Stream surface and stream tube If at any instant time, we draw an arbitrary line in the fluid region and draw the streamlines passing the line, a surface is formed. Such surface is caller streamline surface or stream surface in short. If we consider a closed curve and draw all the streamline passing through it, a stream tube is formed.

Stream surface

Stream tube

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4.6 Integral of Euler Equation along a Streamline In the body force is conservative; the Euler equation can be expressed as:

UPVVVVt

V∇+−∇=×∇×−

•∇+

∂∂ )()()

2(

ρ

rrrrr

For the steady flow 0=∂∂

tVr

Then: )()2

( VVUPVV rrrr

×∇×=−+•

∇ρ

Multiple (dot product) both sides of the above equation by Sd

r, which is an element length alone

a streamline.

SdVVSdUPVV rrrrrr

•×∇×=•−+•

∇ )()2

(ρ

According to vector identity: )()()( CBAACBBAC

rrrrrrrrr×•=×•=×•

If VBVASdC

rrrrrr×∇=== ;; ,

Then 0)()()]([)( =×•×∇=×∇×•=•×∇× VSdVVVSdSdVV

rrrrrrrrr

Therefore, 0)

2(

0)2

(

=−+•

⇒

=•−+•

∇

UPVVd

SdUPVV

ρ

ρrr

rrr

Integrating above equation yields constUPVV=−+

•ρ2

rr

Consider U is only the conservative gravity field, therefore

SHconstgZPVV==++

•ρ2

rr

, where SH is a constant only along a streamline.

- 12/21 -

4.7 Stream Function According to the definition of streamlines, no fluid can cross a stream surface or the walls of a stream tube. By definition, the equation of a streamline is 0=× SdV

vr. Where Sd

vis an elemental length along

a streamline. Vr

is the flow velocity vector.

0=× SdVvr

0)(ˆ)(ˆ)(ˆ

ˆˆˆ

1122213

33111322233321

332211

321

321

=−+−+−=

=×

dqhVdqhVedqhVdqhVedqhVdqhVe

dqhdqhdqhVVVeee

SdVvr

In differential form:

000

112221

331113

223332

=−=−=−


or

constdqh

Vdqh

VdqhV

===33

3

22

2

11

1

⎪⎭

⎪⎬

⎫

=−=−=−

000

112221

331113

223332


⇒ only two independent equations.

For general purpose, in Cartesian coordinate system, the two independent equations can be written as

0),,(),,(),,(0),,(),,(),,(

321

321

=++=++

dzzyxbdyzyxbdxzyxbdzzyxadyzyxadxzyxa

The solutions of the above equations can be expressed as

11 ),,( Czyx =ψ , where 1C is a constant.

22 ),,( Czyx =ψ , where 2C is a constant.

z

x y

P(xi, yi, zi) Sdv

Vr

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It can be seen that the above equation represents a one parameter family of surface. When constant 1C and 2C change their values, the above equations represent different stream surfaces. A streamline in space can be expressed as the curve of intersection of two surfaces. As shown in the figure, along the streamline, the velocity V

r lies in both the

surface of 11 ),,( Czyx =ψ and

22 ),,( Czyx =ψ . Since direction of the gradients of ),,(1 zyxψ and ),,(2 zyxψ (i.e.,

1ψ∇ and 2ψ∇ ) being parallel to the direction that normal to the their respective surfaces, thus, Vr

is normal to 1ψ∇ and 2ψ∇ .

01 =∇• ψVr

and 02 =∇• ψVr

This shows that V

r is normal to the plane formed by the vectors 1ψ∇ and 2ψ∇ . In other words,

Vr

should be parallel to the cross product of 1ψ∇ and 2ψ∇ , i.e.,

33

2

22

2

21

2

33

1

22

1

11

1

321

21

ˆˆˆ

qhqhqh

qhqhqh

eee

V

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∇×∇=

ψψψ

ψψψψψμ

r where μ is a scalar function of position.

Since 0)()( 21 =∇×∇•∇=•∇ ψψμV

r, therefore, μ can be arbitrary scalar to satisfied

0)( =•∇ Vr

μ . For incompressible flows, the continuity equation can be expressed as 0=•∇ V

r, therefore, we

usually choose 1=μ for incompressible flow applications. For compressible flows, the continuity equation be expressed as 0)( =•∇ V

rρ , therefore, we

usually choose ρμ = for compressible flow applications.

Vr Stream line

22 ),,( Czyx =ψ

11 ),,( Czyx =ψ

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4.7.1 Stream Function for 2-D Flows When a flow is called 2-D flow, it means the flow quantities are independent of distance along a certain fixed direction. It we designate Z axis as the direction, we shall have:

0quantitiesany of()≡

∂∂

=∂∂

ZZ.

In Cartesian coordinate system In Cartesian system, we will have:

),(),()0,,(

),,(

yxvvyxuu

vuV

dzdydxSd

===

=r

r

Along a streamline, it can be written as

0),(),(dz

yxvdy

yxudx

==

It immediately follows that 0=dz or constZ = For the steam function 1ψ and 2ψ , we can choose stream function 2ψ is simply z (i.e., z=2ψ ), the 1ψ is the only one unknown function which is denoted as ψ . ψ is a function of x and y only. i.e., ),(1 yxψψ = and z=2ψ . Since 21 ψψμ ∇×∇=V

r, then we have

100

0

ˆˆˆ

ˆ0ˆˆyx

kji

kjviuV∂∂

∂∂

=++=ψψμμμ

r

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

xv

yu

ψμ

ψμ

For incompressible flows, 1=μ ,

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

xv

yu

ψ

ψ

For compressible flows, ρμ = ,

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

xv

yu

ψρ

ψρ

In Cylindrical coordinate system

- 15/21 -

),(),()0,,(

),,(

θθ

θ

θθ

θ

rVVrVVVVV

dzrddrSd

rr

r

===

=r

r


0),(),(dz

rVd

rVdr

r

==θ

θθ θ

It immediately follows that 0=dz or constZ = For the steam function 1ψ and 2ψ , we can choose stream function 2ψ is simply z (i.e., z=2ψ ), the 1ψ is the only one unknown function which is denoted as ψ . ψ is a function of r and θ only. i.e., ),(1 θψψ r= and z=2ψ . Since 21 ψψμ ∇×∇=V

r, then we have

100

0

ˆˆˆ

ˆ0ˆˆθψψμμμθ

θθ ∂∂

∂∂

=++=rr

eee

eeVeVVzr

Zrr

r

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

rV

rVr

ψμ

θψμ

θ

1


⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

rV

rVr

ψθψ

θ

1


⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

rV

rVr

ψρ

θψρ

θ

1

Another Commonly-Used Method to Determine Stream Function:

In a 2-D flow, 22

11

2

1

dqhdqh

VV

=

On the integration of the above equation yields

),,( 1212 qhhfq = or ),,,( 2121 qqhhFF = Let’s call this F as stream functionψ , then ),,,( 2121 qqhhψψ = .

- 16/21 -

Different constant of integration yield different stream lines. Let a, b, c and d represent two streamlines, by definition, not fluid passes ab or cd. Therefore, same mass of fluid must cross ef and gh. If the streamline ab is arbitrarily chosen as a base, every other streamline in the field can be identified by assigning to it a number ψΔ equal to the mass of fluid passing, per second per unit depth perpendicular to the plane containing the base streamline and the streamline in question.

∫∫

∫∫∫

=•=

⋅•=•=•=

−=−=Δ

f

en

f

en

f

en

f

en

f

e

dlVdleV

dleVdAeVAdV

CC

ρρ

ρρρ

ψψψ

ˆ

)1)(ˆ()ˆ(

1212

r

rrrr

Where dl is the elemental length along the line ef. In the limit ef is small, we can write this as :

lVnΔ=Δ ρψ Where lΔ is the normal distance between streamlines.

Or nVl

ρψ≅

ΔΔ

In the limit, 0→Δl , this equation becomes

nlV

llρψψ

=∂∂

=ΔΔ

→Δ 0lim

Thus, the velocity component in any direction is obtained by differencing ψ with respect to distance at right angles to the left of that direction. Therefore:

z

x

y

p1

e

f

g

h a

b

c

d

1ψ2ψ

q1

q2

h2∂ q2

-h1∂ q1

- 17/21 -

221 qh

V∂∂

=ψρ

112 qh

V∂∂

−=ψρ

Accepting the above equation as the equations describingψ , we can show that ψ is constant along a streamline. Proof: Along a stream, 0=× SdV

rrρ by definition, therefore

constd

dqq

dqq

dqhqh

dqhqh

dqhVdqhVdqhVdqhV

=⇒=⇒

=∂∂

+∂∂

⇒

=∂∂

+∂∂

⇒

=−⇒=−

ψψ

ψψ

ψψρρ

0

0

0)()(

00

11

22

1111

2222

12221

112221

Therefore, ψ is a constant on a streamline, that is, streamlines are isolines with constant ψ .

• Since streamlines and stream functions are related to the mass flow rate m& , we can define stream function with continuity equation for steady flows.

• The stream function is a point function which satisfies the equation of continuity for

steady flow identically.

• A necessary condition that a steady flow be physical possible is that a stream function exists.

• For incompressible flows, ψ is changed to ψ to denote that stream function, where ψΔ

denote the volume flow rate. i.e., ll

Vn ∂∂

=∂

∂=

ψρψ )/(

Note: The stream function ψ is defined only for 2-D flow. Stream function ψ exists whether

the flow is rotational or irrotational.

- 18/21 -

4.7.2 Stream Function for Axisymmetric Flows For an axisymmetric flow, the flow quantities are independent of angle, θ , then

0quantitiesany of()≡

∂∂

=∂∂

θθ.

In cylindrical system:

),(),(),0,(

),,(

ZrVVZrVVVVV

dZrddrSd

ZZ

rr

zr

===

=r

rθ


zr VdZrd

rVdr

==0),(θ

ϕ

It immediately follows that 0=θd or const=θ For the steam function 1ψ and 2ψ , we can choose stream function 2ψ is simply z (i.e., θψ =2 ), the 1ψ is the only one unknown function which is denoted as ψ . ψ is a function of r and Z only. i.e., ),(1 Zrψψ = and θψ =2 . Since 21 ψψμ ∇×∇=V

r, then we have

010

0

ˆˆˆ

ˆˆ0ˆ

r

Zr

eee

eVeeVVzr

Zzrr ∂∂

∂∂

=++=ψψμμμ

θ

θ

r

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

=

∂∂

−=⇒

rrV

ZrV

z

r

ψμ

ψμ

1

1


⎪⎪⎩

⎪⎪⎨

⎧

∂∂

=

∂∂

−=⇒

rrV

ZrV

z

r

ψ

ψ

1

1


⎪⎪⎩

⎪⎪⎨

⎧

∂∂

=

∂∂

−=⇒

rrV

ZrV

z

r

ψρ

ψρ

1

1

θ

Z

Y

X r

- 19/21 -

In spherical system:

),(),(

),,(

)sin,,(

ϕϕ

θϕϕ

ϕϕ

θϕ

rVVrVV

VVVV

dRRddRSd

rr

R

==

=

=r

r


0sin

),(),(θϕ

θϕ

ϕ ϕ

dRRV

RdRV

dR

R

==

It immediately follows that 0=θd or const=θ For the steam function 1ψ and 2ψ , we can choose stream function 2ψ is simply z (i.e., θψ =2 ), the

1ψ is the only one unknown function which is denoted as ψ . ψ is a function of R and ϕ only. i.e., ),(1 ϕψψ R= and θψ =2 . Since 21 ψψμ ∇×∇=V

r, then we have

ϕ

ϕψψμμμ

θϕ

θϕϕ

sin100

0

ˆˆˆ

)ˆ0ˆˆ

R

RR

eee

eeVeVV

R

RR ∂∂

∂∂

=++=r

⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

RRV

RVR

ψϕ

μ

ϕψ

ϕμ

ϕ sin1

sin1

2


⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

RRV

RVR

ψϕ

ϕψ

ϕ

ϕ sin1

sin1

2


⎪⎪⎩

⎪⎪⎨

⎧

∂∂

−=

∂∂

=⇒

RRV

RVR

ψϕ

ρ

ϕψ

ϕρ

ϕ sin1

sin1

2

θ

ϕ x

y

z

R r

- 20/21 -

4.8 Irrotational 2-D Ideal Fluid Flow Since flow is 2-D and irrotational ideal flow, then

In Cartesian system, 0=Ωr

⇒ 0=∂∂

−∂∂

=Ωyu

xv

z

Since x

vy

u∂∂

−=∂∂

=ψψ ;

Thus, 0][)()(

2

2

2

2

=∂∂

+∂∂

−=∂∂∂

∂−

∂∂∂

−∂=

∂∂

−∂∂

yxyy

xx

yu

xv ψψ

ψψ

i.e., 02

2

2

2

=∂∂

+∂∂

yxψψ or 02 =∇ ψ Laplace Equation

Laplace equation has solutions which are called as harmonic functions. For 2-D flow

• Any irrotational and incompressible flow has a velocity potential φ and stream function ψ that both satisfy Laplace equation.

• Conversely any solution represents the velocity potential φ or stream function ψ for an irrotational and incompressible flow.

• A powerful procedure for solving irrotational flow problems is to represent φ andψ by linear combinations of known solutions of Laplace equation.

i

N

iiC φφ ∑

=

=1

; i

N

iiCψψ ∑

=

=1

Finding the coefficients iC so that the boundary conditions are satisfied both far from the body and the body surface. Say 1φ and 2φ are solutions of 02 =∇ φ i.e., 0;0 2

21

2 =∇=∇ φφ Then 2211 φφφ AA += is also a solution of the Laplace equation A complicated flow pattern for an irrotational and incompressible flow can be synthesized by adding together a number of elementary flows which are also irrotational and incompressible.

- 21/21 -

The Advantages of using stream function and potential function to solve aerodynamics problems

1. It can replace the nonlinear flow controlling equations (such as N-S equations) by Linear equations (Laplace equation).

2. It reduces the number of unknown numbers, i.e., φψ

ρ

orwvup

⇒

⎪⎪

⎭

⎪⎪

⎬

⎫

3. The solutions of the Laplace equation are adjustable. 4. A complex flow can be decomposed as the summation of many simple flows.

chapter 4 simplification to the conservation...

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