INVISCID FLOW Week 5

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2017 Spring Inviscid Flow 2017 Spring

INVISCID FLOW Week 5

Prof. Hyungmin Park

Multiphase Flow and Flow Visualization Lab.

Department of Mechanical and Aerospace Engineering Seoul National University

Flow Kinematics

o In this chapter, we will study flow kinematics

– Kinematics is concerned with the geometry of motion without regard to the dynamical laws (i.e., referring the force acting on it)

– Flow lines

• Streamlines, Pathlines, Streaklines, and Timelines – Circulation and Vorticity

– Stream tubes and Vortex tubes – Kinematics of vortex lines

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2017 Spring Inviscid Flow

Flow Kinematics – Streamlines

o Three types of flow lines are used frequently for flow-visualization purposes: streamlines, pathlines, streaklines, and timelines. In general, they are all different.

o Streamlines

– lines whose tangents are everywhere parallel to the velocity vector at a given instant of time

– in unsteady flow, it is meaningful to consider only the instantaneous streamlines

o In two-dimensional flow field, by the definition of a streamline, its slope in the x-y plane (dy/dx) must be equal to that of the velocity vector (v/u), so the equation of streamline becomes

3

dy v dx = u

Flow Kinematics – Streamlines

o In case of three-dimensional flow field,

– By integrating this equation for fixed time t, we get z = z(x, y)

o One of the ways to find a solution of above equation is to introduce a new parameter s

– At some point in space, s = 0

– s is increasing along the streamline

, ,

Or, , ,

dy v dz w dz w dx u dy u dy v

dy dx dz dx dz dy dx dy dz

v u w u w v u v w

= = =

= = = Þ = =

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Flow Kinematics – Streamlines

– To find out a streamline which passes (x₀, y₀, z₀), integrate the above equation and use the initial condition of (x, y, z) = (x₀, y₀, z₀) @ s = 0.

o (Example) Consider a two-dimensional flow field defined by

5

ⁱ _i

( , )

_i

dx dy dz dx

ds u x t

u v w ds

\ = = = Þ =

(t is fixed)

(1 2 ), , 0 u = x + t v = y w =

(1 2 )

1 2

(1 2 ),

t s

,

s

dx dy

x t y

ds ds

x c e

⁺

y c e

= + =

= =

Let’s assume the streamline passes

through the point (1, 1) at time t = 0

,

Then @ 0

s s

x e y e y x t

= =

Flow Kinematics – Streamlines

o Streamlines are difficult to generate experimentally in unsteady flow – One should mark a great many particles and notes their direction of

motion during a very short time interval

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Flow Kinematics – Pathlines

o A pathline is a line traced out in time by a given fluid particle as it flows.

o Since the particle under consideration is moving with the fluid at its local velocity, pathlines must satisfy the equations

7

( , )

i

i i

dx u x t dt =

(1 2 ), , 0 u x = + t v = y w =

(1 )

1 2

(1 2 ),

t t

,

t

dx dy

x t y

dt dt

x c e

⁺

y c e

= + =

= =

(1 ) 1 ln

, Then

t t t

y

x e y e

x y

+ +

= =

=

Let’s assume the streamline passes through the point (1, 1) at time t = 0

Flow Kinematics – Pathlines

o How can we obtain pathlines?

– One way is to take a photograph, with very long time exposure, of a small reflective or fluorescent seed particle immersed in a flow.

– This photograph would contain an illuminated curve indicating the particle position at any time during the interval over the exposure time.

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Flow Kinematics – Streaklines

o A streakline is a line traced out by a neutrally buoyant marker fluid that is continuously injected into a flow field at a fixed point in space. The marker fluid may be smoke (in air) or a dye (in water).

o A particle of the marker fluid that is at the location (x, y, z) at time t must have passed through the injection point (x₀, y₀, z₀) at some earlier time t = t. Then the time history of this particle may be obtained by solving the equations for the pathline subject to the initial conditions that (x, y, z) = (x₀, y₀, z₀) at t = t. Then as t takes on all possible values in the range -¥ £ t

£ t, all fluid particles on the streakline will be obtained.

o Streaklines, easily generated experimentally with smoke, dye, or bubble releases, are relatively difficult to compute analytically.

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Flow Kinematics – Streaklines

(1 2 ), , 0 u x = + t v = y w =

(1 )

1 2

(1 2 ),

t t

,

t

dx dy

x t y

dt dt

x c e

⁺

y c e

= + =

= =

(1 ) (1 ) 1 ln

, If 0,

t t t

y

x e y e

t x y

t t t

+ - + -

-

= =

Let’s assume the streamline passes through the point (1, 1) at time t = t

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Flow Kinematics – Streaklines

11

(1 2 ), , 0 u x = + t v = y w =

Lines passes through the point (1, 1)

Flow Kinematics

o Note that streamlines and timelines are instantaneous lines, while the pathlines and streaklines are generated by the passage of time

o In a steady flow,

– the fluid velocity never changes magnitude or direction at any point, every particle which comes along repeats the behavior of its earlier neighbors

– thus, the streamlines, pathlines, and streaklines are identical

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Flow Kinematics – Timelines

o A timeline is a set of fluid particles that form a line at a given instant.

o Experimentally, timelines can be generated using a hydrogen bubble wire

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Flow Kinematics – Illustrated Examples

Hele-Shaw flow (potential flow) Van Dyke, Album of Fluid Motion (1982)

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Flow Kinematics – Illustrated Examples

15 Turbulent boundary

layer (TBL)

Laminar boundary layer (LBL)

The NCFMF Book of Film Notes (1972)

Flow around a circular cylinder, ReD= 10⁴ Van Dyke, Album of Fluid Motion (1982)

Flow around an inclined slender body Van Dyke, Album of Fluid Motion (1982)

Flow Kinematics – Illustrated Examples

Flow around a flapping wing of a fruitfly Shyy et al. (2010)

Flow around a reciprocating flat plate Lee et al. (2013)

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Flow Kinematics – Circulation and Vorticity

o Vorticity (w) of a fluid element is defined as the curl of its velocity vector

o That is, the vorticity is equal to twice the anti-symmetric part of the deformation-rate tensor

o Vorticity is the tendency for elements of the fluid to spin (local rotation) o It should be noted that a fluid element may travel on a circular streamline

while having zero vorticity – Rotational flow (w_{¹ 0)} – Irrotational flow (w= 0)

– Should be distinguished from a circular motion!

• Rotational/Irrotational circular motion

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w = Ñ ´ u

j k j

2

i ijk kj

k j k

u u u

x x x R

w = - e ^¶ ¶ = ^æ ç ç è ^¶ ¶ - ^¶ ¶ ^ö ÷ ÷ ø =

Flow Kinematics – Circulation and Vorticity

o The circulation (^G)contained within a closed contour in a body of fluid is defined as the integral around the contour of the component of the velocity vectorthat is locally tangent to the contour

o where dl represents an element of the contour. The integration is taken counterclockwise around the contour, and the circulation is positive if this integral is positive

G = ò u dl ×

G

u dl

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Flow Kinematics – Circulation and Vorticity

o Relationship between the circulation and vorticity

o which A is the area defined by the closed contour around which the circulation is calculated and n is the unit normal to the surface

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( )

A A

u dl u ndA w ndA

G = ò × = ò Ñ ´ × = ò ×

Stoke’s Theorem

A

u dl

w

n

dA

Flow Kinematics – Stream Tubes & Vortex Tubes

o Imagine a set of streamlines starting at points that form a closed loop.

o These streamlines form a tube that is impermeable since the walls of the tube are made up of

streamlines, and there can be no flow normal to a streamline (by definition). This tube is called a streamtube.

o Frommass conservation, we see that for a steady, one-dimensional flow, the mass-flow rate is

constant along a stream tube. In a constant density flow, therefore, the cross-sectional area of the stream tube gives information on the local velocity.

o If the cross section of a stream tube is

infinitesimally small, it is usually referred to as a stream filament

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Flow Kinematics – Stream Tubes & Vortex Tubes

o A vortex line is a line whose tangents are everywhere parallel to the vorticity vector.

o The series of vortex lines defined by the closed contour form a vortex tube.

o A vortex tube whose area is infinitesimally small is usually referred to as a vortex filament.

o We will discuss the vortex tubes more later when we study the “vorticity equation”

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Flow Kinematics – Kinematics of Vortex Lines

o Divergence of the curl of any vector is identically zero, i.e., the vorticity is divergence free

– there can be no sources or sinks of vorticity in the fluid itself – vortex lines must either form closed loops or terminate on the

boundaries of the fluid

o The continuity equation for incompressible flow o Integrate this over some volume V gives

w 0 Ñ × =

0 Ñ × = u

( ) 0

V s

u dV u n dS

Ñ × =

× =

ò

Gauss Theorem

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Flow Kinematics – Kinematics of Vortex Lines

o Now consider the surface s to be the entire outer surface of an element of a stream tube (or stream filament), including the ends. Then, since u·n = 0 on the walls of the streamtube by definition, then

o where Q₁and Q₂are volume flow rate o Now, going back to the vorticity, we can get

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1 2

( ) ( ) 0

0

A

u n dS

A

u n dS

Q Q

× + × =

- + =

ò ò

( ) 0

V s

dV n dS

w w

Ñ × =

× =

ò ò

Flow Kinematics – Kinematics of Vortex Lines

o That is, circulation at each cross-section of a vortex tube is the same

o Divergence-free vorticity means that vortex tubes must terminate on themselves, at a solid boundary or at a free surface.

o Smoke rings terminate on themselves, while a vortex tube in a free surface flow may have one end at the solid boundary forming the bottom and the other end at the free surface.

1 2

( ) ( ) 0

0

A

w × n dS +

A

w × n dS =

-G + G = G = G

ò ò

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Flow Kinematics – Kinematics of Vortex Lines

25 Smoke Ring

(www.4chandata.org)

Special Forms of Governing Equations

o In this chapter, we will study alternative forms of basic governing equations

– Kelvin’s Theorem: irrotational flow

– Bernoulli Equation: integral of Euler equation under specific conditions – Crocco’s Equation: relation between the entropy to vorticity

– Vorticity Equation: rotational flow

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Kelvin’s Theorem

o For an inviscid fluid in which the density is constant, or in which the pressure depends on the density alone, and for which any body forces that exist are conservative, the vorticityof each fluid particle will be preserved

– µ = 0 (inviscid)

– r = constant or p = p(r)

– conservative body force (per unit mass):

(G: scalar function)

27

j j

f G x

= ¶

¶

1 1

j j

k j

j j

k

k j j

j

j j

u u p

u f

t x x

u u p G

t u x x x

Du p G

Dt x x

r r r

r r

¶ + ¶ = - ¶ +

¶ ¶ ¶

¶ + ¶ = - ¶ + ¶

¶ ¶ ¶ ¶

¶ ¶

= - +

¶ ¶

Euler’s equation

valid for an inviscid fluid subjected to only conservative body forces

Kelvin’s Theorem

o Rate of change of vorticity for a given fluid element ( )

j j

j j j j

Du D dx

D D

u dx dx u

Dt Dt Dt Dt

é ù

G = = ê + ú

ë û

ò ò

( _j) _j _j _j

k j

k

D dx Dx x x

d d u du

Dt Dt t x

¶ ¶

æ ö

= çè ÷ø= çè ¶ + ¶ ÷ø= Eulerian

j jk k

x x d

¶ =

¶ 1

1 ( ) 2

j

j j j j j j j

j j

D Du p G

dx u du dx dx u du

Dt Dt x x

dp dG d u u

r

é ù

G ¶ ¶

= êë + úû= êêë- ¶ +¶ + úúû

é ù

= ê- + + ú

ë û

ò ò

ò

because G and u_jare assumed to be single-valued

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Kelvin’s Theorem

o We consider a system 1) density is constant:

2) p = g(r)

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D dp

Dt r

\ G = -

ò

1 0

D dp

Dt r

G = -

ò

= pressure is single-valued '( ) , D g'( ) 0

dp g d d

Dt

r r r r

r

= G = -

ò

= density is single-valued

D 0 Dt

\ G = if we follow a given contour as it flows, the total vorticity inside that contour remains same à Kelvin’s Theorem

Thus it may be deduced that the total vorticity may be changed by the action of viscosity, the application of non-conservative body forces, or density variations that are not simply related to the pressure variation.

Kelvin’s Theorem

o Kelvin’s Theorem applies to a simply connected region. That is, for any closed contour in the fluid that contains only fluid, there will be some definite value of the circulation

o A closed contour that originally does not include a body cannot at any subsequent time contain a body such as a two-dimensional airfoil

D 0 Dt

G ¹

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Kelvin’s Theorem

o The principal use of Kelvin’s theorem is in the study of incompressible, inviscid fluid flows.

– If a body is moving through such a fluid, or if a uniform flow of such a fluid passes around a body, then the vorticity far from the body will be zero.

– Then, according to Kelvin’s theorem, the vorticity in the fluid will everywhere be zero, even adjacent to the body.

31 0

w= Ñ ´ =u can be used instead of Euler’s equation.

Bernoulli Equation

o Inviscid flow

o Conservative body force o Steady or Irrotational flow

equations of momentum conservation may be integrated to yield a single scalar equation called the Bernoulli equation

j j 1

k

k j j

u u p G

t u x r x x

¶ + ¶ = - ¶ + ¶

¶ ¶ ¶ ¶

( ) (1 ) ( )

2 (1 )

2

j k

k

u u u u u u u u

x

u u u w

¶ = ×Ñ = Ñ × - ´ Ñ ´

¶

= Ñ × - ´

using a vector identity

1 1

( )

2

u u u u p G

t w

r

¶ + Ñ × - ´ = - Ñ + Ñ

¶ in a vector form

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Bernoulli Equation

o Let’s consider a length element, dl and

o Then, the Euler equation becomes

33

1 1 1 1 1

dl p dl p dp d dp dl dp

r r r r r

æ ö

×ç Ñ ÷= ×Ñ = = = ×Ñ

è ø

ò ò

dl dx dy dz d

x y z

¶ ¶ ¶

×Ñ = + + =

¶ ¶ ¶

1 1

p dp

r r

\ Ñ = Ñ

ò

1 2

u dp

u u G u

t w

r

æ ö

¶¶ + Ñçè

ò

+ × - ÷ø= ´

Bernoulli Equation

o For steady flow

o Therefore, as we flow along a streamline (lagrangian) in steady flow 1

2

1 ( )

2

dp u u G u

u dp u u G u u

r w r w

æ ö

Ñç + × - ÷= ´

è ø

æ ö

×Ñç + × - ÷= × ´

è ø

ò

perpendicular to u

D u u

Dt t

= ¶ + ×Ñ = ×Ñ

¶

1 constant along each streamline 2

dp u u G

r ⁺ ^{× - =}

ò

Bernoulli Equation

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Bernoulli Equation

– In case of steady, irrotational flow, the constants for each streamlines are same

o For irrotational flow

35 1 constant everywhere

2

dp u u G

r ⁺ ^{× - =}

ò

1 0

2

u dp

u u G u

t w

r

æ ö

¶¶ + Ñçè

ò

+ × - ÷ø= ´ =

0 0 w u

f

= Ñ ´ =

Ñ ´ Ñ = u= Ñf

1 0

2

1 0

2

1 0

2

d dp

dt G

d dp

dl G

dt

d dp

d G

dt

f f f

r

f f f

r

f f f

r

æ ö

Ñç + + Ñ ×Ñ - ÷=

è ø

æ ö

×Ñçè + + Ñ ×Ñ - ÷ø=

æ + + Ñ ×Ñ - ö=

ç ÷

è ø

ò ò ò

Bernoulli Equation

– valid for irrotational motion of a fluid in which viscous effects are negligible and in which any body forces are conservative

o Bernoulli equation usually helps to establish the condition of

irrotationality by relating the flow under consideration to a simpler form of the flow far upstream

1 ( )

2

d dp

G F t dt

f f f

+

ò

r + Ñ ×Ñ - =

unsteady Bernoulli constant

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Crocco’s Equation

o Relation between entropy and vorticity – Inviscid flow

– No body force – Incompressible flow

o Pressure and density will be replaced by enthalpy and entropy

37 Euler’s Equation

1 1

( )

2

u u u u p

t w

r

¶ + Ñ × - ´ = - Ñ

¶

1 1

de pd dq pd Tds

e h p

r r

r

æ ö æ ö

= - ç ÷+ = - ç ÷+

è ø è ø

= -

1

dh d p pd Tds

r r

æ ö æ ö

- ç ÷= - ç ÷+

è ø è ø

Crocco’s Equation

o Then, the original Euler equation becomes 1

p dp

d pd

r r r

æ ö= æ ö+

ç ÷ ç ÷

è ø è ø

1 dp Tds dh

\-r = -

Using dl d, 1 p T s h

×Ñ = - Ñ = Ñ - Ñr (1 )

2

( 1 )

2

u u u u T s h

t

u T s h u u u t w

w

¶ + Ñ × - ´ = Ñ - Ñ

¶

´ + Ñ = Ñ + × +¶

¶ Crocco’s Equation

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Crocco’s Equation

o Under the condition of steady, adiabatic flow – Energy equation for invisicd, adiabatic flow

– Euler equation without a body force

39 Dh Dp

Dt Dt r =

0

(1 ) 2

( 1 ) ,

2

Du p

Dt

u Du u p Dt

D u u u p

Dt

D Dp Dh p

h u u u p

Dt Dt Dt t

r r r

r r

= -Ñ

× = - ×Ñ

\ + × = - ×Ñ = ¶

¶

0

1 h = +h 2u u×

Crocco’s Equation

– for steady flow,

o For steady, adiabatic flow, the Crocco’s equation becomes

– Usually when the h₀is constant along each streamline, it is constant everywhere: dh₀/dn = 0

0

0, 0 constant

Du p

Dt

Dh p

Dt t h r

r

= -Ñ

= ¶ = =

¶

u´ + Ñ = Ñw T s h0 since h₀is constant along a streamline, Ñh₀is perpendicular to the streamline; so is u ´ w.

ds dh0

U T

dn dn

W + = valid for steady, adiabatic flow of an inviscid fluid in which there are no body forces

ds 0

U T

W + dn =

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Crocco’s Equation

o If s is constant, W must be zero (isentropic flows are irrotational) o If W is zero, s must be constant (irrotational flows are isentropic)

– This is true, in general, only for steady flows of inviscid fluids in which there are no body forces and in which the stagnation enthalpy is constant

41 ds 0

U T

W + dn =

Vorticity Equation

o Vorticity equation is useful in the study of viscous flows in incompressible fluids

2

( ) ( )

(1 ) ( ) ( )

2

( )

( ) ( )

u p

u u u

t

u p

u u u u u

t t u

u u

t

r n

w w n w

w w w n w

¶ + ×Ñ = -Ñ + Ñ

¶

¶ + Ñ × - ´ Ñ ´ = -Ñ + Ñ

¶

¶ - Ñ ´ ´ = Ñ

¶

¶ + ×Ñ = ×Ñ + Ñ

¶

N-S equation, constant density, viscosity

taking curl

Vorticity Equation

( ) ( ) ( ) ( ) ( )

0 0

u u u u u

u

w w w w w

w

Ñ ´ ´ = Ñ × - Ñ × - ×Ñ + ×Ñ

Ñ × = Ñ × =

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Vorticity Equation

o In a two-dimensional flow,

o Vorticity and velocity fields may be obtained without any knowledge of the pressure field

o By taking divergence of N-S equation,

43

2

( ) ( )

( )

u u

t t u

w w w n w

w w n w

¶ + ×Ñ = ×Ñ + Ñ

¶

¶ + ×Ñ = Ñ

¶

2 2 1 2

( ) ( ) ( )

2

p w w u u u u

Ñ r = × + × Ñ - Ñ ×

diffusion equation

Poisson equation