Chapter 3 Kinematics of Fluid Motion

Kinematics of Fluid Motion

Streamlines

3.1 Steady and Unsteady Flow, Streamlines, and Stream tubes 3.2 One-, Two-, and Three-Dimensional Flows

3.3 Velocity and Acceleration

3.4 Circulation, Vorticity, and Rotation Objectives

- treat kinematics of idealized fluid motion along streamlines and flowfields - learn how to describe motion in terms of displacement, velocities, and

accelerations without regard to the forces that cause the motion

- distinguish between rotational and irrotational regions of flow based on the flow property vorticity

다루는 학문

• 유체정역학(Fluid statics): 유체의 상대적인 운동이 없는 경우를 다루는 학문; 전단력이 작용하지 않고 수직력만 작용

• 유체동역학(Fluid dynamics):

• 유체운동학(Fluid kinematics): 유체의 운동을 일으키는 힘을 제 외하고 운동(변위, 유속, 가속도 등)만을 다루는 다루는 학문

( )

d dV dm

F mV m V ma

dt dt dt

= = + =

∑ ^{    }

Joseph Louis Lagrange (1736-1813), Italian mathematician

1) Lagrangian views

~ Each fluid particle is labeled by its spatial coordinates at some initial time.

~ Then fluid variables (path, density, velocity, and others) of each individual particle are traced as time passes.

~ used in the dynamic analyses of solid particles

- We cannot easily define and identify particles of fluid as they move around.

- A fluid is a continuum, so interactions between parcels of fluid are not easy to describe as are interactions between distinct objects in solid mechanics.

- The fluid parcels continually deform as they move in the flow.

as function of time

~ follow a particular particle through the flow field → path line (유적선)

t = t

_{( , , ) a b c}

t = t _{( , , ) x y z}

At , coordinates (position) of a particle At , position of a particle

1

( , , , ) x = f a b c t

2

( , , , ) y = f a b c t

3

( , , , )

z = f a b c t

u x

t

= ∂

∂

2 x 2

u x

a t t

∂ ∂

= =

∂ ∂

v y

t

= ∂

∂

2 y 2

v y

a t t

∂ ∂

= =

∂ ∂

w z

t

= ∂

∂

2 z 2

w z

a t t

∂ ∂

= =

∂ ∂

~ the position is plotted as a function of time

= trajectory of the particle → path line

~ since path line is tangent to the instantaneous velocity at each point along the path, changes in the particle location over an infinitesimally small time are given by

; ;

dx = udt dy = vdt dz = wdt

dx; dy dz

u v w

dt dt dt

= = =

1 dx dy dz dt

u = v = w =

(3.1a) (3.1b)

~ attention is focused on particular points in the space filled by the fluid

~ motion of individual particles is no longer traced

→ A finite volume called a control volume (flow domain; 검사체적) is defined, through which fluid flows in and out.

~The values and variations of the velocity, density, and other fluid variables are determined as a function of space and time within the control volume.

→ flow field

( , , , ) V   = v x y z t

Acceleration field

( , , , ) a = a x y z t

Define the pressure field as a scalar field variable ( , , , )

p = p x y z t where

x y z

V = ue + ve + we

, ,

x y z

e e e  

= unit vectors

(E4) (E3) (E2) (E1)

(

)

V = u v w = u x y z t e +v x y z t e + w x y z t e

(E5)

▪Difference between two descriptions

Imagine a person standing beside a river, measuring its properties.

Lagrangian approach: he throws in a prove that moves downstream with the river flow

Eulerian approach: he anchors the probe at a fixed location in the river

~ Eulerian approach is practical for most fluid engineering problems (experiments).

Logging positions

<GPS floater>

Advection

9

4 2

3

1 8 7 6 5 10

12 13

11

14 17 15

16

1820 24

23

19 21 25

22 29

2826

27 30

9

2 3 8

7 6

5 10

12 13

11

14 17

15

16 18

20 23

19 21 25

22 29 28

26

27 30

Injection Point

Diffusion

Lagrangian measurements

→ In Eulerian view, two types of flow can be identified.

1) Steady flow (정상류)

~ The fluid variables at any point do not vary (change) with time.

~ The fluid variables may be a function of a position in the space.

→ non-uniform flow

→ In Eulerian view, steady flow still can have accelerations (advective acceleration; 이송가속도).

• 가속도=국부가속도+이송가속도

Q

t

~ The fluid variables will vary with time at the spatial points in the flow.

▪ Fig. 3.1

when valve is being opened or closed → unsteady flow when valve opening is fixed → steady flow

t Q

Let fluid variables (pressure, velocity, density, discharge, depth) = ^F

F 0 t

∂ =

∂

F 0 t

∂ ≠

∂

F F F 0

x y z

∂ ∂ ∂

= = =

∂ ∂ ∂

F 0 x

∂ ≠

∂

→ steady flow

→ unsteady flow

→ uniform flow

→ non-uniform (varied) flow

1) Streamline (유선)

~ The curves drawn at an instant of time in such a way that the tangent at any point is in the direction of the velocity vector at that point are called instantaneous streamlines, and they continually evolve in time in an unsteady flow.

in the direction of the fluid velocity at any point.

→ In an unsteady (turbulent) flow, path lines are not coincident with the instantaneous streamlines.

2) Path line (유적선)

~ trajectory of the particle 3) Streak line (유립선)

~ current location of all particles which have passed through a fixed point in space

from a smoke stack or other discharge.

~ In steady flow, Lagrangian path lines are the same as the Eulerian streamlines, and both are the same as the streak lines, because the streamlines are then fixed in space and path lines, streak lines and streamlines are tangent to the steady velocities.

→ In a steady flow, all the particles on a streamlines that passes through a point in space also passed through or will pass through that point as well.

~ In an unsteady flow, the path lines, the streak lines, and the instantaneous streamlines are not coincident.

(∆ →t 0) [Re] How to shoot flow lines?

1) streamline: shoot bunch of reflectors instantly

2) path line: shoot only one reflector with long time exposure

3) streak line: shoot dye injecting from on slot with instant exposure

At high velocity

[Ex] A fluid flow has the following velocity components;

Find an equation for the streamlines of this flow.

1 / , 2 / . u = m s v = x m s

Sol.: 2

1 ( )

dy v x

dx = =u a

Integrating (a) gives

y = x

+ c

1 dx dy dz dt

u = v = w =

~ may be treated as if isolated from the adjacent fluid

~ many of the equations developed for a small stream tube will apply equally well to a streamline.

~ aggregation of streamlines drawn through a closed curve in a steady flow forming a boundary across which fluid particles cannot pass because the velocity is always tangent to the boundary

No flow in and out

dF F F dx F dy F dz dt t x dt y dt z dt

∂ ∂ ∂ ∂

= + + +

∂ ∂ ∂ ∂

F F F F

u v w

t x y z

∂ ∂ ∂ ∂

= + + +

∂ ∂ ∂ ∂

total

derivative local advective (convective)

derivative derivative

t = 0

∂

F F F 0

u v w

x y z

∂ ∂ ∂

+ + =

∂ ∂ ∂

• steady flow:

• uniform flow:

~ All fluid particles are assumed uniform over any cross section.

~ The change of fluid variables perpendicular to (across) a streamline is negligible compared to the change along the streamline.

~ powerful, simple

~ pipe flow, flow in a stream tube

- average fluid properties are used at each section

( ) F = f x

Actual velocity

~ flow fields defined by streamlines in a single plane (unit width)

~ flows over weir and about wing - Fig. 3.4

~ assume end effects on weir and wing is negligible

( , )

v = f x z

~ The flow fields defined by streamlines in space.

~ axisymmetric three-dimensional flow - Fig. 3.5

→ streamlines = stream surfaces

( , , )

v = f x y z

Consider one-dimensional flow in a streamline coordinates (유선좌표계)

Select a fixed point 0 as a reference point and define the displacement of a fluid particle along the streamline in the direction of motion.

→ In time dt the particle will cover a differential distance ds along the streamline.

v ds

= dt

- magnitude of velocity:

where s = displacement (변위)

- direction of velocity: tangent to the streamline

a

a

2) Acceleration,

- acceleration along (tangent to) the streamline = - acceleration (normal to) the streamline =

a 

s r

a  = a s  + a n 

s

dv dv ds dv

a v

dt ds dt ds

= = =

2 r

a v

= − r ← particle mechanics

r s

where = radius of curvature of the streamline at

(3.2) (3.3)

2

s n

a a s a n dv v

v s n

ds r

= +

= +

  

 

[Re] Streamline coordinates

→ Particle moves in a circle with a constant speed.

∆v

OPP′ P QQ′ ′

- direction of : pointing inward, approximately toward the center of circle Apply similar triangles and

PP' v

r v

∴ = ∆ ^p

r

(along a radius inward toward the center of the circle) v v t

v r

∆ ∆

∴ =

v v2

t r

∆ =

∆

2

lim0 t

v v a ^{∆ →} t r

= ∆ =

∆

2 r

a a v

= − = − r

a

=

d dt

ω = θ

v = r ω

2

( )

a r r

r

ω ω

= − = −

3 m s v = x + y

Along a straight streamline,

Calculate velocity and acceleration at the point (8.6)

P(8, 6)

s , a v dv

= ds

a

= − v

/ r

r 0 a = 1) The streamline is straight. →

2) The displacement s is given as

s = x

+ y

Therefore,

v = 3 s 10

s = → v = 3(10 ) = 3 m s 0 3 (3 ) 9 90 m s

s

a v dv s s s

ds ′

= = = =

At (8.6) →

1.04 . vs = m s

The fluid at the wall of the tank moves along the circular streamline with a constant tangential velocity component,

Calculate the tangential and radial components of acceleration at any point on the streamline.

y

x v_s = 1.04 m/s

as

s 0 a v dv

= ds = 1) ds

Because tangential velocity is constant,

ar

2 2

(1.04) 2

0.541 m sec

r 2 a v

= − r = − = − 2)

→ directed toward the center of circle

The velocities are everywhere different in magnitude and direction at different points in the flow field and at different times.

→ three-dimensional flow in Cartesian coordinates At each point, each velocity has components u, ^v, ^w which are parallel to the x-, ^y-, and z- axes.

In Eulerian view,

( , , , ) ( , , , )

( , , , ) u u x y z t v v x y z t w w x y z t

=

=

=

and time as

, ,

dx dy dz

u v w

dt dt dt

= = = (3.4)

where x, ^y, and z are the actual coordinates of a fluid particle that is being tracked

→ The velocity at a point is the same in both the Eulerian and the Lagrangian view.

The acceleration components are

, ,

x y z

du dv dw

a a a

dt dt dt

= = = (3.5)

• Total (substantial, material) derivatives (App. 6)

u u u u

du dt dx dy dz

t x y z

∂ ∂ ∂ ∂

= + + +

∂ ∂ ∂ ∂

v v v v

dv dt dx dy dz

t x y z

∂ ∂ ∂ ∂

= + + +

∂ ∂ ∂ ∂

w w w w

dw dt dx dy dz

t x y z

∂ ∂ ∂ ∂

= + + +

∂ ∂ ∂ ∂

• Substituting these relationships in Eq. (3.5) yields

x

du u u u u

a u v w

dt t x y z

∂ ∂ ∂ ∂

= = + + +

∂ ∂ ∂ ∂

y

dv v v v v

a u v w

dt t x y z

∂ ∂ ∂ ∂

= = + + +

∂ ∂ ∂ ∂

z

dw w w w w

a u v w

dt t x y z

∂ ∂ ∂ ∂

= = + + +

∂ ∂ ∂ ∂

(3.6)

convective acceleration (이송가속도) local acceleration (국부가속도)

t t t

∂ ∂ ∂

For steady flow, , however convective acceleration is not zero.

• 2-D steady flow

(i) Cartesian coordinate; P(x, y)

r = + xi y j

  

v  = + ui  v j 

( , ) v = v x y

dx ;

u = dt dy

v = dt

- horizontal - vertical

x y

a a i a j

= dt = +

u 0 t

∂ =

∂

v 0 t

∂ =

∂

x

du u u

a u v

dt x y

∂ ∂

= = +

∂ ∂

y

dv v v

a u v

dt x y

∂ ∂

= = +

∂ ∂

( , ) P r θ (ii) Polar coordinate

θ

s

r

s = _{r =}

= radian =

where arc length, radius ^x

y

vr

= dt - radial

t

ds d

v r r

dt dt

θ ω

= = = - tangential

ds = rdθ ^d

dtθ ω=

(3.7b) (3.7a)

cos , sin x = r θ y = r θ

2 2

, arc tan y

r x y

θ x

= + =

[Re] Conversion

• Total derivative in polar coordinates

r r

r

v v

dv dr d

r θ

θ

∂ ∂

= +

∂ ∂

t t

t

v v

dv dr d

r θ

θ

∂ ∂

= +

∂ ∂

1 1

r r r r r r r

r r t

dv v dr v d v v s v v

v v v

dt r dt dt r r t r r

θ

θ θ θ

∂ ∂ ∂ ∂ ∂ ∂ ∂

= + = + = +

∂ ∂ ∂ ∂ ∂ ∂ ∂

t t t t t

1

r t

dv v dr v d v v

v v

dt r dt dt r r

∂ ∂ θ ∂ ∂

= + = +

∂ ∂θ ∂ ∂θ

r rr rt

a = a + a

a

v

r

a

v

r

where = acceleration due to variation of in - direction;

= acceleration due to variation of in - direction

1 1 2

r r r r r t

r t r t t r t

dv v v d v v v

a v v v v v v

dt

ω

= − = + − = + −

∂ ∂θ ∂ ∂θ

r r r 1

r t

dv v v

v v

dt r θ r

∂ ∂

= +

∂ ∂

vt

ω = r

a

a

t tt tr

a = a + a

t t

1

t r r t

dv v v v v

a v v v

dt ω ^∂ r r ^∂ r

= + = + +

∂ ∂θ

tt tr

a a

1.04 m s

vt = r = 2 m [IP 3.3] p. 99

For circular streamline along which , (radius of curvatures)

, , , , , ,

t r

x y t r

u v v v a a a a

 

 

 

Calculate at

Determine P(x, y) 2 cos 60 1

x = ^ =

2 sin 60 3

y = ^ =

1) Velocity

• Polar coordinate 1.04 m s vt =

r 0 v dr

= dt =

Apply similar triangles

: : ,

v u

= r y _{r = x}

_{+ y}

_{= 2} sin 60

(1.04)sin 60 0.90 m s

t

t

u v y v

r

= − = −

= − = −





cos 60

(1.04) cos 60 0.52 m s

t

t

v v x v

= r =

= =





• Cartesian coordinate

1.04 1.04 1.04 1.04

x

u u y y x y

a u v

x y r x r r y r

∂ ∂ ∂   ∂  

= ∂ + ∂ = − ∂ − + ∂ − 

2 2

(1.04) 1.082

x 4 x

r

= − = −

1.04 1.04 1.04 1.04

y

v v y x x x

a u v

x y r x r r y r

∂ ∂ ∂   ∂  

= ∂ + ∂ = − ∂  + ∂  

1.082 4 y

= −

(1, 3),

P (1) 0.27 m s

x 4

a = − = −

1.082 2

3 0.47 m s

y 4

a = − = −

At ,

2 2

(1.04) 2

0.54 m s 2

r r t

r r t

v v v

a v v

r r r

∂ ∂

= + − = − = −

∂ ∂θ

1.04 (1.04) 0 m s2

t t r t

t r t

v v v v

a v v

r r θ r r θ

∂ ∂ ∂

= + + = =

∂ ∂ ∂

• Polar coordinate

→ direction toward the center of circle

2 2 2 2

(0.27) (0.47) 0.073 0.22 0.29

x y

a + a = − + = + =

2 2

( 0.54) 0.29 a

= − =

2 2 2 2

r t x y

a a a a

∴ + = +

vortex

3.4.1 Circulation (순환)

As shown in IP 3.3, tangential components of the velocity cause the fluid in a flow a swirl (와류).

→ A measure of swirl can be defined as circulation.

Γ

• Circulation,

= line integral of the tangential component of velocity around a closed curve fixed in the flow (circle and squares)

Vortex (와류)

A flow field in which the streamlines are concentric circles

1) Irrotational vortex 2) Rotational vortex

( cos ) dΓ = V a dl

( cos )

Cd V α dl V dl

Γ =

 ∫

 ∫

 ∫

(3.8)

dl

where = elemental vector of size dl and direction tangent to the control surface at each point

[Re] vector dot product

cos

a b   ⋅ = ab α

→ proceed from A counterclockwise around the boundary of the element

along cos 0 along cos 0

d dx dy

AB BC

Γ ≅   +  

   

 

mean velocity mean velocity

along dxcos180 along dycos180

CD DA

   

+  +  

   

 

mean velocity

along cos 0

2

dx u u dy dx

AB y

   ∂ 

= −

   ∂ 

 



mean velocity

along cos180 ( )

2

dx u u dy dx

CD y

   ∂ 

= + −

   ∂ 

 



2 2

u dy v dx

d u dx v dy

y x

 ∂   ∂ 

Γ ≅  − ∂  +  + ∂  2 2

u dy v dx

u dx v dy

y x

 ∂   ∂ 

− + ∂  − − ∂ 

Expanding the products and retaining only the terms of lowest order (largest magnitude) gives

2 2

u dxdy v dxdy

d udx vdy

y x

∂ ∂

Γ = − + +

∂ ∂ 2 2

u dxdy v dxdy

udx vdy

y x

∂ ∂

− − − +

∂ ∂

v u

d dxdy

x y

∂ ∂  Γ = ∂ − ∂ 

where dx dy is the area inside the control surface

velocity gradient → shear stress

~ measure of the rotational movement

~ differential circulation per unit area enclosed

d v u

dxdy x y

ξ = ^Γ = ^∂ − ^∂

∂ ∂ (3.10)

[Cf] continuity: ^{u v 0}

x y

∂ ∂

+ =

∂ ∂

For polar coordinates

t t r

v v v

r r r

ξ θ

∂ ∂

= + −

∂ ∂ ^(3.11)

AB

If the fluid element tends to rotate, two lines will tend to rotate also.

→ For the instant their average angular velocity can be calculated.

Consider counterclockwise rotation for vertical line

V

ds dudt dt

d du

dy dy dy

θ = = =

2 2

u dy u dy dt u

u u dt

y y dy y

 ∂   ∂  ∂

= − + ∂   − − ∂  = − ∂

V V

d u

dt y

ω θ ^∂

∴ = = −

∂

ω =

where rate of rotation



2 2

H

v dx v dx dt v

d v v dt

x x dx x

θ = +^^ + ^∂∂ ^{ }  − − ^∂∂ ^^ = +^∂∂

H H

d v

dt x

ω = θ = ^∂

∂

Consider average rotation

1 1

2( ^{V H} 2

v u

x y

ω = ω + ω ) = ^^∂∂ − ^∂∂ ^

v u 2 x y

ξ = ^∂ − ^∂ = ω

∂ ∂

(3.12)

ξ ≠ 0

ξ =0 rotational flow (회전류) ~ flow possesses vorticity →

irrotational flow (비회전류) ~ flow possesses no vorticity, no net rotation

= potential flow (velocity potential exists) →

• Actually flow fields can possess zones of both irrotational and rotational flows.

free vortex flow → irrotational flow, bath tub, hurricane, morning glory spillway forced vortex flow → rotational flow, rotating cylinder

Rotational flow

← shear force

Irrotational flow

← gravity, pressure force

vt =ωr v_r = 0

ω

the equations and

→ Forced vortex

→ A cylindrical container is rotating at an angular velocity .

→ rotational flow (forced vortex) possessing a constant vorticity over the whole flow field

→ streamlines are concentric circles

t t r

v v v

r r r

ξ θ

∂ ∂

= + −

∂ ∂

( ) r (0) 0 2 0

r r r r

ξ ω ω ω ω ω

θ

∂ ∂

= + − = + − = ≠

∂ ∂

2

1 2 c

u U y

b

 

=  − 

 

[IP 3.5] When a viscous, incompressible fluid flow between two plates and the flow is laminar and two-dimensional, the velocity profile is

parabolic, .

2D Poiseuille flow

τ

Calculate and (rotation)

No slip condition for real fluid

x y b

∂ ∂

1 1

( )

2 ^{V H} 2

v u

x y

ω = ω +ω = ^ ^∂∂ − ^∂∂ ^ ^{2 2}

1 2

2

c c

U

U y y

b b

 

 

= − −   = 

du

τ µ

2

2 _c

du U

dy b y

τ µ

µ

τ ^{= −} 2 µω ^{= −} µξ

1)

2)

→ rotation and vorticity are large where shear stress is large.

Due: 1 week from today

1. (Prob. 3.3)

A fluid particle moves so that in the Lagrangian frame of reference x = 3 t, y = 9 t², and z = 27 t³. Find the velocity and acceleration of the particle for times from zero (t = 0) to five seconds (t ^{= 5).}

2. (Prob. 3.5)

Calculate the accelerations in the flow of problem 3.1.

1 / ; 2 /

u = m s v = x m s

Sketch the following 2D flowfields (streamline equation) and derive general expressions for their components of acceleration :

(a) u = 4, υ = 3; (b) u = 4, υ = 3x; (c) u = 4y, υ = 0; (d) u = 4y, υ = 3;

(e) u = 4y, υ = 3x; (f) u = 4y, υ = 4x; (g) u = 4y, υ = -4x; (h) u = 4, υ = 0;

(i) u = 4, υ = -4x; (j) u = 4x, υ = 0; (k) u = 4xy, υ = 0 4. (Prob. 3.10)

Calculate the vorticity for the flow in problem 3.1.

1 / ; 2 /

u = m s v = x m s

For the velocity profiles shown below, derive expressions for the vorticity and plot them.

For the free vortex flow the velocities are υ_t = 5/r and υ_r = 0. Assume that lengths are in metres and times are in seconds. Plot the streamlines of this flow and calculate the accelerations and vorticity. Are there any interesting points in the flow?