Lecture 15

Equations of Continuity and Motion (6/6)

15.1 Vortex Motion Problems

Objectives

- Derive 3D equations of continuity and motion

- Derive Navier-Stokes equation for Newtonian fluid - Study solutions for simplified cases of laminar flow

- Derive Bernoulli equation for irrotational motion and frictionless flow - Study solutions for vortex motions

 Vortex

- fluid motion in which streamlines are concentric circles

• Forced vortex

- Rotational cylinder

• Free vortex

- Typhoon, drain hole vortex

For steady flow of an incompressible fluid, apply Navier-Stokes equations in cylindrical coordinates

Assumptions:

0 v_ 

h

  0 t

 



0; 0; 0

r z

v v v

z



  

 p 0



 



p p

z h

 

  

1) r -comp.

1  

rv

r r



 ¹   v   v

r 

z

 

 

   0

1  

0 v 0

r v

 

 

    

 

Navier-Stokes Eq.: Eq. (11.9)

v

 ^ t



r r

v v

r

 



v v

r





 



2

r z

v v

r v z





 



 

 

 

1

r

p rv

r  r r r

  

  

  

2

2 2

1 v

r 

     

    

 

2 v r



 



2 2

v

z

 

  g

 

  

 

 

 

2

1

v p

r r





 





2) -comp.

v t

 ^

  v

v v v

r r

  



 

  

v v

r



 v

v z

 



  

 

1 p r 

  

 ¹   ^{rv 1}

^v

r r r r

 

 

    

        

2 v

r 

 



2 2

v z



   g

 

  

 

 

 

3) z -comp.

 

0 rv

r r r





 

       

v

 ^ t

  v

v

v v

r r





  

 

v v

r



v

v z

 



 

 

 

1

p v

z  r r r r

  

  

  

2

2 2

1 v

r 

  

   

 

2 2

v

z

 

  g

 

  

 

 

 

1 1

0 p

p

g g

z h

 

 

     

 

Integrate  -Eq. 15.2 b) w.r.t. r

Integrate again

1

 

C 1 rv

r r

 



1

 

rC rv

r

 



2

1 2

1 2

2 ( )

2 ( )

r C C rv A

C C

v r B

r





 

 

need 2 BCs (15.3)

1  

0 rv

r r r

   

      

→ hydrostatic pressure distribution

p g

h  

    



p    h

(15.4)

15.1.2 Forced Vortex - rotational flow

Consider cylindrical container of radius R is rotated at a constant angular

velocity W about a vertical axis Substitute BCs into Eq. (15.3)

i)

2 2

0, 0

( ) : 0 0 0

r v

A C C





    

1

1

,

( ) : 2

2

r R v R

B R C R C



 W

 W    W

ii)

Eq. (B) becomes

2

v

2 W r r

  W

2 2

2

.: r 1 p

r Eq

r r

p r

r





W 

 



   W



. : p z Eq

h 

   



(15.6)

(15.7)

Incorporate B.C. to decide C₃ Consider total derivative dp

Integrate once

p p

dp dr dh rdr dh

r h  

 

   W 

 

2 2

2

p  W  r   h  C

0;

r  h  h

p  p

0

0

p    h  C  C

 p

  h

 

0 0

2

p  p   ^W r   h  h

At free surface

p  p

2 2

0

2

h h r

g

  W

→ paraboloid of revolution

(15.8)

(15.9)

• Rotation components in cylindrical coordinates

→ rotational flow

→ Forced vortex is generated by the transmission of tangential shear stresses.

Eq. (10.15):

vorticity

   

1 1

2

1 1

2 2

r z

v v v

r r r

r r

r r

 

 

 

 

          W 

 

     W    W  W  W 2 

2 0

  W 

Free vortex: drain hole vortex, tornado, hurricane, morning glory spillway

For irrotational flow,

2

2

p v

h g

 ^{ }

^

Differentiate w.r.t r

1 1

v 0

p h

r r g v r

 



     

  

p v

r v r

 

  

  

 

coincides with h

z

0, 1

h h h

r  z

  

    

    

 

(A)

Eq (15.2a): r -Eq. of N-S Eq.

Equate (A) and (B)

Integrate using separation of variables

p v

r r



 



v v

v

v r v

r r r

  

 

 ^  ^

    

 

1 1

v r

v

   

r

 

(C)

ln v

  ln r  C

 

ln v

 ln r  ln v r

 C

v r

 C

v^  r

[Cf] Forced vortex

v

 W r

(D)

(15.10)

(15.6)

(B):

 

2 2

4

3 3

p v v r C

r r r r

 

  

   



• Total derivative

2 4

3

p p C

dp dr dh dr dh

r h  r 

 

   

 

p

h



  

 (D)

Integrate once

2 4

2 5

2

p C h C

 r 

   

B.C.:

r   : h  h

p  p

Substitute B.C. into Eq. (E)

0 0 5

p    h  C C

 p

  h

 

0 0 2

2

p p h h C

  r

   

(E)

(15.11)

[Cf] Forced vortex: 0 ^{2 2}





p  p  W  2 r   h  h

0 2

4

0 2

2 h h C

  gr

0

2

h h r

g

  W

 Circulation

 

2 2

4 0 4

0

2 0

q ds

v rd

 C 

 C

        

ds  rd 

v r_  C4

(15.12)

(15.9)

→ Even though flow is irrotational, circulation for a contour enclosing the origin is not zero because of the singularity point.

 Vorticity component _z

1

z

v

 r



  



v v

r r





 



Substitute C⁴ v^  r

4 4 4 4

2 2 2 0

z

C C C C

r r r r r



is rotational (forced vortex) when drain in the tank bottom is suddenly closed.

→ Rankine combined vortex

→ Fluid motion is ultimately dissipated through viscous action.

• Stream function, y

4 2

C 

 

4

2 v C

r r r



y



 

  



ln ln

2 2

dr r K r

y r

 

 

   



where = vortex strength

1. (6-4) Consider an incompressible two-dimensional flow of a viscous fluid in the -plane in which the body force is due to gravity. (a) Prove that the divergence of the vorticity vector is zero. (This expresses the conservation of vorticity, . (b) Show that the Navier- Stokes equation for this flow can be written in terms of the vorticity as . (This is a “diffusion”

equation and indicates that vorticity is diffused into a fluid at a rate which depends on the magnitude of the kinematic viscosity.) Note that is the substantial derivative.

Due: 2 weeks from today

xy

 0

  

d 2

dt   

d dt



q i j k









y z

  ^  ^

 

u w

z x

  ^  ^

 

2. (6-5) Consider a steady, incompressible laminar flow between parallel plates as shown in Fig. 6-4 for the following conditions: a =0.03 m,

U =0.3 m/sec,   0.476 N·sec/m², =625 N/m³ (pressure increases in + x -direction). (a) Plot the velocity distribution, u(z), in the z -

direction. Use Eq. (6.24) (b) In which direction is the net fluid motion? (c) Plot the distribution of shear stress t_zx in the z -direction.

/ p x

 

3. (6-7) An incompressible liquid of density  and viscosity  flows in a thin film down glass plate inclined at an angle a to the horizontal. The

thickness, a , of the liquid film normal to the plate is constant, the velocity is everywhere parallel to the plate, and the flow is steady. Neglect viscous shear between the air and the moving liquid at the free surface. Determine the variation in longitudinal velocity in the direction normal to the plate, the shear stress at the plate, and the average velocity of flow.

4. (6-11) Consider steady laminar flow in the horizontal axial direction through the annular space between two concentric circular tubes. The radii of the inner and outer tube are r₁ and r₂, respectively. Derive the expression for the velocity distribution in the direction as a function of viscosity, pressure gradient , and tube dimensions. p / x

6. (6-21) The velocity variation across the radius of a rectangular bend (Fig.

6-22) may be approximated by a free vortex distribution _ r = const.

Derive an expression for the pressure difference between the inside and outside of the bend as a function of the discharge Q, the fluid density , and the geometric parameters R and b , assuming frictionless flow.

 a / 2)(x ² + 2y – z ²), where a is an arbitrary constant greater than zero. (a) Find the equation for the velocity vector .

(b) Find the equation for the streamlines in the xz (y = 0) plane.

(c) Prove that the continuity equation is satisfied.

q  iu  jv  kw