Lecture 10

Equations of Continuity and Motion (1/6)

10.1 Continuity Equation

10.2 Stream Function in 2-D, Incompressible Flows 10.3 Rotational and Irrotational Motion

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

fluid dynamics, the three-dimensional expressions for conservation of matter and the momentum equations of motion are developed.

conservation of mass → continuity eq.

conservation of momentum → eq. of motion → Navier-Strokes eq.

x y z

  

[Re]

Infinitesimal (differential) control volume ( ) → point form Finite control volume – arbitrary CV → integral form

Apply principle of conservation of matter to the differential CV

→ sum of net flux = time rate change of mass inside C.V.

1) mass flux per unit time (mass flow)

mass vol

Q u A

time  time  

    

= flux in –flux out

∙ net flux through face perpendicular to y -axis

∙ net flux through face perpendicular to z -axis

( u ) ( u )

u y z u x y z x y z

x x

 

 ^  ^{  }

                  

( v )

x y z y



     



( w )

x y z z



     



( x y z ) t 

    



Thus, equating (A) and (B) gives

( ) ( ) ( )

( ) u v w

x y z x y z x y z x y z

t x y z

  

                    

   

( ) ( )

LHS x y z x y z x y z

t t t

  

  

           

  

Since C.V. is fixed →

0

t 

 LHS x y z

t



     



Cancelling terms makes

( ) ( ) ( )

u v w 0

t x y z

   

   

   

   

0 t q

 

    



→ Continuity Eq. for compressible fluid in unsteady flow (point form) (10.1)

(  q ) q   q

      

I II

gradient divergence

(I): q ( ui vj wk ) i j k

x y z

  

 ^{    }

            

u v w

x y z

  

  

  

  

(II): u v w

q x y z

     ^   ^    ^  ^  ^  

  ^{q u v w u v w}

x y z

x y z

   

 ^{         }

               

u v w 0

u v w

t x y z x y z

     ^{ }

      

        

        

d dt



d u v w 0

dt    ^   ^  x  ^  y  ^  z ^   

( ) 0

d q

dt      

(10.2a)

(10.2b)

0

t q

 

    



d dx dy dz dt t x dt y dt z dt

        

   

   

u v w

t x y z

   

   

   

   

1) For steady-state conditions

t 0

  

 

Then (10.1) becomes

( ) ( ) ( )

( ) 0

u v w

x y z q

   

  

     

  

2) For incompressible fluid (whether or not flow is steady)

d 0

dt

  

Then (10.2) becomes

u v w 0 x y z q

  

     

  

(10.4)

d u v w 0

dt    ^   ^  x  ^  y  ^  z ^   

radial

, - u r

azimuthal

, - v 

axial

, - w z

For compressible fluid of unsteady flow

1 ( ) 1 ( ) ( )

ur v w 0

t r r r z

   



   

   

   

For incompressible fluid

1 ( ) 1

ur v w 0

r r r  z

  

  

  

( )

0, 0, v 0

t r z

    

 

    

    

    

1 ( )

ur w 0

r r z

 

  

 

r, u

z, w

In 2D flowfield, define stream function yx, y) as

u v 0

x y

 

 

 

u y

y

 



v x

y

 

 ^(10.8)

(10.7)

• Then, we can have a simplified equation to determine only one unknown function.

→ Thus, continuity equation is satisfied.

2) Apply stream function to the equation for a stream line in 2-D flow Eq. (2.10):

Substitute (10.7) into (10.10)

(10.9)

2 2

=0

u v

x y x y y x x y x y

y y y y

 

                                  

0

vdx  udy 

= =0

dx dy d

x y

y y y

 

  

y = constant (10.12)

→ The stream function is constant along a streamline.

Substitute (10.7) into (10.13)

qdn udy vdx

   

qdn dy dx d

y x

y y y

 

   

 

→ Change in y (dy between adjacent streamlines is equal to the volume rate of flow per unit width (m²/s).

 Stream function in cylindrical coordinates radial

v

r y



  



azimuthal

v

r

y

 



the direction of flow is in the negative x- direction, from right to left.

 Motion of fluid: displacement, rotation

1

v v x v t

x v

t x x

              

       

 

 

  

1) The time rate of rotation of x -face about z -axis

Assume the rate of rotation of fluid element x and y about z-axis is positive when it rotates counterclockwise.

2) The time rate of rotation of y -face about z - axis

3) The net rate of rotation = average of rotations of x -and y -face

1

u u y u t

y u

t y y

        

       

 

   

  

1

z

2

v u

x y

  ^   ^   ^  ^  

1

x

2

w v

y z

  ^   ^   ^  ^  

1

y

2

u w

z x

  ^   ^   ^  ^  

1) Rotation

1 1 1

2 2 2

w v u w v u

i j k

y z z x x y

  ^   ^   ^  ^    ^   ^   ^  ^    ^   ^   ^  ^  

 

1 1

2 q 2 curl q

   

2 2 2

x y z

      

a) Irrotational flow → Ideal fluid

0

   q

x y z

0

     

, ,

w v u w v u

y z z x x y

     

  

     

b) Rotational flow → Viscous fluid

0

   q

Rotational flow

Irrotational flow

Rotational flow

2 curl q q

      

 Rotation in cylindrical coordinates

1 1 2

z r

v v

r z





 

 

       

1 2

r z

v v

z r



 ^   ^   ^  ^  

1 1

2

r z

v v v

r r r

 

 

 

 

         

(10.18)

G = line integral of the tangential velocity component about any closed contour S

10.3.2 Circulation

G   q ds 

2 2 2 2

u dy v dx u dy v dx

d u dx v dy u dx v dy

y x y x

           

G                            

v u

x y dxdy

   

       

v u

d dxdy

x y

    G        

 

2

z

A A A

v u

dA dA q dA

x y 

   

G               

circulation G  0 (if there is no singularity vorticity source).

[Re] Fluid motion and deformation of fluid element translation

Motion

rotation

linear deformation Deformation

angular deformation