Boundary Layers

Hydromechanics VVR090

ppt by Magnus Larson; revised by Rolf L Jan 2014, Feb 2015, Feb 2017

SYNOPSIS

1. Boundar Layer on a Flat Plate

2. Von Karman momentum integral equation

3. Laminar Boundary Layer along a Flat Plate

4. Drag Coefficient for Smooth, Flat Plates

5. Examples/Problems

Figure numbers and Equation numbers refer to

Vennard and Street : Elementary Fluid Mechanics

1. Boundar Layer on a Flat Plate

Boundary layer: the zone in which the velocity profile is

governed by frictional action

V0 = free stream velocity (m/s)

δ = boundary layer (m)

Fig. 13.6

Boundary layer characterized by a Reynolds number:

3900 laminar flow

3900 turbulent flow

oVR

R

R

Viscous sublayer always exists close to the surface.

Mathematical implications of boundary layer:

x = locally tangent to any point along the surface of body

y = locally normal to any point along the surface of body

2. Von Karman momentum integral equation

• Apply momentum equation for control volume ABCD.

• Height of control volume extends beyond the edge of the boundary layer (to

the outer flow).

• At edge of the boundary layer: po(x), Vo(x) (known).

• Small curvature along body.

Conservation of mass:

Conservation of momentum:

2

0

h

oo y h y h

dp dh u dy uv

dx dx

0

0

h

y h

dudy v

dx

Develop conservation of mass equation:

0

h

o o

dv udy

dx

Develop conservation of momentum equation:

2

0

h

oo o o o

dp dh u dy V v

dx dx

Eq. 13.2

Eq. 13.3

Combine conservation of mass and momentum:

2

0 0

h h

oo o

dp d dh u dy V udy

dx dx dx

Euler equation for the outer flow:

0

oo o

o

o oo o

dpV dV

dp dVV

dx dx

Eq. 13.4

2

0 0

h h

oo o o o

dVd du dy V udy V h

dx dx dx

Combining:

Define displacement thickness = δ1

1

0

h

o o o oV V u dy

Flow rate with and without boundary layer

0

o oudy V

1

0

1

h

o o

udy

V

(the distance by which the boundary layer should be displaced to compensate

for the reduction in flow rate on account of boundary layer formation)

Eq. 13.5

Eq. 13.7

In the same manner, define momentum thickness:

2

0

1

h

o o o

u udy

V V

Substitute in and in equation 13.5 and express

o in terms of a local friction coefficient cf:

2

2

fc d

dx

(constant density, no pressure gradients)

21

Eq. 13.7

Eq. 13.11

3. Laminar Boundary Layer along a Flat Plate

Assume parabolic velocity profile:

2

2

2

oV y yu

2

2

2

2

15

2

2 15

22

15

fo

o

oo o

c d

V dx

VdV

dx

Eq. 13.13

2

2

22

15

15

2

30

oo

o

x

VdV

dx

x

V

x R

Integrate the equation:

28

2 15

oo

x

V

R

Substitute in in shear stress expression:/d dx

Eq. 13.16

Local friction coefficient:

8

15f

x

cR

Mean friction coefficient along the plate:

0

1 32

15

x

f f

x

C c dxx R

Eq. 13.17

Eq. 13.18

Relationship between and :

2

30x

RR

= 3900 = 500,000

(transition between laminar and turbulent

conditions in the boundary layer)

R R x

R R x

Eq. 13.19

4. Drag Coefficient for Smooth, Flat Plates

21

2f f oD C V A A: surface area of plate

Fig. 13.9

5. Examples / problems

Example I: Boundary layer and drag on ship model

A ship model 1.5 m long and with a draft of 0.15 m is towed at a

velocity of 0.3 m/s in a basin containing water at 16 C. The model

scale is 1:64.

Assuming that one side of the immersed portion of the hull may

be approximated by a smooth flat plate (1.5 m x 0,15 m), estimate

the frictional drag of the hull and the thickness of the boundary

layer at the stern of the model if the boundary layer is

a) laminar and

b) turbulent.

c) If the measured total drag of the model is 0.178 N, estimate the

total drag of the prototype.