Boundary layer

..perhaps the hardest place to use Bernoulli’s equation

(so don’t)

Drag on a surface – 2 types• Pressure stress / form drag

• Shear stress / skin friction drag

• A boundary layer forms due to skin friction

Boundary layer – velocity profile

• Far from the surface, the fluid velocity is unaffected.

• In a thin region near the surface, the velocity is reduced

• Which is the “most correct” velocity profile?

…this is a good approximation near the “front” of the plate

Boundary layer growth

• The free stream velocity is u0, but next to the plate, the flow is reduced by drag

• Farther along the plate, the affect of the drag is felt by more of the stream, and because of this

• The boundary layer grows

Boundary layer transition• At a certain point, viscous forces become to small

relative to inertial forces to damp fluctuations

• The flow transitions to turbulence• Important parameters:

– Viscosity μ, density ρ– Distance, x– Velocity UO

• Reynolds number combines these into one number

xUxU

Re OOx

First focus on “laminar” boundary layer• A practical “outer edge” of the boundary

layer is where u = uo x 99%

• Across the boundary layer there is a velocity gradient du/dy that we will use to determine τ

• Let’s look at the growth of the boundary layer quantitatively.

• The velocity profiles grow along the surface

• What determines the growth rate and flow profile?

Blasius

• Resistance of fluid to change in velocity due to viscous forces depends on…?– Velocity, UO

– Viscosity, μ– Density, ρ– Position, x

• The Reynolds number• Blasius (1908) derived:

y

fU

u

O

xRe

xg

δ(x)

OU

u

y

xUxU

Re OOx

OOx U

x5

U

xg

Re

xg

B L thickness & fluid properties

xUxU

Re OOx

Blasius, cont’d: Surface shear stress a

• Shear stress

δ(x)

OU

u

y

y

u

y

u

y

fRex

Uu

y x

O

332.0

Rex

Uu

y0yx

O

2/12/1

2/3O

0y x

U332.0

y

u

XO

2/12/1

2/3O

0yRe

x

U332.0

x

U332.0

Boundary layer transition• At a certain point, viscous forces become to small

relative to inertial forces to damp fluctuations

• The flow transitions to turbulence• Important parameters:

– Viscosity μ, density ρ– Distance, x– Velocity UO

• Reynolds number combines these into one number

xUxU

Re OOx

How does BL transition?

Next focus on “turbulent” boundary layer

• A practical “outer edge” of the boundary layer is (still) where u = uo x 99%

• Across the boundary layer there is a velocity gradient as well as variations in the velocity that determine τ

Turbulence: average & fluctuating velocity

The velocity profile concerns the mean velocity. The fluctuating part contributes to the internal stress for high Re flow.

Three zones in the turbulent BL

Viscous sub-layer

• The velocity in the viscous sub-layer is linear

Log law of the wall

Majority of the boundary layer

• 105 < Re < 107

Shear stress, thickness of turbulent boundary layer

(somewhat empirical)

X

2

2O

0y Reln

455.0

2

U

7/1X

2O

Of

Re

027.0

2Uc

7/1XRe

x16.0

7/1X

2O

0y Re

027.0

2

U

X22

O

Of

Re06.0ln

455.0

2Uc

Laminar Turbulent Induced

δ

cf

FS

Cf

Laminar, Turbulence, Induced Turbulence

0

L

0x

dxB

2UBL

F2

O

S

2U 2

O

O

5/1

L

2

O

Re

058.0

2

U

5/1

XRe

x37.0

X

2

2

O

Re06.0ln

455.0

2

U

XRe

x5

X

2

O

Re

664.0

2

U

7/1

XRe

x16.0

5/1

XRe

058.0

X

2 Re06.0ln

455.0

XRe

664.0

5/1

L

2

ORe

074.0UBL

2

1

LL

2

2

O

Re

1520

Re06.0ln

523.0

* UBL2

1

L

2

ORe

33.1UBL

2

1

5/1

LRe

074.0 LL

2 Re

1520

Re06.0ln

523.0

LRe

33.1

Example 9.8 from textAssume that a boundary layer over a smooth, flat plate is laminar at first and then becomes turbulent at a critical Reynolds number of 5 x 105. If we have a plate 3 m long x 1 m wide, and if air at 20°C and atmospheric pressure flows past this plate with a velocity of 30 m/s what will be the average resistance coefficient Cf for the plate? Also, what will be the total shearing resistance of one side of the plate and what will be the resistance due to the turbulent part and the laminar part of the boundary layer?What is the answer to the same questions if the boundary layer is “tripped” by some sufficiently large roughness element that the boundary layer is turbulent from the beginning?

Example