DAY 19: Boundary Layer

flat plate boundary layer: in blue we highlight the region of the flow where velocity is influenced by the presence of the solid surface

flat plate : let us neglect the shape of the leading edge for now

Boundary layer – velocity profile

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

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

• Layer of fluid in the proximity of a boundary (condition)

• Zoom: in this layer of fluid we observe a velocity profile

slow growth: d/dx << d/dy v << u

Boundary layer growth

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

• Farther in x along the plate, the effect of the drag is felt by a larger region of the stream (viscous effects), and because of this the boundary layer grows

• Fluid friction on the surface is associated with velocity reduction along the boundary layer

0y

ody

du

x

Local stress & total force, skin friction

• This is different from the case of a Couette flow, where the gradient is defined by the two boundary conditions (thin film approx.)

• but there is more trouble…

0y

ody

du

Boundary layer transition to turbulence

At a certain distance along a plate, viscous forces become to small relative to inertial forces to damp fluctuations

Picture of boundary layer from text

note that as du/dy decreases in x the shear stress decreases as well

thickness of the boundary layer defined such to include 99% of the velocity variation

BLACKBOARD the laminar boundary layer 19A,B,C,D

Goal : keep laminar regime on the airfoil, to reduce drag. 98%..so far, so what is the problem? CONTROL

0

)(

y

oxdy

xdu

B L thickness in laminar region & fluid properties

xUxURe OO

x

Oxu

x5

Re

x5

Blasius solution viscosity

Boundary layer transition • How can we solve problems for such a complex system?

• We can think about key parameters and possible dimensionless numbers

• Important parameters: – Viscosity μ, density ρ

– Distance, x

– Velocity uO

• Reynolds number combines these into one number

xuxuRe OO

x

δ(x)

0y

What is turbulence ?

turbulence is a state of fluid motion where the velocity field is : highly 3D, varying in space and time , hardly predictable, non Gaussian, anisotropic but somehow statistically organized

coherent structures

The mean velocity profile in the smooth wall turbulent boundary layer : 1) viscous sublayer

the velocity varies linearly, as a Couette flow (moving upper wall). Thus, the shear stress is constant: 𝜏0

𝜏 = 𝜇𝑑𝑢

𝑑𝑦 𝑢 =

𝜏0 𝑦

𝜇

scaling near wall turbulence

We can define a velocity scale u* = 𝜏

𝜌 [m/s] characteristic of near wall turbulence

u* = shear velocity or friction velocity we can rewrite the linear profile in the viscous sublayer as

where 𝜐

𝑢∗ is a length scale (very small, remember 𝜐 =O(10-5 10-6) m2/s,

while u* is a fraction (~5-10%) of the undisturbed velocity U0

𝑢

𝑢 ∗=

𝑦𝑢 ∗

𝜐

we already have 2 velocity scales: 1) u* 2) U0

How many length scale ?

1) 𝜐

𝑢∗

2) 𝛿

𝛿 boundary layer height

viscous sublayer continued

How thick is the viscous sublayer ? it depends on the boundary layer... yes/no? as u* and 𝜐 define the viscous length scale, we can represent the extension of the viscous sublayer in terms of multiples (5-10) of the viscous scale (viscous wall units)

𝛿𝜐 = 5 𝜐

𝑢∗

Note that as u* 𝛿𝜐 : the viscous sublayer becomes thinner Note: roughness protrusion (fixed physical scale) may emerge from the viscous sublayer and change the near wall structure of the flow

𝛿𝜐

The mean velocity profile in the smooth wall turbulent boundary layer : 2) the logarithmic region

here is another velocity scale standard deviation or r.m.s. velocity velocity scale of the energy containing eddies

The mixing length theory: fluid particles with a certain momentum are displaced throughout the boundary layer by vertical velocity fluctuation. This generate the so called Reynolds stresses think about the complication as compared to LAMINAR case

𝜏 = −𝜌𝑢′𝑣′

𝜏 = 𝜇𝑑𝑢

𝑑𝑦

𝜏 = −𝜌𝑢′𝑣′

If we know the stress, we can obtain by integration the velocity profile

mixing length assumption (Prandtl: 𝑢′ = 𝑙 𝑑𝑢

𝑑𝑦 )

What does it mean?

A displaced fluid parcel (towards a faster moving fluid) will induce a negative velocity u’ ~ v’ such that 𝜏 = −𝜌𝑢′𝑣′ = 𝜌𝑙2 𝑑𝑢/𝑑𝑦 2

l represent the scale of the eddy responsible for such fluctuation

very important: we also assume that the size of the eddies l varies with the height l=ky : very reasonable, farther from the wall eddies are larger (attached eddy)

Laminar flow : only viscous “friction”

𝜏 = 𝜇𝑑𝑢

𝑑𝑦

Turbulent flow : small viscous “friction” as compared to momentum transfer by eddies

𝜏 = −𝜌𝑢′𝑣′

However at the small scales at any instant, viscosity still matters (cannot be neglected)

we thus have 𝜏 = −𝜌k2 y2 𝑑𝑢/𝑑𝑦 2

with u* = 𝜏

𝜌

integrating we obtain : 𝑢

𝑢∗=

1

𝑘ln

𝑦𝑢∗

𝜐 +C

Logarithmic law of the wall !!! where u* depends on the flow and the surface k is the von Karman constant(?)=0.395-0.415 (k=0.41 is a good number) C is the smooth wall constant(?) of integration (C=5.5 is a good number)

note that is a rough wall boundary layer 𝑢

𝑢∗=

1

𝑘ln

𝑦

𝑦0

where y0 is the aerodynamic roughness length: it is a measure of aerodynamic roughness, not geometrical (surface) roughness relating with y0 is complicate

The mean velocity profile: where is it valid ? from about 60 viscous wall units to about 15% of he boundary layer height

it makes sense that the extension of the log layer has to be determined by both inner scaling and outer scaling

Laminar and Turbulent BL

• Analytical results Empirical results

BL growth

𝛿 𝑥 =5𝑥

𝑅𝑒1/2 𝛿 𝑥 =0.16𝑥

𝑅𝑒1/7

shear stress coefficient

𝑐𝑓 =𝜏0

1/2𝜌(𝑈0) =

0.664

𝑅𝑒1/2 𝑐𝑓 =0.027

𝑅𝑒1/7 and many others

assuming a 1/7 power law velocity distribution u/U0 = (y/ 𝛿)1/7

Re=xU0/ 𝝊 as the distance x increases cf decreases

Note that a different set of formula exist for the full plate (averaged over the length L)

figure_09_07

QUESTIONS?

Laminar Turbulent Induced

δ(x)

cf

FS

Cf

Laminar, Turbulence, Induced Turbulence

x0

L

0x

dxB

2UBL

F2

O

S

2U2

O

O

7/1XRe

x16.0

XRe

x5

f

2O c

2

U

7/1

XRe

x16.0

7/1XRe

027.0

X

2 Re06.0ln

455.0

XRe

664.0

f

2o

2

U*Area C

7/1LRe

032.0

LL

2 Re

1520

Re06.0ln

523.0

LRe

33.1

xUxURe OO

x

LU

Re OL