6. Laminar and turbulent boundary layers

6 . Laminar and turbulent boundary layers

6.1 Some introductory ideas

Flow boundary layer ‰

A boundary layer of thickness δ ƒ Ludwig Prandtl (1875-1953) made the mathematical description of the boundary in 1904

6. Laminar and turbulent boundary layers

δ = fn(u∞ , ρ , μ , x) δ

6.1 Some introductory ideas

x

= fn (Re x ) Re x ≡

δ x College of Energy and Power Engineering

6 . Laminar and turbulent boundary layers

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6.1 Some introductory ideas

College of Energy and Power Engineering

6 . Laminar and turbulent boundary layers

Flow boundary layer ‰

=

ρu ∞ x u ∞ x = μ ν 4.92 Re x

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6.1 Some introductory ideas

Flow boundary layer Reynolds experiment

Osborne Reynolds (1842~1912) and his laminarturbulent flow transition experiment.

ƒ Laminar

ƒ Transitional

ƒ Transitional / turbulent

Recritical ≡

ρ D (uav )crit = 2100 μ College of Energy and Power Engineering

6 . Laminar and turbulent boundary layers

ƒ Turbulent JHH

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6.1 Some introductory ideas

College of Energy and Power Engineering

6 . Laminar and turbulent boundary layers

Flow boundary layer ‰

u∞ xcrit

ν

‰

6.1 Some introductory ideas

The thermal boundary layer during the flow of cool fluid over a warm plate ∂T −kf

≅ 5 × 105

conduction into the fluid

∂T ∂y

Nu L = 5

=− y =0

(Tw − T∞ ) kf / h

ƒ Nusselt Number T

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= h(Tw − T∞ )

∂y y = 0  

kf / h

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Thermal boundary layer

Boundary layer on a long, flat surface with a sharp leading edge

Re x critical ≡

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L

⎛ T −T ⎞ ∂⎜ w ⎟ ⎝ Tw − T∞ ⎠ ⎛ y⎞ ∂⎜ ⎟ ⎝ L⎠

=

hL = Nu L kf

y / L =0

δ t′ ƒ Nu is inversely proportional to the thickness of the thermal b.l. College of Energy and Power Engineering

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6 . Laminar and turbulent boundary layers

6.2 Laminar incompressible boundary layer on a flat surface

Conservation of mass-The continuity equation ‰

A steady, incompressible, two-dimensional flow field ƒ Velocity field u = ui + vj + wk ƒ continuity equation

∂u ∂v + =0 ∂x ∂y ƒ

streamlines, constant ψ .

v=−

6.2 Laminar incompressible boundary layer on a flat surface

∂ψ ∂x

6 . Laminar and turbulent boundary layers

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6.2 Laminar incompressible boundary layer on a flat surface

6 . Laminar and turbulent boundary layers

Example 6.1 ‰ ‰

∂ψ ∂x

and u = y

ƒ by integrating

∂ψ ∂y

0=− x

∂ψ ∂x

‰

∂ψ ∂y

and u∞ = y

x

ƒ comparing these equations

fn（x）=constant fn（y）=u∞ y +constant

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6.2 Laminar incompressible boundary layer on a flat surface

∂u ∂v ∂w + + =0 ∂x ∂y ∂z

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6.2 Laminar incompressible boundary layer on a flat surface

p ≠ fn ( y )

6 . Laminar and turbulent boundary layers

‰

∂τ yx ⎞ ⎛ ⎛ ∂τ yx ∂p ⎞ ∂p ⎞ ⎛ dy ⎟ dx − τ yx dx + pdy − ⎜ p + dx ⎟ dy = ⎜ − ⎟ dxdy ⎜τ yx + ∂y ∂x ⎠ ⎝ ⎝ ⎠ ⎝ ∂y ∂x ⎠ ƒ Newton’s law of viscous shear

‰

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6.2 Laminar incompressible boundary layer on a flat surface

Rate of x-momentum increase ⎛ 2 ∂ρ u 2 ⎞ ⎛ ∂ρ uv ⎞ dx ⎟ dy − ρ u 2 dy + ⎜ u ( ρ v) + dy ⎟ dx − ρ uvdx ⎜ ρu + ∂x ∂y ⎝ ⎠ ⎝ ⎠ ⎛ ∂ρ u 2 ∂ρ uv ⎞ =⎜ + ⎟ dxdy ∂y ⎠ ⎝ ∂x Momentum eq.

∂ ⎛ ∂u ⎞ ∂p ∂ρu 2 ∂ρuv ⎜μ ⎟− = + ∂y ⎜⎝ ∂y ⎟⎠ ∂x ∂x ∂y

∂u ∂y

⎡ ∂ ⎛ ∂u ⎞ ∂p ⎤ ⎢ ⎜ μ ⎟ − ⎥ dxdy ⎣ ∂y ⎝ ∂y ⎠ ∂x ⎦ JHH

ƒ

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ƒ In the x direction

τ yx = μ

ƒ

∂u ∂u