boundary layers and internal flow

What is a boundary layer?

A thin region near a surface where viscous effects are significant and velocity changes from wall velocity to freestream velocity.

What is the no-slip condition?

The fluid velocity at a solid surface equals the velocity of the surface.

What equation relates viscous shear stress to velocity gradient?

τ = μ(∂u/∂y)

What does ∂u/∂y represent?

The rate at which flow velocity increases in the wall-normal direction.

Define the local Reynolds number for a boundary layer.

Rex = (ρ∞V∞x)/μ∞

How does boundary layer thickness change downstream?

It starts at zero at the leading edge and grows with downstream distance.

What causes transition from laminar to turbulent boundary layer?

Increasing Reynolds number until a critical Reynolds number is reached.

Compare laminar and turbulent boundary layers.

Laminar layers are smooth with little mixing; turbulent layers have strong mixing and fuller velocity profiles.

Which boundary layer has higher wall shear stress?

A turbulent boundary layer.

Why are turbulent boundary layers more resistant to separation?

They possess greater momentum near the wall due to mixing.

How is boundary layer thickness δ commonly defined?

The distance where local velocity reaches 99% of edge velocity (u = 0.99Ue).

What is a favorable pressure gradient?

∂p/∂x < 0; pressure decreases downstream and helps prevent separation.

What is an adverse pressure gradient?

∂p/∂x > 0; pressure increases downstream and promotes separation.

What is flow separation?

When the boundary layer detaches from the surface due to strong adverse pressure gradients.

How does surface roughness affect boundary layers?

It can trigger earlier transition from laminar to turbulent flow.

What effect does boundary layer separation have on an airfoil?

It decreases lift and increases drag.

Define internal flow.

Flow whose downstream development is confined or altered by surrounding boundaries.

Give examples of internal flows.

Pipe flow, duct flow, hydraulic systems, blood vessels, HVAC ducts.

How do internal and external flows differ regarding boundary layers?

In internal flow, boundary layers grow inward and eventually fill the entire flow field.

What is the entrance region of a pipe flow?

The region where boundary layers are growing and the velocity profile is developing.

What is the potential core?

The inner region not yet affected by viscous effects in the entrance region.

What defines fully developed flow?

Boundary layers have merged and the velocity profile no longer changes downstream.

Laminar entrance length relation?

Le/D ≈ 0.05 ReD

Turbulent entrance length relation?

Le/D ≈ 4.4 ReD^(1/6)

How is Reynolds number for pipe flow defined?

ReD = ρVavgD/μ

What happens to pressure in a fully developed pipe flow?

It continuously decreases due to viscous friction losses.

Pressure drop equation for internal flow?

Δp = (1/2)ρVavg²(L/D)f

What is the friction factor for laminar pipe flow?

f = 64/ReD

What is head loss?

A length-based measure of pressure loss due to friction.

Relationship between head loss and pressure drop?

hL = Δp/(ρg)

Head loss equation for horizontal, fully developed flow?

hL = (Vavg²/2g)(L/D)f

Why is pressure drop important?

It determines the pumping power required to maintain flow.

General pumping power equation?

P = QΔp = QρghL

State Poiseuille’s Law for laminar flow rate.

Q = (Δp·πD⁴)/(128μL)

How can pumping power often be reduced?

Increase pipe diameter or reduce velocity/viscosity.

What is hydraulic diameter?

Dh = 4Ac/Pwetted

Why use hydraulic diameter?

To analyze non-circular ducts and pipes using pipe-flow correlations.

Laminar pipe flow Reynolds number range?

Re ≤ 2300

Turbulent pipe flow Reynolds number range?

Re > 4000

What determines friction factor in turbulent flow?

Reynolds number and relative roughness (ε/D).

What is the Darcy-Weisbach friction factor?

The friction factor used in turbulent pipe-flow pressure-loss calculations.

What equation is commonly used for turbulent friction factor?

The Colebrook-White equation.

What tool summarizes turbulent friction-factor data?

The Moody Chart.

What are the key steps in solving internal flow problems?

Find Dh, Vavg, Re, relative roughness, friction factor, pressure drop, then pumping power.