MAE130C - Laminar Boundary Layer Theory

What are the three primary assumptions required to apply Boundary Layer Theory?

1) Uniform, irrotational flow over a 2D flat plate (meaning the only spatial directions are x and y)

2) irrotational flow outside the flow is steady/does not vary with time

3) flow is incompressible and not acted upon by body forces

Define Boundary layer thickness in physical terms

The distance from the wall to the point in which local fluid velocity reaches 99% of the external fluid velocity and viscous shear forces still exist (also known as the 99% thickness)

Why is the pressure gradient across the thickness of the boundary layer (∂p/∂y) assumed to be zero?

When referring to the thin layer approximation, all terms except for the y-pressure gradient go to zero. This is because the u(∂v/∂x), v(∂v/∂y), and ν(∂^2v/∂x^2+∂^2v/∂y^2) terms go to zero as 'x' terms are much larger than 'y' terms, including derivatives by OoM analysis

x terms are much larger than y terms, everything goes to zero except y pressure gradient

In boundary layer theory, what does a "Zero-Pressure-Gradient" (ZPG) flow imply regarding the behavior of the external velocity U(x)?

ZPG flow implies that the rate of change of the external velocity, dU/dx, is zero (meaning external velocity remains constant along the surface in the direction of the flow)

implies rate of change of external velocity is zero , ext vel remains const along surface in direction of flow

According to the thin-layer approximation, we assume the velocity gradient in the y-direction (∂/∂y) is much larger than the gradient in the x-direction (∂/∂x). Provide a physical justification for this scaling.

The velocity gradient across the layer (∂u/∂y) is much larger than the velocity gradient along the flow, and this is caused by the existence of a thin, high shear region that forces the flowing fluid to match the stationary plate velocity

Explain the "Thin-Layer Approximation." Why must L ≫ δ?

An approximation that states that at Reynolds numbers above 1000, changes in a variable with respect to y (1/δ) and are much greater than those with respect to x (1/l). Therefore, L must be much larger than δ because by OoM analysis.

In the context of the outer flow, what does "Inviscid and Irrotational" allow us to use?

An external velocity that does not vary with axial distance or potential flow

What is the physical meaning of the displacement thickness?

The loss of the mass flow rate inside the boundary layer. It is equivalent to the external flow being displaced from a wall by a distance δ* (applies to any incompressible boundary layer)

What is the physical meaning of the momentum thickness?

The flux of the momentum deficit that is proportional to skin friction and drag (applies to any incompressible boundary layer)

For a flat plate (ZPG), how does the wall shear stress (τw) behave as x increases?

As x increases, wall shear stress decreases

Why does the boundary layer grow in the downstream direction?

Viscosity "diffuses" momentum, and diffusion naturally spreads over distance (and time), causing the boundary layer to grow.

What is meant by the shape factor H? What does a high H value indicate about the flow?

H measures the shape of the velocity profile inside the boundary layer—how quickly velocity rises from zero at the wall to the free-stream value. A high H value means that the velocity profile is more 'peaked' or less full or there is more low-velocity fluid near the wall

What is the "Similarity Hypothesis" used in the Blasius solution?

Some quantity or quantities has the same basic shape, but stretches or shrinks according to some scaling

In the Blasius equation, why is the coordinate transformation η = y*sqrt(U/νx) necessary?

If the velocity profile only depends on η, the amount of independent variables governing the velocity profile's evolution goes from two to one, making the solution for boundary layer thickness easier to solve for using the Blausius ODE

Describe the effect of a Favorable Pressure Gradient (dp/dx < 0) on the boundary layer profile.

The external flow accelerates, meaning that the boundary layer thickness will increase due to increased momentum and 'fuller' velocity profile

Describe the effect of an Adverse Pressure Gradient (dp/dx > 0) on the boundary layer profile.

The external flow decelerates, meaning that the boundary layer thickness will decrease due to decreased momentum and 'less full' velocity profile

What physical event occurs when (∂u/∂y)y=0 = 0 in the presence of an adverse pressure gradient?

Boundary layer separation