Exam 2 - Chapter 9

Question 1

What characterizes problems that lead to boundary layers in fluid dynamics?

Answer 1

They involve equations where the term with the highest derivative is multiplied by a very small coefficient, such as viscosity.

Question 2

Why can’t viscous terms be neglected even when Reynolds number is large?

Answer 2

Because near solid boundaries, velocity gradients are large, making viscous effects significant within a thin region—the boundary layer.

Question 3

What happens when the highest-order derivative term is ignored in a differential equation?

Answer 3

The equation loses one degree of freedom, making it impossible to satisfy all boundary conditions.

Question 4

In the simplified ODE εy’’ + ay’ + by = 0, what does ignoring εy’’ cause?

Answer 4

It leads to a first-order equation that can satisfy only one boundary condition, missing the behavior near the boundary.

Question 5

What are the two roots of the characteristic equation in the small ε limit?

Answer 5

λ₁ = -b/a and λ₂ = -a/ε; the second corresponds to a rapidly decaying exponential near the boundary (the boundary layer).

Question 6

What does the term e^{-a x / ε} represent physically?

Answer 6

A rapidly decaying solution confined near the boundary; it enforces the wall boundary condition.

Question 7

What is the outer solution in boundary layer theory?

Answer 7

The inviscid solution obtained by neglecting the small viscous term, valid outside the boundary layer.

Question 8

What is the boundary layer (inner) solution?

Answer 8

The solution near the wall where viscous effects are important and gradients are large.

Question 9

How is the boundary layer variable η defined?

Answer 9

η = x/δ or y/δ, a scaled coordinate within the thin region where velocity changes rapidly.

Question 10

What determines the boundary layer thickness δ in general?

Answer 10

The requirement that the viscous term is comparable to the advective term, giving δ ~ ε or δ/L ~ Re⁻¹ᐟ².

Question 11

What is the approximate scaling of boundary layer thickness with Reynolds number?

Answer 11

δ/L ~ Re⁻¹ᐟ², meaning thinner boundary layers at higher Reynolds numbers.

Question 12

What is the boundary layer solution for the simple ODE example?

Answer 12

y_BL = (Be^{b/a})(1 - e^{-a x / ε}), matching zero at x = 0 and outer solution at larger x.

Question 13

What condition matches the inner and outer solutions?

Answer 13

The value of the inner solution far from the wall equals the value of the outer solution at the same point: y_BL(η → ∞) = y_outer(x₀).

Question 14

What are the 2D steady incompressible Navier–Stokes equations?

Answer 14

u uₓ + v u_y + (1/ρ)pₓ = ν(uₓₓ + u_yy); v uₓ + v v_y + (1/ρ)p_y = ν(vₓₓ + v_yy); uₓ + v_y = 0.

Question 15

What coordinate system is used for 2D boundary layer analysis near a surface?

Answer 15

x is parallel to the wall, y is perpendicular to it (positive away from the wall).

Question 16

How does the order of magnitude of v compare to u within the boundary layer?

Answer 16

v ≈ U δ / L, much smaller than u ≈ U.

Question 17

What scaling gives the boundary layer thickness from x-momentum balance?

Answer 17

U²/L ~ νU/δ² ⇒ δ² ~ νL/U ⇒ δ/L ~ Re⁻¹ᐟ².

Question 18

Why can ∂²u/∂x² be neglected compared to ∂²u/∂y² in the BL?

Answer 18

Because velocity gradients in y are much steeper than in x (δ ≪ L).

Question 19

What simplification does neglecting uₓₓ cause in the BL equations?

Answer 19

It changes the system from elliptic (NS) to parabolic, allowing it to be solved by marching downstream in x.

Question 20

What happens to the pressure gradient term ∂p/∂y inside the BL?

Answer 20

It becomes zero (∂p/∂y = 0), meaning pressure is uniform across the layer.

Question 21

What does ∂p/∂y = 0 imply physically?

Answer 21

Pressure at a given x is the same from the wall to the outer edge of the boundary layer.

Question 22

How is pressure p(x) determined in boundary layer analysis?

Answer 22

From the inviscid flow outside the BL, using Bernoulli’s equation and the known outer velocity U(x).

Question 23

What effect does the BL have on lift?

Answer 23

None directly—pressure distribution (and hence lift) remains the same as in inviscid flow unless separation occurs.

Question 24

What effect does the BL have on drag?

Answer 24

It introduces shear stress τ = μ(∂u/∂y)₍wall₎, producing viscous drag.

Question 25

When can the BL affect lift?

Answer 25

When separation occurs, altering the pressure distribution and causing stall.

Question 26

What are the boundary conditions for the 2D steady BL equations?

Answer 26

u = v = 0 at y = 0 (no-slip); u → U(x) as y → ∞.

Question 27

Why can’t v → 0 as y → ∞?

Answer 27

Because continuity requires vertical velocity to adjust for horizontal velocity deficit within the BL.

Question 28

What do the simplified BL equations look like?

Answer 28

u uₓ + v u_y + (1/ρ)pₓ = ν u_yy, ∂p/∂y = 0, uₓ + v_y = 0.

Question 29

What determines the outer velocity U(x)?

Answer 29

It is fixed by the inviscid flow solution outside the BL.

Question 30

How is the pressure gradient ∂p/∂x within the BL determined?

Answer 30

It is the same as in the outer flow, obtained from Bernoulli’s law using U(x).

Question 31

For a flat plate in uniform flow, what are U(x) and ∂p/∂x?

Answer 31

U(x) = constant = U∞; ∂p/∂x = 0, since the outer flow is uniform.

Question 32

What boundary and initial conditions are used for the flat-plate BL problem?

Answer 32

At x=0: initial condition (leading edge); at y=0: u=v=0; as y→∞: u→U∞.

Question 33

Does including the BL alter inviscid lift or drag conclusions?

Answer 33

Lift remains unchanged (pressure field same); drag is introduced through viscous shear stress.