Boundary layer thickness, displacement thickness, momentum thickness: Question bank

Symbols used throughout (all built in the parent):

$U$ — free-stream speed (fluid far from the wall).
$u = u (y)$ — local speed at height $y$ above the wall.
$δ$ — thickness where $u = 0.99 U$ .
$δ^{*}$ — displacement thickness (mass-flux deficit).
$θ$ — momentum thickness (momentum-flux deficit).
$H = δ^{*} / θ$ — shape factor.

True or false — justify

The boundary layer has a sharp, physical edge at

y = δ

False — the velocity approaches

U

only asymptotically, so

δ

is a convention (the 99% cutoff), not a real wall in the fluid.

For any non-trivial profile,

δ^{*}

is smaller than

δ

(and in general

δ^{*} \leq δ

True — the integrand

1 - u / U

is strictly less than 1 wherever the fluid moves at all, so the accumulated slab is thinner than

δ

; equality

δ^{*} = δ

occurs only in the degenerate case

u = 0

for all

y < δ

(a step profile), which is why the universal statement is

δ^{*} \leq δ

The order is always

θ < δ^{*} < δ

True for any physically reasonable (non-degenerate) profile — the extra factor

u / U < 1

θ

's integrand shrinks it below

δ^{*}

, and both are deficits so they sit inside

δ

; only pathological step profiles collapse the strict inequalities.

Doubling the free-stream speed

U

doubles the momentum thickness

θ

False —

θ

is a length built from the dimensionless profile

u / U

; the

U

's cancel, so it depends on the profile shape and

δ

, not directly on

U

's magnitude.

A larger shape factor

H

means the boundary layer is closer to separating.

True — as the profile becomes flatter/inflected near the wall (adverse pressure gradient),

δ^{*}

grows relative to

θ

, so rising

H

is an early-warning flag for Boundary layer separation.

For a stationary wall,

u (0) = 0

is an assumption we make for convenience.

False — it is the physical No-slip condition enforced by Viscosity: fluid molecules at the surface genuinely have zero relative velocity to the wall.

δ^{*}

measures how far the fluid is displaced downstream.

False — it measures how far the outer inviscid streamlines are pushed outward (away from the wall) to make room for the missing mass flow.

Integrating the

δ^{*}

formula to

\infty

always gives exactly the same number as integrating to

δ

False in general — since

u \to U

only asymptotically, the tail beyond

δ

carries a small but nonzero deficit area; the two integrals agree exactly only if we explicitly assume

u (y) = U

for

y > δ

(the standard modelling convention), otherwise they differ by that tiny tail.

θ

is directly related to the drag force on the wall.

True — the momentum-integral (von Kármán) relation shows the wall shear stress equals

ρ U^{2} \frac{d θ}{d x}

for a flat plate, so the streamwise growth of

θ

is the Skin friction drag per unit width (see the "Why" section below for the one-line derivation).

Spot the error

"Momentum deficit is

ρ (U - u)

, so

θ = \int (1 - u / U) d y

— same as

δ^{*}

The error is dropping a factor of

u

: momentum flux carries the mass flowing (

ρ u

) times the velocity gap

(U - u)

, giving

ρ u (U - u)

, so

θ

has the extra

u / U

weight that makes it smaller than

δ^{*}

"To get

δ^{*}

we normalise the deficit by

ρ U^{2}

Wrong normaliser —

δ^{*}

is a mass deficit, so we divide by the ideal mass flux

ρ U

, not

ρ U^{2}

; the

ρ U^{2}

belongs to the momentum balance for

θ

"Since all three thicknesses have units of length,

δ = δ^{*} + θ

Units matching does not imply an additive law; the three measure different conserved quantities (geometry, mass, momentum) and there is no such sum — only the inequality

θ < δ^{*} < δ

"The boundary layer stops growing once it reaches

δ = 5 x / R e_{x}

at that

x

That formula is

δ

at each

x

; it keeps growing with

x

because

δ \propto x / U x / ν \propto x

— the layer thickens continuously downstream.

"For the sine profile

\int_{0}^{δ} sin (π y /2 δ) d y = δ

, so

δ^{*} = 0

The integral is

2 δ / π \approx 0.64 δ

, not

δ

, giving

δ^{*} = δ (1 - 2/ π) \approx 0.363 δ

— a nonzero, positive deficit as it must be.

"A turbulent boundary layer has

H \approx 3

, higher than laminar, because it is thicker."

Reversed — turbulent profiles are fuller (velocity rises steeply near the wall), so their

H

is lower (~1.3–1.4) than the laminar Blasius

H \approx 2.59

; thickness

δ

is a separate matter from shape.

Why questions

Why do we need three thicknesses instead of just

δ

Because "where the layer ends" is fuzzy and geometric; engineers need precise measures of the two conserved quantities the wall steals — mass flux (

δ^{*}

) and momentum flux (

θ

) — which

δ

alone cannot give.

Why does

θ

weight the deficit by

u / U

but

δ^{*}

does not?

Mass flux depends only on speed once (

ρ u

), so

δ^{*}

sees the deficit once; momentum flux depends on speed twice (

ρ u \cdot u

), so

θ

must include one factor of the actual mass-carrying speed

u / U

Why is

θ

"directly related to drag" — what is the one-line reason?

Take a control volume over the plate: the net momentum flux out equals the total drag force (Newton's 2nd law for the fluid). That momentum flux deficit is exactly

ρ U^{2} θ

, so its rate of change downstream,

τ_{w} = ρ U^{2} \frac{d θ}{d x}

, is the wall shear stress — integrate

τ_{w}

over the plate and you get Skin friction drag.

Why is the choice of 99% (rather than 95% or 99.9%) essentially harmless for

δ^{*}

and

θ

Because near

y = δ

the deficit integrands are already almost zero, so shifting the cutoff barely changes the integrals —

δ^{*}

and

θ

are far less sensitive to the convention than

δ

is.

Why does the Blasius solution give

δ \propto x

and not

\propto x

Diffusion of momentum away from the wall (via Viscosity) spreads like

time

; the fluid's residence time near the plate grows as

x / U

, so the diffused thickness grows as

x / U \cdot ν \propto x

Why is a higher Reynolds number associated with a thinner boundary layer at a given

x

δ = 5 x / R e_{x}

: larger

R e_{x}

means inertia dominates viscosity, so momentum diffuses less far from the wall in the time available, keeping the slowed region thin.

Why can

δ^{*}

be interpreted as pushing streamlines outward?

Because the missing mass flow near the wall must go somewhere — the outer inviscid flow is displaced away from the wall by exactly the thickness

δ^{*}

of ideal

U

-flow that would carry that missing mass.

Edge cases

y = 0

(the wall itself), what is the integrand of

δ^{*}

It equals

1 - u / U = 1 - 0 = 1

, its maximum value — the deficit is largest right at the wall where the fluid is fully stuck.

What does the integrand

\frac{u}{U} (1 - u / U)

θ

equal at both

y = 0

and

y = δ

Zero at both ends — at

y = 0

the factor

u / U = 0

, and at

y = δ

the factor

1 - u / U \approx 0

; the momentum deficit peaks somewhere in the middle of the layer.

Why does integrating

δ^{*}

\infty

usually give the same value as to

δ

— and when does it not?

In modelling we assume

u = U

for

y > δ

, making the tail integrand exactly zero; but for the true asymptotic profile

u \to U

only as

y \to \infty

, so a tiny nonzero tail area exists — negligible in practice, but not literally zero unless the flat-tail assumption is imposed.

For an idealised inviscid ("perfect") fluid with no No-slip condition, what are

δ

δ^{*}

, and

θ

All zero — with no viscosity the fluid slips freely,

u = U

everywhere, there is no deficit, and no boundary layer forms at all.

If a plate is infinitely long, does

δ

grow without bound?

In the pure laminar formula it grows as

x

forever, but physically the layer transitions to turbulence past a critical

R e_{x}

, changing the growth law — so the laminar formula stops applying.

What happens to

θ

right at the point of Boundary layer separation?

The wall shear stress drops to zero and the near-wall flow reverses; the simple deficit picture breaks down because

u

goes negative near the wall, so the profile-based integrals no longer describe an attached layer.

For a step profile (

u = 0

for

y < δ

, then

u = U

), what are

δ^{*}

and

θ

δ^{*} = δ

(full deficit up to

δ

, so equality — the degenerate case where

δ^{*} \leq δ

becomes an equals) and

θ = 0

(the integrand

\frac{u}{U} (1 - \frac{u}{U})

vanishes wherever

u

0

U

) — an unphysical limit showing

θ

needs a gradual profile.

Can

δ^{*}

ever be negative?

Only if

u > U

somewhere (fluid faster than free-stream, as in wall jets or strong favourable gradients); for a standard flat-plate boundary layer

u \leq U

, so

δ^{*} \geq 0