Foundations — Boundary layer thickness, displacement thickness, momentum thickness

This page assumes you know nothing. Before we can read a single formula from the parent note the parent topic, we must earn every symbol it uses. Read top to bottom — each block leans on the one above it.

1. The wall, the fluid, and a coordinate to describe them

Picture a flat plate (a wall) lying flat, with fluid — air or water — flowing over it from left to right. We need two directions to talk about anything:

• $x$ = distance along the plate, measured from the leading edge (the front tip). "How far downstream are we?"
• $y$ = distance straight up from the wall, perpendicular to it. "How high above the surface are we?"

Why the topic needs it: every quantity — velocity, thickness — is measured at a height $y$ and at a downstream position $x$. Without these two rulers we cannot say where anything is.

2. Speed: $u$, the free-stream $U$, and the picture of a velocity profile

Two speeds appear constantly. Keep them straight — one is a capital $U$, one is lowercase $u$.

Because $u$ depends on height, we write it as $u(y)$ — read "$u$ of $y$", meaning "the speed you find at height $y$". If you plot $u$ horizontally against $y$ vertically, you get a curve called the velocity profile.

Why the topic needs it: all three thicknesses are computed from the shape of this $u(y)$ curve. The gap between $u$ and $U$ at each height is the "deficit" the wall has caused.

3. The no-slip condition — why the bottom arrow is zero

This is not obvious — you might expect the fluid to slide freely. But molecules right at the surface cling to it. See No-slip condition for the full story.

Why the topic needs it: no-slip is the reason a boundary layer exists at all. Without it, $u$ would equal $U$ everywhere and there would be nothing to measure.

4. Viscosity $\nu$ — the "stickiness" that spreads the slowing

The wall slows the fluid touching it. But why does the slowing spread upward to layers that aren't touching the wall? Because fluid layers drag on each other — this internal friction is viscosity.

See Viscosity for where $\nu$ comes from. Also note $\rho$ (Greek "rho") = density, the mass packed into each cubic metre of fluid (units $\text{kg}/{\text{m}}^3$). We need $\rho$ because the parent note tracks mass and momentum, and both scale with how much stuff is there.

Why the topic needs it: viscosity is the messenger that carries the wall's "slow down!" order outward, layer by layer, building the profile in §2.

5. Reynolds number $R{e}_x$ — is the flow smooth or messy?

The wall's slowing (viscosity) fights against the fluid's tendency to keep charging ahead (inertia). The ratio of these two decides how the boundary layer behaves.

The subscript $x$ reminds us it grows as you march downstream (bigger $x$). See Reynolds number for the full meaning.

Why the topic needs it: the growth law $\delta \approx 5x/\sqrt{R{e}_x}$ (from Blasius solution, where $\delta$ is defined in §6) is written entirely in terms of $R{e}_x$. You cannot read that formula without knowing what $R{e}_x$ is.

6. Boundary layer thickness $\delta$ — where does the layer "end"?

We keep mentioning $\delta$. Time to earn it. The trouble: $u$ climbs toward $U$ but never quite reaches it — it approaches $U$ only as $y\to \infty$ (asymptotically). So "where does the boundary layer stop?" has no sharp answer. Engineers fix this by picking a convention.

Why the topic needs it: $\delta$ is the ruler for the layer's height. It sets the upper limit of the integrals (§8) — beyond $\delta$ the deficit is essentially zero — and it is what the Blasius growth law $\delta \approx 5x/\sqrt{R{e}_x}$ predicts.

7. The integral sign $\int$ — adding up infinitely many thin layers

The parent note computes thicknesses with expressions like ${\int }_0^{\infty }(\cdots )dy$. This scares beginners, but the idea is simple.

• ${\int }_0^{\infty }$ is the same idea with $b$ replaced by $\infty$ — "keep summing slices all the way up, to infinity."
• But beyond the boundary layer edge $\delta$ (§6), $u\approx U$, so a deficit like $(1-u/U)$ becomes $(1-1)=0$. Adding zeros changes nothing. That is why the parent note says you may stop the sum at $\delta$ instead of $\infty$ — the tail contributes nothing.

Why the topic needs it: ${\delta }^{\ast }$ and $\theta$ are defined as integrals — they are totals of a per-slice deficit across the whole layer.

8. Mass flux and momentum flux — what we are actually summing

The two deficits (mass and momentum) sound abstract. Here is the physical meaning of one thin ribbon of height $dy$ and width $1$ (per unit width):

Why the topic needs it: ${\delta }^{\ast }$ measures a mass-flux deficit (one $u$), $\theta$ measures a momentum-flux deficit (two $u$'s). Getting the number of $u$'s right is the single most common source of errors.

9. The three thicknesses, in one glance

Now every symbol in the parent's formulas is earned — all three thicknesses side by side:

$\delta :u(y=\delta )=0.99U\text{(geometric edge, §6)}$

${\delta }^{\ast }={\int }_0^{\infty }(1-\frac{u}{U})dy,\theta ={\int }_0^{\infty }\frac{u}{U}(1-\frac{u}{U})dy$

• $\delta$ is a height read off the profile (a 99% cutoff), not an integral — that is why it has no $\int$.
• $1-u/U$ = the fractional velocity gap at height $y$ (how far short of full speed the fluid runs there). It is $1$ at the wall, $0$ at the top.
• $u/U$ = the fraction of full speed present — the weight that counts "how much mass is actually moving here."

Downstream, the wall's drag builds up and can eventually stall the flow entirely — that is Boundary layer separation, and the total wall friction it feeds is Skin friction drag. Both are read off from $\theta$. (The parent note also combines ${\delta }^{\ast }$ and $\theta$ into a single ratio called the shape factor — you will meet it there; you do not need it yet.)

Prerequisite map

Read it bottom-up: coordinates and speeds build the profile; no-slip (from viscosity) forces its shape; the 99% cutoff gives $\delta$; ribbons + flux let us sum deficits into ${\delta }^{\ast }$ and $\theta$; the Reynolds number $R{e}_x$ sizes $\delta$.

Equipment checklist

Cover the right side and test yourself — you are ready for the parent note only if every line comes instantly.