Similarity — geometric, kinematic, dynamic; Reynolds similarity: Foundations

The parent note throws around $L$ , $V$ , $ρ$ , $μ$ , $ν$ , "shear stress", "dimensionless", "force ratio", and $Re$ as if you already own them. Here we earn each one, in an order where every new thing leans only on the ones before it.

1. Length $L$ — the "how big" of the flow

The picture: the figure below shows a full-size car and a half-size model of it. A red double-arrow marks $L_{p}$ (the prototype's length) and a violet double-arrow marks $L_{m}$ (the model's length) — whichever length you pick as your "ruler", every other length is compared to it.

Figure — Similarity — geometric, kinematic, dynamic; Reynolds similarity

Figure s01 — Alt-text: a large magenta car labelled $L_{p}$ beside a violet half-size copy labelled $L_{m}$ ; both lengths carry double-headed arrows, and an orange label states $L_{r} = L_{m} / L_{p} = 1/2$ .

Why the topic needs it. Similarity is about scaling — shrinking a real object to a model. You cannot say "1/5th the size" unless you have picked one length to measure "size" by. That chosen length is $L$ , and the ratio of the two versions is the scale factor.

2. Velocity $V$ — the "how fast"

The picture: an arrow drawn on the moving fluid. Its length = speed, its direction = which way the fluid goes. This arrow is what "velocity" looks like.

Why the topic needs it. Forces in a flow depend on speed (fast air pushes harder than slow air). To scale a flow, we must track its speed — so $V$ enters every formula. The scale of velocities is $V_{r} = V_{m} / V_{p}$ .

3. Force and its unit, the Newton — before we weigh anything

Why the topic needs it. Everything ahead — inertia force, viscous force, gravity force, shear stress $τ$ , drag $F_{D}$ — is measured in newtons or pascals. Defining them from base units now means no symbol ( $N$ , $Pa$ ) ever appears unexplained later.

4. Density $ρ$ — the "how heavy per box"

The picture: a $1 m \times 1 m \times 1 m$ box full of the fluid, sitting on a scale. The number the scale reads is $ρ$ .

Why the topic needs it. Newton's law is force = mass × acceleration. To get the mass of a chunk of fluid we need to know how heavy the fluid is per unit volume — that's $ρ$ . Heavier fluid ⇒ more inertia ⇒ harder to push around.

5. Gravitational acceleration $g$ — the "how hard things fall"

The picture: an apple let go, its downward-speed arrow growing longer each second by the same amount $g$ .

Why the topic needs it. Fluids have weight, and weight is mass × $g$ . Whenever a flow has a free surface (waves on water, a ship's wake, a spillway), gravity pulls that surface back down and shapes the flow. To measure the gravity force we must know $g$ . It appears in the gravity-force estimate below and in the Froude number.

6. Viscosity $μ$ — the "how sticky/gooey"

To make this precise we first need one more picture: shear.

Figure s02 — Alt-text: stacked horizontal layers of fluid above a navy wall; magenta velocity arrows grow longer with height $y$ (zero at the wall, largest at the top), and an orange $y$ -axis and a caption mark the velocity gradient $d V / d y$ as how fast the arrows lengthen.

Why the topic needs it. The whole point of Reynolds similarity is a contest between inertia (fluid's push from its momentum) and viscosity (fluid's internal friction slowing it). Without $μ$ and $τ$ there is no "friction" side of the contest to measure.

7. "Dimensionless" — why some numbers have no units

The picture: think of a ratio like "twice as tall." That "2" is the same whether you measured in feet or metres — the units cancelled in the division. Dimensionless numbers are exactly those cancel-everything ratios.

Why the topic needs it. Similarity's magic sentence is "match the dimensionless number and the two flows are the same." That only works because a dimensionless number is unit-free — so a small model and a giant prototype, measured on totally different scales, can honestly share the same value.

8. Force ratios — the heart of dynamic similarity

Formula Building the inertia force

$F_{i} \sim ρ V^{2} L^{2}$ step by step What we do: inertia force = mass × acceleration (Newton's law), sized up by dimensions. ( $\sim$ means "scales like" — same size, ignoring exact factors like $\frac{1}{2}$ .)

Mass: a fluid chunk of size $L$ has volume $\sim L^{3}$ , so mass $\sim ρ L^{3}$ .
Acceleration — why $V^{2} / L$ ? Acceleration is change in speed ÷ time taken. As the fluid sweeps past the body it changes its speed by an amount of order $V$ over the distance $L$ it takes to cross the body. The time to cross that distance at speed $V$ is $t \sim L / V$ . So $acceleration \sim \frac{change in speed}{time} \sim \frac{V}{L / V} = \frac{V ^{2}}{L} .$ That is the only timescale available — the crossing time $L / V$ — which is why the characteristic acceleration comes out as $V^{2} / L$ (think: a particle decelerating over a distance $L$ ).
Multiply: $F_{i} \sim ρ L^{3} \cdot \frac{V ^{2}}{L} = ρ V^{2} L^{2}$ .

The figure below draws all this as a literal tug-of-war: inertia pulling one way, viscosity (and, separately, gravity) pulling back. The ratio of the two pulls is the dimensionless number we care about.

Figure s03 — Alt-text: a rope tug-of-war; a magenta block labelled inertia $F_{i} \sim ρ V^{2} L^{2}$ pulls left, a violet block labelled viscous $F_{v} \sim μ V L$ pulls right; orange captions give $Re = F_{i} / F_{v} = ρ V L / μ$ and $Fr = F_{i} / F_{g} = V^{2} / (g L)$ , with notes that big Re means inertia wins and small Re means viscosity wins.

Why the topic needs both. Different flows are ruled by different contests. Submerged bodies and pipes → inertia vs viscosity → match $Re$ . Free-surface flows → inertia vs gravity → match $Fr$ . Each ratio is dimensionless, so it can be matched between model and prototype.

9. How it all feeds the topic

Once these foundations are in place, head to the parent: Similarity — geometric, kinematic, dynamic; Reynolds similarity (index 2.2.27). Related deeper stops: Reynolds Number, Drag Coefficient, Froude Number, Buckingham Pi Theorem, Navier-Stokes Equations, Boundary Layer.

Equipment checklist

Cover the right side and test yourself.

What does the subscript

m

p

mean?

m

= model (small test piece),

p

= prototype (real full-size thing).

What is the scale factor

L_{r}

The ratio of model length to prototype length,

L_{r} = L_{m} / L_{p}

What is one newton in base SI units?

1 N = 1 kg \cdot m / s^{2}

— the push that accelerates

1

kg by

1

m/s each second.

What is one pascal?

1 Pa = 1 N / m^{2} = 1 kg / (m \cdot s^{2})

— force spread over area.

What are the units of density

ρ

Kilograms per cubic metre,

kg / m^{3}

What is

g

and its value/units?

Gravitational acceleration,

\approx 9.81 m / s^{2}

— how fast falling speed grows each second.

In plain words, what is viscosity

μ

The internal "grip" between fluid layers — how strongly one layer drags the next.

What does the velocity gradient

d V / d y

measure?

How fast the flow speed changes as you move crosswise, away from the wall.

Write Newton's law of viscosity.

τ = μ \frac{d V}{d y}

(shear stress = viscosity × velocity gradient).

Why does the characteristic acceleration scale as

V^{2} / L

Speed changes by

\sim V

over the crossing time

t \sim L / V

, so

a \sim V / (L / V) = V^{2} / L

Derive why the viscous force scales as

μ V L

Force = stress × area

\sim (μ V / L) \times L^{2} = μ V L

How does kinematic viscosity

ν

relate to

μ

ν = μ / ρ

, units

m^{2} / s

What does "dimensionless" mean?

A pure number with all units cancelled, the same in any measuring system.

Why do we compare force RATIOS, not raw forces?

Raw forces differ hugely between model and prototype, but their ratios can be made equal.

Write

Re

four equivalent ways.

Re = \frac{F _{i}}{F _{v}} = \frac{ρ V ^{2} L ^{2}}{μ V L} = \frac{ρ V L}{μ} = \frac{V L}{ν}

Write the Froude number and when to use it.

Fr = \frac{F _{i}}{F _{g}} = \frac{V ^{2}}{g L}

; use it for free-surface flows where gravity competes with inertia.

What is the drag force

F_{D}

The along-flow push the fluid exerts on the body (pressure + shear together), in newtons.

Where does the

\frac{1}{2}

C_{D}

come from?

From the dynamic pressure

\frac{1}{2} ρ V^{2}

, i.e. kinetic energy per unit volume (

\frac{1}{2} m V^{2}

per volume).

What does a large

Re

tell you physically?

Inertia dominates over viscosity — flow tends toward turbulence.

Define the drag coefficient

C_{D}

C_{D} = \frac{F _{D}}{\frac{1}{2} ρ V ^{2} A}

, a dimensionless measure of drag.