Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M: Foundations

This page assumes you have seen none of the notation in the parent note. We build each symbol from a picture before it is ever allowed to appear in a formula.

1. Pressure $P$ — the push of the gas

The picture. Imagine countless tiny gas molecules bouncing off a wall. Each bounce is a tiny shove. Add up all the shoves on one square metre of wall and divide by that area — that number is the pressure.

Figure — Isentropic flow tables — P - P₀, T - T₀, ρ - ρ₀ as functions of M

Why the topic needs it. The isentropic tables are ratios like $P / P_{0}$ . Before we compare "flowing pressure" to "stopped pressure," we must know what pressure is: a push made by molecular motion.

2. Temperature $T$ — the jiggle energy

The picture. In the figure above, the speed of the random bouncing arrows is temperature. Pressure counts the wall-hits; temperature measures how energetic each molecule's dance is.

3. Density $ρ$ and specific volume $v$ — how crowded the gas is

The picture. Same box, more molecules crammed in ⇒ higher density $ρ$ ⇒ smaller room per kilogram $v$ . Squeeze air and $ρ$ rises while $v$ shrinks.

Why the topic needs both. The third table is $ρ / ρ_{0}$ — that "how much did it pile up" measure. But the derivations (enthalpy, isentropic law) are cleaner written per kilogram, so they use $v$ . Knowing $v = 1/ ρ$ lets you switch between the two freely.

4. The gas constants $R$ , $c_{v}$ , $c_{p}$ , and $γ$

We need these four gas quantities before we can even write the speed of sound, so we build them here first.

The picture. Think of $γ$ as the stiffness of the gas-spring. A stiffer spring ( $γ$ larger) heats up more when you squeeze it. Every exponent in the tables is built from $γ$ .

Recall Where do the exponents

$1, \frac{1}{γ - 1}, \frac{γ}{γ - 1}$ come from? All three are just $γ$ arranged differently. For air: $1$ (temperature), $\frac{1}{0.4} = 2.5$ (density), $\frac{1.4}{0.4} = 3.5$ (pressure).

5. Velocity $V$ and the speed of sound $a$

The picture. Two different motions live in the same gas:

$V$ = the whole crowd marching forward together.
$a$ = a rumour (pressure ripple) racing through the crowd.

Why this tool and not another? We need one fixed "yardstick speed" that the gas itself provides, so we can say whether the flow is "fast" or "slow" in an absolute sense. Sound speed is that natural yardstick — it is set by the gas's own temperature. See Speed of Sound and Mach Number.

6. Mach number $M$ — the ONE variable

The picture. $M = 0.5$ : the crowd marches at half the rumour-speed. $M = 1$ : crowd keeps pace with its own rumour (sonic). $M = 2$ : crowd outruns any rumour (supersonic) — disturbances can't warn the gas ahead.

7. Internal energy $u$ , enthalpy $h$ , and the " $+ \frac{1}{2} V^{2}$ " idea

The picture. A see-saw. Speed the gas up ( $V$ big) and thermal energy $h$ (hence $T$ ) must drop. Stop it ( $V = 0$ ) and all the motion energy pours back into thermal energy — that is why stopped gas is hotter.

8. "Stopping the flow gently" — isentropic & adiabatic

The picture. Slowing gas down two ways:

Gently (isentropic): a smooth ramp — energy is stored cleanly, nothing wasted.
Violently (a shock): a wall — some energy is scrambled into extra disorder (entropy) and lost.

9. Stagnation (total) properties $P_{0}$ , $T_{0}$ , $ρ_{0}$

The picture. Point your palm into a fast stream of air. Right at your palm's centre the air can't move — it stagnates. There it is hotter, denser, and pushes harder than the free stream. Those are $T_{0}, ρ_{0}, P_{0}$ .

The star superscript $*$ (as in $P^{*}$ , $T^{*}$ ) is just the stagnation-ratio evaluated at $M = 1$ — the "throat"/critical values used in Converging-Diverging Nozzle & Choking and Area-Mach Relation A/A*.

10. Reading the notation on the tables

Symbol	Say it as	Means
$P / P_{0}$	"P over P-naught"	local pressure ÷ stagnation pressure (a fraction $\leq 1$ )
$v$	"specific volume"	volume of one kilogram, $v = 1/ ρ$ ( $m^{3} / kg$ )
$M^{2}$	"M squared"	Mach number times itself
$()^{γ / (γ - 1)}$	"to the power gamma over gamma-minus-one"	raise the bracket to that exponent
$x^{- n}$	"x to the minus n"	$1/ x^{n}$ , a reciprocal
$P^{}, T^{}$	"P-star, T-star"	the critical value at $M = 1$ (nozzle throat / choking conditions)

Prerequisite map

The diagram below reads top-to-bottom: the three raw properties ( $P, T, ρ$ ) feed the ideal-gas law; temperature and $γ$ build the speed of sound; sound plus flow velocity build the Mach number; temperature builds internal energy, which with the gas law and $γ$ builds enthalpy; enthalpy plus velocity give the energy see-saw, which with isentropic stopping defines the stagnation state; finally Mach number, $γ$ , the gas law and the stagnation state together produce the isentropic flow tables. In words: raw properties → gas relations → Mach number & energy → stagnation → tables.

"Ideal gas law" is the rule $P = ρR T$ from Section 4 that ties pressure, density and temperature together.

Equipment checklist

I can state, in plain words, what pressure is in molecular terms

Total wall-shove from bouncing molecules, per unit area (Pa = N/m²).

I know why temperature must be in Kelvin here

Ratios of temperature only make sense on an absolute scale starting at zero jiggle.

I can define density and specific volume and relate them

ρ

= mass/volume (kg/m³);

v

= volume/kg (m³/kg);

v = 1/ ρ

I can state the ideal-gas law in both forms

P = ρR T

and equivalently

P v = R T

;

R

ties pressure, density and temperature together (air: 287 J/kg·K).

I can explain why

u = c_{v} T

At constant volume no work is done, so all heat raises internal energy:

d u = c_{v} d T

; integrating gives

u = c_{v} T

I can explain why

c_{p} - c_{v} = R

Constant-pressure heating must also do expansion work; that extra work per kelvin equals

R

I can derive

c_{p} = γ R / (γ - 1)

from

γ = c_{p} / c_{v}

and

c_{p} - c_{v} = R

Sub

c_{v} = c_{p} / γ

into

c_{p} - c_{v} = R

, factor

c_{p} (γ - 1) / γ = R

, solve.

I know the constant-specific-heat assumption and when it fails

Calorically perfect gas:

c_{p}, c_{v}, γ

fixed; fails at high

T

(vibration) where

γ

drops.

I can sketch where

a = γ R T

comes from

a^{2} = d P / d ρ

for an isentropic ripple; with

P \propto ρ^{γ}

this gives

γ P / ρ = γ R T

I can tell

V

(bulk flow) apart from

a

(speed of a pressure ripple)

V

= whole crowd marching;

a

= rumour racing through it.

I can write the Mach number and state its sign

M = V / a \geq 0

(

V

is a speed magnitude; only

M^{2}

appears, so direction is irrelevant).

I can state the limiting behaviour of the ratios

M \to 0

: all ratios

\to 1

(nothing spent);

M \to \infty

: all ratios

\to 0

(everything dumps into heat).

I can build

h

from

u

and show

h = c_{p} T

h = u + P v = c_{v} T + R T = c_{p} T

for a perfect gas.

I can explain the energy see-saw and its origin

First law for adiabatic, no-shaft-work flow gives

h + \frac{1}{2} V^{2} =

constant; thermal and motion energy trade.

I know what "isentropic" means and its defining relation

Adiabatic + reversible;

P v^{γ} =

const; the tables' exact link fails if energy is scrambled.

I can define a stagnation property and read the

*

symbol

Value reached by stopping flow to

V = 0

isentropically (subscript 0);

*

marks the critical value at

M = 1