Isentropic Flow Relationships

Fluid Mechanics · FE Reference Handbook section

Fluid Mechanics

16 formulas

10 exam-style examples

~60 min

All Fluid Mechanics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Isentropic Flow Relationships within Fluid Mechanics. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what isentropic flow relationships describes physically and when it applies.
State every one of the 16 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early.

Lecture

Why this section exists. Isentropic Flow Relationships is the part of Fluid Mechanics that lets you connect a pipeline, jet or submerged surface to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as continuity plus energy, with one head-loss or force term. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Crane lowering a steel plate girder onto bridge bearings while ironworkers guide it. — Photo 1. Where this shows up in practice: isentropic flow relationships.
Capstone Studio instructional photograph

Fluid Mechanics — Isentropic Flow Relationships: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a pipeline, jet or submerged surface. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 16 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

T ^ k - 1h	Quantity produced by "T ^ k - 1h" — read its definition and unit from the handbook line directly above the equation.
d 2n	Quantity produced by "d 2n = d t2 n" — read its definition and unit from the handbook line directly above the equation.
P1	Quantity produced by "P1 = T1 1" — read its definition and unit from the handbook line directly above the equation.
T0	Quantity produced by "T0 = T + V2" — read its definition and unit from the handbook line directly above the equation.
h1 + 21	Quantity produced by "h1 + 21 = h2 + 22" — read its definition and unit from the handbook line directly above the equation.
T	Quantity produced by "T = + 2" — read its definition and unit from the handbook line directly above the equation.
P0 c T0 m^k - 1h	Quantity produced by "P0 c T0 m^k - 1h" — read its definition and unit from the handbook line directly above the equation.
P	Quantity produced by "P = T = b1 + k -" — read its definition and unit from the handbook line directly above the equation.
1 : Ma2l^k - 1h	Quantity produced by "1 : Ma2l^k - 1h" — read its definition and unit from the handbook line directly above the equation.
t0 c T0 m^k - 1h	Quantity produced by "t0 c T0 m^k - 1h" — read its definition and unit from the handbook line directly above the equation.
t	Quantity produced by "t = T = b1 + k -" — read its definition and unit from the handbook line directly above the equation.
A	Quantity produced by "A = 1 SS 2 WW" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

In an ideal gas for an isentropic process, the following relationships exist between static properties at any two points in the flow.
k k
P2 t
The stagnation temperature, T0, at a point in the flow is related to the static temperature as follows:
2 : cp
Energy relation between two points:
V2 V2
Vennard, J.K., Elementary Fluid Mechanics, 6th ed., J.K. Vennard, 1954.
The relationship between the static and stagnation properties (T0, P0, and ρ0) at any point in the flow can be expressed as a
function of the Mach number as follows:
1 k - 1 : Ma2
k k
1 1
Compressible flows are often accelerated or decelerated through a nozzle or diffuser. For subsonic flows, the velocity decreases
as the flow cross-sectional area increases and vice versa. For supersonic flows, the velocity increases as the flow cross-sectional
area increases and decreases as the flow cross-sectional area decreases. The point at which the Mach number is sonic is called
the throat and its area is represented by the variable, A*. The following area ratio holds for any Mach number.
A* Ma SS 1_ + i WW
S 2 k 1 W
T X
where
A ≡ area [length2]

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships

Air (k = 1.4) with stagnation conditions T₀ = 406 K and p₀ = 572 kPa flows isentropically at M = 1.8. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 406 K, p₀ = 572 kPa
M = 1.8
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (1.8)^{2} = 1.6480$
Static temperature
$T = 406/1.6480 = 246.4 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 572/ (1.6480)^{3.5} = 99.6 k P a$
Density ratio
$ρ_{0} / ρ = (1.6480)^{2.5} = 3.487$

Answer: T = 246.4 K, p = 99.6 kPa, ρ₀/ρ = 3.487

Why the other options are there

p = 3,287 kPa (ratio inverted)
T = 669.1 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 2

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (2)

Air (k = 1.4) with stagnation conditions T₀ = 551 K and p₀ = 346 kPa flows isentropically at M = 1.7. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 551 K, p₀ = 346 kPa
M = 1.7
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (1.7)^{2} = 1.5780$
Static temperature
$T = 551/1.5780 = 349.2 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 346/ (1.5780)^{3.5} = 70.1 k P a$
Density ratio
$ρ_{0} / ρ = (1.5780)^{2.5} = 3.128$

Answer: T = 349.2 K, p = 70.1 kPa, ρ₀/ρ = 3.128

Why the other options are there

p = 1,708 kPa (ratio inverted)
T = 869.5 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 3

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (3)

Air (k = 1.4) with stagnation conditions T₀ = 579 K and p₀ = 368 kPa flows isentropically at M = 2.3. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 579 K, p₀ = 368 kPa
M = 2.3
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (2.3)^{2} = 2.0580$
Static temperature
$T = 579/2.0580 = 281.3 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 368/ (2.0580)^{3.5} = 29.4 k P a$
Density ratio
$ρ_{0} / ρ = (2.0580)^{2.5} = 6.076$

Answer: T = 281.3 K, p = 29.4 kPa, ρ₀/ρ = 6.076

Why the other options are there

p = 4,602 kPa (ratio inverted)
T = 1,192 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 4

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (4)

Air (k = 1.4) with stagnation conditions T₀ = 403 K and p₀ = 522 kPa flows isentropically at M = 1.3. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 403 K, p₀ = 522 kPa
M = 1.3
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (1.3)^{2} = 1.3380$
Static temperature
$T = 403/1.3380 = 301.2 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 522/ (1.3380)^{3.5} = 188.4 k P a$
Density ratio
$ρ_{0} / ρ = (1.3380)^{2.5} = 2.071$

Answer: T = 301.2 K, p = 188.4 kPa, ρ₀/ρ = 2.071

Why the other options are there

p = 1,446 kPa (ratio inverted)
T = 539.2 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 5

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (5)

Air (k = 1.4) with stagnation conditions T₀ = 595 K and p₀ = 396 kPa flows isentropically at M = 1.8. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 595 K, p₀ = 396 kPa
M = 1.8
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (1.8)^{2} = 1.6480$
Static temperature
$T = 595/1.6480 = 361.0 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 396/ (1.6480)^{3.5} = 68.9 k P a$
Density ratio
$ρ_{0} / ρ = (1.6480)^{2.5} = 3.487$

Answer: T = 361.0 K, p = 68.9 kPa, ρ₀/ρ = 3.487

Why the other options are there

p = 2,275 kPa (ratio inverted)
T = 980.6 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 6

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (6)

Air (k = 1.4) with stagnation conditions T₀ = 336 K and p₀ = 437 kPa flows isentropically at M = 0.9. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 336 K, p₀ = 437 kPa
M = 0.9
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (0.9)^{2} = 1.1620$
Static temperature
$T = 336/1.1620 = 289.2 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 437/ (1.1620)^{3.5} = 258.4 k P a$
Density ratio
$ρ_{0} / ρ = (1.1620)^{2.5} = 1.456$

Answer: T = 289.2 K, p = 258.4 kPa, ρ₀/ρ = 1.456

Why the other options are there

p = 739.1 kPa (ratio inverted)
T = 390.4 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 7

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (7)

Air (k = 1.4) with stagnation conditions T₀ = 400 K and p₀ = 408 kPa flows isentropically at M = 1.6. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 400 K, p₀ = 408 kPa
M = 1.6
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (1.6)^{2} = 1.5120$
Static temperature
$T = 400/1.5120 = 264.6 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 408/ (1.5120)^{3.5} = 96.0 k P a$
Density ratio
$ρ_{0} / ρ = (1.5120)^{2.5} = 2.811$

Answer: T = 264.6 K, p = 96.0 kPa, ρ₀/ρ = 2.811

Why the other options are there

p = 1,734 kPa (ratio inverted)
T = 604.8 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 8

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (8)

Air (k = 1.4) with stagnation conditions T₀ = 409 K and p₀ = 565 kPa flows isentropically at M = 2.0. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 409 K, p₀ = 565 kPa
M = 2.0
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (2.0)^{2} = 1.8000$
Static temperature
$T = 409/1.8000 = 227.2 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 565/ (1.8000)^{3.5} = 72.2 k P a$
Density ratio
$ρ_{0} / ρ = (1.8000)^{2.5} = 4.347$

Answer: T = 227.2 K, p = 72.2 kPa, ρ₀/ρ = 4.347

Why the other options are there

p = 4,421 kPa (ratio inverted)
T = 736.2 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 9

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (9)

Air (k = 1.4) with stagnation conditions T₀ = 321 K and p₀ = 852 kPa flows isentropically at M = 0.5. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 321 K, p₀ = 852 kPa
M = 0.5
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (0.5)^{2} = 1.0500$
Static temperature
$T = 321/1.0500 = 305.7 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 852/ (1.0500)^{3.5} = 718.3 k P a$
Density ratio
$ρ_{0} / ρ = (1.0500)^{2.5} = 1.130$

Answer: T = 305.7 K, p = 718.3 kPa, ρ₀/ρ = 1.130

Why the other options are there

p = 1,011 kPa (ratio inverted)
T = 337.1 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Example 10

Isentropic stagnation-to-static property ratios — Isentropic Flow Relationships (10)

Air (k = 1.4) with stagnation conditions T₀ = 565 K and p₀ = 614 kPa flows isentropically at M = 1.3. Compute the static temperature, static pressure and the density ratio ρ₀/ρ.

Given

T₀ = 565 K, p₀ = 614 kPa
M = 1.3
k = 1.4

Find

T, p and ρ₀/ρ

Start with the thinking

Every isentropic ratio is a power of the same bracket 1 + (k−1)M²/2.
Static values are always below stagnation values in a flowing gas.

Step-by-step solution

Formula
$\frac{T _{0}}{T} = 1 + \frac{k - 1}{2} M^{2}$
Bracket
$1 + 0.2 (1.3)^{2} = 1.3380$
Static temperature
$T = 565/1.3380 = 422.3 K$
Formula
$\frac{p _{0}}{p} = (1 + \frac{k - 1}{2} M^{2})^{k / (k - 1)}$
Substituting
$p = 614/ (1.3380)^{3.5} = 221.6 k P a$
Density ratio
$ρ_{0} / ρ = (1.3380)^{2.5} = 2.071$

Answer: T = 422.3 K, p = 221.6 kPa, ρ₀/ρ = 2.071

Why the other options are there

p = 1,701 kPa (ratio inverted)
T = 756.0 K (multiplied instead of divided)

Reference: FE Reference Handbook — Fluid Mechanics → Isentropic Flow Relationships

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a pipeline, jet or submerged surface, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Isentropic Flow Relationships contains 16 relations; you must be able to find this page in under 15 seconds.
Exam style: continuity plus energy, with one head-loss or force term.
Unit rule: γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

γ = 62.4 lb/ft³ or 9.81 kN/m³; convert psi to feet of head early
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.