Prandtl-Meyer expansion calculator
At Mach 2 in air, with γ of 1.4, the flow has turned about 26.38° from sonic conditions and the Mach angle is 30°. Enter Mach number and heat capacity ratio to get the expansion angle in degrees and radians and the turning still available. Free in your browser.
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Worked example
- Mach
- 2
- Gamma
- 1.4
- Expansion angle deg
- 26.3797608134
- Expansion angle rad
- 0.4604136821
- Mach angle deg
- 30
- Nu max deg
- 130.4540768505
The Prandtl-Meyer expansion function is the isentropic turning angle a calorically perfect gas must take to pass from sonic conditions to a chosen Mach number. Compressible-flow homework, nozzle design, and method-of-characteristics sketches all need that angle in degrees together with the post-expansion Mach number, without rearranging inverse trigonometric identities by hand. This calculator accepts a finite initial Mach number at or above one and a heat capacity ratio γ greater than one, evaluates the textbook closed-form Prandtl-Meyer integral, and returns the expansion angle in degrees and radians, the Mach number, the Mach-wave angle, the vacuum limit ν_max, and the remaining turning still available before vacuum.
What the Prandtl-Meyer expansion angle means and which two numbers you send
The Prandtl-Meyer function ν(M) is the isentropic turning angle, measured from a sonic reference, that a calorically perfect gas must execute to reach Mach number M. In a centered expansion fan a uniform supersonic stream turns around a convex corner through a continuous family of Mach waves: the flow remains isentropic, Mach number rises, static pressure and temperature fall, and the turning between an upstream Mach number M1 and a downstream Mach number M2 is exactly ν(M2) minus ν(M1).
Evaluating ν at a single Mach number therefore answers two standard questions at once: how large is the expansion angle from sonic conditions to that Mach, and what post-expansion Mach belongs to that angle. Search intent clusters around Prandtl-Meyer expansion calculator, Prandtl-Meyer function ν(M), expansion fan angle, supersonic convex corner, Mach angle μ, and heat capacity ratio gamma. People want a numeric angle they can pin against Anderson, John, or NASA tables, a wave-diagram sketch, or a method-of-characteristics notebook — not only a reminder that the inverse-tangent formula exists.
The two inputs are the initial Mach number M, which must be at least one, and the heat capacity ratio γ equals Cp over Cv, which must be strictly greater than one. For diatomic air at ordinary temperatures γ is 1.4; for a monatomic gas it is 5/3.
FAQ
How do I compute the Prandtl-Meyer expansion angle from Mach number?
Send a finite mach at or above 1 and a heat capacity ratio gamma greater than 1 (1.4 for air).
What heat capacity ratio should I use for air?
A monatomic gas uses 5/3. Gamma must be strictly greater than 1 or the identity is undefined.
What is the difference between the expansion angle ν(M) and the Mach angle?
ν(M) is the isentropic turning from sonic conditions to Mach M. The Mach angle μ = arcsin(1/M) is the wave angle of an infinitesimal Mach wave relative to the local flow; both are returned.
How do I find the turning angle between two Mach numbers?
Evaluate the expansion angle at each Mach number and subtract: the turn from the upstream to the downstream Mach equals the downstream angle minus the upstream one, because the flow stays isentropic through the fan.
What does the remaining expansion angle mean?
It is how much more the flow could still turn before reaching the vacuum limit, the largest angle possible for that gas. At Mach 2 with γ of 1.4 about 104° of turning remain.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/prandtl-meyer-expansion \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mach":2,"gamma":1.4}'const res = await fetch("https://api.kit.forhosting.com/phys/prandtl-meyer-expansion", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mach": 2,
"gamma": 1.4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/prandtl-meyer-expansion",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mach": 2,
"gamma": 1.4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/prandtl-meyer-expansion", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mach":2,"gamma":1.4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mach":2,"gamma":1.4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/prandtl-meyer-expansion", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mach": 2,
"gamma": 1.4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.prandtl_meyer_expansion",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Subsonic Mach numbers are rejected as invalid_input because an expansion fan of this type does not exist below Mach one. Successful API calls cost $0.002.
What the Prandtl-Meyer expansion angle means and which two numbers you send
Delivering mach, gamma, expansion_angle_deg, expansion_angle_rad, mach_angle_deg, nu_max_deg, remaining_expansion_deg, formula, and method as JSON makes the result easy to paste into notes, diff in automated tests, or reuse across languages without reimplementing square roots and arctangents. The returned mach is the post-expansion Mach number associated with ν(M) from a sonic start, so the angle and the Mach number travel together in one payload.
Input contract, the closed-form identity, and error behaviour
If gamma is one or less the square-root terms in the identity are undefined or imaginary, so that case is rejected as well. Example: mach equals two and gamma equals 1.4 yields an expansion angle of about 26.38 degrees, a Mach angle of exactly 30 degrees, and a remaining turn of about 104 degrees before vacuum. Values are rounded to twelve decimal places so goldens compare byte-for-byte across JavaScript engines. There is no unit conversion, no real-gas table, and no inversion for Mach given a wall angle; this is the definitional Prandtl-Meyer evaluation from M and γ. POST a JSON body with two required finite numbers: mach, the initial Mach number at or above one, and gamma, the heat capacity ratio strictly greater than one. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, non-numeric text, NaN, and Infinity fail with invalid_input naming the field, and failed validation is not billed. Aliases are accepted: M, m1, mach_number, mach1, and mach_in for mach; g, gam, kappa, heat_capacity_ratio, and ratio_of_specific_heats for gamma. On success the body echoes mach and gamma, then reports expansion_angle_deg and expansion_angle_rad from the textbook identity ν(M) equals the square root of (γ plus one) over (γ minus one), times arctan of the square root of ((γ minus one) over (γ plus one) times (M squared minus one)), minus arctan of the square root of (M squared minus one). Companion fields are mach_angle_deg equal to arcsin of one over M in degrees, nu_max_deg the vacuum turning limit, remaining_expansion_deg equal to ν_max minus ν(M), plus formula and method labels.
Where a Prandtl-Meyer expansion API earns its keep
Classrooms and compressible-flow labs use a Prandtl-Meyer expansion calculator so learners can check ν(M) and the Mach angle for arbitrary supersonic Mach numbers and heat capacity ratios without sign errors in the inverse tangents. Failed validation does not consume balance. Exam-prep generators need a deterministic oracle that marks Mach below one as invalid_input instead of silently returning a NaN that a buggy grader might treat as success. Keep Mach at or above one, keep gamma above one, pin the rounded expansion_angle_deg in tests, and you get reproducible isentropic turning angles for homework, LMS plugins, and edge workers without standing up a full gas-dynamics library of your own.
Integration use cases
Compressible-flow homework and lab checks
Students and graders confirm ν(M) and the Mach angle from a supersonic Mach number and γ, with a clear invalid_input when Mach is below 1.
Nozzle, inlet, and wave-diagram sketches
Design notes and method-of-characteristics worksheets pull expansion_angle_deg, mach, nu_max_deg, and remaining_expansion_deg as stable JSON instead of re-typing inverse tangents.
Deterministic fixtures for gas-dynamics tests
Pipelines that seed unit tests or interactive science demos call a pure Prandtl-Meyer oracle rather than reimplementing ν(M) in every language.
Technical questions
What happens if the Mach number is below 1?
The task fails with invalid_input. The Prandtl-Meyer expansion function is undefined below 1: a subsonic stream cannot form this isentropic centered expansion fan, and the call is not billed.
What does the Prandtl-Meyer expansion calculator cost?
The browser tool is free. API requests use prepaid balance at $0.002 per successful call; invalid input is not billed.