Knowledge card
Free-stream Temperature Condition
Mathematical Modelling — Fluid Dynamics & Heat TransferContinuum mechanics; boundary-layer similarity theory
Symbol
The far-field condition driving the dimensionless temperature to zero.
In words:equalszero.
Open full derivation chain: 2 stepsDerivation
- 1
Open Step 1: Given / definition · The far field is at the ambient temperature: the thermal boundary layer decays to T∞ — the dime…
The far field is at the ambient temperature: the thermal boundary layer decays to T∞ — the dimensional far-field condition to reduce.Given / definition Defining ruleThe starting relation of a derivation — the definition or established equation every later step transforms; it justifies no change itself.Level 0 · more availableBrief — named quantitiesAbout levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
- ★
Open Step 2: Similarity transformation · Insert the θ-map: with T=T∞+(T_r−T∞)θ and nonzero prescribed excess T_r−T∞, the dimensionless t…
Insert the θ-map: with T=T∞+(T_r−T∞)θ and nonzero prescribed excess T_r−T∞, the dimensionless temperature must vanish.Similarity transformation Defining ruleu(x, y, t) = U(x, t)·F(η), η = y/δ(x, t)A change of variables that collapses several independent variables into one similarity coordinate, reducing a PDE to an ODE; each substituted term splits into a dimensional scale times a dimensionless group.Substitution · θ Operand nodeDimensionless Temperature θ(η)The scaled temperature (one at the wall scale, zero in the free stream) — the unknown of the energy ODE; its definition is composed on demand.Substitution: The operand (a map or model) replaces the source directly.Open term-change ledger: 1 records
Open record 1: T(∞) → θ(∞)
Consumes:L0Produces:L0The θ-map T=T∞+(T_r−T∞)θ turns T(∞)=T∞ into (T_r−T∞)θ(∞)=0; because the prescribed reference excess is nonzero, the condition passes to θ(∞).θ(∞)Resulting expressionFree-stream TemperatureThe dimensionless temperature evaluated in the free stream.Level 0 · more availableBrief — named quantitiesAbout levels
Why the levels load one at a time
Only level 0 travels with this article. The complete expansion is several megabytes, so deeper levels are fetched one at a time when you ask for them — and how deep it goes is not known until you get there.
To go deeper still, open any quantity in the reading as its own concept card — it carries its own derivation and its own depth ladder.
How many levels remain is not yet known — the depth is discovered one level at a time.
Equation Workspace
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Calculator
Equation Workspace
Set known values, calculate unknowns, inspect dependencies, and run numerical solves.
Equation workspace
Set the values you know, then calculate the remaining quantities in Free-stream Temperature Condition.
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Domain Analysis
Unit status, dimensional trails, per-step checks, and custom unit expressions.
Step-by-step unit check (2 steps)
Step 1 given
Expand checked this step.Step 2 similarity_transform
Expand checked this step.
Try your own — the unit calculator
Ever wondered whether a formula “adds up”? This tool answers one plain question: do the units match? Pick a couple of quantities (distance, time, a speed…), join them with ×, ÷, +, or =, and press Run. It works out the resulting unit — for example distance ÷ time gives a speed, LT⁻¹ (metres per second) — and flags anything that can’t be right, like adding a length to a time.
Reads as: θ(∞)=0
“θ(∞)=0” is analysed by expanding its full defining equation (its children and their relations), not as a bare symbol. Use Check this concept’s units above for the same result.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Mathematical Analysis
Object type, lawful operations, conditions, comparisons, and derivation-step checks.
Check one step at a time (2 steps)
Step 1 — given
Step 2 — similarity transform
Dimension I
Definition
What it is — and what it is not
- What it is
- S1The dimensional far-field condition: T → T∞ as y → ∞.
- S1.1essenceT(∞) set equal to T∞.
- S1The dimensional far-field condition: T → T∞ as y → ∞.
- Wisdoms
- S2The far-field condition in similarity variables: θ(∞) = 0.
- S2.1essenceθ(∞) set equal to 0.
- S2The far-field condition in similarity variables: θ(∞) = 0.
Dimension II
In practice
How to deal with it
No practical guidance recorded yet.