Dimensionless groups reference
A durable lookup for the scaling groups that appear across fluid mechanics, heat transfer, and aerothermal experiments, with definitions, typical use, and pointers to the full derivations in the fluid mechanics curriculum.
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Fluid mechanics
Heat transfer
Not sure where to start? 4 places to go
Start with your question
Why visitors arrive: You need a quick lookup for Reynolds, Mach, Nusselt, or similarity checklists.
Your question: Which dimensionless groups apply to my flow or heat-transfer problem?
You may also be asking
- Where is the full dimensional analysis lesson?
- How do I get fluid properties?
- How are groups used in experiments?
Where to go next
- Dimensional analysis module: Derivation of Pi groups
- Air properties calculator: Inputs for Re, Pr, Ma
- Experimental measurement article: Similarity in campaigns
- Curriculum index: Broader learning path
When to reach for a Pi group
Dimensionless numbers compress competing physical effects into ratios. They are how wind-tunnel tests claim relevance to engine conditions, how probe calibrations are transferred between facilities, and how low-order heat-exchanger models stay traceable.
- Derivations: Dimensional analysis module and practice problems
- Measurement context: Experimental aerothermal measurement article · Uncertainty fundamentals
- Properties: Air properties calculator for Re, Pr, and Ma inputs
Fluid mechanics
| Group | Definition | Physical meaning | Typical use |
|---|---|---|---|
| Reynolds Re | \(\mathrm{Re} = \dfrac{\rho U L}{\mu}\) | Inertia ÷ viscous forces | Laminar/turbulent transition; dynamic similarity in pipe and external flows; probe calibration transfer |
| Mach Ma | \(\mathrm{Ma} = \dfrac{U}{a}\) | Flow speed ÷ sound speed | Compressibility; choking; high-speed tunnel matching |
| Froude Fr | \(\mathrm{Fr} = \dfrac{U}{\sqrt{g L}}\) | Inertia ÷ gravity | Free-surface flows; wave phenomena |
| Weber We | \(\mathrm{We} = \dfrac{\rho U^2 L}{\sigma}\) | Inertia ÷ surface tension | Jet breakup; capillary-dominated flows |
| Stokes Stk | \(\mathrm{Stk} = \dfrac{\mu U}{\rho g D^2} = \dfrac{\mathrm{Fr}^2}{\mathrm{Re}}\) (form varies) | Viscous ÷ gravitational effects on particles | Particle settling; related to Fr/Re ratios, see curriculum problem set |
| Euler Eu | \(\mathrm{Eu} = \dfrac{\Delta p}{\rho U^2}\) | Pressure forces ÷ inertia | Cavitation; pressure-loss coefficients in internal flows |
Heat transfer
| Group | Definition | Physical meaning | Typical use |
|---|---|---|---|
| Nusselt Nu | \(\mathrm{Nu} = \dfrac{h L}{k_{\mathrm{fluid}}}\) | Convective ÷ conductive transport at a wall | Convective HTC correlations; heat-exchanger segment sizing |
| Prandtl Pr | \(\mathrm{Pr} = \dfrac{c_p \mu}{k} = \dfrac{\nu}{\alpha}\) | Momentum diffusivity ÷ thermal diffusivity | Coupling velocity and thermal boundary layers; property evaluation from air properties |
| Péclet Pe | \(\mathrm{Pe} = \mathrm{Re}\,\mathrm{Pr}\) | Advection ÷ diffusion of energy | Thermal entrance lengths; conjugate problems |
| Biot Bi | \(\mathrm{Bi} = \dfrac{h L}{k_{\mathrm{solid}}}\) | External convection ÷ internal conduction in a solid | Lumped-capacitance validity; transient thermography interpretation, see HPT thermography note |
| Stanton St | \(\mathrm{St} = \dfrac{h}{\rho U c_p}\) | Heat transfer ÷ enthalpy flux | Boundary-layer heat transfer; Reynolds analogy contexts |
Similarity checklist for experiments
Before claiming a model test represents a prototype, state which Pi groups are matched and which are relaxed, and document the consequence in the uncertainty budget.
| Test type | Usually matched | Often relaxed | Read next |
|---|---|---|---|
| Incompressible internal flow | Re, geometry (scale) | Absolute pressure if density weakly varying | Viscous flow module |
| High-speed external aerodynamics | Re, Ma, γ (gas) | Wall temperature if adiabatic assumption holds | CFD-experiment validation |
| Convective heat transfer | Re, Pr, boundary-layer state | Bi in the solid if lumped model justified | Thermal & additive design |
| Pneumatic probe calibration | Re at calibration point, Mach regime | Facility-specific polynomial bias, see probe uncertainty note | Instrumentation theme |
Related resources
Part of
This page sits within the broader knowledge structure on lucasrey.com:
- Engineering - Central knowledge platform - tools, curriculum, notes, and research assets.
- Curriculum - Structured fluid mechanics modules from first principles.
Connected work
Related content from the same research and engineering work:
- Technical notes index
- Curriculum index
- Dimensional analysis module
- Practice problems
- Dimensionless groups reference - Referenced from Pneumatic probes overview
- Dimensionless groups reference - Referenced from Measurement uncertainty fundamentals
- Dimensionless groups reference - Referenced from Additive heat-exchanger low-order models
Part of Engineering
This page is part of the engineering knowledge platform on lucasrey.com.
- Engineering hub - Full knowledge platform index
- Engineering notes - Reference material and methods documentation - instrumentation and turbomachinery clusters.
- Pneumatic probes overview - 3-hole vs 5-hole selection, calibration workflow, and facility transfer - orientation before instrument-specific derivations.
- 3-hole probe uncertainty - Uncertainty propagation for multi-hole pneumatic probes.
- HPT transient thermography - Transient IR data-reduction and uncertainty workflow for ECAT turbine campaigns.
- Engineering notes - Reference material and methods documentation - instrumentation and turbomachinery clusters.
- Fluid mechanics curriculum - Index and learning path across modules.
- Dimensional analysis - Pi groups, similarity, and model testing.
- Engineering calculators - Air properties and pipe flow (live); probe uncertainty and heat-exchanger screening (planned).
- Air properties calculator - Dry-air thermophysical properties - density, viscosity, conductivity, and specific heats for aerothermal sizing (200 to 2000 K).
- Experimental aerothermal measurement - Methods article on traceability, similarity, and facility transfer - links instrumentation, curriculum, and tools.
- Instrumentation & measurement - Pneumatic probing, calibration transfer, and uncertainty propagation.
- Thermal & additive design - Low-order heat-exchanger design and additive manufacturing constraints.
- Technical notes index
- Curriculum index
- Dimensional analysis module
Engineering knowledge platform
More tools, curriculum, notes, and research from the same body of work:
- Engineering - Central knowledge platform - tools, curriculum, notes, research, and applied engineering work.
Where to go from here
Ways to take this further, in the order they usually happen.
- Get air properties at your own conditions - Conductivity, viscosity, specific heat, gamma and Prandtl for dry air from a temperature alone, and density, thermal diffusivity and kinematic viscosity once you add a pressure, each beside the correlation it came from and its valid range
- The rest of the engineering work - Tools, notes and the software behind them, indexed by topic
- How an engagement is scoped - Defined remit, timeline and deliverables, and which path suits which kind of work
- Discuss a piece of technical work - Scoped advisory, aerothermal review, instrumentation design and R and D roles
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