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
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.
Reference
Similarity
Fluid mechanics
Heat transfer
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 St | \(\mathrm{St} = \dfrac{\mu U}{\rho g D^2}\) (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 | \(\mathrm{Nu} = \dfrac{h L}{k}\) | 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
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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 Teaching
- Dimensionless groups reference - Referenced from Pneumatic probes overview
- Dimensionless groups reference - Referenced from Measurement uncertainty fundamentals
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 (live); probe uncertainty and related tools (planned).
- Air properties calculator - Dry-air thermophysical properties - density, viscosity, conductivity, and specific heats for aerothermal sizing (200–2000 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.