ashfordeOU Dimensional Analysis (cross-cutting/units-atmos/dimensional-analysis)
Use when the task is dimensional analysis of an engineering relation: the dimensional homogeneity of an equation, the dimensionless groups of a problem from the Buckingham Pi theorem, the similarity numbers (Reynolds, Mach, Froude) at a test or flight condition, and the scaling of wind tunnel model test results to full scale under dynamic similarity.
Domain quick reference
Dimensional analysis works with dimension exponent vectors over the SI base dimensions, in the order mass M, length L, time T, temperature Theta, electric current I, amount of substance N, luminous intensity J. Most aerospace relations only need M, L, T.
- Homogeneity check: every term of a valid equation carries the same dimension vector. Bernoulli's equation terms p, 0.5 rho v^2, and rho g h all carry M L^-1 T^-2, so check_homogeneity returns True; a term such as rho v (M L^-2 T^-1) breaks the equation. Worked: check_homogeneity([("p", (1, -1, -2)), ("0.5 rho v^2", (1, -1, -2)), ("rho g h", (1, -1, -2))]) returns (True, (1.0, -1.0, -2.0)).
- Buckingham Pi theorem: with n variables and a dimension matrix of rank r, exactly n - r independent dimensionless groups exist. The rank is the number of dimensionally independent variables, found by Gaussian elimination of the dimension matrix; each Pi group is a null-space vector of that matrix. Worked: sphere drag with F, D, rho, V, mu over M, L, T (5 variables, rank 3) gives n_pi = 2, and the group set spans Re = rho V D / mu (exponents 0, 1, 1, 1, -1) and the drag coefficient F / (rho V^2 D^2) (exponents 1, -2, -1, -2, 0).
- Similarity numbers (SI inputs, all positive): Reynolds Re = rho v l / mu with rho in kg/m3, v in m/s, l in m, mu in Pa.s; Mach M = v / a; Froude Fr = v / sqrt(g l) for free-surface flows. Worked: rho 1.225 kg/m3, v 80 m/s, chord 2.0 m, mu 1.781e-5 Pa.s gives Re = 1.10e7; the same v with a = 340.3 m/s gives M = 0.235; v 5 m/s over l 2.5 m gives Fr = 1.01.
- Dynamic similarity: matching a dimensionless group between model and
full scale keeps the corresponding physical effect proportional.
For Reynolds matching, V_model = V_proto * (L_proto / L_model) *
(nu_model / nu_proto). Worked: a 1:10 model (scale_ratio 10) in the
same fluid needs V_model = 10 * 80 = 800 m/s to match a full-scale
80 m/s condition, which is usually impractical; the same fluid and
Mach constraint force partial similarity, so tunnels raise density
(pressurized or cryogenic operation) to raise Re at acceptable
speed. Force scaling with the same dimensionless force coefficient:
F_proto = F_model * (L_proto / L_model)^2 * (rho_proto / rho_model)
- (V_proto / V_model)^2. Worked: a 12.5 N model drag at Re-matched conditions in the same fluid scales to F_proto = 12.5 * 100 * 1.0 * (1/10)^2 = 12.5 N full scale.
Workflow
- Name the physical relation and list the variables with their dimension exponent vectors over the base dims in use (usually M, L, T).
- Run check_homogeneity(terms) on the equation terms; a False verdict means a term has the wrong dimensions, fix the equation before any further computation.
- Build the variables dict and call buckingham_pi(variables, base_dims=("M", "L", "T")); read rank, n_pi, and the pi_groups, then name each group from its exponents (Re, a drag coefficient, a lift coefficient, and so on).
- Compute the similarity numbers for the model and full-scale conditions with reynolds_number(rho, v, l, mu), mach_number(v, speed_of_sound), and froude_number(v, l, g) where relevant.
- For a model test, find the required model speed with required_model_speed(scale_ratio, prototype_speed, kinematic_viscosity_ratio) and check it against the tunnel capability; then convert a measured model load to full scale with force_scaling(force_model, scale_ratio, density_ratio, velocity_ratio).
- Record the Pi groups and the matched (and unmatched) similarity parameters in the test plan; a result is only similarity-valid for the parameters that were actually matched.
Pitfalls
- Routing to unit-conversion instead: converting a value between unit systems (psi to Pa, kt to m/s, speed to Mach) routes to cross-cutting/units-atmos/unit-conversion; that leaf never checks dimensional consistency and never forms Pi groups. If the question is "is my equation homogeneous" or "what are the dimensionless groups", route here.
- Routing to temperature-conversion or isa-atmosphere: scale conversion (K, C, F, R) is temperature-conversion; atmosphere properties (rho, T, mu, speed of sound) come from isa-atmosphere. This leaf consumes those values but does not produce them.
- Routing to the numerics leaves: regression, integration, root finding, and the other numerics leaves operate on numbers without tracking dimensions; a pure data-fitting task routes to least-squares-regression, not here.
- Repeating variables that are not dimensionally independent: the repeating set must span the rank of the dimension matrix, otherwise the Pi groups are not independent and the theorem undercounts.
- Wrong characteristic length: Re uses the length scale of the flow (chord for a wing, diameter for a sphere); swapping them changes Re by the length ratio and mis-states the flow regime.
- Sign errors in group exponents: mu enters Re with exponent -1; a flipped sign makes the group dimensionful.
- Matching Re alone in compressible flow: when Mach is significant both Re and M must match, which one fluid cannot usually satisfy with a geometric scale model; state which parameter is unmatched.
- Reversing the scale ratio: scale_ratio is L_proto / L_model, so a 1:10 model has scale_ratio 10, not 0.1.
- Assuming same-fluid force scaling: the velocity and density ratios must be carried explicitly; the 12.5 N result above is special to Re-matched, same-fluid tests.
Behavior contract (gate 3)
The homogeneity check, Buckingham Pi group formation, similarity numbers, and dynamic similarity scaling are exercised by the gate 3 contract test: scripts/test_dimensional_analysis.py against scripts/dimensional_analysis_logic.py (stdlib unittest, offline). Run: python3 scripts/test_dimensional_analysis.py
Compliance
- Standards referenced, not reproduced: SEP-2640 frames the skill packaging and delivery practice only. Dimensional analysis itself is fundamental mathematics (the Buckingham Pi theorem, similarity parameters) and is not governed by a certification standard; the worked cases are textbook engineering calculations, summary-only per standards-map.yaml.
- compliance: STANDARDS-REF, gated: false.