Model-free analysis of physical equations by tracking the units of measurement (M, L, T, Θ, I, N, J) of each variable. Core theorem: Buckingham 1914 π-theorem — a physical relation F(q_1,…,q_n)=0 among n dimensioned quantities with…
dimensional-analysis
Buckingham 1914 π-theorem: n−r independent dimensionless groups
Buckingham 1914 (Phys. Rev. 4:345) synthesis and generalisation of Rayleigh's 1877 earlier work: any physical equation F(q_1,…,q_n) = 0…
Reynolds-number similarity: laminar ↔ turbulent transition at Re ~ 2300
Reynolds 1883 (Phil. Trans. Roy. Soc. 174:935) injected dye into pipe flow and identified a threshold flow speed above which laminar…
Rayleigh-Bénard convection onset at dimensionless threshold Ra_c
Rayleigh 1916 (Phil. Mag. 32:529) linear-stability analysis of a horizontal fluid layer heated from below identifies a dimensionless…
Pipe-flow π-count: 5 variables, rank 3 → 2 π-groups
Textbook application of the Buckingham π-theorem to pipe-flow pressure drop. Variables and their dimensions: Δp [ML⁻¹T⁻²], ρ [ML⁻³], V…
Reynolds-number value at reference state and its laminar/turbulent ratio
Elementary dimensional check of the Reynolds-number definition at a chosen reference state. With V = L = 1 (consistent units) and ν =…
Rayleigh-Bénard free-free critical Rayleigh number Ra_c = 27π⁴/4
Closed-form result for the free-free (stress-free) Rayleigh-Bénard problem. Rayleigh 1916's linear-stability analysis of the Boussinesq…
Kolmogorov -5/3 law: E(k) = C_K * epsilon^{2/3} * k^{-5/3} via scaling-group dimensional analysis
Kolmogorov -5/3 turbulence energy-spectrum framework. Setup: homogeneous isotropic turbulence in the inertial range - far from the integral…
Nusselt-Rayleigh scaling law: Nu = C_N * Ra^{1/3} for high-Ra turbulent Rayleigh-Benard convection
Nusselt-Rayleigh scaling framework for turbulent Rayleigh-Benard convection. Setup: a fluid between two horizontal plates at temperatures…
Vaschy-Buckingham Pi-theorem: n variables with k independent dimensions reduce to n-k dimensionless groups
Vaschy-Buckingham Pi-theorem framework (structured statement). Setup: a physical relation f(q_1, q_2, ..., q_n) = 0 among n dimensional…
Theorem: Kolmogorov E(k) = C_K*epsilon^(2/3)*k^(-5/3) unit check L^3 T^{-2}
Theorem (Kolmogorov dimensional-exponent consistency): for the spectral energy density E(k) = C_K * epsilon^(2/3) * k^(-5/3), the exponent…
Theorem: Nusselt-Rayleigh Nu = C_N*Ra^(1/3) has log-slope d(ln Nu)/d(ln Ra) = 1/3
Theorem (Nusselt-Rayleigh logarithmic-slope invariant): for the Malkus scaling Nu = C_N * Ra^(1/3), the logarithmic slope is the…
Theorem: Reynolds number Re = rho*U*L/mu is dimensionless; exponent sums (M,L,T) = (0,0,0)
Theorem (Vaschy-Buckingham Reynolds-number dimensionless verification): Reynolds number Re = rho * U * L / mu where [rho] = M L^{-3}, [U] =…
Buckingham Pi theorem (1914)
Buckingham 1914: any physically meaningful equation involving N variables with K independent dimensions reduces to N-K dimensionless Pi…
Rayleigh method (1899)
Rayleigh 1899: assume product-of-powers form Q = prod x_i^{a_i}; match dimensions of [Q] with [x_i^{a_i}]; predates Buckingham; used for…
Reynolds number (1883)
Reynolds 1883: Re = rho U L / mu; ratio inertial vs viscous forces; transition to turbulence Re ~ 2300 in pipe; basis of fluid-similarity…
Planck natural units (1899)
Planck 1899: l_P = sqrt(h G/c^3) ~ 1.6e-35 m; t_P ~ 5.4e-44 s; m_P ~ 2.2e-8 kg; T_P ~ 1.4e32 K; sets natural-units scheme using only h, c,…
Similitude (Froude / Mach / Weber / Bond)
Froude 1868 Fr = U/sqrt(g L); Mach 1887 Ma = U/c_s; Weber 1893 We = rho U^2 L / sigma; Bond Bo = rho g L^2 / sigma; suite of dimensionless…
Anomalous dimensions (RG)
Wilson 1971-1975 RG: anomalous-dimension gamma(g) modifies engineering-dimension d at fixed-point; eta_critical = 2 gamma(g*); explains…
Rayleigh method (1899)
Lord Rayleigh 1899 dimensional-method; modern modern foundational text + Kolmogorov-1941 turbulence + 2024 ML-symbolic-regression discovery.
Buckingham π detail (1914)
E Buckingham 1914 + Vaschy 1892 Π-theorem; modern modern foundational text + 2024 ML-driven dimensional-discovery + non-dim group…
Reynolds (1883)
O Reynolds 1883 Re=ρUL/μ; modern modern foundational text + transition-Re + 2024 ML-DNS + Reynolds-stress modeling RANS+LES hybrids.
Strouhal (1878)
V Strouhal 1878 St=fL/U; modern modern foundational text + von-Kármán-vortex-shedding + 2024 modern bio-inspired wind-energy harvesting.
EFT-dimensional (Weinberg 1979)
S Weinberg 1979 EFT power-counting Λ-cutoff; modern modern foundational text + chiral-PT + Wilsonian-RG + 2024 ML-discovered EFT.
Sedov-Taylor (1946)
L Sedov 1946 + G Taylor 1950 self-similar blast-wave; modern modern foundational text + supernova-remnant + 2024 hypersonic-impact-blast.