Fluid Mechanics, Dimensional Analysis & Tow-Tank / Wind-Tunnel Testing
Froude vs Reynolds Scale-Model Similitude Scaling Formula
Calculate Tow-Tank Froude-Scaled Model Speed V_proto / SQRT(Lambda), Cubic Force Scaling F_proto / Lambda^3, Propulsion Power Scaling P_proto / Lambda^3.5, and Unpressurized Reynolds-Similitude Speed V_proto * Lambda.
See why ship tow tanks and wind tunnels cannot satisfy Froude and Reynolds similitude simultaneously!
Experimental fluid mechanics relies on Buckingham Pi dynamic similitude between a full-scale prototype and a 1:Lambda scale model (Lambda = L_proto / L_model)! Free-surface ship hulls, spillways, and offshore platforms are governed by gravity waves—requiring equal Froude Number Fr = V / SQRT(g * L), which dictates a SLOWER Tow-Tank Model Speed V_model(Fr) = V_proto / SQRT(Lambda), Model Drag Force F_model = F_proto / Lambda^3 (same water density), and Model Tow Power P_model = (F_proto * V_proto) / Lambda^3.5! Conversely, matching viscous boundary-layer Reynolds Number Re = V * L / nu in the same unpressurized fluid requires a FASTER Wind-Tunnel Speed V_model(Re) = V_proto * Lambda—proving why a 1:25 scale model cannot satisfy both laws at once without a pressurized or cryogenic wind tunnel!
Verify on a 1:25 Tow-Tank Hull Campaign—Length_Scale_Ratio Lambda = 25 (B2), FullScale_Ship_Speed V_proto = 15.0 m/s (B3), and FullScale_Wave_Drag F_proto = 500.0 kN (500,000 N, B4): Froude-Matched Tow-Tank Speed is 15 / SQRT(25) = 3.00 m/s (E2); Froude-Scaled Model Drag Force in the same-density basin is 500,000 / (25^3) = 500,000 / 15,625 = 32.00 N (E3); Froude-Scaled Model Tow Power is (500,000 * 15) / (25^3.5) = 7,500,000 / 78,125 = 96.00 W (E4); whereas Unpressurized Same-Fluid Reynolds Matching would demand 15 * 25 = 375.00 m/s (supersonic in air and cavitating in water, E5)!
| Similitude Scaling Metric (Scale Lambda=25, V_proto=15 m/s, F_proto=500 kN) | Formula | Verified Output |
|---|---|---|
| Froude Tow-Tank Speed (m/s) | V_proto / SQRT(Scale_Ratio_Lambda) | 3.00 m/s (15.00 @ 1:1 Scale) |
| Froude Model Drag Force (N) | (F_proto_kN * 1000) / (Lambda^3) | 32.00 N (500,000 N @ 1:1) |
| Froude Model Tow Power (W) | (F_proto_N * V_proto) / (Lambda^3.5) | 96.00 W |
| Unpressurized Reynolds Speed | V_proto * Scale_Ratio_Lambda | 375.00 m/s (Incompatibility Proof) |
Froude-Similitude Model Tow-Power Scaling (Lambda^3.5) Formula
Validate Scale_Ratio Lambda >= 1, V_proto > 0, and F_proto_kN > 0!
=LET(lm_s, B2, vp_m, B3, fp_k, B4, IF(OR(lm_s < 1, vp_m <= 0, fp_k <= 0), "Invalid Similitude Scale Input", ROUND((fp_k * 1000 * vp_m) / (lm_s ^ 3.5), 2)))Full-Scale 1:1 Identity Verification Check (Set Scale Ratio Lambda = 1)
Set Scale_Ratio B2 = 1: Froude Speed E2 and Reynolds Speed E5 both equal 15.00 m/s, Model Force E3 equals 500,000.00 N, and Power E4 equals 7,500,000.00 W (7.50 MW)!
=IF(ISTEXT(E4), E4, ROUND(B3 * B2, 2))Try the example
Use the original values or change them to test your understanding.