Wind-Tunnel Turbulence & Experimental Aerodynamics, L.V. King (1914) CTA Anemometry
King’s Law CTA Hot-Wire Anemometer Velocity Formula
Calculate Active Wire Overheat Difference (K), Bearman-Corrected Squared Bridge Voltage E²corr (V²), True Compensated Flow Velocity (m/s), and Uncorrected Raw Velocity (m/s).
Closed-return wind-tunnel air warming +27 K during a run tricks an un-compensated 520 K CTA hot-wire probe into reading 24.6 m/s when true airspeed is 36.6 m/s (-33% error)!
**Louis Vessot King (`1914`)** established the boundary-layer heat-transfer law for **Constant-Temperature Hot-Wire Anemometers (`CTA`)**, where a `100`-kHz servo Wheatstone bridge holds a `5`-um tungsten filament at constant Overheat Temperature `T_w` (`K`): at Calibration Air Temperature `T_0` (`K`), **Squared Bridge Voltage obeys `E^2 = A + B * (U^n)`** (`with King’s theoretical exponent n = 0.50`, or calibrated **`n ~= 0.45`** by Collis & Williams)! Yet if test-section air warms or cools to **`T_f` (`K`)**, convective heat loss scales with `(T_w - T_f)`—so before inverting King’s Law, **Bearman’s Temperature-Drift Correction `E_{corr}^2 = E_{meas}^2 * ((T_w - T_0) / (T_w - T_f))`** (`V²`) MUST be applied to recover **True Fluid Velocity `U_{true} = ((E_{corr}^2 - A) / B)^(1 / n)`** (`m/s`)!
Verify on a `5`-um Tungsten Boundary-Layer Probe at Calibration Temperature (`T_f = T_0 = 295` K)—Measured_Bridge_Voltage `E` = 2.36 V (B2), King_Zero-Flow_Intercept `A` = 1.68 V² (B3), King_Slope `B` = 0.92 `V²/(m/s)^n` (B4), Calibrated_Exponent `n` = 0.45 (B5), Sensor_Wire_Temp `T_w` = 520 K (B6), Calibration_Fluid_Temp `T_0` = 295 K (B7), and Actual_Tunnel_Temp `T_f` = 295 K (B8): Active Overheat `T_w - T_f` equals `225.0` K (E2), Temperature-Corrected `E_{corr}^2` equals `5.570` V² (E3), and both True Velocity (E4) and Raw Uncorrected Velocity (E5) agree at **`24.62` m/s**!
| King’s Law (1914) CTA Anemometer KPI (E=2.36 V, A=1.68 V², B=0.92, n=0.45, Tw=520 K) | Formula | Verified Output |
|---|---|---|
| Active Wire Overheat Tw - Tf (K) | Wire_Temp_Tw_K - Actual_Fluid_Tf_K | 225.0 K @ 295 K (198.0 K @ 322 K tunnel) |
| Temp-Corrected E²corr (V²) | (Bridge_E_V ^ 2) * ((Tw - T0) / (Tw - Tf)) | 5.570 V² (6.329 V² drift-compensated) |
| True Compensated Velocity (m/s) | ((Corrected_E2 - King_A) / King_B) ^ (1 / n) | 24.62 m/s (36.60 m/s TRUE airspeed!) |
| Uncorrected Raw Velocity (m/s) | (((Bridge_E_V^2) - King_A) / King_B) ^ (1 / n) | 24.62 m/s (24.62 m/s = -32.7% ERROR!) |
L.V. King (1914) & Collis-Williams CTA Temperature-Compensated Velocity Formula
Validate `Intercept A >= 0.1` V², `Slope B > 0`, `Exponent n` in `[0.30, 0.65]`, `Wire_Temp T_w` in `[350, 900]` K, `Cal/Actual Fluid Temps` in `[240, T_w - 40]` K, and `E^2 > A` & `E_{corr}^2 > A`!
=LET(em_v, B2, az_v, B3, bs_k, B4, nx_k, B5, tw_k, B6, tz_k, B7, tf_k, B8, ec_s, (em_v ^ 2) * ((tw_k - tz_k) / (tw_k - tf_k)), IF(OR(az_v < 0.1, bs_k <= 0, nx_k < 0.3, nx_k > 0.65, tw_k < 350, tw_k > 900, tz_k < 240, tz_k > tw_k - 40, tf_k < 240, tf_k > tw_k - 40, (em_v ^ 2) <= az_v, ec_s <= az_v), "Invalid HotWire Input", ROUND((((em_v ^ 2) - az_v) / bs_k) ^ (1 / nx_k), 2)))Un-Chilled Wind-Tunnel Heating (`T_f = 322` K = `48.9°C` at Same `2.36` V) 33%-Error Check
Let fan friction warm recirculating wind-tunnel air to `B8 = 322.0` K (`Overheat` E2 shrinks `12%` to `198.0` K) while bridge voltage still reads `2.36` V: Because `1 / n = 2.222` amplifies small voltage-squared shifts nonlinearly, Uncorrected Raw Velocity E5 stays stuck at `24.62` m/s while **True Airspeed E4 is actually `36.60` m/s** (`E_{corr}^2 = 6.329` V² in E3)—proving an RTD temperature probe MUST always accompany every CTA hot-wire channel!
=IF(ISTEXT(E5), E5, LET(ec_s, (B2 ^ 2) * ((B6 - B7) / (B6 - B8)), ROUND(((ec_s - B3) / B4) ^ (1 / B5), 2)))Try the example
Use the original values or change them to test your understanding.