Tethered Balloon Wind Sensor — theory

Low-level winds from a tethered pilot balloon. Based on C. P. Hoult, Tethered Pilot Balloon Bias Errors (rev. 1), 2013, Kite Catenary, 2006, and the CSULB wind-sensor reports of M. F. Buharie, S. Mohamed and M. Samir.

Open the calculatorBias-error memo (PDF)

Overview

Wind compensation needs the wind profile at launch time. The classical method — free pilot balloons tracked by two theodolites — is beyond a student budget, and free balloons rise erratically. A cheaper way uses a single helium balloon on a thin tether: the wind pushes it downwind until the tether leans at an angle set by the ratio of the balloon’s lift to its drag. An elevation sensor (such as a SkyScout) at the winch reads that angle and the azimuth. This works up to about 1 km; above that, winds come from the forecast winds aloft. The measured and forecast winds, resolved north–south and east–west, go into the trajectory simulation for wind compensation.

Force balance

The balloon of diameter D has buoyancy B=\rho g\pi D^3/6 in the air of density \rho at its height, and weight W (skin plus gas). The gas is filled on the ground, so its weight is W_g=\frac{p_G}{R_gT_G}\,g\,\frac{\pi D^3}{6}. In steady wind the tether lines up with the resultant of the net lift B-W and the drag \tfrac12\rho V^2C_DS, so

\tan\Theta=\frac{B-W}{\tfrac12\rho V^2C_DS}\quad\Rightarrow\quad V=\sqrt{\frac{2(B-W)\cot\Theta}{\rho C_DS}}

where S=\pi D^2/4. The calculator takes the air density at the balloon from the standard troposphere at the site elevation plus the balloon height h=d\sin\Theta. The response curve (elevation against wind speed) shows why a 3 ft balloon suits winds up to about 7 m/s: smaller balloons are blown down at modest winds, larger ones barely lean in light air. The sphere’s drag coefficient falls from about 0.47 to 0.1–0.2 above the drag crisis (Reynolds number ≈ 3×10⁵), which a large balloon in strong wind can reach.

Tether bias corrections

Strictly, the angle in the force balance is the tether angle at the balloon, not the one measured at the winch. Gravity sags the tether and the wind bows it downwind; both make the angle at the winch steeper, so the raw formula overestimates the wind.

Sag. The catenary solution for a flexible cable gives the change of angle between the ends, \Delta\Theta=W_T\cos\Theta/2T, with the tether weight W_T and tension T=(B-W)\csc\Theta:

\Delta\Theta_W=\frac{W_T\sin2\Theta}{4(B-W)}

Drag. The cross-flow component V\sin\Theta drags each element of tether of radius r; over a deployed length d in a uniform wind, D_T=C_d\rho rdV^2\sin^2\Theta, and

\Delta\Theta_D=\frac{C_d\rho rdV^2\sin^3\Theta}{2(B-W)}

Both are small, so they add, and the cotangent in the force balance is corrected to first order:

\cot(\Theta+\Delta\Theta)\simeq\cot\Theta\left[1-\frac{W_T+C_d\rho rdV^2\tan\Theta\sin\Theta}{2(B-W)}\right]=\cot\Theta\,(1-b-c)

The corrected speed is therefore V_c=V\sqrt{1-b-c}, with V from the uncorrected balance. The tether cross-flow drag coefficient is about 1.92 at the low Reynolds number (≈ 56) of a 0.011 in line.

Wind components

The balloon drifts downwind, so its azimuth \psi from the winch is the direction the wind blows towards. The north and east components for the trajectory simulation are V_N=V_c\cos\psi and V_E=V_c\sin\psi.

Is the reading good? Verdict and fixes

Three things decide whether a reading can be trusted.

  • Not blown down. Near the sphere’s drag crisis the drag stops growing with wind speed, so the calibration curve folds back: CSULB found it double-valued once the zenith angle passes about 75°, i.e. below about 15^\circ elevation (Samir et al., p. 6; the balloon size was chosen from sphere drag data on p. 3). The verdict is unsafe there. The balloon Reynolds number Re=\rho VD/\mu (viscosity at 15 °C, an estimate) is shown with a note past 3\times10^5, the smooth-sphere drag crisis; it is not judged, since CSULB’s measured curve stayed usable to about 30 ft/s for its 3 ft balloon.
  • The angle error. Since V\propto\sqrt{\cot\Theta}, an angle error \delta\Theta gives
\frac{\delta V}{V}=\frac{\delta\Theta}{\sin2\Theta}

smallest at 45°. With your accuracy and largest accepted error e_{max}, the good angles are \Theta_{lo}\le\Theta\le90^\circ-\Theta_{lo} with \sin2\Theta_{lo}=\delta\Theta/e_{max} (and never below 15°).

  • The tether correction of eq. (5) is first order, so the verdict is marginal when it exceeds the correction you accept. The longest line within it is found by bisection on the tether length.

When the angle is outside the band, the balloon diameter that puts the same raw wind just inside the band (30° when blown down, 60° when too high) is solved by bisection: the elevation from eq. (1), \tan\Theta=(B-W)/(\tfrac12\rho V^2C_DS), rises with the diameter. The bands are this site’s choices except the 15° limit, which is from the CSULB report.

Verification

The tests reproduce the field tool’s wind speeds, bias terms, corrected speeds and heights for both of its readings when given its own buoyancy, weights and densities. They also reproduce the memo’s worked example (3 ft balloon, 600 ft of tether at 56.44°, raw speed 10.708 ft/s, tether-weight term 0.01685), its helium weight of 0.15 lb, the component split and a sinking-balloon case.

Notes on the method

  • Tether-drag term of the memo’s example. The memo’s numerical example puts the normal velocity V\sin\Theta in place of V in eq. (5). Eq. (5) already contains the \sin^2\Theta of the normal velocity, so this counts it twice: the example quotes 0.0882 where eq. (5) gives 0.127. The calculator (like the field tool) uses eq. (5) as written; the example then overestimates the wind by about 7.5 %, not 5.4 %.
  • Gas weight. The field tool computes the helium weight from pressure in pascals and volume in cubic feet in one formula, which roughly doubles it (0.31 lb instead of 0.15 lb for a 3 ft balloon). The calculator computes the gas mass consistently; the field tool’s numbers are reproduced in the tests only when its own weights are supplied.
  • Tether length. Enter the tether paid out directly; the winch odometer calibration of the original field tool is specific to its winch.
  • Steady, uniform wind. Gusts move the balloon; average several readings.

References

  1. C. P. Hoult, “Tethered Pilot Balloon Bias Errors (rev. 1),” RS&T, 2013. View in library
  2. C. P. Hoult, “Kite Catenary,” RS&T, 2006. View in library
  3. C. P. Hoult, “Winch Equations,” RS&T, 2008. View in library
  4. M. Samir and M. F. Buharie, “A New Wind Sensor for Rocket Launches,” CSULB. View in library
  5. M. F. Buharie and S. Mohamed, “A New Wind Sensor for Rocket Launches,” AIAA student paper, 2008. View in library
  6. R. T. Jones, “Effects of Sweepback on Boundary Layer and Separation,” NACA Report 884, 1947.
  7. S. F. Hoerner, Fluid-Dynamic Drag, 1965.