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Bulletin No. 23


rev. 11/89

D. H. Lenschow and P. Spyers-Duran

Note: At this time NCAR operates only one aircraft, a C-130Q Hercules (tail number N130AR). NCAR operated the three aircraft mentioned in this Bulletin when it was written in 1989. The basic equipment and instrumentation, equations and principles discusssed here are still pertinent and being used at our facility.



    NCAR presently operates three aircraft equipped with inertial navigation systems (INS) and a gust probe or a radome with pressure ports for air motion measurements. These aircraft (King Air, Sabreliner, and Electra) can be instrumented to measure both mean horizontal wind and turbulent fluctuations of the three air velocity components. The components are obtained by subtracting the velocity of the airplane, measured by the INS, from the velocity of the air, measured by the radome or gust probe. Since the flow measurements are in the aircraft frame of reference, these measurements are first converted to the earth's frame of reference by using the three airplane attitude angles measured by the INS, and are then subtracted from the airplane velocity components, which are obtained from integrated accelerometer outputs (corrected for Coriolis accelerations) in an earth-based frame of reference. Most applications of air-motion sensing systems involve either

    (1) measurement of the mean horizontal wind, which is useful mainly for synoptic or mesoscale meteorological experiments, or
    (2) measurement of high-frequency fluctuations of all three velocity components, which are used mainly to calculate turbulence statistics such as variances, and turbulence fluxes of heat, moisture, momentum, and turbulence energy.



    In this section we derive the equations for calculating the air velocity components from the INS and the flow-sensing probes. The derivation is similar to Axford's (1968). However, the terminology and coordinate systems are modified to conform to standard aeronautical engineering usage (Etkin, 1959) for the aircraft coordinate system and to standard inertial navigation (Kayton and Fried, 1969) and meteorological usage for the air velocity in the local earth coordinate system.

    The velocity of the air with respect to the earth, V = iu + jv + kw, is obtained by adding the velocity of the aircraft with respect to the earth, vp, and the velocity of the air with respect to the aircraft, va. If all the measurements are made at the same point, we have

    Equation 2.1:  vector v = vector v sub p + vector v sub a

    The components of va are measured with sensors mounted on the aircraft, either on a boom forward of the aircraft to reduce the effects of the distortion of the airflow by the aircraft or alternatively, from pressure ports on a radome to take advantage of the pressure differences produced by asymmetric flow about the aircraft's nose. On the NCAR aircraft the components of vp are obtained from the integrated outputs of accelerometers mounted on the stabilized platform of an INS. Alternatively, vp could be obtained from radiation transmitting and receiving devices, such as Doppler radars or radar altimeters, or from radio or radar navigation techniques.

    If the aircraft velocity is obtained from integrated accelerometer outputs, the velocity is measured in an inertial frame of reference. In an earth-based coordinate system, we have

    Equation 2.2: derivative dv sub p/dt = a - (omega sub p + omega sub e) x vp + g


    vector a is the measured acceleration
    vector omega sub e and vector omega sub p are the angular velocities of the earth and platform, respectively; and
    vector g is the gravitational acceleration (which includes the centripetal acceleration).

    Since the accelerometers may not be collocated with the air velocity sensors, it may be necessary to add the term vector OMEGA sub p cross R to (2.2), where vector OMEGA sub p is the angular acceleration of the aircraft and R = iX + jY + kZ is the distance from the accelerometers to the air velocity sensors. Thus, (2.1) can be combined with the integral of (2.2) and rewritten as

    Equation 2.3:  vector v = vectors v sub a + v sub p + OMEGA sub p cross R

    In the general case, the measured components of va and vp may not be in the local earth coordinate system. The airflow sensors, for example, are usually rigidly connected to the aircraft fuselage. Therefore, it is necessary to know the angles between the coordinates of the airflow sensors and the local earth coordinates. The measured velocity components can then be rotated by means of the appropriate angular transformation to the local earth axes so that the rotation angles of the platform gimbals are equivalent to the aircraft attitude angles. From the inside platform gimbal outward, i.e., from the stabililzed accelerometer and gyro cluster outward to the aircraft frame, the order of the rotations is as follows: first, a rotation greek psi (true heading), about the z-axis of the earth coordinate system; second, a rotation greek theta (pitch), about the y'-axis, which is the y-axis rotated in the horizontal plane by the angle greek psi (true heading) ; and third, a rotation greek PHI (roll), about the x'-axis, which is the x-axis in the aircraft coordinate system. The aircraft axes are shown in Fig. 2.1; a right-handed angular rotation is positive. The symbols and axes have been defined to correspond to standard aerodynamic conventions (Etkin, 1959).

    Fig. 2.1. Illustration drawing.  Top:  Coordinate systems used in deriving the air velocity equations.  Bottom: Airplane attitude angles and axes used.

    Fig. 2.1. Top: Coordinate systems used in deriving equations for calculating the air velocity components. Bottom: Airplane attitude angles and axes used in equations for calculating the air velocity components.

    In matrix notation, the transformation of the air velocity from aircraft (x', y', z') to platform (x, y, z) coordinates {same as (b) coordinates in Fig. 2.1} is given by

    Equation 2.4:  matrix u sub i = matrix T sub i,j x matrix u prime sub j


    Equation 2.5:  Expansion of matrix T sub i,j


    Equation 2.6:  Further expansion of matrix T sub i,j


    The terms of vector OMEGA sub p cross vector R must also be transformed to the local earth coordinate system. X is more than an order of magnitude larger than Y and Z on all the NCAR aircraft installations, so that terms involving Y and Z can be neglected. Setting X = L, we have

    Equation 2.7:  Define matrix R sub i = matrix T sub i,j using the L (X) element with Y and Z set to zero

    Similarly for vector OMEGA sub p we have

    Equation 2.8:  OMEGA sub p sub i = matrix functions involving attitude angles and their rates of change

    Thus, in vector notation,

    Equation 2.9:  vector OMEGA cross R = L x vector functions (3 orthogonal) of attitude angles and their rates of change

    The angle of attack, greek alpha (angle of attack), is the angle of the airstream with respect to the aircraft in the aircraft's vertical plane, with greek alpha (angle of attack) positive in the downward direction; the angle of sideslip, greek beta (angle of sideslip), is the angle of the airstream with respect to the aircraft in the aircraft's horizontal plane, with clockwise (looking from above) rotation positive, as shown in Fig. 2.1. The air velocity components with respect to the aircraft are calculated by first correcting the measured true airspeed for angle of attack and sideslip sensitivities (See Section 4.); the magnitude of the corrected true airspeed is defined as Ua. We then rotate the airflow vector to the airplane coordinate system. If we first rotate through the angle greek alpha (angle of attack) to the horizontal plane, then rotate to the longitudinal aircraft axis, the components of air velocity in the aircraft frame of reference are -Ua D-1 along the x'-axis, -Ua D-1 tan greek beta (angle of sideslip), along the y'-axis, and -Ua D-1 tan greek alpha (angle of attack) along the z'-axis, where D = (1 + tan2 greek alpha (angle of attack) + tan2 greek beta (angle of sideslip))1/2

    [This differs from the expressions given by Lenschow (1972). For small flow angles, however, the magnitudes of the differences are negligible. We are grateful to A. Weinheimer and J. Leise, who each independently pointed out the error and corrected the previous derivation.]

    So far, the equations have been derived using standard aeronautical engineering terminology. In converting to a meteorological and inertial navigation frame of reference, we make the following changes: first, the sign of vertical velocity is changed so that w is positive in an upward direction; second, greek psi (true heading), which now corresponds to the true heading, is measured from the y-axis, or from north, and the z-axis now points east, as shown in Fig. 3. As a result, greek psi (true heading)(new) = greek psi(old) - 90°, v(new) = - v(old), and w(new) = - w(old). The final equations used for calculating the air velocity with respect to the earth are:

    Equation 2.10:  Final equations for calculating the three wind components u, v and w

    These are the exact equations that are used in the standard NCAR data processing routines for calculating the air velocity. For purposes of illustration, however, it can be shown that for approximate straight and level flight (i.e., when the pilot or autopilot attempts to keep the aircraft level, but air velocity fluctuations still cause perturbations in the aircraft velocity and attitude angles) many of the terms in (2.4) are negligible.

    First we assume that terms involving the separation distance, L, are small. This is generally true if the separation distance is less than about 10 m and the aircraft is not undergoing a pilot-induced pitching maneuver. Next we make the following small-angle approximations: for the horizontal components, u and v, we assume that the cosines of greek alpha (angle of attack), greek theta (pitch) and greek PHI (roll) are unity; that terms that involve the products of sines of two of the above angles are negligible; and that tan (greek alpha (angle of attack), greek beta (angle of sideslip)) math symbol--approximately equals sin (greek alpha (angle of attack), greek beta (angle of sideslip)). For the vertical component w, we make the same assumptions for the angles greek beta (angle of sideslip), greek theta (pitch) and greek PHI (roll). Equations (2.10) then reduce to a form commonly used for approximate calculations of the three velocity components,

    Equation 2.11:  simplified equations for calculating the 3-dimensional wind components

    We can estimate the required accuracy of attitude and airstream incidence angle measurements from (2.11). In order to obtain the mean horizontal wind, the angle (greek psi (true heading) + greek beta (angle of sideslip)) must be measured accurately. For a mean wind accuracy of 0.5 m/s and an aircraft speed of 100 m/s, this angle must be measured with an absolute accuracy of 0.005 radians, or 0.06°. A heading angle of this accuracy is difficult to measure with a standard aircraft magnetic compass. An INS, however, can measure true heading to well within this accuracy.

    A reasonable figure for short-term velocity accuracy necessary to estimate turbulence fluctuations is 0.1 m/s. At an airplane speed of 100 m/s, the required angular accuracy for the first terms on the right side of (2.11) is 0.001 radians, or 0.060°. This can be a difficult requirement to meet. Fortunately, an INS has even more stringent accuracy requirements than this for accurate navigation, so attitude angles of this short-term accuracy can be measured.





    Regardless of where instruments are located and how carefully they are calibrated, errors are likely to be present in the measured variables. Ground tests are not useful for calculating velocity-related errors. Wind tunnel tests are difficult and prohibitively expensive for exact simulation of flight conditions. Therefore, in-flight calibrations play an important role in estimating errors and correcting aircraft measurements.

    Because of upwash ahead of the aircraft, the airflow angles (attack and sideslip) and airspeed measured at the tip of a nose boom may differ considerably from the actual values that would be measured far away from the aircraft. The upwash affects not only the sensitivity, but also the zero offset of angle measurements, which, therefore, must also be determined from in-flight calibrations.

    Maneuvers used for this purpose involve changes in aircraft speed and attitude angles. The following list summarizes several maneuvers used on NCAR aircraft equipped with an INS and the information that can be obtained from them; examples of these maneuvers are shown in Fig. 4.1:

    Figure 4.1a.  Examples of reverse heading and airspeed maneuvers

    Fig. 4.1a. Examples of reverse heading and airspeed maneuvers used to check the quality of air velocity measurements. The lateral and longitudinal velocity components are measured with respect to the aircraft; therefore, the measured wind should change sign, but not amplitude, after the 180° turn, if the wind field remains constant and is measured without error. An error in airspeed will result in a difference in the amplitude of only the longitudinal component before and after the turn, while an error in the sideslip angle will similarly affect only the lateral component, which simplifies correction procedures. The airspeed maneuver modulates the attack and pitch angles; if pitch angle is measured accurately, the error in attack angle a can be determined by comparing the vertical wind component with respect to the airplane (Ua sin greek alpha (angle of attack)) with the vertical wind component with respect to the earth (w). In this example, there is little correlation between the two, so the fluctuations in w are presumed to be due to turbulence rather than an inaccurate measurement of greek alpha (angle of attack). The airspeed maneuver can also be used to estimate airspeed-dependent errors in other variables and the temperature recovery factor.

    Figure 4.1b.  Examples of pitch and yaw maneuvers

    Fig. 4.1b. Examples of pitch and yaw maneuvers. The pitch maneuver is used as an overall check on the accuracy of the w measurement; in this example, there is little modulation of w during the pitching maneuver, which implies that fluctuations in w are measured accurately. Similarly, the yaw maneuver is used as an overall check on the lateral (with respect to the aircraft) component; again there is little modulation of u and v (in geographic coordinates) by the yawing maneuver.



    As with all physical measuring systems, there are limitations, both in the accuracy of the measurement and in the bandwidth (range of wavelengths) over which air velocity can be measured. Furthermore, the limitations are somewhat different for different velocity components. The horizontal velocity error, for example, has two components: a cyclic component with a period of 84.4 min and a nonperiodic component that usually increases with time. A typical navigational accuracy of the INS is approximately 0.5 m/s. Because of the periodic component, the velocity error for a position error rate of 0.5 m/s is usually larger than 0.5 m/s/h (or 1 kt/h). Exact numbers cannot be specified, since sources of error are multifarious. Some of these are the following:

In the vertical direction, the airplane velocity error is limited by using the measured pressure altitude as a long-term reference. Since this is a relatively accurate and stable reference, the vertical velocity of the airplane can be measured to better than 0.1 m/s if horizontal pressure gradients are not unusually large. The wind component along the longitudinal axis of the airplane is obtained from the difference between the airplane true airspeed and the INS-measured longitudinal airplane velocity. Since the airplane speed is usually several times the wind speed, this involves a small difference of two large numbers (the faster the airplane flies, the larger the numbers). Because of the upstream effects of the air flowing around the airplane and the sensor-mounting structures, the Pitot-static and static pressure measurements used in the true airspeed calculation must be empirically corrected. This is done by: