Aircraft-Based Flux Density Measurements

Abstract

Thiso chapter presents aircraft-based methods of measuring the flux densities of sensible and latent heat, carbon dioxide, ozone, nitrous oxide, methane, and other trace gases. The main techniques and sensors that are used to measure flux densities with an aircraft are briefly described. Factors that affect the accuracy of those flux density measurements are discussed, including analysis techniques, run lengths, sampling heights, surface and environmental conditions, and data quality assessment. The use of aircraft-based flux density measurements to evaluate the representativeness of tower-based flux measurements is examined. The versatility of aircraft to act as sensor platforms under a wide range of conditions is demonstrated using several interesting examples. Future potential research directions are mentioned.

Appendix

1.1 True Airspeed by the Flight Mechanical Angles α and β

To obtain a proper solution of (48.7), we start in the aerodynamic coordinate system ADONS (index a) of the flight. The ADONS is defined as the system in which the airspeed vector has the components

$$\boldsymbol{v}_{\mathrm{tas,a}}=\left(-\left|\boldsymbol{v}_{\mathrm{tas}}\right|,0,0\right),$$

and is therefore called the airspeed-fixed coordinate system. A rotation of the ADONS about the vertical (lift) axis z_a by the angle −β

$$\mathbf{T}_{\mathrm{ea}}(-\beta)=\begin{pmatrix}\cos\beta&-\sin\beta&0\\ \sin\beta&\cos\beta&0\\ 0&0&1\end{pmatrix}$$

(48.B9)

and then about the transverse axis (cross-force) axis y_a by the angle α

$$\mathbf{T}_{\boldsymbol{fe}}(\alpha)=\begin{pmatrix}\cos\alpha&0&-\sin\alpha\\ 0&1&0\\ \sin\alpha&0&\cos\alpha\end{pmatrix}$$

(48.B10)

yields the description of the airspeed vector in the ACONS [48.98],

$$\begin{aligned}\displaystyle\boldsymbol{v}_{\mathrm{tas}}&\displaystyle=\begin{pmatrix}u_{\mathrm{tas}}\\ v_{\mathrm{tas}}\\ w_{\mathrm{tas}}\end{pmatrix}=\mathbf{T}_{\mathrm{fe}}(\alpha)\mathbf{T}_{\mathrm{ea}}(-\beta)\boldsymbol{v}_{\mathrm{tas,a}}\\ \displaystyle&\displaystyle=-\left|\boldsymbol{v}_{\mathrm{tas}}\right|\begin{pmatrix}\cos\alpha\cos\beta\\ \sin\beta\\ \sin\alpha\cos\beta\end{pmatrix}\;.\end{aligned}$$

(48.B11)

The angle α is positive for nose-lifting rotations, while β is positive for rotations to the port side. Both the angles and the order of rotation have their origins in wind-tunnel experiments. The first rotation T_ea leads to the experimental coordinate system (index e) that is cross-axis fixed. The components of the true airspeed vector v_tas in the ACONS are:

• u_tas, which is oriented along the roll axis of the aircraft and positive in the flight direction

• v_tas, which is oriented along the pitch axis of the aircraft and positive in the starboard direction

• w_tas, which is oriented along the normal (or yaw) axis and positive downwards.

1.1.1 True Airspeed by the FHP Angles $\tilde{\alpha}$ and $\tilde{\beta}$

The outputs of the FHP (Fig. 48.2) are pressure differences that are used to calculate v_tas in spherical coordinates in the ACONS:

• $\tilde{\alpha}$ (air flow from below gives a positive $\tilde{\alpha}$)

• $\tilde{\beta}$ (air flow from starboard gives a positive $\tilde{\beta}$)

• |v_tas| (the norm of the true airspeed vector).

Fig. 48.B7

(a) Aircraft attitude angles Θ and Ψ in the MONS and air-flow angles α, $\tilde{{\beta}}$ in the ACONS: (b) top view, (c) side view, (d) front view (after [48.99] © American Meteorological Society, used with permission)

The flow angles $\tilde{\alpha}$ and $\tilde{\beta}$ should not be confused with the rotation angles α and β that are commonly used in flight mechanics and wind-tunnel experiments [48.98].

There is a well-defined relationship between the two sets of angles,

$$\cos\tilde{\alpha}=\frac{u_{\mathrm{tas}}}{\sqrt{u_{\mathrm{tas}}^{2}+w_{\mathrm{tas}}^{2}}}=\cos\alpha$$

(48.B12)

and [48.61]

$$\tan\tilde{\beta}=\frac{v_{\mathrm{tas}}}{u_{\mathrm{tas}}}=\frac{\tan\beta}{\cos\alpha}\;.$$

(48.B13)

Using only the measurable angles $\tilde{\alpha}$ and $\tilde{\beta}$ (where $\tilde{\alpha}\equiv\alpha$), the expression for the true airspeed vector equals the well-known definition in [48.99],

$$\boldsymbol{v}_{\mathrm{tas}}=-\frac{\left|\boldsymbol{v_{\mathrm{tas}}}\right|}{D}\begin{pmatrix}1\\ \tan\tilde{\beta}\\ \tan\alpha\end{pmatrix}\;,$$

(48.B14)

where the normalization factor

$$D=\sqrt{1+\tan^{2}\alpha+\tan^{2}\tilde{\beta}}$$

(48.B15)

(see also [48.100, 48.59, 48.61]).

1.1.2 Rotation into the MONS

The attitude of the aircraft can be described by the Eulerian angles Θ (pitch), Φ (bank or roll), and Ψ (yaw, heading, or azimuth; see Fig. 48.B7). Transformation from the ACONS to the MONS is performed via three sequential rotations [48.100, 48.98],

$$\mathbf{T}_{1}(\Phi)=\begin{pmatrix}1&0&0\\ 0&\cos\Phi&-\sin\Phi\\ 0&\sin\Phi&\cos\Phi\end{pmatrix}$$

(48.B16)

(i. e., about the x_f axis),

$$\mathbf{T}_{2}(\Theta)=\begin{pmatrix}\cos\Theta&0&\sin\Theta\\ 0&1&0\\ -\sin\Theta&0&\cos\Theta\end{pmatrix}$$

(48.B17)

(pitching about the y_f axis), and

$$\mathbf{T}_{3}(\Psi)=\begin{pmatrix}\cos\Psi&-\sin\Psi&0\\ \sin\Psi&\cos\Psi&0\\ 0&0&1\end{pmatrix}$$

(48.B18)

(yawing about the z_f axis). The permutation

$$\mathbf{T}_{4}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&-1\end{pmatrix}$$

(48.B19)

is necessary to transform into the MONS, which is defined by the following meteorological wind vector components:

• u, which is positive eastwards

• v, which is positive northwards

• w, which is positive upwards.

Finally, $\mathbf{M}_{\mathrm{mf}}=\mathbf{T}_{4}\cdot\mathbf{T}_{3}\cdot\mathbf{T}_{2}\cdot\mathbf{T}_{1}$ is inserted into (48.7), and the components of the meteorological wind v can then be calculated.

1.2 True Airspeed, Temperature, and Pressure

The true airspeed |v_tas| of the aircraft in relation to the stagnant air causes a significant discrepancy between the total temperature T_tot and the total air pressure p_tot (both located at the tip of the FHP) on the one hand and the intrinsic properties T and p (the static temperature and pressure) of the undisturbed atmosphere on the other. The total pressure at the tip of the FHP is

$$p_{\mathrm{tot}}=p+\Updelta p_{\mathrm{q}}\;,$$

(48.B20)

where Δp_q is the dynamic pressure increment due to the airspeed. This can be determined from (48.B20) by measuring p in a hole in the aircraft fuselage at a location where the flow is parallel to the aircraft skin, similar to a Prandtl (Pitot static) tube.

Energy conservation in a compressible flow,

$$\frac{u_{1}^{2}}{2}+c_{p}T_{1}=\frac{u_{2}^{2}}{2}+c_{p}T_{2}$$

(48.B21)

(where c_p is the specific heat capacity of air, u₁ = |v_tas|, u₂ = 0, T₁ = T, and T₂ = T_tot), leads to the true airspeed as a function of the temperature difference,

$$\left|\boldsymbol{v_{\mathrm{tas}}}\right|^{2}=2c_{p}\left(T_{\mathrm{tot}}-T\right)$$

(48.B22)

(see also [48.100]). Since the intrinsic air temperature T is not measured, T is replaced using the equation for an ideal fluid undergoing an adiabatic process,

$$\frac{T}{T_{\mathrm{tot}}}=\left(\frac{p}{p_{\mathrm{tot}}}\right)^{\kappa}\;,$$

(48.B23)

where the Poisson number

$$\kappa=\frac{R_{\mathrm{L}}}{c_{p}}=\frac{\gamma-1}{\gamma}\;.$$

(48.B24)

Here, $R_{\mathrm{L}}=c_{p}-c_{V}={\mathrm{287}}\,{\mathrm{J{\,}(kg{\,}K)^{-1}}}$ and γ = c_p ∕ c_V, where c_V = 718 J (kg K)⁻¹ is the specific heat at constant volume. Finally, the true airspeed can be obtained without the intrinsic temperature via

$$\left|\boldsymbol{v_{\mathrm{tas}}}\right|^{2}=2c_{p}T_{\mathrm{tot}}\left[1-\left(\frac{p}{p_{\mathrm{tot}}}\right)^{\kappa}\right].$$

(48.B25)

Remaining measurement uncertainties are treated by including calibration and installation coefficients that can only be determined through comparison with a pressure sensor that is not influenced by the flow field around the aircraft, for instance with a trailing cone on a calibration flight [48.101].

1.2.1 Using the Mach Number

Applying the definition of the Mach number (the ratio of the true airspeed to the speed of sound) [48.102, 48.103],

$$\mathrm{Ma}^{2}=\frac{2}{\gamma-1}\left[\left(\frac{p_{\mathrm{tot}}}{p}\right)^{\kappa}-1\right],$$

(48.B26)

to (48.B25) transforms it into

$$\left|\boldsymbol{v_{\mathrm{tas}}}\right|^{2}=\gamma R_{\mathrm{L}}T\mathrm{Ma}^{2}\;,$$

(48.B27)

and applying (48.B26) to (48.B23) transforms it into

$$\frac{T_{\mathrm{tot}}}{T}=1+\frac{\gamma-1}{2}\mathrm{Ma}^{2}\;.$$

(48.B28)

1.2.2 Recovery Factor

In practice, the temperature cannot be measured at the same location as the true airspeed vector v_tas (located at the stagnation point at the tip of the FHP). Also, the heat produced by the dynamic pressure increment immediately starts to diffuse into the surrounding air. The temperature T_m measured somewhere in the vicinity of the FHP is smaller than T_tot but still larger than the intrinsic temperature T. The ratio [48.103, 48.104, 48.99]

$$r_{\mathrm{c}}=\frac{T_{\mathrm{m}}-T}{T_{\mathrm{tot}}-T}\leq 1$$

(48.B29)

is referred to as the recovery factor. It depends on the sensor and is usually a function of the true airspeed (but not the air moisture). Inserting (48.B29) into (48.B23) gives the static (intrinsic) temperature of the air,

$$T=\frac{T_{\mathrm{m}}}{r_{\mathrm{c}}\left[\left(\frac{p_{\mathrm{tot}}}{p_{\mathrm{s}}}\right)^{\kappa}-1\right]+1}$$

(48.B30)

(in contrast to [48.99, equation 15]). It is also advisable to apply the recovery factor to the true airspeed calculation in (48.B27), yielding

$$\left|\boldsymbol{v_{\mathrm{tas}}}\right|^{2}=\frac{2c_{p}T_{\mathrm{m}}}{r_{\mathrm{c}}+\frac{2}{\mathrm{Ma}^{2}(\gamma-1)}}$$

(48.B31)

(unlike [48.99, equation 16]).

If the recovery factor r_c is not known (i. e., not provided by the manufacturer of the sensor), the pressure p_T at the location of the temperature probe has to be measured using an additional pressure probe. The static temperature is then calculated using (48.B23)

$$T=T_{\mathrm{m}}\left(\frac{p_{\mathrm{s}}}{p_{T}}\right)^{\kappa}\;.$$

(48.B32)

This method was applied in the fast temperature probe of the Helipod [48.105]. The true airspeed is then calculated by inserting (48.B32) into (48.B27).

1.3 Flow Angles

The following description addresses the application of typical FHPs, which (in addition to the static pressure) only measure differential pressures [48.106]. Air flow systems involving more or less than five holes [48.104, 48.107, 48.108] or a single turning sensor [48.109] for differential pressure measurement are treated in a very similar way. In larger probes (e. g., pressure holes in the aircraft fuselage, as used in the Space Shuttle and the F-18 High Angle of Attack Research Vehicle), it is possible to measure individual absolute pressures and determine flow angles even more accurately [48.110]. The description of outdated systems such as vanes [48.102, 48.103, 48.111, 48.2] exceeds the limits of this analysis.

The local wind vector in the ACONS is determined from the dynamic pressure increment Δp_q and the pressure differences between opposite pressure holes in the FHP (i. e., the pressure differences in the horizontal plane Δp_β = P2 − P4 and the vertical plane Δp_α = P1 − P3, where p_j denotes the individual holes of the FHP and the central hole is denoted P0; see Fig. 48.2). The pressure differences Δp_α and Δp_β increase as the flow angles α and $\tilde{\beta}$ increase, but they also depend on the airspeed (and therefore on the dynamic pressure increment Δp_q and the Mach number) and the air density ρ (and therefore the altitude z). In general, this can be expressed via

$$\varphi=F\left(\Updelta p_{\varphi},\Updelta p_{\mathrm{q}},z\right)\quad\text{where }\varphi=\alpha,\tilde{\beta}\;.$$

(48.B33)

Here, F denotes a functional relationship.

Usually, the influences of the airspeed and the altitude can be considered by weighting the pressure difference by the dynamic pressure increment. The simplest assumption is therefore

$$\varphi=\frac{1}{K_{\mathrm{FHP}}}\ \frac{\Updelta p_{\varphi}}{\Updelta p_{\mathrm{q}}}\;,$$

(48.B34)

where the calibration coefficient κ_FHP considers any disturbance of the airstream by the FHP (and also by the entire aircraft fuselage) and local stream effects directly at the pressure hole.

It is difficult to determine the dynamic pressure increment Δp_φ and the total pressure p_tot (48.B20) since the stagnation point is usually located somewhere between the holes of the FHP, so it cannot be measured directly. Approximating the total pressure by the pressure p₅ measured at the central hole of the FHP would lead to a wind vector measurement that is very sensitive to the aircraft attitude, wind speed, and wind direction (see also [48.109]).

It is understood that any offset angles α₀ or β₀ due to either bias in the pressure transducers or asymmetry of the FHP must be quantified in a laboratory, a wind tunnel, or in flight tests beforehand. Calibration routines for both wind-tunnel experiments and flight maneuvers can be found in the literature [48.101, 48.102, 48.103, 48.110, 48.112, 48.113, 48.114, 48.59, 48.60].

The well-known FHP made by Goodrich Sensor Systems (formerly Rosemount) provides an additional pressure difference

$$\Updelta p_{\mathrm{ref}}=\mathrm{P0}-\mathrm{P2}$$

(48.B35)

between one of the horizontal holes and the central hole. The dynamic pressure increment is then estimated using

$$\Updelta p_{\mathrm{q}}\approx\Updelta p_{\mathrm{ref}}+\frac{1}{2}\Updelta p_{\beta}\;,$$

(48.B36)

and the flow angles are determined via (48.B34) with K_FHP set to 0.088 for airspeeds below 0.6 Ma [48.115].

It should be noted that Δp_ref refers only to the horizontal plane (i. e., the stagnation point is assumed to be located somewhere along the line connecting the two opposite holes #2 and #4).

An improvement on the Rosemount method requires an additional differential pressure measurement between the central hole and one of the holes in the vertical plane (#1 or #3), resulting in two disjunct equations of the same type as (48.B34),

$$\begin{aligned}\alpha & =\frac{1}{K_{\mathrm{FHP},\alpha}}\frac{\Updelta p_{\alpha}}{\Updelta p_{\mathrm{ref},\alpha}+\frac{1}{2}\Updelta p_{\alpha}}\end{aligned}$$

(48.B37)

$$\begin{aligned}\tilde{\beta} & =\frac{1}{K_{\mathrm{FHP},\beta}}\frac{\Updelta p_{\beta}}{\Updelta p_{\mathrm{ref},\beta}+\frac{1}{2}\Updelta p_{\beta}}\;.\end{aligned}$$

(48.B38)

It is obvious that this method represents only a slight rather than a fundamental improvement on the usual Rosemount method, since no consistent dynamic pressure increment can be determined. This is mainly due to the general strategy of using a Cartesian approach to solve a rotationally symmetric problem. More sophisticated and complex methods have also been published [48.60, 48.61, 48.62].