a. Wind tunnel and experimental setup

The wind tunnel consists of a horizontal Plexiglas tube that is 2 m long and 0.2 m in diameter. A sketch of the setup is shown in Fig. 1, in which the flow direction is from right to left. The ratio between tube length l and tube diameter d that is necessary for fully developed turbulence can be approximated by l/d ≈ (A)Re^B, with the flow Reynolds number Re = ud/ν and two constants A = 0.6 and B = 0.25 for turbulent flows (Bohl 1998). Here u is the mean flow velocity, ν = μ/ρ_a = 1.5 × 10⁻⁵ m² s⁻¹ is the kinematic viscosity of air, μ is the dynamic viscosity of air (1.81 × 10⁻⁵ kg m⁻¹ s⁻¹), ρ_a is the air density, and d is the tube diameter. With u = 10 m s⁻¹, it follows that Re ∼ 10⁵ and (A)Re^B ∼ 11. Therefore, l₂/d ∼ 8 and the conditions for fully developed turbulence are nearly met. This conclusion is also supported by power spectral analysis (discussed in Fig. 8).

The spray in the wind tunnel is created by an injector nozzle. The basic operating principle and location of this nozzle is shown in Fig. 1. The injector nozzle is designed to create a spray with a median droplet diameter of 10 μm. Purified water was used to protect the wire of the HWA from any further material remaining at the surface. The tube to the water reservoir can be closed by a manual valve and since no additional compressed air passes through this tube, this valve has no influence on the flow speed. Thus, identical flow situations with or without spray can be established. To increase the flow speed, two additional nozzles consisting of 1/4-in. tubes are placed side by side with the mixing nozzle. The maximum pressure for the nozzles is 5 bars each. For the 1/4-in. tubes the pressure can be roughly adjusted to vary the flow speed. However, it was found that stable flow conditions exist only for completely opened valves; that is, experiments with three different stable flows could be performed. The three different flow cases are denoted in the text by subscripts describing the valve settings (cf. Table 1).

b. Hot-wire anemometer

Turbulence was measured with a one-dimensional HWA (Dantec Dynamics A/S, Denmark, type 54T30) that was located about l₂ = 160 cm behind the inlet of the wind tunnel (cf. Fig. 1). The HWA system is a constant temperature anemometer with a platinum-plated tungsten wire (type 55P01) with a diameter of 5 μm and an overall length of around 3 mm, whereas the sensing part is only 1.25 mm long. Essentially, an HWA measures the convective heat transfer between the heated wire and the flow that is described by the Nusselt number Nu = C + (D)Re^1/2 (King’s law) with the Reynolds number Re = ud_HW/ν (Goldstein 1996; Bruun 1995). Here, C and D are empirical constants and d_HW is the diameter of the hot wire. The electrical power input P is ∼V²_p ∼ Nu; therefore, the output voltage V_p of the HWA is ∼Re^1/4 ∼ u^1/4. The bandwidth of the system is 10 kHz.

To calibrate the HWA over the range of the three different flow speeds, a few measurements for varying speeds were made simultaneously with a Pitot tube placed slightly behind the hot wire, so as not to disturb the turbulence measurements. The sampling frequency f_s was set at 1 kHz for all devices. A correlation plot of the flow velocity u_Pitot measured with the Pitot tube and the HWA output V_HW is shown in Fig. 2. To reduce the scatter, the hot-wire signal was averaged over nonoverlapping flow velocity ranges of 1 m s⁻¹. A fit with a third-order polynomial was made to estimate the calibration coefficients.

c. Characterization of the spray

To study the effects of droplets on the HWA signal it was necessary to characterize the spray under the three flow conditions. A particle volume monitor (PVM-100A) was placed at the end of the wind tunnel to measure the liquid water content (LWC) of the spray (cf. Fig. 1). An internal low-pass filter for antialiasing was set at around 100 Hz. Details of this optical probe can be found in Gerber et al. (1994).

The droplet size distribution ΔN/Δd_d is derived from measurements performed with an M-Fast Forward Scattering Spectrometer Probe (FSSP; Schmidt et al. 2004), which was located at the end of the tube (instead of the PVM). The integral over ΔN/Δd_d gives the droplet number concentration N. Figure 3 depicts the size distributions for the three different flow states. For the two situations with the highest flow velocities (110 and 111), both distributions have nearly the same shape with a mean diameter around 9 μm and a long tail toward large droplet sizes. The ratio of both distributions and number concentrations (∼2) is similar to the ratio of the flow velocities. For the third case with low flow velocities, the number concentration reaches very high values and the M-Fast FSSP is near saturation. Thus, the M-Fast FSSP measurements for this case should be interpreted with caution.

d. Influence of droplet–hot-wire collisions on the velocity signal

When a droplet impacts the hot wire, the system reacts so as to evaporate the water and maintain the wire at a constant temperature. As a result a sudden spike in the voltage is observed. This spike must be distinguished from voltage variations due to turbulent velocity fluctuations. Essentially, the result is a continuously sampled signal interrupted by random spikes. The usefulness of the velocity signal is inversely related to the fraction of signal lost because of droplet–hot-wire collisions. In the following we attempt to estimate that fraction.

1) Impact rate of droplets on the hot wire

The particle Stokes number for a 10-μm droplet interacting with a hot wire is

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where ρ_w is the density of liquid water. This Stokes number is much higher than the critical Stokes number required for impaction (St ∼ 0.3); therefore, we can safely use the geometric cross section in determining the droplet–hot-wire collision rate.

The hot wire has a cross section σ for a typical cloud droplet with an average diameter of d_d = 10 μm of σ = L(d_HW + d_d) ≈ 1.25 mm × (5 μm + 10 μm) ≈ 1.9 × 10⁻⁸ m². With a droplet number concentration of N = 675 cm⁻³, the mean free path is 1/(Nσ) ≈ 8 cm. Furthermore, with a mean flow velocity of u ≈ 10 m s^–1 we obtain an average arrival time of τ = 1/(Nσu) ≈ 8 ms. In fact, if the droplets are randomly distributed in the turbulent flow, we expect that the probability density function for the (random) droplet interarrival time t is exponential, p(t) = τ⁻¹ exp(−t/τ). This is observed to be a reasonable approximation, as illustrated in Fig. 4, which shows the probability density function (PDF) of interarrival times under flow conditions 110. As a result of the skewness of the exponential function, we expect there will be a significant number of velocity data segments that remain uninterrupted for times several times larger than τ (e.g., see Fig. 4 where it can be seen that approximately 1% of the interarrival times are ≥5τ.). Thus, the natural tendency of random droplet arrivals to be clustered improves the ability to extract meaningful data.

2) Duration of a droplet impact in the hot-wire signal

If we wish to know the fraction of useable signal we must estimate the typical duration of a voltage spike in the hot-wire signal due to a droplet impact. We proceed by considering the time required for a spherical droplet to evaporate after coming into contact with the hot wire. Of course, after a collision the droplet may no longer be a sphere, but we expect the analysis to give a satisfactory estimate for the purpose of a scale analysis.

The ratio of energy required to evaporate a droplet with radius r_d and the energy required to increase the droplet temperature by approximately ΔT = 80 K (e.g., from a laboratory temperature around 20°C to T_w = 100°C) is α = L_υ/(c_pΔT) = 2.2 × 10⁶ J kg⁻¹ (4200 J kg⁻¹ K⁻¹ × 80 K) ⁻¹ ≈ 7, where L_υ is the latent heat of vaporization for water and c_p is the specific heat capacity at constant pressure for water. Therefore, as a first-order approximation we consider only the evaporation time scale of a droplet with a uniform, constant temperature of 100°C. The evaporation of a droplet is described by (e.g., Pruppacher and Klett 1997)

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with the diffusivity for water vapor D_υ = 0.211(T/T₀)^1.94(p₀/p) × 10⁻⁴ ≈ 3.6 × 10⁻⁵ m² s⁻¹ (with T₀ = 273 K, T = 373 K, and p₀/p ≈ 1), the universal gas constant R = 8.31 J mol⁻¹ K⁻¹, the molecular weight of water M_w = 18 × 10⁻³ kg mol⁻¹, the saturation water vapor pressure at droplet temperature E(T_W = 100°C) ≈ 10⁵ Pa, and the ambient water vapor pressure e_∞ at ambient temperature T_∞. The ambient temperature is much lower than the droplet temperature and, therefore, e_∞ (T_∞) ∼ E(T_w), such that the second term in the brackets of Eq. (2) is negligible. Since the right-hand side of Eq. (2) is constant {C = [M_wD_υ/ρ_wR][E(T_w)/T_w] ≈ 2.1 × 10⁻⁸ m² s⁻¹}, the integration reveals the time T necessary for the complete evaporation of the droplet of initial radius γ_d:

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Because of the large relative speeds between the air and the hot wire, the evaporation time is slightly smaller because of “ventilation,” or advection of the vapor field away from the droplet by the flow. The Reynolds number–dependent ventilation coefficient (e.g., Pruppacher and Klett 1997) does not exceed 1.5 for our flow conditions. For droplet sizes commonly observed in this experiment (e.g., 1–10 μm) our analysis suggests that evaporation times are approximately in the range 1 ≤ T ≤ 10 ms. Actual voltage spikes resulting from droplet impact are observed to be on the order of 0.5 ms (cf. Fig. 6). We speculate that the difference is due to enhanced evaporation due to the increase in surface area when droplets wet the hot wire. Nevertheless, the analysis provides an upper bound for the duration of voltage spikes resulting from droplet–hot-wire collisions.

This section may be summarized by specifying the fraction of time during which the velocity signal is influenced by droplet evaporation. The fraction can be estimated as the ratio of the droplet evaporation time T and the mean interarrival time τ:

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From Eq. (4) it can be seen that for a constant u, ϕ varies approximately as Nd³_d, which is proportional to the liquid water content. It follows, therefore, that in order to maximize the fraction of useful turbulent velocity signal from a given hot-wire system, the product LWC × u should be minimized.

e. Hot-wire measurements and despiking method

To describe and test the despiking method, we choose flow and cloud conditions similar to those that would likely be encountered with low-true-airspeed cloud research platforms (see section 3). Specifically, we begin with flow conditions for state 110, with a mean flow of u = 9 m s⁻¹ and a mean LWC = 0.8 g m⁻³. The complete time series is 30 s long. The first 12 s were without droplets; for the last 18 s the spray was added. Figures 5a–c show the time series of the calibrated hot-wire signal u_HW, the LWC, and the first time derivative a_HW = ∂_tu_HW, respectively. Two subsequences labeled “Part I” and “Part II” are used for further comparison of drop-free and cloudy conditions. The spikes of the impacting droplets are most evident in the time series of a_HW (cf. Fig. 5c).

A typical signal of a droplet spike (cf. spike in the dashed box in Fig. 5c) is enlarged in Fig. 6. The signal has a duration of approximately 0.5 ms (five samples); the amplitude change of Δu = 20 m s⁻¹ corresponds to an acceleration of nearly 10⁵ m s⁻², which is about one order of magnitude higher than the maximum values observed for the drop-free flow. Therefore, the spikes can be clearly distinguished from natural turbulence and a threshold can be defined. The signal during the droplet impact is then linearly interpolated for a duration of five samples (cf. dashed line in Fig. 6).

To derive a criterion for the threshold, PDFs of u_HW and a_HW are calculated and depicted in a semilogarithmic plot in Fig. 7. For smaller values, u_HW can be well approximated with a Gaussian fit whereas the wings of a_HW are closer to an exponential curve, in agreement with theory (Frisch 1995). For positive velocities larger than around 6 m s⁻¹ an increased probability of u_HW is found because of the spikes. For the derivative a_HW this effect is symmetric (cf. Fig. 6), and the influence of the spikes appears as a sharp change in the slope, which is clarified by the dashed lines. This slope change is used to define the threshold ±a_t for the despiking algorithm. For this case, a_t = 1.5 × 10⁴ m s⁻² is chosen.

To illustrate the effectiveness of this simple despiking algorithm, power spectra F(kη) are calculated as a function of the nondimensional parameter kη and shown in Fig. 8, where k is the wavenumber and η is the Kolmogorov microscale. The spectra are calculated for Part I (drop free) and Part II (raw signal and despiked signal) individually. In general all three spectra follow a −5/3 slope and drop off at around kη = 0.05, where dissipation becomes effective. However, it can be clearly seen that the spectrum of the data including the spikes drops off much less rapidly than for the drop-free case (Part I). This effect is significantly decreased by the despiking algorithm. For kη < 0.03, there is no difference between the spectra including drops and the despiked spectra.

f. Hot-wire measurements for three different flow states

To confirm the utility of the despiking algorithm over a broad range of cloud conditions, it is tested for the two additional wind tunnel flow states described in section 2a. The mean parameters for all three flow cases are summarized in Table 1. For this investigation the setup was slightly changed, resulting in 20% increased mean flow velocities compared with the calibration (cf. Fig. 2). With the total number of detected spikes N_tot and the total integration time T_i, the droplet number concentration N_HW = N_totσuT_i can be estimated and compared with the concentration derived from FSSP data (N_FSSP). The results are also included in Table 1. For the two cases with higher flow rates (110 and 111), the N_HW is about 20%–25% lower than the concentration observed from FSSP measurements. There are two possible explanations for this disagreement. First, the collision efficiency between the hot wire and droplets is smaller than one and, second, the signal of small droplets is too low and therefore is interpreted as natural turbulence. The latter argument is supported by the fact that the spectrum of the despiked data lies above the spectrum of the drop-free data. For the low flow case (010), the hot wire “detected” 5 times more droplets than counted with the M-Fast FSSP. In this case, the definition of the threshold was difficult and also high accelerations in the drop-free part were interpreted as droplet spikes. Here, the despiking algorithm reduces natural variance and consequently the spectrum of the despiked data is below the spectrum of the drop-free data over a wide frequency range. On the one hand, a lot of droplet spikes would remain in the time series when increasing the threshold; therefore, a compromise had to be found. On the other hand, as described in section 2c, the M-Fast FSSP was near saturation and coincidence counts led to an underestimation of N_FSSP.

The ratio ϕ′ in Table 1 is equal to Eq. (4), but the factor N d²_d(d_HW + d_d) is approximated by (LWC) (4/3 πρ_w)⁻¹. The direct measured ratio ϕ_m = N_totT_m/T_i is the ratio between the total time affected by droplet strikes and the integration time with T_m = 0.5 ms as the mean observed evaporation time (e.g., the observed length of the voltage spikes due to droplet impact).

In Fig. 9, the power spectral densities (PSDs) F(kη) (hereafter called “spectra”) for all three datasets are shown as a function of kη. The spectra of the three cases are vertically shifted for clearer presentation. For the high flow case (111), nearly no differences can be found between the spectrum of the data including droplets and the despiked data. The spectrum follows a −5/3 slope for inertial subrange behavior over a range of nearly 2 decades and starts to drop off at kη = 10⁻¹ because of the influence of dissipation. Most obvious is the effect of despiking for the low flow case (010) where the spectrum of the droplet-free part drops off significantly at kη = 10⁻¹, whereas the spectrum of the data including droplets is contaminated at dissipation scales. However, for this case all 3 spectra exhibit much more scatter than observed for the other 2 cases, and the slope in the inertial subrange is slightly steeper than −5/3. The spectra of the despiked and drop-free data agree at least qualitatively.

The first 4 statistical moments are calculated for the hot-wire signal and the central difference ΔHW = (HW_i+1 − HW_i−1)/2 before and after the despiking algorithm was applied. All moments are summarized for all three datasets in Table 2. The first moment (e.g., the mean value μ) is not influenced by droplet impacts. The mean values differ slightly from the values given in Table 1 since the moments are estimated from the second part (including droplets) of the record only. For higher moments the influence of the despiking algorithm becomes more obvious. In particular, for the kurtosis of the hot-wire signal of set “010,” the despiking leads to a reduction from K ≈ 8 to K ≈ 3, which is the theoretical value for a Gaussian distribution, which illustrates the efficiency of the despiking algorithm.

Figure 10 shows the time series of the despiked data and the removed spikes (shifted for a better presentation). For set 010, the despiked data still include a few spikes due to coincidences. If the time between two spikes is too short, the linear interpolation fails and ends within the following spike. Since the LWC and the droplet number concentration for set 010 are quite unrealistic for most atmospheric cloud types, we use the despiking algorithm without taking coincidence into account. Compared to Fig. 5 for set “110,” the onset of the spray can hardly be detected in the despiked data in Fig. 10.