## Abstract

The ultrafast aircraft thermometer, built for measuring temperature in clouds at flight speeds up to 100 m s^{−1}, employs a 2.5-*μ*m-thick platinum-coated tungsten wire as a sensing element. When temperature increases, the wire resistance increases. The changes are amplified by an electronic system. Temperature measurements made in a wind tunnel and during flights show noise that is related to the von Kármán vortex street generated behind the shield that protects the sensing element against the impact of cloud droplets. To reduce both the level of turbulence and the amount of water collected on the shield, suction is applied through the slits in its sides. The effect of suction on the flow field is twofold. First, at the Reynolds numbers that the thermometer is operated suction eliminates aerodynamic disturbances. Second, suction diverts the inner part of the boundary layer into the slit. This inner part is a region of strong shear and, therefore, a region where intensive viscous heating takes place. When the suction is *on* much of the air that is heated in the boundary layer in the front part of the shield is removed through the slits and never reaches the sensor. To study the role of the shield with suction and *confirm its chosen shape*, two-dimensional (2D) direct numerical simulations (DNSs) are performed of the airflow and of the trajectories of droplets of various sizes and initial positions. The influence on the temperature distribution of the irreversible dissipation of energy due to air viscosity is also examined. This is found to have a small but measurable effect. The effects associated with sampling and processing of the analog signal obtained from the sensing wire are discussed. The results herein quantitatively explain the nature of the measured aerodynamic noise.

## 1. Introduction

The ultrafast aircraft resistance thermometer constructed in the Institute of Geophysics, Warsaw University, has been improved during the last few years (Haman 1992; Haman et al. 1997, 2001). The sensing element of the resistance thermometer is a 2.5-*μ*m-thick platinum-coated tungsten wire (Fig. 1). When the temperature increases, the resistance of the metal wire increases too. Low heat capacity of the wire ensures fast response to temperature variations in the small-scale structures of cloud turbulence. An electronic system precisely monitors this resistance and provides temperature measurements. The device measures the temperature of cloudless air and of warm clouds with a 10-kHz sampling frequency. During a flight at the speed of 100 m s^{−1} this corresponds to a spatial resolution of the order of 1 cm. The streamlined shield protects the sensing wire against cloud droplets. The shield may act in a passive or in an active mode. In the second case, the shield sucks air and water collected on its surface through the suction slits. In both cases the shield introduces in the measurement area aerodynamic disturbances that are interpreted by Haman et al. (2001) as vortex shedding from the shield. The thermometer records the resulting fluctuations of temperature as noise (Fig. 2). When suction is turned *on* the level of noise is considerably smaller than in the passive mode. Moreover, the average temperature is 0.3 K higher due to the increased average pressure in the wake. Recently, the temperature recordings made in a low-turbulence wind tunnel with an airspeed of 80 m s^{−1} showed temperature fluctuations with an average amplitude of ±0.2 K with a few kilohertz frequency corresponding to the scale of turbulence of the order of centimeters (Haman et al. 2001).

The ultrafast aircraft resistance thermometer is one of the few devices specially designed for the airborne atmospheric turbulence studies. Among them is the thermometer and hot-wire anemometer mounted on a radio-controlled aircraft model flying at speeds of about 30 m s^{−1}, which had the averaging scale of about 10 cm (Kukharets and Tsvang 1998). Another airborne ultrasonic anemometer–thermometer recorded with 1-kHz frequency at speeds up to 110 m s^{−1}, which also gave spatial resolution of the order of 10 cm (Cruette et al. 2000).

The designers of the fast thermometers aim at measurements with millimeter spatial resolution. Such an improvement will be possible with new designs of both the electronics and the shield of the thermometer. To design a shield that will allow low noise measurements at frequencies higher than 10 kHz, we have to understand the nature of the observed aerodynamic disturbances. In this study we analyze turbulence in the neighborhood of the sensing element. To study the role of the shield with suction we perform 2D direct numerical simulations (DNSs). We present the results of the numerical simulations of the flow around the shield with and without suction. The possibilities of direct visualization of the small density fluctuations in the atmosphere and, in particular, in the wake of the thermometer’s shield, have also been investigated (Rosa et al. 2004).

The dynamics of the clear airflow around the shield is simulated using FEATFLOW, a finite-element software for the incompressible Navier–Stokes equations, developed by S. Turek of the University of Heidelberg (information online at www.featflow.de). The package solves the Navier–Stokes equations in velocity–pressure variables. To calculate the temperature changes we first make an adiabatic approximation expressing them in terms of pressure fluctuations, and then we calculate the additional effect of heating through the irreversible energy dissipation due to viscosity.

It should be stressed that all of our computations are two-dimensional, while 3D features are known to be present in the considered range of Reynolds numbers. However, here we focus on the dominant effect of suction on the von Kármán vortices or their “seeds,” that is, the weak instabilities in the elongated ‘formation region’ (Williamson 1996). Our 2D computations show that those are suppressesed by suction, and this is likely to persist when the 2D constraint is relaxed and the flow is fully three-dimensional.

The paper is organized as follows. In section 2 we describe the parameters used in the FEATFLOW simulations. Section 3 contains the results of the simulations and the analysis of the effect of viscous heating. In section 4 we analyze the frequency of the aerodynamic noise, while in section 5 we explain the presence of several different frequencies in the noise spectrum. In section 6 we study the dependence of the temperature measurements on the velocity of the aircraft. In section 7 we show the computations of the motion of water droplets in the simulated flow around the thermometer shield with and without suction. Finally, in section 8 we offer some conclusions.

## 2. Simulations of the small-scale turbulence in the airflow

The existence of the small-scale turbulence generated by the streamlined shield of the thermometer’s sensing element is illustrated by the plots of the noisy temperature measurements (Fig. 2). To understand the source of the measured noise we perform 2D simulations of the flow of clear air in a small region around the shield. The FEATFLOW software is an efficient tool for modeling turbulent airflow that is governed by the incompressible Navier–Stokes equations. The assumption of incompressibility is justified for the flow with velocity well below Mach 1. The virtue of FEATFLOW lies in the possibility of a flexible definition of the locally anisotropic spatial meshes on which the discrete velocity and pressure values are calculated.

### a. Boundary conditions

The boundary conditions listed in Table 1 describe the details of the clear airflow in the neighborhood of the thermometer’s sensing element located behind the shield that is to be optimized. The symbols used in Table 1 are explained in Fig. 3. At the inlet we assume a uniform flow with the airspeed of 80 m s^{−1}. In the simulations we analyze two cases that were tested in the experiments (Haman et al. 2001). When the suction is turned *off*, both normal and tangent components of the air velocity are equal to zero on the whole surface of the shield (and so are the *u* and *υ* components in the *x* and *y* directions, respectively). The outlet pressure drop, as shown in Table 1, also vanishes on three outlet borders Γ_{out}. When the suction is turned on we apply a constant static pressure drop of 10^{4} Pa at the suction slits of the shield. It is the only boundary condition that distinguishes the two considered cases.

### b. Mesh generation

The simulations of the small-scale turbulence have been done on several meshes, some examples of which are shown in Fig. 4. The mesh generation begins with an approximation of the shape of the streamlined shield with linear functions. In the subsequent trials, the cells of an initial coarse mesh of certain geometry are subdivided to increase the spatial accuracy of the simulations. The subdivision is automatically continued until the estimated accuracy reaches a desired level.

The grid elements are quadrilateral. The pressure in each element is constant, while the velocity is approximated by quadratic functions of the coordinates. The nodes of the velocity field are located at the midpoints of the elements’ edges.

### c. Numerical method

The multigrid projection method solves the linear systems that are obtained by applying a finite-element approximation to the Navier–Stokes equations within an accuracy that depends on space and time discretization. In our case, the method guarantees high efficiency and robustness at high Reynolds numbers (Turek 1997, 1999). The time derivatives are discretized using a Crank–Nicolson algorithm with an adaptive time step in the range (*t*_{min} = 0.001 ms; *t*_{max} = 0.005 ms). Several tests made with various time steps, proved that our fully time-dependent problem is solved satisfactorily. According to Turek (1997), for a nonstationary flow the discrete projection method is considerably faster than the coupled solution technique.

## 3. Results of the simulations of the small-scale turbulence

### a. Pressure and vorticity distributions in the absence of suction

Four snapshots of time evolution of the small-scale turbulence shown in Fig. 5 are simulated on a fine mesh generated from the coarse grid shown in Fig. 4a. The pressure distribution shown in Fig. 5a is quantized into 25 levels. The isolines of the streamfunction, that is, the streamlines, show the instantaneous direction of the flow (Fig. 5b).

The pressure fluctuations in the wake are related to the distribution of vorticity generated in the boundary layer of the shield and shed from it. In Fig. 6 we show a grayscale plot of the instantaneous distribution of vorticity (for the color version of this plot, see information online at www.igf.fuw.edu.pl/Rosa/featflow.htm). At the Reynolds number of 4900, the wake at the location of the sensing wire is only weakly unstable, so the von Kármán vortex street is not well developed there. The sensing wire is within the *formation region*, while the fully developed von Kármán street may be expected to appear farther downstream, outside of the simulation region (Williamson 1996). However, the traces of individual vortices can be noticed. They are responsible for the temperature fluctuations that are recorded by the sensor that detects the pressure drops in the cores of the passing vortices. It is worth understanding the structure and dynamics of the wake vortices because they can, in principle, be identified not only by the characteristic frequencies associated with the advection of the entire street (essentially determined by the Strouhal number), but, also, when the measurement techniques are refined, by the patterns of the pressure depressions in individual events.

In Fig. 7 we show the pressure fluctuations computed on two initial grids with different resolutions. To illustrate the differences between the meshes we plot the results on the type-A mesh (used in most simulations) and the type-E mesh that show the biggest discrepancies. The simulations reveal that modeling the temporal pressure variations, when suction is off, is sensitive to the details of the computational grid. In general, the magnitude of the fluctuations is in the same range as that inferred from the temperature measurements. However, its exact value depends on the type of mesh. Understanding the geometrical characteristics of the wake flow helps to decide which type of mesh is appropriate in this case.

### b. Pressure distribution and velocity field when suction is on

The primary purpose of applying suction through the slits in the sides of the shield was to remove a water film forming on the solid surface when the shield is hit by cloud droplets. Otherwise, the film would spread to the rear stagnation point, where new droplets would form, detach under the action of a local strain and, possibly, hit the sensing element. Those secondary droplets are less frequent than the primary ones, many of which are simply deflected by the shield, but nevertheless must be avoided. Their formation is stopped, or at least greatly reduced, when the oncoming water film is sucked in through the slits.

Applying suction has another important effect that significantly improves the operation of the thermometer. The convective instabilities that may start to develop in the boundary layer upstream of the slits are sucked in. Our simulations show that suction stabilizes the downstream part of the boundary layer and also the wake. The noise-reduction effect of suction is prominent in the wind tunnel temperature measurements. This is partly a result of the suppression of vortex shedding and the associated pressure fluctuations. Later we will discuss another mechanism that is different from the modifications of the pressure field, through which suction directly affects the measured temperature.

In Fig. 8 we show the pressure and the velocity fields when the free-stream airspeed equals 80 m s^{−1}. In the vicinity of the slits suction accelerates the fluid to 120 m s^{−1}. An apparent small asymmetry is due to the approximation of the shield’s outline by a piecewise linear shape.

In Fig. 9 we show the pressure at the location of the sensor as a function of time for different meshes. The suction is turned *on* and it stabilizes the flow. The pressure fluctuations are greatly reduced, and so are the associated temperature fluctuations (to be discussed later), which is in agreement with the experimental observations. For *t* > 0.34 ms, the differences between the meshes are negligible.

There seems to be little systematic difference between computations on different grids. In the following we present mostly the results obtained on type-A meshes.

### c. Adiabatic temperature changes

Our numerical solutions of the Navier–Stokes equation yield the distribution of pressure in the vicinity of the sensing element. When the suction is off, the pressure behind the shield shows sustained periodical variations with amplitude of about 400 Pa (Fig. 7). When the suction is turned on, the flow, after an initial transient lasting about 0.34 ms, stabilizes, and no fluctuations are observed. To compare our numerical simulations with the temperature measurements of Haman et al. (2001) we convert our computed pressure distributions into temperature. As a first approximation we assume the adiabatic relation.

This is obtained by solving the energy equation. In the absence of irreversible processes the temperature equation

where *p* is air pressure, *T* is air temperature, *t* is time, *C _{p}* ≈ 1005.5 J kg

^{−1}K

^{−1}is the specific heat of air at constant pressure,

*ρ*is air density,

*p*=

*ρRT*,

*R*= 287 J kg

^{−1}K

^{−1}is the gas constant for dry air, and

*D*/

*Dt*≡ ∂/∂

*t*+

**v**· ∇ is the material derivative with velocity vector

**v**= (

*u, υ*), gives the adiabatic relation

where Δ*T* is the temperature variation of the small-scale structure of air turbulence resulting from the change of pressure Δ*p* and air density *ρ* = 1.17 kg m^{−3} at a wind tunnel temperature of 302 K. Therefore, the pressure fluctuations of the order of Δ*p* = ±200 Pa correspond, in the adiabatic approximation, to the temperature variations with amplitude of about Δ*T* = ±0.17 K.

The temperature fluctuations so calculated qualitatively agree with the experimental data shown in Fig. 2. In the case without suction, the obtained amplitude of the temperature variations ±0.17 K is in agreement with the wind tunnel measurements. With the suction turned on, the simulations faithfully reproduce the suppression of the fluctuations, although the experimental data still exhibit some residual noise at the 0.1-K level. We can then see that, even in the adiabatic approximation, we always obtain qualitative, and without suction even quantitative, agreement between the numerical simulations and the wind tunnel experiments. Nevertheless, we will now discuss in greater detail the magnitude and role of the energy dissipation due to viscosity.

### d. Validity of the adiabatic assumption

In the regions of strong shear, that is, in a boundary layer on the shield’s surface, viscous heating becomes significant. Its rate is equal to

where *e _{ij}* is the strain tensor

and *μ* = 1.47 × 10^{−5} kg m^{−1} s^{−1} is the dynamic viscosity of air. In Fig. 10 we show the grayscale plots of log(Φ). The range of Φ is so large that the logarithmic scale is necessary.

The total amount of heat released in a fluid element reaching the sensor is given by the Lagrangian integral over the path line of the element ∫^{θ}_{0} Φ *dt*, where *θ* is the time that the element spends in the region of strong shear. We have computed this integral for three selected particles of fluid, and the path lines are shown in Fig. 11. The path equations,

were integrated along with the time stepping of the Navier–Stokes equation by a subroutine added to the FEATFLOW code.

Figure 11 shows three particle path lines. When the suction is turned off (Fig. 11a) the flow is unsteady and the fluid particles approach the sensor along various paths. The path lines depend on the time when the element arrived at the front of the shield. Two such paths are shown in Fig. 11a. They must be computed simultaneously with the time stepping of the Navier–Stokes solver by a built-in additional subroutine. The accuracy is, again, quite important, although the error of the total heat release calculation is much harder to estimate, because it involves computing statistics over an ensemble of particle paths.

When the suction is on (Fig. 11b) the flow is steady and, therefore, there is only one trajectory reaching the sensor (except for some numerical imperfections the flow is top-down symmetric, so the mirror image of this trajectory also hits the sensor). Every fluid element that passes the sensor, no matter at what time it arrived at the front of the shield, must have followed this path. Therefore, under the steady flow assumption, the cumulative effect of viscous heating can be computed by integrating the energy equation along this single path. This is justified when we can neglect the diffusive heat exchange with the neighboring fluid elements. The path can, in principle, be calculated by integrating Eq. (4) backward in time with a precomputed, steady velocity field obtained from the Navier–Stokes solver. We calculate it by iterating the forward-in-time integrations that give better accuracy.

The accuracy of the path computation is important because the viscous heating has a very steep gradient in the boundary layer, so even a small deflection of the trajectory will critically affect the temperature rise at the sensor. If the trajectory runs farther from the shield surface, the cumulative heat release will be underestimated. If the fluid element comes too close, it will be sucked into a slit.

In Fig. 12 we show Φ as a function of time for the three trajectories (Fig. 11). We notice that without suction (Fig. 12a) the cumulative heat release and, therefore, the temperature fluctuations considerably vary from one trajectory to another. Therefore, apart from the temperature fluctuations due to the pressure variations, we may also expect a significant contribution from the wavy thermal wake.

Integrating Φ over the trajectories of Fig. 11a we obtain ∫ Φ_{1 }*dt* ≈ 123 J m^{−3} for the first and ∫ Φ_{2 }*dt* ≈ 17 J m^{−3} for the second. For the trajectory shown in Fig. 11b with the suction-off case the integral is equal ∫ Φ_{3 }*dt* ≈ 17 J m^{−3}. To calculate the temperature rise we integrate the equation

and obtain

For the three trajectories we obtain Δ*T*_{1} = 0.122 K, Δ*T*_{2} = 0.017 K, and Δ*T*_{3} = 0.017 K.

It is instructive to compare these values with the adiabatic temperature variations in the absence of suction (Fig. 7). The magnitudes of the two effects may be comparable and, in the absence of suction, one must not neglect the irreversible processes.

## 4. Fourier analysis of the computed pressure fluctuations

The pressure fluctuations obtained from the simulations made on the type-A grid (Fig. 4a) refined to 19 712 elements with the suction off have a Fourier spectrum shown in Fig. 13. There are just two frequencies: 13.4 and 26.8 kHz. The first is the frequency of vortex shedding from the shield. The origin of the second frequency will be explained later.

There is a limitation on the frequency response of the measuring system of the ultrafast aircraft thermometer because of the thermal inertia of the sensing wire (Haman et al. 2001), which determines the time constant *τ* of the thermometer. Its estimated value is (0.7–1.4) × 10^{−4} s (Haman et al. 1997), which corresponds to a frequency range of 14–7 kHz. Allowing for the factor-of-2 departures from this rough estimate of *τ*, at least at the low end, we see that the wire can respond to both frequencies found in the simulations. The problem of recovering those components of the signal that are affected by the finite response time of the instrument is an important issue in many airborne systems and has been discussed before (McCarthy 1973; Inverarity 2000).

The spectrum of the computed fluctuations contains only two frequencies, while the spectrum of the signal recorded in the experiments has many peaks. In the simulations we have direct access to the pressure field, while the experimental data are, in fact, sequences of either current or voltage readings at the output of some electronic processing system. An important part of this processing is the discretization or sampling of a continuous signal. We can use our numerical simulations to test the effect of sampling on the frequency spectrum. To this end we prepare a *simulated sensor* signal. At each time step we save the value of pressure at the location of the sensor and then perform a spline interpolation between the time steps to obtain a model-continuous signal. This signal is then sampled with frequencies of 9 and 10 kHz, which are the sampling frequencies used in the experiments. The power spectra of sampled signals, thus obtained, are shown in Fig. 14.

The difference between the spectra is striking. None shows the original 13.4- and 26.8-kHz high frequencies, but all contain “ghost frequencies” that are strongly dependent on the sampling frequency, whose origin can be traced. Those peaks are due to aliasing or frequency shift equal to a multiple of the sampling frequency. Their position is precisely determined by the formula (Lyons 1997)

where *f _{i}* is the original (input) frequency (13.4 or 26.8 kHz),

*k*is an integer, and

*f*is the observed (output) frequency.

_{o}We now show the numerical results and an analytic formula describing the dependence of the basic frequency on the flight speed. The FEATFLOW simulations show that the recorded frequencies are linear functions of the velocity imposed at the inlet, that is, the speed of the aircraft. The dimensionless measure of the frequency of vortex shedding *f* is the Strouhal number (St):

This is the product of *f* and the time scale of advection past the body of size *D* in a flow with velocity *U*. The Strouhal number depends on the shape of the body, possibly on the angle of attack, and on the Reynolds number (Re). The following empirical formula describes the relation St(Re) for the flow around a circular cylinder in the range of 1.6 × 10^{3} < Re < 1.5 × 10^{5} (Norberg 2003):

where *x* = log(Re/1.6 × 10^{3}).

In Fig. 15 the frequency–velocity relation [Eq. (8)] that is obtained for the shield from the FEATFLOW simulations (squares and diamonds) is compared with that for a circular cylinder with a Strouhal number St = 0.1853, calculated using the theoretical Eq. (9) (dashed line). For the aircraft speed *U* = 80 m s^{−1}, we have the Reynolds number based on the shield width *D*, Re = (*ρUD*/*μ*), which equals 4900, thus, Eq. (9) gives for the shield St = 0.1504. The simulations were done for 10 values of the inlet velocity. The diamonds and squares correspond to the lower and higher frequencies, respectively, similar to those existing in the power spectra of pressure fluctuations (see Fig. 13). The solid lines are lines of the best fit. Obtained results show that both frequencies are linear functions of the inlet velocity, which is in agreement with the theoretical Eq. (8).

We may conclude that even when the original continuous signal is dominated by two frequencies or, as we will later explain, one basic frequency and its double, the spectrum of sampled and processed data will exhibit a complex pattern. With the fundamental frequency *f* = 13.4 kHz, all peaks seen in Fig. 14 can be found in the spectrum of our artificial, resampled signal. This could, in principle, be reversed. Assuming that the peaks are aliases of one basic frequency, a rough estimate that we know, we could, from their pattern, reconstruct that basic frequency from each set of real experimental data. However, the direct procedure is easier, and for that we need to find from the FEATFLOW simulations the basic frequency of the von Kármán vortex street corresponding to the particular thermometer geometry and the actual flight speed.

## 5. Explanation of two or more frequencies

The simulations show that the spectrum of pressure at the location of the sensor always has two peaks corresponding to a basic frequency and its double (Fig. 13). The basic frequency, as we explained in section 2, is the frequency of vortex shedding from the shield. The origin of the double frequency can be understood by looking at the pressure distribution along a cross streamline of a section passing through the point where the sensor is located. In Fig. 16 we show the time dependence of this one-dimensional pressure distribution. The distance between the centers of vortices, that is, the white patches in Fig. 16b, determines the basic frequency (13.4 kHz). The wire sensor residing on the centerline picks up two signals from the two sequences of vortices, one on each side of the centerline. They have an opposite sense of rotation, but the pressure distributions in both sequences are the same. Therefore, the wire sensor records two signals of the same amplitude and the same basic frequency, but with phases shifted by *π*. The signal from each vortex sequence is periodic but *not* harmonic and, therefore, such a sum of the two signals clearly looks like a signal with double the basic frequency (two harmonic signals shifted by *π* would, obviously, cancel out).

The farther we are from the centerline, the weaker the double-frequency component. Hence, any possible cross-stream offset of the sensing element would strongly affect the relative height of the two peaks in the spectrum of the pressure fluctuations. The “spoof peaks” that appear in the spectrum of the sampled signal due to aliasing will also be affected. Therefore, precision in the positioning of the sensing element relative to the shield is very important. This precision is, naturally, always finite and a small misalignment can never be ruled out, so each thermometer ought to be calibrated in a wind tunnel if the recorded frequencies are to be properly understood.

## 6. Dependence of pressure and its fluctuations on the flow speed

The velocity of the aircraft on which the thermometer is mounted does vary. In our simulations we have also tested the dependence of pressure and its fluctuations on the *speed* of the axially oncoming airstream (the angle of attack set firmly to 0°). This dependence needs to be understood if we try to distinguish between the long-time variations in the recorded temperature due to uneven flight speed and those attributable to the real changes in the ambient air temperature along the flight path.

In Figs. 17a and 17b we show the nondimensionalized average pressure at the sensor versus the Reynolds number when the suction is off and on, respectively. This relation, at least in the range of Reynolds numbers considered here, appears to be a constant function; that is, the average pressure drop in the wake is a quadratic function of speed.

We obtain, in both cases,

The computed values of the proportionality constant are *C*_{off} = 0.021, *C*_{on} = 0.012, with the suction off and on, respectively.

We have also tested the dependence of the amplitude of pressure fluctuations on the speed of the aircraft. The relative fluctuations at a given point are defined as

where *p _{i}* is the value of pressure at the

*i*th time step and the overbar denotes a running average over 800 time steps. When the suction is off the relative fluctuations appear to be proportional to the aircraft’s speed, as shown in Fig. 18. Calculated values of the relative pressure fluctuations for inlet velocities in the range of 50–150 m s

^{−1}and the suction off are well approximated by the following relation:

*Dp*= −5.2 · 10

^{−5}Re + 0.697. This corresponds to the dimensional formula Δ

*p*= 0.57

*U*− 17.42, where

*U*is in meters per second and

*p*is in pascals. No fluctuations can be seen in the simulations when the suction is turned on.

## 7. Droplets

Droplets suspended in a cloud through which the thermometer is moving from time to time hit the sensing element. A signature of every such collision is a short-lasting drop of about 1.5 K in the recorded temperature caused by the wet-bulb effect (Haman et al. 2001). The measurements become hard to interpret when they are plagued by too many collisions and, hence, there is need for a shield. It considerably reduces the collision frequency but does not eliminate them altogether. The precise computation of the collision probability density function is a very difficult task, involving the physics of the two-phase contact and the full 3D free-surface hydrodynamic simulations of a deforming droplet moving in a boundary layer. Here we simulate the motion of individual droplets of various sizes in the Stokes approximation. Our prime interest is in assessing the effect of suction on the droplets’ trajectories.

The droplets are considered small enough for the surface tension to keep them spherical throughout their motion. The Stokes approximation holds on two conditions. First, that the *droplet Reynolds number* is small. This condition means that the motion of the droplet *relative* to the ambient air is slow. The second condition is that in the vicinity of the droplet the scale of the spatial variations of the air velocity is much larger than the droplet radius *a*. This means that in the frame of reference anchored at the droplet’s center, the ambient airflow tends to a uniform stream as we move out of the center. Equivalently, it means that the flow in the local frame of reference moving with the local air velocity is the flow driven by a slowly moving sphere in an otherwise stationary fluid. If the first condition is also satisfied, then this flow is a Stokes flow and, therefore, the force that the fluid exerts on the droplet is a linear function of the relative velocity given by the Stokes formula (Batchelor 1967). Therefore, the trajectory of the droplet **x**(*t*) is a solution of the following system of ordinary differential equations:

where *m* is the mass of the droplet, **v*** _{d}* is the velocity of the droplet,

*a*is its radius, and

**v**(

**x**,

*t*) is the velocity that air would have at the location of the droplet’s center if the droplet was not there. For a droplet of water in air we have C ≈ 6 (Batchelor 1967, section 4.9).

To simplify the fourth-order Runge–Kutta algorithm for solving Eq. (13), we have first interpolated the velocity field **v**(**x**, *t*), originally given on an irregular finite-element mesh, to a square grid of 200 × 100.

The a posteriori checks show that the first condition is well satisfied. In the inlet plane the Reynolds number of droplets of all sizes (based on the droplet’s relative velocity) is equal to zero and increases along their trajectories, peaking, for most droplets, inside the boundary layer on the shield’s surface at about 0.1 for the smallest and 30 for the biggest droplets. Thus, we can accept that typically the droplet Reynolds number remains small along the entire trajectory, so that the departures from the Stokes flow are insignificant. Occasionally, however, on short stretches of the droplet trajectory the Reynolds number may attain values as high as 100 when the droplet is briefly accelerated in the cross stream near the suction slit.

The second assumption is not satisfied when the droplet is close to the surface of the shield. Therefore, when computing the trajectories of the droplets that pass the shield at distances comparable or less than their radii, we are likely to make an error in deciding whether the droplet will hit the shield and stick to it, or be carried past it. We have assumed that a droplet sticks to the surface when it comes closer than *a*.

In Fig. 19 we show the computed trajectories of 20 droplets starting at the inlet plane and equally spaced in the cross-stream (spanwise) direction. The droplets start with the same velocity as the oncoming airstream (no relative motion). When the suction is off (left column), larger droplets are deflected by the shield and do not hit the sensor, but the small ones can be entrained in the von Kármán vortices and follow a rather erratic path going through the point where the 2.5-*μ*m-thick wire sensor is located. When the suction is on (right column), large droplets again are unlikely to hit the sensor, but the smaller ones come very close as the suction deflects the boundary layer in the wake toward the shield’s symmetry axis. This means that small droplets are likely to hit the sensor when any small external perturbations, for example, small deviations of the angle of attack, appear in the oncoming stream.

## 8. Conclusions

We have made 2D simulations of the flow and the motion of the cloud droplets around the shield of the sensing element of an utrafast thermometer developed at the Institute of Geophysics, Warsaw University, for airborne measurements of cloud microphysics. The thermometer is mounted on aircrafts and records the temperature inside and outside of the clouds. It has such a small time constant of the order of 10^{−4} s that, with the aircraft speed of about 80 m s^{−1}, the resolution of the temperature field mapping can be as low as centimeters or even milimeters. The ultrafine resistive wire of which the sensing element is built is capable of registering temperature variations that are considerably smaller than 0.1 K. At this level of sensitivity the wire detects the hydrodynamic perturbations of the oncoming stream due to a shield, placed in front of it, that is intended to protect the wire mainly against the impact of cloud droplets that spoil the data when they wet the sensing wire, causing a short drop in the temperature recording. As droplets hit the shield, a water film on its surface is removed by suction through the slit in its surface. Our aim was to understand the physics of the recorded temperature fluctuations when no suction is applied and then to assess how suction affects those fluctuations and what effect it has on the motion of droplets. Although the parameters of the problem, for example, the shape of the shield or the range of the aircraft speeds, were specific to a particular instrument, the issues considered, like the influence of suction on the wake dynamics, are of a general nature.

Our simulations demonstrate that, in general, turning the suction on improves the quality of the recorded temperature data because it reduces noise and decreases, in ideal conditions, the probability of droplets colliding with the sensing wire.

The effect of suction on the flow field is twofold. First, at the Reynolds numbers that the thermometer is operated, suction practically eliminates the von Kármán vortices in the region where the sensing wire is located. In the absence of any upstream disturbances the wake of the shield is stable and there are virtually no adiabatic temperature fluctuations associated with the pressure drops in the cores of the vortices shed from the shield and passing the sensor. Second, suction diverts the inner part of the boundary layer into the slit. This inner part is a region of strong shear and, therefore, a region where intensive viscous heating takes place. When the suction is on, much of the air that is heated in the boundary layer in the front part of the shield is removed through the slits and never reaches the wake. Our computations show that without suction this thin layer of warmer air appears in the undulating wake, thus, increasing the amplitude of the recorded fluctuations.

The suction also affects the probability of droplets colliding with the sensor in two ways. The first is the removal of water from the shield’s surface. This dramatically improves the operation of the thermometer as it reduces, or even eliminates, the water flowing along the surface to the rear stagnation point, and then detaching and forming secondary droplets well positioned to hit the sensor. The second, weaker effect is, like the effect on pressure fluctuations, the stabilization of the wake. On the one hand, this eliminates the wandering of small droplets in the von Kármán vortices (such wandering droplets may hit the sensor). On the other hand, the suction deflects the droplets *toward* the axis of the wake, thus, reducing the width of the droplet-free “channel” behind the shield. Any oncoming small perturbation or even a small change of the angle of attack is then more likely to send a droplet toward the sensor.

We have also established the general dependence of the *average pressure drop* and of the amplitude of *pressure fluctuations* on the aircraft’s speed. This helps to assess the systematic temperature shift and the level of noise and, therefore, to plan future experiments.

The simulations enabled us to properly understand the complexity of the spectrum of the recorded signal. If the system recorded a *continuous* signal, as we can do in the simulations, then the spectrum obtained in the flow without suction would be dominated by two frequencies—the basic frequency of the vortex shedding and its double. The two frequencies need to be computed so that the signal processing and data acquisition system can be optimized to ensure the least possible attenuation of these two frequencies. In fact, the system records a sampled signal and, therefore, the spectrum contains additional peaks, which are the effect of aliasing of the true frequencies. The simulations clarify the pattern of this aliasing, and this is essential for the analysis of the experimental data. To understand the small scales of cloud turbulence we need to “clean” the signal by removing *all* frequencies that are associated with the instruments we use, both the “true” ones and the “spoof” frequencies due to aliasing.

Further investigations of the spectra are needed. They would require 3D hydrodynamical simulations to verify whether the spectrum in the central part of the wake is still dominated by two frequencies when the flow is 3D and the end effects are taken into account. We believe, however, that the main conclusions drawn from our present 2D simulations are robust and will remain valid even if the 3D effects are accounted for. The 3D features are certainly present in the considered range of Reynolds numbers, but the “coarse grained” flow pattern is still that of periodically shed vortices (Williamson 1996). In 3D simulations, like in 2D ones, suction is likely to suppress those primary vortices, thus, eliminating the main “contaminant” of the temperature recordings. High-resolution 3D simulations are needed to verify this conjecture and to check whether the 3D features, like streamwise vortices and loops, are also suppressed. Even if this is not the case, their contributions to the temperature spectrum is likely to be much weaker and in a much higher frequency range.

## Acknowledgments

We gratefully acknowledge a free FEATFLOW license received from Professor Stefan Turek of the University of Heidelberg. We are indebted to Professor Jacek Rokicki of Warsaw Technical University for valuable discussions and for suggesting the use of FEATFLOW. We thank Professor Szymon Malinowski for his helpful comments on experimental details and an anonymous referee for illuminating remarks on the cylinder wake dynamics.

This work was supported by the Polish Ministry of Scientific Research and Information Technology Grant 3 T10C 045 27 and European Union Grant EVK2-CT-2002-80010 CESSAR.

## REFERENCES

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## Footnotes

*Corresponding author address:* Bogdan Rosa, Institute of Geophysics, Warsaw University, Pasteura 7, 02-093 Warsaw, Poland. Email: rosa@igf.fuw.edu.pl