# On Neglecting Free-Stream Turbulence in Numerical Simulation of the Wind-Induced Bias of Snow Gauges

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}, demonstrating that adjustment curves based on the simplifying assumption of uniform free-stream conditions do not accurately portray the wind-induced bias of snow measurements.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Calculation of a Realistic Wind Turbulence Intensity

^{©}, Newcastle, UK) from the Nafferton (UK) field test site (see Figure 1a) were analyzed. Data were composed of 38 minutes of high-frequency (20 Hz) wind measurements (Figure 1b). The moving average (Figure 1b) was calculated by testing different dimensions of the moving window, N, and checking that the Reynolds average approach was satisfied [20]. This requires that the mean values of the turbulent fluctuations $\overline{{u}^{\prime}\left(\text{}t\right)}$, $\overline{{\mathrm{v}}^{\prime}\left(\text{}t\right)}$ and $\overline{{w}^{\prime}\left(\text{}t\right)}$ in all directions (x, y, z) must be null. A trial-and-error procedure was adopted: after choosing a tentative value for N, the wind measurements were divided in wind classes (defined by their central value U

_{ref}from 1 to 5.5 m s

^{−1}with bin size of 0.5 m s

^{−1}), based on their average magnitude. For each wind class, the mean of the turbulent fluctuations was calculated, and the N value associated with the minimum of the average fluctuations was chosen (resulting in N = 125). Finally, the relative turbulence intensity values (${I}_{u}$, ${I}_{v}$ and ${I}_{w}$) for each wind class were calculated as follows:

#### 2.2. CFD Simulations

_{a}= 1.2 × 10

^{−5}m

^{2}s

^{−1}and density ρ

_{a}= 1.3 kg m

^{−3}at a reference environmental temperature T

_{a}= 0 °C. For each configuration, at the inlet of the computational domain (y–z plane), the undisturbed wind speed, U

_{ref}, was imposed parallel to the x-axis and it was maintained uniform and constant in time, while a null gradient condition was set for pressure. At the outlet y–z plane, opposite to the inlet, a null gradient condition for the velocity and the atmospheric pressure was imposed, while the lateral surfaces of the domain were set as symmetry planes. The ground and the gauge body were assumed as impermeable surfaces with a no-slip condition. In all computational cells, initial conditions were imposed equal to U

_{ref}for the velocity and equal to zero for the relative pressure.

#### 2.3. Particle Tracking and Collection Efficiency

^{−1}m

^{−3}) as a function of the particle diameter d (mm), over the specified range of diameters. In this work, the PSD formulation proposed in ref [25] for dry snow was used, having the characteristic exponential function proposed by ref [26], and obtained by fitting experimental observations:

^{−1}m

^{−3}is the intercept and Λ mm

^{−1}is the slope of the linear form of the curve in a semi-log plot.

_{0}and a relationship for Λ as a function of the rainfall intensity (RI mm h

^{−1}) was found, as follows [26]:

^{−1}]) are expressed as follows [25]:

## 3. Results

^{−1}). As expected, along the two horizontal directions (x and y) the relative turbulence intensity values (${I}_{u}$, ${I}_{v}$) were very similar, while for the vertical direction (z) the relative turbulent intensity values $({I}_{w}$) were lower (Figure 5). In all cases, the relative turbulence intensity values decreased with increasing wind speed and seemed to have an asymptotical constant limit, although data for high wind speed classes were too few to be conclusive; this behavior is in line with literature observations, albeit referring to higher elevation measurements [19].

_{ref}= 6 m s

^{−1}, at the inlet surface (upwind y, z plane of the computational domain) as a boundary condition. When the airflow overtook the three obstacles, its mean velocity magnitude decelerated and assumed the value of 2.5 m s

^{−1}, which became the new reference wind speed (U

_{ref}) for the turbulent free-stream configuration. Thanks to the scalability (low Reynolds dependence) of the mean flow fields under uniform free-stream conditions [11], the CFD results were rescaled by assuming U

_{ref}= 2.5 m s

^{−1}, to make the results comparable.

_{ref}= 2.5 m s

^{−1}, the desired level of turbulence as measured at the Nafferton (U.K.) field test site as ${I}_{u}$ = 0.25.

_{mag}/U

_{ref}) of the instantaneous flow velocity in the horizontal plane (x,y) at the gauge collector elevation is reported. Along the transversal direction (y/D), the flow field started to become uniform many diameters away from the collector. As shown in Figure 9, the transversal component of the flow velocity (U

_{y}/U

_{ref}) revealed the typical characteristics of vortex shedding which occurred with a regular frequency and with alternate opposite velocity directions.

^{−1}.

_{ref}= 2.5 m s

^{−1}are plotted as a function of SI in Figure 10, where an inversion of the sign of the relative difference between the two curves is evident at about 10 mm h

^{−1}. Beneath that threshold, neglecting the free-stream turbulence in the simulation would lead to an overestimation of the CE, with a resulting underestimation of the wind-induced error of precipitation measurements. The opposite occurs beyond 10 mm h

^{−1}, with an underestimation of the CE that leads to an overestimation of the wind-induced error. Note that the typical snowfall rates are well below that threshold, and the impact of neglecting the free-stream turbulence intensity may lead to an overestimation of the collection efficiency of about 30% at SI = 1 mm h

^{−1}and U

_{ref}= 2.5 m s

^{−1}.

^{−1}, particles smaller than 2 mm account for about 99.7% of the total number of particles in the unit volume of the atmosphere, which results in about 82.8% of the equivalent water volume. At 10 mm h

^{−1}, however, these values become equal to 77% of the total number of particles and 7% of the equivalent water volume.

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

^{©}and are available from the authors with the permission of EML

^{©}.

## Acknowledgments

^{©}) from the Nafferton (U.K.) field test site and this work was developed as partial fulfilment of the PhD thesis of the first author. Special thanks to the Italian Air Force for providing access to the high-performance computing resources at the ECMWF, within the activities of the CIMO/WMO Lead Centre of Precipitation Intensity through the High-Computing ISCRA class C project “Aerodynamic Response of Precipitation Gauges Immersed in a Turbulent Wind Field”.

## Conflicts of Interest

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**Figure 1.**(

**a**) The Nafferton (U.K.) field test site with the 3D Sonic anemometer and EML

^{©}aerodynamic rain gauges installed at the ground surface, and (

**b**) high-frequency wind measurements (black line) and moving average with N = 125 (grey line).

**Figure 2.**(

**a**) Portion of the geometric setup with the three pillars positioned upstream of the gauge, and the wind direction indicated by the white arrow. (

**b**) Horizontal section (x,y plane) and gradual refinement of the computational mesh close to one sample pillar at a generic elevation.

**Figure 3.**(

**a**) Refinement region around the Geonor gauge body in the central vertical section (x,z plane at y = 0) and (

**b**) gradual refinement close to its surface in the horizontal section at the gauge collector elevation (x,y plane at z = −2 m).

**Figure 4.**Mean values and standard deviations of turbulent fluctuations (left-hand axis) and sample size (right-hand axis) for each wind class, measured by a 3D sonic anemometer at the Nafferton (U.K.) field test site.

**Figure 5.**Relative turbulence intensity for the three Cartesian directions for each wind class, measured by a 3D sonic anemometer at the Nafferton (U.K.) field test site.

**Figure 6.**Decrease of the three numerical relative turbulence intensity profiles at the gauge collector elevation along the spatial domain between the position of the obstacles (x/D = −70) and the gauge (x/D = 0).

**Figure 7.**Normalized magnitude (U

_{mag}/U

_{ref}) of the instantaneous flow velocity in the vertical plane (x,z) at y/D = 0 for the turbulent free-stream conditions.

**Figure 8.**Normalized magnitude of the instantaneous flow velocity (U

_{mag}/U

_{ref}) in the horizontal plane (x,y) at the gauge collector elevation for the turbulent free-stream conditions.

**Figure 9.**Normalized transversal component (U

_{y}/U

_{ref}) of the instantaneous flow velocity in the horizontal plane (x,y) at the gauge collector elevation for the turbulent free-stream conditions.

**Figure 10.**CE curves as a function of SI, obtained under uniform and turbulent free-stream conditions at U

_{ref}= 2.5 m s

^{−1}.

**Figure 11.**Percentage number and volume of particles having a diameter less than 2 mm within a unit volume of the atmosphere, based on the assumed particle size distribution (PSD).

**Table 1.**Catch ratios obtained for each solid particle size d at U

_{ref}= 2.5 m s

^{−1}using the LPT model and based on the large eddy simulation (LES) mean airflow fields for the Geonor precipitation gauge under uniform and turbulent free-stream conditions.

d (mm) | 0.25 | 0.5 | 0.75 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|---|---|

Uniform | 0.190 | 0.393 | 0.449 | 0.492 | 0.590 | 0.659 | 0.708 | 0.734 | 0.757 | 0.787 | 0.787 |

Turbulent | 0.164 | 0.164 | 0.216 | 0.289 | 0.590 | 0.718 | 0.777 | 0.820 | 0.849 | 0.869 | 0.892 |

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**MDPI and ACS Style**

Cauteruccio, A.; Colli, M.; Lanza, L.G.
On Neglecting Free-Stream Turbulence in Numerical Simulation of the Wind-Induced Bias of Snow Gauges. *Water* **2021**, *13*, 363.
https://doi.org/10.3390/w13030363

**AMA Style**

Cauteruccio A, Colli M, Lanza LG.
On Neglecting Free-Stream Turbulence in Numerical Simulation of the Wind-Induced Bias of Snow Gauges. *Water*. 2021; 13(3):363.
https://doi.org/10.3390/w13030363

**Chicago/Turabian Style**

Cauteruccio, Arianna, Matteo Colli, and Luca G. Lanza.
2021. "On Neglecting Free-Stream Turbulence in Numerical Simulation of the Wind-Induced Bias of Snow Gauges" *Water* 13, no. 3: 363.
https://doi.org/10.3390/w13030363