## 1. Introduction

The winds near the surface of Mars are an important focus of study in the exploration of Mars. They impact the safety of the delivery of robotic or manned missions to the surface of Mars. They also are the principal agent of geologic change on the planet in the present time. Without a hydrosphere, like on Earth, the bulk of the transport of water around Mars seasonally and on longer time scales may occur in the atmosphere, controlled by the winds at the surface. While water is the important radiative constituent in the earth’s atmosphere, dust plays that role for Mars. Dust lifting is poorly understood for Mars and is controlled by the near-surface winds. All of these reasons suggest why we need to be able to measure the winds at the surface of Mars, and yet this is a difficult problem to address, considering the constraints imposed by the delivery of a scientific package to the surface of a distant planet.

Planetary landers are typically highly constrained in cost, power, volume, and mass. The latter two variables are often the most challenging when it comes to designing an instrument that can accurately measure the winds without interference from the lander itself. On Earth, a common rule of thumb when siting a meteorological station is to place it at least 10 times as far away from any other obstacles as the height of the instrument, and often measurements are made from 10-m towers (Egan and Baldelli 2009). Such a large tower is highly unlikely for a typical planetary lander; a 1-m meteorology mast is more feasible. Even with this more modest height, a separation of the instrument from the rest of the lander by approximately 10 m would be prohibitively difficult as well. Consequently, meteorological measurements made from planetary landers are a compromise where the instrument cannot be sited in an ideal location, as one would ensure on Earth, to avoid interference of the measurements. However, it may be possible to select how best to locate the instruments within the constraints that the realities of the mission impose, and still get the best data possible. Additionally, it may be possible to understand the nature of the perturbations that the proximity to the lander may induce, and thus correct the observations for those perturbations afterward. The goal of this study is to address these following two issues: 1) how to position a wind sensor near a lander to minimize the flow perturbations caused by lander interference within the engineering constraints on the possible separation distance, and 2) how to correct the measurements for these perturbations to recover the original unperturbed flow characteristics.

The motivation for this study came from the development of a novel anemometer for use on Mars (Banfield and Dissly 2005) that will allow fast (>10 Hz), precise (<5 cm s^{−1}), and 3D measurements of the wind in this low-density atmosphere. Wind measurements were carried out on the Viking landers (Tillman et al. 1994), Mars Pathfinder (Sullivan et al. 2000), and the Phoenix Lander (Taylor et al. 2008). They gave valuable results, but the simplicity of both the instruments and approaches used had numerous drawbacks. One of them was that, because of the relaxation time of the instruments, only the mean velocities were measured; a significant fraction of the turbulent quantities were averaged over time despite the fact that these are a key component to understanding of the dynamics of the atmosphere. The new instrument is based on the terrestrial sonic anemometers used in boundary layer research on Earth, but it uses more sophisticated signal processing and low acoustic impedance transducers to optimize performance in the thin Martian atmosphere. Like the terrestrial sonic anemometers, the fast response, high-precision 3D capability and open sensing volume allow for measurements of not only the mean winds in the boundary layer of Mars, but also the turbulent eddies that advect past the instrument. The ability to resolve these fluctuations opens the possibility to directly measure the transports through the boundary layer of momentum, heat, and trace species (e.g., water) resulting from the eddies and the turbulence. It is in the context of this more sophisticated instrument that we recognized the need to identify how to place such an instrument on a Martian lander in an optimal sense, and, further, how to correct its measurements for the perturbations induced by the proximity to the lander.

There is essentially no in-depth research published on this topic. On Earth, this problem does not arise because it is nearly always possible to locate the anemometer away from significant obstacles. In the general case, some theoretical approaches (Batchelor and Proudman 1954; Hunt 1973; Wyngaard 1981) have been proposed, but they are based on assumptions that make them of little interest in the case considered here. An attempt was made to model the influence of the lander on the measurements for the Viking mission (Tillman et al. 1994), but it was restricted to the mean velocity using a potential flow model. A more recent study was conducted by Moat et al. on the airflow distortion over merchant ships. This study used computational fluid dynamics (CFD; Moat et al. 2006a,b) and wind tunnel experiments (Moat et al. 2004), but only the mean quantities were studied, and the scaling of the problem makes it difficult to find reliable information for the problem of interest here.

This article reports on the study of the flow around a Martian lander using computational fluid dynamics. The simulations were conducted with two simplified representations of a lander and used a Reynolds stress model to compute the flow around the lander. The results of these simulations are used to identify the critical spacing and position of the anemometer relative to the lander to ensure reliable measurements. The optimal position derived does not, however, take into account components such as antennas, which may be present on real Martian landers and may affect the wind patterns. The article is divided in three sections: the first one concerns the characteristics of Mars and the behavior of the wind velocity in the boundary layer; the second describes the simulations that were conducted to understand the influence of the lander on the flow; and in the last one, the results of the simulations are discussed and an optimal position for the anemometer is determined.

## 2. Martian atmospheric conditions

### a. Characteristic quantities of Mars atmosphere

Several characteristics of the Martian atmosphere have to be known to conduct the simulations. The following numerical values are considered to represent the atmosphere of Mars at ground level and were used for the simulations.

The mean molecular mass of the atmosphere *M* was determined by Seiff and Kirk (1977). The pressure *P* and temperature *T* data acquired by the Viking Lander 1 and 2 missions during descent were used along with the hydrostatic equation to determine the mean molecular mass *M* = −(*RT*/*Pg*)*dP*/*dz*, where *R* is the universal gas constant, *g* = 3.693 m s^{−2} is the local acceleration resulting from gravity, and *z* is the altitude. The mean molecular mass that is used for the simulation, *M* = 43.34 g mol^{−1}, is derived from the data of Viking Lander 2 because it agrees well with the molecular mass derived from the atmospheric composition data (Kieffer et al. 1992) Viking Lander 1 and 2 surface pressures at landing were, respectively, 762 and 781 Pa. Because surface pressures have large variations depending, for example, on the altitude and season, the value *P* = 800 Pa, which represents the correct order of magnitude at the surface of Mars, is used for this work.

Temperature, like pressure, displays a lot of variability depending, for example, on the latitude and season. The value chosen for the simulations, *T* = 217 K, is, according to Leovy (1969), the planetary equilibrium temperature, which corresponds to the average temperature on Mars given by the National Aeronautics and Space Administration (NASA; Williams 2007). This value is coherent with the measures made by the Viking landers (Hess et al. 1977, their Fig. 6), is confirmed by the Mars Pathfinder (Schofield et al. 1997), and is also in line with the measures made by the Phoenix Lander (Holstein-Rathlou et al. 2010). This value does not, of course, capture the variability of the temperature that has been observed on the three missions mentioned, but it should be noted that temperature is not a parameter that is directly used in the simulations because no thermal effects are considered. It is only used to estimate the dynamic viscosity of the Martian atmosphere at ground level.

*μ*is the dynamic viscosity at temperature

*T*,

*μ*

_{0}is the reference dynamic viscosity at reference temperature

*T*

_{0}, and

*C*is Sutherland’s constant. Because carbon dioxide is the main component of the Mars atmosphere, its viscosity will be used to describe the flow around the lander. The numerical values of the constants for carbon dioxide are

*μ*

_{0}= 1.480 × 10

^{6}Pa s,

*T*

_{0}= 293.15 K, and

*C*= 133.333 K. The dynamic viscosity is, therefore,

*μ*

_{CO2}(217 K) = 1.0996 × 10

^{−5}Pa s.

Assuming that the atmosphere of Mars is an ideal gas, the density and the kinematic viscosity are *ρ* = 0.019217 kg m^{−3} and *ν* = 5.7218 × 10^{−4} m^{2} s^{−1}, respectively. The value for the density derived by the equation describing an ideal gas is in agreement with the data from the two Viking missions (Seiff and Kirk 1977) and from Mars Pathfinder (Schofield et al. 1997).

### b. Model for the atmospheric boundary layer

The velocity at the surface of Mars is described by the vector field **U** (*x*, *y*, *z*), which is the velocity of the flow at position *x***x** + *y***y** + *z***z**, where (**x**, **y**, **z**) is an orthonormal basis such that **x** is the direction of the mean inlet velocity and **z** points up. The components of **U** are *U*, *V*, and *W* such that **U** = *U***x** + *V***y** + *W***z**. The notation 〈*A*〉 refers to the ensemble average of the quantity *A* and for each component of the velocity, the fluctuations *u*, *υ*, and *w* are defined by *U* = 〈*U*〉 + *u*, *V* = 〈*V*〉 + *υ*, and *W* = 〈*W*〉 + *w*. The quantities 〈*uu*〉, 〈*υυ*〉, 〈*ww*〉, 〈*uυ*〉, 〈*uw*〉, and 〈*υw*〉, which are not equal to zero, are referred to as Reynolds stresses.

*z*/

*δ*< 0.1, with

*δ*the altitude at which the speed is uniform, the mean inlet profile can be described by a simple logarithmic expression (Prandtl 1925),

*z*

_{0}is called the “roughness length” and

*u** is the “friction velocity.” The quantity

*ρ*(

*u**)

^{2}is a measure of the stress because

*ρu**

^{2}= −

*ρ*〈

*uw*〉 for the surface layer, and

*κ*is a nondimensional constant whose value is 0.41.

Values of the parameters *z*_{0} and *u** were measured on Mars during the Mars Pathfinder mission by the imager for Mars Pathfinder (IMP) windsock experiment (Smith et al. 1997). The IMP allowed direct measurement of the wind profile for the first time because no such capability was available on the Viking landers. The device was composed of three different windsocks made of aluminum (for a total mass of 39 g) at different altitudes of 33.1, 62.4, and 91.6 cm above the solar panel, with the more sensitive windsocks being the ones nearer to the ground. The wind speed at the altitude of the windsocks was measured through the deflection of the windsock. To link the angle of deflection to the wind speed each windsock was calibrated in the Mars Surface Wind Tunnel (MARSWIT) operated by Arizona State University at the NASA Ames Research Center (Seiff et al. 1997). During the mission, wind profile information was harvested through a 100-s, 12-image sequence, which was activated from 1 to 4 times per mean solar day on Mars (sol). The device simplicity’s drawback is the complexity of data reduction, especially with light wind. The experimental data were reported and analyzed by Sullivan et al. (2000) by averaging over periods of approximately constant wind. The values measured for *u** and *z*_{0} are the conditions 1–4 in Table 1.

Figure 1 shows that the data from the Mars Pathfinder mission do not cover the plane (*u**, *z*_{0}) extensively. Therefore, four sets of parameters were added in order to capture the different regimes that may exist. The additional values, 5–8 in Table 1, are not meant to have a physical meaning but simply provide a more complete coverage of the (*u**, *z*_{0}) plane and possibly account for Martian conditions that were not measured; simulations were conducted for these eight sets of parameters. For six of the eight sets of parameters (1, 2, 3, 4, 5, and 6), the value of the mean velocity at the 1-m elevation is between 7.8 and 9.2 m s^{−1}. For sets 7 and 8, the mean velocities at 1-m elevation are respectively 3.1 and 23.1 m s^{−1}. This is consistent with the dynamical structure of the Martian lower atmosphere developed by Gierasch and Goody (1968) and with the measurements made by the Phoenix Lander reported by Holstein-Rathlou et al. (2010) and by the two Viking landers (Hess et al. 1977). Finally, by considering these values for the wind speed and a size of the lander of *L* = 1 m, the order of magnitude of the Reynolds number for the flow is approximately 10^{4}.

## 3. The computational fluid dynamics simulations

### a. Software

The simulations were carried out using the commercially available software Gambit for the geometry and mesh generation and Fluent for the CFD part. Fluent was used to compute the three-dimensional, steady-state solution using a Reynolds stress model with standard coefficients (Pope 2000, chapter 11). A pressure-based solver was used with a compressible flow. The pressure equation was discretized using PRESTO. The momentum, turbulent kinetic energy, turbulent dissipation rate, and Reynolds stress equations were discretized using a second-order scheme, and the SIMPLEC (Van Doormaal and Raithby 1984) pressure–velocity coupling was used.

A Reynolds stress model was chosen to solve the equations of fluid dynamics because it offers a compromise between simpler models, such as the *k*–*ε* model, and direct Navier–Stokes (DNS) simulation. It requires much less computing time than DNS, which is an important point because several computations were conducted to test various inlet profiles and lander shapes. However, because it solves the transport equations for each of the six Reynolds stresses and for the dissipation *ε*, the Reynolds stress model gives information on the structure of the turbulence.

The simulations were conducted in Swanson Laboratory in the Sibley School of Mechanical and Aerospace Engineering at Cornell University. The computers used were Dell Precision 690 MiniTower with a Dual Core Xeon Processor (2.00 GHz, 4-MB L2 Cache 1333 MHz) and 8-GB RAM running 64-bit Windows XP Operating System. The total number of simulations carried out is 24 for a total amount of computer time of 48 days.

### b. Computation domain

The computation domain Ω is a rectangular parallelepiped defined by (units: m): (*x*, *y*, *z*) in [−10, 20] × [−10, 10] × [0, 10]. For *x* = −10 m, a velocity inlet boundary condition is applied. For *x* = 20 m, a pressure outlet condition is used with the Martian atmospheric pressure (800 Pa). For *z* = 0 m, which represents the ground, a no-slip wall condition is chosen. Finally, for the other boundaries, a symmetry condition is used, which corresponds to a wall at which the no-slip condition is not applied.

Simulations were carried out with two types of landers—one modeled by a half-sphere and one by a half-cube—in order to study the influence of the lander shape on the simulations results. A sphere of radius 1 m was chosen and the size of the cube is such that the cube is exactly contained within the sphere. This led to an edge length *a* = 2/^{3}. The domain used for the spherical lander contains 2 853 028 cells and the one for the cubic lander has 2 902 064 cells. For these two domains, as well as for the one introduced in the next paragraph, grid independence has been achieved. This has been demonstrated by comparing the numerical solutions on domains with different numbers of grid points. Therefore, the results presented in this article do not depend on the grids used for the simulations.

The center of the reference frame, that is, (*x*, *y*, *z*) = (0, 0, 0), corresponds to the center of the spherical lander. The domain is described by three quantities, *r*, *θ*, and *ϕ*, which are, respectively, the distance from the center of the domain, the elevation angle, and the azimuth angle, as described by Fig. 2.

### c. Simulation process

Before simulating the flow on the grid described above, the profile of the inlet quantities were determined. The mean velocities are known analytically because 〈*V*〉 and 〈*W*〉 are assumed to be zero for all altitudes and 〈*U*〉 is assumed to follow the logarithmic law given by Eq. (2). To determine the profiles of the six Reynolds stresses that correspond to a given set of values *u** and *z*_{0}, a simulation with no obstacle over 500 m was performed for each set of parameters on an auxiliary domain, defined by (units: m) (*x*, *y*, *z*) in [−490, 10] × [−10, 10] × [0, 10]. At the steady state, which is assumed to be reached at the end of the auxiliary domain, the Reynolds stresses profiles correspond to the velocity profile that was put at the inlet of the domain. The auxiliary domain, which is 500 m long, contains 800 000 cells. At the inlet of the auxiliary domain, the logarithmic mean velocity profiles are used, but default values are used for the turbulent quantities. The comparison of the profiles at *x* = 0 m and at *x* = 10 m showed that the steady state is reached for every set of parameters. The profiles at *x* = 0 m and at *x* = 10 m in the auxiliary domain were used as inlet profiles for the simulations with the lander and as profiles compared to the flow around the lander, respectively.

## 4. Results and discussion

The goal of this study was to identify positions around the lander such that, when the flow distorted by the lander is measured, the anemometer gives reliable information on the unperturbed flow. To do so, in the first part this section explores the behavior of the perturbed flow in the plane of symmetry (*ϕ* = 0°). This gives an optimal elevation angle *θ* at which the perturbation is minimized. It also sets a lower bound on the distance from the lander such that the measurements are reliable. In the second part the flow is studied around the lander at this particular elevation angle and the influence of *ϕ* is discussed.

A quantity called “relative difference” is used in the discussion. It is defined by (*A*′ − *A*)/*A*, where *A* is a given quantity measured at a given position without the lander and *A*′ is the same quantity at the same position but with the lander.

The first observation that has to be made on the results concerns the independence of the distortion with respect to the inlet profiles. For each of the eight sets of parameters studied, the relative difference of a given quantity has approximately the same value for a given shape of the lander and for a given position. [This is illustrated by Figs. 4 and 6 (right) and Fig. 8 for the quantities ‖**U**‖ and 〈*uw*〉.] This observation is important because it means that the results that are shown in the following sections are robust, in the sense that they do not depend very much on the inlet parameters. Therefore, in the rest of this article, results from the simulations using parameter set 1 will be mainly used in the figures presented and in the discussion, except where explicitly stated otherwise. It must be kept in mind, however, that the seven other sets of parameters yield similar results, which support the general conclusions drawn.

The other observation that can be made concerning the results is the independence concerning the shape of the lander. It appears that, for *r* ≥ 1.2 m, the results exhibit the same behavior for the sphere and for the cube, except the cube appears to be a sphere of smaller radius. As a result, the spherical lander will be considered in the following discussion because this shape will provide more restrictive constraint on the position of the anemometer.

### a. In the plane of symmetry

Figure 3 shows the dependence of the mean velocity magnitude on *θ* for distances from the center of the lander between 1.2 and 2.4 m. For the two shapes, the flow exhibits the same type of behavior: at approximately the elevation of *θ* = 55°, the relative difference is 0 whatever the distance from the lander. Elevations with a value larger than 90° are discussed in the next paragraph in which variations of the azimuth angle *ϕ* are considered. To build on this observation, Fig. 4 shows the domain on the (*r*, *θ*) plane where the relative difference of the mean velocity is smaller than 2% for the spherical lander. It shows that if the anemometer is placed upwind at an elevation *θ* of approximately 55°, the velocity magnitude measured will be the same, within 2%, as the velocity magnitude without the lander for all the inlet profiles.

In term of constraint on the position of the anemometer, it shows that the optimal position in the plane of symmetry is constrained by *θ* = 55°. Considering the same quantity in planes defined by *ϕ* ≠ 0° confirms that this constraint on the angle theta is optimal. Figure 5, for example, shows the behavior of the relative difference of the velocity magnitude in the plane defined by *ϕ* = 90°. In this specific plane, the relative difference is nearly constant for all elevations and, contrary to the behavior in the plane of symmetry (Fig. 3), there is no angle *θ* for which the relative difference is 0. Therefore, no optimal elevation angle can be found when the wind is perpendicular to the position of the anemometer.

A constraint on the distance of the anemometer from the center of the lander can be derived by considering the Reynolds stresses. Figure 6 shows the behavior in the plane of symmetry of the relative difference of the quantity 〈*uw*〉, which is a measure of the stress in the surface layer. The right plot shows that the results are independent of the inlet profile because all of the curves are superimposed. When *r* is greater than 1.8 m, the values of all of the relative differences for the Reynolds stresses, except for 〈*υw*〉, are stable for angles *θ* around 55°. The relative difference for the quantity 〈*uu*〉, 〈*υυ*〉, 〈*ww*〉, 〈*uυ*〉, and 〈*uw*〉 are approximately 7%, −4%, 38%, −40%, and 42% respectively at the location *r* = 1.8 m, *θ* = 55°, and *ϕ* = 0°. This means that, even if the flow is distorted by the lander, the unperturbed quantities can be computed from the measured quantities using the values of the relative differences.

### b. At a constant elevation

The results on the conic surface defined by *θ* = 55° are now considered. Figure 7 shows the variation of the mean velocity magnitude with *ϕ*. The plots show a bell shape with a maximum at approximately *ϕ* = 90°. This means that the mean flow is less distorted upwind than on the side of the lander. To evaluate this effect, Fig. 8 shows the domain in the (*r*, *ϕ*) plane where the mean velocity relative difference is smaller than 5% for the spherical lander. For this value of the relative difference with *r* = 1.8 m, the limit on *ϕ* to ensure good measurements is 80° in the worst case (set of parameters 8), with no limit in the other cases. Figure 9 shows how the direction of the wind speed is modified by the presence of the lander. The two curves have expected shapes and are quite symmetric around *ϕ* = 90°. The wide range of variation for *θ _{U}* and

*ϕ*can be used to determine the direction of the wind.

_{U}Concerning the Reynolds stresses, the quantities 〈*uu*〉, 〈*υυ*〉, and 〈*ww*〉 display little variation with *ϕ*. At the distance *r* = 1.8 m, the relative difference for 〈*uu*〉 varies between 0% and 10%. It is maximum upwind and downwind (*ϕ* = 0° and 180°) and reaches its minimum on the side of the lander (*ϕ* = 90°). The relative difference for 〈*υυ*〉 varies around 0% with an amplitude of a few percent. This means that the value for the relative difference for 〈*υυ*〉 given above can be taken as true all around the lander as long as *r* ≥ 1.8 m. The relative difference for 〈*ww*〉 is larger than for the two previous stresses, but is still bounded. At *r* = 1.8 m, its value is between 19% and 39%. The relative difference of 〈*uw*〉 decreases with *ϕ*, from 42% to 0% (Fig. 10). This is a large variation but if *ϕ* is known, then it is possible, as for all of the other Reynolds stresses except 〈*uυ*〉 and 〈*υw*〉, to know precisely the value of the relative difference and compute its value without the lander.

### c. Recommendation on the position of the anemometer

The optimal elevation has been shown to be *θ* = 55°. As for the distance from the center of the lander, a minimal distance of 1.8 m offers reliable measurements and allows one to determine the undisturbed values of the quantities of interest from the values measured with the lander, except for the quantities 〈*uυ*〉 and 〈*υw*〉.

This position has been derived for the two shapes considered in this article, which have a characteristic size of one meter. However, the generalization of this result to different lander sizes can be considered. Indeed, the blockage effect of the volume of the lander can be expected to be the primary source of perturbations. This effect is essentially an inviscid phenomenon and should therefore scale with the lander size. On the contrary, viscous effects can be expected to induce only higher-order perturbations. These considerations lead to the conclusion that the anemometer should be placed at a minimum distance of 0.8 times the characteristic size of the lander.

## 5. Conclusions

Understanding the flow behavior around a lander under Martian conditions has proved to be valuable in assessing the validity of measurements made by an anemometer placed at various positions relative to a lander. Simulations were conducted for two shapes—spherical and cubic—with eight different inlet profiles representing a range of turbulence conditions in the Martian boundary layer. The simulations’ results allowed the determination of positions of the anemometer around the lander, where the perturbations resulting from the lander in the wind speed measurements are minimized. The behavior of the flow around the lander was shown to not depend heavily on its shape and on turbulence conditions. It means that the constraints derived on the position of the anemometer are robust.

For this study, the conducted simulations solved the Reynolds stress equations, which allowed the turbulent behavior of the flow to be captured under conditions of neutral atmospheric stability. However, no study has been made on thermal influence over the flow in the presence of the lander. The gradient of temperature that exists at the surface of Mars varies from strongly stable at night to highly unstable in the afternoon. A next step in the simulation process is to include this effect and understand how the flow behaves with the presence of a lander under different amounts of atmospheric static stability. This can be done using the same approach as the one presented in this article.

Concerning the shape of the lander, the simulations showed that it has little influence on the results. However, they do not take into account components such as antennas. Therefore, more precise simulations could be carried out when a more precise shape for a lander is available. Finally, the results presented in this report come only from simulations. To complement this study, wind tunnel experiments reproducing Martian conditions, in term of dimensionless quantities and inlet profiles, would be interesting to conduct.

## Acknowledgments

The authors would like to acknowledge helpful discussions with Lance R. Collins, Peter J. Gierasch, Anthony Toigo, and Zellman Warhaft. This research was supported by NASA PIDDP Grant NNX08AN03G, “Martian Acoustic Anemometer.”

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Parameters for the inlet velocity profile; 1–4: values of *u** and *z*_{0} in the Martian boundary layer measured during Mars Pathfinder mission (Sullivan et al. 2000); 5–8: additional sets of values for which simulations were conducted.