**6. Boom aerodynamics**

MEDA wind sensors are a new design based on the wind REMS sensor [2, 3] that was embarked on the Mars Science Laboratory (MSL) Curiosity rover [4] currently sending data from Mars atmosphere. The behavior of thermal wind sensors installed over the boom of the Mars rover is similar to hot wire, described in earlier paragraph. They are based on a group of four sensitive surface hot film sensors (named dices A, B, C, and D) installed in the same plane (see **Figure 2**). Sensors are

refrigerated when the wind is blowing over each dice, and consequently, an electric power must be apply to keep temperature constant on the dice (constant operation reference temperature CTA). The value of electric power gives a measurement of the airspeed on each dice and the direction from blowing. Dices situated in front the flow are cooled more than those located in back positions, since the wind heats up as the first ones are cooled and heat is transferred to the airstream. A comparative analysis of the electric power required for each of the four dices will give us an

Finally, a joint calibration of the four dices allows us to know the wind speed and

**Figure 3** shows the set-up for functional tests of rover wind sensors. A simplified version of hot film sensor, similar to the real sensor was implemented over a flat plate. A specially designed bed fabrication in extruded polystyrene was used to support the plate with hot film sensors in order to perform the functional tests by wind tunnel testing to verify the wind endurance. During this test campaign, several values of airspeed were blown by wind tunnel and the air was flowing on the sensors refrigerating them. Readings of electric signals from dices electronic chip were acquired to verify the correct behavior of the wind sensors, and its integrity was verified when the flow was blowing over them. A laser Doppler anemometer

the incidence angle over the sensor. A more detailed description is in Ref. [4].

indication of the angle of incidence of the air flow.

was used as an airspeed standard.

*Mars Exploration - A Step Forward*

**Figure 2.**

**Figure 3.**

**74**

*Mars rover wind sensors during functional tests.*

*Rover wind sensor boom.*

The rover wind sensors are installed on the boom surface. Booms are located in the vertical mast over the upper rover central box surface (see **Figures 1** and **2**). Two booms are at the same height and contained in the same horizontal plane, with an angular offset of 120°. The boom external geometry is similar to these of the Pitot-static tube. This is a classical instrument for measuring airspeed in aerodynamics and fluid mechanics that has proven its efficiency for more than a century. This tube has a hemispherical head followed by a cylindrical body with axisymmetric geometry so that the longitudinal flow is coming to the wind sensor without detachment. On the other hand, rover booms have a hemispherical head followed by a cylindrical body with polygonal faces where thermal sensors are installed (see **Figure 2**). In this section, a theoretical analysis of two simplified cases is presented: longitudinal axisymmetric flow and cross flow over the boom.

### **6.1 Longitudinal axisymmetric flow**

When the wind is blowing in longitudinal axisymmetric direction, the head of the boom receives the flow without detachment. A first approximation to this kind of flow can be studied by considering the incompressible potential flow on the axisymmetric Rankine half-body of revolution. This flow is produced when a uniform stream of velocity *U*<sup>∞</sup> is flowing over a three-dimensional point source of strength *M* located at the origin. This situation is depicted in **Figure 4**, for a half-body of revolution with radius *a*.

A stagnation point denoted by P is produced by the upstream source point when both singularities are equal in velocity. The location of this point is calculated as follows:

$$U\_{\infty} = \frac{M}{4\pi d^2},\tag{14}$$

and finally, the distance to point P is

$$d = \frac{1}{2} \sqrt{\frac{M}{\pi U\_{\infty}}}.\tag{15}$$

The velocity potential *Φ* corresponding to the superposition of a uniform stream and a source located in the origin is given by the following expression:

$$\Phi(\mathbf{x}, \mathbf{r}) = U\_{\infty}\mathbf{x} - \frac{M}{4\pi R} \tag{16}$$

where *<sup>R</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> <sup>p</sup> .

**Figure 4.** *Axial flow over wind sensor boom.*

On the other hand, the Stokes stream function *Ψ* for this flow is given by,

$$
\Psi^{\Psi}(r,z) = \frac{M}{2} \left( 1 - \frac{\varkappa}{R} \right) + \pi r^2 U\_{\infty}. \tag{17}
$$

*u x*ð Þ ,*r U*<sup>∞</sup>

*Aerodynamics of Mars 2020 Rover Wind Sensors DOI: http://dx.doi.org/10.5772/intechopen.90912*

> *u x*ð Þ ,*r U*<sup>∞</sup>

¼ 1 þ

**Figure 5** shows the graph of nondimensional velocity *u/U*<sup>∞</sup> versus the nondimensional distance *x/2a*. The velocity is growing from the stagnation point located at *χ<sup>P</sup>* ¼ �1*=*4 until unity nondimensional velocity that is reached when curve cross the origin (*χ* ¼ 0). After that, a peak is produced, and the freestream velocity value is got for a longitudinal distance *x/2a = 1.5. S*o that, when the flow is coming to a boom section located at a distance of approximately equal to three times the radius (*x = 3a*), the wind flows with the freestream velocity, and the hot film sensor will be refrigerated by the same manner as in nonperturbed conditions.

Wind flowing transversal to the boom (boom in cross-flow) can be studied as a first approximation by the two-dimensional complex potential flow over a circular

variable, *ζ*, as follows:

**6.2 Transversal flow**

**Figure 5.**

**Figure 6.**

**77**

*Potential flow of a cylinder.*

cylinder, as represented in **Figure 6**.

*Nondimensional wind velocity on rover boom in axial flow.*

¼ 1 þ

1 16

And simplifying, we can find an expression that only depends on the unique

*χ* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>8</sup>*ζ*<sup>2</sup> � <sup>1</sup> � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup> � <sup>4</sup>*ζ*<sup>2</sup> q� �

*<sup>χ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ζ</sup>*<sup>2</sup> � �<sup>3</sup> <sup>q</sup> *:* (27)

<sup>64</sup> *:* (28)

The half-body surface stream function is equal to the flow rate coming from the inner of the half-body. Taking into account that it is a body of revolution, transversal area is *<sup>A</sup>* <sup>¼</sup> *<sup>π</sup>a*<sup>2</sup> and the flow coming from the inner body, crossing the circular area denoted as *A* (see **Figure 4**), is *AU*∞, exactly the strength of the point source (*<sup>M</sup>* <sup>¼</sup> *<sup>π</sup>a*2*U*∞), and the distance *<sup>d</sup>* is computed being *d = a/2.*

The dividing streamline represents the surface body, and it is obtained when the stream function is equal to the source strength, *M*,

$$M = \frac{M}{2} \left( 1 - \frac{\varkappa}{R} \right) + \pi r^2 U\_{\infty}. \tag{18}$$

And the corresponding velocity is

$$u(\mathbf{x}, r) = \frac{\partial \Phi(\mathbf{x}, r)}{\partial \mathbf{x}} = U\_{\infty} + \frac{M}{4\pi} \frac{\mathbf{x}}{\left(\mathbf{x}^2 + r^2\right)^{3/2}} \tag{19}$$

$$w(\mathbf{x}, r) = \frac{\partial \Phi(\mathbf{x}, r)}{\partial r} = \frac{M}{4\pi} \frac{r}{(\mathbf{x}^2 + r^2)^{3/2}}. \tag{20}$$

Velocity over body surface is computed taking into account that both variables *x* and *r* are related by the body stream function as follows:

$$\frac{M}{2}\left(1+\frac{\varkappa}{R}\right) = \pi r^2 U\_{\infty} \tag{21}$$

Resulting in

$$r^2 = \frac{a^2}{2} \left( 1 + \frac{x}{\sqrt{x^2 + r^2}} \right) \tag{22}$$

Taking nondimensional variables defined as

$$\chi = \frac{\mathfrak{x}}{2a} \tag{23}$$

$$
\zeta = \frac{r}{2a}.\tag{24}
$$

Both variables are related by the stream function as follows:

$$8\zeta^2 = 1 + \frac{\chi}{\sqrt{\chi^2 + \zeta^2}}.\tag{25}$$

The variable *χ* results in a function of *ζ*, given by,

$$\chi^2 = \frac{\left(8\zeta^2 - 1\right)^2}{16\left(1 - 4\zeta^2\right)}.\tag{26}$$

Finally, the longitudinal velocity component is given by the following expression:

*Aerodynamics of Mars 2020 Rover Wind Sensors DOI: http://dx.doi.org/10.5772/intechopen.90912*

$$\frac{u(\varkappa,r)}{U\_{\infty}} = 1 + \frac{1}{16} \frac{\varkappa}{\sqrt{\left(\varkappa^2 + \zeta^2\right)^3}} \cdot \tag{27}$$

And simplifying, we can find an expression that only depends on the unique variable, *ζ*, as follows:

$$\frac{u(\varkappa,r)}{U\_{\infty}} = 1 + \frac{(8\zeta^2 - 1)\sqrt{\left(1 - 4\zeta^2\right)}}{64}.\tag{28}$$

**Figure 5** shows the graph of nondimensional velocity *u/U*<sup>∞</sup> versus the nondimensional distance *x/2a*. The velocity is growing from the stagnation point located at *χ<sup>P</sup>* ¼ �1*=*4 until unity nondimensional velocity that is reached when curve cross the origin (*χ* ¼ 0). After that, a peak is produced, and the freestream velocity value is got for a longitudinal distance *x/2a = 1.5. S*o that, when the flow is coming to a boom section located at a distance of approximately equal to three times the radius (*x = 3a*), the wind flows with the freestream velocity, and the hot film sensor will be refrigerated by the same manner as in nonperturbed conditions.

#### **6.2 Transversal flow**

On the other hand, the Stokes stream function *Ψ* for this flow is given by,

<sup>1</sup> � *<sup>x</sup> R* � �

The half-body surface stream function is equal to the flow rate coming from the inner of the half-body. Taking into account that it is a body of revolution, transversal area is *<sup>A</sup>* <sup>¼</sup> *<sup>π</sup>a*<sup>2</sup> and the flow coming from the inner body, crossing the circular area denoted as *A* (see **Figure 4**), is *AU*∞, exactly the strength of the point source

The dividing streamline represents the surface body, and it is obtained when the

þ *πr* 2

> *M* 4*π*

*x*

*r*

<sup>1</sup> � *<sup>x</sup> R* � �

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>U</sup>*<sup>∞</sup> <sup>þ</sup>

Velocity over body surface is computed taking into account that both variables *x*

¼ *πr* 2

*x* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> <sup>p</sup> � �

∂Φð Þ *x*,*r <sup>∂</sup><sup>r</sup>* <sup>¼</sup> *<sup>M</sup>* 4*π*

1 þ *x R* � �

1 þ

*<sup>χ</sup>* <sup>¼</sup> *<sup>x</sup>* 2*a*

*<sup>ζ</sup>* <sup>¼</sup> *<sup>r</sup>* 2*a*

<sup>8</sup>*ζ*<sup>2</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>χ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>χ</sup>*<sup>2</sup> <sup>¼</sup> <sup>8</sup>*ζ*<sup>2</sup> � <sup>1</sup> � �<sup>2</sup>

Finally, the longitudinal velocity component is given by the following

þ *πr* 2

*U*∞*:* (17)

*U*∞*:* (18)

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> ð Þ<sup>3</sup>*=*<sup>2</sup> (19)

*<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*<sup>2</sup> ð Þ<sup>3</sup>*=*<sup>2</sup> *:* (20)

*U*<sup>∞</sup> (21)

*:* (24)

*<sup>χ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>p</sup> *:* (25)

16 1 � <sup>4</sup>*ζ*<sup>2</sup> � � *:* (26)

(22)

(23)

*M* 2

*Ψ*ð Þ¼ *r*, *z*

(*<sup>M</sup>* <sup>¼</sup> *<sup>π</sup>a*2*U*∞), and the distance *<sup>d</sup>* is computed being *d = a/2.*

*<sup>M</sup>* <sup>¼</sup> *<sup>M</sup>* 2

∂Φð Þ *x*,*r*

stream function is equal to the source strength, *M*,

*u x*ð Þ¼ ,*r*

*w x*ð Þ¼ ,*r*

and *r* are related by the body stream function as follows:

*M* 2

*r* <sup>2</sup> <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> 2

Both variables are related by the stream function as follows:

The variable *χ* results in a function of *ζ*, given by,

Taking nondimensional variables defined as

And the corresponding velocity is

*Mars Exploration - A Step Forward*

Resulting in

expression:

**76**

Wind flowing transversal to the boom (boom in cross-flow) can be studied as a first approximation by the two-dimensional complex potential flow over a circular cylinder, as represented in **Figure 6**.

**Figure 5.** *Nondimensional wind velocity on rover boom in axial flow.*

**Figure 6.** *Potential flow of a cylinder.*

The complex potential *w z*ð Þ corresponding to a cylinder with radius *a* is given by the superposition of a uniform stream and a doublet [12]

$$w(z) = U\_{\infty} \left( z + \frac{a^2}{z} \right) \tag{29}$$

where *z* is the complex variable defines as follows:

$$z = \varkappa + i\mathfrak{y}.\tag{30}$$

Real and imaginary parts of *w* define the potential of velocity *ϕ* and the stream function *ψ*, respectively,

$$\phi = U\_{\infty} \left( r + \frac{a^2}{r} \right) \cos \theta \tag{31}$$

$$
\psi = U\_{\infty} \left( r - \frac{a^2}{r} \right) \sin \theta. \tag{32}
$$

over the body wall needs zero velocity in the wall and the outer velocity (*U*∞) in the boundary layer limit. The boundary layer is a thin layer where the effects of viscous forces are of the same order of the inertial forces. The Reynolds number of flow outer boundary layer is larger than unity, and consequently, the inertial forces are larger than the viscous. When the wind is flowing on the boom surface, the boundary layer is growing downstream and the turbulence and adverse pressure gradients produce the separation of the boundary layer and the detachment of flow, as it can be observed in **Figure 8**. In this situation, wind is flowing over booms in very complex conditions to be simulated analytically and the flow coming to each side of boom depends on the incidence angle of wind, so that thermal sensors are operating in a no ideal condition and heat transfer mechanisms are far from a theoretical environment. It is because the booms are calibrated isolated in order to characterize the output electrical signals from sensor when they are subjected to different wind

*Real flow around a circular cylinder: Upstream cylinder (left) and behind the cylinder (right).*

*Aerodynamics of Mars 2020 Rover Wind Sensors DOI: http://dx.doi.org/10.5772/intechopen.90912*

The external surface of rover is not designed following aerodynamics criteria. The main body of this vehicle is a box with rectangular section, supported by six wheels. A vertical mast is erected on the upper surface of the main box body. The instruments of MEDA are installed over this mast. Two booms are located perpendicular to the axis mast. The rover vehicle is larger than other elements as mast and booms, and the flow viewed by small devices is mainly affected by the presence of

conditions.

**Figure 7.**

the rover.

**79**

**Figure 8.**

**7. Rover vehicle aerodynamics**

*Boundary layer over the boom.*

The conjugate velocity is obtained directly by derivation as follows:

$$\frac{dw}{dz} = U\_{\infty} \left( 1 - \frac{a^2}{z^2} \right) = u - iv. \tag{33}$$

This expression can be simplified by making use of the Euler formula *<sup>z</sup>*�<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*�<sup>2</sup>*e*�*i*2*<sup>θ</sup>* <sup>¼</sup> *<sup>a</sup>*�<sup>2</sup>ð Þ *cos* <sup>2</sup>*<sup>θ</sup>* � *i sin* <sup>2</sup>*<sup>θ</sup>* , and the velocity components over the cylinder surface are the followings:

$$u = 2U\_{\infty} \sin^{2}\theta \tag{34}$$

$$v = -2U\_{\infty} \sin \theta \cos \theta. \tag{35}$$

The square of velocity magnitude is given by

$$\left|V\right|^2 = \mathfrak{u}^2 + \mathfrak{v}^2\tag{36}$$

$$\left|V\right|^2 = U\_{\infty}^2 \left[ (\mathbf{1} - \cos 2\theta)^2 + (\sin 2\theta)^2 \right] \tag{37}$$

$$\left|V\right|^2 = 2U\_{\infty}^2(1 - \cos 2\theta). \tag{38}$$

And finally, the modulus of velocity over the cylinder surface is

$$|V| = 2\,\,U\_{\infty}\sin\theta.\tag{39}$$

When the angle is zero, the velocity is zero, corresponding to a stagnation point, but when angle is 90°, the velocity is 2 *U*∞, rising to the maximum value of velocity.

The potential flow approximation is valid only in face front the flow, but it is not valid in the wake because the flow is detached. **Figure 7** shows a comparison view between attached flow upstream cylinder (left) and detached flow behind the cylinder (right).

#### **6.3 Boundary layer**

Potential flow is used as a first estimation of flow parameters, but it is not a real flow because viscosity is not present in these flow models. The no-slip condition

*Aerodynamics of Mars 2020 Rover Wind Sensors DOI: http://dx.doi.org/10.5772/intechopen.90912*

The complex potential *w z*ð Þ corresponding to a cylinder with radius *a* is given by

Real and imaginary parts of *w* define the potential of velocity *ϕ* and the stream

*a*2 *r* � �

*r* � �

*z*2 � �

This expression can be simplified by making use of the Euler formula *<sup>z</sup>*�<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*�<sup>2</sup>*e*�*i*2*<sup>θ</sup>* <sup>¼</sup> *<sup>a</sup>*�<sup>2</sup>ð Þ *cos* <sup>2</sup>*<sup>θ</sup>* � *i sin* <sup>2</sup>*<sup>θ</sup>* , and the velocity components over the cylinder

*<sup>u</sup>* <sup>¼</sup> <sup>2</sup>*U*<sup>∞</sup> *sin* <sup>2</sup>

<sup>∞</sup> ð Þ <sup>1</sup> � *cos* <sup>2</sup>*<sup>θ</sup>* <sup>2</sup> <sup>þ</sup> ð Þ *sin* <sup>2</sup>*<sup>θ</sup>* <sup>2</sup> h i

When the angle is zero, the velocity is zero, corresponding to a stagnation point, but when angle is 90°, the velocity is 2 *U*∞, rising to the maximum value of velocity. The potential flow approximation is valid only in face front the flow, but it is not valid in the wake because the flow is detached. **Figure 7** shows a comparison view between attached flow upstream cylinder (left) and detached flow behind the

Potential flow is used as a first estimation of flow parameters, but it is not a real flow because viscosity is not present in these flow models. The no-slip condition

*ϕ* ¼ *U*<sup>∞</sup> *r* þ

*<sup>ψ</sup>* <sup>¼</sup> *<sup>U</sup>*<sup>∞</sup> *<sup>r</sup>* � *<sup>a</sup>*<sup>2</sup>

The conjugate velocity is obtained directly by derivation as follows:

*dz* <sup>¼</sup> *<sup>U</sup>*<sup>∞</sup> <sup>1</sup> � *<sup>a</sup>*<sup>2</sup>

*a*2 *z* � �

*z* ¼ *x* þ *iy:* (30)

*cosθ* (31)

*sinθ:* (32)

¼ *u* � *iv:* (33)

*θ* (34)

*v* ¼ �2*U*∞*sinθcosθ:* (35)

j j *<sup>V</sup>* <sup>2</sup> <sup>¼</sup> *<sup>u</sup>*<sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>2</sup> (36)

j j *V* ¼ 2 *U*∞*sinθ:* (39)

<sup>∞</sup>ð Þ 1 � *cos* 2*θ :* (38)

(29)

(37)

*w z*ð Þ¼ *U*<sup>∞</sup> *z* þ

the superposition of a uniform stream and a doublet [12]

where *z* is the complex variable defines as follows:

*dw*

The square of velocity magnitude is given by

j j *<sup>V</sup>* <sup>2</sup> <sup>¼</sup> *<sup>U</sup>*<sup>2</sup>

j j *<sup>V</sup>* <sup>2</sup> <sup>¼</sup> <sup>2</sup>*U*<sup>2</sup>

And finally, the modulus of velocity over the cylinder surface is

function *ψ*, respectively,

*Mars Exploration - A Step Forward*

surface are the followings:

cylinder (right).

**78**

**6.3 Boundary layer**

**Figure 7.** *Real flow around a circular cylinder: Upstream cylinder (left) and behind the cylinder (right).*

over the body wall needs zero velocity in the wall and the outer velocity (*U*∞) in the boundary layer limit. The boundary layer is a thin layer where the effects of viscous forces are of the same order of the inertial forces. The Reynolds number of flow outer boundary layer is larger than unity, and consequently, the inertial forces are larger than the viscous. When the wind is flowing on the boom surface, the boundary layer is growing downstream and the turbulence and adverse pressure gradients produce the separation of the boundary layer and the detachment of flow, as it can be observed in **Figure 8**. In this situation, wind is flowing over booms in very complex conditions to be simulated analytically and the flow coming to each side of boom depends on the incidence angle of wind, so that thermal sensors are operating in a no ideal condition and heat transfer mechanisms are far from a theoretical environment. It is because the booms are calibrated isolated in order to characterize the output electrical signals from sensor when they are subjected to different wind conditions.

**Figure 8.** *Boundary layer over the boom.*
