CFD Analysis of Natural Convection Performance of a MMRTG Model Under Martian Atmospheric Conditions

Abstract

Understanding the thermal behaviour of radioisotope generators under Martian conditions is essential for the safe and efficient operation of planetary exploration rovers. This study investigates the heat transfer and flow mechanisms around a simplified full-scale model of the Multi-Mission Radioisotope Thermoelectric Generator (MMRTG) by means of Computational Fluid Dynamics (CFD) simulations performed with ANSYS Fluent 2023 R1. The model consists of a central cylindrical core and eight radial fins, operating under pure CO₂ at a pressure of approximately 600 Pa, representative of the Martian atmosphere. Four cases were simulated, varying both the reactor surface temperature (373–453 K) and the ambient temperature (248 to 173 K) to reproduce typical diurnal and seasonal scenarios on Mars. The results show the formation of a buoyancy-driven plume rising above the generator, with peak velocities between 1 and 3.5 m/s depending on the thermal load. Temperature fields reveal that the fins generate multiple localized hot spots that merge into a single vertical plume at higher elevations. The calculated dimensionless numbers (Grashof ≈ 10⁵, Rayleigh ≈ 10⁵, Reynolds ≈ 10², Prandtl ≈ 0.7, Nusselt ≈ 4) satisfy the expected range for natural convection in low-density CO₂ atmospheres, confirming the laminar regime. These results contribute to a better understanding of heat dissipation processes in Martian environments and may guide future design improvements of thermoelectric generators and passive thermal management systems for space missions.

1. Introduction

The study of the celestial bodies that make up the solar system has always been a great challenge for mankind. To this end, owing to technology, instruments such as telescopes, satellites and space vehicles have been built to explore the solar system and make great discoveries in the field of astronomy. Since the turn of the century, interest in Mars has grown notably, driven by the discovery of evidence of liquid water on its surface and the consequences this may have. From the first Viking mission [1] to Mars in 1975 to the recently launched Rover Perseverance in 2021 [2], massive progress has been made in characterising the red planet. Perseverance is the most advanced rover NASA has sent to Mars to date, equipped with the most cutting-edge technology [3]. Mars rovers are equipped with numerous instruments that allow them to analyse the environment around them and to take geological samples from the surface they are travelling on (Figure 1).

Recent studies have investigated the aerodynamic behaviour of various devices in the Martian atmosphere, for example, rotor-based craft such as rectangular-blade rotors under low-density carbon dioxide (CO₂) (Huang et al.) [4] and propeller systems adapted to the ultra-low Reynolds number environment of Mars (Zhang et al.) [5]. These studies focus primarily on lift, drag, and flow separation characteristics of flying vehicles in the rarefied Martian atmosphere. Furthermore, other investigations have also evaluated the flow field and aerodynamic interference around rover-mounted wind sensors, recognising how rover geometries disturb ambient measurement and environmental flows. Rodriguez-Sevillano et al. [6] compare the aerodynamics of the Mars rover 2020 wind sensors obtained in a water channel with those obtained in a wind tunnel. On the other hand, García-Magariño et al. [7] studied the aerodynamics of the Mars Rover 2020 wind sensors using a Particle Image Velocimetry (PIV) system in order to obtain the correction functions of the incident wind speed as a function of its direction. However, the existence of other instruments has an influence on the aerodynamics of the rover itself, and this influence could be important when processing and analysing the data sent by the rover during its service life on the red planet.

For this reason, it could be interesting to study the aerodynamic link to natural convection heat transfer around a radioisotope thermoelectric generator (MMRTG) [8] which is incorporated in Mars rover vehicles. The MMRTG converts heat from natural radioactive plutonium into electrical power for the rover’s engineering systems and scientific payload. Understanding external thermal dissipation under Martian atmospheric conditions is critical to ensure stable and reliable power output. Previous studies have shown that the heat released by the MMRTG evolves with seasonal and diurnal cycles, affecting overall rover thermal management [9]. On the other hand, it has been evaluated that buoyant flows in the complex Martian boundary layer significantly affect the convective efficiency of heat dissipation systems [10]. Several works have explored natural convection from heated bodies under low-pressure CO₂ conditions, either via experimental testing or Computational Fluid Dynamics (CFD) approaches, highlighting the relevance of Prandtl and Grashof numbers in low-density regimes [11,12]. More recently, Bardera et al. [13] studied the thermal convection generated by a heated flat plate under Martian environmental conditions, offering insights into flow behaviour. This configuration, while simplified, can be considered analogous to the heat transfer from a single fin of the MMRTG.

The present work extends this line of research by analysing the complete cylindrical core of the MMRTG together with its eight radial fins, under realistic Martian atmospheric conditions and in full scale. Four distinct operating cases are evaluated using CFD simulations in a commercial CFD software (ANSYS Fluent 2023 R1), varying both the reactor surface temperature and the ambient temperature, to reproduce representative thermal scenarios such as day/night cycles and operational conditions of the reactor. All simulations are conducted with the pressure-based solver, incorporating the thermophysical properties of Martian CO₂ at ~600 Pa [14].

The novelty of this study lies in bridging the gap between idealized models of isolated surfaces and the realistic three-dimensional thermal behaviour of a complete MMRTG under Martian atmospheric conditions. Unlike previous works that focused on simplified flat-plate [13] or two-dimensional analyses, the present research provides full-scale CFD simulations that capture the coupled influence of the cylindrical core and radial fins on natural convection. These results are not only relevant for scientific understanding but also provide practical insight for optimizing and enhancing heat dissipation in future radioisotope thermoelectric generators used in planetary missions.

2. Theoretical Background

2.1. Natural Convection

Natural convection, also referred to as free convection, is a heat-transfer mechanism driven by buoyancy forces that arise from density gradients in a fluid exposed to temperature differences. When a solid surface is heated, the adjacent fluid layer decreases its density and rises under the influence of gravity, while cooler and denser fluid descends to replace it, creating a convective flow. Unlike forced convection, where flow is influenced by external sources, natural convection is exclusively governed by thermal gradients and gravitational acceleration. Its efficiency depends mainly on two factors: the effects of buoyancy, associated with the Grashof number, and thermal diffusion, which is represented by the Prandtl number. These parameters determine whether the flow remains laminar or transitions to turbulence. It is important to mention that this process plays a key role in many terrestrial and extra-terrestrial applications, ranging from electronic cooling to planetary thermal control systems [15,16].

In Martian atmospheric conditions, natural convection becomes particularly challenging due to the very low ambient pressure and the predominance of CO₂, which is associated with a reduced fluid density and a significant alteration in buoyancy flow structures. Several works have shown that the start and strength of natural convection are drastically limited under these conditions, making radiative heat transfer relatively more important compared to Earth [17]. Nevertheless, studies of heated surfaces and simplified geometries, such as flat plates or cylinders, demonstrate that buoyant flows can still develop and affect the heat dissipation of critical components like the MMRTG of the Rover vehicles [10]. Understanding this phenomenon under Martian conditions is therefore essential for predicting the thermal behaviour of space systems and ensuring their long-term reliability.

2.2. Governing Equations

Before defining the equations that are involved in the problem, it has been considered of interest to define the main dimensionless numbers that will appear in this study in order to define easily the different scenarios of the problem. Additionally, in fluid mechanics it is highly recommended to work with this kind of parameter to facilitate the interpretation of the equations [18]. The dimensionless parameters that will be used during this work are the following ones:

• The Reynolds number quantifies the relative importance of inertial to viscous forces in a fluid flow. It is defined as follows:

$$Re={\displaystyle \frac{uL}{\nu }}$$

(1)

where $u$ is a characteristic velocity, $L$ a characteristic length and ν the kinematic viscosity. Low Reynolds numbers correspond to laminar regimes dominated by viscous effects, whereas high values indicate the existence of turbulence [18]. Furthermore, the Reynolds number can also be expressed with the buoyant velocity $u_r$ taking into account buoyancy effects. It is obtained from dimensional analysis, scaling with the thermal expansion coefficient ($\beta$), gravity ($g$) and imposed temperature difference between the surface temperature $\left(T_s\right)$ and the ambient temperature ($T_a$):

$$u_r=\sqrt{g\beta L(T_s-T_a)}$$

(2)

thus, the Reynolds number becomes:

$$Re=\sqrt{{\displaystyle \frac{g\beta L^3(T_s-T_a)}{{\nu }^2}}}=\sqrt{Gr}$$

(3)

where Gr is the Grashof number, which compares buoyancy to viscous forces. This formulation highlights the strong link between buoyancy-driven flows and the Reynolds number, showing that in thermal convection Re is not an independent parameter but directly connected to the Grashof number [15]. The following expressions shows the combination between these parameters associated with different thermal convection cases:

$${Re}^2\gg Gr Forced convection$$

(4)

$${Re}^2~Gr Mixed convection$$

(5)

$${Re}^2\ll Gr Free convection$$

(6)

• The Prandtl number is the ratio between the kinematic viscosity and the thermal diffusivity ($\alpha$). Both of them are intrinsic properties of a fluid.

$$Pr={\displaystyle \frac{\nu }{\alpha }}$$

(7)

• The Rayleigh number is the product of the Grashof and Prandtl numbers:

$$Ra=Gr·Pr={\displaystyle \frac{g\beta L^3(T_s-T_a)}{\nu \alpha }}$$

(8)

which is the principal parameter governing the onset and regime of natural convection. Below a critical Ra, heat transfer is mainly conductive; above it, convection becomes significant and turbulent natural convection develops [16].

• The Nusselt number expresses the relationship between the heat flow exchanged by convection and the heat flow exchanged by conduction. This parameter can be defined as follows:

$$Nu={\displaystyle \frac{h·L}{k}}$$

(9)

where ℎ is the convective heat-transfer coefficient, L is the characteristic length of the geometry (in this case, the MMRTG diameter), and k is the thermal conductivity of CO₂ under Martian conditions which is estimated to be 1.4 × 10⁻² W·m⁻¹·K⁻¹ for a temperature of 200 K, according to the correlations of Huber et al. [19]. In this study, ℎ is obtained from the average wall heat flux $q$ and the temperature difference between the MMRTG surface and the surrounding atmosphere:

$$h={\displaystyle \frac{q}{T_s-T_a}}$$

(10)

The problem of thermal convection can be described by the conservation equations of mass, momentum, and energy for a Newtonian fluid with variable density in the buoyancy term. Under the Boussinesq approximation, the dimensionless equations are:

$${\nabla }^{*}{\overrightarrow{V}}^{*}=0$$

(11)

$${\displaystyle \frac{D{\overrightarrow{V}}^{*}}{D{t}^{*}}}={\displaystyle \frac{\overrightarrow{g}\beta L(T_s-T_a)}{u_r^2}}{T}^{*}+{\displaystyle \frac{1}{{Gr}^{1/2}}}{\nabla }^{*2}{\overrightarrow{V}}^{*}$$

(12)

$${\displaystyle \frac{D{T}^{*}}{D{t}^{*}}}={\displaystyle \frac{1}{Pr·{Gr}^{1/2}}}{\nabla }^{*2}{T}^{*}$$

(13)

where ${\overrightarrow{V}}^{*}={\displaystyle \raisebox{1ex}{$\overrightarrow{V}$}\!\left/ \!\raisebox{-1ex}{$u_{ref}$}\right.}$ and $t^{*}={\displaystyle \raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$L/u_{ref}$}\right.}$ are the dimensionless velocity and time. The dimensionless coordinates are expressed as $x^{*}={\displaystyle \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}$, $y^{*}={\displaystyle \raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}$ and $z^{*}={\displaystyle \raisebox{1ex}{$z$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}$. The parameter $T^{*}$ is defined as follows:

$$T^{*}={\displaystyle \frac{T-T_a}{T_s-T_a}}$$

(14)

2.3. Martian Atmosphere

To perform the simulations, it is essential to characterise the Martian atmospheric environment. Unlike Earth, Mars presents a markedly different gaseous composition which is linked with an altered thermophysical behaviour, offering a striking example of the variety of planetary atmospheres encountered within the solar system. Its atmosphere lacks the nitrogen–oxygen mixture typical of Earth and is instead dominated by carbon dioxide (CO₂), which governs most of its thermodynamic properties. Several works have assessed whether Martian CO₂ behaves as a Newtonian fluid, ensuring the applicability of conventional experimental and numerical approaches as a continuum media [20]. The validity of the continuum assumption under Martian conditions was verified by computing the Knudsen number $Kn=\lambda /L$, where λ is the mean free path of CO₂ molecules and L a characteristic length. For a mean free path $\lambda \approx 1.3\times {10}^{-5}$ m at 600 Pa and 210 K, and the characteristic length defined as the reactor diameter ($d=0.27$ m), the Knudsen number obtained is $Kn\approx 4.8\times {10}^{-5}$, confirming that the flow remains in the continuum regime ($Kn<{10}^{-2}$) [18].

The Martian atmosphere is also extremely complex, with pressures and densities much lower than those found on Earth [17]. As a result, convective heat transfer is considerably weaker under Martian conditions than in terrestrial environments. Furthermore, the planet experiences large thermal gradients (from 173 to 273 K), rapid fluctuations in temperature and wind velocity, and the presence of suspended dust particles during storm events [21]. Among the most distinctive phenomena are the so-called dust devils, vortical structures in which hot air masses ascend while colder fluid descends, promoting vertical mixing and efficient redistribution of dust.

In addition, Martian gravity (approximately one third that of Earth) plays a key role in natural convection processes. Although weaker, many studies have shown that classical boundary-layer theories developed for terrestrial conditions remain valid on Mars [20].

Table 1. Main thermophysical and environmental parameters of the Martian atmosphere.

Parameter	Approximate Value
Surface pressure	~600 Pa
Surface temperature	173 to 273 K
Atmospheric composition	~95.3% CO2, 2.7% N2, 1.6% Ar, traces of O2, CO, H2O
Density at surface ($\rho$)	0.011–0.018 kg/m3
Dynamic viscosity ($\mu$)	~1.0–1.2 × 10−5 Pa·s
Thermal conductivity ($k$)	~0.01–0.02 W/(m·K)
Gravity acceleration ($g$)	3.71 m/s2

summarizes the characteristic parameters of the Martian atmosphere [19].

The dimensionless numbers defined in Equations (3), (7) and (8) can be estimated considering the values of the above parameters (

Table 2. Characteristic ranges of dimensionless numbers governing natural convection in the Martian atmosphere.

Dimensionless Number	Typical Range of Values on Mars
Reynolds ($Re$)	102–103
Prandtl ($Pr$)	∼0.7
Grashof ($Gr$)	102–106
Rayleigh ($Ra$)	102–106
Nusselt ($Nu$)	3–5

) and other studies such as Sun, Y et al. [12] and Bardera et al. [13].

2.4. Radiative and Convective Heat Transfer Under Martian Conditions

The thermal energy released by the MMRTG into the Martian environment is governed mainly by the combined effects of radiation and natural convection. The steady-state surface energy balance can be expressed as follows:

$${\dot{q}}_{total}={\dot{q}}_{conv}+{\dot{q}}_{rad}$$

(15)

where:

$${\dot{q}}_{conv}=h·\left(T_s-T_a\right) \left[{\displaystyle \raisebox{1ex}{$\mathrm{W}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.}\right]$$

(16)

$${\dot{q}}_{rad}=\epsilon ·\sigma ·\left({T_s}^4-{T_a}^4\right) \left[{\displaystyle \raisebox{1ex}{$\mathrm{W}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2$}\right.}\right]$$

(17)

Here, $\epsilon$ is the surface emissivity, $\sigma =5.67\times {10}^{-8} {\displaystyle \raisebox{1ex}{$\mathrm{W}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^2·{\mathrm{K}}^4$}\right.}$ is the Stefan–Boltzmann constant. Equation (15) represents the classical energy balance at the surface of a radiating body [22,23].

Due to the extremely low atmospheric density on Mars, natural convection is very weak. Consequently, thermal radiation dominates the heat dissipation process from hot surfaces, such as the MMRTG. Several studies have confirmed that radiative heat losses in the Martian environment exceed convective losses by several orders of magnitude, even under moderate temperature differences [24,25].

The convective heat transfer coefficient $h$ can be related to the dimensionless Nusselt number. To estimate the natural convection heat-transfer around the MMRTG, the Churchill and Chu correlation [26] was adopted, which is valid for horizontal surfaces and cylinders over a wide range of Rayleigh numbers:

$$h=Nu·\left({\displaystyle \frac{k}{L_c}}\right)={\left(0.60+{\displaystyle \frac{0.387·{Ra}^{1/6}}{{\left[1+{\left(0.559/Pr\right)}^{9/16}\right]}^{8/27}}}\right)}^2·\left({\displaystyle \frac{k}{L_c}}\right)$$

(18)

This expression provides a smooth transition between laminar and turbulent regimes of natural convection and is particularly suitable for cylindrical geometries such as the MMRTG, enabling the estimation of ℎ for Martian CO₂ using known thermophysical properties presented in Table 1 and dimensionless numbers defined in Section 2.2.

2.5. MMRTG Model

The geometric model of the Multi-Mission Radioisotope Thermoelectric Generator (MMRTG) was developed using CATIA V5R21 to provide a simplified yet representative configuration for CFD simulations under Martian atmospheric conditions. The model consists of two main components:

• Central cylindrical core: representing the heat source region of the MMRTG.
• Eight radial fins: uniformly distributed around the cylindrical surface, functioning as extended surfaces that increase the effective heat dissipation area.

The MMRTG geometry employed in this study was designed to reproduce the approximate dimensions and configuration of the real units integrated into NASA’s Curiosity and Perseverance rovers [8]. The eight-finned structure and the main characteristic ratios between fin height, thickness, and cylinder length were selected to provide a representative baseline for analysing natural convection under Martian atmospheric conditions. The central cylinder is characterized by its diameter d and length L. Each fin is described by its height $h_f$, thickness $t_f$ and length $L_f$ measured radially from the cylinder wall. In

Table 3. Main dimensions of the MMRTG model.

Parameter	Symbol	Value (m)
Cylinder Diameter	d	0.27
Cylinder length	L	0.68
Fin length	$L_f$	0.66
Fin height	$h_f$	0.20
Fin thickness	$t_f$	0.01

, the values of these parameters are summarized.

Figure 2 shows two different views of the model of the MMRTG and the parameters defined above.

2.6. Study Cases

All simulations presented in this work correspond to natural convection scenarios, in which buoyancy forces induced by temperature gradients are the only driving mechanism of fluid motion. The scope of this study is to analyse the thermal interaction between the MMRTG and the Martian atmosphere under realistic operating conditions. To this end, two parameters were varied:

• Ambient temperature of Mars (${\mathit{T}}_{\mathit{a}}$), chosen within the representative diurnal and seasonal range observed on the planet (from 173 K during cold nights to 273 K during daytime conditions) [14].
• Surface temperature of the MMRTG (${\mathit{T}}_{\mathit{s}}$) which typically ranges between 323 K and 473 K, depending on the heat generation and electrical load demand [27].

By combining two representative values of each parameter, four distinct cases were defined. These cover operational scenarios corresponding to cold night conditions, warmer daytime conditions, and different MMRTG thermal loads. These scenarios are presented in

Table 4. Four natural convection study cases under Martian atmospheric conditions.

Case	${\mathit{T}}_{\mathit{s}}$ (MMRTG)	${\mathit{T}}_{\mathit{a}}$ (Ambient)	$∆\mathit{T}$	Conditions
1	373 K	223 K	150	Sunset conditions at mid-latitudes
2	423 K	173 K	250	Large thermal contrast in a nocturnal scenario
3	453 K	248 K	205	Peak reactor performance, warmer daytime atmosphere
4	423 K	273 K	150	High ambient temperature and reactor at elevated load

.

3. CFD Setup

3.1. Computational Domain

To simulate accurately the thermal behaviour of the MMRTG under Martian atmospheric conditions, a computational fluid dynamics (CFD) model was developed. The simulations were carried out with ANSYS Fluent [28], using a control volume sufficiently large to minimize the influence of boundary conditions on the flow around the generator. The computational domain was constructed as a rectangular enclosure, with the MMRTG model strategically positioned to ensure flow symmetry and to allow buoyancy-driven plumes to develop without artificial confinement effects.

The computational domain boundaries were defined to reproduce a free natural-convection environment under Martian conditions. The lower horizontal face was set as a pressure inlet, imposing the ambient temperature and static pressure corresponding to the Martian atmosphere (≈600 Pa). The upper horizontal face acted as a pressure outlet, allowing the heated CO₂ to exit the domain without artificial recirculation. The lateral faces ($S_1, S_2, S_3$ and $S_4$) were assigned as symmetry boundaries to reduce confinement and ensure plume development along the vertical direction.

All MMRTG external surfaces (cylinder and fins) were modelled as isothermal walls at the reactor surface temperature defined for each study case. This assumption is justified by estimating the Biot number defined as $Bi={\displaystyle \raisebox{1ex}{$h·L_c$}\!\left/ \!\raisebox{-1ex}{$k_{Al}$}\right.}$ [16]. Using the maximum convective heat transfer coefficient obtained in this study ($h=0.241 {\displaystyle \raisebox{1ex}{$\mathrm{W}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{K}·{\mathrm{m}}^2$}\right.})$, a characteristic length defined as the fin half-thickness ($L_c=0.5·t_f=0.005 \mathrm{m}$), and assuming a high-conductivity material such as an aluminium alloy ($k_{Al}=150 {\displaystyle \raisebox{1ex}{$\mathrm{W}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{K}·\mathrm{m}$}\right.}$), the resulting Biot number is $Bi\approx 8\times {10}^{-6}$. Since $Bi\ll 0.1$, internal conduction resistance is negligible compared to the fluid-side convection resistance, validating the isothermal wall assumption.

This configuration ensures physical consistency with natural-convection setups commonly adopted in previous CFD studies under low-density CO₂ atmospheres, such as Bardera et al. [13]. Figure 3 shows the control volume with the main dimensions.

3.2. Mesh Strategy

Regarding the meshing process, the computational domain was discretized using an unstructured tetrahedral grid. A global element size of 5.0 × 10⁻² m was imposed in the far field, while local refinement was applied around the MMRTG geometry (cylinder and fins) to capture steep thermal and velocity gradients. The local cell size near the solid surfaces was reduced down to 3.0 × 10⁻³ m, with additional refinement of approximately 5.0 × 10⁻⁴ m in high-curvature regions. Inflation layers (ten layers, growth rate 1.2) were included on all MMRTG surfaces to adequately resolve the thermal boundary layer.

To ensure mesh-independent results, a grid-sensitivity analysis was performed using four different meshes with increasing resolution, ranging from coarse to extra-fine. For each mesh configuration, key grid-quality metrics (number of cells, average skewness, minimum orthogonal quality and near-wall growth parameters) were evaluated and are summarized in

Table 5. Grid-quality metrics and thermal parameters for the four mesh configurations.

Mesh ID	Total Elements	Max. Skewness	Min. Orthogonal Quality	Max. Aspect Ratio	Min. Element Quality
1	$6.97\times {10}^6$	0.255	0.745	1.93	0.817
2	$12.07\times {10}^6$	0.243	0.757	1.88	0.825
3	$18.12\times {10}^6$	0.238	0.761	1.87	0.829
4	$24.13\times {10}^6$	0.239	0.760	1.86	0.828

.

All meshes exhibit good geometrical quality according to standard CFD criteria. The maximum skewness values remain below 0.26, which ensures accurate interpolation and numerical stability. The minimum orthogonal quality is consistently higher than 0.74, confirming that cell alignment and orthogonality are well preserved across the domain. The maximum aspect ratio is below 2, indicating nearly isotropic elements even in boundary-layer regions. The minimum element quality exceeds 0.81 for all cases, well above the recommended threshold of 0.1 for reliable convergence. These results confirm that all meshes fulfil the recommended ranges for high-fidelity CFD simulations [29].

The influence of grid refinement on the simulation results was assessed by comparing two representative global quantities: the area-averaged heat flux ($q$) and the area-averaged Nusselt number ($Nu$) over the MMRTG surfaces. The variation in these quantities with the total number of elements is presented in Figure 4, showing that both parameters asymptotically converge beyond approximately 18 million elements. Between Mesh 3 and Mesh 4, the variation in the area-averaged total heat flux and the average Nusselt number falls below 1%, indicating that further refinement. Therefore, the selected mesh (Figure 5) was deemed sufficiently fine to ensure accurate prediction of the natural convection flow and thermal fields under Martian conditions.

3.3. Solver

To assess the influence of turbulence modelling under the low-density Martian environment, cases were initially solved using both a laminar flow model and Shear Stress Transport (SST) $k$−$\omega$ turbulence model [30,31]. The results obtained for key thermal variables, such as the area-averaged heat flux and Nusselt number, showed negligible differences between both approaches (less than 2% variation). This confirms that the flow regime around the MMRTG operates within the laminar or transitional range, consistent with the relatively low Rayleigh numbers computed for all study cases. Nevertheless, the (SST) $k$−$\omega$ model was selected for the final simulations presented in this work, since it provides enhanced near-wall resolution and robustness in predicting buoyancy-driven flows [32,33], ensuring stable convergence for all configurations under Martian atmospheric conditions. The working fluid was defined as pure CO₂, consistent with the real Martian atmosphere, and the operating pressure was fixed at 600 Pa to replicate average conditions at the surface of Mars. The fluid density was evaluated using the ideal-gas law, which enables accurate computation of buoyancy effects through direct coupling between temperature and density.

The governing equations for continuity, momentum, and energy were solved with second-order discretization schemes to improve numerical accuracy. Convergence was monitored by tracking the residuals of all governing equations (set to 10⁻⁶) [34]. Under these criteria, all simulations converged satisfactorily and the $y^{+}$ value obtained was $y^{+}<0.1$ for the four study cases.

4. Results

In this section, the results of the CFD simulations performed for the four study cases summarized in Table 4 are presented. The analysis focuses on the thermal and flow fields generated by the MMRTG under different Martian atmospheric conditions. Specifically, the results include temperature contour maps that illustrate the development of the buoyant thermal plume, temperature profiles along different sections to quantify heat dispersion in the surrounding medium, and velocity fields associated with the free-convective motion induced by the reactor.

On the one hand, Figure 6 presents the temperature contours for the four cases, represented on two orthogonal planes (xz and yz) that intersect the symmetry plane of the MMRTG model. In all cases, the heated cylindrical core and radial fins act as the primary sources of thermal energy, generating a buoyant plume that extends vertically under Martian conditions. The xz-plane highlights the longitudinal development of the thermal wake, whereas the yz-plane emphasizes the influence of the finned geometry on the distribution of the hot jet.

In Case 1 ($Tₛ=373 \mathrm{K}, Tₐ=223 \mathrm{K}$), corresponding to sunset conditions, a relatively moderate thermal plume can be observed, with a confined region of heated flow rising above the MMRTG model. Case 2 ($Tₛ=423 \mathrm{K}, Tₐ=173 \mathrm{K}$) provide the largest temperature gradient (ΔT = 250 K), producing a stronger buoyancy effect and a more intense plume that extends vertically with higher thermal contrast. Case 3 ($Tₛ=453 \mathrm{K}, Tₐ=248 \mathrm{K}$) represents the scenario with the highest reactor temperature. The resulting plume is larger and more energetic, although the reduced temperature difference with respect to the ambient moderates the contrast in the far field. Finally, Case 4 ($Tₛ=423 \mathrm{K}, Tₐ=273 \mathrm{K}$) corresponds to the warmest atmospheric condition. Here, the plume is still present but the overall contrast is reduced due to the smaller ΔT, resulting in smoother temperature gradients in the surrounding fluid. Overall, the four cases clearly illustrate the vertical alignment of the plumes which confirms the dominance of gravity-driven flow for these cases in the Martian atmosphere.

Figure 7 shows the temperature distribution along the transversal direction y at different axial positions z above the MMRTG, for each of the four operating cases considered. Four sections have been selected: the first one located close to the generator surface ($z_1$), and the subsequent ones ($z_2, { z}_3$ and $z_4$) separated by a constant distance equal to the reactor diameter (d). This configuration allows for an evaluation of the evolution of the thermal plume as it rises along the vertical direction.

At the closest section to the MMRTG ($z_1$), the temperature distribution exhibits multiple local maximums. In particular, three distinct peaks can be identified, which correspond to the alignment of the heated fins with the measurement axis. This evidences the strong influence of the geometry on the near-field temperature field. As the axial distance increases, these local peaks progressively merge into a single central value of temperature located at $y=0$. Furthermore, the intensity of the maximum temperature decreases, and the plume becomes broader and smoother due to mixing with the surrounding colder CO₂. Comparing the different cases, Case 2 shows the largest gradients close to the reactor, driven by the strong thermal difference ($∆T=250 \mathrm{K}$), while Case 4 presents more uniform profiles because of the relatively warm ambient temperature.

On the other hand, velocity contours induced by the temperature gradient between the MMRTG model and the environment are presented in Figure 8.

As expected, the buoyant flow induced by the thermal gradients forms a vertical plume above the MMRTG, which starts near the fins and the immediate wake region. The maximum velocities obtained in these simulations lie in the range of 1–3 m/s, depending on the operating case. Case 2, with the largest temperature difference ($∆T=250 \mathrm{K}$) produces the strongest plume, with velocities peaking close to 3.5 m/s, whereas Case 1 and Case 4 show more moderate values around 1.5–2.0 m/s. Case 3, corresponding to the highest reactor temperature, also develops a vigorous plume, though slightly less intense than Case 2 due to the warmer ambient conditions that reduce buoyancy.

In order to provide a quantitative characterization of the thermal behaviour of the MMRTG under Martian conditions, the relevant dimensionless parameters were calculated for each of the four operating cases. The characteristic thermophysical properties of carbon dioxide were evaluated at the film temperature, defined as $T_f= 0.5·(T_s+T_a)$, which represents the average between the reactor surface temperature and the ambient atmospheric temperature [35].

Using this approach ensures consistency with standard heat transfer methodologies and allows for a comparison with previous studies of natural convection in low-density CO₂ atmospheres. The Reynolds, Grashof, Rayleigh, and Prandtl numbers were calculated following Equations (1)–(8). These parameters are presented in

Table 6. Dimensionless parameters for each case analysed.

Case	${\mathit{T}}_{\mathit{s}}\left(\mathbf{K}\right)$	${\mathit{T}}_{\mathit{a}}\left(\mathbf{K}\right)$	${\mathit{T}}_{\mathit{f}}\left(\mathit{K}\right)$	${\mathit{u}}_{\mathit{r}}(\mathbf{m}/\mathbf{s})$	$\mathit{P}\mathit{r}$	$\mathit{G}\mathit{r}$	$\mathit{R}\mathit{e}$	$\mathit{R}\mathit{a}$
1	373	223	298	1.13	0.74	4.63 × 105	681	3.45 × 105
2	423	173	298	1.46	0.74	7.72 × 105	879	5.75 × 105
3	453	248	351	1.21	0.77	3.89 × 105	624	3.00 × 105
4	423	273	348	1.04	0.77	2.91 × 105	539	2.24 × 105

, together with the estimated characteristic buoyant velocity.

To quantify the thermal performance of the MMRTG under Martian atmospheric conditions, the results obtained from the CFD simulations were post-processed to extract the area-averaged values of key heat-transfer parameters.

Table 7. Comparison between Nusselt number obtained from CFD and from theoretical definition.

Case	${\mathit{T}}_{\mathit{s}}\left(\mathbf{K}\right)$	${\mathit{T}}_{\mathit{a}}\left(\mathbf{K}\right)$	$∆\mathit{T}$	$\mathit{q}\left(\frac{\mathit{W}}{{\mathit{m}}^2}\right)$	$\mathit{h}\left(\frac{\mathit{W}}{{\mathit{m}}^2·\mathit{K}}\right)$	$\overline{\mathit{N}\mathit{u}}$	$\mathit{N}\mathit{u}$
1	373	223	150	28.293	0.189	3.642	3.638
2	423	173	250	60.235	0.241	4.653	4.647
3	453	248	205	37.443	0.183	3.527	3.523
4	423	273	150	24.392	0.163	3.140	3.136

summarizes, for each of the four study cases, the total heat flux released by the MMRTG surfaces and the corresponding area-averaged Nusselt number derived from the simulations. In addition, the theoretical Nusselt number expected for natural convection was estimated, allowing for a direct comparison between the numerical predictions and the analytical formulation (Equation (9)). This analysis provides a clear assessment of the influence of temperature gradients on buoyancy-driven flow under low-density Martian conditions.

The calculated heat-transfer coefficients ℎ are extremely low, ranging from 0.163 to 0.241 W/m²·K, consistent with the weak natural convection expected in the low-density CO₂ atmosphere of Mars.

Finally, the convective (${\dot{q}}_{conv}$) and radiative (${\dot{q}}_{rad}$) heat fluxes obtained from CFD simulations (using the Discrete Ordinates (DO) radiation model) are analysed and compared with the theoretical estimations derived from the energy balance equations introduced in Section 2.4. Figure 9 presents a bar chart comparing the theoretical and CFD results for each case study.

The results show excellent agreement between the theoretical and CFD values, particularly for the radiative component, where both results are nearly identical. For convection, a small deviation is observed, mainly due to the simplified estimation of the convective coefficient $h$ derived from the adopted empirical correlation (Equation (18)). Furthermore, the simulations revealed that the inclusion of radiative effects influences only the total heat flux, without significantly altering the velocity or temperature contours in the computational domain. This behaviour confirms that, under the low-density Martian atmosphere, radiation mainly acts as a surface heat dissipation mechanism, whereas the flow field and temperature distribution remain governed by convective transport.

5. Conclusions

The present work has numerically investigated the thermal and flow behaviour of a generic MMRTG model under Martian atmospheric conditions, considering four representative operating scenarios that combine different reactor surface and ambient temperatures. All cases were solved using CFD with a pressure-based solver, reproducing the thermophysical properties of CO₂ at ~600 Pa.

The robustness of the numerical results was assessed to ensure reliability of the simulations. Numerical uncertainty from the grid refinement study (Section 3.2) was found to be minimal: the variation in both the area-averaged Nusselt number and the total heat flux between Mesh 3 (18.12 million elements) and Mesh 4 (24.13 million elements) was less than 1%, indicating mesh convergence. Model-form sensitivity was evaluated by comparing the final SST $k-\omega$ turbulence model results with those obtained using a purely laminar flow model. The comparison showed negligible differences, less than 2% variation in the key thermal variables. Finally, input sensitivity to the thermal gradient is evident across the four test cases: the plume strength and the total heat flux scale directly with the imposed temperature difference ($\Delta T$), which is the primary driving force of the natural convection flow.

The results show that the MMRTG generates a persistent thermal plume, whose structure and intensity strongly depend on the temperature difference between the reactor surface and the environment. Higher reactor surface temperatures and lower ambient conditions lead to stronger buoyant forces, enhancing both the plume extension and the induced velocities. The analysis of temperature contours revealed localized heating around the fins and a gradual narrowing of the plume with increasing distance from the source, whereas the velocity fields highlighted the upward acceleration of the flow near the base followed by stabilization at higher elevations. The cross-sectional profiles confirmed the role of the fins in shaping the plume close to the reactor, producing three distinct peaks in the thermal distribution at short distances that progressively merge into a single value of temperature.

The evaluation of the dimensionless parameters confirmed the consistency of the numerical predictions with typical Martian natural convection regimes. In all four cases, the Prandtl number remained close to unity, reflecting the comparable momentum and thermal diffusivities of carbon dioxide under low-pressure conditions. The calculated Grashof and Rayleigh numbers were in the range reported for buoyancy-driven flows on Mars, ensuring that free convection is the dominant transport mechanism. Moreover, the Reynolds numbers and buoyant velocities estimated were consistent with the values expected in CO₂ atmospheres. Finally, for the four cases, the Nusselt number shows excellent agreement with the area-averaged Nusselt obtained from CFD simulations. This consistency confirms that the simulated convective heat transfer behaviour follows the expected scaling for buoyancy-driven flow around a cylindrical body under low-pressure CO₂ conditions typical of Mars.

Including the radiative effect in the simulations, the results confirm that the radiative heat flux clearly dominates over the convective, being roughly one order of magnitude higher across all studied cases. On average, radiative heat transfer accounts for over 90% of the total dissipated energy, confirming that radiation is the primary thermal transport mechanism for the MMRTG under Martian atmospheric conditions.

In summary, this study bridges the gap between simplified models of heated plates and the realistic three-dimensional operation of MMRTG systems. The methodology and results may serve as a reference for future design improvements of radioisotope thermoelectric generators and for a possible experimental test under Martian conditions, which allows us to validate the results extracted from CFD analysis.