CFD analysis of Space x's starship's re-entry phase
Northeastern University: Utilizing Solidworks, ANSYS Fluent, and Matlab, I performed a CFD analysis of Space X's Starship Rocket upon its re-entry stage into the atmosphere. Pulling pressure profiles and Velocity fields, I was able to evaluate the effectiveness of Starship's bellyflop technique.
Introduction
This simulation models the reentry stage of the SpaceX Starship throughout its belly flop phase. Within this stage the rocket rotates into a horizontal position in order to increase its air resistance, i.e. drag. This is very uncommon since wings are a design element to induce lift, where the Starship manipulated this feature to “stall”, falling at a 90 degree angle through the atmosphere. In doing so, less rocket fuel is required to slow down the rocket, reducing to a low enough speed where a short flip-and-landing burn technique is used to touch down gently on the landing pad. The Starship is the most powerful launch vehicle yet to be built and the first vehicle to demonstrate total reusability, resulting in no damage to the vehicle or the boosters when landing. The belly flop phase is controlled using four flaps located on the back end of the ship. The force on these flaps, as well as the belly of the “beast” is extremely large when balancing the opposing drag forces and gravitational forces on this body. For clarity, the Starship weighs around 100 tons and a material of mostly stainless steel for low altitude flights, which includes the test flights so far for improving the belly flop and landing methods. The 3D body used in this fluent simulation was created in SolidWorks using figure [1] of the Starship with the corresponding dimensions.
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Since drag force is the only force opposing gravity when the Starship is in free-fall corresponding to the “stall” belly flop phase, the simulations ran outputted drag force and drag coefficients for varying velocity with constant planetary environments in Earth and Mars leading to a simulation in transient state, and varying angles in the z and y direction correlated to re-entry angles with constant velocity leading to a simulation in steady state. Varying attack angles also account for the minimal, slow oscillation from nose to tail when the Starship is balancing drag forces per location on the ship’s underbelly to orient itself in the ideal “belly flop” position. This should not occur in the x direction since the ship is symmetrical along this axis.
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Equation (1) is used to calculate the drag force of Starship at freefall:
where FD is the drag force on the object, is the density of fluid, v is the speed of the object relative to the fluid, CD is the drag coefficient, and A is the cross sectional area of the object. For computing the drag force for varying velocity in two different planetary environments, the cross sectional area and vertical drag coefficient is constant, the velocity is varying with gravity, and the density is varying with pressure. Re-computing drag force for varying fall angles relative to the y and z direction but with a constant velocity and planetary environment, the density, velocity, and drag coefficient are constants, varying drag force only as a function of cross sectional area.
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The varying velocity for two different planetary environments was determined using a constant acceleration of 9.8ms2 and 3.7ms2 for Earth and Mars, respectively. Even though gravity is not perfectly constant on these two planets, the simulation run is performed assuming perfectly constant acceleration leading to a linear change in velocity with respect to time. The gravitation formula to determine change in acceleration dependent on distance from the planet's core is defined by equation (2): where G is the gravitational constant, m1 and m2 are the mass of objects 1 and 2 respectively, and r is the distance between the masses. The density, as stated before, varies when the simulation is run in a transient state, aka varying velocity from constant acceleration. Density is a function of atmospheric pressure, temperature, and humidity. Simplified from the Ideal Gas Law, density for atmospheric applications, taking into account density as a function of pressure and temperature is equal to equation (3): where R is the universal gas constant and T is the temperature of the air, all dependent on altitude. Therefore, a changing altitude results in a changing pressure, correlating to a change in temperature, and therefore a changing R value. The R value is dependent on the type of fluid, which also changes with humidity. As seen from this formula, there are many varying parameters for simulating the re-entry of the Starship into the atmosphere. Since this simulation is focusing on the belly flop method of re-entry and the effect of drag during this, our simulation is simplified with varying velocity and varying fall angle for the first and second simulation, respectively.
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It is also important to note that the mach number has not been accounted for in this simulation, and is roughly around mach 25 when returning from orbit. Mach 10 to mach 25 NASA defines as high supersonic speed. Extensive heat transfer would also occur due to extremely fast speeds causing intense friction forces at the convection interface [1].
Figure 1: Dimensions of the Starship from SpaceX CAD drawing
SOFTWARE DESCRIPTION
SolidWorks was used in 3D modeling the Starship, being imported to ANSYS Workbench for the geometry of the fluent modeling, using ANSYS meshing software to produce cells in the 3D model and then using a fluent solver to run the simulation. The parameters were initialized in the ANSYS Workbench itself.
PROCEDURE
There were two different simulation set-ups in this ANSYS project, one with constant velocity and varying fall angles in Earth’s atmosphere, and one with constant atmospheric conditions for both Earth and Mars with varying velocity, taking into account changing acceleration per planet. The geometry and mesh is at a constant throughout this simulation, which is referenced further in the walk-through video submitted with this paper.
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For the simulation varying velocity for the two differing planetary atmospheres, transient state was chosen in the simulation to account for constant acceleration. The most important steps in the ANSYS setup was creating a named expression of VelMag = 9.8 [m/s^2]*Time for Earth and VelMag = 3.7 [m/s^2]*Time for Mars. This takes into account the changing velocities with respect to time for constant downward acceleration per planet. For set-up, the type of fluid used in the simulation will also change depending on the atmosphere, with Earth’s atmosphere being 78% nitrogen, 21% oxygen, and 1% consisting of random gasses. Mars’ atmosphere consists of 95% carbon dioxide, 2.8% molecular nitrogen,and 2% argon. For this simulation, “air”, the default fluid, was used for Earth with a density of 1.225 [kg/m^3] and viscosity of 1.7894*10^-5 [kg/m-s], and carbon dioxide with a density of 0.02 [kg/m^3] and viscosity of 9.82*10^-5 [kg/m-s] was used for Mars. The rest of the alterations to the simulation were done for the solution and results similarly to the second simulation, conducted in the walk-through video posted with this paper.
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The second simulation ran for fall angles from 90 degrees attack angle, aka the ideal belly flop orientation, to 45 degrees attack angle. This simulation focuses on the re-entry of the Starship and the “rocking” motion of the ship resulting from differing force values from the nose to tail of the ship. This was performed in the simulation by keeping the velocity magnitude constant, therefore parameterizing the velocity components with trigonometric relations in the x and y axes. The cos and six values were pasted into the parameters of the simulation, resulting in 9 different simulations. This simulation is run in steady state conditions and a constant fluid of air at a density of 1.225 [kg/m^3] and viscosity of 1.7894*10^-5 [kg/m-s]. Everything else besides set up for this simulation is consistent with the previous simulation, and is further explained in the walk-through video submitted with this paper.
RESULTS
Below are the figures showing the pressure and velocity results per mesh cell for the Starship from the varying velocity in different planetary environments. It is important to note that the contour scales located to the right of the simulated object are not equal for Earth and Mars.
Figure 2: Bottom pressure contour on Mars (left) and Earth (right)
Higher pressure is found around the outside of the figure [2] for the Starship free falling in Earth’s atmosphere. Negative pressure is shown in yellow and light green for Earth, with positive pressure being darker green and blue located more in the center of the Starship body. The starship’s pressure contour distribution on Mars is similar to Earth’s, yet at a lower scale, implying lower pressures for both positive and negative from a much lower density and gravity.
Figure 3: Top pressure contour on Mars (left) and Earth (right)
As said previously for the top pressure contour on these two planets, the pressure spectrum for Mars is much lower than the spectrum for Earth shown in the figure [3] above. One can also see that there is a positive pressure in the center of the cylindrical portion of the body, and a lower pressure on the outsides of the cylindrical portion. This could potentially be because of vortexes being created from high velocities on the sides of the bodies.
Figure 4: Side velocity contour on Mars (left) and Earth (right)
Figure [4] shows that there is a very high velocity in the sides and front of the Starship where there is less surface area acting normal to the fluid velocity. The maximum velocity on Mars is less than the maximum velocity on Earth from the difference in gravity. The side fins act as a wall for the velocity to collide into, causing a stagnation point and altering fluid to then wrap around the object. This causes lower velocity at the fins and above the fins.
Figure 5: Back velocity contour on Mars (left) and Earth (right)
The back velocity contour shown in figure [5] shows similar results to figure [4]. It shows a constant velocity until collision with the Starship, and then a reduction in velocity around the belly of the ship, and an increase in velocity on the outside of the fins where the fluid was directed to flow once colliding. There is also a vortex on the top of the fins from the high velocities on the outsides of the fins.
Figure 6: Drag force (top) and change in drag (bottom) plots with respect to Mars (left) and Earth (right) atmospheres
Figure [6] above analyzes the force of drag applied on Starship as a function of velocity for both Mars and Earth conditions. In both atmospheres, the drag force plots display a near exponential relationship with velocity, as would be expected from equation 1. The plots analyzing change in drag with respect to time over velocity are near linear with slight perturbation.
Figure 7: Bottom pressure contour for 90° (left) and 45° (right) attack angle
Figure 8: Top pressure contour for 90° (left) and 45° (right) attack angle
Figure [8] also shows a smaller maximum pressure at 45° than 90°. There is still positive pressure experienced in the center of the “barrel” from a vortex around the body, but it is much less in the 45° tilt than the 90° tilt. This vortex is also shifted to the nose of the body when there is less than a 90° tilt. There is also negative pressure experienced on the nose of the Starship at the 45° tilt, where this is not visible in the belly flop position. The
Figure 9: Side velocity contour for 90° (left) and 45° (right) attack angle
In figure [9] above, the velocity contour on the side of the Starship’s body is shown. When the Starship is dropping at 90°, the velocity vector is similar to that of the previous simulation. The velocity contour of the 45° tilt shows the zero velocity locations at a slight tilt from the angle that the body interferes with the fluid. Instead of being a normal plane intruding on the motion of the fluid, the fins/wings are now tilted. The back end also has increasing velocity where the tail is stopping more fluid from having a larger surface area, therefore decreasing the fluid velocity.
Figure 10: Bottom velocity contour for 90° (left) and 45° (right) attack angle
The figure [10] above has a constant velocity coming into the ship, with increased velocity on the outskirts of the fins and a smaller velocity being above the body and the fins. This is from a vortex being created from the high velocity on the outside of the fins.
Figure 11: Drag forces per angle of attack against iterations of simulations
Figure [11] portrays convergence of the X, Y, and Z-components of drag force as the simulation progresses. Beginning with an initialization of 300 iterations to get closer to a converged result for each chase, the simulation then inputs the parameters and continues the calculations. The Y and Z-components of drag were able to converge to near constant results, as is displayed by the distinct, nearly horizontal lines. The X-component, however, was not able to, as it has a chaotic and random nature.
Figure 12: Drag force as a function of angle of attack
Taking the last point from each data set, Figure [12] displays the directional components of drag as a function of degrees of angle of attack. The Y-component of drag portrays the best relationship to degrees with a smooth line. This relationship is estimated to be trigonometric and periodic.
CONCLUSION
As a result of the above findings, the belly flop method would not be effective within Mars conditions. Comparing the drag profiles within Figure [6], considerably higher drag forces are experienced on Earth at the same velocity. At over 100 degrees of magnitude greater, Earth’s atmosphere has the ability to considerably slow the descent of SpaceX’s Starship rocket upon its re-entry even with its larger acceleration of gravity.
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When performing the belly flop, it is essential to maintain a perpendicular orientation with respect to the direction of motion. This creates the highest value of drag force creating the best possible reduction in velocity.
APPENDICES
[1] “SpaceX starship prototype exploded, but it's still a giant leap towards Mars,” Department of Engineering, 04-Jan-2021. [Online]. Available: http://www.eng.cam.ac.uk/news/spacex-starship-prototype-exploded-it-s-still-giant-leap-towards-mars. [Accessed: 14-Dec-2022].
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[2] Starship specs - weight, volumes, etc. [Online]. Available: https://forum.nasaspaceflight.com/index.php?topic=50049.0. [Accessed: 14-Dec-2022].