A NUMERICAL STUDY OF PARACHUTE INFLATION BASED ON A MIXED METHOD

Li YU, Prof PhD Date of birth: 1969. Education: Nanjing University of Aeronautics and Astronautics. 2006–PhD (Eng.). Affiliations and functions: member of Safety and Rescue Committee in CSAA; member of Return and Re-entry Committee in CSA; editor of “Spacecraft Recovery & Remote Sensing”. Research interests: ἀuid structure interaction, ADS design, environmental control systems. Publications: over 30 research papers. Present position: professor and doctoral tutor in College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics.


Introduction
A p arachute i s a n im portant aer odynamic de celerator and is widely used in aviation, aerospace, weaponry, and other areas. ἀ e working style is simple, but t he inflation i s a t ypical interaction of s tructure a nd fluid t hat is a complex transient and nonlinear process (Yu, Ming 2007;Potvin et al . 2011). A t present, p arachute desig n is m ainly b ased o n em pirical a nd s emi-empirical f ormulas. ἀ e t raditional desig n needs a l arge number of physical tests to verify. However, this approach not only consumes a lo t o f m oney b ut a lso ext ends t he desig n cycle, which is not helpful for explaining the parachute inflation. ἀ erefore, numerical simulation b egan to b e applied for its economy and flexibility.
Fluid-structure in teraction (FS I) m ethods, w hich are a pplied in aer odynamic de celerator sys tem (ADS) research, have developed rapidly over the past few years. ἀ e r epresentatives a re a rbitrary L agrangian-Eulerian (ALE) method (Coquet et al. 2011;Tutt et al. 2011), the immersed b oundary (IB) m ethod (K im, P eskin 2009), etc. (Kenji 2012;P otvin et al. 2011). Of t hese methods, the ALE m ethod c an f ully co nsider fa bric co ntact a nd material p ermeability a nd h as b een a pplied in ac tual design. However, this method consumes a large amount of co mputing r esources, a nd t he t otal n umber o f e lements must be controlled within some range. Moreover, since this method applied in most engineering practices is based on a laminar model (Coquet et al. 2011;Tutt et al. 2011), t he results are rougher in c alculating high Re number flow field.
CFD is another major simulation method, but how to get the canopy shape is a key problem. Canopy shape was generated from CAD software in most studies (Cao, Jiang 2007;M cQuilling et al. 2011; N oetscher, C harles 2011) a nd wa s p rocessed a s a r igid b ody w ithout p ermeability. P revious CFD m ethods h ave b ig differences from the actual engineering.
In t his w ork, a C9 p arachute, a t ypical flat p arachute, is simulated by an FSI and CFD m ethod. Firstly, the f olded p arachute inflating in a n infinite m ass c ase is simulated u sing LS-DYNA b ased o n a n ALE m odel. ἀ en the inflated canopy shape is exported. Finally, the flow around this shape is simulated by using CFX based on porous media and the -k ε turbulence model.

Finite element model
ἀ e C9 p arachute is made of MIL-c-7020 type III fa bric (Calvin 1984), a nd the parameters of the model are shown in table 1. ἀi s m odel i s c alculated in a n infinite mass c ase based on the ALE method (the case in which deceleration effect can be negligible is called infinite mass case; otherwise it is called finite mass case; the latter not only considers the effect of flow field and structure but also considers t he flight c haracteristics o f t he p arachute; the l atter n eeds a w ider co mputational do main a nd i s more sensitive to coupling coefficients than the former). A penalty function is applied to process the fabric contact. ἀ e p rinciples a nd f ormulas des cribing t he ALE method can be found in related papers (Souli et al. 2000;Casadei et al. 2001). Figure 1 shows the finite element model based on ALE des cription. ἀ e lin es a nd c anopy a re co mpletely straightened, and the connection point of lines is fixed. ἀ e c anopy looks like '*' from a bove. ἀ e c anopy and Abstract. ἀ e C9 parachute was the research object in this work and was studied by using a fluid-structure interaction method and CFD m ethod. An arbitrary Lagrangian-Eulerian method, a k ind of fluid-structure interaction method, was used to simulate the inflation process. ἀ e dynamic relationship between canopy shape and flow field was obtained. ἀ e c anopy s hape in a s table p hase wa s exp orted a nd wa s t ransformed into t he p orous m edia do main. ἀ en the flow around the canopy shape was simulated by the CFD method we used based on the k-ε turbulence model. ἀ e experiments verified t he acc uracy of structural change and the feasibility of t he p orous media model. ἀ e arbitrary Lagrangian-Eulerian method not only can obtain the dynamic results of structure and flow field but also can provide a m ore accurate bluff body for further CFD analysis. ἀ e CFD method based on porous media and the turbulence model can obtain more detailed and accurate flow field results, which can be used as a co mplement to fluid-structure interaction analysis. ἀi s mixed method can improve the accuracy of analysis and be useful for other permeable fabric research.
Keywords: aerodynamic decelerator system, inflation process, fluid-structure interaction, porous media, parachute. lines are meshed by triangular elements (20,228) and bar elements (2,356). ἀ e hexahedral elements (147,392) are used to mesh the flow field. ἀ e canopy (Lagrangian description) and fluid domain (Eulerian description) interpenetrate with each other. ἀ e inlet boundary of the flow field is set as normal velocity inlet with a value of 80 m/s, and the others are shown in figure 2. a) canopy and lines b) parachute and flow field

Numerical results
ἀ e b ottom o f t he c anopy i s inflated first, a nd a ' bottleneck' arises at the lower middle p osition of the canopy (Figs 3 a nd 4). As more air enters the canopy more quickly, the 'bottleneck' effect is aggravated and gradually moves to the top of the canopy (Fig. 5). M oreover, the velocity vector and the position of the high-pressure zone show that the air has difficulty flowing through the 'bottleneck' (Figs 3-5). S tress is therefore concentrated on the 'bottleneck' position. ἀ e v ent exp ands t ransiently w hen t he ' bottleneck' moves to the top of the canopy (Fig. 6), and then the canopy has the classical 'squid' state (the 'squid' state is also called the 'bulb' state in some literature) (Wang 1997). ἀ e fully inflated area gradually expands to the bottom of the canopy, and inflation is completed at last (Figs 7 a nd 8). After the 'squid' state, the stress concentrates on the 'bulge' stably, and velocity and pressure remain stable.

Comparison with experiment
ἀ e numerical results are compared with a related experiment ( Fig. 9) in t his paper. Both shape changes are similar; the 'bottleneck' moves from the bottom to the top and the non-inflating part is relaxed. ἀ e numerical results and experiment indicate that the essence of the 'bottleneck' is that the flow of air into the canopy is blocked. ἀ e 'bottleneck' effect only blocks the flow of air into the canopy, but does not restrain canopy movement.
where µ i s fluid v iscosity, pem K i s t he p ermeability coefficient o f t he m edium, loss K i s t he r esistance-loss coefficient, ρ is the density of the fluid, v′ is the p ermeability velocity, and e is the thickness of the medium. To v erify t he f easibility a nd acc uracy o f t he CFD method based on the porous media model, a verification model is established according to a reference (Aquelet et al. 2006;Jia et al. 2009). ἀ e model and boundary conditions a re s hown in figure 10, a nd t he differences a re shown in t able 3 (t he model in A quelet's work is called Model A for short, and the model established in this section is called Model B for short). Table 4 shows the comparison of results in the same working conditions.
It can be seen that the relative errors of Model B are smaller than Model A and are controlled within 5%. ἀ e precision of Model B is stable and not affected by velocity change. ἀ erefore the porous media model can be used to simulate canopy permeability.

Model processing
At 0.879 s, t he p arachute has b een inflated and is in a stable phase (Fig. 8). In this section, the canopy elements at that moment are exported from ALE results. ἀ e geometry is regenerated from the shell elements. ἀ en the geometry is cleaned up; s ome unnecessary fabric folds are removed (Fig. 11).    . 11. Geometry after processing ἀ e parameters in t able 5, w hich describe the canopy shape, indicate that the geometry is the same as the actual shape. ἀ e details such as the bulge caused by airflow a re c learly des cribed, a nd t he entire c anopy lo oks like a b owl with a p etal-like edge rather than a sm ooth hemisphere.
According t o t he g eometry a bove (Fig . 11), t he canopy i s m eshed b y t riangular e lements (51,503). ἀ en t hree layers o f p rismatic elements (154,509) a re dragged b ased o n t hose t riangular e lements. ἀ ese prismatic elements are set as the porous media domain. ἀ e fluid domain, w hich surrounds t he p orous media domain, is meshed by tetrahedral elements (1,467,126). Figure 12 s hows t he m odel u sed in CFD sim ulation. ἀ e boundary conditions are the same as the test model (Fig. 10).
ἀ e incompressible steady-state simulation is solved by the fully coupled method. ἀ e turbulence model applied in the CFD simulation is the k-ε model. ἀ e principles and descriptions of the formulas of the k-ε model can be found in related works (Lin et al. 2005). Figure 13 shows the velocity vector and pressure contour of the CFD results.

Results and analysis
From the results above, there exists a smaller velocity vector in porous media domain, which indicates weak airflow through the canopy surface. ἀ e weak airflow and the airflow around the canopy produce a small eddy near the canopy surface. ἀ e direction is opposite to the big eddy that is produced in the upper flow field of the parachute.
Compared w ith t he r esults in figure 8, t he r esults of the CFD m ethod are more detailed and more accurate. Moreover, the drag coefficient, an important design parameter of ADS, is calculated based on this equation: where d F is drag force, ρ is fluid density, v is velocity of airflow, and o S is the area of the canopy. Table 6 s hows t he dra g c haracteristics. ἀ e dra g coefficient c alculated b ased o n t he CFD m ethod i s in agreement w ith t he exp erimental d ata, w hile t he va lue based on ALE is about 22.5% higher than the upper limit of the experimental value. a) the L agrangian m esh (des cribing t he c anopy) and E ulerian m esh (des cribing t he fluid domain) interpenetrate with each other in the ALE method, and the nodes on the interface need not be merged. ἀi s kind of pre-process is easy, but the mesh cannot be refined according to canopy shape. Since b ody-fitted mesh is applied in t he CFD method, which can be refined according to the s hape o f t he c anopy, t he CFD m ethod c an capture more flow field details; b) the ALE m ethod n eeds r e-mapping a lmost every time step in t ransient calculation, and the amount of calculation is large. ἀ e total number of elements is limited by hardware conditions in engineering practice. However, this limitation is smaller in s teady-state calculation based on the CFD method; c) at present, t he ALE m ethod in m ost en gineering applications and in this work is based on the laminar model. ἀ e accuracy would be affected in calculating high Re number flows.

Conclusions
In t his w ork, t he inflation p rocess in a n infinite m ass case was simulated by the ALE m ethod. ἀ en the canopy shape in stable phase was exported for further flowaround analysis based on the CFD method. ἀ e conclusions are as follows: a) the ALE m ethod solves the fabric contact problem based on a p enalty function and considers the fabric permeability. ἀ e inflation process of a folded parachute can be simulated more accurately. Moreover, the model pre-process is simple; b) the A LE r esults c an p rovide c anopy s hape f or further a nalysis. ἀ e g eometry exp orted f rom shell elements is more natural and realistic than the g eometry g enerated f rom CAD s oftware, which c an im prove t he acc uracy o f n umerical calculation; c) porous media domain w ith a cer tain t hickness can simulate the fabric p ermeability. Flow field results b ased o n t his m odel ar e different w ith those based on the traditional rigid model; d) based on the same bluff body, the flow field results of t he CFD m ethod are more det ailed and accurate t han ALE r esults. ἀ erefore t he CFD method c an be a c omplementary analysis f or getting more accurate aerodynamic parameters. However, the produce conditions of 'bottleneck' effect and how to use the CFD method to simulate the flow around t he un expanded c anopy (Figs 3-7