Abstract.
Immersed boundary (IB) methods with adaptive mesh refinement (AMR) techniques are assessed for atmospheric entry applications, including effects of chemical nonequilibrium (CNE) and gas-surface interactions (GSI). The performance of a conservative cut-cell and two non-conservative ghost-cell IB methods is assessed in comparison with analytical solutions, data from literature, and results obtained with a reference solver that operates on body-fitted grids. All solvers use the same external thermochemistry library so that all observed differences can be attributed to the underlying numerical methods.
Results from eight benchmark cases are reported. Four cases are selected to verify the implementation of chemistry, transport properties, catalytic boundary conditions, and shock capturing. Four validation cases consider blunt geometries with adiabatic/isothermal and inert/catalytic/ablative boundary conditions. Overall, the results obtained with the IB solvers are in very good agreement with the reference data. Discrepancies arise with ghost-cell methods for cases with large temperature or concentration gradients at the wall and are attributed to mass conservation errors. Only a strictly conservative cut-cell IB method is on par with body-fitted grid methods.
key points:
- The paper evaluates immersed boundary (IB) methods for simulating hypersonic flows relevant to atmospheric entry conditions. Specifically, it assesses the accuracy of IB methods for applications with strong thermal gradients and gas-surface interactions.
- It compares results from two IB solvers (INCA using a cut-cell method, CHESS using a ghost-cell method) against a reference body-fitted solver US3D. To isolate the effects of the numerical methods, all solvers are coupled to the same external thermochemistry library Mutation++.
The immersed boundary (IB) method is a way to handle complex geometries when using Cartesian grids, rather than body-fitted meshes. The key aspects are:
- The Cartesian grid does not conform to the boundaries. The solid object simply intersects the mesh.
- The boundary conditions are imposed by modifying the solution in cells near the boundary.
- This avoids costly mesh generation for complex shapes.
There are two main approaches:
- Ghost-cell method:
- Boundary conditions are enforced by extrapolating the solution to ghost cells inside the solid object.
- Fast to implement, but not strictly conservative. Errors can occur for problems with large gradients at boundaries.
- Cut-cell method:
- The Cartesian cells are cut by the boundary to conform to the shape.
- The solution in the irregularly shaped cells is modified to satisfy conservation.
- More complex implementation, but provides sharp representation of boundaries.
The paper tested both these IB techniques to assess their accuracy compared to body-fitted grids:
- INCA uses a cut-cell approach with careful treatment of conservation.
- CHESS uses a ghost-cell approach.
Benchmarks
The benchmarks showed cut-cell IB can match body-fitted results, while ghost-cell had limitations in some cases. This demonstrates the importance of conservation for IB methods applied to hypersonic flow problems.
- A set of 8 benchmark cases are used for code verification and validation. This includes 0D and 1D cases to verify chemistry and transport models, as well as 2D blunt body cases with different surface conditions (adiabatic, isothermal, catalytic, ablative). The 8 benchmark cases to evaluate the immersed boundary methods include:
4 verification cases:
- 0D reactor - verifies chemical source term implementation with a 5-species air model.
- 1D diffusion - verifies transport properties with temperature gradient causing diffusion.
- 1D catalytic diffusion - verifies catalytic boundary condition implementation.
- 1D shocktube - verifies shock capturing using a Riemann problem.
4 validation cases:
- 2D cylinder, inert adiabatic wall - validates chemical nonequilibrium modeling.
- 2D cylinder, inert isothermal wall - validates surface heat flux predictions.
- 2D cylinder, catalytic isothermal wall - adds exothermic catalytic reactions.
- 2D ablating geometry - validates implementation of ablation and blowing.
The cylinder cases use conditions from an experimental study by Knight et al. The ablating case is based on a plasma wind tunnel experiment by Helber et al.
The benchmark cases consider effects relevant to atmospheric entry - non-equilibrium chemistry, thermal gradients, catalysis, and ablation. The simple 0D and 1D cases verify the models and methods. The 2D cases then validate the accuracy for curved geometries and surface conditions representative of hypersonic flows.
Overall, these benchmarks provide code verification and validation for immersed boundary methods on problems with features important for simulating atmospheric entry conditions. The variety of conditions also helps identify limitations of different numerical approaches.
- Overall, the conservative cut-cell IB method in INCA matched the body-fitted solver very well for all test cases. The ghost-cell IB method in CHESS showed inaccuracies in predicting surface heat fluxes for cases with cold isothermal or ablative walls.
- These deficiencies are attributed to mass conservation errors inherent in ghost-cell IB methods, which are not strictly conservative. Additional tests corroborated that the errors appear independent of implementation details.
- The benchmark cases established here can serve for future code verification and validation of IB methods for simulating hypersonic flows with gas-surface interactions relevant to atmospheric entry.
The main codes being verified and validated in the paper were:
- US3D - A finite-volume body-fitted mesh solver developed by NASA/Minnesota. This served as the reference code.
- INCA - An immersed boundary solver using a cut-cell approach on Cartesian grids with adaptive mesh refinement.
- CHESS - An immersed boundary solver using a ghost-cell approach.
The paper looked at several metrics to evaluate the performance and accuracy of the immersed boundary solvers compared to the body-fitted reference:
- For 0D and 1D verification cases - Matching of transient species concentrations and distributions against analytical solutions.
- Temperature and species profiles along stagnation streamlines.
- Surface pressure distributions.
- Surface heat flux predictions.
- Contour plots of flow variables (temperature, species concentrations).
- Shock stand-off distance.
- Ablation mass blowing rate profiles.
The comparisons showed excellent agreement between the cut-cell IB solver (INCA) and body-fitted results for all metrics on the test cases.
The ghost-cell IB solver (CHESS) also matched well for adiabatic cases, but showed significant discrepancies for surface heat fluxes and blowing rates for cooled/ablating conditions. This was attributed to lack of strict conservation in the ghost-cell approach.
Overall, the variety of verification and validation metrics assessed the key physics modeling and numerical accuracy required for simulating hypersonic flows with boundary layer and gas-surface interaction effects.
Hypersonicc Flow Theory
The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluids. They are named after the French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes, who developed them in the early 1800s.
The Navier-Stokes equations express the conservation of mass and momentum for Newtonian fluids. They can be written in three dimensions as follows:
Continuity equation:
∂ρ/∂t + ∇·(ρu) = 0
Momentum equations:
ρ(∂u/∂t + u·∇u) = -∇p + μ∇²u + ρf
where:
- ρ is the fluid density
- u is the fluid velocity vector
- p is the fluid pressure
- μ is the fluid viscosity
- f is a body force per unit volume
The continuity equation states that the mass of fluid entering any given volume must be equal to the mass of fluid leaving that volume. The momentum equations state that the rate of change of momentum of a fluid element is equal to the sum of the forces acting on that element.
The Navier-Stokes equations are a fundamental tool in fluid mechanics and are used to model a wide range of fluid flows, including:
- Airflow around aircraft and spacecraft
- Water flow in rivers and lakes
- Blood flow in the human body
- The motion of the Earth's atmosphere and oceans
The Navier-Stokes equations are very complex and cannot be solved analytically for most real-world problems. However, there are a number of numerical methods that can be used to solve the Navier-Stokes equations for specific problems.
The Navier-Stokes equations are one of the most important equations in physics and engineering, and they continue to be an active area of research.
Artifacts
Unfortunately the paper does not mention any artifacts or data being made available. The benchmark cases are described in detail, but the actual simulation results from the solvers tested are not provided.
The paper does say that the benchmark cases were chosen through a collaborative effort between multiple research groups, with the goal of establishing well-defined test cases that could be readily reproduced by others.
However, without access to the raw data or result files, it would be difficult to fully reproduce the comparisons shown in the paper. The descriptions provide enough specifics that the cases could potentially be set up and run independently in other solvers for comparison. But the raw reference data from the US3D, INCA, and CHESS simulations are not openly shared or cited as being available elsewhere.
Overall, while the paper doesn't directly provide any reusable artifacts or data, it does establish a useful set of validation test cases that could be implemented by other research groups to further assess immersed boundary methods for hypersonic flow simulations. But without the reference data, new results could only be compared qualitatively based on the information and plots contained in this paper.
Unfortunately the paper does not provide any information about accessing the source code for US3D, INCA, and CHESS. As far as I could find, these flow solvers do not seem to be open source or freely available.
US3D, which stands for "Unstructured Solver for 3D compressible Navier-Stokes Equations," is a computational fluid dynamics (CFD) software developed collaboratively by NASA (National Aeronautics and Space Administration) and the University of Minnesota. This software is designed for simulating fluid flow and heat transfer in three-dimensional unstructured grids, making it particularly useful for modeling complex aerodynamic and fluid flow phenomena.
US3D is used for a wide range of applications, including the analysis of aircraft and spacecraft aerodynamics, combustion processes, and other fluid flow problems. Its unstructured grid capability allows it to handle complex geometries more effectively than structured grid CFD codes, making it a valuable tool in aerospace engineering and research.
US3D:
- Developed by NASA and University of Minnesota. https://ntrs.nasa.gov/api/citations/20180002133/downloads/20180002133.pdf
- Seems to be restricted for use within NASA.
- The paper cites the user manual but no public download is available.
INCA (Incompressible Navier-Stokes Code for Aeroacoustics), a computational fluid dynamics (CFD) code that was developed at the Delft University of Technology (TU Delft) in the Netherlands, specifically by Professor Philipp Schlatter and his research group, including Dr. Sebastian Hickel. INCA is primarily used for simulating turbulent flows and aeroacoustic phenomena.
INCA is known for its high-fidelity simulations of turbulence and its applications in various research areas, including aerodynamics, aeroacoustics, and fluid dynamics. It has been used to investigate complex flow phenomena and has contributed to our understanding of turbulence and related problems.
INCA: https://www.inca-cfd.com/
- Developed at TU Delft. https://www.inca-cfd.com/
- Used mainly by Hickel's research group.
- Website shows it is a proprietary research code not available to public.
CHESS:
- Developed at Polytechnic University of Bari.
- Closed source. Used by Pascazio's research group.
https://www.researchgate.net/profile/Giuseppe-Pascazio
https://www.dmmm.poliba.it/index.php/en/profile/143-gpascazio
https://users.ba.cnr.it/istp/istpal18/HYMEP/lectures/Pascazio.pdf
https://research.poliba.it/poliba-researchers/giuseppe-pascazio - Paper gives no information on obtaining the code.
So in summary, these solvers seem to be academic research codes that are not openly distributed or downloaded. Likely the developers would need to be directly contacted for collaboration or access. It would be difficult to reproduce these results without their active cooperation.
The paper provides good detail on the numerical methods and models implemented in each code. But without access to the actual programs or source code, reproducibility is limited to recreating the test cases independently.
Computational Resources
The paper does not provide specific details on the computational resources used to run the code benchmarks. However, it does mention:
- The INCA solver was run using resources from the Delft High Performance Computing Centre and the Dutch national supercomputer Cartesius.
- The CHESS solver was run at the Polytechnic University of Bari. Specific resources are not mentioned.
- For US3D, computational resources are not discussed. As a NASA code, it was likely run on NASA supercomputing facilities.
While exact hardware details are not provided, we can make some reasonable assumptions about the scale of resources required:
- The 2D cylinder and ablator validations require solving the Navier-Stokes equations on fine grids to resolve shocks and boundary layers. This implies using high-performance parallel clusters.
- Convergence studies were run to ensure grid-independence, so multiple grid resolutions were evaluated. This further increases computational expense.
- The paper mentions run times of several hours to days for the 2D cases. This suggests using hundreds to thousands of cores to obtain results in a reasonable timeframe.
- INCA's use of adaptive mesh refinement also helps improve resolution and efficiency compared to fixed body-fitted meshes.
Overall, while specifics are not provided, the scale and duration of the simulations indicates these benchmarks relied on state-of-the-art supercomputing resources commonly found at academic/government research centers and facilities. The computational requirements match what would be expected for highly-resolved hypersonic flow simulations. It would seem to be valuable to compare computational resources required and run times for the various codes.
Authors
The paper does not provide full bios or background information on the authors, but based on the author affiliations, we can summarize a bit about their apparent research focus and expertise:
- Ata Onur Başkaya, Stefan Hickel - Aerodynamics Group at Delft University of Technology. Their research likely focuses on computational fluid dynamics and aerodynamics.
- Michele Capriati - Inria, Centre de Mathématiques Appliquées, École Polytechnique. His background seems to be in applied mathematics for aerospace applications.
- Alessandro Turchi - Italian Space Agency. He likely has experience in aerospace engineering and aerothermodynamics.
- Thierry Magin - von Karman Institute for Fluid Dynamics. His research covers gas-surface interactions and hypersonic flows.
So overall, the author list includes experts in aerodynamics, computational fluid dynamics, applied mathematics, aerospace engineering, and hypersonics. They are based at universities, research institutes, and space agencies in the Netherlands, France, Belgium and Italy.
While not their full background, this context suggests the authors have the relevant domain knowledge and computational research experience needed for assessing immersed boundary methods on hypersonic flow problems relevant to atmospheric entry conditions.
Wind Tunnel Validation of Model Data
The paper does not provide extensive details on validation of these codes against wind tunnel experiments. However, some relevant points:
- For the 2D cylinder case, the conditions were matched to experiments carried out in the high enthalpy shock tunnel at DLR by Karl et al. The paper references that these experiments were used for code validation in an earlier study by Knight et al.
- For the ablating geometry case, the conditions were based on plasma wind tunnel experiments by Helber et al at the von Karman Institute. The authors first simulated these experiments to validate their ablation models before creating the simplified 2D case.
- The paper mentions that no ground testing can fully replicate hypersonic flight conditions simultaneously. So computational simulations are still needed to complement wind tunnel testing.
- Validating the immersed boundary techniques was a main focus of the paper rather than comprehensive validation against experiments. The test conditions were taken from relevant wind tunnel studies but new experimental data was not generated.
So in summary, while the benchmark cases were motivated by real wind tunnel experiments, new validation data was not provided in this paper. The authors reference prior studies that validated the body-fitted and immersed boundary codes against specific wind tunnel measurements. But those details are not fully covered in this paper, which focuses more on verifying the IB methods. Additional validation would need to be done to use these solvers for predictive simulations.
Other CFD Codes
Some other notable hypersonic computational fluid dynamics (CFD) codes currently being used for research and development include:
- DPLR (Data-Parallel Line Relaxation) - Developed by NASA, used widely for entry vehicle design.
- LAURA (Langley Aerothermodynamic Upwind Relaxation Algorithm) - Also from NASA, models chemical and thermal nonequilibrium.
- LeMANS - Developed at University of Michigan, optimized for ablative material response.
- USim - CFD code from University of Minnesota, extended from DPLR.
- Rocflu - Density-based CFD code from ONERA in France.
- HYB2D - From RWTH Aachen University, hybrid fluid-particle approach.
These codes see significant use at:
- NASA centers (Ames, Langley, Johnson) and partners like University of Minnesota.
- National research centers like ONERA (France), DLR (Germany), JAXA (Japan).
- Academic groups at universities with strong aerospace/aerophysics research.
- Industry sites across the space sector like SpaceX, Blue Origin, ArianeGroup.
- Defense labs interested in missiles, reentry vehicles, directed energy weapons.
In summary, hypersonic CFD is an active field with a variety of solvers developed and used across academia, government, and industry, especially by groups involved in spacecraft design, atmospheric entry modeling, and high-speed flows research.
The paper itself does not provide direct comparisons between US3D, INCA, CHESS and other hypersonic CFD codes. However, based on my research, some general comparisons can be made:
- DPLR, LAURA, and USim are structured body-fitted grid codes like US3D. They use similar numerical methods and have demonstrated good accuracy for hypersonic flows.
- LeMANS also uses body-fitted grids but has specialized modeling for ablation physics and thermal response. It would be a good comparison for the ablating cases.
- Rocflu uses unstructured grids, which provides flexibility but can have issues resolving boundary layers.
- HYB2D uses a hybrid particle-continuum approach rather than pure Navier-Stokes. Its applicability to the benchmark cases may be limited.
- The immersed boundary solvers INCA and CHESS provide alternatives to body-fitted grid generation. But factors like conservation, accuracy near boundaries, and shock-capturing need to be assessed as done in this paper.
In general, the body-fitted grid codes tend to be more mature and widely validated for hypersonic applications. The benchmarks in this paper help qualify how the IB methods in INCA and CHESS compare to trusted body-fitted approaches like US3D.
Additional code comparisons on these cases could better map out the strengths and weaknesses of different numerical approaches. But this paper provides a good baseline assessment showing the cut-cell IB method can match body-fitted results if conservation is treated carefully.
Here are some open-source CFD codes that are known for their capabilities in simulating hypersonic flows:
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OpenFOAM: OpenFOAM is a widely used open-source CFD toolbox. It includes a variety of solvers and models suitable for simulating hypersonic flows. Users can develop custom solvers or modify existing ones to suit their specific needs. OpenFOAM was first released as open source by OpenCFD Ltd. in 2004. Since that time it has matured to become the leading open source software for Computational Fluid Dynamics. They continue today as the core entity promoting its open, collaborative and progressive development and maintenance, with the declared long-term support of partners in industry, academia and key individuals. See "Hypersonic simulations using open-source CFD and DSMC solvers" for an example useage. See also "Validation of OpenFOAM and hy2Foam in Hypersonic Flow Simulations"
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SU2 (Stanford University Unstructured): SU2 is an open-source CFD code designed for both academic and industrial applications. It has capabilities for hypersonic flow simulations and is known for its flexibility and user-friendly interface.
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DPLR (DPLR Public License Research): DPLR is an open-source CFD solver designed specifically for hypersonic flows. It is developed by NASA and the United States Air Force. It is highly specialized for hypersonic research and may require some expertise to use effectively. The manual on the NASA technical reports server. This software is only available for use by federal employees and contractors to the federal government working on projects where this tool would be applicable.
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XFlow: XFlow is a commercial CFD software, but it has an open-source variant called "XFlow-Express" that can be used for academic and research purposes. It is known for its ability to handle complex flow phenomena, including hypersonic flows.
OpenFOAM uses a finite volume method to discretize the Navier-Stokes equations. The finite volume method is a numerical method for solving partial differential equations by dividing the domain into a set of control volumes and applying the conservation laws to each control volume.
OpenFOAM uses a variety of computational techniques to solve the Navier-Stokes equations for hypersonic flows. These techniques include:
- Explicit time integration: Explicit time integration methods are used to advance the solution in time. Explicit time integration methods are simple to implement, but they can be computationally expensive for hypersonic flows.
- Implicit time integration: Implicit time integration methods are used to advance the solution in time. Implicit time integration methods are more computationally expensive than explicit time integration methods, but they are more stable and can be used for larger time steps.
- Upwind schemes: Upwind schemes are used to discretize the convective terms in the Navier-Stokes equations. Upwind schemes are important for hypersonic flows because they can accurately capture the shock waves that are present in hypersonic flows.
- Turbulence models: Turbulence models are used to model the turbulent effects in hypersonic flows. Turbulence models are important for hypersonic flows because they can accurately predict the aerodynamic forces and heat transfer rates.
OpenFOAM is a powerful tool for solving hypersonic flow problems. It is used by researchers and engineers in a variety of industries, including aerospace, automotive, and energy.
Hypersonic solvers are still quite specialized and not as accessible for general research use compared to subsonic CFD. The OpenFOAM is a good open option for nonequilibrium aerothermodynamics simulations. But most other major codes require specific institutional partnerships or arrangements to access.
Sources:
High fidelity computational fluid dynamics (CFD) codes are developed to model reacting, non-equilibrium flows. The DPLR (Data Parallel Line Relaxation) and US3D codes enable prediction of aerothermal environments to establish thermal protection system (TPS) material requirements for atmospheric entry spacecraft.
C. N. Onyeador, C. J. Waligura, L. Lopez, D. Hoskins, K. M. Sabo, W. L. Harris
Massachusetts Institute of Technology, Cambridge, MA, 02139
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