Speaker
Description
1. Introduction
Computational Fluid Dynamics (CFD) represent the most cost-effective tool to characterize re-entry flows aiming at providing an accurate prediction of engineering inputs (e.g. surface heat fluxes) for the design of TPS and the overall success of space missions. The vast majority of state-of-the-art CFD codes tackling hypersonic applications rely on second-order Finite Volume (FV) that are robust and easy to implement. Unfortunately, achieving an accurate heat flux prediction is still a challenge even on simple geometries, leading to an unoptimized design of the TPS. CFD analysts acknowledge the importance of using well-designed grids to obtain accurate flow results, since mesh/shock misalignment lead to shock instability, noisy post-shock regions and poor heat flux predictions [1]. To overcome those deficiencies, state-of-the-art ATD simulations require demanding mesh generation processes and grid convergence studies resulting in a high run-time and considerable computational effort. All this makes numerical analysis more challenging and time-consuming especially for those kind of applications where high aspect ratio meshes are needed, close to the space vehicle walls, in order to resolve boundary layer features and obtain accurate numerical heat fluxes [2]. Adaptive Mesh Refinement (AMR) represent a robust procedure for improving the quality of the physical results, especially heat flux prediction, due to a local increase of the grid resolution and mesh/shock alignment, at the price of an increased algorithmic complexity.
2. Adaptive Mesh Refinement
In our previous work [3], a robust and efficient r-refinement algorithm (i.e. repositioning of mesh points based on the solution of a pseudo-elastic system of equations) has been developed and already applied to multiple test cases. The existing AMR algorithm works on triangles, isotropic quadrilateral and tetrahedral cells, is fully parallel, implemented as a standalone module and totally physics-independent, letting the user decide which monitor physical quantity to use for driving the adaptation according to the application. In this work, we develop novel concepts and integrate them into the existing AMR algorithms, in order to automatically improve 2D Cartesian high aspect ratio meshes and being able to overcome the unphysical stagnation heat flux peak resulting from state-of-the-art FV simulations on an initially unfitted mesh.
- Semi-torsional (ST) spring for 2D Cartesian meshes
In this work, we propose a new variant of the ST spring analogy that can be applied to 2D quadrilateral meshes. The novel ST combines local physical and geometrical properties.
- Hybrid Spring Analogy (HSA)
The idea for the HSA is to define a certain distance, denoted acceptable distance, from a user-defined boundary in which the mesh will incorporate two different spring concepts. In this work, HSA leads to specify two different dynamic mesh motions: (1) nodes close to the wall boundary are frozen and (2) the others are free to move based on the solution of the associated pseudo-elastic system.
- Smoothing the nodal displacement
The mesh distribution is not uniform and this affects the nodal wall distance computations which is needed within the HSA. We observed that the limit of each wall distance contour crosses some cells and hence, within the same cell, different nodal wall distance values exist. Hence, starting from the limit cells located at the acceptable distance, i.e. interface, a linear interpolation applied to the dynamic mesh motion from blocked nodes to a full movement is proposed.
- Smoothing the spring network
The use of the HSA can be extended to split the mesh into three different spring concepts. The original idea was to use the following three different spring analogies: Blocked nodes close to the wall, ST analogy for the transient and linear spring analogy for the rest of the grid. The latter is used in order to refine the shock only by physics-based gradients. A jump in the stiffness coefficient from the ST spring concept to the linear one is observed and introduces brutal changes in the displacement of the nodes leading the code to crash due to the bad mesh quality. An exponential interpolation is applied to smooth the spring constant from the ST to the linear one.
- Solution interpolation
The AMR is a post-processing step. In this manner, the cell-center value is conserved and moves due to the nodal displacements. Since the r-refinement suffers when dealing with high aspect ratio meshes, the flow field is affected negatively by the adaptation and for some test cases, the shock gets too close to the boundaries, leading to a blow-up of the solution. Hence, an interpolation step after the mesh refinement is implemented to ideally dissociates the computations of the nodal movement and the solution in each cell center value in order to preserve the shock position and the subsonic flow distribution.
3. Hypersonic Flow simulations
The reference test case presented is the hypersonic flow around the Qarman CubeSat. QARMAN is the QubeSat for Aerothermodynamic Research and Measurements on AblatioN of VKI, developed in the framework of the QB50 project. For the present results, dissociated air with five species is considered: $O, N, NO, O_2, N_2$. For the TCNEQ simulation a two-temperatures model is used considering the following free-stream conditions:
- $M = 8.46$
- $p = 39.53 [Pa]$
- $\rho$ = $1.970\cdot 10^{-4} [kg/m^3]$
- $T = 530.3 [K]$
- $T_v = 2343.3 [K]$
In addition the wall is considered isothermal with $T_{w}=1000K$.
Using our new AMR based on the flow density as a monitored flow field variable, the grid is adapted and mesh points migrate toward the bow shock, increasing the grid node density around the shock. This result indicates how promising is the proposed approach for tackling high-aspect ratio Cartesian meshes.
To highlight the improvement of the physical results, COOLFluiD heat flux estimation is computed on the same fitted and unfitted computational grid. When applying our new AMR algorithm, the unphysical heat flux stagnation peak resulting from state-of-the-art FV simulations on an initially unfitted mesh disappears leading to more accurate predictions when applying the newly developed physics-based AMR.
References
[1] Peter Gnoffo and al. Computational Aerothermodynamic Simulation Issues on Unstructured Grids. In 37th AIAA Thermophysics Conference, Fluid Dynamics and Co-located Conferences. AIAA, 2004
[2] Kyu Hong Kim and al. Methods for the accurate computations of hypersonic flows: II. shock-aligned grid technique. JCP, 174(1):81 – 119, 2001.
[3] Firas Ben Ameur and Andrea Lani. Physics-based r-adaptive algorithms for high-speed flows and plasma simulations. reviewed, JCP, 2018.
Summary
When predicting heat fluxes acting on Thermal Protection Systems (TPS) of space vehicles during the hypersonic re-entry phase, state-of-the-art numerical flow solvers do not automatically achieve the required accuracy, even on simple geometries, unless great care is taken when generating the corresponding computational mesh. Within this context, our work proposes an improved mesh adaptation algorithm based on r-refinement (i.e. repositioning of mesh points based on the solution of a pseudo-elastic system of equations) especially designed to handle high-aspect ratio cells, as required by viscous hypersonic flow calculations. The target application for this work is the flow simulation and, in particular, the heat flux prediction around a blunted QARMAN CubeSat geometry. The COOLFluiD aerothermodynamic (ATD) code, i.e. a state-of-the-art second-order implicit Finite Volume solver for thermochemical nonequilibrium flows, is used for all computations, together with linear meshes. Our mesh adaptation algorithm, based upon the concept of hybrid spring analogy in combination with both physics- and geometry-based stiffness definitions, manage to better capture the shock layer properties and shows its potential for automatically tackling the heat flux issue on our representative 2D testcase.
