An official website of the United States government
The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.
more »
« less
CaTSM: A Pseudo‐Spectral Thermodynamically Consistent Model of Compressible Flows
Abstract We introduce a pseudo‐spectral algorithm that includes full compressible dynamics with the intent of simulating near‐incompressible fluids, CaTSM (Compressible and Thermodynamically consistent Spectral Model). A semi‐implicit scheme is used to model acoustic waves in order to evolve the system efficiently for such fluids. We demonstrate the convergence properties of this numerical code for the case of a shock tube and for Rayleigh‐Taylor instability. A linear equation of state is also presented, which relates the specific volume of the fluid linearly to the potential temperature, salinity, and pressure. This permits the results to be easily compared to a Boussinesq framework in order to assess whether the Boussinesq approximation adequately represents the relevant exchange of energy to the problem of interest. One such application is included, that of the development of a single salt finger, and it is shown that the energetic behavior of the system is comparable to the typical canonical development of the problem for oceanographic parameters. However, for more compressible systems, the results change substantially even for low‐Mach number flows.
more »
« less
- PAR ID:
- 10383668
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Advances in Modeling Earth Systems
- Volume:
- 14
- Issue:
- 11
- ISSN:
- 1942-2466
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We investigate the long‐time properties of the two‐dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size ε. Under the classical Miles‐Howard stability condition on the Richardson number, we prove that the system experiences a shear‐buoyancy instability: the density variation and velocity undergo an inviscid damping while the vorticity and density gradient grow as . The result holds at least until the natural, nonlinear timescale . Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles‐Howard spectral stability condition; (B) a variation of the Fourier time‐dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, that is, tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.more » « less
-
The weakly compressible translating hollow vortex pair is examined using the Imai–Lamla formula and a direct conformal mapping approach applied to a Rayleigh–Jansen expansion in the Mach number, 𝑀, taken to be small. The incompressible limit has been studied by Crowdy et al. (Eur. J. Mech. B Fluids 37 (2013), 180–186) who found explicit formulas for the solutions using conformal mapping. We improve upon their results by finding an explicit formula for the conformal map which had been left as an integral in Crowdy et al. (2013). The weakly compressible problem requires the solution of two boundary value problems in an annulus. This results in a linear parameter problem for the perturbed propagation velocity and speed along the vortex boundary. We find that two additional constraints are required to solve for the perturbed parameters. We require that the perturbation in vortex area and the perturbation in centroid separation to vanish. Three possible centroid definitions are given and the results worked out for each case. It is found that the correction to the propagation velocity is always negative, so that the vortex always slows down at first order due to compressible effects. Numerical results are consistent in the limit of small vortex size with the previous result by Leppington (J. Fluid Mech. 559 (2006), 45–55) that the propagation parameter is unchanged to 𝑂(𝑀2).more » « less
-
Abstract Gravity waves (GWs) and their associated multi‐scale dynamics are known to play fundamental roles in energy and momentum transport and deposition processes throughout the atmosphere. We describe an initial machine learning model—the Compressible Atmosphere Model Network (CAM‐Net). CAM‐Net is trained on high‐resolution simulations by the state‐of‐the‐art model Complex Geometry Compressible Atmosphere Model (CGCAM). Two initial applications to a Kelvin‐Helmholtz instability source and mountain wave generation, propagation, breaking, and Secondary GW (SGW) generation in two wind environments are described here. Results show that CAM‐Net can capture the key 2‐D dynamics modeled by CGCAM with high precision. Spectral characteristics of primary and SGWs estimated by CAM‐Net agree well with those from CGCAM. Our results show that CAM‐Net can achieve a several order‐of‐magnitude acceleration relative to CGCAM without sacrificing accuracy and suggests a potential for machine learning to enable efficient and accurate descriptions of primary and secondary GWs in global atmospheric models.more » « less
-
Abstract This paper focuses on a system of three-dimensional (3D) Boussinesq equations modeling anisotropic buoyancy-driven fluids. The goal here is to solve the stability and large-time behavior problem on perturbations near the hydrostatic balance, a prominent equilibrium in fluid dynamics, atmospherics and astrophysics. Due to the lack of the vertical kinematic dissipation and the horizontal thermal diffusion, this stability problem is difficult. When the spatial domain is Ω = R 2 × T with T = [ − 1 / 2 , 1 / 2 ] being a 1D periodic box, this paper establishes the desired stability for fluids with certain symmetries. The approach here is to distinguish the vertical averages of the velocity and temperature from their corresponding oscillation parts. In addition, the oscillation parts are shown to decay exponentially to zero in time.more » « less
