Inviscid Supersonic Flow Research Articles

Neural Network with Local Converging Input for Unstructured-Grid Computational Fluid Dynamics

In recent years, surrogate models based on deep neural networks have been widely used to solve partial differential equations for fluid flow physics. This kind of model focuses on global interpolation of the training data and thus requires a large network structure. The process is both time consuming and computationally costly. In the present study, we develop a neural network with local converging input (NNLCI) for high-fidelity prediction using unstructured data. The framework uses the local domain of dependence with converging coarse solutions as input, thereby greatly reducing computational resource and training time. As a validation case, the NNLCI method is applied to study two-dimensional inviscid supersonic flows in channels with bumps. Different bump geometries and locations are examined to benchmark the effectiveness and versatility of this new approach. The NNLCI method can accurately and efficiently capture the structure and dynamics of the entire flowfield, including regions with shock discontinuities. For a new bump configuration, the method can perform prediction with only one neural network, eliminating the need for repeated training of multiple networks for different geometries. A saving of computing wall time is achieved by several orders of magnitude against the high-fidelity simulation with the same level of accuracy. The demand on training data is modest, and the training data can be allocated sparsely. These features are especially advantageous compared with conventional global-to-global deep learning methods and physics-informed methods.

Registration-based model reduction of parameterized two-dimensional conservation laws

We propose a nonlinear registration-based model reduction procedure for rapid and reliable solution of parameterized two-dimensional steady conservation laws. This class of problems is challenging for model reduction techniques due to the presence of nonlinear terms in the equations and also due to the presence of parameter-dependent discontinuities that cannot be adequately represented through linear approximation spaces. Our approach builds on a general (i.e., independent of the underlying equation) registration procedure for the computation of a mapping Φ that tracks moving features of the solution field and on an hyper-reduced least-squares Petrov-Galerkin reduced-order model for the rapid and reliable computation of the solution coefficients. The contributions of this work are twofold. First, we investigate the application of registration-based methods to two-dimensional hyperbolic systems. Second, we propose a multi-fidelity approach to reduce the offline costs associated with the construction of the parameterized mapping and the reduced-order model. We discuss the application to an inviscid supersonic flow past a parameterized bump, to illustrate the many features of our method and to demonstrate its effectiveness.

Analysis of nonlinear aeroelastic response of curved panels under shock impingements

This study focuses on the fluid–structure interaction of two-dimensional flexible curved panels subjected to an impinging oblique shockwave in inviscid supersonic flow. A fluid–structural coupling solver is employed to simulate the nonlinear aeroelastic responses of four curved panels under different incident shock strengths. The responses, comprising the static deformation, flutter onset, and post-flutter are determined and compared to those of flat panel and shock-free conditions. The curvature effects, shock–shock interaction, as well as incident strengths, have profound influences on the solution. In the absence of incident shock condition, the greater the curvature of the panel, the smaller the critical flutter dynamic pressure. In the condition of shock impingement, for the smaller curvature, strong incident shock can advance the flutter onset, and there exist multiple solutions under different initial conditions; while for the larger curvature, strong incident shock can suppress the flutter onset, and the travel-like oscillation is observed at post-flutter.

On a posteriori error estimation using distances between numerical solutions and angles between truncation errors

The geometric properties of the ensemble of numerical solutions obtained by the algorithms of different inner structure are addressed from the prospects for a posteriori error estimation. The numerical results are presented for the two-dimensional inviscid supersonic flows, containing shock waves. The truncation errors are computed using a postprocessor, the approximation errors are calculated by the subtraction of the numerical and the analytic solutions. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The correlation of the angles between the approximation errors and the corresponding angles between the computable truncation errors is observed in numerical tests and discussed from the viewpoint of the measure concentration effect and the algorithmic randomness. The analysis of the truncation errors’ geometry and the distances between solutions enables the estimation of the approximation error norm on the ensemble of numerical solutions obtained by the independent algorithms.

New Approach of Axisymmetric Compressible Finite-Volume Lattice Boltzmann Method for Numerical Simulation of Supersonic Inviscid Flow

A new approach of the axisymmetric compressible lattice Boltzmann method (LBM) has been developed for the numerical simulation of supersonic inviscid flow using the finite volume method. The circular function idea has been used for capturing the compressibility effect in the supersonic flow field. In this study, the axisymmetric LBM equations based on the circular function have been derived for the first time and presented in detail and appropriate axisymmetric boundary conditions have been applied for the 2-dimensional, 13-velocity, 2-energy-level lattices. For validation of developed code, two supersonic axisymmetric flows have been simulated around the hemisphere nose and 60° blunt body. A comparison of the obtained results with some valid empirical data shows the accuracy of the developed numerical algorithm. Also, the results of 2D and axisymmetric simulations have been studied to evaluate the effects of the new derived axisymmetric formulation on flow macroscopic properties.

A high-order accurate unstructured spectral difference lattice Boltzmann method for computing inviscid and viscous compressible flows

In the present work, the spectral difference lattice Boltzmann method (SDLBM) is implemented on unstructured meshes for the solution methodology to be capable of accurately simulating the compressible flows over complex geometries. Both the inviscid and viscous compressible flows are computed by applying the unstructured SDLBM. The compressible form of the discrete Boltzmann–BGK equation with the Watari model is considered and the solution of the resulting system of equations is obtained by applying the spectral difference method on arbitrary quadrilateral meshes. The accuracy and robustness of the unstructured SDLBM for simulating the compressible flows are demonstrated by simulating four problems that are steady inviscid supersonic flow past a bump, steady inviscid subsonic flow over the two-element NACA 4412-4415 airfoil with and without the ground effect, steady viscous transonic flow around the NACA 0012 airfoil and unsteady viscous subsonic flow past two side-by-side cylinders. The results obtained by applying the unstructured SDLBM are in good agreement with those of the available high-order accurate Euler/Navier-Stokes solvers and also the experimental data. The present study introduces the unstructured SDLBM as an appropriate inviscid and viscous compressible LBM flow solver for accurately simulating fluid flows over practical problems.

The characteristic boundary condition in pressure-based methods

This article deals with the implementation of the characteristic boundary condition, also known as the pressure far field boundary condition, in segregated pressure-based flow solvers. This boundary condition applies the Riemann invariants to determine the flow variables at domain inlets and outlets. The newly developed method is implemented in a cell-centered unstructured grid code following a finite volume discretization. Testing is performed by solving the following problems: turbulent flow over a flat plate; inviscid transonic flow over a circular are bump; inviscid supersonic flow over three equidistant circular arc bumps; turbulent flow over an axisymmetric transonic bump; turbulent flow over a NACA 0012 airfoil at 10° angle of attack; and turbulent flow over the three-dimensional ONERA M6 wing at 3.06° angle of attack. Predictions with the current pressure-based solver are compared with similar ones generated using a density-based method and with experimental data. Results are in excellent agreement.

Some aspects of numerical modeling of inviscid supersonic flow in a duct with a central wedge

On the example of the supersonic inviscid flow in a duct with a central wedge, the problem of increased errors of the solution arising in the vicinity of the fractures of a streamlined surface is investigated. The effectiveness of various ways to improve the solution quality is evaluated. Calculations based on the MUSCL approach for increasing the approximation order are performed using the finite-volume unstructured SINF/Flag-S code developed at the Peter the Great St. Petersburg Polytechnic University. The data obtained with this code are compared with the results of calculations performed using the ANSYS Fluent.

Comparing the results of analytical method and numerical simulation method for supersonic flow over main wing of S-125 Neva/Pechora missle


 
 
 This paper compares the results of Computational Fluid Dynamic (CFD) and analytical method when applying to the case of supersonic flow over main wing of S-125 Neva/Pechora surface-to-air missle. The theory of two-dimensional, steady, inviscid supersonic flow over oblique surface, equations of relations between Mach number, pressure, temperature of flow in front and behind of shock wave, details of MATLAB algorithm and Spalart-Allmaras turbulent model will be displayed in this paper. Meshing process had been done by Meshing module and numerical simulation process had been done by Fluent module of ANSYS software. The results of two methods were presented in graphs of Mach number, pressure, temperature distributions along the chord line of wing. The reasons of differences between two results will be presented at the end of this paper. 
 
 

On the theory of supersonic inviscid flow separation in gasdynamic problems

A general schematic flow representation that explains the mechanism of inviscid gas separation in time-dependent and three-dimensional gas flows is presented. The scenario of gas flow separation from a body surface or a mixing layer is described as a vortex which induces in the flowfield a velocity opposing to that of the main flow, thus decelerating it. Within the framework of this scenario the analytical conditions of separation are obtained for conical and self-similar gas flows which coincide with the results of experimental and numerical simulations.

Theoretical solutions to three-dimensional asymmetrical shock/shock interaction

This paper explores theoretical solutions to the three-dimensional (3D) shock/shock interaction induced by 3D asymmetrical intersecting compression wedges in supersonic inviscid flows. For Mach interactions, an analytical method known as spatial-dimension reduction, which transforms the problem of 3D steady shock/shock interaction into a two-dimensional (2D) pseudo-steady problem on cross sections, is used to obtain the solutions in the vicinity of the Mach stem. The theoretical solutions include the pressure, temperature, density, Mach number behind the Mach stem, and total pressure recovery coefficient. Numerical simulations are performed to validate the theoretical results. Here, the NND scheme is employed by solving 3D inviscid Euler equations, and good agreements are obtained. The asymmetry of the flow characteristics induced by the wedge angle and sweep angle are thoroughly discussed.

Investigations of Three-Dimensional Shock/Shock Interactions over Symmetrical Intersecting Wedges

This study explored inviscid supersonic corner flows induced by three-dimensional symmetrical intersecting compression wedges by introducing the spatial dimension reduction theoretical approach to transform the three-dimensional steady shock/shock interaction problem into a two-dimensional pseudosteady problem; this method allows not only wave configurations, which include regular reflection and Mach reflection, to be determined accurately, but also flowfield characteristics, which include density, temperature, pressure, and total pressure recovery coefficient near the regular reflection point (or in the vicinity of the Mach reflection triple point), as well as the location and the strength of the Mach stem. Theoretical results were compared to numerical simulation (performed by solving three-dimensional inviscid Euler equations with an non-oscillatory and non-free-parameters dissipative finite difference scheme) and analyzed thoroughly. The effects of inflow Mach number, sweep angle, and wedge angle on flowfield parameters and wave configurations were also considered. The influence of sweep angle is negligible, but the effects of Mach number and wedge angle are significant.

Optimal Forebody Shape for Minimum Drag in Supersonic Flow

In this work, a simple and efficient numerical approach to determine the shape of the minimum-drag axisymmetric forebody in inviscid supersonic flow with an attached shock constraint has been described. Taylor–Maccoll equation in conjunction with the tangent cone method is employed to estimate the pressure drag coefficient which is also chosen as the cost function. The forebody geometry is parameterized using a Non-Uniform Rational B-Splines (NURBS) curve whose control points are the design variables for optimisation using the steepest descent algorithm. Numerical studies demonstrate that the optimal forebody geometry for a given length and base radius has as much as 15 % lesser drag, depending on the Mach number than a cone of the same fineness ratio and that the convergence to the optimal solution exhibits a relatively weak Mach-number dependence.

Investigation of numerical viscosities and dissipation rates of second-order TVD-MUSCL schemes for implicit large-eddy simulation

Monotonically integrated large-eddy simulation (MILES) approach utilizes the dissipation inherent to shock-capturing schemes to emulate the role played by explicit subgrid-scale eddy diffusivity at the high-wavenumber end of the turbulent energy spectrum. In the current study, a novel formulation is presented for quantifying the numerical viscosity inherent to Roe-based second-order TVD-MUSCL schemes for the Euler equations. Using this formulation, the effects of numerical viscosity and dissipation rate on implicit large-eddy simulations of turbulent flows are investigated. At first, the three-dimensional (3-D) finite-volume extension of the original Roe's flux, including Roe's Jacobian matrix, is presented. The fluxes are then extended to second-order using van Leer's MUSCL extrapolation technique. Starting from the 3-D Roe-MUSCL flux, an expression is derived for the numerical viscosity as a function of flux limiter and characteristic speed for each conserved variable, distance between adjacent cell centers, and a scaling parameter. Motivated by Thornber et al. [16] study, the high numerical viscosity inherent to TVD-MUSCL schemes is mitigated using a z-factor that depends on local Mach number. The TVD limiters, along with the z-factor, were initially applied to the 1-D shock-tube and 2-D inviscid supersonic wedge flows. Spatial profiles of numerical viscosities are plotted, which provide insights into the role of these limiters in controlling the dissipative nature of Roe's flux while maintaining monotonicity and stability in regions of high gradients. Subsequently, a detailed investigation was performed of decaying homogeneous isotropic turbulence with varying degrees of compressibility. Spectra of numerical viscosity and dissipation rate are presented, which clearly demonstrate the effectiveness of the z-factor both in narrowing the wavenumber range in which dissipation occurs, and in shifting the location of dissipation peak closer to the cut-off wavenumber. The present z-factor approach of limiting numerical viscosities is shown to perform as well as, or even better in some cases, than the Thornber et al. approach of directly limiting velocity jumps at a cell interface.

Modeling and Simulation of Supersonic Flow in 2D Nozzle Using MacCormack’s and Upwind Methods

In this work, two-dimensional inviscid supersonic flow in nozzle has been investigated using CFD schemes and characteristics method. The employed scheme is MacCormack’s finite volume method. Our own code CHARMAC, was written using MATLAB environment. Standard boundary conditions and the grid parameters were considered to solve the problem. Before analyzing the flow by CFD method, we obtained the ideal nozzle geometry using the method of characteristics for a 2D divergent Nozzle. Using 2D nozzle flow relations, an optimal throat area is found. At the end we compare the results with the advection upstream splitting Method (Ausm) and Fluent.

Numerical modelling of the entrainment of particles in inviscid supersonic flow

The interaction between particles situated in close proximity and moving at supersonic speeds is investigated computationally. The simplest case of the motion of a single particle travelling behind a lead particle is used to elucidate the role of aerodynamic forces in the motion of a group of particles. The effect of the following parameters on the drag and lift forces acting on each of two particles of equal diameter in proximity is investigated: the free-stream Mach number, and the axial and lateral displacements of the trailing particle. The two-dimensional flow field is numerically simulated using an unsteady Euler CFD code to find the steady-state drag and lift coefficients for both particles. Three static zones of aerodynamic influence in the wake of the lead particle are identified, which are denoted as the entrainment, lateral attraction, and ejection zones. A non-dimensional representation of the zones of influence is given. It is shown that the dynamic entrainment of particles can occur even when the path of the trailing particle originates outside the entrainment and lateral attraction zones.

Aerodynamic Computations Using a Finite Volume Method with an HLLC Numerical Flux Function

A finite volume method for the simulation of compressible aerodynamic flows is de- scribed. Stabilisation and shock capturing is achieved by the use of an HLLC consistent numerical flux function, with acoustic wave improvement. The method is implemented on an unstructured hybrid mesh in three dimensions. A solution of higher order accuracy is obtained by reconstruc- tion, using an iteratively corrected least squares process, and by a new limiting procedure. The numerical performance of the complete approach is demonstrated by considering its application to the simulation of steady turbulent transonic flow over an ONERA M6 wing and to a steady inviscid supersonic flow over a modern military aircraft configuration.

A New Spatial Discretization Strategy of the Convective Flux Term for the Hyperbolic Conservation Laws

:In this work, a new spatial discretization scheme for flows governed by the hyperbolic conservation laws is proposed. The spatial discretization involves the concept of classical particle velocity upwinding (PVU) for the convective flux term in the hyperbolic conservation laws. The novelty of the approach lies in the use of the fluid particle velocity or the entropy wave speed at the cell interface to ascertain the upwind direction. The cell face convective fluxes are obtained from a first order or a second order upwind biased interpolation, depending on whether the cell under consideration lies in the vicinity of a discontinuity or in a region of steep gradients in the solution. The discontinuities or regions of steep gradients are detected by employing a smoothness indicator function as employed in some of the earlier studies. The proposed spatial discretization strategy has been combined with a two step, second order explicit time integration strategy for the application to the solution of the unsteady Euler/Navier-Stokes equations in the strong conservation form. Test cases involving two 1-D Riemann problems, three 2-D inviscid supersonic flow problems and a 2-D viscous supersonic flow problem, have been employed to establish the validity of the procedure and to assess the performance of the proposed strategy. The proposed PVU scheme performs quite favorably in comparison to conventional schemes. From the point of view of implementation, particularly in multidimensional scenarios, this strategy offers a good balance of accuracy and simplicity.

Basic function scheme of polynomial type

A new numerical method named as basic function method is proposed. It can directly discretize differential operators on unstructured grids. By expanding the basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the second-order polynomial as a basic function and applying the flux splitting method and the combination of central and upwind schemes to suppress non-physical fluctuation near shock waves, a second-order basic function scheme of polynomial type is proposed to solve inviscid compressible flows numerically. Numerical results of typical examples for two-dimensional inviscid compressible transonic and supersonic steady flows indicate that the new scheme has high accuracy and high resolution for shock waves. Combined with the adaptive remeshing technique, satisfactory results can be obtained.

Computation of Inviscid Supersonic Flows Around Cylinders and Spheres With the V-SGS Stabilization and YZβ Shock-Capturing

The YZβ shock-capturing technique was introduced originally for use in combination with the streamline-upwind/Petrov–Galerkin (SUPG) formulation of compressible flows in conservation variables. It is a simple residual-based shock-capturing technique. Later it was also combined with the variable subgrid scale (V-SGS) formulation of compressible flows in conservation variables and tested on standard 2D test problems. The V-SGS method is based on an approximation of the class of SGS models derived from the Hughes variational multiscale method. In this paper, we carry out numerical experiments with inviscid supersonic flows around cylinders and spheres to evaluate the performance of the YZβ shock-capturing combined with the V-SGS method. The cylinder computations are carried out at Mach numbers 3 and 8, and the sphere computations are carried out at Mach number 3. The results compare well to those obtained with the YZβ shock-capturing combined with the SUPG formulation, which were shown earlier to compare very favorably to those obtained with the well established OVERFLOW code.