Two-dimensional Supersonic Flow Research Articles

Assessment of Multi-Physical Fields Reconstruction Approaches in Three-Dimensional Supersonic Flow with Shocks

The paper evaluates the application of several multi-physical fields reconstruction approaches based on the Tomographic Particle Image Velocimetry (PIV) in three-dimensional supersonic flows with shock waves, including the pressure, temperature, and density. The MacCormack method and the Flux Vector Splitting method with outstanding performance in two-dimensional supersonic flow are extended into three-dimensional forms. The conventional Poisson method is also considered because of its widespread application and high accuracy in subsonic and transonic flow. All of these approaches are established by the conservative Navier-Stokes equations and solved by the time-marching iteration process, combined with the perfect gas law and adiabatic flow assumption. The performances are evaluated by numerical velocimetry data. To simulate the typical three-dimensional features, the cases of uniform freestream at Mach 0.78 and 3 past a cone with a 20° half-angle are selected to obtain a sufficient spanwise velocity component. The results confirm the feasibility of the PIV-based reconstruction methods in the conical flow field. The conventional Poisson method performs well only in the subsonic case while the Flux Vector Splitting method has a better performance in supersonic flow, including higher accuracy, stability, and efficiency, with a low-level root-mean-square error of 0.449% and local maximum relative error of 1%.

Acoustic emission due to the interaction between shock and instability waves in two-dimensional supersonic jet flows

An analytical model is developed to study the sound produced by the interaction between shock and instability waves in two-dimensional supersonic jet flows. The jet is considered to be of vortex-sheet type and two-dimensional Euler equations are linearized to determine the governing equations for shock and instability waves and their interaction. Pack's model is used to describe shock waves, while instability waves are calculated using spatial stability analysis. The interaction between shock and instability waves can be solved analytically by performing Fourier transform and subsequently using the method of steepest descent. Sound produced by the interaction between the instability wave and a single shock cell is studied first, after which that due to a number of cells follows. We find that the model developed in this study can correctly predict the frequencies of the fundamental screech tone and its first and second harmonics. We show that the predicted sound directivity, even from a single shock cell, is in good agreement with experimental data. In particular, this model shows the strongest noise emission close to the upstream direction but the emitted noise starts to rapidly decay as the observer angle approaches $180^\circ$ , which is in accordance with experimental results; this suggests that the effective noise from a single shock cell is far from of the monopole type as assumed in the classical Powell's model. We find that the noise directivity is very sensitive to the local growth rate of the instability waves and the noise is generated primarily through the Mach wave mechanism.

МЕТОД ПОСТАНОВКИ ГРАНИЧНЫХ УСЛОВИЙ ПРИ ЧИСЛЕННОМ РЕШЕНИИ ЗАДАЧ ГАЗОДИНАМИКИ В ОБЛАСТЯХ СЛОЖНОЙ ФОРМЫ

The question of setting boundary conditions for a two-dimensional supersonic gas flow in accessible forms is considered. An alternative method of setting boundary conditions on the body surface is proposed. The results of the calculation of double Mach reflection of a shock wave from a flat wall and the gas flow inside of liquid-propellant rocket engine using alternative method are presented and analyzed. The conclusion about the possibility of applying the new method in further calculations is made.

The effect of dust particles on the evolution of planar and non-planar shock wave in two-dimensional supersonic flow of van der Waals gas

The effect of dust particles on the evolution of planar and non-planar shock wave in two-dimensional supersonic flow of van der Waals gas

Two-dimensional pseudo-steady supersonic flow around a sharp corner for the generalized Chaplygin gas

<abstract><p>In this paper, the expansion problem which arises in a two-dimensional (2D) isentropic pseudo-steady supersonic flow expanding into vacuum around a sharp corner for the generalized Chaplygin gas is studied. This expanding problem catches the interaction of an incomplete centered simple wave with a backward planar rarefaction wave and the interaction of a non-planar simple wave with a rigid wall boundary of the 2D self-similar Euler equations. Using the methods of characteristic decompositions and invariant regions, we get the hyperbolicity in the wave interaction domains and prior $ C^{1} $ estimates of solutions to the two interaction problems. It follows the global existence of the solution up to infinity of the gas expansion problem.</p></abstract>

Concave Bump for Impinging-Shock Control in Supersonic Flows

In the present study, a novel concave bump for impinging-shock control in two-dimensional supersonic flows is investigated. An analytical method for preliminary bump design based on a generalized shape of a shock-canceling bump has been developed and verified numerically. An extensive proof-of-concept study was performed at a freestream Mach number ranging from 2.5 to 5.0 for shock-generator angles varying from 6 to 12 degrees. It could be demonstrated that a concave bump designed for a given flow-deflection angle is capable of significantly reducing the size of the separation bubble as well as the total pressure losses throughout the Mach number range investigated. The achievable gains depend on the Mach number, the flow-deflection angle, and the relative impingement position of the incident shock front on the bump. The highest values of separation-length reduction (up to 100%), momentum thickness reduction (up to 31%), and pressure recovery factor increase (up to 33%) were obtained at the optimum shock impingement position for the largest deflection angle studied. The concave bump is less effective, and in some cases even disadvantageous, when the incident shock wave does not optimally strike the bump crest.

The formation of shock wave in a two-dimensional supersonic planar and axisymmetric non-ideal gas flow with magnetic field

The formation of shock wave in a two-dimensional supersonic planar and axisymmetric non-ideal gas flow with magnetic field

Black-Box Function Aerodynamic Topology Optimization Algorithm via Machine Learning Technologies

The objective of this research is to construct an efficient global topology optimization method using machine learning technologies. In the conventional design process of mechanical design, the conceptual design is the earliest stage of the design process; and it is carried out based on the designer’s own idea/experience. Therefore, it is difficult to obtain innovative and high-quality concepts overwhelming the designer’s knowledge since the earliest stage of the design process has the largest impact. In this research, therefore, the black-box function aerodynamic topology optimization algorithm via machine learning technologies (FANTOM) is developed to overcome the problem. In the FANTOM approach, topology optimization problems are solved using/combining two efficient global optimization methods developed by the authors: the efficient global optimization method for discontinuous optimization problems with infeasible regions using classification method, and the efficient global optimization method via clustering/classification methods and exploration strategy. In the present approach, topological optimal designs can be obtained only by setting an objective function and constraint conditions. The validity of the FANTOM approach is demonstrated in an inviscid drag minimization problem at a two-dimensional supersonic flow condition, which provides an optimal topology as the Busemann biplane airfoil. Executing topology optimizations with the variation in freestream Mach number, it is also demonstrated that the FANTOM approach can explore topological optimal designs robustly.

Automatic detection of oblique shocks and simple waves in schlieren images of two-dimensional supersonic steady flows

This paper presents a technique to automatically detect flow structures recorded in schlieren images of supersonic flows. The method provides position and local slope of the detected lines. Salient flow structures considered are shock waves, expansion fans and Mach waves. If the local flow direction is known, the technique provides local Mach number measurement. Moreover, the peculiarities of schlieren technique make the algorithm suitable for Mach measurements in flows where direct velocity measurements are challenged by difficult insemination and for which calibrated aerodynamic pressure probes are not available. The technique is applied to two test cases. In the first case study, the local Mach number is measured at the axis of a supersonic nozzle operated with siloxane vapor MDM. In the second test case, supersonic flow of MDM was analyzed around a diamond shaped airfoil providing the Mach number upstream/downstream oblique shocks/expansion fans and the slope of oblique shocks/fan edges.

Peculiarities of the flows forming in processes of an impulse starting of a supersonic wind tunnel with different diffusers

The peculiarities of supersonic unsteady flows forming in the processes of an impulse starting of a wind tunnel including a fore-chamber, a nozzle, a diffuser, and an exhaust tank are considered. The fore-chamber is separated from the flow duct by a thin breakable diaphragm. Before the wind tunnel starting, the gas contained in the exhaust tank is pumped out down to a very small pressure, and then the high-pressured working gas is fed into the fore-chamber. Upon reaching some value of this pressure, the diaphragm “instantaneously” ruptures, and the working gas starts exhausting through the nozzle: a rapid unsteady process of the wind tunnel starting arises. The numerical simulation of the flows forming at the impulse starting of the simplest experimental gas-dynamic facility has been carried out; the facility is arranged with a nozzle forming at its exit a two-dimensional supersonic flow with the Mach number of 2.9, and with replaceable diffusers having different relative areas of the throat. The numerical computations of two-dimensional unsteady flows have been carried out using the Reynolds-averaged Navier—Stokes equations and the SST k-ω turbulence model. The flow patterns computed numerically are compared with the data of the optical visualization of the flow obtained in the experimental gas-dynamic facility in the process of its starting.

Single-mode flutter of an elastic plate in the presence of the boundary layer

The flutter of an elastic plate in a viscous supersonic gas flow is investigated. The influence of viscous perturbations of the boundary layer on the single-mode flutter is studied considering two different types of a boundary layer profile: the generalized convex boundary layer profile and the profile with a generalized inflection point. It is shown that in the case of the convex layer for thick boundary layers the plate is fully stabilized. In the case of a profile with the generalized inflection point, the thickening of the layer first yields the increase of the growth rates of the perturbations, and the growth is greater than in the inviscid approximation. Numerical simulation of two-dimensional supersonic gas flows with boundary layers (that can have a destabilizing effect on the flutter of an elastic surface) over different curved surfaces is performed. The flow patterns over them are studied: regions of these surfaces, over which stable boundary-layer profiles with a generalized inflection point are formed, are found, and their boundaries are obtained.

Two-dimensional supersonic flow over concave surfaces

Planar, constant curvature wedge configurations are experimentally and numerically studied in order to determine conditions relating to singly curved shock waves and the effects of varying surface curvatures on shock geometry and downstream flow. Post-shock flow compression is seen to be an important factor in determining conditions on the reflecting surface. A symmetrical configuration of constant radii, concave wedges, similar to that of an internal compression intake, was experimentally investigated. Three models were tested with surface curvature radii of 160, 120, and 80 mm, each at Mach numbers of 3.0, 3.2, and 3.4 and at two separation distances. In addition, the applicability of planar curved shock theory (CST) is briefly examined for the conditions immediately behind the shock and the following streamline geometry calculated using a Taylor series expansion. Streamline shapes, as predicted by CFD and CST, agree exactly on the downstream surface of the shock waves. Agreement deteriorates with downstream distance as well as with increasing surface curvature.

Global Topology Optimization in Two-Dimensional Supersonic Flow

Global Topology Optimization in Two-Dimensional Supersonic Flow

Effect of Small Bluntness on Formation of Görtler Vortices in a Supersonic Compression Corner Flow

The influence of small cylindrical bluntness of the leading edge of a flat plate on formation of spatial structures in a nominally two-dimensional supersonic compression corner flow at the Mach number M∞ ≈ 8 and a laminar state of the undisturbed boundary layer is studied by the method of temperature-sensitive paints. Streamwise vortices are found in the region of reattachment of the separated flow in a wide range of Reynolds numbers (0.15 · 106–2.55 · 106) for various angles of flow deflection and plate lengths. It is demonstrated that the existence of these vortices induces spanwise oscillations of the heat transfer coefficient; the amplitude of these oscillations may reach 30%. The maximum deviations of the Stanton number reaching 80% are observed in the case with significant roughness of the leading edge of the flat plate. Both the maximum Stanton numbers in the reattachment region and the amplitude of spanwise oscillations of the Stanton number induced by streamwise vortices are found to decrease significantly in the case of small bluntness of the leading edge. Solutions of three-dimensional Navier–Stokes equations are obtained for some test conditions. The computed results are in good agreement with experimental data, which points to a significant stabilizing effect of small bluntness on the intensity of streamwise vortices.

Design of a three-dimensional scramjet nozzle considering lateral expansion and geometric constraints

A new method based on quasi two-dimensional supersonic flow and maximum thrust theory to design a three-dimensional nozzle while considering lateral expansion and geometric constraints is presented in this paper. To generate the configuration of the three-dimensional nozzle, the inviscid flowfield is calculated through the method of characteristics, and the reference temperature method is applied to correct the boundary layer thickness. The computational fluid dynamics approach is used to obtain the aerodynamic performance of the nozzle. Results show that the initial arc radius slightly influences the axial thrust coefficient, whereas the variations in the lateral expansion contour, the length and initial expansion angle of the lower cowl significantly affect the axial thrust coefficient. The three-dimensional nozzle designed by streamline tracing technique is also investigated for comparison to verify the superiority of the new method. The proposed nozzle shows increases in the axial thrust coefficient, lift, and pitching moment of 6.86%, 203.15%, and 642.86%, respectively, at the design point, compared with the nozzle designed by streamline tracing approach. In addition, the lateral expansion accounts for 22.46% of the entire axial thrust, while it has no contribution to the lift and pitching moment in the proposed nozzle.

Model reduction for supersonic cavity flow using proper orthogonal decomposition (POD) and Galerkin projection

The reduced-order model (ROM) for the two-dimensional supersonic cavity flow based on proper orthogonal decomposition (POD) and Galerkin projection is investigated. Presently, popular ROMs in cavity flows are based on an isentropic assumption, valid only for flows at low or moderate Mach numbers. A new ROM is constructed involving primitive variables of the fully compressible Navier-Stokes (N-S) equations, which is suitable for flows at high Mach numbers. Compared with the direct numerical simulation (DNS) results, the proposed model predicts flow dynamics (e.g., dominant frequency and amplitude) accurately for supersonic cavity flows, and is robust. The comparison between the present transient flow fields and those of the DNS shows that the proposed ROM can capture self-sustained oscillations of a shear layer. In addition, the present model reduction method can be easily extended to other supersonic flows.

Entropic lattice Boltzmann model for gas dynamics: Theory, boundary conditions, and implementation.

We present in detail the recently introduced entropic lattice Boltzmann model for compressible flows [N. Frapolli et al., Phys. Rev. E 92, 061301(R) (2015)PLEEE81539-375510.1103/PhysRevE.92.061301]. The model is capable of simulating a wide range of laminar and turbulent flows, from thermal and weakly compressible flows to transonic and supersonic flows. The theory behind the construction of the model is laid out and its thermohydrodynamic limit is discussed. Based on this theory and the hydrodynamic limit thereof, we also construct the boundary conditions necessary for the simulation of solid walls. We present the inlet and outlet boundary conditions as well as no-slip and free-slip boundary conditions. Details necessary for the implementation of the compressible lattice Boltzmann model are also reported. Finally, simulations of compressible flows are presented, including two-dimensional supersonic and transonic flows around a diamond and a NACA airfoil, the simulation of the Schardin problem, and the three-dimensional simulation of the supersonic flow around a conical geometry.

Supersonic flow past a flat lattice of cylindrical rods

Two-dimensional supersonic laminar ideal gas flows past a regular flat lattice of identical circular cylinders lying in a plane perpendicular to the free-stream velocity are numerically simulated. The flows are computed by applying a multiblock numerical technique with local boundary-fitted curvilinear grids that have finite regions overlapping the global rectangular grid covering the entire computational domain. Viscous boundary layers are resolved on the local grids by applying the Navier–Stokes equations, while the aerodynamic interference of shock wave structures occurring between the lattice elements is described by the Euler equations. In the overlapping grid regions, the functions are interpolated to the grid interfaces. The regimes of supersonic lattice flow are classified. The parameter ranges in which the steady flow around the lattice is not unique are detected, and the mechanisms of hysteresis phenomena are examined.

Gas-kinetic unified algorithm for hypersonic aerothermodynamics covering various flow regimes solving Boltzmann model equation

How to solve hypersonic aerothermodynamics and complex flow mechanism covering various flow regimes from high rarefied free-molecular flow of outer-layer space to continuum flow of near-ground is one of the frontier basic problems in the field of fluid physics. In this work, the unified Boltzmann model equation based on the molecular velocity distribution function is presented for describing complex hypersonic flow transport phenomena covering all flow regimes by physics analysis and model processing of the collision integral to the Boltzmann equation. The discrete velocity ordinate method is developed to simulate complex flows from low Mach numbers to hypersonic flight, and the gas-kinetic coupling-iteration numerical scheme is constructed directly to solve the evolution and updating of the molecular velocity distribution function by employing the unsteady time-splitting method and the NND finite-difference technique. Then, the gas-kinetic unified algorithm (GKUA) is presented to~simulate the three-dimensional hypersonic aerothermodynamics and flow problems around space vehicles covering various flow regimes from free-molecule to continuum. To verify the accuracy and reliability of the present GKUA and simulate gas thermodynamic transport phenomena covering various flow regimes, firstly, the two-dimensional supersonic flows around a circular cylinder are simulated in the continuum regime of Kn= 0.0001 and in the high rarefied regime of Kn= 0.3 through the comparison between the Navier-Stokes (N-S) solution and the direct simulation Monte Carlo (DSMC) result, respectively. It is indicated that the GKUA can exactly converge to the N-S solution in the continuum flow regime, and the computed results of the GKUA are consistent with the DSMC simulation with a small deviation of 0.45% in the high rarefied flow regime. Then, the three-dimensional complex hypersonic flows around reusable satellite shape are studied as one of the engineering applications of the GKUA with a wide range 0.002 Kn 1.618 of the free-stream Knudsen numbers and different Mach numbers during re-entering Earth atmosphere with the flying altitudes of 110-70~km. The computed results are found to be in high resolution of the flow fields and in good agreement in a deviation range of 0.27%-8.56% by comparison among the relevant reference data, DSMC and theoretical predictions. The complex flow mechanism, flow phenomena and changing laws of hypersonic aerothermodynamics are revealed for spacecraft re-entry into the atmosphere, and the effects of rarefied gas and wall temperature on the aerothermodynamics characteristics of re-entry satellite shape are compared and analysed with different Knudsen numbers and wall temperature ratios of Tw/T = 1.6, 10 and 15.6. It is validated that the non-dimensional heat flux coefficient in the rarefied transitional flow regime is higher than that of the continuum and near-continuum flow regimes, the high wall temperature results in the enlarging amplitude of temperature variation on the stagnation line and the serious effect on the heat flux of the stagnation point, and wall temperature becomes lower, the heat flux coefficient of wall surface becomes bigger, and the friction force and pressure coefficients decrease. The non-equilibrium level of flow velocity slip and temperature jump on the surface of space vehicle becomes severer, and the stronger heat transfer effect between the space vehicle and the gas flow is produced as the Mach number or Knudsen number of the free-stream flow increases. It can be realized from this study that the gas-kinetic unified algorithm directly solving the Boltzmann model velocity distribution function equation may provide an important and feasible way that complex hypersonic aerothermodynamic problems and flow mechanisms from high rarefied free-molecule to continuum flow regimes can be solved effectively and reliably.

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.