Two-dimensional Compressible Flow Research Articles

On the existence of simple waves for two-dimensional non-ideal magneto-hydrodynamics

Abstract In this article, a method called characteristic decomposition is used to show the presence of simple waves for the two-dimensional compressible flow in a non-ideal magneto-hydrodynamics system. Here, a steady and pseudo-steady state magneto-hydrodynamics system is considered, and we provide a characteristic decomposition of the flow equations in both systems. This decomposition ensures the presence of a simple wave adjacent to a region of constant state for the system under consideration. Further, this result is extended as an application of the characteristic decomposition in a pseudo-steady state, and we prove the existence of a simple wave in a full magneto-hydrodynamics system by taking the vorticity and the entropy to be constant along the pseudo-flow characteristics. These results extend the fundamental theorem proposed by Courant and Friedrichs for a reducible system (R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, New York, Interscience Publishers, Inc., 1948, p. 464). A motivational work was carried out for an ideal gas by Li et al. (“Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations,” Commun. Math. Phys. Math. Phys., vol. 267, no. 1, pp. 1–12, 2006) and for a non-ideal gas by Zafar and Sharma (“Characteristic decomposition of compressible Euler equations for a non-ideal gas in two-dimensions,” J. Math. Phys., vol. 55, no. 9, pp. 093103–093112, 2014], [M. Zafar, “A note on characteristic decomposition for two-dimensional euler system in van der waals fluids,” Int. J. Non-Linear Mech., vol. 86, pp. 33–36, 2016].

On immiscibility preservation conditions of material interfaces in the generic five-equation model

Interfaces separating pure materials and mixtures tend to be severely smeared with interface-capturing methods for compressible multi-material flows, necessitating the requirement of various interface-sharpening techniques. However, these techniques have various problems related to consistency, conservation, and thermodynamic compatibility. In this work, we derive a general theoretical formulation of interface-sharpening techniques for the generic five-equation model. This theoretical formulation is not only conservative in mass, momentum, and total energy but also asymptotically compatible with the thermodynamic mixture laws of the mixture model upon which it is constructed, and is independent of various specific numerical algorithms. We further propose a general numerical method to solve this theoretical formulation. The proposed method is consistent and conservative, and it prevents spurious errors at the interfaces. Examples of one- and two-dimensional multimaterial compressible flow problems, including shocks and interfaces, are considered to verify the analysis and demonstrate the efficiency of the method.

A Vertex-Centered Arbitrary Lagrangian-Eulerian Finite Volume Method with Sub-Cells for Two-Dimensional Compressible Flow

A Vertex-Centered Arbitrary Lagrangian-Eulerian Finite Volume Method with Sub-Cells for Two-Dimensional Compressible Flow

Implication of injection angle and jet-to-cross flow ratio on the combustion characteristics of a scramjet combustor

The present work deals with the implication of injection angle and jet-to-cross flow ratio on the combustor performance of a cavity-based scramjet combustor. Unsteady, two-dimensional compressible flow through the scramjet combustor is considered to comprehend the effects of Mach number (Ma), jet-to-cross flow ratio ([Formula: see text]) and injection angle (α) on the combustion characteristics. Results are presented for the following range of parameters: 2.0 ≤ Ma ≤ 2.7; 15° ≤ α ≤ 60° and 0.5 ≤ [Formula: see text] ≤ 1.2. Intense recirculation zone is observed at θ = 45° irrespective of inlet Mach number which are quintessential for proper mixing. Further, the size of the recirculation zone is the largest when jet-to-cross flow ratio is equal to 0.8 acting as a precursor for enhanced mixing performance. Moreover, static pressure reaches an optimum value at injection angle of 45° beyond which further increase would result in thermal choking. Our study reveals that mixing is improved with an increase in Mach number (Ma). It is observed that the size of the recirculation zone is strongly dependent on the jet-to-cross flow pressure ratio along with the injection angle. To put it in a nutshell, the injection angle of 45° along with a jet-to-cross flow pressure ratio of 0.8 is conducive the optimal performance in scramjet combustor.

Implementation of IDEAL algorithm based on Delaunay triangular mesh for 2D-compressible flows

PurposeTo present the detailed implementation processes of the IDEAL algorithm for two-dimensional compressible flows based on Delaunay triangular mesh, and compare the performance of the SIMPLE and IDEAL algorithms for solving compressible problems. What’s more, the implementation processes of Delaunay mesh generation and derivation of the pressure correction equation are also introduced.Design/methodology/approachProgramming completely in C++.FindingsFive compressible examples are used to test the SIMPLE and IDEAL algorithms, and the comparison with measurement data shows good agreement. The IDEAL algorithm has much better performance in both convergence rate and stability over the SIMPLE algorithm.Originality/valueThe detail solution procedure of implementing the IDEAL algorithm for compressible flows based on Delaunay triangular mesh is presented in this work, seemingly first in the literature.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Structural stability of transonic shock flows with an external force

This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shock solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed by Weng and Xin [Sci. Sinica Math., 49 (2019), pp. 307–320] to deal with the transonic shock problem.

Rapid and sparse reconstruction of high-speed steady-state and transient compressible flow fields using physics-informed graph neural networks

Explosion flow fields are characterized by shock waves with varying intensity and position (i.e., explosive loads), which are the primary causes of structural damage. Accurate and rapid prediction of explosive loads is crucial for structural blast-resistant design and daily security management. While existing empirical models and numerical simulation methods can capture the propagation characteristics of explosive shock waves, high-precision simulation requires a massive computational workload, which is insufficient to meet the fast computational demands of various explosive scenarios. To address this contradiction, this study constructed a sparse reconstruction model for two-dimensional explosion fields based on machine learning algorithms. The model utilizes sparse observational data to establish a mapping relationship to the distribution of the entire flow field. The model is built by a physics-informed graph neural network (PIGN). The graph neural network is employed to associate node features, while the physical network is utilized to control model convergence, aiming to enhance model performance. Using the constructed dataset, the PIGN model was tested. Performance and generalization capabilities of the model were assessed by comparing its results with numerical simulation. This evaluation analyzed the relative error distribution and error statistical results of the reconstructed flow field. The results indicate that the PIGN model can effectively reconstruct explosion fields, with an average error in the reconstructed flow field below 4%. Furthermore, when the number of probe points reaches 10, the average error of the flow field reconstructed by the model is close to 6%. This model not only provides a highly reliable distribution of explosion overpressure and pressure-time variations but also, with a well-trained model, accomplishes flow field reconstruction within 1 ms. It offers a novel approach for achieving rapid and reasonable prediction of explosion fields or two-dimensional compressible flow fields.

A cluster analysis-based shock wave pattern recognition method for two-dimensional inviscid compressible flows

Compressible flows typically exhibit multiple shock waves which interact with each other, making the detection of these shock waves crucial for various aspects of flow studies including construction of high-order numerical schemes (e.g., shock-fitting), adaptive grid refinement, and flow visualization. This study aims to effectively identify and localize multiple shock waves and their interaction points in two-dimensional inviscid steady and unsteady flows. A novel shock wave pattern recognition method based on cluster analysis is proposed, including three processes. First, a series of grid-cells located at the transition zones of captured shock waves are extracted using a shock wave detection approach based on local flow variation. Subsequently, these grid-cells are grouped into numerous clusters using the classical K-means clustering algorithm, with categorization based on nearest neighbor features. Finally, a strategy is introduced to merge relevant adjacent clusters and further localize the points where shock waves interact. The Bézier curve fitting technique is then employed to obtain the high-quality shock-lines. Several numerical cases demonstrate that this method achieves high localization accuracy for shock-lines while being minimally affected by grid type and scale variations. Moreover, it enables clear and effective identification of the shock interaction patterns in both steady and unsteady flows, providing an effective visualization means for analyzing the motion and evolution of shock wave configurations.

A parameter-free staggered-grid Lagrangian scheme for two-dimensional compressible flow problems

The present study aims to develop a parameter-free staggered-grid Lagrangian scheme that avoids the empirical parameters needed for artificial viscosity and anti-hourglass force. The artificial viscosity is equal to the pressure jump, which is constructed by the relationship between pressure and velocity. The anti-hourglass force is established using pressure compensation, which is the difference in pressure between sub-cells belonging to the same primary cell. The scheme maintains the conservation of total mass, momentum, and energy. Numerical experiments show that the scheme is highly robust for extreme flow problems with different fine meshes such as Sedov, Noh, Saltzmann triple point and Rayleigh–Taylor instability. The robust parameter-free scheme is well suited for multi-physics problems and engineering applications.

Ideal and non-ideal planar compressible fluid flows in radial equilibrium

Two-dimensional compressible flows in radial equilibrium are investigated in the ideal dilute-gas regime and the non-ideal single-phase regime close to the liquid–vapour saturation curve and the critical point. Radial equilibrium flows along constant-curvature streamlines are considered. All properties are therefore independent of the tangential streamwise coordinate. A differential relation for the Mach number dependency on the radius is derived for both ideal and non-ideal conditions. For ideal flows, the differential relation is integrated analytically. Assuming a constant specific heat ratio $\gamma$ , the Mach number is a monotonically decreasing function of the radius of curvature for ideal flows, with $\gamma$ being the only fluid-dependent parameter. In non-ideal conditions, the Mach number profile also depends on the total thermodynamic conditions of the fluid. For high molecular complexity fluids, such as toluene or hexamethyldisiloxane, a non-monotone Mach number profile is admissible in single-phase supersonic conditions. For Bethe–Zel'dovich–Thompson fluids, non-monotone behaviour is observed in subsonic conditions. Numerical simulations of subsonic and supersonic turning flows are carried out using the streamline curvature method and the computational fluid dynamics software SU2, respectively, both confirming the flow evolution from uniform flow conditions to the radial equilibrium profile predicted by the theory.

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.

Evolution behavior of cavitation bubble in pure Sn liquid medium with narrow gap under low-amplitude ultrasound

In this study, a numerical model of cavitation bubble in the narrow-gap pure Sn liquid medium was established by two-dimensional compressible multiphase flow simulation. The effects of the pressure amplitude and the gap size on the shape, size and position of the cavitation bubble were investigated. The calculation results showed that the cavitation bubble in the narrow-gap soldering seam could exist stably after experiencing two stages of the nonlinear oscillation and the near-wall oscillation with the low-amplitude ultrasound and moved directionally on the metal substrate surface. When the pressure amplitude increased or the gap size decreased, the directional motion rate of the cavitation bubble increased and the shape of the bubble was elliptical due to the confinement effect of the substrate wall. The ultrasonic degassing mechanism of the narrow-gap soldering seam under the action of exponential decay ultrasonic vibration was analyzed by comparing the fluid pressure and velocity field variations. The flow field in the center of the soldering seam vibrated stronger than that of the peripheral regions, which could promote the outward motion of the cavitation bubble. Within the calculation time of 0.002 s, the maximum horizontal motion distance of bubble in the narrow-gap soldering seam was 1.13 mm.

Shock location effect on direct contact condensation stability

Direct contact condensation (DCC) is an efficient way of transferring heat between fluids’ vapor and liquid phases by mixing them directly rather than using a device like a heat exchanger. However, in the field, DCC may produce destructive and unwanted vibrations due to the instability of the gaseous plume condensing in the process fluid under certain conditions. Previous research on this topic has focused on correlating the flow conditions of the liquid and vapor phases to engineering quantities, such as the heat transfer coefficient, plume length, stability, and the dominant frequency exhibited by the DCC process. The present study investigates the impact of nozzle geometry on plume stability. It was motivated by the drastic difference observed experimentally between the stability of two identically designed nozzles with slightly different microfeatures. Tests were performed with several nozzle geometries, including converging/diverging, straight, and converging nozzles. The results from these experiments show that a nozzle with a supersonic flow at the exit plane is more likely to be stable than one with a subsonic exit flow. It is, therefore, possible to correlate the pressure ratio across the nozzle and nozzle geometry to the stability of the plume. This understanding allows a DCC nozzle geometry to be analyzed and designed for high stability with relatively simple one- and two-dimensional compressible flow modeling tools capable of resolving only the single-phase, compressible flow in the nozzle itself

Global regularity of solutions to the 2D steady compressible Prandtl equations

<abstract><p>In this paper, we study the global $ C^{\infty} $ regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary $ y = 0 $ for any $ x > 0 $.</p></abstract>

Towards high-accuracy deep learning inference of compressible flows over aerofoils

The present study investigates the accurate inference of Reynolds-averaged Navier–Stokes solutions for the two-dimensional compressible flow over aerofoils with a deep neural network. Our approach yields networks that learn to generate precise flow fields for varying body-fitted, structured grids by providing them with an encoding of the corresponding mapping to a canonical space for the solutions. We apply the deep neural network model to a benchmark case of incompressible flow at randomly given angles of attack and Reynolds numbers and achieve an improvement of more than an order of magnitude compared to previous work. Further, for transonic flow cases, the deep neural network model accurately predicts complex flow behavior at high Reynolds numbers, such as shock wave/boundary layer interaction, and quantitative distributions like pressure coefficient, skin friction coefficient as well as wake total pressure profiles downstream of aerofoils. The proposed deep learning method significantly speeds up the predictions of flow fields and shows promise for enabling fast aerodynamic designs.

Flow process and energy release of hydrogen in fluorine

Hydrogen is a clean energy source and fluorine is a strong oxidant. The chemical reaction between the two will make the utilization of hydrogen energy significantly improved, which is a new way to develop and utilize hydrogen energy. In this study, the ANSYS Fluent software is used to solve the reaction kinetics process, which has great advantages in the simulation of combustion and chemical reaction flow, and Computational fluid dynamics (CFD) adopts numerical methods to solve the Navier-Stokes (N–S) partial differential equation. Here, the CFD code ANSYS Fluent is performed to calculate the two-dimensional compressible reaction flow, therefore, we use the methods to explore the flow process and reaction energy release of hydrogen in fluorine and a new ways of hydrogen energy development is explored. The initial stage of the explosion for F2 filling into H2/air in the pipeline is volatile. Chemical reactions for F2 filling into H2/air in the pipeline interact with flows for F2 filling into H2/air in the pipeline. Energy release of hydrogen in fluorine depends on the flow and mixing process. When F2 is filled into a premixed H2/air in the pipeline with a length of 0.9 m and a diameter of 0.1 m, the explosion pressure reaches 0.307MPa–0.408 MPa and the explosion temperature reaches 2698 K–3107 K. The explosion reaction is very rapid and only takes about 40 ms to reach equilibrium. The higher the incident pressure at the pressure inlet, the smaller the peak pressure and temperature are, when the reaction reaches equilibrium.

2차원 압축성 유동 해석자의 OpenACC를 이용한 GPU 활용

A two-dimensional compressible flow solver was ported to GPU using OpenACC. The performance of this program was tested in three cases: viscous flow on a flat plate, inviscid transonic flow in a channel with a bump, and inviscid transonic flow around n0012 airfoil. The GPU program running on RTX 3090 showed up to 20 times of speedup compared to the original program using a single core of Ryzen 5950X, and up to 10 times of speedup compared to the program that uses all resources on the CPU. Among the computation steps of the program, implicit residual smoothing was the worst performing step on the GPU. To address this problem, skipping residual smoothing and using the Cyclic Reduction algorithm were tried as alternatives. These approaches improved speedup in almost every case, increasing the maximum speedups to 24 times compared to the single core program and 11 times compared to the 16-core program. The results show that GPU is a suitable tool for accelerating CFD solvers due to its memory bandwidth and parallel computing capability and using OpenACC is a viable method for porting programs to GPU.

The Influence of Limiters on Davis-Yee and Harten-Yee TVD Schemes

TVD schemes have many selections of limiters, but the recommendation of limiters for specific cases is not available in the literature. This study focuses on incorporating two flux limiters as the extension of the TVD schemes proposed by Harten-Yee and Davis-Yee and extends the test case on external flows, blunt-body. The method used in this study is Harten-Yee Upwind TVD and Davis-Yee Symmetric TVD scheme with different limiter functions to simulate cases for two-dimensional compressible flow. The results show that all the limiter functions can capture shock waves when the flow passes through the geometry at Mach number . The flow features such as bow shock, oblique shock, shock wave reflection, interaction, and expansion wave can all be captured in the case of the bump in a channel and wedge. While in the case of the external supersonic flow passing through the blunt-body, the presence of a bow shock was captured. We discovered that Davis-Yee limiter number 2 performs significantly better than other proposed Davis-Yee and Harten-Yee limiters for the case in this study. Therefore, the Davis-Yee Upwind TVD method is recommended to be applied for the identical case on the expansion of this study

Two-stage fourth-order subcell finite volume method on hexahedral meshes for compressible flows

As an extension of the two-stage fourth-order subcell finite volume (SCFV) method that we developed for two-dimensional compressible flows [C. Zhang et al., “Two-stage fourth-order gas kinetic solver based compact subcell finite volume method for compressible flows on triangular meshes,” Phys. Fluids 33, 126108 (2021)], this study continues our efforts toward three-dimensional (3D) simulations on hexahedral meshes. The two components of subcell divisions and two-stage fourth-order time stepping are utilized to improve efficiency and enhance compactness, which are crucial for 3D simulations. In particular, the current method subdivides each cell into a set of subcells or control volumes (CVs) to increase the degrees of freedom for high-order reconstruction, which involves only face-neighboring cells. For traditional finite volume (FV) methods, high-order reconstruction is performed on each CV individually. In contrast, the reconstruction of SCFV is shared by a set of CVs belonging to the same cell, which can be much more efficient and compact. Moreover, the SCFV framework is combined with the high-order flux evolution by adopting a robust and time-dependent gas-kinetic flux solver and an efficient two-stage fourth-order temporal discretization. The multi-stage Runge–Kutta (RK) method is thus avoided. The coupling of inviscid and viscous terms in the gas-kinetic flux enables us to directly simulate viscous flows. To capture shocks, a limiting procedure by hierarchical reconstruction is developed for effectively preserving the accuracy in smooth flow regions and suppressing numerical oscillations near flow discontinuities. Several benchmark cases are tested. The high-order accuracy and efficiency of this scheme are validated and compared to the k-exact FV method and the traditional Riemann solver combined with a multi-stage RK method. In particular, the simulation of the supersonic Taylor–Green vortex problem demonstrates the good performance of this scheme in compressible turbulence with the presence of shock waves.