Three-dimensional Compressible Flow Research Articles

Discretely Nonlinearly Stable Weight-Adjusted Flux Reconstruction High-Order Method for Compressible Flows on Curvilinear Grids

To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.

Coupled Modeling of Sea Surface Launch Flow and Multi-Body Motion

To simulate the launching process of missile complex flow, movement, and constraint states, a multifield coupling model is put forward based on a computational fluid dynamics (CFD) method. In this coupled model, a CFD method is used to solve the three-dimensional compressible transient flow, and the six-degree motion of the launching platform is considered, and the virtual contact method is used to deal with the constraint states of the guideway and the slider. The active force and moment are applied to the launching platform to simulate its rolling, pitching, and heaving motions under the 5-level waves. Collision detection is carried out through the minimum clearance distance between the slider and the guideway, and the contact force is handled by a modified Herz collision model. In the problem of launching a missile from the water surface, the change characteristics of the flow field, the load response characteristics, and the relative motion laws of the missile and the launching platform during the catapulting process are investigated. The results show that the motion laws of the projectile and the launch tube in the constrained direction are the same, and the established coupling model is able to simulate the launch separation process of the missile in the constrained state. In addition, the effect of wind load on the missile ejection process is analyzed using the coupled model.

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

Numerical investigation on the use of various blade tips for the helicopter's main rotor blade in forward flight regarding aerodynamic performance and HSI noise considerations

The goal of this research is to investigate aerodynamic performance and high-speed impulsive (HSI) noise of various blade tips for the helicopter's main rotor in forward flight. For this purpose, three-dimensional compressible flow field around the rotor blades is numerically simulated by the solution of the unsteady Reynolds averaged Navier-Stokes (URANS) equations with the "SST k-ω" turbulence model. Solution is quantified by the calculation of torque, drag, and thrust coefficients of CQ, CD, CT, and L/D ratio. HSI noise on the rotor plane is assessed by the calculation of sound pressure level at certain points of the flow field using the Ffowcs Williams-Hawking (FW-H) equation. Blade tips are eleven and include anhedral, dihedral, Eagle and combinations of them, and also Blue Edge and rectangular. The combinations are inspired by the tip configurations of the fixed wings. In this regard effects of the tip geometry parameters including curvature and height of its parts have been considered as well. All of the tips are examined for two flight conditions of Caradonna and Tung two-blade rotor with fixed pitch angle and time-varying pitch angles. Based on the present results, it is found that blade tips of dihedral family do not improve the aerodynamic performance of the helicopter blade. Also, addition of anhedral part to the blade tip improves the L/D ratio. According to the present acoustic analysis, it was seen that after the Blue Edge, the Eagle tip stands in order of priority. For the condition of Caradonna and Tung two-blade rotor with tip Mach number of 0.8 and advance ratio of 0.2, HSI noise of the Eagle-tip blade decreases by 2.39 % compared to that of the two-blade rotor with rectangular tip. However, the aerodynamics performance of the Eagle tip is much better than that of the Blue Edge.

An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media

An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media

Consistent lifting relations for the initialization of total-energy double-distribution-function kinetic models.

A lifting relation connecting the distribution function explicitly with the hydrodynamic variables is necessary for the Boltzmann equation-based mesoscopic approaches in order to correctly initialize a nonuniform hydrodynamic flow. We derive two lifting relations for Guo etal.'s total-energy double-distribution-function (DDF) kinetic model [Z. L. Guo etal., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704], one from the Hermite expansion of the conserved and nonconserved moments, and the second from the O(τ) Chapman-Enskog (CE) approximation of the Maxwellian exponential equilibrium. While both forms are consistent to the compressible Navier-Stokes-Fourier system theoretically, we stress that the latter may introduce numerical oscillations under the recently optimized discrete velocity models [Y. M. Qi etal., Phys. Fluids 34, 116101 (2022)10.1063/5.0120490], namely a 27 discrete velocity model of the seventh-order Gauss-Hermite quadrature (GHQ) accuracy (D3V27A7) for the velocity field combined with a 13 discrete velocity model of the fifth-order GHQ accuracy (D3V13A5) for the total energy. It is shown that the Hermite-expansion-based lifting relation can be alternatively derived from the latter approach using the truncated Hermite-polynomial equilibrium. Additionally, a relationship between the order of CE expansions and the truncated order of Hermite equilibria is developed to determine the minimal order of a Hermite equilibria required to recover any multiple-timescale macroscopic system. Next, three-dimensional compressible Taylor-Green vortex flows with different initial conditions and Ma numbers are simulated to demonstrate the effectiveness and potential issues of these lifting relations. The Hermite-expansion-based lifting relation works well in all cases, while the Chapman-Enskog-expansion-based lifting relation may produce numerical oscillations and a theoretical model is developed to predict such oscillations. Furthermore, the corresponding lifting relations for Qi etal.'s total energy DDF model [Y. M. Qi etal., Phys. Fluids 34, 116101 (2022)10.1063/5.0120490] are derived, and additional simulations are performed to illustrate the generality of our approach.

An Adaptive Mesh Refinement–Rotated Lattice Boltzmann Flux Solver for Numerical Simulation of Two and Three-Dimensional Compressible Flows with Complex Shock Structures

An adaptive mesh refinement–rotated lattice Boltzmann flux solver (AMR-RLBFS) is presented to simulate two and three-dimensional compressible flows with complex shock structures. In the method, the RLBFS, which has a strong shock-capturing capability and can effectively eliminate the shock instability phenomenon, is applied to solve the flow filed by reconstructing the fluxes at each cell interface adaptively with the mesoscopic lattice Boltzmann model. To locally and dynamically improve the resolution of intricate shock structures and optimize the required computational resources, a block-structured adaptive mesh refinement (AMR) technique is introduced. The validity and effectiveness of the proposed method are confirmed through a range of two and three-dimensional numerical cases, including the shock tube problem, the four-wave Riemann problem, explosion within a rectangular box, and the vorticity induced by a shock. The results obtained using the AMR-RLBFS exhibit excellent agreement with published data and demonstrate high accuracy in capturing complex shock structures. The computational efficiency of the AMR-RLBFS can be also improved significantly compared to the RLBFS on uniform grids. Furthermore, the numerical outcomes underscore the capability of the AMR-RLBFS to eliminate shock instability effects while efficiently capturing a broader spectrum of small-scale vertical structures. These findings highlight the ability of AMR-RLBFS to improve the computational efficiency and capture intricate shock structures effectively, making it a valuable tool for studying a wide range of compressible flows from aerodynamics to astrophysics.

Flow irreversibility and heat transfer effects on turbine efficiency

The quantification of the flow irreversibilities is crucial to developing future turbomachinery. Traditionally, design correlations and efficiencies are based on comparisons to isentropic relations to quantify the deviation from an “ideal” case. However, isentropic relations typically assume one-dimensional adiabatic flow, representing a significant departure from the actual situation in turbines operating with large levels of heat transfer. In this paper, the aerothermal losses and power generation in reversible processes are derived for three-dimensional compressible flow considering heat transfer effects. A new definition of the power potential in a reversible process is proposed that can be used to perform detailed budgeting of the losses. The proposed new loss framework can be adopted locally, which enables the precise identification of the loss source. New aerodynamic and aerothermal efficiency expressions are proposed to assess shaft power extraction in combination with heat exchange. The new method enables designers to identify the regions with high loss generation in steady and also in unsteady flow conditions, offering new avenues to evaluate the loss generation mechanisms during the design phase. Therefore, our new tool is essential to developing reduced-order models for the design of future fluid machinery.

Global Existence and Decay Rate of Smooth Solutions for Full System of Partial Differential Equations for Three-Dimensional Compressible Magnetohydrodynamic Flows

We focus on the global existence andLp−Lqrates of convergence for the compressible magnetohydrodynamic equations inR3. We prove the global existence of smooth solutions using the standard energy method under the condition that the initial data are close to a constant equilibrium state inH3. Rates of convergence for the solution inLqnorm with2≤q≤6and its first- and second-order derivatives inL2norm are obtained, if the initial data belong toLpwith1≤p≤65.

Development of three-dimensional rotated lattice Boltzmann flux solver for the simulation of high-speed compressible flows

A three-dimensional (3D) Rotated Lattice Boltzmann Flux Solver (RLBFS) is proposed and analyzed with matrix-based linear stability theory for simulating compressible flows in a wide range of Mach numbers. The 3D RLBFS applies the finite volume method to discrete the Navier-Stokes equations and evaluates its fluxes at each cell interface. To improve numerical stability, the convective fluxes are obtained in a hybrid way by using the D1Q4 lattice Boltzmann model in two perpendicular directions decomposed by the outer normal vector of each cell interface. The viscous fluxes are computed in a conventional way using the second-order central scheme. The stability performance and order accuracy of the proposed 3D RLBFS are examined by using the matrix-based stability theory and L2 errors on different meshes respectively. It is shown that the 3D RLBFS is stable even at high Mach number and has the second-order accuracy in space. The reliability and capability of the proposed method is further evaluated by simulating several challenging compressible flow problems, including subsonic flow over the DLR-F4, transonic flow over the ONERA M6 wing, supersonic flow around NACA0012 wing, hypersonic flow over the Viking lander capsule, hypersonic flow over a hemisphere and hypersonic flow around a blunt nose double cone. The obtained numerical results are in good agreement with experimental and/or numerical data published in the literature, indicating that the present method provides a reliable and effective tool for simulating practical three-dimensional compressible flow problems in aeronautics and astronautics.

Optimal temporal decay rates for the compressible viscoelastic flows

For the Cauchy problem of the three-dimensional compressible viscoelastic flows, we establish the optimal temporal decay rates of the all-order spatial derivatives of the global strong solution in the weaker initial condition. The main novelty of this paper is that the optimal decay estimates of the highest-order derivatives of the solution is obtained by using spectral analysis and energy method, which can be considered as the further investigation to [X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal. 45 (2013) 2815–2833] with only the lower-order derivative estimates.

An Evaluation of Accuracy and Efficiency of a 3D Adaptive Mesh Refinement Method with Analytical Velocity Fields

Appropriate mesh refinement plays a vital role in the accuracy and convergence of computational fluid dynamics solvers. This work is an extension of the previous work that further demonstrates the accuracy of the 3D adaptive mesh refinement method by comparing the accuracy measures between the ones derived from the analytical fields and those identified by the refined meshes. The adaptive mesh refinement method presented in this study is based on the law of mass conservation for three-dimensional incompressible or compressible steady fluid flows. The assessment of the performance of the adaptive mesh refinement method considers its key features such as drawing closed streamline and identification of singular points, asymptotic planes, and vortex axis. Several illustrative examples of the applications of the 3D mesh refinement method with a multi-level refinement confirm the accuracy and efficiency of the proposed method. Furthermore, the results demonstrate that the adaptive mesh refinement method can provide accurate and reliable qualitative measures of 3D computational fluid dynamics problems.

A positivity-preserving and conservative high-order flux reconstruction method for the polyatomic Boltzmann–BGK equation

In this work, we present a positivity-preserving high-order flux reconstruction method for the polyatomic Boltzmann–BGK equation augmented with a discrete velocity model that ensures the scheme is discretely conservative. Through modeling the internal degrees of freedom, the approach is further extended to polyatomic molecules and can encompass arbitrary constitutive laws. The approach is validated on a series of large-scale complex numerical experiments, ranging from shock-dominated flows computed on unstructured grids to direct numerical simulation of three-dimensional compressible turbulent flows, the latter of which is the first instance of such a flow computed by directly solving the Boltzmann equation. The results show the ability of the scheme to directly resolve shock structures without any ad hoc numerical shock capturing method and correctly approximate turbulent flow phenomena in a consistent manner with the hydrodynamic equations.

Stabilization effect of elasticity on three-dimensional compressible vortex sheets

We are concerned with the vortex sheet solutions for the three-dimensional compressible isentropic elastic flows. This is a nonlinear hyperbolic problem with a characteristic free boundary. Compared with the analysis in two dimensions, this added dimension leads to more complicated frequency interactions between the effects of elasticity and the fluid velocity, making the stability analysis more challenging. Through a very delicate examination of the Lopatinskiĭ determinant of the linearized boundary value problem, necessary and sufficient conditions are established for the linear stability of the planar vortex sheet solutions. These conditions are closely related to the geometric properties of the elastic deformation gradient and provide the first stability criterion justifying the stabilization effect of elasticity on the compressible vortex sheets in the three-dimensional elastodynamics. In contrast to the two-dimensional isentropic elastic fluids, we find that the stability can only hold in the subsonic region for the three-dimensional vortex sheets.

Evolution of tornado-like vortices in three-dimensional compressible rectangular cavity flows

The spatial structure and time evolution of tornado-like vortices in a three-dimensional cavity are studied by topological analysis and numerical simulation. The topology theory of the unsteady vortex in the rectangular coordinate system (Zhang, Zhang & Shu, J. Fluid Mech., vol. 639, 2009, pp. 343–372) is generalized to the curvilinear coordinate system. Two functions $\lambda (q_1,t)$ and $q(q_1,t)$ are obtained to determine the topology structure of the sectional streamline pattern in the cross-section perpendicular to the vortex axis and the meridional plane, respectively. The spiral direction of the sectional streamlines in the cross-section perpendicular to the vortex axis depends on the sign of $\lambda (q_1,t)$ . The types of critical points in the meridional plane depend on the sign of $q(q_1,t)$ . The relation between the critical points of the streamline pattern in the meridional plane and that in the cross-section perpendicular to the vortex axis is set up. The flow in a three-dimensional rectangular cavity is numerically simulated by solving the three-dimensional Navier–Stokes equations using high-order numerical methods. The spatial structures and the time evolutions of the tornado-like vortices in the cavity are analysed with our topology theory. Both the bubble type and spiral type of vortex breakdown are observed. They have a close relationship with the vortex structure in the cross-section perpendicular to the vortex axis. The bubble-type breakdown has a conical core and the core is non-axisymmetric in the sense of topology. A criterion for the bubble type and the spiral type based on the spatial structure characteristic of the two breakdown types is provided.

Study of the phase resonance phenomenon in Francis turbine

In hydraulic machinery, the phenomenon of rotor-stator interaction will produce periodic pressure waves in the stationary cascade. These pressure waves have the same phase, and will spread and superimpose in the volute. When the pressure waves propagate and superpose, the pressure pulsation with large amplitude will be formed. This phenomenon is called phase resonance, which will threaten the safe and stable operation of the machinery. Therefore, it is necessary to study the rotor-stator interference phenomenon and analyze the risk of phase resonance. In this paper, a high head Francis turbine with phase resonance in engineering is analyzed, which is also verified by the existing theoretical analysis. The phase resonance in the turbine is analyzed by using one-dimensional hydro acoustic model and three-dimensional compressible flow CFD simulation method, and the results are consistent with the actual engineering phenomenon. In order to study the measures to improve the phase resonance phenomenon, this paper carries out the simulation calculation from changing the rotation speed, wave speed and guide vane-runner blade number combination. The results show that adjusting the above parameters can reduce the amplitude of large pressure pulsation caused by the phase resonance in different degrees, and changing the number combination of guide vanes and runner blades is the most effective and feasible measure.

Large-eddy simulations of self-excited thermoacoustic instability in a premixed swirling combustor with an outlet nozzle

Reducing the footprint of greenhouse gases and nitrogen oxides (NOx) emissions from combustion systems means that they have been operating under lean or ultra-lean fuel–air premixed conditions. Under such conditions, self-excited large-amplitude pulsating thermoacoustic instabilities may occur, characterized by deafening combustion noises and even “violent” structural vibrations, which is, therefore, highly undesirable in practice. By conducting chemical reaction-thermodynamics-acoustics-swirling flow coupling investigations, we have numerically explored the generation and mitigation mechanisms of self-excited pulsating oscillations in a methane-fueled swirling combustor in the presence and absence of an outlet nozzle. Hence, a large-eddy simulation was performed on a fully three-dimensional compressible flow via an open-source platform, OpenFOAM. Furthermore, a thorough assessment was made to understand the fundamental physics of the interaction of the swirling flame, either constructively or destructively, to the acoustic pressure perturbations by examining the local Rayleigh criterion/index. A further explanation was made on implementing the outlet nozzle that can mitigate such periodic pulsating combustion via attenuating the fuel fraction fluctuations, vortices processing, and changing temperature field. It was also found that the dominant pulsating mode is switched from the 1/4 standing-wave wavelength mode to the 3/4 wavelength mode. Finally, more physical insights were obtained by conducting a proper orthogonal decomposition analysis on the energy distribution between the thermoacoustic modes.

A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows

High-order implicit shock tracking (fitting) is a new class of numerical methods to approximate solutions of conservation laws with non-smooth features, e.g., contact lines, shock waves, and rarefactions. These methods align elements of the computational mesh with non-smooth features to represent them perfectly, allowing high-order basis functions to approximate smooth regions of the solution without the need for nonlinear stabilization, which leads to accurate approximations on traditionally coarse meshes. The hallmark of these methods is the underlying optimization formulation whose solution is a feature-aligned mesh and the corresponding high-order approximation to the flow; the key challenge is robustly solving the central optimization problem. In this work, we develop a robust optimization solver for high-order implicit shock tracking methods so they can be reliably used to simulate complex, high-speed, compressible flows in multiple dimensions. The proposed method integrates practical robustness measures into a sequential quadratic programming method that employs a Levenberg-Marquardt approximation of the Hessian and line-search globalization. The robustness measures include dimension- and order-independent simplex element collapses, mesh smoothing, and element-wise solution re-initialization, which prove to be necessary to reliably track complex discontinuity surfaces, such as curved and reflecting shocks, shock formation, and shock-shock interaction. A series of nine numerical experiments—including two- and three-dimensional compressible flows with complex discontinuity surfaces—are used to demonstrate: 1) the robustness of the solver, 2) the meshes produced are high-quality and track continuous, non-smooth features in addition to discontinuities, 3) the method achieves the optimal convergence rate of the underlying discretization even for flows containing discontinuities, and 4) the method produces highly accurate solutions on extremely coarse meshes relative to approaches based on shock capturing.

On the Cauchy problem of the full Navier-Stokes equations for three-dimensional compressible viscous heat-conducting flows subject to large external potential forces

On the Cauchy problem of the full Navier-Stokes equations for three-dimensional compressible viscous heat-conducting flows subject to large external potential forces

Refined blow-up criteria for the three-dimensional viscous compressible flows with large external potential force and general pressure

Refined blow-up criteria for the three-dimensional viscous compressible flows with large external potential force and general pressure