Simulation Of Compressible Flows Research Articles

Characteristic-based and interface-sharpening algorithm for high-order simulations of immiscible compressible multi-material flows

The present work focuses on the simulation of immiscible compressible multi-material flows with the Mie–Grüneisen-type equation of state governed by the non-conservative five-equation model [1]. Although low-order single fluid schemes have already been adopted to provide some feasible results, the application of high-order schemes (introducing relatively small numerical dissipation) to these flows may lead to results with severe numerical oscillations. Consequently, attempts to apply any interface-sharpening techniques to stop the progressively more severe smearing interfaces for a longer simulation time may result in an overshoot increase and in some cases convergence to a non-physical solution occurs. This study proposes a characteristic-based interface-sharpening algorithm for performing high-order simulations of such flows by deriving a pressure-equilibrium-consistent intermediate state (augmented with approximations of pressure derivatives) for local characteristic variable reconstruction and constructing a general framework for interface sharpening. First, by imposing a weak form of the jump condition for the non-conservative five-equation model, we analytically derive an intermediate state with pressure derivatives treated as additional parameters of the linearization procedure. Based on this intermediate state, any well-established high-order reconstruction technique can be employed to provide the state at each cell edge. Second, by designing another state with only different reconstructed values of the interface function at each cell edge, the advection term in the equation of the interface function is discretized twice using any common algorithm. The difference between the two discretizations is employed consistently for interface compression, yielding a general framework for interface sharpening. Coupled with the fifth-order improved accurate monotonicity-preserving scheme [2] for local characteristic variable reconstruction and the tangent of hyperbola for the interface capturing scheme [3] for designing other reconstructed values of the interface function, the present algorithm is examined using some typical tests, with the Mie–Grüneisen-type equation of state used for characterizing the materials of interest in both one- and two-dimensional spaces. The results of these tests verify the effectiveness of the present algorithm: essentially non-oscillatory and interface-sharpened results are obtained.

Scattering to Higher Harmonics for Quarter-Wave and Helmholtz Resonators

The nonlinear response of acoustic resonators is investigated over a broad range of frequencies and amplitudes. Helmholtz resonators with a symmetric neck and an asymmetric neck, respectively, as well as quarter-wave resonators are considered. Describing functions for impedance and the reflection coefficient of a Helmholtz resonator at various sound pressure levels are determined from compressible flow simulation and validated against experimental data. The particular focus of the present study is the nonlinear scattering to higher harmonics. For the Helmholtz resonator with a symmetric neck, a distinct pattern in the amplitudes of the higher harmonics is observed, where the odd harmonics dominate the response, whereas the even harmonics are almost negligible. Such an “odd-harmonics-only” pattern, which was observed previously in an experiment at the orifices, is explained by a quasi-steady analysis based on the Bernoulli equation, assuming a symmetric flow pattern at the neck. For the Helmholtz resonator with an asymmetric neck, it is observed in computational fluid dynamics simulations that even harmonics contribute noticeably to the resonator response, such that the odd-harmonics-only pattern is less pronounced. For the markedly asymmetric geometry of the quarter-wave resonator, the second harmonic is dominant and the odd-harmonics-only pattern vanishes completely. The quasi-steady analysis is extended successfully to also describe nonlinear scattering to higher harmonics for asymmetric configurations and flow patterns. Overall, the scattering to higher harmonics remains on a moderate level, even at very high excitation levels for the Helmholtz resonator configurations. For the quarter-wave resonator, the scattering is more pronounced and contributes perceptibly to the response at high excitation amplitudes.

Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21–23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.

Numerical simulation of multi-dimensional inviscid compressible flows by using TV-HLL scheme

This paper deals with the numerical solution of inviscid compressible flows. The three-dimensional Euler equations describing the mentioned problem are presented and solved numerically with the finite volume method. The evaluation of the numerical flux at the interfaces is performed by using the Toro Vazquez-Harten Lax Leer (TV-HLL) scheme. An essential feature of the proposed scheme is to associate two systems of differential equations, called the advection system and the pressure system. It can be implemented with a very simple manner in the standard finite volume Euler and Navier–Stokes codes as extremely simple task. The scheme is applied to some test problems covering a wide spectrum of Mach numbers, including hypersonic, low speed flow and three-dimensional aerodynamics applications.

Determination of Acoustic Impedance for Helmholtz Resonators Through Incompressible Unsteady Flow Simulations

The present work proposes and assesses a methodology based on incompressible computational fluid dynamics simulations to study the acoustic behavior of Helmholtz resonators under a large range of excitation amplitudes. It constitutes an alternative approach to the more widespread one based on compressible flow simulations to analyze the nonlinear regime of Helmholtz resonators. In the present methodology, the resonator is decomposed into its two main components: an assumed incompressible orifice neck and a compressible backing volume. The transfer impedance of the single orifice is obtained by means of an incompressible solver of the flow equations without turbulence modeling, whereas an analytical model accounts for the compliance of the gas in the backing cavity. The proposed methodology is compared for validation purposes to both numerical results of the full compressible equations and experimental data for the complete resonator at different sound pressure levels. A good agreement between the results of the two numerical approaches could be achieved. Numerical results match also fairly well with experimental data, but a systematic overprediction of the resistance by simulations is observed. Accounting for the presence of microrounded edges, presumably present due to manufacturing processes, allows a better agreement between numerical and experimental results.

Simulation of compressible two-phase flows with topology change of fluid–fluid interface by a robust cut-cell method

We develop a robust cut-cell method for numerical simulation of compressible two-phase flows with topology change of the fluid–fluid interface. In cut cell methods the flows can be solved in the finite volume framework and the jump conditions at the interface are resolved by solving a local Riemann problem. Therefore, cut cell methods can obtain interface evolution with high resolution, and at the same time satisfactorily maintain the conservation of flow quantities. However, it remains a challenge for the cut cell methods to handle interfaces with topology change or very high curvature, where the mesh is not sufficiently fine to resolve the interface. Inappropriate treatment could give rise to either distorted interface advection or unphysical oscillation of flow variables, especially when the regularization process (e.g. reinitialization in the level set methods) is implemented. A robust cut-cell method is proposed here, with the interface being tracked by a level set function. The local unphysical oscillation of flow variables in the presence of topology change is shown to be greatly suppressed by using a delayed reinitialization. The method can achieve second-order accuracy with respect to the interface position in the absence of topology changes of interface, while locally degrading to first-order at the interface region where topology change occurs. Its performance is examined through a variety of numerical tests, such as Rayleigh collapse, shock-bubble interaction, and shock-induced bubble collapse in water. Numerical results are compared against either benchmark solutions or experimental observations, and good agreement has been achieved qualitatively and/or quantitatively. Finally, we apply the method to investigating the collapse process of two tandem bubbles in water.

CFD-based shape optimization of steam turbine blade cascade in transonic two phase flows

In this study CFD-based shape optimization of a 3D nozzle and a 2D turbine blade cascade is undertaken in the presence of non-equilibrium condensation within the considered flow channels. A two-fluid formulation is used for the simulation of unsteady, turbulent, supersonic and compressible flow of wet steam accounting for relevant phase interaction between nucleated liquid droplets and continuous vapor phase. An in-house CFD code is developed to solve the governing equations of the two phase flow and was validated against available experimental data. Optimization is carried out in respect to various objective functions. It is shown that nucleation rate and maximum droplet radius are the best suited target functions for reducing thermodynamic and aerodynamic losses caused by the spontaneous nucleation. The maximum increase of 2.1% in turbine blade efficiency is achieved through shape optimization process.

A Hybrid Lattice Boltzmann Flux Solver for Simulation of Viscous Compressible Flows

AbstractIn this paper, a hybrid lattice Boltzmann flux solver (LBFS) is proposed for simulation of viscous compressible flows. In the solver, the finite volume method is applied to solve the Navier-Stokes equations. Different from conventional Navier-Stokes solvers, in this work, the inviscid flux across the cell interface is evaluated by local reconstruction of solution using one-dimensional lattice Boltzmann model, while the viscous flux is still approximated by conventional smooth function approximation. The present work overcomes the two major drawbacks of existing LBFS [28–31], which is used for simulation of inviscid flows. The first one is its ability to simulate viscous flows by including evaluation of viscous flux. The second one is its ability to effectively capture both strong shock waves and thin boundary layers through introduction of a switch function for evaluation of inviscid flux, which takes a value close to zero in the boundary layer and one around the strong shock wave. Numerical experiments demonstrate that the present solver can accurately and effectively simulate hypersonic viscous flows.

GPU‐accelerated direct numerical simulations of decaying compressible turbulence employing a GKM‐based solver

SummaryGas Kinetic Method‐based flow solvers have become popular in recent years owing to their robustness in simulating high Mach number compressible flows. We evaluate the performance of the newly developed analytical gas kinetic method (AGKM) by Xuan et al. in performing direct numerical simulation of canonical compressible turbulent flow on graphical processing unit (GPU)s. We find that for a range of turbulent Mach numbers, AGKM results shows excellent agreement with high order accurate results obtained with traditional Navier–Stokes solvers in terms of key turbulence statistics. Further, AGKM is found to be more efficient as compared with the traditional gas kinetic method for GPU implementation. We present a brief overview of the optimizations performed on NVIDIA K20 GPU and show that GPU optimizations boost the speedup up‐to 40x as compared with single core CPU computations. Hence, AGKM can be used as an efficient method for performing fast and accurate direct numerical simulations of compressible turbulent flows on simple GPU‐based workstations. Copyright © 2016 John Wiley & Sons, Ltd.

High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows

Restrictions on the maximum allowable time step of explicit time integration methods for direct and large eddy simulations of compressible turbulent flows at high Reynolds numbers can be very severe, because of the extremely small space steps used close to solid walls to capture tiny and elongated boundary layer structures. A way of increasing stability limits is to use implicit time integration schemes. However, the price to pay is a higher computational cost per time step, higher discretization errors and lower parallel scalability. In quest for an implicit time scheme for scale-resolving simulations providing the best possible compromise between these opposite requirements, we develop a Runge–Kutta implicit residual smoothing (IRS) scheme of fourth-order accuracy, based on a bilaplacian operator. The implicit operator involves the inversion of scalar pentadiagonal systems, for which efficient parallel algorithms are available. The proposed method is assessed against two explicit and two implicit time integration techniques in terms of computational cost required to achieve a threshold level of accuracy. Precisely, the proposed time scheme is compared to four-stages and six-stages low-storage Runge–Kutta method, to the second-order IRS and to a second-order backward scheme solved by means of matrix-free quasi-exact Newton subiterations. Numerical results show that the proposed IRS scheme leads to reductions in computational time by a factor 3 to 5 for an accuracy comparable to that of the corresponding explicit Runge–Kutta scheme.

A Lagrangian finite element method for 3D compressible flow applications

We discuss a Lagrangian approach for the simulation of 3D compressible flows on non-structured tetrahedral meshes. The formulation is nodal-based, in the sense that kinematic and thermodynamic variables are all interpolated with continuous P1 polynomials. The equations are solved with an explicit time-marching scheme without stabilizing terms and with the inclusion only of a shock-capturing viscosity. Several examples featuring shock propagations and mixing of fluids are addressed, with particular emphasis on the stability of the approach and on possible strategies for mesh update.

On the extension of LES methods from incompressible to compressible turbulent flows with application to turbulent channel flow

The objective of the present work is to validate the compressible Large-Eddy Simulation (LES) models implemented in the in house parallel unstructured CFD code TermoFluids. Our research team has implemented and tested several LES models over the past years for the incompressible regimen. In order to be able to solve complex turbulent compressible flows, the models are revisited and modified if necessary. In addition, the performance of the implemented hybrid advection scheme is an issue of interest for the numerical simulation of turbulent compressible flows. The models are tested in the well known turbulent channel flow problem at different compressible regimens.

A scalable fully implicit method with adaptive time stepping for unsteady compressible inviscid flows

The class of fully implicit methods is drawing more attention in the simulation of fluid dynamics for engineering community, due to the allowance of large time steps in extreme-scale simulations. In this paper, we introduce and study a scalable fully implicit method for the numerical simulations of unsteady compressible inviscid flows governed by the compressible Euler equations. In the method, a cell-centered finite volume scheme together with the local Lax-Friedrichs (LLF) formula is used for the spatial discretization, and a backward differentiation formula is applied to integrate the Euler equations in time. The resultant nonlinear system at each time step is then solved by a parallel Newton-Krylov method with a domain decomposition type preconditioner. To improve the performance of the proposed method, we introduce an adaptive time stepping method which adjusts the time step size according to the initial residual of Newton iterations. Therefore, the proposed fully implicit solver overcomes the often severe limits on the time steps associated with existing methods. Numerical experiments validate that the approach is effective and robust for the simulations of several compressible inviscid flows. We also show that the newly developed algorithm scales well with more than one thousand processor cores for the problem with tens of millions of unknowns.

A novel weighting switch function for uniformly high‐order hybrid shock‐capturing schemes

SummaryHybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem‐dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non‐oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high‐order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh‐order hybrid compact‐reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth‐order WENO scheme, and it has seventh‐order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh‐order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.

A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids

In this work, a spectral difference lattice Boltzmann method (SDLBM) is developed and applied for an accurate simulation of two-dimensional inviscid compressible flows on structured grids. The compressible form of the discrete Boltzmann–BGK equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid. Here, the 2D compressible Lattice Boltzmann (LB) model proposed by Watari is applied. The spectral difference (SD) method is implemented for the solution of the LB equation in which the particle distribution function is stored at the solution points while the fluxes are computed at the flux points for calculating the particle distribution function. For time accurate solutions, the fourth-order Runge–Kutta scheme is used to discretize the temporal term in the LB equation. Note that the procedure of implementing the SD method to solve the LB equation is nearly the same as that procedure developed in the literature for the solution of the Euler equations. The accuracy and robustness of the present solution methodology are demonstrated by simulating different benchmark compressible flow problems. A sensitivity study is also conducted to evaluate the effects of the numerical parameters and the grid size/distribution on the accuracy and performance of the solution. At first, three inviscid compressible flow problems, namely, the stationary isentropic vortex, the shock tube and the shock–vortex interaction are solved by using the SDLBM to demonstrate the accuracy and robustness of the present solution methodology. Results computed for these test cases are in good agreement with the analytical and the available numerical solutions. To more assess the accuracy and robustness of the SDLBM, a third-order finite volume LBM (FVLBM) is also developed and the solutions obtained by these two methods for the test cases simulated are thoroughly compared with each other. It is demonstrated that the SDLBM is more accurate and robust compressible inviscid flow solver that can be used for evaluating the other compressible LB-based flow solvers. As the final problem, the supersonic flow past a bump is simulated to demonstrate the accuracy and robustness of the SDLBM in reaching to a steady-state solution.

A Switch Function-Based Gas-Kinetic Scheme for Simulation of Inviscid and Viscous Compressible Flows

AbstractIn this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.

An improved ghost‐cell immersed boundary method for compressible flow simulations

SummaryThis study presents an improved ghost‐cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level‐set method to farther image points, incorporating a higher‐order extra/interpolation scheme for the ghost‐cell values. A sensor is introduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement is used to improve the representation of the geometry efficiently in the Cartesian grid system. The improved ghost‐cell method is validated against four test cases: (a) double Mach reflections on a ramp, (b) smooth Prandtl–Meyer expansion flows, (c) supersonic flows in a wind tunnel with a forward‐facing step, and (d) supersonic flows over a circular cylinder. It is demonstrated that the improved ghost‐cell method can reach the accuracy of second order in L1 norm and higher than first order in L∞ norm. Direct comparisons against the cut‐cell method demonstrate that the improved ghost‐cell method is almost equally accurate with better efficiency for boundary representation in high‐fidelity compressible flow simulations. Copyright © 2016 John Wiley & Sons, Ltd.

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.

A high-order adaptive Cartesian cut-cell method for simulation of compressible viscous flow over immersed bodies

A new adaptive finite volume conservative cut-cell method that is third-order accurate for simulation of compressible viscous flows is presented. A high-order reconstruction approach using cell centered piecewise polynomial approximation of flow quantities, developed in the past for body-fitted grids, is now extended to the Cartesian based cut-cell method. It is shown that the presence of cut-cells of very low volume results in numerical oscillations in the flow solution near the embedded boundaries when standard small cell treatment techniques are employed. A novel cell clustering approach for polynomial reconstruction in the vicinity of the small cells is proposed and is shown to achieve smooth representation of flow field quantities and their derivatives on immersed interfaces. It is further shown through numerical examples that the proposed clustering method achieves the design order of accuracy and is fairly insensitive to the cluster size. Results are presented for canonical flow past a single cylinder and a sphere at different flow Reynolds numbers to verify the accuracy of the scheme. Investigations are then performed for flow over two staggered cylinders and the results are compared with prior data for the same configuration. All the simulations are carried out with both quadratic and cubic reconstruction, and the results indicate a clear improvement with the cubic reconstruction. The new cut-cell approach with cell clustering is able to predict accurate results even at relatively low resolutions. The ability of the high-order cut-cell method in handling sharp geometrical corners and narrow gaps is also demonstrated using various examples. Finally, three-dimensional flow interactions between a pair of spheres in cross flow is investigated using the proposed cut-cell scheme. The results are shown to be in excellent agreement with past studies, which employed body-fitted grids for studying this complex case.

Development of discrete gas kinetic scheme for simulation of 3D viscous incompressible and compressible flows

The sphere function-based gas kinetic scheme (GKS), which was presented by Shu and his coworkers [23] for simulation of inviscid compressible flows, is extended to simulate 3D viscous incompressible and compressible flows in this work. Firstly, we use certain discrete points to represent the spherical surface in the phase velocity space. Then, integrals along the spherical surface for conservation forms of moments, which are needed to recover 3D Navier–Stokes equations, are approximated by integral quadrature. The basic requirement is that these conservation forms of moments can be exactly satisfied by weighted summation of distribution functions at discrete points. It was found that the integral quadrature by eight discrete points on the spherical surface, which forms the D3Q8 discrete velocity model, can exactly match the integral. In this way, the conservative variables and numerical fluxes can be computed by weighted summation of distribution functions at eight discrete points. That is, the application of complicated formulations resultant from integrals can be replaced by a simple solution process. Several numerical examples including laminar flat plate boundary layer, 3D lid-driven cavity flow, steady flow through a 90° bending square duct, transonic flow around DPW-W1 wing and supersonic flow around NACA0012 airfoil are chosen to validate the proposed scheme. Numerical results demonstrate that the present scheme can provide reasonable numerical results for 3D viscous flows.