Harten-Lax-van Leer-Contact Research Articles

Numerical simulation of the atomization of liquid transverse jet in supersonic airflow

The present study provides a numerical method for liquid jet atomization in supersonic gas crossflow. Compressibility of the gas and incompressibility of the liquid are considered. High-order accurate weighted essentially non-oscillatory schemes and the Harten–Lax–van Leer contact approximate Riemann solver are used for gas flows. Liquid flow is simulated by the Chorin projection method. The motion of the sharp interface between the gas and liquid is simulated by the volume of fluid method. In order to verify the accuracy of the numerical method, numerical and experimental results for the droplet breakup in the supersonic gas flow are compared. The method is employed to simulate the liquid jet atomization in the supersonic gas crossflow. According to numerical results, the breakup process is analyzed for four different stages. The discussion for the effect of the Mach number for the gas crossflow on the liquid jet atomization is given.

Impact of numerical hydrodynamics in turbulent mixing transition simulations

Underresolved simulations are unavoidable in high Reynolds (Re) and Mach (Ma) number turbulent flow applications at scale. Implicit large-Eddy simulation (ILES) often becomes the effective strategy to capture the dominating effects of convectively driven flow instabilities. We evaluate the impact of three distinct numerical strategies in simulations of transition and turbulence decay with ILES: the Harten–Lax–van Leer (HLL) Riemann solver applying Strang splitting and a Lagrange-plus-Remap formalism to solve the directional sweep—denoted split; the Harten–Lax–Van Leer-Contact (HLLC) Riemann solver using a directionally unsplit strategy and parabolic reconstruction—denoted unsplit; and the HLLC Riemann solver using unsplit and a low-Ma correction (LMC)—denoted unsplit*. Three case studies are considered: (1) a shock tube problem prototyping shock-driven turbulent mixing, (2) the Taylor–Green Vortex (TGV) prototyping transition to turbulence, and, (3) an homogeneous isotropic turbulence (HIT) case, focusing on the impact of discretization on transition and decay from fixed well-characterized initial conditions. Significantly more accurate predictions are provided by the unsplit schemes, in particular, when augmented with the LMC. For given resolution, only the unsplit schemes predict the turbulent mixing transition after reshock observed in the shock tube experiments. Relevant comparisons of ILES based on Euler and Navier–Stokes equations addressing potential occurrence of low-Re regimes in the applications are presented. Unsplit* schemes are instrumental in allowing to capture the spatial development of the TGV flow and its validation at prescribed Re with significantly less resolution. HIT analysis confirms higher simulated turbulence Re and increased small-scale content associated with the unsplit discretizations.

Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness

This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some technical decompositions as well as several key properties of the admissible states and HLLC flux, rigorous positivity-preserving analyses are carried out. It is proven that the resulting well-balanced DG schemes, coupled with strong stability preserving time discretizations, satisfy a weak positivity property, which implies that one can apply a simple existing limiter to effectively enforce the positivity-preserving property, without losing high-order accuracy and conservation. The proposed methods and analyses are applicable to the Euler system with general equation of state. Extensive one- and two-dimensional numerical tests demonstrate the desired properties of these schemes, including the exact preservation of the equilibrium state, the ability to capture small perturbation of such state, the robustness for solving problems involving low density and/or low pressure, and good resolution for smooth and discontinuous solutions.

The Sod gasdynamics problem as a tool for benchmarking face flux construction in the finite volume method

The Finite Volume Method in Computational Fluid Dynamics to numerically model a fluid flow problem involves the process of formulating the numerical flux at the faces of the control volume. This process is important in deciding the resolution of the numerical solution, thus its quality. In the current work, the performance of different flux construction methods when solving the one-dimensional Euler equations for an inviscid flow is analyzed through a test problem in the literature having an exact (analytical) solution, which is the Sod problem.The work considered twenty two flux methods, which are: exact Riemann solver (Godunov), Roe, Kurganov-Noelle-Petrova, Kurganov-Tadmor, Steger-Warming Flux Vector Splitting, van Leer Flux Vector Splitting, AUSM, AUSM+, AUSM+−up, AUFS, five variants of the Harten-Lax-van Leer (HLL) family, and their corresponding five variants of the Harten-Lax-van Leer-Contact (HLLC) family, Lax-Friedrichs (Lax), and Rusanov.The methods of exact Riemann solver and van Leer showed excellent performance. The Riemann exact method took the longest runtime, but there was no significant difference in the runtime among all methods.

Flow-field analysis and pressure gain estimation of a rotating detonation engine with banded distribution of reactants

The flow-field structure and pressure gain performance of a rotating detonation engine with banded distribution of reactants have been studied using two-dimensional numerical simulations. The reactants are premixed H2/Air mixture. An unsteady reacting flow solver named rhoHLLCFoam is developed based on the open source software OpenFOAM. Unsteady Reynolds Averaged Navier-Stokes (RANS) equations are solved with second order accuracy in space and time with Harten-Lax-van-Leer-Contact (HLLC) Riemann scheme. The solver resolves the combustion phenomena through finite rate chemistry reaction model with Arrhenius form of reaction rate by using Ó Conaire scheme. After checking the reliability of the solver, two sets of cases with various inlet-area ratios (ψ) and equivalence ratios (ϕ) are conducted. The result shows that with ψ<1.0, the reactants in front of detonation waves present a discretely banded distribution which causes a series of reverse compression waves in flow-field. This paper estimates the specific impulse and specific thrust of combustion chamber. It's shown that these parameters increase with the promotion of ψ. By calculating the area-averaged stagnation pressure along axial direction of combustion chamber, the pressure gain ratio (η) of the rotating detonation engine is estimated. The result suggests that η decreases dramatically with the reduction of ψ. In order to achieve pressure gain, ψ must be greater than 0.60. Moreover, the equivalence ratio should be around unity to obtain higher value of η.

Numerical simulation of droplet formation in different microfluidic devices

This study aims to assess the droplet formation process in microfluidic devices using an all Mach-number multi-phase flow solver. Harten-Lax-van Leer-Contact (HLLC) Riemann solver is implemented for solving the discretized equations while Tangent of Hyperbola for INterface Capturing (THINC) method is applied to reduce the excessive diffusion of the method at the interface. The multi-block strategy is implemented in the flow solver to improve its capability of simulating flows in more complicated, multi-port droplet formation geometries. First, the computational performance of the numerical solver is validated through simulating the benchmark Rayleigh-Taylor instability problem and the droplet formation in a planar flow-focusing geometry. The comparison of numerical results with available analytical/experimental data indicates a precise prediction of flow features. Then, the droplet generation process in coflowing devices with Newtonian liquid is numerically studied. Finally, the shear-thinning effects of the dispersed and continuous phase on the drop formation characteristics are investigated. It is shown that deviation of the continuous phase from Newtonian behavior has a strong impact on droplet size, speed, and generation rate. On the other hand, assuming the dispersed phase as non-Newtonian does not strongly affect the aforementioned properties of the droplet formation regime.

Numerical Simulation of Flood Inundation in a Small-Scale Coastal Urban Area Due to Intense Rainfall and Poor Inner Drainage

: This study presents the numerical simulation and analysis of the characteristics of the flood inundation in a small-scale coastal urban area due to the intense rainfall and poor inner drainage from the tidal level rise occurring during a typhoon. The employed numerical model is a two-dimensional finite volume model with a well-balanced HLLC (Harten–Lax–Van Leer contact) scheme. The target area is a coastal urban area downstream of the Gohyun river; which is located in Geoje City of Kyungsangnam Province, Korea. This area was significantly damaged by flood inundation due to the heavy rainfall and significant increase in the tidal level during Typhoon “Maemi”, which occurred in September 2003. The numerical model used in this study is verified using the flood inundation traces observed in the selected urban area. Moreover; the characteristics of the flood inundation based on the change in the river inflow due to the increase or decrease in the intensity of the possible heavy rainfall that may occur in the future are simulated and analyzed.

An implementation of the weighted essential non-oscillatory scheme for numerical approximations of the pressureless gas dynamics equations

In this study, the Weighted Essential Non-Oscillatory (WENO) scheme is implemented for numerical approximations of the pressureless gas dynamics equations. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver serves as a basic building block with the fifth order WENO scheme involving the first order Lax-Friedrichs scheme as a flux limiter for the numerical stability. In particular, the WENO scheme suppressed the oscillatory around the delta shock waves and the flux limiter guaranteed the positivity of densities around the vacuum. Lastly, numerical testes for one-dimensional problems are presented for capturing the delta shock waves and vacuum.

Effect of heat transfer and geometry on micro-thruster performance

Coupled Navier-Stokes and Direct Simulation Monte Carlo (NS-DSMC) simulations of gas flow in micro-nozzles for various wall thermal conditions and geometrical aspects are presented. Micro-thrusters employed in miniature spacecraft and microsatellites experience substantial changes in wall thermal conditions. This can influence the internal boundary layer development and the exit plume structure of a micro-nozzle. These changes in flow physics differ with the nozzle divergence angle and the proximity of a similar nozzle in the cluster. Continuum solvers often fail to analyze the micro-nozzles operating in vacuum conditions as the flow in micro-nozzles experiences continuum, transitional, and rarefied regimes. Coupled NS-DSMC solver is an effective alternative that can simulate the non-equilibrium effects in a micro-nozzle flow field. A steady solution of the entire flow field has been obtained using a non-linear Harten-Lax-van Leer-Contact (HLLC) scheme based finite volume solver with a higher order slip boundary condition. Continuum breakdown regions are identified based on the gradient-length local (GLL) Knudsen number condition. This initial steady solution on the flow transition boundary is implemented in the DSMC solver as a Dirichlet boundary condition. The present computations are useful in calibrating micro-propulsion controllers to adapt to the substantial momentum changes associated with various nozzle wall thermal conditions and the proximity of similar nozzles in the cluster.

Effect of heat transfer on an angled cavity placed in supersonic flow

Experimental and numerical studies on the effect of heat transfer to an open cavity placed in supersonic crossflow are presented. Cavity floor is subjected to the desired amount of heat flux and the consequent effects in the flow dynamics inside the cavity and its neighborhood are systematically assessed. Cavity flowfield has been visualized using high-speed Schlieren imaging and pressure on the cavity floor has been measured using both steady and unsteady pressure transducers. Two-dimensional unsteady compressible turbulent flow field has been numerically simulated using a Harten-Lax-van Leer-Contact (HLLC) scheme based unstructured finite volume method solver. The numerical method has been validated using both experimental data available in the literature as well as the wall pressure measured in the present study. Significant changes in flow dynamics such as the growth of the recirculation region, shearlayer oscillation, shearlayer impingement on the aft wall of the cavity, and frequency of cavity induced oscillations have been observed. Experimental observations are well complemented by the numerical studies and could reveal the physics of alteration in cavity flow dynamics with the heat addition.

Numerical analysis of hypersonic thermochemical non-equilibrium environment for an entry configuration in ionized flow

A theoretical methodology for thermochemical non-equilibrium flow combing with the HLLC (Harten-Lax-van Leer Contact) scheme was applied to study the hypersonic thermochemical non-equilibrium environment of an entry configuration in ionized flow. A two-temperature controlling model was utilized and the Gupta’s 11 species (N2, O2, NO, O, N, NO+, N2+, O2+, N+, O+, e−) thermochemical non-equilibrium model was taken. Firstly, numerical calculations of hypersonic thermochemical non-equilibrium environments for different aerodynamic shapes were carried out to verify the reliability of the method above. Then, the method was used to research the effects of ionization and wall catalysis on the hypersonic thermochemical non-equilibrium environment of the entry configuration in ionized flow. The shock stand-off distance can be reduced by thermochemical reactions but doesn’t continue to decrease significantly when ionization occurs. The shock stand-off distance calculated by the 11 species model is 4.2% smaller than that calculated by the 5 species (N2, O2, NO, O, N) thermochemical non-equilibrium model without considering ionization. Ionization reduces wall heat flux but increases wall pressure a little. The effect of ionization on aerothermal loads is greater than that of aerodynamic loads. The thermochemical reactions of electrons and ions catalyzed at the wall increase wall heat flux significantly but make a small change in wall pressure. The maximum wall heat flux obtained by only considering the electrons and ions catalyzed at the partially catalytic wall condition is 11.8% less than that calculated at the super-catalytic wall condition.

Numerical experiments of flux difference splitting methods with high resolution scheme for supersonic flows

In this work, we have carried out the assessment of a high resolution scheme for unsteady compressible flow. For high order spatial accuracy, we have used fifth order weighted essentially non oscillatory (WENO) scheme. This scheme is applied to four flux difference splitting (FDS) methods: Harten-Lax-van-Leer (HLL), Roe solver, Harten-Lax-van Leer-Contact (HLLC), and Rusanov methods. We have compared results of these flux schemes with each other. WENO scheme is used for the reconstruction of left and right state variable across the cell interface for high resolution. The reconstruction procedure is performed in terms of primitive variables instead of conservative variable, in order to avoid spurious oscillation. We have considered two test cases: shock wave reflection and supersonic viscous flow over a flat plate, to access the performance of FDS schemes. An explicit third order TVD Runge-Kutta method is used for advancement of solution in time. The present results are compared with available numerical solutions. WENO-HLLC has good shock capturing capabilities as compare to WENO-Roe, WENO-HLL and WENO-Rusanov methods. It also provides best results inside and outside the boundary layer.

The HLLC Riemann solver

The HLLC (Harten–Lax–van Leer contact) approximate Riemann solver for computing solutions to hyperbolic systems by means of finite volume and discontinuous Galerkin methods is reviewed. HLLC was designed, as early as 1992, as an improvement to the classical HLL (Harten–Lax–van Leer) Riemann solver of Harten, Lax, and van Leer to solve systems with three or more characteristic fields, in order to avoid the excessive numerical dissipation of HLL for intermediate characteristic fields. Such numerical dissipation is particularly evident for slowly moving intermediate linear waves and for long evolution times. High-order accurate numerical methods can, to some extent, compensate for this shortcoming of HLL, but it is a costly remedy and for stationary or nearly stationary intermediate waves such compensation is very difficult to achieve in practice. It is therefore best to resolve the problem radically, at the first-order level, by choosing an appropriate numerical flux. The present paper is a review of the HLLC scheme, starting from some historical notes, going on to a description of the algorithm as applied to some typical hyperbolic systems, and ending with an overview of some of the most significant developments and applications in the last 25 years.

A preconditioned solver for sharp resolution of multiphase flows at all Mach numbers

A preconditioned five-equation two-phase model coupled with an interface sharpening technique is introduced for simulation of a wide range of multiphase flows with both high and low Mach regimes. Harten-Lax-van Leer-Contact (HLLC) Riemann solver is implemented for solving the discretized equations while tangent of hyperbola for interface capturing (THINC) interface sharpening method is applied to reduce the excessive diffusion of the method at the interface. In this work, preconditioning technique is used in a system of equations including viscous and capillary effects. Several one- and two-dimensional test cases are used to evaluate the performance and accuracy of this method. Numerical results demonstrate the efficiency of preconditioning in low Mach number flows. Comparisons between results of preconditioned and conventional system highlight the necessity of using preconditioning technique to reproduce main characteristics of low-speed flow regimes. Additionally, preconditioned systems transform to the conventional systems at high Mach number flows thus exhibiting the same level of accuracy as the standard flow solver. Therefore, the preconditioned compressible two-phase method can be used as an all-speed two-phase flow solver accounting for capillary and viscous stresses.

Comparison of Shallow Water Solvers: Applications for Dam-Break and Tsunami Cases with Reordering Strategy for Efficient Vectorization on Modern Hardware

We investigate in this paper the behaviors of the Riemann solvers (Roe and Harten-Lax-van Leer-Contact (HLLC) schemes) and the Riemann-solver-free method (central-upwind scheme) regarding their accuracy and efficiency for solving the 2D shallow water equations. Our model was devised to be spatially second-order accurate with the Monotonic Upwind Scheme for Conservation Laws (MUSCL) reconstruction for a cell-centered finite volume scheme—and be temporally fourth-order accurate using the Runge–Kutta fourth-order method. Four benchmark cases of dam-break and tsunami events dealing with highly-discontinuous flows and wet–dry problems were simulated. To this end, we applied a reordering strategy for the data structures in our code supporting efficient vectorization and memory access alignment for boosting the performance. Two main features are pointed out here. Firstly, the reordering strategy employed has enabled highly-efficient vectorization for the three solvers investigated on three modern hardware (AVX, AVX2, and AVX-512), where speed-ups of 4.5–6.5× were obtained on the AVX/AVX2 machines for eight data per vector while on the AVX-512 machine we achieved a speed-up of up to 16.7× for 16 data per vector, all with singe-core computation; with parallel simulations, speed-ups of up to 75.7–121.8× and 928.9× were obtained on AVX/AVX2 and AVX-512 machines, respectively. Secondly, we observed that the central-upwind scheme was able to outperform the HLLC and Roe schemes 1.4× and 1.25×, respectively, by exhibiting similar accuracies. This study would be useful for modelers who are interested in developing shallow water codes.

Solving the Riemann problem for realistic astrophysical fluids

We present new methods to solve the Riemann problem both exactly and approximately for general equations of state (EoS) to facilitate realistic modeling and understanding of astrophysical flows. The existence and uniqueness of the new exact general EoS Riemann solution can be guaranteed if the EoS is monotone regardless of the physical validity of the EoS. We confirm that: (1) the solution of the new exact general EoS Riemann solver and the solution of the original exact Riemann solver match when calculating perfect gas Euler equations; (2) the solution of the new Harten-Lax-van Leer-Contact (HLLC) general EoS Riemann solver and the solution of the original HLLC Riemann solver match when working with perfect gas EoS; and (3) the solution of the new HLLC general EoS Riemann solver approaches the new exact solution. We solve the EoS with two methods, one is to interpolate 2D EoS tables by the bi-linear interpolation method, and the other is to analytically calculate thermodynamic variables at run-time. The interpolation method is more general as it can work with other monotone and realistic EoS while the analytic EoS solver introduced here works with a relatively idealized EoS. Numerical results confirm that the accuracy of the two EoS solvers is similar. We study the efficiency of these two methods with the HLLC general EoS Riemann solver and find that analytic EoS solver is faster in the test problems. However, we point out that a combination of the two EoS solvers may become favorable in some specific problems. Throughout this research, we assume local thermal equilibrium.

The effect of concentration gradients on deflagration-to-detonation transition in a rectangular channel with and without obstructions – A numerical study

Explosions in homogeneous reactive mixtures have been widely studied both experimentally and numerically. However, in accident scenarios, mixtures are usually inhomogeneous due to the localized nature of most fuel releases, buoyancy effects and the finite time between release and ignition. It is imperative to determine whether mixture inhomogeneity can increase the explosion hazard beyond what is known for homogeneous mixtures. The present numerical investigation aims to study flame acceleration and transition to detonation in homogeneous and inhomogeneous hydrogen-air mixtures with two different average hydrogen concentrations in a horizontal rectangular channel. A density-based solver was implemented within the OpenFOAM CFD toolbox. The Harten–Lax–van Leer–Contact (HLLC) scheme was used for accurate shock capturing. A high-resolution grid is provided by using adaptive mesh refinement, which leads to 30 grid points per half reaction length (HRL). In agreement with previous experimental results, it is found that transverse concentration gradients can either strengthen or weaken flame acceleration, depending on average hydrogen concentration and channel obstruction. Comparing experiments and simulations, the paper analyses flame speed and pressure histories, identifies locations of detonation onset, and interprets the effects of concentration gradients.

Large time step HLL and HLLC schemes

We present Large Time Step (LTS) extensions of the Harten-Lax-van Leer (HLL) and Harten-Lax-van Leer-Contact (HLLC) schemes. Herein, LTS denotes a class of explicit methods stable for Courant numbers greater than one. The original LTS method (R.J. LeVeque, SIAM J. Numer. Anal. 22 (1985) 1051–1073) was constructed as an extension of the Godunov scheme, and successive versions have been developed in the framework of Roe's approximate Riemann solver. In this paper, we formulate the LTS extension of the HLL and HLLC schemes in conservation form. We provide explicit expressions for the flux-difference splitting coefficients and the numerical viscosity coefficients of the LTS-HLL scheme. We apply the new schemes to the one-dimensional Euler equations and compare them to their non-LTS counterparts. As test cases, we consider the classical Sod shock tube problem and the Woodward-Colella blast-wave problem. We numerically demonstrate that for the right choice of wave velocity estimates both schemes calculate entropy satisfying solutions.

Study of total variation diminishing (TVD) slope limiters in dam-break flow simulation

A two-dimensional (2D) dam-break flow numerical model was developed based on the finite-volume total variation diminishing (TVD) and monotone upstream-centered scheme for conservation laws (MUSCL)-Hancock scheme, which has second-order accuracy in both time and space. A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver was used to evaluate fluxes. The TVD MUSCL-Hancock numerical scheme utilizes slope limiters, such as the minmod, double minmod, superbee, van Albada, and van Leer limiters, to prevent spurious oscillations and maintain monotonicity near discontinuities. A comparative study of the impact of various slope limiters on the accuracy of the numerical flow model was conducted with several dam-break examples including wet and dry bed cases. The numerical results of the superbee and double minmod limiters agree better with the theoretical solution and have higher accuracy than other limiters in one-dimensional (1D) space. The ratio of the downstream water depth to the upstream water depth was used to select the proper slope limiter. For the 2D numerical model, the superbee limiter should not be used, owing to significant numerical dispersion.

Numerical study of a planar shock interacting with a cylindrical water column embedded with an air cavity

This paper performs a numerical study on the interaction of a planar shock wave with a water column embedded with/without a cavity of different sizes at high Weber numbers. The conservative-type Euler and non-conservative scalar two-equations representing the transportation of two-phase properties consist of the diffusion interface capture models. The numerical fluxes are computed by the Godunov-type Harten-Lax–van Leer contact Riemann solver coupled with an incremental fifth-order weighted essentially non-oscillatory (WENO) scheme. A third-order total variation diminishing (TVD) Runge–Kutta scheme is used to advance the solution in time. The morphology and dynamical characteristics are analysed qualitatively and quantitatively to demonstrate the breakup mechanism of the water column and formation of transverse jets under different incident shock intensities and embedded-cavity sizes. The jet tip velocities are extracted by analysing the interface evolution. The liquid column is prone to aerodynamic breakup with the formation of micro-mist at later stages instead of liquid evaporation because of the weakly heating effects of the surrounding air. It is numerically confirmed that the liquid-phase pressure will drop below the saturated vapour pressure, and the low pressure can be sustained for a certain time because of the focusing of the expansion wave, which accounts for the cavitation inside the liquid water column. The geometrical parameters of the deformed water column are identified, showing that the centreline width decreases but the transverse height increases nonlinearly with time. The deformation rates are nonlinearly correlated under different Mach numbers. The first transverse jet is found for a water column with an embedded cavity, whereas the water hammer shock and second jet do not occur under the impact of low intensity incident shock waves. The $x$-velocity component recorded at the rear stagnation point can remain unchanged for a comparable time after a declined evolution, which indicates that the downstream wall of the shocked water ring somehow moves uniformly. It can be explained that the acceleration of the downstream wall is balanced by the trailing shedding vortex, and this effect is more evident under higher Mach numbers. The increased enstrophy, mainly generated at the interface, demonstrates the competition of the baroclinic effects of the shock wave impact over dilatation.