Numerical assessment of a shock‐detecting sensor for low dissipative high‐order simulation of shock–vortex interactions

Considering the importance of high‐order schemes implementation for the simulation of shock‐containing turbulent flows, the present work involves the assessment of a shock‐detecting sensor for filtering of high‐order compact finite‐difference schemes for simulation of this type of flows. To accomplish this, a sensor that controls the amount of numerical dissipation is applied to a sixth‐order compact scheme as well as a fourth‐order two‐register Runge–Kutta method for numerical simulation of various cases including inviscid and viscous shock–vortex and shock–mixing‐layer interactions. Detailed study is performed to investigate the performance of the sensor, that is, the effect of control parameters employed in the sensor are investigated in the long‐time integration. In addition, the effects of nonlinear weighting factors controlling the value of the second‐order and high‐order filters in fine and coarse non‐uniform grids are investigated. The results indicate the accuracy of the nonlinear filter along with the promising performance of the shock‐detecting sensor, which would pave the way for future simulations of turbulent flows containing shocks. Copyright © 2014 John Wiley & Sons, Ltd.     

High‐order shock capturing schemes for turbulence calculations

This work investigates high‐order central compact methods for simulating turbulent supersonic flows that include shock waves. Several different types of previously proposed characteristic filters, including total variation diminishing, monotone upstream‐centered scheme for conservation laws, and weighted essentially non‐oscillatory filters, are investigated in this study. Similar to the traditional shock capturing schemes, these filters can eliminate the numerical instability caused by large gradients in flow fields, but they also improve efficiency compared with classical shock‐capturing schemes. Adding the nonlinear dissipation part of a classical shock‐capturing scheme to a central scheme makes the method suitable for incorporation into any existing central‐based high‐order subsonic code. The amount of numerical dissipation to add is sensed by means of the artificial compression method switch. In order to improve the performance of the characteristic filters, we propose a hybrid approach to minimize the dissipation added by the characteristic filter. Through several numerical experiments (including a shock/density wave interaction, a shock/vortex interaction, and a shock/mixing layer interaction) we show that our hybrid approach works better than the original method, and can be used for future turbulent flow simulations that include shocks. Copyright © 2009 John Wiley & Sons, Ltd.     

A higher‐order unsplit 2D direct Eulerian finite volume method for two‐material compressible flows based on the MOOD paradigms

A higher‐order unsplit multi‐dimensional discretization of the diffuse interface model for two‐material compressible flows proposed by R. Saurel, F. Petitpas and R. A. Berry in 2009 is developed. The proposed higher‐order method is based on the concepts of the Multidimensional Optimal Order Detection (MOOD) method introduced in three recent papers for single‐material flows. The first‐order unsplit multi‐dimensional Finite Volume discretization presented by SPB serves as foundation for the development of the higher‐order unlimited schemes. Specific detection criteria along with a novel decrementing algorithm for the MOOD method are designed in order to deal with the complexity of multi‐material flows. Numerically, we compare errors and computational times on several 1D problems (stringent shock tube and cavitation problems) computed on 2D meshes with the second‐ and fourth‐order MOOD methods using a classical MUSCL method as reference. Several simulations of a 2D shocked R22 bubble in the air are also presented on Cartesian and unstructured meshes with the second‐ and fourth‐order MOOD methods, and qualitative comparisons confirm the conclusions obtained with 1D problems. These numerical results demonstrate the robustness of the MOOD approach and the interest of using more than second‐order methods even for locally singular solutions of complex physics models. Copyright © 2014 John Wiley & Sons, Ltd.     

Implicit large eddy simulation using the high‐order correction procedure via reconstruction scheme

We investigate implicit large eddy simulation of the Taylor–Green vortex, Comte‐Bellot–Corrsin experiment, turbulent channel flow and transitional and turbulent flow over an SD7003 airfoil using the high‐order unstructured correction procedure via reconstruction (CPR) scheme, also known as the flux reconstruction scheme. We employ P1 (second‐order) to P5 (sixth‐order) spatial discretizations. Results show that the CPR scheme can accurately predict turbulent flows without the addition of a sub‐grid scale model. Numerical dissipation, concentrated at the smallest resolved scales, is found to filter high‐frequency content from the solution. In addition, the high‐order schemes are found to be more accurate than the low‐order schemes on a per degree of freedom basis for the canonical test cases we consider. These results motivate the further investigation and use of the CPR scheme for simulating turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

Discretization of transport equations on 2D Cartesian unstructured grids using data from remote cells for the convection terms

This paper presents a new finite volume discretization methodology for the solution of transport equations on locally refined or unstructured Cartesian meshes. The implementation of the cell‐face values of the dependent variables enables the employment of data from remote cells and thus the use of higher‐order differencing schemes. It also results in simple and flux‐conservative multiple‐scale stencils for the discretization of the governing equations. The latter are finally cast into a generalized form that does not depend on the local mesh structure. The performance of the numerical model is demonstrated on some classical 2D problems using various gridding techniques and a bounded second‐order upwind scheme. A stable and efficient behaviour of the algorithm is observed in all test cases. The results indicate that the combination in the present model of both local grid refinement and second‐order discretization can produce substantially more accurate solutions than each of the above techniques alone, for the same computational effort. The method is also applicable to turbulent flows and can be easily extended to three‐dimensions. Copyright © 2003 John Wiley & Sons, Ltd.     

High‐order strand grid methods for low‐speed and incompressible flows

A novel high‐order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low‐order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation‐by‐parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high‐order preconditioned method, while turbulent body‐of‐revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.     

Large eddy simulations of wall‐bounded flows using a simplified immersed boundary method and high‐order compact schemes

This paper presents a solution algorithm based on an immersed boundary (IB) method that can be easily implemented in high‐order codes for incompressible flows. The time integration is performed using a predictor‐corrector approach, and the projection method is used for pressure‐velocity coupling. Spatial discretization is based on compact difference schemes and is performed on half‐staggered meshes. A basic algorithm for body‐fitted meshes using the aforementioned solution method was developed by A. Tyliszczak (see article “A high‐order compact difference algorithm for half‐staggered grids for laminar and turbulent incompressible flows” in Journal of Computational Physics) and proved to be very accurate. In this paper, the formulated algorithm is adapted for use with the IB method in the framework of large eddy simulations. The IB method is implemented using its simplified variant without the interpolation (stepwise approach). The computations are performed for a laminar flow around a 2D cylinder, a turbulent flow in a channel with a wavy wall, and around a sphere. Comparisons with literature data confirm that the proposed method can be successfully applied for complex flow problems. The results are verified using the classical approach with body‐fitted meshes and show very good agreement both in laminar and turbulent regimes. The mean (velocity and turbulent kinetic energy profiles and drag coefficients) and time‐dependent (Strouhal number based on the drag coefficient) quantities are analyzed, and they agree well with reference solutions. Two subfilter models are compared, ie, the model of Vreman (see article “An eddy‐viscosity subgrid‐scale model for turbulent shear flow: algebraic theory and applications” in Physics and Fluids) and σ model (Nicoud et al, see article “Using singular values to build a subgrid‐scale model for large eddy simulations” in Physics and Fluids). The tests did not reveal evident advantages of any of these models, and from the point of view of solution accuracy, the quality of the computational meshes turned out to be much more important than the subfilter modeling.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist

We propose a pressure‐based unified solver for gas‐liquid two‐phase flows where compressible and incompressible flows coexist. Unlike the original thermo–Cubic Interpolated Propagation Combined Unified Procedure (CIP‐CUP) method proposed by Himeno et al (Transactions of the Japan Society of Mechanical Engineers, Series B, 2003), we split the advection term of the governing equations into a conservation part and into the rest. The splitting of advection term has two advantages. One is the high degree of freedom in choosing discretization schemes such as central‐difference schemes, upwind schemes, and Total Variation Diminishing (TVD) schemes. The other is the ease of implementation on unstructured grids. The advantages enable the analyses of various flows such as turbulent and supersonic ones in actual complicated boundaries. Therefore, the solver is useful for practical analyses. The solver was validated on the following test cases: subsonic single‐phase flows, incompressible single‐phase turbulent flows, and incompressible gas‐liquid two‐phase flows. With unstructured grids, we obtained the equivalent results as the ones with structured grids. After the validations, subsonic jet impinging on a water pool was calculated and compared with experimental results. It was confirmed that the calculated results were consistent with the experimental ones.     

Numerical simulation of turbulent free surface flow with two‐equation k–ε eddy‐viscosity models

This paper presents a finite difference technique for solving incompressible turbulent free surface fluid flow problems. The closure of the time‐averaged Navier–Stokes equations is achieved by using the two‐equation eddy‐viscosity model: the high‐Reynolds k–ε (standard) model, with a time scale proposed by Durbin; and a low‐Reynolds number form of the standard k–ε model, similar to that proposed by Yang and Shih. In order to achieve an accurate discretization of the non‐linear terms, a second/third‐order upwinding technique is adopted. The computational method is validated by applying it to the flat plate boundary layer problem and to impinging jet flows. The method is then applied to a turbulent planar jet flow beneath and parallel to a free surface. Computations show that the high‐Reynolds k–ε model yields favourable predictions both of the zero‐pressure‐gradient turbulent boundary layer on a flat plate and jet impingement flows. However, the results using the low‐Reynolds number form of the k–ε model are somewhat unsatisfactory. Copyright © 2004 John Wiley & Sons, Ltd.     

A robust multi‐grid pressure‐based algorithm for multi‐fluid flow at all speeds

This paper reports on the implementation and testing, within a full non‐linear multi‐grid environment, of a new pressure‐based algorithm for the prediction of multi‐fluid flow at all speeds. The algorithm is part of the mass conservation‐based algorithms (MCBA) group in which the pressure correction equation is derived from overall mass conservation. The performance of the new method is assessed by solving a series of two‐dimensional two‐fluid flow test problems varying from turbulent low Mach number to supersonic flows, and from very low to high fluid density ratios. Solutions are generated for several grid sizes using the single grid (SG), the prolongation grid (PG), and the full non‐linear multi‐grid (FMG) methods. The main outcomes of this study are: (i) a clear demonstration of the ability of the FMG method to tackle the added non‐linearity of multi‐fluid flows, which is manifested through the performance jump observed when using the non‐linear multi‐grid approach as compared to the SG and PG methods; (ii) the extension of the FMG method to predict turbulent multi‐fluid flows at all speeds. The convergence history plots and CPU‐times presented indicate that the FMG method is far more efficient than the PG method and accelerates the convergence rate over the SG method, for the problems solved and the grids used, by a factor reaching a value as high as 15. Copyright © 2003 John Wiley & Sons, Ltd.     

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

A high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.     

On the performance of WENO/TENO schemes to resolve turbulence in DNS/LES of high‐speed compressible flows

High‐speed compressible turbulent flows typically contain discontinuities and have been widely modeled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1.5 and Re_τ = 222 is conducted using OpenSBLI, a high‐order finite difference computational fluid dynamics framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against the WALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The sixth‐order adaptive TENO scheme is found to produce comparable results to those obtained with nondissipative fourth‐ and sixth‐order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model.     

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

In this article, we present two improved third‐order weighted essentially nonoscillatory (WENO) schemes for recovering their design‐order near first‐order critical points. The schemes are constructed in the framework of third‐order WENO‐Z scheme. Two new global smoothness indicators, τ_L3 and τ_L4, are devised by a nonlinear combination of local smoothness indicators (IS_k) and reference values (IS_G) based on Lagrangian interpolation polynomial. The performances of the proposed schemes are evaluated on several numerical tests governed by one‐dimensional linear advection equation or one‐ and two‐dimensional Euler equations. Numerical results indicate that the presented schemes provide less dissipation and higher resolution than the original WENO3‐JS and subsequent WENO3‐N scheme.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

A high‐order backward forward sweep interpolating algorithm for semi‐Lagrangian method

Conventional semi‐Lagrangian methods often suffer from poor accuracy and imbalance problems of advected properties because of low‐order interpolation schemes used and/or inability to reduce both dissipation and dispersion errors even with high‐order schemes. In the current work, we propose a fourth‐order semi‐Lagrangian method to solve the advection terms at a computing cost of third‐order interpolation scheme by applying backward and forward interpolations in an alternating sweep manner. The method was demonstrated for solving 1‐D and 2‐D advection problems, and 2‐D and 3‐D lid‐driven cavity flows with a multi‐level V‐cycle multigrid solver. It shows that the proposed method can reduce both dissipation and dispersion errors in all regions, especially near sharp gradients, at a same accuracy as but less computing cost than the typical fourth‐order interpolation because of fewer grids used. The proposed method is also shown able to achieve more accurate results on coarser grids than conventional linear and other high‐order interpolation schemes in the literature. Copyright © 2017 John Wiley & Sons, Ltd.