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.     

Simulation of multiple shock–shock interference using implicit anti‐diffusive WENO schemes

Accurate computations of two‐dimensional turbulent hypersonic shock–shock interactions that arise when single and dual shocks impinge on the bow shock in front of a cylinder are presented. The simulation methods used are a class of lower–upper symmetric‐Gauss–Seidel implicit anti‐diffusive weighted essentially non‐oscillatory (WENO) schemes for solving the compressible Navier–Stokes equations with Spalart–Allmaras one‐equation turbulence model. A numerical flux of WENO scheme with anti‐diffusive flux correction is adopted, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of first‐order dissipative methods. Experimental flow fields of type IV shock–shock interactions with single and dual incident shocks by Wieting are computed. By using the WENO scheme with anti‐diffusive flux corrections, the present solution indicates that good accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Computed surface pressure distribution and heat transfer rate are also compared with experimental data and other computational results and good agreement is found. Copyright © 2009 John Wiley & Sons, Ltd.     

A high‐resolution scheme for compressible multicomponent flows with shock waves

A simple methodology for a high‐resolution scheme to be applied to compressible multicomponent flows with shock waves is investigated. The method is intended for use with direct numerical simulation or large eddy simulation of compressible multicomponent flows. The method dynamically adds non‐linear artificial diffusivity locally in space to capture different types of discontinuities such as a shock wave, contact surface or material interface while a high‐order compact differencing scheme resolves a broad range of scales in flows. The method is successfully applied to several one‐dimensional and two‐dimensional compressible multicomponent flow problems with shock waves. The results are in good agreement with experiments and earlier computations qualitatively and quantitatively. The method captures unsteady shock and material discontinuities without significant spurious oscillations if initial start‐up errors are properly avoided. Comparisons between the present numerical scheme and high‐order weighted essentially non‐oscillatory (WENO) schemes illustrate the advantage of the present method for resolving a broad range of scales of turbulence while capturing shock waves and material interfaces. Also the present method is expected to require less computational cost than popular high‐order upwind‐biased schemes such as WENO schemes. The mass conservation for each species is satisfied due to the strong conservation form of governing equations employed in the method. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A high‐order finite difference method for numerical simulations of supersonic turbulent flows

This paper describes a new class of three‐dimensional finite difference schemes for high‐speed turbulent flows in complex geometries based on the high‐order monotonicity‐preserving (MP) method. Simulations conducted for various 1D, 2D, and 3D problems indicate that the new high‐order MP schemes can preserve sharp changes in the flow variables without spurious oscillations and are able to capture the turbulence at the smallest computed scales. Our results also indicate that the MP method has less numerical dissipation and faster grid convergence than the weighted essentially non‐oscillatory method. However, both of these methods are computationally more demanding than the COMP method and are only used for the inviscid fluxes. To reduce the computational cost for reacting flows, the scalar equations are solved by the COMP method, which is shown to yield similar results to those obtained by the MP in supersonic turbulent flows with strong shock waves. Copyright © 2011 John Wiley & Sons, Ltd.     

One‐dimensional shock‐capturing for high‐order discontinuous Galerkin methods

Discontinuous Galerkin methods have emerged in recent years as an alternative for nonlinear conservation equations. In particular, their inherent structure (a numerical flux based on a suitable approximate Riemann solver introduces some stabilization) suggests that they are specially adapted to capture shocks. However, numerical fluxes are not sufficient to stabilize the solution in the presence of shocks. Thus, slope limiter methods, which are extensions of finite volume methods, have been proposed. These techniques require, in practice, mesh adaption to localize the shock structure. This is is more obvious for large elements typical of high‐order approximations. Here, a new approach based on the introduction of artificial diffusion into the original equations is presented. The order is not systematically decreased to one in the presence of the shock, large high‐order elements can be used, and several linear and nonlinear tests demonstrate the efficiency of the proposed methodology. Copyright © 2012 John Wiley & Sons, Ltd.     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.     

Unsteady shock‐fitting for unstructured grids

An unstructured, shock‐fitting algorithm, originally developed to simulate steady flows, has being further developed to make it capable of dealing with unsteady flows. The present paper discusses and analyses the additional features required to extend to unsteady flows, the steady algorithm. The properties of the unsteady version of this novel, unstructured shock‐fitting technique, are tested by reference to the inviscid interaction between a vortex and a planar shock: a comparative assessment of shock‐capturing and shock‐fitting is made for the same test problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimized sixth‐order monotonicity‐preserving scheme by nonlinear spectral analysis

In this paper, sixth‐order monotonicity‐preserving optimized scheme (OMP6) for the numerical solution of conservation laws is developed on the basis of the dispersion and dissipation optimization and monotonicity‐preserving technique. The nonlinear spectral analysis method is developed and is used for the purpose of minimizing the dispersion errors and controlling the dissipation errors. The new scheme (OMP6) is simple in expression and is easy for use in CFD codes. The suitability and accuracy of this new scheme have been tested through a set of one‐dimensional, two‐dimensional, and three‐dimensional tests, including the one‐dimensional Shu–Osher problem, the two‐dimensional double Mach reflection, and the Rayleigh–Taylor instability problem, and the three‐dimensional direct numerical simulation of decaying compressible isotropic turbulence. All numerical tests show that the new scheme has robust shock capturing capability and high resolution for the small‐scale waves due to fewer numerical dispersion and dissipation errors. Moreover, the new scheme has higher computational efficiency than the well‐used WENO schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

The advantages of using high‐order finite differences schemes in laminar–turbulent transition studies

This paper presents various finite difference schemes and compare their ability to simulate instability waves in a given flow field. The governing equations for two‐dimensional, incompressible flows were solved in vorticity–velocity formulation. Four different space discretization schemes were tested, namely, a second‐order central differences, a fourth‐order central differences, a fourth‐order compact scheme and a sixth‐order compact scheme. A classic fourth‐order Runge–Kutta scheme was used in time. The influence of grid refinement in the streamwise and wall normal directions were evaluated. The results were compared with linear stability theory for the evolution of small‐amplitude Tollmien–Schlichting waves in a plane Poiseuille flow. Both the amplification rate and the wavenumber were considered as verification parameters, showing the degree of dissipation and dispersion introduced by the different numerical schemes. The results confirmed that high‐order schemes are necessary for studying hydrodynamic instability problems by direct numerical simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

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

Hybrid 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.     

Evaluation of some high‐order shock capturing schemes for direct numerical simulation of unsteady two‐dimensional free flows

The present study addresses the capability of a large set of shock‐capturing schemes to recover the basic interactions between acoustic, vorticity and entropy in a direct numerical simulation (DNS) framework. The basic dispersive and dissipative errors are first evaluated by considering the advection of a Taylor vortex in a uniform flow. Two transonic cases are also considered. The first one consists of the interaction between a temperature spot and a weak shock. This test emphasizes the capability of the schemes to recover the production of vorticity through the baroclinic process. The second one consists of the interaction of a Taylor vortex with a weak shock, corresponding to the framework of the linear theory of Ribner. The main process in play here is the production of an acoustic wave. The results obtained by using essentially non‐oscillatory (ENO), total variation diminishing (TVD), compact‐TVD and MUSCL schemes are compared with those obtained by means of a sixth‐order accurate Hermitian scheme, considered as reference. The results are as follows; the ENO schemes agree pretty well with the reference scheme. The second‐order accurate Upwind‐TVD scheme exhibits a strong numerical diffusion, while the MUSCL scheme behavior is very sensitive to the value on the parameter β in the limiter function minmod. The compact‐TVD schemes do not yield improvement over the standard TVD schemes. Copyright © 2000 John Wiley & Sons, Ltd.     

Implementation of high‐order compact finite‐difference method to parabolized Navier–Stokes schemes

The numerical solution to the parabolized Navier–Stokes (PNS) and globally iterated PNS (IPNS) equations for accurate computation of hypersonic axisymmetric flowfields is obtained by using the fourth‐order compact finite‐difference method. The PNS and IPNS equations in the general curvilinear coordinates are solved by using the implicit finite‐difference algorithm of Beam and Warming type with a high‐order compact accuracy. A shock‐fitting procedure is utilized in both compact PNS and IPNS schemes to obtain accurate solutions in the vicinity of the shock. The main advantage of the present formulation is that the basic flow variables and their first and second derivatives are simultaneously computed with the fourth‐order accuracy. The computations are carried out for a benchmark case: hypersonic axisymmetric flow over a blunt cone at Mach 8. A sensitivity study is performed for the basic flowfield, including profiles and their derivatives obtained from the fourth‐order compact PNS and IPNS solutions, and the effects of grid size and numerical dissipation term used are discussed. The present results for the flowfield variables and also their derivatives are compared with those of other basic flow models to demonstrate the accuracy and efficiency of the proposed method. The present work represents the first known application of a high‐order compact finite‐difference method to the PNS schemes, which are computationally more efficient than Navier–Stokes solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Development of temporal and spatial high‐order schemes for two‐fluid seven‐equation two‐pressure model and its applications in two‐phase flow benchmark problems

Current existing main nuclear thermal‐hydraulics (T‐H) system analysis codes, such as RALAP5, TRACE, and CATHARE, play a crucial role in the nuclear engineering field for the design and safety analysis of nuclear reactor systems. However, two‐fluid model used in these T‐H system analysis codes is ill posed, easily leading to numerical oscillations, and the classical first‐order methods for temporal and special discretization are widely employed for numerical simulations, yielding excessive numerical diffusion. Two‐fluid seven‐equation two‐pressure model is of particular interest due to the inherent well‐posed advantage. Moreover, high‐order accuracy schemes have also attracted great attention to overcome the challenge of serious numerical diffusion induced by low‐order time and space schemes for accurately simulating nuclear T‐H problems. In this paper, the semi‐implicit solution algorithm with high‐order accuracy in space and time is developed for this well‐posed two‐fluid model and the robustness and accuracy are verified and assessed against several important two‐phase flow benchmark tests in the nuclear engineering T‐H field, which include two linear advection problems, the oscillation problem of the liquid column, the Ransom water faucet problem, the reversed water faucet problem, and the two‐phase shock tube problem. The following conclusions are achieved. (1) The proposed semi‐implicit solution algorithm is robust in solving two‐phase flows, even for fast transients and discontinuous solutions. (2) High‐order schemes in both time and space could prevent excessive numerical diffusion effectively and the numerical simulation results are more accurate than those of first‐order time and space schemes, which demonstrates the advantage of using high‐order schemes.     

Comparison of c‐space and p‐space particle tracing schemes on high‐performance computers: accuracy and performance

The present paper presents a comparison of four different particle tracing schemes which were integrated into a parallel multiblock flow simulation program within the frame of a co‐visualization approach. One p‐space and three different c‐space particle tracing schemes are described in detail. With respect to application on high‐performance computers, parallelization and vectorization of the particle tracing schemes are discussed. The accuracy and the performance of the particle tracing schemes are analyzed extensively on the basis of several test cases. The accuracy with respect to an analytically prescribed and a numerically calculated velocity field is investigated, the latter in order to take the contribution of the flow solver's error to the overall error of the particle traces into account. Performance measurements on both scalar and vector computers are discussed. With respect to practical CFD applications and the required performance especially on vector computers, a newly developed, improved c‐space scheme is shown to be comparable to or better than the investigated p‐space scheme. According to accuracy the new c‐space scheme is considerably more advantageous than traditional c‐space methods. Finally, an application to a direct numerical simulation of a turbulent channel flow is presented. Copyright © 2002 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.     

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.     

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

In the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.     

Shock‐capturing with natural high‐frequency oscillations

This paper explores the potential of a newly developed conjugate filter oscillation reduction (CFOR) scheme for shock‐capturing under the influence of natural high‐frequency oscillations. The conjugate low‐ and high‐pass filters are constructed based on the principle of the discrete singular convolution (DSC), a local spectral method. The accuracy and resolution of the DSC basic algorithm are accessed with a one‐dimensional advection equation. Two Euler systems, the advection of an isotropic vortex flow and the interaction of shock–entropy wave are utilized to demonstrate the utility of the CFOR scheme. Computational accuracy and order of approximation are examined and compared with the literature. Some of the best numerical results are obtained for the shock–entropy wave interaction. Numerical experiments indicate that the CFOR scheme is stable, conservative and reliable for the numerical simulation of hyperbolic conservation laws. Copyright © 2003 John Wiley & Sons, Ltd.