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JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page. Skip to main contentSkip to article ScienceDirect * Journals & Books * * Search RegisterSign in * Access through your institution * Purchase PDF Search ScienceDirect ARTICLE PREVIEW * Abstract * Introduction * Section snippets * References (58) * Cited by (1) * Recommended articles (6) COMPUTERS & MATHEMATICS WITH APPLICATIONS Volume 117, 1 July 2022, Pages 299-311 ON THE APPLICATION OF HIGHER-ORDER BACKWARD DIFFERENCE (BDF) METHODS FOR COMPUTING TURBULENT FLOWS Author links open overlay panelM.R. Nived, Sai Saketha Chandra Athkuri, Vinayak Eswaran Show more Add to Mendeley Share Cite https://doi.org/10.1016/j.camwa.2022.05.007Get rights and content ABSTRACT The Backward Differentiation Formulae (BDF) of orders one to six are implemented in an in-house three-dimensional finite-volume (FVM) compressible flow code that operates on unstructured meshes. The lower-upper symmetric Gauss-Seidel (LUSGS) implicit relaxation technique based on the splitting of convective Jacobians is used to obtain a variable-order stable implementation in the solver. The temporal order of accuracy of the implemented schemes is verified using the Method of Manufactured Solutions (MoMS). Practical engineering flow problems are simulated to investigate the operational stability of BDF methods so that these can be used as a cheaper alternative to multi-stage methods in RANS, hybrid RANS-LES, and LES implementations. Steady-state flow simulations of supersonic flow over a flat plate and high Reynolds number flow over a NACA-0012 airfoil show that algorithm with BDF schemes of orders 1-5 is stable at high CFL numbers (700-1000) and produces a converged solution. Stokes' second problem and a high-resolution delayed detached eddy simulation (DDES) of flow over a 3D circular cylinder using the BDF1-BDF6 methods demonstrate the use of these higher-order temporal schemes in unsteady laminar and turbulent flows. It is concluded that BDF methods of orders 3-5 can be practically employed to achieve higher levels of temporal accuracy in flow simulations when the level of spatial accuracy is also high (ENO/WENO schemes, spectral methods, etc.) in hybrid RANS-LES, LES or DNS. Even without higher-order spatial accuracy, the same BDF schemes can be used in Adaptive Time-stepping (ATS) methods to obtain prescribed temporal accuracy efficiently by algorithmically switching between the different schemes. INTRODUCTION Over the past few decades, a tremendous growth in the development and availability of computational power has resulted in high-resolution flow simulations on complex geometries becoming more affordable. Challenging flow problems involving laminar-turbulent transition and massive separation at high Reynolds numbers (Re) have been simulated using Unsteady Reynolds Averaged Navier-Stokes (URANS) solvers, Large Eddy Simulation (LES), and hybrid RANS-LES methods, all of which involve unsteady flow simulation. While steady-state problems can be computed economically using lower-order methods, the unsteady simulation of turbulence necessitates the use of higher-order accurate schemes in space and time. Lower numerical error in higher-order schemes allows the use of relatively coarser meshes and larger time-step sizes to efficiently compute unsteady turbulent phenomena. However, higher-order schemes often require significantly greater computational effort compared to lower-order ones, affecting the overall cost-feasibility of turbulent flow simulations. The class of implicit Backward Differentiation Formulae (BDF) schemes [1] possesses a unique advantage of being single-stage and is only marginally costlier with increasing order of accuracy, but limited by stability considerations up to an order of six. In this paper, we present an implicit solver that uses standard BDF schemes to operate from temporal orders of one to six, and investigate its accuracy and stability in turbulent flows on unstructured meshes. Implicit time integration methods are expensive compared to explicit methods like the fourth-order Runge-Kutta (RK4) scheme in the computational effort needed to advance the solution by a single time-step. The explicit RK4 method has comparatively low numerical error, but the Courant-Friedrichs-Lewy (CFL) stability restriction (Courant number ≤1.0) on the time-step size results in much larger overall simulation times than required for accuracy alone. On the other hand, implicit methods allow the use of larger time-step sizes that reduce the marching time needed to reach steady-state and are widely used in industrial CFD solvers for false transient solution of steady-state cases. However, in transient flow simulations, for the use of larger time-steps in implicit schemes a higher order of accuracy is essential to resolve the smaller time scales that are characteristic of turbulent flows. Implicit time integration techniques are broadly classified as: * • Multi-stage (single-step) methods which compute the solution at the new unknown level from the immediately previous solution, using multiple intermediate stages of global matrix inversion, * • Multi-step (single-stage) methods which directly integrate for the solution at the unknown level using a stencil containing solutions from several previous time-levels. Some of the popular single-step methods used by the Computational Fluid Dynamics (CFD) community include implicit θ schemes [2], the Crank-Nicolson method [3], generalized-α schemes [4], implicit Runge-Kutta (IRK) methods [5], and Rosenbrock methods [6]. As these methods are single-step schemes, they are self-starting and do not require any special algorithm to initiate higher-order time marching. The set of generalized-α schemes are utmost second-order accurate, limiting the computational order of accuracy despite having multiple stages. The IRK and Rosenbrock set of methods can provide higher-order accurate solutions and can be used to obtain results with a lower level of numerical error much more efficiently compared to second-order schemes. Among the IRK methods, Singly Diagonally Implicit Runge–Kutta (SDIRK) methods [7] are attractive due to their simplicity in implementation as they have a constant Jacobian over all intermediate stages of time integration, as demonstrated by Chen [8]. Computational efficiency and convergence studies of multi-stage schemes have been carried out by Jothiprasad et al. [9], Montlaur et al. [10], and Holst et al. [11]. However, all the above methods need multiple stages (levels) of intermediate time integration to be performed to obtain the solution at a new time-level. This results in linear systems which are much larger than that of single-stage schemes, resulting in an onerous increase in computational expense for high resolution turbulent flow simulations computed in three-dimensions (3D). Multi-step methods predict the solution at the next time-level using additional solutions at previous time-levels. The main advantage of these methods is that they need only a single stage of linear system inversion, irrespective of the order of the scheme. The storage of solution at multiple time-levels only marginally increases the overall memory requirement of simulations. The Backward Differentiation Formulae (BDF) represent a family of multi-step time-integration methods which was developed by Curtiss and Hirschfelder [12]. A variable order version of the BDF methods with increasing time-step sizes to reduce solution time for initial value problems of ODEs was later popularized by Gear [13] in 1971. Although BDF schemes of orders one and two (BDF1 and BDF2) are widely used in literature as they are unconditionally stable, the stability limitations of BDF of higher orders (BDF3-BDF6) have resulted in them being less investigated. A comparison study of 2-3 stage IRK methods and variable step-size BDF schemes (order one to six), conducted by Hay et al. [14] on a Finite Element 3D Navier-Stokes solver for deforming domains, underlines the superior computational efficiency of the BDF schemes used in their respective stability regions. An efficient higher-order BDF algorithm for rapidly varying time-step sizes was devised by Brayton et al. [15] in 1972. Adaptive time-step/variable-order BDF algorithms have been implemented and validated on Navier-Stokes solvers in a few recent articles [16], [17], [18]. In this paper, we investigate the accuracy and stability of higher-order BDF methods for practical engineering flows, using an in-house unstructured grid compressible flow RANS solver named PRAVAHA [19], [20]. This implicit solver has been equipped with a first-order Adaptive Time-Stepping (ATS) BDF algorithm for compressible flows developed by Kalkote et al. [17], [18]. They have validated the solver for unsteady viscous flows using a non-preconditioned all-speed algorithm that is stable and convergent even for flows in the incompressible regime. In this work, the implicit algorithm of the solver is extended to higher-order BDF methods, using CFL-based time-stepping. The implicit flux Jacobians are computed using an approximate form of the flux vectors to improve the stability in higher-order time integration. The temporal order of accuracy of the implementation of higher-order schemes in the solver is verified using the method of manufactured solutions (MoMS), which is commonly used to obtain the spatial order using a series of nested meshes. The implicit formulation that uses the higher-order BDF is then employed to simulate unsteady turbulent flow over a circular cylinder in addition to two steady-state flow problems and one 1D unsteady laminar flow case. The applicability of the present work is twofold. First, it provides a stable variable-order (one to six) implementation of the BDF on a turbulent flow solver that can be easily extended to a higher-order version of the Adaptive-Time stepping scheme (ATS) of Kalkote et al. [17]. The higher-order ATS solver can then presumably compute unsteady flow simulations efficiently, by using the largest possible stable time-step size based on local error control. And secondly, the higher-order BDF solver can be used to resolve the required scales of turbulent fluctuations in Direct Numerical Simulation (DNS), LES and hybrid RANS-LES solutions where higher accuracy spatial discretization is employed, at a much lesser computational expense. These applications will be explored in future studies with the in-house solver. The organization of the paper is as follows. In Section 2, we describe the governing differential equations and the spatial discretization used in the in-house solver. The implicit algorithm is then detailed in the next section, followed by the calculation of approximate convective and viscous implicit Jacobians. In Section 4, we present a technique to evaluate the order of accuracy of the BDF implementation using the Method of Manufactured Solutions (MoMS) by means of computing the local and global truncation errors. The results of laminar supersonic flow over a flat plate and high Reynolds number (Re) flow over an airfoil are presented in Section 5, followed by Stokes' second problem and a 3D hybrid RANS-LES flow simulation of a circular cylinder. SECTION SNIPPETS BASE SOLVER AND SPATIAL DISCRETIZATION In this work, an in-house cell-centered finite-volume compressible flow solver–PRAVAHA, which operates on unstructured meshes, is employed to implement and investigate the efficiency of higher-order BDF methods. Previous papers based on the solver [20], [21], [22] describe the spatial discretization used and demonstrate its capabilities as a steady-state RANS solver. As the temporal accuracy of the solution needs to be preserved for computing unsteady flows, we use an integral conservative form IMPLICIT TIME-INTEGRATION High-resolution flow simulations of turbulence demand time-accurate integration strategies to accurately resolve the characteristic time scales of turbulent flows. Explicit schemes are often preferred due to their simplicity in computing the solution at a new time-level from known values at previous time-levels. This also makes the parallelization of the solver straightforward. However, the CFL stability limitation of explicit schemes results in a drastic reduction in the time-step size due to VERIFICATION OF HIGHER-ORDER ACCURACY OF BDF METHODS Before carrying out actual flow simulations, we now establish the accuracy of the BDF methods in the unstructured grid solver. In this work, we use the MoMS in a rather unconventional way to verify the temporal order of accuracy of the implicit time-integration in the in-house solver. RESULTS AND DISCUSSION In this section, we study the performance and stability of the higher-order BDF schemes of orders 1–6 implemented in an in-house unstructured grid solver in obtaining solutions of engineering flow problems at a reasonable computational expense. Four different test cases are considered in various Mach regimes with Reynolds numbers (Re) ranging from 5000 to 6 million. The cases simulated include steady-state solutions for a flat plate and airfoil, unsteady viscous flow above an oscillating plate CONCLUSIONS In this work, higher-order BDF methods are utilized to perform the temporal discretization in an in-house unstructured grid compressible flow solver. The BDF methods of orders 3-6 are not commonly used in CFD flow solvers in spite of being computationally cheaper than multi-stage schemes due to their lack of A- and L-stability. A variable-order algorithm that employs classical BDF schemes of orders 1-6 along with the implicit Jacobian computations is presented in this paper. Stability concerns ACKNOWLEDGEMENTS The first author is funded by the Research Scholar Program (RSP) of Tata Consultancy Services (TCS) Cycle '16. The second author is supported by the core research grant no. CRG/2020/000901 of the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India. The authors sincerely acknowledge the efforts of Nikhil Kalkote and Ashwani Assam and who had a crucial role in the development of the in-house implicit solver. REFERENCES (58) * J. Chabassier et al. INTRODUCTION AND STUDY OF FOURTH ORDER THETA SCHEMES FOR LINEAR WAVE EQUATIONS J. COMPUT. APPL. MATH. (2013) * K.E. 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Kalkote et al. TOWARDS DEVELOPING AN ADAPTIVE TIME STEPPING FOR COMPRESSIBLE UNSTEADY FLOWS INT. J. NUMER. METHODS HEAT FLUID FLOW (2019) N. Kalkote et al. ACCELERATION OF LATER CONVERGENCE IN A DENSITY-BASED SOLVER USING ADAPTIVE TIME STEPPING AIAA J. (2019) V. Sharma et al. DEVELOPMENT OF ALL SPEED THREE DIMENSIONAL COMPUTATIONAL FLUID DYNAMICS SOLVER FOR UNSTRUCTURED GRIDS View more references CITED BY (1) * COMPUTATION OF DRAG CRISIS OF A CIRCULAR CYLINDER USING HYBRID RANS-LES AND URANS MODELS 2023, Ocean Engineering Show abstract Numerical simulation of flow past a circular cylinder across the “drag-crisis” region is extremely challenging for turbulence models because the boundary layer undergoes laminar–turbulent transition and variable-locus separation. We investigate the SA-DDES hybrid model along with two variants, namely, SA-kLES and SA-ILES, based on Spalart–Allmaras (SA) model, and include for comparison the SA-BCM transition and the SA-URANS models, for Re ranging from 50,000 to 5 million, using an in-house unstructured grid solver. All hybrid RANS-LES models produced clearly turbulent-like behavior, as evident from the Q-criterion, while the URANS models did not. A decline in the drag coefficient is noticed in all the turbulence models, but not the sharp decrease observed experimentally, with one exception: the SA-BCM transition model, which predicted the drag coefficients much closer to the experiments. The hybrid RANS-LES models outperformed the URANS SA-BCM model only in the fully turbulent trans-critical region and better represent the physics in the wake region for all Reynolds numbers studied. All the hybrid RANS-LES models produced similar results, suggesting comparatively equal performance in predicting separated flows. We believe that the performance of a hybrid model for mid-range Reynolds numbers will be greatly enhanced if the model is equipped to handle the laminar–turbulent transition. RECOMMENDED ARTICLES (6) * Research article ASSESSMENT OF THE FLAT-TOP STABLE GFEM FOR FREE VIBRATION ANALYSIS Computers & Mathematics with Applications, Volume 117, 2022, pp. 271-283 Show abstract The stable Generalized Finite Element Method, SGFEM, has been used for numerical stabilization of the generalized finite element method during enrichment process of shape functions. However, some troubles persist in dynamic analysis. The flat-top strategy is an alternative to hold up the SGFEM but, despite its good performance in quasi-static problems, for dynamic problems some points remain opened, such as the accuracy of natural frequencies, the computational effort to reach a desired degree of precision (for each natural frequency), and the conditioning of mass and stiffness matrices. The main objective of this work is to assess the performance of the flat-top SGFEM in the context of free vibration analysis Based in many numerical simulations, the sensitivity of the method is verified for different parameters, such as, the size of the flat-top partition of unity (PU) intervals, the smoothness of enriched functions, and the stabilization parameter. Despite the flat-top SGFEM improves the numerical stability for modal analysis, it is shown that the convergence rates can be degraded. * Research article A STAGGERED CELL-CENTERED DG METHOD FOR THE BIHARMONIC STEKLOV PROBLEM ON POLYGONAL MESHES: A PRIORI AND A POSTERIORI ANALYSIS Computers & Mathematics with Applications, Volume 117, 2022, pp. 216-228 Show abstract In this paper, a staggered cell-centered discontinuous Galerkin method is developed for the biharmonic problem with the Steklov boundary condition. Our approach utilizes a first-order system form of the biharmonic problem and can handle fairly general meshes possibly including hanging nodes, which favors adaptive mesh refinement. Optimal order error estimates in L2 norm can be proved for all the variables. Moreover, the approximation of the primal variable superconverges in L2 norm to a suitably chosen projection without requiring additional regularity. Residual type error estimators are proposed, which can guide adaptive mesh refinement to deliver optimal convergence rates even for solutions with singularity. Numerical experiments confirm that the optimal convergence rates in L2 norm can be achieved for all the variables. Moreover, all the provided residual type error estimators show the desired results. In particular, the numerical results demonstrate that the proposed scheme on a polygonal approximation of the disk works well for the classic Babuška example. * Research article ERROR ESTIMATES OF SECOND-ORDER BDF GALERKIN FINITE ELEMENT METHODS FOR A COUPLED NONLINEAR SCHRÖDINGER SYSTEM Computers & Mathematics with Applications, Volume 122, 2022, pp. 117-125 Show abstract This paper aims to study two kinds of second-order BDF finite element schemes for a coupled nonlinear Schrödinger system. The proposed schemes are semi-implicit schemes where nonlinear terms are linearized by the implicit-explicit method and explicit method, respectively. We prove that error estimates in L2-norm are unconditionally optimal which means that there is no restriction of the time step and mesh size. Finally, numerical results are given to illustrate theoretical analysis. * Research article GOAL-ORIENTED A POSTERIORI ERROR ESTIMATION FOR THE BIHARMONIC PROBLEM BASED ON AN EQUILIBRATED MOMENT TENSOR Computers & Mathematics with Applications, Volume 117, 2022, pp. 312-325 Show abstract In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment tensor. Then, an abstract unified framework for the goal-oriented a posteriori error estimation is derived based on the equilibrated moment tensor and the potential reconstruction that provides a guaranteed upper bound for the error of a numerical approximation for the goal functional. In particular, C0 interior penalty and discontinuous Galerkin finite element methods are employed for the practical realisation of the estimators. Numerical experiments are performed to illustrate the effectivity of the estimators. * Research article ERROR ANALYSIS OF THE SECOND-ORDER BDF FINITE ELEMENT SCHEME FOR THE THERMALLY COUPLED INCOMPRESSIBLE MAGNETOHYDRODYNAMIC SYSTEM Computers & Mathematics with Applications, Volume 118, 2022, pp. 110-119 Show abstract In this paper, we consider the initial-boundary value problem for the three-dimensional incompressible magnetohydrodynamic coupled heat equation through the Boussinesq approximation. A finite element fully discrete scheme is proposed for approximating and solving this coupled system numerically, where the second-order extrapolation scheme is used for the discretization of time derivative terms and the mixed finite element method is used for the spatial discretization. Furthermore, we use the mini finite element to approximate the velocity and pressure, and use piecewise linear finite elements to approximate the magnetic field and temperature. We prove the unconditional stability of fully discrete scheme and derive the optimal second-order convergent accuracy in both time and spatial directions. Finally, numerical results are presented to illustrate the second-order convergence rates by taking the CFL condition of order 1. * Research article FREE VIBRATION ANALYSIS OF LAMINATED RECTANGULAR PLATES WITH VARYING THICKNESS USING LEGENDRE-RADIAL POINT INTERPOLATION METHOD Computers & Mathematics with Applications, Volume 117, 2022, pp. 187-205 Show abstract The purpose of this paper is to investigate the free vibration characteristics of laminated rectangular plate with varying thickness (LRPVT) using the meshfree strong form method. The theoretical formulation for free vibration analysis is based on the first order shear deformation theory (FSDT), and the governing equations for the LRPVT are derived using Hamilton's principle. In order to approximate the displacement function in the governing equations, a novel Legendre-radial point interpolation method (LRPIM) shape function using combined basis of thin plate spline (TPS) radial functions and Legendre polynomials is presented. The boundary conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness values. The accuracy and reliability of the LRPIM shape function are verified by comparing with the results of the literatures and finite element software ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of the LRPVT with various geometry and boundary conditions are presented through numerical examples. View full text © 2022 Elsevier Ltd. All rights reserved. * About ScienceDirect * Remote access * Shopping cart * Advertise * Contact and support * Terms and conditions * Privacy policy We use cookies to help provide and enhance our service and tailor content and ads. By continuing you agree to the use of cookies. Copyright © 2023 Elsevier B.V. or its licensors or contributors. ScienceDirect® is a registered trademark of Elsevier B.V. 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