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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
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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.




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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
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   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.


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