Viscous and hyperviscous filtering for direct and large-eddy simulation

doi:10.1016/j.jcp.2021.110115

Journal of Computational Physics

Volume 431, 15 April 2021, 110115

https://doi.org/10.1016/j.jcp.2021.110115 Get rights and content

Highlights

•
A new solution filtering technique for direct and large-eddy simulation is proposed.
•
The novelty is that both the molecular and artificial dissipations are represented.
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Only coefficients have to be coded in a conventional finite-difference routine.
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This is an efficient alternative to implicit time integration of the viscous term.
•
Connections are established with spectral vanishing viscosity for implicit LES.

Abstract

This work is dedicated to the solution filtering technique for performing direct and large-eddy simulation. It is shown that this approach is equivalent to the use of spectral viscosity as a possible ersatz of subgrid-scale modelling. In the framework of finite-difference schemes, the filter operator can be designed to ensure time consistency while easily controlling the level and scale selectivity of the dissipation thus introduced. Then, a new family of filter schemes is developed in order to represent both the molecular and artificial dissipations. The resulting viscous filter operator is straightforward to implement through a simple modification of its coefficients that depend on the molecular/artificial viscosity and the time step. A definitive advantage in terms of computational efficiency is obtained for computational configurations where the time step is restricted by the Fourier condition, as a simple alternative to implicit time integration of the viscous term. Numerical tests clearly show that this viscous filtering method is flexible, accurate and numerically stable at large Fourier number.

Introduction

In the context of large-eddy simulation (LES), spatial filtering can refer to different concepts and techniques, as a potential source of misunderstanding between users and developers in this field of computational fluid dynamics. The essential meaning of spatial filtering is related to the LES formalism itself as a tool of decomposition between the large-scale component of the solution and its residual small-scale component designated as the subgrid-scale (SGS) part. This first concept has been widely developed with the purpose to derive the governing equations of the large-scale motion while introducing the unknown SGS tensor in the framework of a closure problem. For the general background based on this way to define the LES problem, the reader is referred to the textbooks [1], [2], [3] which also include the presentation of the most popular SGS models. In this study, this primitive meaning of spatial filtering is not addressed. The related SGS modelling is also not considered, avoiding the difficult questions about the way to account for this “implicit filtering” in the SGS model itself. For the most popular SGS model, namely the Smagorinsky model, the spatial filtering, to which the solution is actually subjected, is a model feature. This feature can only be known a posteriori, as an output of the LES, with a major role of the numerical error for the filtering actually obtained [4].

In this study, we are interested in the technique that consists in the application of spatial filtering every time step on the solution during the time advancement of the governing equations. This technique is sometimes referred as “solution filtering” [5] or as “relaxation filtering” [6], [7]. It can also be designated as “explicit filtering”, but this term is potentially confusing with the SGS modelling approach where the non-linear convective terms, based on the implicitly filtered (i.e. large-scale) solution, are themselves filtered with a given operator to define an alternative SGS tensor in a modified closure problem [8], [9], [10]. Here, for simplicity, the term “solution filtering” is preferred to clearly express that the filter operation is applied directly on the solution without any reference to other formalisms or non-linearity treatments. We also only focus on the systematic application of the filter every time step, even if a periodic application every n time steps can also be an option to control the resulting artificial dissipation while saving computational time [11], [12], [13], [14].

The solution filtering approach is shared by a wide community in the field of LES as well as in direct numerical simulation (DNS), see for instance [15], [16], [11], [17], [18], [19]. When used in DNS, it can be viewed as a way to control the development of numerical oscillations, due for instance to aliasing errors, when the viscous term is not strong enough to ensure this control [20]. Typically, it enables simulators to perform DNS at marginal resolution while improving the physical realism of their solutions which can be virtually free from small-scale spurious oscillations. When this idea is pushed further using coarse computational mesh at high Reynolds number, the filtering solution strategy is referred as an ersatz of SGS modelling in a fuzzy formalism mainly based on considerations about the expected functional role of the SGS uncaptured by the computational mesh. Viewed in this perspective, this approach may be classified in the field of implicit LES, the solution filtering corresponding to an artificial regularisation process applied throughout the calculation. Since its very beginning through the MILES approach [21], implicit LES has become very popular with the development of a wide range of technics to ensure regularisation (see for instance the collective book [22]). In this work, the goal is not to propose a new approach for implicit LES but to develop a new technique that can be put in this perspective while being firstly useful for DNS.

To introduce artificial dissipation for regularisation purposes, the most popular method is to differentiate the convective term using upwind schemes. An alternative is to boost artificially the viscous term through an overestimation of the second derivatives at small scales, as proposed by [23] where a comparison with the traditional upwind-based strategy is presented for a one-dimensional (1D) linear convection/diffusion equation. In [23], and in a more detailed way in [4], it has been shown that this type of numerical dissipation is the finite-difference counterpart of spectral vanishing viscosity (SVV) [24], [25], [26], [27]. In this paper, also based on a finite-difference framework, the links between SVV, boosted second derivative and filtering are clarified to open the way for a new technique of solution filtering.

The manuscript is organized as follows. In section 2, a generic finite-difference operator is defined with a classic definition of its coefficient relations up to a given order of accuracy. Then, the role of solution filtering is described in section 3 by considering a 1D model equation where the filtering operation can be clearly expressed inside the time advancement. In this simplified framework, an equivalent SVV is introduced in section 4 as a function of the filter transfer function. Based on this equivalence, the strength and the scale-selectivity of the filter can be controlled through the scaling of the scheme coefficients while ensuring the time consistency of the resulting regularisation operator. This principle is extended in section 5 in order to incorporate the molecular dissipation into the filter scheme. In section 6, the numerical accuracy and stability of this new type of filtering, called “viscous filtering”, is analysed through spatial and temporal convergence tests. Then, DNS/LES results are presented in section 7 to assess the viscous filtering strategy in demanding computational configurations involving fully developed turbulence. The major advantage of viscous filtering, in terms of numerical stability, is clearly exhibited, as an efficient alternative strategy to time implicit integration of the viscous term. The main conclusions are summarized in section 8 while drawing perspectives for further developments. Finally, an appendix is provided to clearly establish the close link between viscous filter and second derivative finite-difference schemes.

Section snippets

Filtering scheme and transfer function

Here, the filtering approach is considered in the framework of finite-difference schemes. For simplicity, the discretization is based on a regular mesh and the formalism is presented in 1D through the basic filtering operator expressed as $α_{f} {\hat{f}}_{i - 1} + {\hat{f}}_{i} + α_{f} {\hat{f}}_{i + 1} = a_{f} f_{i} + b_{f} \frac{f_{i - 1} + f_{i + 1}}{2} + c_{f} \frac{f_{i - 2} + f_{i + 2}}{2} + d_{f} \frac{f_{i - 3} + f_{i + 3}}{2} + e_{f} \frac{f_{i - 4} + f_{i + 4}}{2}$ with its associated transfer function $T (k Δ x) = \frac{a_{f} + b_{f} \cos (k Δ x) + c_{f} \cos (2 k Δ x) + d_{f} \cos (3 k Δ x) + e_{f} \cos (4 k Δ x)}{1 + 2 α_{f} \cos (k Δ x)}$ where $f_{i} = f (x_{i})$ are the values of a generic function $f (x)$ on the nodes $x_{i} =$

Functional role of solution filtering

Despite its name, as already mentioned in the introduction section 1, this type of filtering operator has no direct connection with the filtering framework of the LES formalism. To make clear this essential point, let us consider $\frac{d u}{d t} = λ u$ as a simple model equation of the DNS/LES governing equations. The complex variable λ can be defined to consider a generic convection/diffusion equation with $λ = - i c k - ν k^{2}$ where ν is the molecular viscosity, c is the convective velocity and $u (k, t)$ is the solution

Equivalence with spectral vanishing viscosity

The comparison between the time advancement (13) with its exact counterpart (14) leads to the conclusion that the deviation of T from 1 corresponds to the numerical error introduced, on purpose, by the filtering. It is easy to show that (13) also corresponds to the exact time integration of (11) with $λ = - i c k - (ν + ν_{s}) k^{2}$ where $ν_{s}$ can be interpreted as the SVV associated with the filtering. By the exact identification $T = \exp (- ν_{s} k^{2} Δ t)$ , the expression of $ν_{s}$ can be obtained with $ν_{s} (k Δ x) = - \frac{\ln T (k Δ x)}{k^{2} Δ t}$

Viscous filtering

In this work, we propose a new approach where the filter is designed to provide both the molecular and artificial dissipations. Thanks to this feature, the viscous term can be removed from (11) by using $λ = - i c k$ . In this way, both the artificial and molecular dissipation have to be included in T. To enable this feature, Taylor's expansions must be performed by reference to the viscous kernel to obtain order conditions between the coefficients $(α_{f}, a_{f}, b_{f}, c_{f}, d_{f}, e_{f})$ . More precisely, the purpose is to

Accuracy analysis

To validate the present viscous filtering procedure while exhibiting its main accuracy features, the standard convection/diffusion equation $\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = ν \frac{\partial^{2} u}{\partial x^{2}}$ is solved where $u (x, t)$ is the solution expressed in the physical space. With the initial condition $u (x, 0) = \exp [- {(\frac{x - L_{x} / 2}{σ_{x}})}^{2}]$ this equation has an exact solution with $u (x, t) = \sqrt{\frac{σ_{x}^{2}}{σ_{x}^{2} + 4 ν t}} \exp [- \frac{{(x - c t - L_{x} / 2)}^{2}}{σ_{x}^{2} + 4 ν t}]$ where $L_{x}$ is the computational domain considered. The problem is solved using periodicity in x while considering a narrow Gaussian

DNS/LES results

In this section, the present viscous filtering technique is used to perform DNS/LES of two academic turbulent flows. The corresponding scheme has been implemented in the finite-difference code Incompact3d that numerically solves the incompressible Navier-Stokes equations. This code is massively parallel and 6th-order accurate in space when free-slip or periodic boundary conditions are used, as in the present study. The computational mesh is Cartesian with $n_{x} \times n_{y} \times n_{z}$ nodes regularly distributed in

Conclusion and discussion

In this study, a new approach of spatial filtering is proposed for solving the governing equations of fluid mechanics. The general framework of this approach is the concept of solution filtering which consist in the application of a spatial filter operator every step of the time advancement. The method is developed in the context of finite-difference schemes as a simple spatially discrete integrating factor commonly used in spectral Fourier methods for computational fluid dynamics [33], [34].

CRediT authorship contribution statement

Eric Lamballais: Conceptualization, Formal analysis, Methodology, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. Rodrigo Vicente Cruz: Formal analysis, Methodology, Software, Validation, Visualization, Writing – review & editing. Rodolphe Perrin: Formal analysis, Validation, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was granted access to the HPC resources of TGCC/CINES under the allocation A0052A07624 made by GENCI. The authors are indebted to Damien Biau for his insightful remarks on the topics considered here.

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In this note, the importance of spectral properties of viscous flux discretization in solving compressible Navier–Stokes equations for turbulent flow simulations is discussed. We studied six different methods, divided into two different classes, with poor and better representation of spectral properties at high wavenumbers. Both theoretical and numerical results have revealed that the method with better properties at high wavenumbers, denoted as $α$ -damping type discretization, produced superior solutions compared to the other class of methods. The proposed compact $α$ -damping method converged towards the direct numerical simulation (DNS) solution at lower grid resolution compared with the other class of methods and is, therefore, a better candidate for high fidelity large-eddy simulations (LES) and DNS studies.

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Viscous and hyperviscous filtering for direct and large-eddy simulation

Highlights

Abstract

Introduction

Section snippets

Filtering scheme and transfer function

Functional role of solution filtering

Equivalence with spectral vanishing viscosity

Viscous filtering

Accuracy analysis

DNS/LES results

Conclusion and discussion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgement

J. Comput. Phys.

J. Comput. Phys.

Comput. Fluids

Comput. Math. Appl.

Phys. Fluids

J. Comput. Phys.

Comput. Fluids

Fluid Dyn. Res.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.

J. Comput. Phys.

Comput. Fluids

SoftwareX

Int. J. Heat Mass Transf.

Comput. Fluids

Comput. Methods Appl. Mech. Eng.

Elements of Direct and Large-Eddy Simulation

Large Eddy Simulation of Incompressible Flow: An Introduction

Large-Eddy Simulation of Turbulence

Filter shape dependence and effective scale separation in large-eddy simulations based on relaxation filtering

Comput. Fluids

An attempt to assess the quality of large eddy simulations in the context of implicit filtering

Flow Turbul. Combust.

Large eddy simulations of transitional round jets: influence of the Reynolds number on flow development and energy dissipation

Phys. Fluids