High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy
Introduction
For fundamental flows in academic configuration, the usefulness of highly accurate numerical schemes for DNS/LES is fully recognized. For very simplified geometries, in terms of accuracy and computational efficiency, the most spectacular gain is obtained using spectral methods based on Fourier or Chebyshev representation [5]. The various combinations between these two types of spectral discretization have allowed numerous authors to consider efficiently some fundamental turbulent problems in the context of incompressible flows. Among these fundamental problems, one could cite homogeneous turbulence (full Fourier representation), transitional or turbulent channel flow (mixed Fourier–Chebyshev representation) or turbulence in a cavity (full Chebyshev representation), etc. Unfortunately, for fundamental problems in slightly more complex geometry, the full spectral approach in Fourier or Chebyshev space is not appropriate. Furthermore, despite the recent progress in computational fluid dynamics (CFD) methods, the use of sophisticated meshes combined with high-order schemes remains a challenging task that requires important numerical developments to accommodate accuracy and efficiency. For a general presentation of this ambitious numerical strategy, see [8] where the authors focus on collocation and spectral elements for the simulation of incompressible flows.
To better understand a wide range of transitional or turbulent phenomena [25], many DNS/LES studies of fundamental flows in non-academic geometries remain to be performed. The term non-academic means here that the geometry is not as complex as in a real industrial configuration, while not being simple enough to allow the straightforward use of high-order methods, spectral or not. To treat favourably that kind of flows, this paper presents an intermediate approach leading to a compromise between accuracy of spectral methods and versatility of industrial codes. To our knowledge, the potential of such an incompressible Navier–Stokes solver has not been investigated in the literature. In a complementary way to collocation and spectral elements, the high-order strategy considered in this paper is based on finite difference schemes implemented on a Cartesian mesh, the main advantages of this specific numerical configuration being its simplicity and efficiency. It will be shown that such a numerical code can be significantly more versatile than a fully spectral code while offering in practice a comparable accuracy for the DNS of turbulent flows. Furthermore, to increase the potential of the code, the interest of its combination with an immersed boundary method (IBM) will also be evaluated, so that any solid wall geometry can be freely imposed despite the use of a simple Cartesian mesh.
It is well known that in terms of numerical development and computational efficiency, the enforcement of incompressibility can be critical, especially for high-order methods. In this paper, using a projection method, a fully spectral Poisson solver is proposed, in the three spatial directions, even if the boundary conditions do not seem to suit well for a Fourier representation. The advantages of this technique are discussed in terms of computational efficiency, accuracy and simplicity. Several validations are proposed to show the potential of this approach through the use of high-order schemes, the boundary condition treatment and the choice of the mesh organisation (collocated or partially staggered configuration). The results are analysed in order to target the general context for which quasi-spectral accurate schemes could be really attractive.
The organisation of the paper is as follows. After an overview of the general framework of present numerical methods in Sections 2 General framework of the numerical method, 3 Spatial discretization of convective and viscous terms, the treatment of the pressure is detailed in Section 4 where the benefit of the modified spectral formalism is argued. Furthermore, an extension toward non-regular meshes is proposed in Section 5. Finally, a set of formal and practical validations is presented and analysed in Section 7 to evaluate the actual benefit of present high-order method.
Section snippets
Governing equations
The governing equations are the forced incompressible Navier–Stokes equationswhere is the pressure field (for a fluid with a constant density ) and the velocity field. In these forced Navier–Stokes equations, the forcing field is used through an immersed boundary method or sometimes to perform numerical tests where analytical solutions are available thanks to a relevant definition of . More details about the exact expression of
Spatial discretization of convective and viscous terms
Let us consider a uniform distribution of nodes on the domain with for . The approximation of values of the first derivative of the function can be related to values by a finite difference scheme of the formBy choosing and , this approximation is sixth-order accurate while having a so-called “quasi-spectral behaviour” [24] due to its capabilities to represent accurately a
Spatial discretization of the pressure
It is well known that the treatment of incompressibility is a real difficulty to obtain solutions of incompressible Navier–Stokes equations. The unavoidable solving of the Poisson Eqs. (9), (11) introduced by the fractional step method can be computationally very expensive, especially when high-order schemes are used in combination with iterative techniques. This point is so delicate that in terms of computational cost, the use of second-order schemes could be finally more efficient. Based on
Extension toward a stretched mesh in one direction
At this stage, the overall procedure for the discretization of the pressure is valid only for a regular mesh in the three spatial directions. This constraint can be overcome through the use of a mapping easily usable in physical space, the basic principle being based on an equally spaced mesh in the mapped space. However, expressed in spectral space, a general mapping can introduce a serious difficulty in terms of algorithm and computational cost. Indeed, the main problem is linked to the use
Concluding remarks and computational cost
As the strict equivalence between the spatial discretization in physical and spectral spaces is always maintained, the present technique allows the verification of the incompressibility condition up to the machine accuracy. In practice, this property seems to be important in terms of stability. If the matching between the physical and spectral discretization is not strictly satisfied, the resulting residual divergence is found to increase during the calculation whatever its initial level. For
Validation and results
In this section, six computational configurations are considered, with different sets of boundary conditions and different flow motions. In the Section 7.1, periodic or free-slip conditions are used for preliminary verifications in 2D problems. Then, the ability of the present code to treat satisfactory no-slip boundary conditions (despite the use of a modified spectral approach for the pressure) is evaluated in Section 7.2 for a stationary flow. Note that in the same section, the error
Discussion and conclusion
A numerical strategy suitable for DNS/LES of incompressible flows is presented in this paper. The approach proposed is based on a mixed method where the spatial differentiation is mainly performed in physical space whereas the treatment of the Poisson equation is carried out in spectral space. The simplicity of the method is presented through basic rules that allow the use of high-order schemes via an equivalent formulation in physical and spectral spaces using the concept of the modified wave
Acknowledgments
Calculations were carried out at IDRIS, the computational centre of the CNRS. We are grateful to Jalel Chergui and Jean-Marie Teuler for their very precious help in developing specific and optimized FFT routines. We acknowledge Damien Biau for his helpful remarks during the revision and David Froger for his support for the DNS of turbulent channel flow. We also wish to thank Werner Kramer and Herman Clercx for having kindly provided us their data.
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