An axis treatment for flow equations in cylindrical coordinates based on parity conditions
Introduction
Solving the governing equations of fluid motion in cylindrical coordinates is computationally efficient for axisymmetric flows, such as pipe flows, round jets and wakes, etc. The use of the cylindrical coordinate system, with z, r and θ denoting the streamwise, radial and azimuthal directions, respectively, is advantageous in two ways: firstly an efficient Fourier spectral method can easily be employed for the spatial discretization of the θ-direction. More importantly, the (z, r, θ) coordinate system can be mapped to curvilinear cylindrical coordinates (ξ, η, θ), requiring only four two-dimensional metric terms, rather than nine three-dimensional metric terms when performing a mapping in all coordinate directions (e.g. [1]). This reduces algorithmic operations and the memory requirement for simulations with a given grid size significantly, with in particular the latter being an important factor for high-performance computing on current bandwidth-limited systems. However, one of the challenges of using cylindrical coordinates is an accurate and stable treatment of the singularity at r = 0 for non-axisymmetric flows. Over the last decades several methods have been proposed for treating the coordinate singularity. For example, l’Hopital’s rule has been applied to the singular terms in the governing equations, using one-sided finite difference schemes to solve the new equations at the centerline [2]. However, this approach could lead to loss of accuracy when the one-sided scheme is less accurate than the interior scheme and adds to the computational cost, in particular when using compact schemes. More recently, methods using a Cartesian grid inset at the axis [3], series expansions at the axis [4], or the introduction of a new radial coordinate that excludes the singularity [5] have been introduced and used successfully. Numerous other methods have been applied to the axis singularity, some of them listed in Mohseni and Colonius [5]. It should be mentioned that the efficiency of a particular method might vary depending on whether the spatial discretization is accomplished using a pseudo-spectral, finite volume, or other method. For example, in the context of spectral-Galerkin methods, extra pole conditions [6], a change of dependent variables [7], or a radial discretization of Fourier coefficients that depends on the Fourier mode [8] can be adopted.
Here, a method that differs from the above mentioned approaches is presented that is computationally efficient, accurate and stable method. The novel axis-treatment is based on parity conditions for each azimuthal Fourier mode and variable. Crucially, no new radial coordinate needs to be defined and the axis line can be included in the numerical solution of the original set of equations.
The paper is organized as follows. Section 2 is concerned with the derivation of parity conditions of scalars and individual components of the velocity vector. In Section 3 it is demonstrated how the axis treatment based on parity conditions is implemented into finite difference schemes at and close to the axis. A convergence study of the novel method and two test cases focusing on non-axisymmetric flows are presented in Section 4 before concluding in Section 5.
Section snippets
Approach
The present approach is valid for transport equations in cylindrical coordinates, given they are represented in azimuthal Fourier modes. Taking the Fourier series representation of a function ϕ(z, r, θ) asthe Navier–Stokes or Euler equations can be written for each Fourier mode aswhere and are the azimuthal Fourier modes of U, A, B, C and D as given in, for example, Appendix A of Sandberg and Fasel [9].
Implementation of axis treatment into flow equations
With all parity states of each variable and flow equation determined, a finite difference representation of the flow equations in the axial and radial directions can be implemented for each azimuthal Fourier mode.
Analysis
The proposed axis treatment is implemented into a newly developed multi-block compressible Navier–Stokes solver in generalized cylindrical coordinates. The novel solver uses various fourth-order accurate finite difference schemes and sixth-order accurate filters for the radial and axial derivatives. In the spanwise, or azimuthal, direction, a pseudo-spectral approach using the FFTW3 libraries is employed. For the time integration a low-storage fourth-order Runge–Kutta scheme is used [12]. To
Conclusion
A novel axis treatment for flow equations in cylindrical coordinates represented in azimuthal Fourier modes has been derived. The approach is based on parity conditions for each flow variable or equation. It was shown that scalar quantities and the axial velocity component have even parity while the radial and azimuthal velocity components have odd parity. The resulting symmetry property on each azimuthal Fourier mode for each variable was then used to construct compact and standard
Acknowledgments
This work was supported by the Royal Academy of Engineering/EPSRC Research fellowship (EP/E504035/1). The author is grateful to Charles Burke for his initial contributions and valuable discussions.
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