Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier–Stokes equations
Introduction
Numerous techniques for the solution of viscous incompressible Navier–Stokes equations can generally be classified into two broad categories: global and local methods. Global methods encompass spectral and pseudospectral methods which make use of the whole computational domain for the calculation of derivatives and quadrature formulas [1], [2], [3]. These methods have the advantage that they converge exponentially towards the exact solution as the number of modes is increased and hence have been very widely used in numerical calculations demanding high accuracy for a wide range of length scales, such as direct numerical simulation of turbulence. The main drawback of these methods is their inability to handle complicated boundary conditions and flow in complex geometries. Besides, clustering of collocation nodes at the boundary, in the case of Chebyshev spectral methods, also leads to very strict stability restrictions for long time integration [1], [3], [4]. Spectral element [5] and domain decomposition techniques have been quite successful in extending the applicability of spectral methods to some complicated flow problems. However, such methods are computationally expensive and are relatively difficult to implement. Unlike these global methods, local methods such as finite difference, finite volume and finite element methods compute the derivatives using neighboring nodes and are much more robust in handling complex boundary conditions and complicated geometries. The main disadvantage of these methods is their slow convergence towards the actual solution with grid refinement which necessitates use of many more grid points to achieve a desired accuracy level, when compared to global methods.
Compact higher order finite difference schemes [6], [7], [8] provide an effective way of combining the robustness of finite difference schemes and the accuracy of spectral methods. The computation of derivatives in compact finite differences is implicit in the sense that the derivative values at a particular node are computed not only from the function values but also from the values of the derivative at the neighboring nodes. Such an approach yields a global scheme without sacrificing the advantage of low computational cost and robustness of a scheme on a local stencil, since solution of the resulting multidiagonal sparse system can be carried out very efficiently. Compared to the finite difference schemes of the same order of accuracy, compact schemes utilize a smaller stencil and give better resolution especially at higher wavenumbers. Extensive study and discussion of the resolution characteristics of the higher order compact schemes on a uniform grid was carried out by Lele in [8]. Since then compact schemes have attained wide popularity in solving various problems involving incompressible and compressible flows [9], [10], [11], [12], [13], [14], [15], [16], [17]. In spite of this the application of higher order compact schemes has been restricted due to the instability of high-order boundary closures. An extensive review of the issues involving the instability of high-order boundary closures on uniform grids along with a discussion of the underlying concepts has been given in [18].
An alternative point of view adopted in [19], [20], [21], [22], [23], [24], [25] is to relate the instability associated with high-order boundary closures to the large oscillations of high-order interpolating polynomials (Runge phenomena) near the boundaries of a uniform grid. A Chebyshev interpolation polynomial utilizes crowding of collocation nodes at the boundary to suppress Runge phenomena. An extension of this idea to high-order compact schemes on non-uniform grid, with more grid points clustered at the boundary, was shown to yield stable boundary closures with the same order of accuracy as the interior [21]. Instead of using a grid transformation which remains susceptible to boundary instability an alternative route of deriving compact schemes from the interpolation polynomial was adopted in [21]. It was found that in accordance with the eigenvalue analysis, the computations for one and two-dimensional wave equations remained stable on a sufficiently stretched grid by suppressing oscillations due to Runge phenomena. The high-order schemes were subsequently applied to a two-dimensional linear convection diffusion equation in order to demonstrate their stability and accuracy. Note that the idea of using high-order interpolation polynomials has been utilized in [22], [23], [24], [25] to develop a spectral finite volume method on unstructured grids. However, our present approach differs from these previous works utilizing high-order finite difference schemes on Chebyshev collocation nodes in that we use high-order compact schemes on a stretched non-uniform grid [4], [21] which avoids the severe stability restrictions on time step size. In addition, our high-order compact schemes are constructed on a single domain, though in principle the idea can be extended to multiple domains.
The purpose of this paper is to extend the higher order compact schemes which were developed for linear model equations in [21] to incompressible Navier–Stokes equations in two dimensions. Compared to linear two-dimensional convection diffusion equation, the incompressible Navier–Stokes equations involve non-linear convective terms and an additional constraint of incompressibility. Hence, it is not clear if the high-order compact schemes on non-uniform grids, which are stable for linear wave and convection-diffusion equation, will maintain their stability and high accuracy for the incompressible Navier–Stokes equations. Previous investigations employing compact schemes for numerical solution of the Navier–Stokes equations have usually been limited to orders of accuracy of four [9], [10], [12], [15], [16], [17] and six [14]. In order to circumvent the problem of instability of high-order uniform compact schemes the accuracy of boundary closure schemes was lowered in [10]. Another approach is to use filtering in order to remove modes that are not resolved properly. However, filtering adds dissipation to the numerical scheme and this often results in a loss of sharp flow features in the computed solution [3]. In contrast to all these techniques, our aim in this work is to employ very high-order (4th–20th-order) collocated compact schemes to the incompressible Navier–Stokes equations without filtering. The overall order of the schemes is kept uniform throughout the computational domain by constructing boundary closure schemes which have the same order as the interior.
In this paper we are concerned with the two-dimensional incompressible driven cavity flow, convection in a differentially heated cavity with adiabatic top and bottom walls, and flow over an impulsively started cylinder. These problems have been widely used as test cases for the assessment and validation of numerical methods. Various different formulations that have been used successfully in the past to solve two-dimensional incompressible driven cavity problem include vorticity–streamfunction [29], [30], [31], [32], [33], velocity–pressure [27], [28], [36], [38], [39], vorticity–velocity [12], [40], [41], [42], [43], and pure streamfunction [15] formulation. The two most widely used techniques in order to handle the incompressibility constraint have been the fractional step approach and the influence matrix technique (cf. [1]). The influence matrix technique of Kleiser and Schumann [38] makes use of the principle of superposition for linear problems to compute a divergence free velocity field. This method has been used successfully in the past for the numerical solution in both the primitive variable [39] and the vorticity–streamfunction formulation [1]. The advantages and disadvantages of each formulation along with the issues involved in their implementation have been discussed in [37]. Due to its relative ease of implementation we use the vorticity–streamfunction formulation in our current work. However, in principle the schemes can be applied to the primitive variable formulation using either the influence matrix technique or the projection method for enforcing divergence free velocity field.
The paper is organized as follows: the details of the governing equations along with the desingularization technique employed to compute accurate numerical solutions are described in Section 2. The high-order compact schemes on non-uniform grids, with the same order for the boundary closures as for the interior schemes are described in Section 3. The temporal discretization scheme together with the influence matrix technique for computing the boundary values of vorticity, and Helmholtz/Poisson solver are described in Section 4. Results from numerical experiments showing the accuracy and stability of high-order compact schemes on a non-uniform grid with more grid points clustered at the boundary, when used to solve driven cavity flow, a convection benchmark problem and uniform flow past a circular cylinder, are presented in Section 5 before the conclusions.
Section snippets
Governing equations in vorticity–stream function formulation
We consider the incompressible Navier–Stokes equations for the unsteady two-dimensional flows in terms of the variables vorticity ω and stream function ψ given byalong with appropriate initial and boundary conditions. Here, J is the Jacobian with respect to the x − y coordinates and represents the non-linear term:Re is the Reynolds number defined as Re = UL/ν, with U, L and ν representing a characteristic velocity, a characteristic
Very high-order compact schemes on non-uniform grids
The numerical approximation of the governing Eq. (4), with the boundary conditions (5), is carried out on a computational grid such as the one shown in Fig. 1(b). The calculation of the first and the second derivatives in the spatial variables x and y, for the convective and viscous terms, is accomplished through the application of the high-order compact schemes on a non-uniform grid. We employ the non-uniform grid compact schemes for the first and second derivatives derived in [21] for
Time discretization
The temporal discretization of the governing Eq. (4) considered in the present work is semi-implicit and utilizes Adam–Bashforth explicit scheme for the non-linear Jacobian terms and second-order implicit Backward Differentiation scheme for the diffusion terms. The viscous term is treated implicitly in order to avoid the strong stability restriction. This combination of schemes has been widely used in obtaining incompressible flow solutions and is generally referred to as the AB/BDI2 scheme [1]
One-dimensional wave equation
In order to study the resolution characteristics of the high-order compact finite difference schemes on non-uniform grids we solve the one-dimensional wave equationwith initial and boundary conditions given byfor various values of wavenumber ω. As shown in [21], the high-order schemes are stable for the wave equation computations only if the grid stretching parameter α is prescribed in accordance with the eigenvalue analysis (Table 2). The
Conclusions
In this paper compact finite difference schemes over non-uniform grids with very high-orders of accuracy ranging from four to 20 are implemented to solve two-dimensional viscous incompressible flow. Two methods for the numerical solution of Poisson and Helmholtz equations using high-order compact schemes on non-uniform grid are developed. It is shown that the high-order non-uniform grid compact schemes with order of accuracy of up to 20, along with the boundary closures of the same order as the
Acknowledgments
The first author thanks Professor V.K. Dhir of the MAE Department at UCLA for support during this research. The second and third authors are sponsored by the Air Force Office of Scientific Research, USAF, under AFOSR Grant No. FA 9550-04-1-0029, monitored by Dr. John Schmisseur.
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