Pure Streamfunction Formulation Research Articles

A high‐order pure streamfunction method in general curvilinear coordinates for unsteady incompressible viscous flow with complex geometry

AbstractIn this paper, a high‐order compact finite difference method in general curvilinear coordinates is proposed for solving unsteady incompressible Navier‐Stokes equations. By constructing the fourth‐order spatial discretization schemes for all partial derivative terms of the pure streamfunction formulation in general curvilinear coordinates, especially for the fourth‐order mixed derivative terms, and applying a Crank‐Nicolson scheme for the second‐order temporal discretization, we extend the unsteady high‐order pure streamfunction algorithm to flow problems with more general non‐conformal grids. Furthermore, the stability of the newly proposed method for the linear model is validated by von‐Neumann linear stability analysis. Five numerical experiments are conducted to verify the accuracy and robustness of the proposed method. The results show that our method not only effectively solves problems with non‐conformal grids, but also allows grid generation and local refinement using commercial software. The solutions are in good agreement with the established numerical and experimental results.

A physics-preserving pure streamfunction formulation and high-order compact solver with high-resolution for three-dimensional steady incompressible flows

In this paper, a pure streamfunction high-order compact (HOC) difference solver is proposed for three-dimensional (3D) steady incompressible flows. A physics-preserving pure streamfunction formulation is first introduced for the steady 3D incompressible Navier–Stokes (NS) equations without in-flow and out-flow boundary conditions, where the divergence of streamfunction ∇ · ψ is maintained in the convective and the vortex-stretching terms together in the nonlinear term of equations to reduce the physics-informed loss. Moreover, taking the streamfunction-vector components and their first-order partial derivatives as unknown variables, some fourth-order compact schemes are suggested for the partial derivatives that appear in the streamfunction formulation, and a high-resolution HOC scheme is introduced for approximating the pure third-order partial derivatives in the convective term. Meanwhile, a new HOC scheme is proposed for the first-type boundary conditions of the streamfunction. Finally, a fourth-order compact difference scheme and its algorithm are established for the 3D steady incompressible NS equations in the streamfunction form, subject to no in-flow and out-flow boundary conditions. Several numerical examples are carried out to validate and prove the accuracy, convergence, and efficiency of the proposed new method. Numerical results reveal that the proposed method not only can achieve fourth-order accuracy but also has excellent convergence, high-resolution, and low computational cost at higher Reynolds number.

A parallel high-order compact scheme for the pure streamfunction formulation of the 3D unsteady incompressible Navier–Stokes equation

The high-order compact scheme for the pure streamfunction formulation of the Navier-Stokes equation (Yu and Tian, 2019) is extended to unsteady problems. The unsteady term is discretized by the Crank-Nicolson scheme. A new boundary scheme is also established for symmetrical boundaries to reduce the computational cost of symmetrical flow problems. The parallelization of the present scheme is realized through a boundary approximation approach which maintains the accuracy of calculations for the first- and second-order derivatives on the interior boundaries. The multigrid method for solving the governing equations is parallelized as well. These means are essential to feasible large-scale simulations. Test on a problem with an analytical solution proves that the proposed algorithm is fourth-order accurate in space and second-order in time. Several numerical experiments on lid-driven cavity problems with various configurations and Reynolds numbers up to 3200 are carried out. Results show that the present scheme is efficient and robust enough to resolve fine flow structures and temporal characteristics with moderate grid sizes and time steps.

A hybrid [formula omitted]-[formula omitted] HOC approach for surface tension driven flows in level set framework

In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or the stream-function vorticity form, we recast the governing equations into the biharmonic form of the N–S equations which carries stream-function ψ as the only variable to be computed. The coupling of this formulation with level set (LS) enables the simulation of two phase flows by the core computation of only the single variable ψ free from discontinuities in the whole physical domain. In the process, we also propose a novel hybrid HOC algorithm for post-processing the pressure on either side of the interface including the jump across it. The robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves, static bubble and oscillating bubble at equilibrium in a fluid at rest, and deforming circular drops in shear flows. We compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends. Our approach is proved to be very efficient in terms of its easy implementation, relatively low computational time and high order of accuracy.

A unified approach for embedded boundary conditions for fourth‐order elliptic problems

SummaryAn efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier–Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche's approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B‐splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter. Copyright © 2014 John Wiley & Sons, Ltd.

A Two-Phase Flow Solver for Incompressible Viscous Fluids, Using a Pure Streamfunction Formulation and the Volume of Fluid Technique

Accurate multi-phase flow solvers at low Reynolds number are of particular interest for the simulation of interface instabilities in the co-processing of multilayered material. We present a two-phase flow solver for incompressible viscous fluids which uses the streamfunction as the primary variable of the flow. Contrary to fractional step methods, the streamfunction formulation eliminates the pressure unknowns, and automatically fulfills the incompressibility constraint by construction. As a result, the method circumvents the loss of temporal accuracy at low Reynolds numbers. The interface is tracked by the Volume-of-Fluid technique and the interaction with the streamfunction formulation is investigated by examining the Rayleigh-Taylor instability and broken dam problem. The results of the solver are in good agreement with previously published theoretical and experimental results of the first and latter mentioned problem, respectively.

Second kind integral equation formulation for the modified biharmonic equation and its applications

A system of Fredholm second kind integral equations (SKIEs) is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions. Such boundary value problem arises naturally when the incompressible Navier–Stokes equations are solved via a semi-implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream-function formulation. The advantages of such an approach (Greengard and Kropinski, 1998) [14] are two fold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. Our construction, though similar to that of Farkas (1989) [10] for the biharmonic equation, is modified such that the SKIE formulation has low condition numbers for large values of the parameter. We illustrate the performance of our numerical scheme with several numerical examples. Finally, the scheme can be easily coupled with standard fast algorithms such as FFT, fast multipole methods (Greengard and Rokhlin, 1987) [15], or fast direct solvers (Ho and Greengard, 2012; Martinsson and Rokhlin, 2005) [17,25] to achieve optimal complexity, bringing accurate large-scale long-time fluid simulations within practical reach.

Recent Developments in the Pure Streamfunction Formulation of the Navier-Stokes System

In this paper we review fourth-order approximations of the biharmonic operator in one, two and three dimensions. In addition, we describe recent developments on second and fourth order finite difference approximations of the two dimensional Navier-Stokes equations. The schemes are compact both for the biharmonic and the Laplacian operators. For the convective term the fourth order scheme invokes also a sixth order Pade approximation for the first order derivatives, using an approximation suggested by Carpenter-Gottlieb-Abarbanel (J. Comput. Phys. 108:272---295, 1993). We also introduce the derivation of a pure streamfunction formulation for the Navier-Stokes equations in three dimensions.

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

In this paper we continue the study, which was initiated in (Ben-Artzi et al. in Math. Model. Numer. Anal. 35(2):313---303, 2001; Fishelov et al. in Lecture Notes in Computer Science, vol. 2667, pp. 809---817, 2003; Ben-Artzi et al. in J. Comput. Phys. 205(2):640---664, 2005 and SIAM J. Numer. Anal. 44(5):1997---2024, 2006) of the numerical resolution of the pure streamfunction formulation of the time-dependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in the last three afore-mentioned articles, to fourth order accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators. The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly in the time-stepping scheme. The approximation of the convective term is compact in the no-leak boundary conditions case and is nearly compact (thirteen points stencil) in the case of general boundary conditions. However, we stress that in any case no unphysical boundary condition was applied to our scheme. Numerical results demonstrate that the fourth order accuracy is actually obtained for several test-cases.

Convergence of a Compact Scheme for the Pure Streamfunction Formulation of the Unsteady Navier–Stokes System

This paper is devoted to the analysis of a new compact scheme for the Navier–Stokes equations in pure streamfunction formulation. Numerical results using that scheme have been reported in [M. Ben‐Artzi et al., J. Comput. Phys., 205 (2005), pp. 640–664]. The scheme discussed here combines the Stephenson scheme for the biharmonic operator and ideas from box‐scheme methodology. Consistency and convergence are proved for the full nonlinear system. Instead of customary periodic conditions, the case of boundary conditions is addressed. It is shown that in one dimension the truncation error for the biharmonic operator is $O(h^4)$ at interior points and $O(h)$ at near‐boundary points. In two dimensions the truncation error is $O(h^2)$ at interior points (due to the cross‐terms) and $O(h)$ at near‐ boundary points. Hence the scheme is globally of order four in the one‐dimensional periodic case and of order two in the two‐dimensional periodic case, but of order 3/2 for one‐ and two‐ dimensional nonperiodic boundary conditions. We emphasize in particular that there is no special treatment of the boundary, thus allowing robust use of the scheme. The finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications.

A pure-compact scheme for the streamfunction formulation of Navier–Stokes equations

A pure-streamfunction formulation is introduced for the numerical simulation of the two-dimensional incompressible Navier–Stokes equations. The idea is to replace the vorticity in the vorticity-streamfunction evolution equation by the Laplacian of the streamfunction. The resulting formulation includes the streamfunction only, thus no inter-function relations need to be invoked. A compact numerical scheme, which interpolates streamfunction values as well as its first order derivatives, is presented and analyzed. A number of numerical experiments are presented, including driven and double driven cavities, where the Reynolds numbers are sufficiently large, leading to symmetry breaking of asymptotic solutions.

An Integral Equation Approach to the Incompressible Navier--Stokes Equations in Two Dimensions

We present a collection of methods for solving the incompressible Navier--Stokes equations in the plane that are based on a pure stream function formulation. The advantages of this approach are twofold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. The disadvantage is that the solution of a nonlinear fourth-order partial differential equation is required. By recasting this partial differential equation as an integral equation, we avoid the ill-conditioning which hampers finite difference and finite element methods in this environment. By using fast algorithms for the evaluation of volume integrals, we are able to solve the equations using O(M) or O(M log M) operations, where M is the number of points in the discretization of the domain.