Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations

Abstract

Divergence-conforming B-splines are developed for application to the incompressible Navier–Stokes equations on geometrically mapped domains. These enable smooth, pointwise divergence-free solutions and thus satisfy mass conservation in the strongest possible sense. Semi-discrete methods based on divergence-conforming B-splines are shown to conserve linear and angular momentum and satisfy balance laws for energy, vorticity, enstrophy, and helicity. These are geometric structure-preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate. The methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior.

Introduction

The unsteady incompressible Navier–Stokes equations are infused with vast geometric structure, evidenced by a wide array of balance laws for momentum, angular momentum, energy, vorticity, enstrophy, and helicity. These balance laws are considered to be of prime importance in the evolution of laminar and turbulent flow structures [38], [39], [40], [41], and they are even believed to play a role in the regularity of Navier–Stokes solutions [8], [32]. The key to unlocking much of the geometric structure of Navier–Stokes flow is precisely its volume-preserving nature, yet most numerical methods only satisfy the incompressibility constraint in an approximate sense. Consequently, such methods do not obey many fundamental laws of physics. In particular, semi-discretizations which conserve momentum are typically guaranteed to balance energy if and only if the incompressibility constraint is satisfied pointwise. This is especially concerning as energy plays a fundamental role in numerical stability [35].

In this paper, we present new divergence-conforming B-spline semi-discretizations for the unsteady Navier–Stokes problem. These semi-discretizations are motivated by the theory of isogeometric discrete differential forms [12], [13] and extend the steady Navier–Stokes discretizations presented in [23] to unsteady Navier–Stokes flows. As incompressibility is satisfied pointwise, these semi-discretizations replicate the geometric structure of the unsteady Navier–Stokes equations and admit discrete balance laws for momentum, angular momentum, energy, vorticity, enstrophy, and helicity. In this sense, our semi-discretization scheme may be thought of as a structure-preserving or mimetic discretization procedure for the unsteady Navier–Stokes equations. We impose no-penetration boundary conditions strongly and no-slip boundary conditions weakly using Nitsche’s method. This enables our method to handle boundary layers without resorting to stretched meshes [6], [7]. This also allows our discretization procedure to naturally default to a conforming approximation of Euler flow in the limit of vanishing viscosity and to possess both energy and helicity as inviscid invariants. The proposed semi-discretizations are extended to general mapped geometries of scientific and engineering interest using divergence- and integral-preserving transformations for velocity and pressure fields respectively. In addition to all the features mentioned above, a recent paper of Guermond [24] suggests that our semi-discretizations converge to physically relevant weak solutions satisfying a local (in space–time) energy balance. It is not known at this time whether or not such a convergence property holds for spectral semi-discretizations.

The use of B-splines in the numerical analysis of unsteady Navier–Stokes flow has already been conducted with much success. The novelty of the method presented here is simply the use of tensor-product B-splines that are capable of exactly satisfying the incompressibility constraint. In the Direct Numerical Simulation (DNS) community, a common method of choice in simulating wall-bounded flows is the use of Fourier spectral discretizations in periodic directions and B-splines in wall-normal directions [33], [34], [37]. In this setting, B-splines are often preferred over polynomial-based spectral discretizations due to their high resolving power, easy implementation of boundary conditions, and ability to employ stretched grids. Recently, Bazilevs et al. studied the turbulence problem in a series of papers using NURBS-based isogeometric analysis in conjunction with a Variational Multiscale (VMS) methodology. In these papers, it was found that the increased continuity of splines led to enhanced numerical results [1], [3], [4], [6], [7]. It is believed that much of this success can be attributed to the spectral-like properties of B-splines. In Fig. 1, we have plotted the phase errors associated with one-dimensional k-method NURBS (which in this setting reduce to B-splines of maximal continuity) and $C^0$ finite element discretizations of the first-order wave equation. Note that the phase error associated with the quadratic NURBS solution is much smaller than that associated with the quadratic finite element solution. Indeed, it can be shown that the phase error for $C^{p-1}$ NURBS solutions scales like $O(h^{2p+2})$ while the phase error for $C^0$ finite element solutions scales like $O(h^{2p})$ (see Chapter 9 of [15]). Recently, the theory of Kolmogorov n-widths has been utilized to shed more light on the approximation properties of B-splines [20]. In this study, it was revealed that B-splines are much more accurate on a per degree-of-freedom basis than standard finite elements and possess similar approximation properties to that of a spectral basis. We believe that by combining the spectral-like properties of B-splines with the preservation of the geometric structure of the unsteady Navier–Stokes equations, our semi-discretization procedure may become a useful tool for both engineering analysis and the mathematical study of the unsteady Navier–Stokes equations.

An outline of this paper is as follows. In the following section, we present some basic notation. In Section 3, we recall the unsteady Navier–Stokes problem subject to homogeneous Dirichlet boundary conditions. In Section 4, we briefly review B-splines, the basic building blocks of our new discretization technique, and in Section 5, we define the B-spline spaces which we will utilize to discretize velocity and pressure fields. In Section 6, we present our semi-discrete variational formulation for the steady Navier–Stokes problem, prove well-posedness and a discrete energy inequality, and present an a priori error estimate. In Section 7, we derive various balance laws for our semi-discretizations, including balance of linear and angular momentum, energy, vorticity, enstrophy, and helicity. In Section 8, we present numerical results illustrating the advantages of our semi-discretization procedure on three benchmark problems: two-dimensional Taylor–Green vortex flow, alternating cylindrical Couette flow, and three-dimensional Taylor–Green vortex flow. Each of these problems is sensitive to preservation of conserved quantities and the growth and decay of functionals associated with geometrical structure of the flow. In Section 9, we draw conclusions. Before proceeding, it should be mentioned that we do not consider any artificial diffusion mechanisms or subgrid turbulence models in this paper. As such, the semi-discretizations presented here should only be utilized if the flow field is sufficiently resolved by the spatial mesh. That being said, standard Large Eddy Simulation models can be utilized in conjunction with the proposed semi-discretizations to capture fine-scale turbulent effects on coarse meshes.