3 Features of a basic numerical approach

Abstract

We formulate the basic numerical approach to simulate the Navier–Stokes equations, both for compressible as well as incompressible fluids. We present two different numerical representations of the solution that both give rise to large systems of ordinary differential equations. In particular, we consider the method of lines to obtain a discretization that can directly comply with the main conservation properties of the Navier–Stokes equations. Moreover, we consider finite element discretization. We also describe and illustrate the main aspects that influence the quality of a numerical solution, such as the order of accuracy of the scheme, stability and convergence. Finally, we turn to computational characteristics of the discrete problem and present global templates within which compressible and incompressible flow solvers can be formulated.

3.1 Introduction

In this chapter, we consider a number of general aspects that play a role in the simulation of turbulent flows. In the previous chapters, we identified the basic Navier–Stokes equations and sketched some properties of turbulent solutions. In total, a system of partial differential equations needs to be solved. The equations governing compressible fluid flow contain first-order derivatives with respect to time t and first and second-order derivatives with respect to the spatial coordinates. In general, these equations can be written in the form

where U is a vector representing the dependent variables and N is a nonlinear differential operator that contains the spatial derivatives and possible source terms and body force densities. Specifically, for compressible flow U=[ρ,ρu,e] with ρ the density, u the velocity and e the energy density respectively. Since we are interested in unsteady solutions integration over time plays an important role, alongside spatial discretization. Global features of the numerical methods will be illustrated in this chapter, while time integration is considered in the next chapter, and Chapter 5 is devoted to spatial discretization. For simplicity, we will consider the problem of performing a direct numerical simulation and identify extensions required for large-eddy simulation whenever appropriate.

Discretization

Since analytical solutions to the Navier–Stokes equations are not available in most cases of interest, it is obvious that a numerical treatment of these equations is required. In order to obtain a formulation of the flow problem that can be treated on a computer, a ‘discrete’ formulation needs to be obtained, through a method of ‘discretization’. Roughly speaking, this implies that the desired solution is represented in a ‘finite’ way. As an example, one may represent the solution by the collection of its values on a large number of spatial locations and times. This approach requires the introduction of the ‘computational grid’, usually a grid covering the physical flow-domain only. An example of such an approach is a finite volume discretization [128] to which we turn momentarily. Likewise, one could represent the solution with respect to some suitable set of basis functions and consider the problem of finding the appropriate time-dependent coefficients of the solution with respect to this set of basis functions. Examples of this second representation of numerical solutions are a spectral method [24] or a finite element approach. In all these formulations, the time variable is treated differently from the spatial variables, although direct space–time discretizations can also be formulated.

Convergence: consistency and stability

When studying and developing numerical methods for partial differential equations the central requirement is convergence. By this, we imply that the numerically obtained solution will monotonically become a better approximation of the analytical solution if the spatial and temporal resolution of the numerical representation are suitably increased. Two important aspects are consistency and stability to which we will return shortly. For linear problems, these concepts are sufficient to ensure convergence in an appropriate norm as is stated in the Lax equivalence theorem [154]. This theorem ensures that, if the computational effort spent on approximating a solution is suitably increased, then the numerical solution will indeed become better, and we gradually approach the analytical solution by following a proper refinement path. The ‘rate’ at which convergence is achieved is characterized by the accuracy of the discretization. Although such a general equivalence theorem is missing for nonlinear problems, traditionally the accuracy and stability of the discretization are important characteristics of a numerical method also in this nonlinear case.

Next to the accuracy and stability of a discretization, the desire to maintain the basic properties of the continuous governing equations can be adopted as guiding principles to formulate suitable numerical methods. An example of such structure preservation is the conservation property expressed by (1.9). If this property also holds for the discrete representation of the flow problem, it is felt that a better discretization is arrived at, compared to the case in which only accuracy requirements are invoked. How many rigorous properties, e. g., symmetries [219], of the continuous Navier–Stokes equations can (and should) be incorporated in a particular numerical method is a priori not clear. Various choices have been made in the literature that contribute to the large variety of numerical methods that are nowadays available or still under development.

In this part of the book, which consists of the next three chapters, we do not aim to give an exhaustive treatment of numerical methods for partial differential equations. There are good specialized books for that. Rather, we try to provide sufficient detail to appreciate the essential differences and capabilities of the various methods and their computational consequences. This should help to decide which (type of) numerical method is most suitable for studying a specific flow phenomenon, taking features such as robustness of the algorithms, computational costs, dynamical implications for possibly under-resolved flow features and geometrical complexity into account.

The organization of this chapter is as follows. In Section 3.2 we will introduce the basic framework of discretization and show that a large system of ordinary differential equations emerges from the system of partial differential equations. Elements that determine the quality of the resulting numerical solution will be considered in Section 3.3. In Section 3.4 we formulate templates for compressible and incompressible flow solvers.

3.2 Basic approaches to discretization

In this section, we consider two important discretization approaches for the Navier–Stokes equations. For convenience, we focus on the compressible Navier–Stokes equations and consider the general initial-boundary value problem. First, we turn to the ‘method of lines’. Then, we consider an approach based on the introduction of a suitable set of basis functions that enables determining an approximate series expansion of the solution. In both approaches, a large system of ordinary differential equations arises, which forms the basis for the numerical simulation.

Physical and function space discretizations

The compressible Navier–Stokes equations can concisely be written as

where U denotes the ‘state-vector’, Fj the total flux-vector in the xj direction and V the flow-domain with boundary ∂V. This system of partial differential equations determines the evolution of an initial condition U(x,0)=U0(x) and requires in addition the specification of suitable boundary conditions which we denote by B(u)=b where B usually contains various types of boundary treatments and b represents external forcing and imposed flow variables. Typical boundary conditions include Dirichlet, e. g., u=b, Neumann, e. g., ∂xu=b, or mixed conditions, e. g., a Robin boundary condition u+a∂xu=b.

This initial boundary value problem can be discretized in various ways. We distinguish between physical space and function space discretization, although...