Introduction

From previous discussions, it is apparent that it is possible to solve sets of differential equations numerically for practical parameter ranges, by solving PDEs with stringent requirements of resolving wide ranges of spatial and temporal scales. In the literature, one comes across a number of efforts which solve Navier–Stokes equation for turbulent flows at moderate Re. Also, large RAM machines are now available at desk-top, allowing one to solve problems with large number of grid points. This encourages search for faster methods; faster than the methods discussed in previous chapters. Higher accuracy methods are expected to fulfill this, as these allow solving the same problem with fewer grid points. Conversely, same methods allow solving problems at higher Reynolds number with the same grid. This chapter deals with developments and analyses of higher accuracy methods.

Effectiveness of CFD schemes is generally ranked by the leading order term of the Taylor series expansion used to define the truncation error of any scheme. It is expected that higher the order, better will be the quality of the scheme. While this may appear as a correct expectation, discussions in Chapters 8 and 10 will convince the readers that classification of schemes simply based on order of truncation error is incomplete. In Chapter 10, schemes were assessed based on their behaviour in the spectral plane. The Fourier spectral method provides maximum accuracy and it is seen that higher the order of explicit schemes, better is the representation.

Introduction

We have discussed classical developments, as well as, recent higher accuracy computing methods so far. We have compared various discrete computing methods with the same framework of spectral analysis. In this chapter, we include topics of more recent origin, which have bearing with the central theme of high accuracy computing of flows and wave phenomena.

In Sections 4.7–4.10, we have discussed about length- and time-scales excited in flows from the perspective of waves as building blocks. While future computational activities would be predominantly for DNS, there are many issues to be sorted out. For DNS, error must be the major concern. In Chapters 8–11, we have identified various sources of error with the help of developed error dynamics in Eqn. (8.31) for convection dominated problems. These sources of error are due to numerical stability, phase and dispersion errors. In subsequent chapters, we have added to this list by considering spurious upstream propagating waves (q-waves), the Gibbs' phenomenon, aliasing error etc. We also discuss about better time discretization methods developed with DRP analysis of Chapter 8. This helps solving space–time dependent problems by optimized time integration method which reduces error. This has been presented for high accuracy two-time level, multi-stage Runge–Kutta methods.

Introduction

A beginner in the field of computing is hardly aware of the importance of formulations responsible for the accuracy in scientific computing. In theoretical fluid mechanics it is not important as to which form of Navier–Stokes equation is solved. But, in computing this is paramount, and there is always ongoing debate among practitioners about the superiority of different formulations and numerical methods employed by different schools of thought.

Governing differential equations are obtained by considering a control volume and balancing fluxes of quantities of interest, obtained in the limit of vanishing size of the control volume. This provides a point-by-point description of intrinsic properties of interest. In many methods of computing, this point description of conservation principle is integrated over a finite control volume. Obtaining governing differential equation is described in this way here, for the conservation of mass, momentum and energy. This is followed by discussion on desirability of casting differential equations in conservation form, which is found to be impervious to details of discretization to a great extent, as compared to non-conservation form.

Here, we also note that often one requires to investigate the problem by formulating the governing equation in non-inertial frame. Readers will have no difficulty in appreciating the need for it in weather forecasting. It is also appropriate and convenient for many engineering flows, where one part of the body is in relative motion with respect to other parts, as in problems of aeroelasticity. This is also needed for rigid bodies of arbitrary shape executing time-dependent motion.

Introduction

Initial developments in computing were dominated by two classes of problems: (i) the jury or the boundary value problems – typically classified as elliptic partial differential equation in Chapter 3; (ii) the evolution or the initial–boundary value problems which are represented by parabolic and hyperbolic partial differential equations. In fact, the solution methods for heat equation (a parabolic partial differential equation) were central to the early development of the subject. These classical approaches are discussed in this chapter, with additional insight brought through spectral analysis of the schemes. It is noted that the stability analysis of numerical schemes was developed with respect to heat equation by von Neumann, as described in [41, 53]. This was considered a major milestone in the development of the subject. But, the readers' attention is also drawn to the correct analysis advanced recently, as described in [259] and Chapter 8, with respect to 1D convection equation.

In fluid dynamics, a major milestone was the introduction of boundary layer concept by Ludwig Prandtl in 1904, which dominated fluid dynamics studies. Readers are referred to [209] for details of the development. Boundary layer equation is an example of parabolic partial differential equation.

There are many new developments in scientific computing, in its application to fluid flows and wave phenomena, which warrant their consolidation in a single source, covering some of the key developments. I have been convinced by many students and peers that there is a definitive need for a single source book which deals with topics covered here. I would like to acknowledge their inspiration. My main motivation in writing this manuscript is to communicate something new and powerful as opposed to conventional derivatives of products churned out by existing schools of thought.

However, this book also provides general introduction to computational fluid dynamics (CFD), using well tested classical methods of solving partial differential equations (PDEs) for the sake of completeness. These are to be found in Chaps. 1 to 9 and 13, but re-interpreted using the spectral analysis method introduced in Chaps. 4, 8 and 10. This provides an unity of approach in understanding numerical methods for parabolic, elliptic and hyperbolic PDEs. The spectral analysis tool has been refined in recent years by the author's group, with which disparate methods can be easily compared.

Introduction

Waves are usually associated with hyperbolic partial differential equations. What waves have to do in general with CFD? It is noted earlier that many problems governed by parabolic or elliptic PDEs require numerical algorithms which treat the problems as belonging to hyperbolic system. It is explained later that the numerical characteristics of the equivalent hyperbolic system must include the single characteristic of a parabolic system for the method to be stable and accurate. Similarly, solving an elliptic PDE by iterative methods is equivalent to solving an equivalent time-evolution problem. In fact, it is in this context the eigenvalues and eigenvectors of associated linear algebraic system provide physical justification. Thus, this chapter contains expository material in understanding wave-attributes of many problems which support waves directly as in hyperbolic system or any other system that supports dispersive waves.

Wave motions have the characteristic property that after a signal (information) is observed at one point, a closely related signal (information) may later be observed at a different point. Thus, waves are the means by which information travel in space and time, without significant movement of the medium. Quite often, what is perceived as motion is related to movement of phase and energy. These two motions are characterized by phase speed and group velocity which will be described shortly.

Introduction

Having established the tools for discretization and methods of analysis of the discretized equation, here we explain methods for solving Navier–Stokes equations of steady and unsteady incompressible viscous flows. Versions of Navier–Stokes equation in primitive and derived variables, expressed in inertial and non-inertial frames have been given in Chapter 2. Various versions of pressure Poisson equation are also derived in Chapter 2, which is to be solved for accurate evaluation of loads and detailed pressure distribution.

Navier–Stokes equation is an evolution equation for vorticity, a primary physical quantity of interest for unsteady laminar and turbulent flows. Primarily, vorticity is generated at physical boundaries for wall-bounded flows, as a consequence of no-slip condition. In free shear layers, vorticity is generated by flow instabilities at interfaces in mixing layers and jets; primary instability mechanisms are attributed to Rayleigh–Taylor, Kelvin–Helmholtz, Görtler mechanisms. Hence, VTE is central for analysis and solution of Navier–Stokes equation. Attendant velocity field can be obtained from the solution of SFE for 2D flows. Poisson equations relating velocity and vorticity field can also be solved for 2D and 3D flows in vorticity–velocity formulation. As stated in Chapter 2, vorticity–stream function formulation is preferred over vorticity–velocity formulation, whenever accurate solution is desired.

Introduction

We have discussed in Chapter 11 about classes of high accuracy compact finite difference schemes for spatial discretization, which along with time discretization schemes, can be used for DNS. It was noted using spectral analysis that the performance of any numerical method is best judged by the spatial and temporal resolutions provided by the combined analysis rather than by the formal order of truncation error terms of individually discretized terms. In this respect, a few compact difference methods are preferred over explicit methods, due to computational efficiency and accuracy. Higher efficiency is due to lesser requirements of points due to higher resolution and accuracy is due to implicit satisfaction of physical dispersion relation with larger time step. It was seen that two second order methods proposed [104, 260, 277], provided higher resolution than sixth order methods [2, 360]. In [104, 277], higher resolution schemes were obtained by minimizing L2-norm of solution error, evaluated as difference of numerical solution from exact solution in the Fourier spectral space for solving 1D convection equation.

There is no unique procedure for the Padè schemes, relating derivative(s) and function values at the nodes. While this can provide flexibility in the choice of a method, if sufficient care is not taken, then it may lead to inconsistency. FVM and FEM can be viewed in this respect to provide logical development in obtaining discrete equations based on physical statement of the problem. Both these methods and other integral methods are said to satisfy conservation principles in a weak sense. In FVM, physical conservation processes are fulfilled in a finite control volume. Physical “densities” of conserved properties are related to fluxes entering and leaving the control volume through control surfaces. These fluxes are created and governed by physical convection and diffusion processes, along with pressure gradient and body forces.

Introduction

In Chapter 2, we have derived most general form of conservation laws which are solved in CFD. For continuum flows, these are given by Navier-Stokes equation. A quick look at them will show that these are non-linear partial differential equations. To be more precise, they are linear with respect to the highest derivative terms and such equations are called quasi-linear PDEs. It is possible to classify such equations based on the behavior of their solutions. In this chapter, we classify quasilinear PDEs, so that we can derive specific numerical methods for each equation Type in subsequent chapters. Despite the observation that different numerical methods are chosen for different classes of PDEs, we will also see here and in later chapters, that there is a generality of approach in treating these PDEs, whose solution and error propagate in time in a unified manner. This will be clearly evident even for time-independent problems, which are solved iteratively, as will be demonstrated through an example in Section 3.3.

Classification of Partial Differential Equations

Consider the moving boundary problem as shown below. This class of problems is also known as the propagation problem.

This book aims at covering the foundations of high accuracy computing methods within the framework of Computational Fluid Dynamics (CFD) in an era of rapidly developing and evolving hardware and software.

From the hardware point of view, huge parallel machines with tens of thousands cores are installed at national facilities and research laboratories giving the practioners of scientific computing tools that they could not have dreamt of a decade ago. The advent of Graphical Processing Units (GPUs) also modifies the course of CFD as everyone tries to strain the computational tools to their last bits and extracts the highest speed-up. This is not surprising as one of the unsolved problems in classical physics is the understanding and control of turbulence in nature and technological applications.

From the software viewpoint, the advent of commercial packages including mesh generators, solvers and graphics tools, provide the numericists with appealing users interfaces and deliver numerical results for extremely different and various problems involving complicated geometries, peculiar boundary conditions and complex physics to be captured. This has had a major impact on the CFD community.

A question that is often raised consists in asking “Why should we not use the simplest schemes and run them on millions (billions) of processors?” The problem as we will discover rapidly is that simple schemes are very often too naive and lead to numerical disaster. We cannot assume that our intellectual indolence will be compensated by the computer’s power. At the end of the day, a bad method will produce inconsistent and poor results.