The pressure fluctuations generated by propellers in the wake of hulls are markedly different from those produced in uniform inflow. The flow in the propeller plane abaft a hull varies spatially as well as temporally. Here we deal only with the effects attending spatial variations peripherally and radially as provided by wake surveys which give the averaged-over-time velocity components as a function of r and γ for a fixed axial location. Temporal variations in the components are aperiodic and cannot be addressed until sufficient measurements have been made to determine their frequency spectra. Ultimately, numerical solutions of the Navier-Stokes equations may provide both spatial and temporal aspects of hull wakes.

Here the spatial variations in the axial and tangential components are reflected in the pressure jump Δp which is taken to vary harmonically with blade position angle γ0. Then we discover a coupling between the harmonics of Δp(γo) and the harmonics of the propagation function yielding a plethora of terms all at integer multiples of blade frequency. Graphical results are given for pressure and velocity fields showing the effect of spatial non-uniformity of the inflow.

We have seen in the previous chapter that the pressure field arising from a lifting-surface model of a propeller in a uniform flow is that due to pressure and velocity dipoles distributed over the blade. Both dipole strengths were constant in time since we considered uniform and stationary inflow.

It is well known that the flow abaft of ships is both spatially and temporally varying. This variability arises from the “prior” or upstream history of the flow produced by the action of viscous stresses and hull-pressure distribution acting on the fluid particles as they pass around the ship from the bow to the stern. Thus the blade sections “see” gust patterns which over long term have mean amplitudes but from instant-to-instant change rapidly with time because of the inherent unsteadiness of the turbulent boundary layer.

Our knowledge of the distribution of flow in the propeller disc is almost entirely based on pitot-tube surveys conducted on big (≈ 6 m) models in large towing tanks and in the absence of the propeller. These are termed nominal wake flows. As is well appreciated, pitot-tube measurements provide only long-term averages of the velocity components at various angular and radial locations in the midplane of the propeller. These measurements depend upon the calibration of the pitot-tube in uniform flow whereas the wake flow radially and tangentially has the effect of shifting the stagnation point on the pitot-tube head, a mechanism not operating in the calibration mode. Thus there is a systematic error which is, to the authors' knowledge, not generally corrected. Moreover, wake-fraction (and thrust-deduction) calculations based upon tests with the same model in several large model basins and upon repetitive tests with the same model in the same large model basin, have shown remarkably different results. A similar scatter was also found in results of wake surveys.

Abstract

This article is for those who have already a computer program for incompressible viscous transient flows and want to put a turbulence model into it. We discuss some of the implementation problems that can be encountered when the Finite Element Method is used on classical turbulence models except Reynolds stress tensor models. Particular attention is given to boundary conditions and to the stability of algorithms.

Introduction

Many scientists or engineers turn to turbulence modeling after having written a Navier-Stokes solver for laminar flows.

For them turbulence modeling is an external module into the computer program. Generally, the main ingredients to built a good Navier-Stokes solver are known; this includes tools like mixed approximations for the velocity u and pressure p to avoid checker board oscillations and also upwinding to damp high Reynolds number oscillations; however the problems that one may meet while implementing a turbulence model are not so well known because these models have not been studied much theoretically.

Judging from the literature [3] [11] [12] [15] [19] [22] the most commonly used turbulence models seem to be

• algebraic eddy viscosity models (zero equation models)

• k= ε models (two equations models)

• Reynolds stress models

All three start from a decomposition of u and p into a mean part and a fluctuating part u’. However oscillations are understood either as time oscillations or space oscillations or even variations due to changes in initial conditions. In any case, the decomposition u+u’ is applied to the Navier-Stokes equations.

Introduction

This article contains a summary account of covolume methods for incompressible flows. Covolume methods are a recently developed way to solve both compressible and incompressible flow problems on unstructured meshes. The general idea is to use complementary pairs of control volumes to discretize flux, circulation and other expressions which occur in the governing equations. These complementary volumes (covolumes for short) are related by an orthogonality property which is a basic feature of the covolume approach. One of the simplest mesh configurations which is suitable is the Delaunay- Voronoi mesh pair. This is introduced in the next section. After that we proceed through div-curl systems to the stationary Stokes equations and the Navier-Stokes equations. We will show that for uniform meshes the covolume equations for the stationary Stokes equations specialize to the MAC (staggered mesh) scheme, and that the MAC scheme itself is actually equivalent to a velocity-vorticity scheme. Some numerical results are presented in the last section.

Since this article is intended only as an overview, we will present most of the results in a two dimensional setting. Almost all of the ideas and techniques do generalize nicely to three dimensions but are harder to visualize than in two dimensions. Given our limited aims it would be inappropriate to present proofs of most of the mathematical results. We will refer to the original sources for these and other details.

One of the reasons for introducing covolume methods is to find lower order methods for viscous flows which are free of “spurious mode” problems.

Summary

Our object in this article is to present some aspects of new emerging methods in numerical analysis and their application to Computational Fluid Dynamics. These methods stem from Dynamical Systems Theory.

The attractor describing a turbulent flow is approximated by smooth manifolds and by projecting the Navier-Stokes equations onto these manifolds we obtain new algorithms, the Inertial Projections. These algorithms have proven to be stable, efficient and well suited for long time integration of the equations. They can be implemented with all forms of spatial discretizations, spectral methods, finite elements and finite differences (possibly also wavelets).

Introduction

Algorithms that have been introduced at a time of scarce computing resources may not be well adapted to supercomputing and to the more difficult problems that are tackled at the present time or that we foresee for the near future.

For incompressible fluid mechanics (or thermal convection), by using the full Navier-Stokes equations (NSE) we can, at present time, compute flows at the onset of turbulence: in particular the permanent regime is not anymore a stationary flow. The flow can be time periodic if a Hopf bifurcation has occurred or the permanent regime can be an even more complicated flow. Examples of time dependent flows have been numerically computed in the case of the driven cavity (see e.g. [15], [16], [20], [4] and [34]) and for other types of flows (see e.g. [14], [33] and [30]). Section 12.2 of this article is a brief survey, for the CFD practitioner, of some basic and relevant concepts in Dynamical Systems Theory (behavior for large time of the solutions of the NSE).

Introduction

Physically the Stokes equations model “slow” flows of incompressible fluids or alternatively isotropic incompressible elastic materials. In Computational Fluid Dynamics, however, the Stokes equations have become an important model problem for designing and analyzing finite element algorithms. The reason being, that some of the problems encountered when solving the full Navier-Stokes equations are already present in the more simple Stokes equations. In particular, it gives the right setting for studying the stability problem connected with the choice of finite element spaces for the velocity and the pressure. It is well known that these spaces cannot be chosen independently when the discretization is based on the “Galerkin” variational form. This method belongs to the class of saddle-point problems for which an abstract theory has been developed by Brezzi [1974] and Babuska [1973]. The theory shows that the method is optimally convergent if the finite element spaces for velocity and pressure satisfy the “Babuska-Brezzi” or “inf-sup” condition. In computations the violation of this condition often leads to unphysical pressure oscillations and a “locking” of the velocity field, cf. Hughes [1987]. During the last decade this problem has been studied thoroughly and various velocity-pressure combinations have been shown to satisfy the Babuska-Brezzi condition. Unfortunately, however, it has turned out that many seemingly natural combinations do not satisfy it. (See Girault and Raviart [1986], Brezzi and Fortin [1991], and references therein.)

In this chapter we will review a recent technique of “stabilizing” mixed methods. In this approach the standard Galerkin form is modified by the addition of mesh-dependent terms which are weighted residuals of the differential equations.

Abstract

A numerical algorithm for enforcing the conservation of mass in incompressible flow simulation is discussed and the details of the implementations in terms of standard finite volumes and finite elements are given. It is also demonstrated that both segregated and coupled iterative techniques are applicable and numerical results of test cases for two and three dimensional cavity flows are presented.

Introduction

In many applications, numerical simulations of three dimensional incompressible flows are needed. The problem of satisfying exactly the continuity equation, for these flows, is well known [1] [2].

It is conceivable to update the velocity field using the momentum equations but it is not clear how to update the pressure to conserve mass, since no pressure term appears in the continuity equation. Various methods have been introduced to tackle this problem including the penalty method, the artificial compressibility method, the artificial viscosity methods, the projection and the pressure correction methods. (The discussion, here, is limited to methods based on primitive variables. The vector potential and/or velocity vorticity formulations are not covered, see for example the work of Osswald, Ghia & Ghia [19] which still requires staggered orthogonal grids to enforce mass conservation.)

In the primitive variable methods, either staggered grids are used or the continuity equation is modified. For example in the penalty method of Temam [3], a small term proportional to the pressure is added to the continuity equation, while in Chorin's artificial compressibility method [4], the continuity equation is modified by an artificial time dependent term proportional to the time derivative of the pressure.

Numerical methods for incompressible fluid dynamics have developed to the point at which a survey of the field is both timely and appropriate. A major stimulus to the field has been the large number of applications in which incompressible flows play a crucial role, and this has spurred the interest of numerous computational engineers and mathematicians. The articles which follow provide a reasonably broad view of algorithmic and theoretical aspects of incompressible flow calculations.

It should be noted at the outset that it can be dangerous to define an algorithm for simulating incompressible flows by setting, for example, the density to be constant in a successful compressible flow algorithm. The nature of the pressure as a Lagrange multiplier rather than as a thermodynamic variable as well as the infinite speed of propagation of disturbances and other factors peculiar to incompressible flows, make algorithmic development and implementation in this context a unique undertaking (see Appendix 7A).

Perhaps the first major advance in the application of large scale digital computation to incompressible flows occured in the late 1950s with the introduction of staggered mesh techniques, exemplified, for example, by the Marker-and-Cell (MAC) scheme. The use of staggered meshes in the context of the primitive variable formulation was found to provide a stable discretization of the incompressibility constraint. Shortly thereafter, it was realized that the use of staggered meshes could be avoided by employing the streamfunction-vorticity formulation in which the incompressibility constraint does not explicitly appear. Numerous finite difference algorithms were proposed and used based on this formulation of the Navier-Stokes equations.

Introduction

The rapid development and introduction of new supercomputer systems over the last decade has opened new opportunities for numerical studies of incompressible fluid flows. A new awareness has also developed that the emerging hardware technologies influence both the nature and implementation of effective algorithms to solve these problems. Specific software implementation issues concern such varied questions as code dependencies, locality, round-off errors, storage access and capacity, input-output, workstation interfaces, cache utilization, . . . and are affected significantly by such computer hardware characteristics as the graininess of parallel systems, vector length, instruction conflicts, instruction set design, network access, distribution and paging of memory, to mention but a few. One significant result of this complex environment has been the stimulation of new ideas to make optimal use of the new supercomputer architectures and to achieve both high accuracy and high computational efficiency in the fluid simulations.

In this paper, we shall review some novel methods that are especially well suited for various aspects of incompressible fluid flow simulation studies. The key dynamical feature of these flows is the absence of shock waves, so many of the results to be stated for incompressible flows should also carry over to shock-free flows at moderate Mach numbers. Even within the context of incompressible flows, there is a wealth of dynamical phenomena to be studied, including laminar flows, transition to turbulence, turbulence, free surface flows, heat transfer, particle transport, fluid-structural interactions, and multiphase flows, among other phenomena.

Introduction

The control of fluid motions for the purpose of achieving some desired objective is crucial to many technological applications. In the past, these control problems have been addressed either through expensive experimental processes or through the introduction of significant simplifications into the analyses used in the development of control mechanisms. Only recently have flow control problems been addressed, by scientists and mathematicians, in a systematic, rigorous manner. This interest is quickly expanding so that, at this time, flow control is becoming a very active and successful area of inquiry. For example, recent publications, e.g., [1] [28], provide analyses of various aspects of flow control problems, and include one or more of the following components:

• the construction of mathematical models, invoking minimal assumptions about the physical phenomena;

• the analysis of the mathematical models to answer questions about the existence and regularity of solutions and to derive necessary conditions that optimal controls and states must satisfy;

• the construction and analysis of discretization methods for determining approximate solutions of the optimal control problems, and the rigorous derivation of error estimates; and

• the development of computer codes implementing discretization algorithms, both for the purpose of showing the efficacy of these methods, and also to solve problems of practical interest.

An optimal control or optimization problem is composed of two ingredients: a desired objective and control mechanisms that are used to (hopefully) achieve the desired objective. In a mathematical description of such problems, the desired objective is usually expressed in terms of the extremization of a functional depending on the state of the system, and possibly also on the control mechanisms.

Introduction

The finite element method has been an established method for approximating incompressible flow for a number of years. The primitive variable, velocity/pressure formulation, is the most popular way to implement the method although it is not the only possibility. Much theoretical work has been done to establish convergence and error estimates and there is a large amount of literature on the topic, see for example Temam [1984], Thomasset [1981], Girault and Raviart [1986], Gunzburger [1989]. Effort has been concentrated on two-dimensional flow and although mathematically three-dimensional flow is no more difficult, in practice the current state of computer hardware makes the implementation of three-dimensional elements much more problematic. It is only the availability of modern supercomputers that has allowed the approximation of such flows to be attempted. However, an element and method of solution that works well on a large vector processor may be quite inefficient on a fine-grained parallel computer thus the concept of the “best element” or even a “good element” may be highly dependent on the computer on which it is to be implemented.

Most methods of solution involve at least one iteration of one form or another, the innermost loop being the solution of a set of linear equations. For practical three dimensional flow problems a direct method of solution is unlikely to be a feasible proposition for almost all situations and this inner system of equations will have to be solved iteratively. “How accurately do we need to solve this system?” and “What are the interactions between the various iterations taking place?” are two questions that have to be faced by anyone implementing the finite element method for three-dimensional flow.

Introduction

This book represents a timely review of state-of-the-art algorithms in the many branches of Computational Fluid Dynamics (CFD). Algorithms for a wide range of incompressible flow phenomena are presented ranging from highly turbulent flows to low-speed non-Newtonian flows. Despite the enormous advances of the past ten years it is clear that there are still many interesting new Algorithmic directions evolving.

Much of the material presented herein is from the perspective of the algorithm researcher whose ultimate goal is the development of viable algorithms for the accurate and efficient simulation of real-world CFD problems. In this chapter, I would like to take the opportunity to discuss CFD from a different perspective - that of a commercial CFD package developer and more importantly that of the user of CFD codes. For, in the final analysis, it is the user who must apply the algorithms developed by researchers and packaged in commercial CFD codes. Over the last ten years CFD has evolved to the point where commercial CFD codes have been developed and introduced into the marketplace. These codes are being used with increasing frequency and it is probably fair to state that today the use of CFD in industry is increasingly becoming an accepted analysis tool.

Over the last decade the application of CFD has spread from its original aerospace beginnings to increasing application in a broad spectrum of industries ranging from the more traditional automotive and electronics industries to exciting new applications in the biomedical and food industries. In these industries CFD is still not used in a truly design environment; however, the real payback from the use of CFD will come when it is used on a daily basis as a design tool by design engineers who are not CFD specialists.

Introduction

Vortex methods are a type of numerical method for approximating the solution of the incompressible Euler or Navier-Stokes equations. In general, vortex methods are characterized by the following three features.

1. The underlying discretization is of the vorticity field, rather than the velocity field. Usually this discretization is Lagrangian in nature and frequently it consists of a collection of particles which carry concentrations of vorticity.

2. An approximate velocity field is recovered from the discretized vorticity field via a formula analogous to the Biot-Savart law in electromagnetism.

3. The vorticity field is then evolved in time according to this velocity field.

In the past two decades a number of different numerical methods for computing the motion of an incompressible fluid have been proposed that have the above features. In this article we consider a class of such methods which are based on the work of Chorin [1973, 1978, 1980, and 1982]. Members of this class are related by the manner in which a vorticity field in an inviscid, incompressible flow is discretized and subsequently evolved. It is common practice to use the term vortex method or the vortex method to refer to a member of this class when it is used to model the incompressible Euler equations. One can modify the vortex method and use it to model the incompressible Navier-Stokes equations by adding a random walk. This is known as the random vortex method. One can also replace the random walk by some non-random technique for solving the diffusion equation. Such methods are generally referred to as deterministic vortex methods.