This textbook complements but goes beyond many existing texts that introduce the concept of the finite element method and its numerous applications in virtually all engineering disciplines. The focus of this book branches out to two essential thermal science areas. These are the heat transfer and fluid flow disciplines. First, there is a total of four introductory chapters, which (combined) introduce the general finite element concept. Next, the finite element heat transfer applications are discussed. The heat transfer applications vary from the simple one-dimensional thin-rod problem all the way up to a fully three-dimensional heat-conduction problem with various boundary conditions and internal heat absorption. In the fluid mechanics part of the book, each chapter is focused on a single flow problem category varying in complexity from a simple incompressible potential flow field to the full-scale three-dimensional turbulent flow structure in complex-geometry computational domains. Complex problems such as the perturbation technique (within a finite element framework) are also addressed. Among these, the common factor is in the means with which the finite element model is uniquely adapted to overcome a historically persisting analytical challenge in this and other similar problems under the same classification. While emphasizing the problem's fluid mechanics foundation, including such aspects as the extraction of the computational domain from a larger system and the choice of sound engineering boundary conditions, an equal emphasis is placed on handling the inherent analytical challenge. Also explained in each case is the choice of a suitable finite element formulation, particularly in problems involving convection-dominated flow fields.

The finite element method is a numerical technique for obtaining approximate solutions to a wide spectrum of engineering problems. Although originally developed to study the stresses in complex airframe structures, it has since been extended and applied to a broad field in continuum mechanics. Because of its diversity and flexibility as an analysis tool, this particular technique is receiving much attention in both academia and industry fields.

Although this brief comment on the finite element method answers the question posed by the section heading, it does not give us the operational definition we need to apply the method to a particular problem. Such an operational definition, along with a description of the method fundamentals, requires considerably more than just one paragraph to develop. Hence, the first segment of this book is devoted to basic concepts and fundamental theory. Before discussing more aspects of the finite element method, we should first consider some of the circumstances leading to its inception, and we should briefly contrast it with other numerical techniques.

In more and more engineering situations today, we find that it is necessary to obtain approximate numerical solutions to problems rather than exact closed form solutions. For example, we may want to find the load capacity of a plate that has several stiffeners and odd-shaped holes, the concentration of pollutants during nonuniform atmospheric conditions, or the rate of fluid flow through a passage of arbitrary shape.

Introduction

In this chapter we combine the flow-field time dependency with a fully three dimensional solution domain. The result is a large-size computational model requiring a great deal of computer resources. In view of how involved the problem is, several CPU time-saving techniques are devised and implemented.

Example

The relative motion between the stator and rotor subdomains within an axial (Figure 15.1) or centrifugal turbomachinery stage creates an unsteady-flow field that is periodic in time. Limiting the discussion to the axial turbine stage case, the statorcascade wake pattern around the circumference (see Figure 15.1) not only will shape the rotor flow behavior but also will expose its blades to a pattern of oscillating pressure that may very well lead to premature fatigue failure. In fact, the close proximity of the two (stator and rotor) cascades (Figure 15.2) in a predominantly subsonic flow field places the stator vanes in the same fluctuating-stress environment, but with lesser amplitude by comparison.

The small stator/rotor axial-gap length within a typical turbomachinery stage is a double-edge sword. On the one hand, the smaller the gap the less is the total pressure loss within it. This loss is a natural outcome of the boundary layer growth over the endwalls, which, together with the profile losses, constitutes a significant part of the stage losses. However, a small gap length magnifies the cyclic fluctuations within the rotor subdomain as a result of wake cutting, upstream vortex shedding, and potential flow interaction between the stationary and rotating blade rows.

Introduction

During the past half century, engineering analysis has relied on the traditional finitedifference method to obtain computer-based solutions to difficult flow problems. The progress and success achieved in these pursuits have been, in many cases, noteworthy. Slow viscous flows, boundary layer flows, diffusion flows, and variableproperty flows are just some examples of areas for which analysts have developed refined calculation procedures based on the finite-difference method.

Yet there remains a number of problems for which the finite-difference methods were proven inaccurate. Problems involving complex geometries, multiplyconnected domains, and complicated boundary conditions always pose quite a challenge. Finite element methods can help in alleviating these difficulties but should not be expected to triumph in every case where the finite-difference methods have failed. Instead, the finite element methods offer easier ways to treat complex geometries requiring irregular meshes, and they provide a more consistent way of using higher-order approximations. In some cases, the finite element approach can provide an approximate solution of the same order of accuracy as the finite difference method but at less expenses. Regardless of the method used, the accurate numerical solution of most of the viscous-flow problems requires vast amounts of computer time and data storage, and of course, problems of numerical stability and convergence can occur with either method.

Only since the early 1970s has the finite element method been recognized as an effective means for solving difficult fluid mechanics problems. Literature on the application of finite element methods to fluid mechanics is rapidly increasing, with contributions being made virtually daily.

In this chapter we apply Galerkin's weighted-residual finite element approach to a special category of flow problems. This is where only the through-flow and tangential momentum equations (beside the continuity equations, of course) suffice as the flow-governing equations. This problem is perhaps best represented by the socalled quasi-three-dimensional flow field in analyzing airfoil cascades. Some terms within the finite element formulation are presented and modeled as “source” terms, in analogy with a special problem category in heat conduction. Also, implicit means are used in enforcing the cascade periodicity conditions.

Introduction

In the cascade theory discipline, the basic problem is that of a three-dimensional periodic flow in the blade-to-blade hub-to-casing passage (Figure 12.1). In modeling this flow type, it is crucial to account for such real-flow effects as boundary layer separation, flow recirculation, and trailing-edge mixing losses. Existing numerical models in this area vary in complexity from the potential flow category [1–3] to that of the fully three-dimensional viscous flow field [4, 5]. Compared with the strictly two-dimensional and three-dimensional flow models, the quasi-three-dimensional approach (which is the topic in this chapter) to the cascade flow problem has been recognized as a sensible compromise in terms of both economy and precision. It is, however, the viscous flow version of the problem, under this approach, that is in need of further enhancement, particularly in the area of simulating the hubto-casing flow interaction effects on the blade-to-blade flow field.

Overview

There are many approaches to the solution of linear and nonlinear boundary-value problems, and they range from completely analytical to completely numerical. Of these, the following deserve attention:

1. • Direct integration (exact solution):

2. • Separation of variables

3. • Similarity solutions

4. • Fourier and Laplace transformations

5. • Approximate solutions

6. • Perturbation

7. • Power series

8. • Probability schemes

9. • Method of weighted residuals (MWR)

10. • Finite difference techniques

11. • Ritz method

12. • Finite element method

For a few problems, it is possible to obtain an exact solution by direct integration of the governing differential equation. This is accomplished, sometimes, by an obvious separation of variables or by applying a transformation that makes the variables separable and leads to a similarity solution. Occasionally, a Fourier or Laplace transformation of the differential equation leads to an exact solution. However, the number of problems with exact solutions is severely limited, and most of these have already been solved.

Because regular and singular perturbation methods are primarily applicable when the nonlinear terms in the equation are small in relation to the linear terms, their usefulness is limited. The power-series method is powerful and has been employed with some success, but because the method requires generation of a coefficient for each term in the series, it is relatively tedious. It is also difficult, if not impossible, to demonstrate that the series converges.

Overview

Of the different problem categories in the remainder of this text, this is the simplest and, appropriately, a good starting point. A potential flow field is one where a single field variable suffices and a single flow-governing equation applies. This variable has typically been chosen as either the stream function ψ or the velocity potential ϕ. This apparent simplicity, nevertheless, may (in the larger picture) underestimate the critical role a potential-flow code often plays in a typical cascade-design setting, as well as the inherent analytical difficulty in securing a single-valued flow solution in a multiply-connected domain, with the latter being the focus of this chapter.

Beginning as early as the 1930s, several methods were devised for the problem of potential flow past a cascade of lifting bodies. Some of these methods were based on the use of conformal transformation [1–5], where one or more transformation step(s) are used in mapping the computational domain into a set of ovals or a flatplate cascade [4, 5]. A separate category of analytical solutions [6] is based on the so-called singularity method, whereby sequences of sources and sinks and/or vortices are used to replace the airfoil itself. Next, the streamline-curvature method was established [7] as a viable approach to the airfoil-cascade-flow problem. With advent of the computer revolution came several numerical models of the problem based on the finite-difference method [9], finite element [9–12], and finite-volume [13] computational techniques.

From an analytical viewpoint a flow passage will conceptually suffer one level of multiconnectivity at any point where two streams with two different histories are allowed to mix together.

Evaluation of the surface integrals in the problem of three-dimensional (3D) heat conduction corresponding to the different boundary conditions requires the discretization of the body surfaces into surface finite elements that take the form of surface triangles. A set of natural coordinates would be advantageous in defining these elements, especially if these surfaces are curved.

The natural coordinates are local coordinates that vary in a range between zero and unity. At any of the element's vertices, one of these coordinates has a value of unity, whereas the others are all zeros. Use of these coordinates simplifies the evaluation of integrals in the element's equations. This additional advantage is a consequence of the existing closed-form integration formulas that evaluate these integrals.

The derivation given in this appendix generalizes the natural coordinates' definition for two-dimensional (2D) plane elements, all lying in one plane, to the case in which these plane elements exist in a 3D space. Such generalization was essential because the elements dealt with in the analysis lie on the 3D body-surface segments, which are, in turn, 3D.

Let A represent the area of the triangular element in Figure A.1, with i(xi, yi, zi),j(xj, yj, zj) and k(xk, yk, zk), denoting its vertices.

This model upgrade is intended to embrace a variety of high-speed noncontact seals and bearings traditionally used in gas turbine applications. In this case, the heatenergy exchange between the hardware and the working medium may be far from being negligible. This very fact calls for including a separate energy equation in the set of flow-governing equations cited in Appendix E. The boundary conditions on insertion of this equation involve such variables as the local convection heat transfer coefficient and the local “wall” and flow-stream temperatures.

Given the fact that the rotor-to-housing clearance width is extremely small, the problem of friction choking (in a Fanno-process type of mechanism) is indeed part of this compressible flow problem. Unfortunately, the occurence of this choking type requires external intervention during the flow solution process. During the iterative procedure, where the momentum-equations convection terms are continually linearized, and once the nodal magnitudes of velocity vector are attained, the intervention process begins by computing the corresponding nodal magnitudes of Mach number. These are then examined to see if the Mach number is in excess of unity anywhere in the computational domain (the seal exit station in particular), which is impossible in a subsonic nozzle-like passage. Referring to the simple annular seal in Figure 16.20, the term nozzle here is applicable in the sense that the blockage effect of the boundary layer growth over the solid walls causes, in effect, a streamwise reduction in the cross-flow area, turning what is physically a constant-area passage into a subsonic nozzle in the sense of rising displacement thickness (a fraction of the boundary layer thickness that depends on the boundary-layer velocity profile).

Introduction

Problems in real life do not regularly come in the form of a “given” computational domain in which to solve the flow-governing equations. The fact of the matter is that we are given a large flow domain, and we are interested in finding the flow behavior over just a subregion of it. In separating our subdomain, we should be careful to add the effect of the remainder of the bigger system to our subdomain of interest. Perhaps one of the most illustrative examples of such a situation is the seal segment in the secondary (or leakage) flow passage in the the pump stage shown in Figure 14.1. Note that our focus is on this secondary passage and, in particular, the seal part of it.

Leakage flow in the shroud-to-housing gap of centrifugal pumps has significant performance and rotor-integrity consequences. First, it is the leakage flow rate, as determined by the through-flow velocity component, that is typically a major source of the stage losses. The swirl velocity component, on the other hand, is perhaps the single most predominant destabilizing contributor to the impeller rotordynamic behavior [3]. Control of the through-flow velocity in the clearance gap is often achieved through use of a tight-clearance seal. Suppression of the flow swirl, however, requires a careful design of the leakage passage and/or the use of such devices as the so-called swirl brakes (e.g., [7, 13]) or straightening grooves/ribs in the inner housing surface (e.g., [12]). Unfortunately, an efficient leakage-control device, such as the labyrinth seal, may itself trigger the instability problem of fluid-induced vibration [8].

Because the whirl frequency is already a built-in parameter in the zeroth-order flow problem, it takes only a lateral eccentricity of the rotor axis for complete whirl excitation to materialize. It is important to understand the physical aspects of the flow field under such disturbance.

Consider the case where an observer is “attached” to the origin of the whirling frame of reference (see Figure 16.8) and is whirling with it. To this observer, the distortion of the flow domain is the result of an upward displacement of the housing (and not the rotor) surface. To the same observer, all nodal points in the original finite element discretization model will undergo varied amounts of upward displacement, except for nodes on the rotor surface which will remain undisplaced. As a result, varied amounts of geometric deformations will be experienced by the entire assembly of finite-elements in the rotor-to-housing flow passage (see Figure 16.8). Finally, the observer, whose rotation is at the rate of the whirl frequency Ω, will register a “relative” linear velocity of the housing surface whose magnitude is Ωrh, where rh is the local radius of the housing inner surface at this particular axial location, as shown in Figure 16.8.

With no lack of generality, consider the rotor position in Figure 16.13, where the nodes of the typical element (e) have been displaced by different amounts. The nodal displacements, as shown in the figure, depend on the original ycoordinate of each individual node.