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.

Overview

Since its inception, the finite element method has followed the tracks of other well established computational techniques, particularly the finite-difference method. The fluid mechanics applications remained limited to mostly steady-state flow applications in two-or three-dimensional domains, with one or more complicating real-flow effects (e.g., compressibility, turbulence, etc.) being part of the computational model. Differing in complexity and accuracy, the finite-elements themselves have been taken as fixed-geometry subdomains, and this is the critical point where the following analysis categorically differs.

In many real-life applications, the flow domain itself undergoes small timedependent changes (or distortions) that, in most cases, are periodic. The problem of wing flutter is an example of such a situation in the external aerodynamics discipline. The problem under focus here involves the vibration of a fluid-encompassed rotor in a confined-flow type of arrangement (Figure 16.1) and is known to have a major impact on the system's rotordynamic integrity.

Originally handled via finite-difference techniques, the solution strategy was to repeatedly solve the entire physical problemby marching in time while slightly altering the flow-domain geometry at each time level. Tedious as it was, this approach is hardly economical, nor is it based on prior knowledge of what controls the time increment and, often, what fluid/structure features are to be monitored. Prohibiting solid advances in addressing the problem, many believe, is that it was historically handled in either a fluid-dynamics or a mechanical-vibrations type of approach but not a combination of the two mentalities.