The development of the governing equations in an axisymmetric natural coordi-nate system presented in this chapter requires use of several standard vector operators. Appendix A of Vavra (1960) provides detailed derivations of these operators. Here, the various operators are presented for reference purposes, without derivation. The gradient of any function φ is given by
(3-61)
where →e is a unit vector. The divergence of any vector →V is given by
(3-62)
The curl of any vector →V is given by
(3-63)
The Laplacian of any function φ is given by
(3-64)
In evaluating the convective derivative in Eq (3-15), the following vector identity is useful.
(3-65)
Equation (3-65) is usually written as
(3-66)
where V is the magnitude of the vector →V as given by the dot product.
EXERCISES
3.1 Consider time-steady flow in a rotating coordinate system. Use Eqs.
(3-28) through (3-30) to analyze the flow at stations outside of the (r r r) r r (r r)
blade row passages where the flow can be considered to be axisym-metric, but may have a tangential velocity component. Develop equa-tions governing the variation of Cθ, s and I on stream surfaces.
3.2 For the flow analysis in Exercise 3.1, modify Eq. (3-30) to consider cases where the relative flow angle, β′, is known for all stream sur-faces, where tanβ′ = Wθ / Wm. Repeat for cases where the absolute flow angle, β, is known for all stream surfaces where tanβ = Cθ/ Cm. 3.3 For Eq. (3-66), express V in terms of its components in the three
coor-dinate directions. Derive an equation for 1–
2
→∇V2in terms of V and its derivatives.
3.4 Consider one-dimensional, time-steady flow in a simple annular pas-sage, i.e., κm= 0 and all gradients with respect to n and θ are identi-cally zero. The passage width, b(m), is a function of the meridional coordinate. Derive a set of governing equations for this problem from Eqs. (3-21) through (3-27).
3.5 Consider time-steady one-dimensional flow at the exit of a simple annular passage, with two identical boundary layers on the end-walls.
The boundary layer parameters θ, δ*and δ and the inviscid core flow data ρ, u and P at the passage exit are known. The flow is incompress-ible, i.e., ρ is constant and Pt= P + 1/2ρu2. Develop expressions for the exit mass and momentum flow in terms of the boundary layer and inviscid core flow parameters.
3.6 Assume that the boundary layer flow and inviscid core flow in Exer-cise 3.5 mix instantaneously into a uniform flow with no change in static pressure. By requiring conservation of mass and momentum, show that the total pressure loss between the inviscid core flow and the fully mixed flow is given by
∆Pt=12ρe eu2[(2δ*/ )b2+4θ/ ]b
The traditional approach to axial-flow compressor aerodynamic design was to use various families of airfoils as the basis for blade design. American practice was based on various families defined by the National Advisory Committee for Aeronautics (NACA), the most popular being the 65-series family. British practice often centered about the C-series families, using circular-arc or parabolic-arc camberlines. As design requirements began to favor transonic operation, double-circular-arc blades became popular. The performance characteristics of these air-foil families are well understood due to extensive experimental cascade testing, much of which is available in the literature.
In recent years, use of blades designed for a prescribed surface velocity distri-bution or blade loading style, instead of for predefined airfoil families has become popular. Often, inverse design methods that predict the blade shape required for the desired blade loading are used. As the relation between blade shape and preferred loading styles became better understood, it also became common to use conventional or direct analysis methods in a trial-and-error mode to arrive at the same result. These airfoils have been referred to as prescribed velocity distribution (PDF) blades (Cumpsty, 1989), even though the term con-trolled diffusion airfoils is probably more common today. Although the literature offers general guidelines for these designs, the actual airfoil designs in use are proprietary. In general, the performance characteristics of these airfoils are well known only to the organizations that developed them.
As discussed in the preface to this book, this situation posed a significant com-plication to the goal of providing a complete description of the working design and analysis system. It was quickly recognized that it is no longer possible to write a book that can be directly applied to all of the many proprietary designs in use today. But this is not considered to be a serious limitation. In this writer’s experience, the process of adapting classical blade performance prediction mod-els to a more modern controlled diffusion airfoil design is not particularly diffi-cult, assuming the performance characteristics of the airfoil are known.
This chapter provides a complete description of the more commonly used tra-ditional airfoil families, and Chapter 6 provides a detailed description of the per-formance modeling for these same airfoil families. This ensures that this book is at least complete in the context of classical axial-flow compressor technology.
Chapter 4
AXIAL-FLOW COMPRESSOR
BLADE PROFILES
General concepts from the literature used to guide the development of controlled diffusion airfoils are also briefly reviewed in this chapter.
NOMENCLATURE
a = distance along chord to the point of maximum camber
b = distance normal to the chord line to the point of maximum camber Cl0 = isolated airfoil lift coefficient
c = chord length
d = length defined in Eq. (4-24) i = incidence angle
o = blade throat opening
R = circular-arc radius of curvature s = blade pitch (spacing)
tb = blade maximum thickness x = coordinate along the chord y = coordinate normal to the chord yC = y coordinate at the origin of RC
α = angle of attack
β = flow angle relative to axial direction χ = blade angle relative to the chord line δ = deviation angle
γ = stagger (setting) angle
κ = blade angle relative to the axial direction θ = camber angle
σ = solidity
φ = parameter defined in Eq. (4-30)
Subscripts
C = camberline parameter
L = blade lower or pressure surface parameter U = blade upper or suction surface parameter
1 = blade leading edge parameter 2 = blade trailing edge parameter
4.1 CASCADE NOMENCLATURE
Figures 4-1 and 4-2 illustrate the basic parameters used to describe axial-flow com-pressor blades and cascades. Blades are defined by a mean camberline, y(x), upon which a profile or thickness distribution, tb(x), is imposed. The angles between slopes to the camberline and the chord line at the leading and the trailing edges are designated as χ1and χ2, respectively. The blade camber angle is defined as
(4-1) θ χ= 1+χ2
Axial-Flow Compressor Blade Profiles • 61
FIGURE4-1 Basic Airfoil Geometry
FIGURE4-2 Basic Cascade Geometry
The pitch or the spacing between adjacent blades, s, and the chord length, c, define the cascade solidity, σ, by