Two-Dimensional Cascades
Introduction
The design and performance prediction of axial flow compressors and turbines has been based, in the main, upon measurements of the flow-through two-dimensional cascades of blades. However, to an increasing extent, computational fluid dynamic (CFD) methods are now being used to simulate cascade testing. The flow within a turbomachine is, in general, unsteady and three dimensional. For cascade analysis, the flow across individual blade rows is treated as two dimensional and steady. This approach is appropriate for many compressor and turbine designs and the derived flow characteristics obtained from cascade tests have usually been found to be satisfactory, although laborious to collect.
Reviews of the many types of cascade tunnels, which includes low-speed, high-speed, intermittent blowdown, suction tunnels, are available in the literature, e.g., Sieverding (1985), Baines, Oldfield, Jones, Schulz, King, and Daniels (1982), and Hirsch (1993). The range of Mach numbers in axial flow turbomachines can be considered to extend from M 5 0.1 to 2.5:
i. low speed, operating in the range 20-60 m/s;
ii. high speed, for the compressible flow range of testing.
A typical low-speed, continuous running cascade tunnel is shown in Figure 3.1(a). This linear cascade of blades comprises a number of identical blades, equally spaced and parallel to one another. Figure 3.1(b) shows the test section of a cascade facility for transonic and moderate super- sonic inlet velocities. The upper wall is slotted and equipped for suction, allowing operation in the transonic regime. The flexible section of the upper wall allows for a change of geometry so that a convergent-divergent nozzle can be formed, allowing the flow to expand to supersonic speeds upstream of the cascade.
It is most important that the flow across the central region of the cascade blades (where the flow measurements are made) is a good approximation to two-dimensional flow and that the flow repeats (i.e., is periodic) across several blade pitches. This effect could be achieved by employing a large number of long blades, but then an excessive amount of power would be required to oper- ate the tunnel. With a tunnel of more compact size, aerodynamic difficulties become apparent and arise from the tunnel wall boundary layers interacting with the blades. In particular, and as illustrated in Figure 3.2(a), the tunnel wall boundary layer merges with the end blade boundary
layer, and as a consequence, this blade usually stalls, resulting in a nonuniform flow across the cascade.
In a compressor cascade, the rapid increase in pressure across the blades causes a marked thickening of the wall boundary layers and produces an effective contraction of the flow, as depicted in Figure 3.3. A contraction coefficient, used as a measure of the boundary layer growth through the cascade, is defined by ρ1c1 cos α1/(ρ2c2 cos α2). Carter et al. (1950) quoted values of 0.9 for a good tunnel dropping to 0.8 in normal high-speed tunnels and even less in bad cases.
These are values for compressor cascades; with turbine cascades higher values can be expected, since the flow is accelerating and therefore the boundary layers will not be thickened.
Because of the contraction of the main through-flow, the theoretical pressure rise across a compressor cascade, even allowing for losses, is never achieved. This will be evident since a contrac- tion (in a subsonic flow) accelerates the fluid, which is in conflict with the diffuser action of the cascade.
To counteract these effects, it has been customary (in Great Britain) to use at least seven blades in a compressor cascade, each blade having a minimum aspect ratio (blade span/chord length) of 3.
With seven blades, suction is desirable in a compressor cascade, but it is not usual in a turbine cascade. In the United States, much lower aspect ratios have been commonly employed in compressor cascade testing, the technique being the almost complete removal of tunnel wall boundary layers from all four walls using a combination of suction slots and perforated end walls to which suction is applied. Figure 3.2(b) illustrates the effective application of suction to produce a more uniform flow-field.
For axial flow machines of high hub-tip radius ratios, radial velocities are negligible and the flow may be described as two dimensional. The flow in the cascade is then likely to be a good model of the flow in the machine. With lower hub-tip radius ratios, the blades of a turbomachine will normally have an appreciable amount of twist along their length and a varying space-chord ratio. In such cases, a number of cascade test measurements can be applied to cover the design of the blade sections at a number of radial locations. However, it should be emphasized that, in all cases, the two-dimensional cascade is a simplified model of the flow within a turbomachine, which in reality can include various three-dimensional flow features. For sections of a turbomachine where there are separated flow regions, leakage flows or significant spanwise flows, the cascade model will not be accurate and careful consideration of the three-dimensional effects is required. Further details of three-dimensional flows in axial turbomachines are given in Chapter 6.
Cascade geometry
A cascade blade profile can be conceived as a curved camber line upon which a profile thickness distribution is symmetrically superimposed. In Figure 3.4, two blades of a compressor cascade are shown together with the notation needed to describe the geometry. Several geometric parameters that characterize the cascade are:
Further parameters that are needed to describe the cascade blade shape include its camber line shape, thickness distribution, the radii at the leading and trailing edges, and the maximum thickness to chord ratio, tmax/l.
The camber angle, θ, is the change in angle of the camber line between the leading and trailing edges that equals α0 2 α0 in the notation of Figure 3.4. For circular arc camber lines, the stagger angle is ξ 5 ð1=2Þðα0 1 α0 Þ. The change in angle of the flow is called the deflection, ε 5 α1 2 α2, and in general this will be different to the camber angle due to flow incidence at the leading edge and deviation at the trailing edge. The incidence is the difference between the inlet flow angle and the blade inlet angle:
Compressor blade profiles
The modern approach in compressor design is to use blade profiles designed by the so-called prescribed velocity distribution (PVD) method. In this approach, the designer will select a blade surface velocity distribution and a computational method determines the aerofoil thickness and curvature variation required to achieve the desired aerodynamics. Despite this, many blade designs are still in use based upon geometrically prescribed profiles. The most commonly used geometric families are the American National Advisory Committee for Aeronautics (NACA) 65 Series, the British C Series, and the double circular arc (DCA) or biconvex blade.
The NACA 65 Series blades originated from the NACA aircraft wing aerofoil and were designed for approximately uniform loading. Figure 3.5 compares the profiles of the most widely used blade sections drawn at a maximum thickness to chord ratio of 20%, for the purpose of clarity. In fact, the maximum t/l ratios of compressor blade sections are nowadays normally less than 10%
and often a value of 5% is used because of the superior high Mach number performance that can be achieved with thinner blades. The NACA 65 Series has its maximum thickness at 40%, whereas the C Series is at 30% and the DCA Series is at 50%. These differences have a marked effect on the velocity distributions measured around the blades surfaces. Aerofoils with the maximum thick- ness near the leading edge and, consequently, with a well rounded leading edge have a wide operating range but a poorer high speed performance than blades with a sharp leading edge and the maximum thickness point further back.
The exact details of the different profiles are very well documented, e.g., Mellor (1956), Cumpsty (1989), Johnson and Bullock (1965), Aungier (2003), and it is not thought useful or nec- essary to reproduce these in this book.
The actual blade shape is defined by one of these profile shapes superimposed on a camber line. This can be a simple circular arc although, as shown by Aungier (2003), a parabolic arc allows a more flexible style of blade loading. The blade profile is laid out with the selected scaled thickness distribution plotted normal to the chosen camber line. Correlations for the performance of the dif- ferent styles of compressor aerofoil are discussed within Section 3.5 later in this chapter.
Turbine blade profiles
The shape of turbine blades is less critical than it is in a compressor cascade. However, the designer still needs to exercise some care in the selection of blades to attain good efficiency with highly loaded blade rows. Nowadays, the process of specifying blade row geometry (blade shape, flow angles, and space-chord ratio) is accomplished by computational methods but, ultimately, the designs still need to be backed up by cascade tests. Figure 3.6 shows a photograph of a typical high-speed turbine cascade that is used to represent the aerofoils of a conventional low-pressure turbine within an aero engine. The blade profiles illustrate the high turning and the contraction of the passage flow area within a turbine blade row.
During the early design phase of a turbine, or when cascade results are unavailable, one-dimensional calculations and correlation methods can be used to estimate the blade row performance of turbine blade rows. These are discussed within Section 3.6.