The low pressure (LP) fan and high pressure (HP) compressor are designed in a similar manner. These both suffer from design problems that are exacerbated by low hub/tip (h/t) ratio blades. One such problem is blade twist. We wanted to compare four vortex theory models, including Mixed-Vortex swirl distribution and three constant-work models.

Figure 1: Large blade twist in a two-stage LP fan rotor (Solid Edge ST9, Keyshot 6).

# Blade Stall and de Haller

The fluid (air) in the compressor is energized by the rotor blades. Each blade is curved to provide the proper airflow at each point along the height of the blade. If the blade’s shape is not properly matched to the RPM, radius, and mass flow of the compressor, the airflow can actually leave the surface of the blade. This phenomenon is called stall, and would be rather unfortunate if experienced during a flight. For decades, scientists have studied what makes a blade stall. Various design factors (e.g. diffusion, de Haller number, etc.) offer a clear indication of stall. In 1953, P. de Haller observed that the ratio of incoming and outgoing velocities along the airfoil must not be less than 0.72. While the de Haller number is not the defacto reference to how the air is behaving, it provides a wonderful, early indication when things are going wrong.

# Swirl Distribution

Blade geometry in a rotating compressor is rather complex. Each point along the blade experiences different forces. This is because the further you travel along the blade, away from the hub, the faster the blade is moving. One method of planning compressor blade geometry is to design the system at the mean line, in some cases, the middle of the blades. Blade angles are derived from required velocities of the energized fluid surrounding the moving blades. Once the basic mean blade angles are known, the remainder of the angles along its length must be adjusted to compensate for difference in radius, from hub to tip. Various methods of adjusting the blade angles have been developed. The mathematics of doing this and maintaining radial equilibrium is called swirl distribution and vortex theory.

One such distribution model, the Free-Vortex method was initially observed in our example. This method is an obvious choice because it maintains a constant axial velocity constant through the system (Figure 3). Unfortunately, Free-Vortex airflow at the hub and tip become rather impossible in low h/t ratios. When this happens, major alterations are needed, or a different design criteria altogether.

## Vortex Models Compared

We explored four vortex blade design approaches: Free Vortex, Exponential Vortex, First Power Vortex, and Mixed Vortex distribution models. The first three represent constant work models, which maintain a constant empathy rise along each blade station. The Mixed-Vortex model was suggested by Lewis in 1996, and instead focuses on targeting a comfortable coefficient of work, or blade loading.

Figure 2: De Haller ratio comparison for four swirl distribution models along radial blade stations 0-6 (MathCAD).

Notice in Figure 2 how the Exponential and Free Vortex models stray below the de Haller limit in the stator. The Forced-Vortex model behaves rather well, but then becomes the must problematic in the rotor. The Exponential-Vortex model is the best behaved overall, but still experiences adverse blade angles at the base of the rotor.

In contrast, notice how well behaved the Mixed-Vortex model is. At first glance, the Mixed-Vortex model is an attractive option. Unfortunately, a peek at Figure 3 may change your mind.

Figure 3: Comparison of axial velocities four swirl distribution models along radial blade stations 0-6 (MathCAD).

The Free-Vortex model maintains an ideal constant axial velocity throughout a system. In fact, all three constant-work systems can result in viable axial flow velocities. However, the Mixed-Velocity model, while gradual and smooth, delivers a staggering degree of difference in velocities leaving the rotor. The result is a complex movement of air along the blades in three dimension. Lewis warned that if the axial velocity in any portion of the Mixed-Vortex model drops too low, a flow reversal can occur. If this remains a problem, the only option is to rework the design until a constant-work model performs better, or to increase the h/t ratio. The latter leads to a larger diameter engine.

# Conclusion

All vortex models have their drawbacks, which engineers and scientists continue study. This problem remains one of the rather elusive and frustrating aspects of compressor design. Free Vortex design is a desirable method of blade design, but cannot be employed in every case. Mixed Model Vortex offers an attractive option in our fan design, if the axial velocities can be kept in a more reasonable ratio. We continue to investigate variations in swirl distribution, including relaxed vortex and non-constant work models to find the safest approach with the least amount of deviation from our intended design limits.

# Popular References

Dixon, S L, and C A. Hall. *Fluid Mechanics and Thermodynamics of Turbomachinery*. 6th, and 7th ed., Butterworth-Heinemann/Elsevier, 2010.

Lewis, R I. *Turbomachinery Performance Analysis*. London: Arnold, 1996.

Mattingly, Jack D., et al. *Aircraft Engine Design*. 2^{nd} ed., AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., 2002.

Mattingly, Jack D. *Elements of Propulsion: Gas Turbines and Rockets*. 2^{nd} ed., AIAA Education Series, American Institute of Aeronautics and Astronautics, Inc., 2006.