^{1}

^{2}

^{*}

^{2}

^{2}

This paper presents two optimized rotors. The first rotor is as part of a 3-blade row optimization (IGV-rotor-stator) of a high-pressure compressor. It is based on modifying blade angles and advanced control of curvature of the airfoil camber line. The effects of these advanced blade techniques on the performance of the transonic 1.5-stage compressor were calculated using a 3D Navier-Stokes solver combined with a vortex/vorticity dynamics diagnosis method. The first optimized rotor produces a 3-blade row efficiency improvement over the baseline of 1.45% while also improving stall margin. The throttling range of the compressor is expanded largely because the shock in the rotor tip area is further downstream than that in the baseline case at the operating point. Additionally, optimizing the 3-blade row block while only adjusting the rotor geometry ensures good matching of flow angles allowing the compressor to have more range. The flow diagnostics of the rotor blade based on vortex/vorticity dynamics indicate that the boundary-layer separation behind the shock is verified by on-wall signatures of vorticity and skin-friction vector lines. In addition, azimuthal vorticity and boundary vorticity flux (BVF) are shown to be two vital flow parameters of compressor aerodynamic performance that directly relate to the improved performance of the optimized transonic compressor blade. A second rotor-only optimization is also presented for a 2.9 pressure ratio transonic fan. The objective function is the axial moment based on the BVF. An 88.5% efficiency rotor is produced.

The development of modern high thrust-weight ratio gas turbine engines for aircraft requires a compression system with high total pressure ratio and adiabatic efficiency, but also needs sufficient stall margin for stable and robust operation. If the blade loading for a fan/compressor is increased remarkably in one stage, the Mach Number and pressure gradient may be raised excessively leading to high diffusion and large scale boundary-layer separation may occur due to the shock wave or low momentum fluid accumulation in the corner. Thus the performance of a compressor on or off-design point will not be maintained. An advanced compressor aerodynamic optimization design technique and flow diagnosis method that can modify geometry parametrically, and take into account complex flow physics such as viscous shear, shocks, and vortex structure will aid further development of these compression systems.

Based on this motivation, many advanced aerodynamic design concepts and design system have been developed. In the early 1960s, Wang Zhongqi et al. [

The early studies did not have 3D CFD to help in the design, and required extensive experiments to verify design concepts. More recently, optimization strategies that automate geometry creation with CFD simulation have been developed. Benini [

A turbomachinery aerodynamic optimization design system based on smooth spanwise control of geometry [

Vorticity and vortex structures appear inevitably in compressor viscous flows and heavily influence the aerodynamic performance of the compressor. Therefore the vorticity/vortex dynamics approach can be used to help guide and understand aerodynamic design and flow diagnosis for turbomachinery. It plays a significant role in turbomachinery aerothermodynamics theory and application [

Optimization in the design process of fan/compressors has still not been utilized to its fullest potential. This is largely due to the large design space as well as the issues related to effectively defining actual design processes and precisely stating the optimization statement. Hence, the optimization platform defined by Nemnem et al. [

The essential task of turbomachinery design is to create the geometry so the flow has minimum loss, with proper changes in flow direction or turning while ensuring the stress does not exceed strength and material properties of the structure. In general, a smooth aerodynamic shape is required for most of turbomachinery flow paths. In other words, the geometry curve should have continuous first and second derivatives. Mathematically, the curvature is defined as for a two-di- mensional curve y = f(x), It is also the reciprocal of the radius of curvature on the curve. Based on this formula, if the first and second derivatives are continuous, then the curvature is continuous. The crucial part is the second derivative y’’ in the expression of curvature. Hence the second derivative is used for the airfoil parameters set as described by Nemnem et al. [

To illustrate the relation between the streamline curvature and static pressure gradient, an analysis of a fluid microelement unit in a blade to blade section is used, as shown in

Currently, many parametric curves such as cubic splines, B-splines and Bezier splines have been used to build three-dimensional blade. However, mathematically, the third derivative is not continuous on the node for these spline curves. Therefore, in order to avoid these problems, the cubic B-spline is applied to define the second derivative of the airfoil camber line. It is integrated twice to get points that define the camber line, which is then fourth derivative continuous. A smooth thickness distribution is then applied along with a smooth leading and trailing to generate the airfoil.

The turbomachinery aerodynamic optimization design system used to produce results in this paper is based on Shell script, Python scripts and Fortran executable in Linux framework. There are three main parts of it: the user-defined module, optimization script module and the Dakota optimization module [

optimization variables, parametric geometry, journal files for CFD solver, and mesh generation setting et al. The optimization script module automatically executes the three-dimensional blade geometry builder (3D BGB) [

The blade geometry generator (3D BGB) has many options to vary the 3D blade including smooth spanwise variations of leading and trailing edge angles, blade meanline and thickness, lean and sweep of the stacking axis, and chord. In the optimization of the first rotor, only blade leading and trailing edge angles and the blade meanline shape spanwise has been varied. By not varying the blade stacking, axial chord and stacking, it was felt that a stress analysis would not have to be done as part of a multiobjective optimization. The optimization could be used for aerodynamics only. This means there is a larger design space could be explored in the future, but with inherently more complexity.

The flow is not uniform at inlet and outlet of each blade row. The spanwise non-uniformity must be accounted for at the very least. A 1.5 stage simulation is therefore carried out to ensure the performance of the rotor is maintained within the multistage environment.

From the mid-1970s to the early 1980s, NASA worked with industry Energy Efficient Engine (E^{3}) research program. One industry partner was GE, and the program advanced the design technology for all components that had an impact on engine efficiency. The core compressor that was designed was highly loaded with 10 stages and had a pressure ratio of 22.6. Holloway et al. [

Accordingly, in order to illustrate the advantage of the optimization design and flow diagnosis method in this paper, the IGV and first stage of the E^{3} core compressor from GE has been chosen as the baseline case, and the rotor is optimized and analyzed in detail. ^{3} compressor. The baseline design uses GE E^{3} axial and radial coordinates at the leading edge and trailing edge as well as the metal angles as defined in the report [^{3} geometry, but one base on it. The modifications to the geometry for optimization will be the leading and trailing edge angles as well as modifying the curvature of the meanline. These adjustments have been made sequentially, first angle modification, and then curvature modifications explained in more detail in the next two subsections.

The quadrilateral meshes on the blade surfaces for 1.5-stage compressor is shown in

Parameter | Value |
---|---|

Corrected first rotor tip tangent speed (m/s) | 451.7 |

Corrected design angular velocity (rpm) | 12303 |

Design mass flow rate (kg/s) | 54.4 |

Tip flow coefficient of the first stage | 0.417 |

Tip work coefficient of the first stage | 0.307 |

Relative Mach Number at L.E. of the first rotor tip | 1.356 |

Hub-to-tip ratio at the L.E. of the first rotor | 0.507 |

The number of blades for IGV, Rotor 1 and Stator 1 | 32, 28, 50 |

Clearance at Rotor 1 tip (mm) | 0.5 |

O_{4}H topological structure is applied to three blade rows, and the total hexahedron-mesh cell number is about 1,550,000 for 3 blade rows and the y+ of first grid cell is less than 1 in order to capture the flow details in boundary-layer area.

The design approach for high pressure compressors is to use a multi-fidelity approach starting with meanline, and then an axisymmetric solver with loss models which will give a starting point for a 3D blade shape. The GE E^{3} compressor has been analyzed as described by Turner et al. [

Optimization has the potential to change this approach to design, although the designer must still remain integrally in the loop. This paper demonstrates an automated method of 3D CFD optimization process with parametric geometry manipulation. The genetic algorithm (GA) is effective for a relatively small number of design parameters. The GA approach is not as dependent on having fully converged CFD solutions, as is a gradient-based optimization method. The first optimization was performed to adjust the leading and trailing edge angles. The simulation domain is shown in ^{3} compressor. These parameters modify a B-spline curve that ensures the modified blade is smooth. The IGV and stator 1 are the baseline and all other geometry parameters are kept the same.

Optimization Statement 1:

Maximize: adiabatic efficiency η for 1.5-stage domain;

Constraints: mass flux

total pressure ratio

Design Variables: 10 variables, 5 control points for added flow angle Δβ at L.E. and 5 at the T.E. on rotor1 as shown in

Genetic algorithm (GA) parameters: population size is 12, and the maximum iteration is 100.

Constants: All other parameters for blades are based on E^{3} report [

The efficiency did not increase based on this optimization, but the mass flow increased by a small amount. The amount of angle change, Δβ, at the rotor leading and trailing edge as a function of relative rotor blade height is shown in

The curvature of the airfoil camber line is modified by changing spanwise B-spline control points (similar to the angle adder shown in

Optimization Statement 2:

Maximize: adiabatic efficiency η for 1.5-stage domain;

Constraints: mass flux

total pressure ratio

Design Variables: 24 variables for 4 sections, which have 3 inside points are used to control curvature for the unit chord as seen in

Genetic algorithm (GA) parameters: population size is 12, and the maximum iteration is 70.

Constants: Use all other blade definitions from the result of optimization 1.

A comparison of baseline and optimum blade shapes are shown in

and

The curvature control through the optimization allows for an indirect control of the pressure gradients.

for baseline, optimizations 1 and 2 on the design angular velocity. It demonstrates that the adiabatic efficiency, total pressure ratio and mass flow rate are increased.

These are overall parameters of the 3-blade row block, and having the rotor optimized with the stationary blade rows included allows for the correct matching of the flow field. The adiabatic efficiency of the 3-blade row block increases by 1.45% (delta efficiency), whereas the rotor-only efficiency gain is 1.27% (delta efficiency). The block efficiency gain is higher, implying there is improvement in the stationary blade rows due to better matching. The optimization constraint was to have a higher mass flux than the baseline. The optimum increased the mass flow by nearly 0.6% to 53.97 kg/s for the design back pressure. This is still lower than the E^{3} design intent of 54.4 kg/s. The design report [

Parameters | Baseline | Optimum | Percentage | |
---|---|---|---|---|

Optimized point (3-blade rows) | Efficiency (η) | 86.63% | 88.08% | +1.67% |

PR (π) | 1.797 | 1.800 | +0.223% | |

Mass flow rate (kg/s) | 53.65 | 53.97 | +0.596% | |

Optimized point (rotor-only) | Efficiency (η) | 89.39% | 90.66% | +1.42% |

PR (π) | 1.832 | 1.834 | +0.109% | |

Choking flow rate (kg/s) | 53.98 | 54.27 | +0.537% | |

Stall mass flow rate (kg/s) | 51.19 | 50.40 | −1.54% |

reduced slightly due to the flow turning angle being decreased below 40% span. By contrast it is raised at the outer regions of the span mainly due to the strength of shock wave being decreased as will be described in more detail. The adiabatic efficiency is improved over the entire span mainly due to the smooth variations of the curvature parameters and their link spanwise.

The link between the flow and blade geometry controlling curvature can be seen in the relative Mach Number at 95% span as shown in

shock loss, and a reduction of the boundary-layer separation after the shock and improving throttled range.

The performance of the tip is vital for transonic compressor rotors. The rotor work capability is determined heavily by the tip flow, and the stall margin is largely influenced by the shock structure in this region. A comparison of isentropic Mach Number for optimum and baseline rotor is shown in

The absolute Mach Number profiles as a function of relative blade height at inlet and outlet plane for the stator is shown in

ratio increases, the pressure gradient is appropriately modified and the whole aerodynamic performance of the compressor block is improved remarkably.

Vortex/vorticity dynamics is an important concept in fluid dynamics, which has played an exceptionally crucial role in turbomachinery. It provides novel ideas for turbomachinery aerodynamic design and viscous flow diagnosis by focusing on the derivative/gradient of flow parameters to examine essential flow information and underlying dynamic processes. In this section, the physical mechanisms for boundary-layer separation due to expansion or compression and shear effects, such as shock wave and corner separation are explored from the viewpoint of vorticity dynamics. The physical cause of improving performance of the compressor by controlling curvature will be analyzed in detail, and this relatively new flow diagnosis method for transonic compressors is presented.

The relationship between vorticity and throughflow capability of a compressor will be explored by deducing the expression for azimuthal vorticity ω_{θ}. The discussion is limited to a single axial-flow compressor rotor as shown in _{2} plane that is perpendicular to the z-axis can be expressed as follows:

where, r_{1} and r_{2} are radii for hub and casing respectively. The axisymmetric assumption means

al-flow turbomachinery. Therefore, the Equation (1) is simplified as follows:

Considering the designers main focus on the meridonal flow parameters, an analysis of meridional azimuthal vorticity _{1}, the near-casing zone A_{3} and the main flow core zone A_{2}. In the A_{2} zone, _{1} zone, _{3} zone,

According to Equation (2) and the discussion of _{θ} in the near-casing zone A_{3} and make A_{3} with a higher radius. Alternatively a decrease of negative azimuthal vorticity ω_{θ} in the near-hub zone A_{1} and make A_{1} with lower radius would also increase the mass flow rate. From a traditional fluid dynamics viewpoint, this means the same as reducing the boundary-layer thickness.

The discussion is explained further using _{θ} area is smaller for the optimum case than for the baseline case, which mainly occurs in the near-casing and near-wall of suction side zone for rotor cross section, and in the near-casing and lower spanwise area for stator outlet plane. It illustrates the flow core zone is limited and the boundary-layer thickness is decreased partially for the optimum case compared to the baseline case. Therefore the mass flow rate, total pressure ratio and adiabatic efficiency are improved wholly for the optimum case as already shown in _{z} as shown in

The radial vorticity ω_{r} and skin-friction vector τ lines are shown in _{r} is decreased entirely for the optimum case compared to the baseline case. It illustrates that the shock strength and separated flow

immediately following the shock wave are weakened considerably, while the secondary flow is alleviated by the modification of the curvature on the airfoil camber line. Moreover, the sign of radial vorticity ω_{r} is reversed across the separation bubble. The beginning of the separation bubble has higher negative ω_{r}. The start and end of the separation bubble coincide with the sign change of radial vorticity ω_{r} on the transonic rotor suction side.

The Boundary Vorticity Flux (BVF) is a very important vorticity dynamics parameter since it directly relates to the on-wall physical generation of vorticity [

where,

where, n is unit normal vector for wall, a_{B} is wall accelerator. In general, for Newton fluid such as air, since the viscous coefficient with very small value and

the body force is ignored, then

The axial component of

The literature [

To better understand the boundary-layer separation on the rotor surface, a set of boundary-layer separation criterion has to be introduced from the literature [

Separation zone warning: Not only the skin-friction vector

Separation line criteria: The curvature of vorticity lines reach a maximum (Both for open and closed separation).

Separation watch: Tangent BVF

These boundary-layer separation criteria are sufficiently verified by vector lines of (

High entropy gradient and static pressure gradient is a necessary but not sufficient condition for boundary-layer separation as shown in

The axial moment acted on the fluid by the rotor blades due to pressure is:

which is the torque due to pressure forces. The power is the torque times the angular rotation rate. Using the transformation

and Gauss theorem and Stokes theorem, formula (6) can be expressed as:

Here, _{z} is its axial-component. _{z} contains two parts, of which the first part is surface integration of the second-moment of_{z} will be, through which more beneficial effects would be brought to the pressure rise capability of the rotor blade passage. Therefore, the axial component of BVF _{z} (torque due to pressure forces) as the objective function for optimization of the rotor blade in a ansonic fan. By maximizing the axial moment due only to pressure and not viscous forces is related to the efficiency since this torque is part of the efficiency equation that does useful work. The design parameters are shown in

Maximize: axial-moment M_{z} at the design point for fan rotor blade passage domain;

Constraints: mass flux

Design Variables: 20 variables for 4 sections, which have 3 points used to control curvature for the unit chord;

Genetic Algorithm (GA) parameters: population size is 12, and the maximum iteration is 300;

Constants: position and metal angle at leading edge and trailing edge for rotor in the meridian plane.

The baseline case of the fan rotor has been designed by a Turbomachinery Axisymmetric design code [

Design parameters | Values |
---|---|

Corrected first rotor tip tangent speed (m/s) | 495.32 |

Corrected design angular velocity (rpm) | 21,500.0 |

Design flow rate (kg/s) | 24.30 |

Tip flow coefficient of the rotor | 0.285 |

Tip work coefficient of the rotor | 0.450 |

Relative Mach Number at LE of rotor tip | 1.57 |

Hub/tip ratio at the inlet of the rotor | 0.41 |

The number of rotor blade | 14 |

Clearance at rotor tip (mm) | 0.5 |

the optimized highly-loaded fan rotor has a wide operating range with high adiabatic efficiency and favorable through flow capability along the whole operating line. This aerodynamic performance level is beyond the design of conventional axial fan rotor. Obviously, the optimization based on vorticity dynamics is one of promising novel design concepts.

Parameters | Optimum | Baseline | Difference | Percentage |
---|---|---|---|---|

Efficiency (η) | 88.49% | 87.55% | +0.0094 | +1.07% |

PR (π) | 2.896 | 2.863 | +0.0330 | +1.15% |

Mass flow m dot (kg/s) | 24.586 | 24.197 | +0.3890 | +1.61% |

Stall Margin (SM) | 10.74% | 8.57% | +2.17% | +25.32% |

slightly due to high Mach Number in this area, so the optimization goal is to further control shock wave and secondary flow after the shock wave.

_{z} is beneficial to increase total pressure ratio for the transonic fan rotor.

The Mach Number at 95% span for the design point are shown in

The stacked sections for the optimized blade are shown in

An optimization process has been implemented for two different transonic fans. The first optimization is the first stage plus IGV of a high pressure compressor base on the GE E^{3} design. Only the rotor was modified while keeping the IGV and stator 1 unchanged. The leading edge and trailing edge flow angles in addition to the curvature of the rotor meanline up and down the span were allowed to vary in the optimization process. The parametric geometry generator is connected to a 3D CFD solver and a Single Objective Function Genetic Algorithm was used to drive the optimization. The optimization leads to an increase of adiabatic efficiency of 1.45% with a slight increase in total pressure ratio of 0.206%. Furthermore, the flow rate is increased by 0.552% and stall flow rate is reduced by 1.541% which greatly improve stall margin and operation range. Improved stall margin was not stated goal of the optimization, and its improvement is attributed to simulating all three blade rows at once. Solving all three blade rows at the same time ensures that the rotor is being optimized within a multi- stage influence. In addition, the use of curvature of the meanline and ensuring smoothness spanwise of these parameters allows a small number of parameters to drive aerodynamically appropriate geometry changes through the optimization process.

In addition to traditional spanwise profiles and Mach Number explanations, a vortex/vorticity dynamics technique was used to further understand the improvements in performance. The results indicate that the rotor loading is restructured; the shock and the separated flow behind it together with the corner secondary flow on the rotor suction side have improved with the optimized blade.

The azimuthal vorticity and boundary vorticity flux (BVF) are two important parameters for which the utility has been demonstrated in this paper. The radial vorticity reverses sign on either side of the separation bubble generated by shock on the suction side. The “signature” of the separation bubble with high negative values of radial vorticity is captured with this approach. In addition, the axial derivate of entropy is high and is another significant feature of the transonic compressor rotor flow.

Comparing with the flow parameters themselves, checking their derivation/ gradient is more meaningful for the flow diagnosis, especially for the diagnosis of “focus flow”, because the derivation/gradient for flow parameters can give abundant and crucial flow information for guiding designers to improve the turbomachinery performance.

A second optimization method based on vorticity dynamics theory has also been presented for a 2.9 Pressure Ratio Transonic Fan Rotor. For this case the objective function was the axial moment based on BVF. The results indicate that camber curvature control technology is very helpful to manage shock wave and boundary layer flow. The beneficial effect of the objective function M_{z} based on vorticity dynamics in increasing the total pressure for transonic fan rotor is also verified. A rotor with 88.5% efficiency at 2.9 pressure ratio was produced. The optimization of turbomachinery through-flow geometry linked to vorticity dynamics may provide designers with novel prospects to further improve aerodynamic loading while enhancing the aerodynamic efficiency.

Optimization strategies are still evolving. It is clear that instead of taking the designer out of the loop, it allows the designers job to be one of deciding the objective function(s), constraints, and variables to modify. Understanding improvements is still required, and advanced methods such as vortex/vorticity dynamics become extremely useful.

The present optimization study focused on improving efficiency based solely on modifying the rotor leading and trailing edge angles as well as controlling the curvature of the blade meanline sections. This was done to fully demonstrate the new capability for smooth curvature control and to allow a purely aerodynamic optimization since the blade stacking did not change nor did the area of each blade section. In addition a mixing plane CFD simulation was performed for the 3 blade row solutions, no fillet was modeled in the rotor, and the gaps under the IGV and stator 1 (they are both variable) were not modeled. Improved simulation to account for unsteadiness using harmonic methods and for more detailed geometry would be part of future work.

The current capability of the geometry generator includes smooth variations of the stacking axis to allow for lean and sweep as well as axial chord and thickness. Detailed leading edge designs are also possible. These geometry changes greatly affect the stress on a blade so a Finite Element stress analysis would have to be part of the optimization. This has been part of a multi-objective function optimization in another project where mass of the blade is the other objective function. Stall margin was not an explicit goal in the optimization, but did improve. Operability may have to be part of a formal optimization statement, and current efforts are exploring this. Some additional capability still needs to be added to the geometry generator. This includes non-axisymmetric endwalls using a method similar to the curvature control demonstrated in this paper. In addition, parametric cavity design is already being explored.

This work was supported by National Natural Science Foundation of China, Grant No. 51506036.

Chen, H.L., Turner, M.G., Siddappaji, K. and Mahmood, S.M.H. (2017) Flow Diagnosis and Optimization Based on Vorticity Dynamics for Transonic Compressor/Fan Rotor. Open Journal of Fluid Dynamics, 7, 40-71. https://doi.org/10.4236/ojfd.2017.71004

A Area or zone (m^{2})

(r, θ, z) Cylindrical coordinates with the axial z-direction; when used as subscripts denotes the corresponding components of a vector

(x, y) 2D Cartesian coordinates for general curve

y’ = dy/dx The first derivative for y about x

y” = dy’/dx The second derivative for y about x

C Curvature for general 2D curve

R Radius of Curvature for general 2D curve

(s, n) Curve coordinates for curve coordinates system; when used as subscripts denote corresponding components

p Static pressure on fluid (pa)

u Velocity vector on fluid in inertial reference frame (m/s)

w Velocity vector on fluid in relative reference frame (m/s)

(u, v) Non-dimensional coordinates in basic airfoil plane

m’ Normal coordinate for the streamline

PR Total pressure ratio on compressor

dp/dz Axial gradient of static pressure for fluid

ds/dz Axial gradient of static entropy for fluid

Ma or M Local Mach Number on fluid

M_{z}_{ }Axial Moment due to pressure (N.m)

c Local meridian curve distance (m)

C_{m} Total meridian curve distance (m)

inc Δβ at the leading edge of the blade (˚)

dev Δβ at the trailing edge of the blade (˚)

cp_{l} Control point at leading edge for rotor

cp_{t} Control point at trailing edge for rotor

ρ Density on fluid (kg/m^{3})

π Total pressure ratio on compressor

η Adiabatic efficiency on compressor

τ Skin-friction vector for fluid (Fluid to wall) (m^{3}/s^{2})

ω Vorticity vector of fluid (1/s)

^{2})

Ω Angular velocity of compressor rotor (rpm)

Δβ Deviation on metal angle (˚)

^{2}/s)

m Meridional plane

is Isentropic

f Body force for fluid per unit mass (m^{2}/s^{2})

Submit or recommend next manuscript to SCIRP and we will provide best service for you:

Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc.

A wide selection of journals (inclusive of 9 subjects, more than 200 journals)

Providing 24-hour high-quality service

User-friendly online submission system

Fair and swift peer-review system

Efficient typesetting and proofreading procedure

Display of the result of downloads and visits, as well as the number of cited articles

Maximum dissemination of your research work

Submit your manuscript at: http://papersubmission.scirp.org/

Or contact ojfd@scirp.org