The design of a transonic-fan rotor is optimized using numerical computations of the full three-dimensional Navier-Stokes equations. The CFDRC-ACE multiphysics module, which is a pressure-based solver, is used for the numerical simulation. The code is coupled with simplex optimization algorithm. The optimization process is started from a suitable design point obtained using low fidelity analytical methods that is based on experimental correlations for the pressure losses and blade deviation angle. The fan blade shape is defined by its stacking line and airfoil shape which are considered the optimization parameters. The stacking line is defined by lean, sweep, and skews, while blade airfoil shape is modified considering the thickness and camber distributions. The optimization has been performed to maximize the rotor total pressure ratio while keeping the rotor efficiency and surge margin above certain required values. The results obtained are verified with the experimental data of Rotor 67. In addition, the results of the optimized fan indicate that the optimum design is found to be leaned in the direction of rotation and has a forward sweep from the hub to mean section and backward sweep to the tip. The pressure ratio increases from 1.427 to 1.627 at the design speed and mass flow rate.
Transonic fans are widely used in recent aircraft engines to obtain maximum pressure ratios per stage. High stage pressure ratios are important to reduce the engine weight, size, and operational costs. Performance of transonic fan has reached a high level but further improvements are required by engine manufacturers.
Recently, axial flow fans have been developed to a point, where stage efficiency has exceeded 90%. The goal of the current study is to maximize the total pressure ratio and to satisfy the required efficiency and mass flow rate. High fidelity methods are used for the design optimization of transonic fans of high pressure ratios. Numerical methods offer a feasible approach to solve complex nonlinear optimization problems involving a multitude of design variables and constraints in a systematic and efficient manner. Application of these computational design optimization approaches for fan blade designs can reduce design cost and design cycle and increase efficiency of jet engines [
The problem of the design validation and optimization of transonic compressors using numerical solution of the turbulent flow equations has been under investigation by several researchers [
The objective of the current study is the optimization of the rotor blade shape for a transonic fan to maximize the rotor pressure ratio while satisfying certain geometrical and performance constraints. A parametric study for the fan blade geometry is performed using a preliminary design method for transonic fans to find a suitable starting point for the high fidelity design method [
The numerical simulations are done using the CFDRC-ACE multiphysics module, which is a pressure-based solver. The code solves the time-dependent, Reynolds-averaged Navier-Stokes equations for turbulent, compressible flows using a finite volume, time-marching approach on multizone, structured grids. Spatial accuracy is nominally second-order upwind formulation. Steady flows are simulated through an iterative process using local time stepping. Turbulence is computed using the Standard
For relative inlet Mach numbers in the order of 1.3 and higher, the most important design intent is to reduce the Mach number in front of the passage shock. This is of primary importance due to the strongly rising pressure losses with increasing preshock Mach number and because of the increasing pressure losses due to the shock/boundary layer interaction or shock-induced separation. The reduction of the preshock Mach number can be achieved by zero or even negative curvature in the front part of the blade suction side and by a resulting precompression shock system reducing the Mach number upstream of the final strong passage shock. The thickness is also kept very low, about 2% of chord for the tip section of a transonic fan. Besides inducing energy losses, the presence of shock waves makes transonic compressors particularly sensitive to variations in blade section design. An investigation of cascade throat area, internal contraction, and trailing edge effective camber on compressor performance shows that small changes in mean line angles, and consequently in the airfoil shape and passage area ratios, significantly affect the performance of transonic blade rows [
One of the most important airfoil design parameters affecting the aerodynamics of transonic blading is the chord-wise location of maximum thickness. Good performance is obtained for the lower shock front losses with the finer section which results when the location of the maximum thickness is moved aft. An optimum maximum thickness location is assumed to exist in the range of 55% to 60% of the chord length for transonic fan rotors [
The flow field in a compressor is not influenced by the two-dimensional airfoil geometry. The three-dimensional shape of the blade is also of great importance, especially in transonic compressor rotors where an optimization of shock structure and its interference with secondary flows is required. Many experimental and numerical studies have been done for the design and analysis of three-dimensional shaped transonic blading (e.g., [
An optimization algorithm is used to maximize the total pressure for the fan rotor blades using the CFD-RC package. The rotor blade geometry is parameterized in order to facilitate its handling through the design process. The level of success in parameterizing the blade is dependent on two factors. The first factor is the flexibility and amount of coverage of all possible solutions. The second factor is the compactness of the parameters. The more accurate the description of the blade, the bigger the number of parameters. The optimization process is used to determine the following dimensions: thickness distribution of each section at different radii, camber distribution of each section at different radii, staking line of the different blade sections from hub to tip, blade twist distribution over the staking line.
The prementioned design parameters are the factors affecting blade shape optimization process. They can be classified into two main groups. The first group is the hub-to-shroud (
The hub-to-shroud (meridional) plane group geometry is parameterized as follows: blade inlet and exit hub radius, blade inlet and exit tip radius, equation describing the hub curve in the meridional plane, equation describing the tip curve in the meridional plane.
This classification is based on the direct physical dimensions and their effect on the fan performance. Inlet hub and tip radii together are affecting the inlet area and so the average inlet Mach number. The inlet tip radius affects the peripheral speed and, consequently, the value of relative Mach number at the tip, which should be kept as low as possible to minimize losses in this part. The equations describing the hub and tip curves in the meridional plane can affect the pressure gradient on the hub and tip surfaces, boundary layer growth, and the associated velocity profile. It is difficult to manipulate these parameters efficiently using simple 1D or even 2D models. It should be manipulated using 3D CFD solvers to account for its different impacts on the flow. The meridional plane group is shown in Figure
Hub-to-shroud (meridionalplane) group.
The blade-to-blade plane geometry is parameterized as follows: blades spacing, equation describing the mean blade camber line, equation describing the blade thickness distribution along meridional coordinates, blade chord distribution, stagger angle.
The first parameter (blades spacing) depends on the number of blades and the radius at this location. Its combined effect with the second and third parameters and blade height determines the chocking conditions. The blade camber, thickness distribution, and the stagger angle determine the blade inlet and exit angles. The blade inlet angle affects directly the velocity triangle at the inlet, flow incidence angle, and relative Mach number. The exit blade angle affects the work of the rotor, through affecting the circumferential absolute exit velocity. Figure
Blade-to-blade plane group.
It is a set of geometry parameters that may appear in the two planes described above. It could be parameterized as follows: equation describing the axial position of the section staking point over the radial direction, equation describing the circumferential position of the section staking point over the radial direction.
The first parameter is the staking point axial coordinates, which should cause the blade to sweep back or forward. This sweep has its effect on improving the blade adiabatic efficiency as mentioned by [
Detailed analyses of the measured and calculated flow fields indicate that two mechanisms are primarily responsible for the differences in aerodynamic performance among these rotors. The first mechanism is the change in the radial shape of the passage shock near the casing by the end wall effect, and the second is the radial migration of low momentum fluid to the blade tip region. Similar results are obtained in a parallel investigation which identified the reduced shock/boundary layer interaction, resulting from reduced axial flow diffusion and less accumulation of centrifuged blade surface boundary layer at the tip, as the prime contributor to the enhanced performance with forward sweep [
The second parameter is the staking point circumferential coordinates, which should cause the blade lean. As mentioned in [
Higher performance can be achieved using a proper combination of two orthogonal blade curvatures, that is, the use of a blade curved both axially and tangentially and swept and leaned at the same time as applied in the current study. Figure
Definition of swept and leaned rotor blade geometry.
The blade camber line curve has a major role in the design problem. It describes the blade angle distribution along the meridional path. The inlet and exit blade angles affect directly the work transferred to the fluid. Describing the camber line could be done using polynomial or Bezier curves. Five- or six-point Bezier curve is sufficient to describe a complex curve, where a polynomial of higher order is needed to do the same job. In the present work, three sections are defined; each section is defined using five-point Bezier curve for description of the camber line curve. That is to say, if every point has two coordinates (
Bezier curve describing blade camber line.
The blade thickness distribution along the meridional coordinates could be described using polynomial or Bezier curve. In the present work, this parameter is investigated using seven-point Bezier curve, where the coordinates of the second and six points in the meridional directions represent the leading and trailing edges radiuses. The remaining points are treated as described above in the camber line treatment. Figure
Bezier curve describing blade thickness distribution.
Typical blade thickness and blade camber line in blade-to-blade plane using Bezier curve.
In order to start the optimization for the section thickness and camber distributions, the Bezier control points should be determined. A Bezier curve is defined by a set of control points
There are different methods to find the control points coordinates that accurately represent the section camber and thickness distributions. One method that gives accurate results is that using optimizations algorithm. The algorithms are used to locate the control points with the best fitting to the original curve. Figure
Bezier 5 control points representation for the camber line obtained by PSO.
A fixed value for the tip clearance of 1.5% from blade tip chord is only considered in the current study. The actual behaviour of the rotor blade is affected by the combination of the geometrical parameters together, not by everyone alone. That limits the ability of trusting empirical and simple one-dimensional equations result. This pushes the designer towards the obligatory 3D CFD simulations, which deal with the actual geometry as one unit, combining all the previously mentioned points. The variation of the coordinates of any geometric parameter will lead to a new geometry. Thus, it is easy to manipulate the problem using an optimizer.
The CFD analysis or simulation is highly dependent on the boundary conditions because the flow is internal and the boundary conditions are applied in proximity to the complex flow features. The first objective of this section is to describe the subsonic inflow and outflow boundary conditions that have been implemented into the CFD code and applied for the analysis of flows through transonic fans. The second objective is to validate the utilized CFD code CFD-ACE through a comparison of the results with the previous computational and the experimental studies.
The CFDRC-ACE multiphysics module is a pressure-based solver. It solves the time-dependent, Reynolds-averaged Navier-Stokes equations for turbulent, compressible flows using a finite volume, time-marching approach on multizone, structured grids. Spatial accuracy is nominally second-order upwind formulation. Steady flows are simulated through an iterative process using local time stepping. Turbulence is modeled using the Standard
The computational domain for the rotor is constructed as a rotating domain. The blade row is represented by a single blade passage considering a 3D periodic sector along the whole rotor passage as indicated in Figure
The computational domain and boundary conditions.
Figure
The geometry and mesh of each block is generated using PYTHON script file in the preprocessor of CFDRC package. The mesh used for the model is mainly structured. Figure
Rotor mesh.
The geometry chosen to validate the code is the transonic high-speed axial fan rotor of NASA Rotor 67. This low aspect ratio rotor is the first stage rotor of a two-stage transonic fan designed and tested with laser anemometer measurements at the NASA Glenn Research Center [
At the design mass flow rate, the relative Mach number distribution along the blade-to-blade 50% passage chord length at 50% and 20% span measured from tip section is presented in Figure
Comparisons between the experimental data [
Figure
Comparison between the current computations of the rotor pressure ratio and the measured NASA Rotor 67.
A grid sensitivity study is performed. The objective is to determine the level at which the solution is invariant with the grid size. The grids used in the simulations are generated using characteristic grid spacing,
A factor of safety of
A grid sensitivity study is performed to ensure that the baseline grid has adequate sizes to resolve the solid wall boundary layers and the shock system [
Rotor pressure ratio for different grid sizes.
Number of cells |
|
|
|
|
|
---|---|---|---|---|---|
Normalized 1st grid spacing |
|
|
|
|
|
(stage total pressure ratio) | 1.426 | 1.4317 | 1.4327 | 1.433 | 1.4334 |
Fan rotor structured grid.
Effect of grid spacing on the accuracy of the steady state solution.
Nondimensional mass flow rate convergence history.
The aim of the present study is to obtain an optimum blade geometry for a given preliminary design of the transonic fan with some geometrical and performance constraints. The final task is to combine all the developed modules in association with the simplex optimization algorithm to complete the optimization cycle. Figure
A flowchart for the design optimization process.
The simplex algorithm is used for the optimization process. The algorithm is a direct (nongradient) optimization method and requires only one objective function evaluation per design iteration. The algorithm is robust and is likely to converge. The algorithm is easy to use because it has only three parameters to adjust (initial values of variables, first step size, and minimum and maximum variables values). Some of the algorithm disadvantages are that, as with most algorithms, the algorithm may find a local minimum instead of the global minimum. Different minimum solutions can be found by starting the optimizer at different initial points. Since the simplex algorithm does not use past information to accelerate movement through the design space, convergence can be slow (especially with a large number of design variables).
A simplex is a polygon defined by
New design point in simplex optimization algorithm.
As the algorithm progresses through the design space, two setbacks can occur. The first setback occurs, if the current worst design is created in the previous iteration. If this point is again reflected, the algorithm would bounce back and forth between two configurations. The algorithm instead reflects the second worst point. The simplex moves in a different direction, away from the stall point. An objective function that has a steep valley leading to a local minimum will cause the simplex algorithm to cycle infinitely through the same design points at the rim of the valley. The second setback is that when simplex cycles through the same designs over a period of several iterations, the algorithm is stalled. Reducing the physical size of the simplex allows it to fit into the valley and get closer to the minimum solution. The size reduction is done at the first instance of a repeated design.
An initial value for each design variable must be specified. The optimizer uses initial variable values as a starting guess for the optimization studies. These values will be used to create the first design. To start the optimization process, one must enter a value for the first step. This value essentially sets the geometric size of the simplex and affects the behavior of the algorithm. A good rule of thumb for choosing a value for Delta is 20% of the size of the entire design space. The minimum and maximum values for each design variable are specified. This will bound the optimizer, preventing it from choosing designs that lie outside this range. Constraints are useful for preventing creation of unrealistic geometry or application of unrealistic boundary or volume conditions. The maximum and minimum values were set as ±10% for most of the variables such Bezier points coordinates. This cycle is segmented into main steps that were previously developed and programmed. They are as follows. The low fidelity (preliminary design) is started and the geometry of the new design specified. The simplex optimization algorithm starts with the low fidelity optimal as a baseline of optimization. The geometry construction variables are imported in CFD-GEOM and geometry is constructed. The constructed geometry is passed through the constraint verifications stage. In this stage the geometry is checked to make sure that it satisfies the constraints. Then, the edge grid is generated and all steps are being carried out by executing the developed grid generation module. The boundary condition, initial condition, and solver controls are applied in the simulation module by executing the solver setting module. Then, the analysis objective module executes. Output of this module is a data file containing values, which is the average value of the cost function and the geometric parameters. The simplex optimizer continues to run on the other geometry, and the path of the optimization is stored in a data file to monitor the history of results during the optimization run.
All the above steps are arranged and programmed using the PYTHON language and conducted in the simulation manager module, which is one of the modules in the CFDRC package.
The current transonic rotor under consideration is a first-stage rotor of a three stage fan recently designed [
Baseline data of the selected case.
Requirements | Inlet conditions | Selected parameters | |||
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Mass flow rate | 120 kg/sec | Total inlet temperature | 300 K |
|
<10500 |
Fan pressure ratio | 2.6 | Total inlet pressure | 101.325 kPa | Hub-to-tip ratio | 0.2–0.7 |
Diffusion factor | <0.55 | ||||
Inlet axial Mach number | 0.7 |
Low fidelity design parameters.
Parameter | First stage | Second stage | Third stage | |||
---|---|---|---|---|---|---|
Rotor | Stator | Rotor | Stator | Rotor | Stator | |
Blades number | 25 | 27 | 34 | 35 | 34 | 35 |
Mean radius (m) | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 |
Aspect ratio | 3.45 | 3.15 | 4 | 3.8 | 3.5 | 3.55 |
Blade height | 0.28 | 0.21 | 0.2 | 0.19 | 0.17 | 0.16 |
Hub/tip ratio | 0.39 | 0.49 | 0.52 | 0.53 | 0.57 | 0.6 |
(r.p.m) | 9,800 | — | 9,800 | — | 9,800 | — |
Tip speed (m/s) | 450 | — | 411.3 | — | 398 | — |
|
1.63 | — | 1.15 | — | 1.08 | — |
Pressure ratio | 1.55 | 1.36 | 1.25 | |||
Isentropic efficiency | 0.944 | 0.9337 | 0.9052 |
Low fidelity fan performance map.
The history of the optimization process for the
CFD simulations with and without optimization.
Without optimization | High fidelity optimum design | |
---|---|---|
Pressure ratio | 1.43 | 1.627 |
Isentropic efficiency | 0.82 | 0.842 |
Leaned | Non | In direction of rotation |
Swept | Non | Forward swept (hub to mean) |
Numbers of blades | 25 | 22 |
Variation of the rotor pressure ratio during the optimization process.
The high fidelity design is found to lean toward the direction of rotation. The rotor blade is swept forward from the hub to mean portion of the blade and have a backward swept for the rest of the blade as presented in Figure
Comparison between the blade shape of the high fidelity optimal design and the design without optimization.
It is clear from Figure
Comparison between airfoil shapes with and without optimization at 0%, 50%, and 100% span from the hub.
Similar transonic stages with inlet Mach number of 0.7 to 1.1 limited by a pressure ratio from 1.15 to 1.6 and an isentropic efficiency from 80% to 85% are obtained as indicated by Boyce [
The total pressure contours at the inlet and exit planes of the rotors are presented in Figure
Comparison between total pressure contours for high fidelity and low fidelity designs.
Design without optimization
High fidelity optimal design
The Mach number contours of the high fidelity optimal design at the mean section are compared to those of the low fidelity and the results are presented in Figure
Comparison between high fidelity design and low fidelity design Mach contour at mean section.
Without optimization
High optimal fidelity design
The same observations are shown in Figure
High fidelity design Mach contour near the hub (a) and tip (b) sections.
Near hub section
Near tip section
Close to the tip section, the shock structure is affected by the tip clearance flow. Figure
High fidelity design Mach contour at tip section.
High fidelity design total pressure contours at four meridional planes.
Flow path at different sections near the blade tip leading edge.
90% Span
95%
100%
Figure
Total pressure contour and flow path at the blade tip clearance.
Performance of the high fidelity optimum design during the off-design operation is presented in Figures
Variation of the total pressure ratio with the mass flow rate at different rotational speeds for high fidelity design.
Variation of the isentropic efficiency with the mass flow rate at different rotational speeds.
The same trend is observed for the different operating speeds but the operation range decreases with the increase in rotor speed. This result sets a limit on the range of the operating speed. The operation range measures the stability of the rotor performance. One of the definitions for the surge margin is that defined by Gostelow et al. [
In the current study, the design of a transonic fan is optimized using numerical simulation of the compressible-viscous flow equations and simplex optimization algorithm. The results obtained using the CFDRC code are verified with the experimental data of Rotor 67. A grid sensitivity analysis is performed for the numerical simulations. The cost function of the optimization process is the rotor total pressure ratio. The blade geometry is defined in terms of set of optimization groups describing the section chord, stagger angle, staking position, the section thickness, and camber distributions. The original total number of variables for the three sections is 84. The optimum design is found to be leaned in the direction of rotation and has a forward sweep from the hub-to-mean section and backward sweep to the tip. The pressure ratio increased by 14% at the design speed and mass flow rate. The peak efficiency increments were numerically observed using a blade prevalently curved towards the direction of rotation and slightly backward inclined near the tip. A fewer number of blades is achieved to reduce the rotor weight. The performance of the new design shows a stable operation during a wide range in the off design.
The authors declare that there is no conflict of interests regarding the publication of this paper.