Blade Loading Defined: Fundamentally - the pressure difference between the pressure and suction sides of a blade along a meridional pathline. It directly controls the amount of work done or extracted and reveals problematic flow regions like cavitation or shockwaves.
Spanwise Work Control: The non-dimensionalized angular momentum (rVtheta*), and its distribution from hub to shroud (‘spanwise’). It is an input parameter to the 3D Inverse design method and rooted directly in turbomachinery fluid dynamics.
Streamwise Loading Control: The other key input control in 3d Inverse Design. This defines the distribution of work from leading to trailing edge at the hub and shroud . A blade can be "fore-loaded," "aft-loaded," or "mid-loaded," (or some combination across the span) which manages surface pressure/velocity distribution and mitigates cross-passage gradients and other loss mechanisms..
The Inverse Solver: An iterative process that starts with a flow field and evolves a blade shape that will enforce it.
Shape Parameterization: Because designs are defined by aerodynamic controls rather than multiple geometry controls, the design space is low-dimensional and physics-led, making it highly compatible with Machine Learning through Reactive Response Surface optimization.
What is the difference between Direct Design and 3D Inverse Design?
How does Inverse Design enable shape parameterization and efficient optimization?
What examples of Machine Learning for turbomachinery have used 3D Inverse Design?
Blade loading is the single most important aerodynamic design parameter for turbomachinery. It can be viewed as the pressure difference from one side of the blade to the other along a meridional pathline. As shown in the figure below.
Blade loading is pressure difference from one side of the blade to the other
The pressure on the blade is sampled at some constant span(s) on both sides of the blade, when plotted against meridional distance, a characteristic trace plot is generated which can immediately reveal problematic regions of flow which can affect overall performance. For example:
Figure 2: Blade loading traces that indicate (from L to R) cavitation in a pump, negative incidence in an axial compressor, shockwave impingement in a radial compressor
But as well as indicating the general condition of the flow close to the blade surface, blade loading also defines the amount of work made or extracted by the blade and the distribution of that work from hub to tip. As the pressure difference across the blade is, in fact, the force felt by the blades, then the power generated or absorbed by the machine is ‘Blade force’ x ‘blade speed’. This combination of flow features and machine work input/output is what makes blade loading such a useful and powerful means of controlling blade design. The task for all designers is to create a blade whose loading does not result in issues such as the examples shown above, whilst generating or absorbing the required amount of work as a 3-dimensional solution. There are two approaches to meeting this challenge - Direct Design or Inverse Design methods.
In the direct method the blade is designed starting from the distribution of blade angles at the leading edge and trailing edge, derived from 1D/meanline design tools, or assumed by experience and past designs.The blade angle distribution and thickness is then specified along each meridional pathline and the more spanwise stations used, the more complex the blade shape can be.
Figure 3: Direct design starts from 2D blade profiles, then ‘stacked’ to create a 3D blade, this blade is then analysed or simulated to determine performance
Figure 4: Inverse Design starts from the specification of blade performance - which can be the exit velocity triangle(s). This defines the blade loading - which drives the creation of a 3D blade shape
The problem with direct design is that it takes a lot of parameters to control blade shape with sufficient variety to have a broad design space. The blade shape is the input and blade performance is the output so there is no causal link between the input parameters and the output - we only discover blade performance after analysis and because blade angle distribution can be ‘anything’, then blade performance can be ‘anything’. So direct design encourages a ‘trial-and-error’ design method, which is wasteful of time and resources.
In the Inverse Design method, for both axial and radial turbomachinery, the work co-efficient and blade loading is the input to the blade design. The blade geometry is calculated by the solver to satisfy the inputs and generate a unique 3D blade solution. This means directly controlling the aerodynamic performance of the blade and providing a more logical design approach.
There are 2 key concepts that define and control the 3D Inverse Design method - the spanwise distribution of work, and the meridional distribution of blade loading.
How do we control the spanwise work distribution
The Euler turbomachinery equation links work to the rate of change of angular momentum of the working fluid
If inlet Vtheta =0 , then specific work, W
And we define
Where Rref and Uref are reference length and speed (normally the outer radius and tip speed).
And because:
Power = rotor speed x mass flow rate x ΔrV 𝜽 (from the Euler eqn)
We can directly control the amount and distribution of work between the blade input and outlet by directly controlling the distribution of rVtheta*.
Figure 5: A constant spanwise distribution of rVtheta* implies a free vortex exit swirl distribution at the blade trailing edge - typical for centrifugal turbomachines
Free vortex distribution is generally used in centrifugal machines but in axial turbomachinery where the low peripherical velocities at the hub can create low momentum fluids and recirculation we can directly reduce the work done in the hub region. Of course as we aim to maintain the same work therefore we need to redistribute it to other areas of the blade. In the example below it is increased proportionally in the shroud region. From an aerodynamic perspective this inhibits cross-passage flow migration the hub and it is typically recommended for axial rotor and stator configurations. However, in rotor only applications, particularly open axial propeller fans and pumps, this may lead to increased velocities at the shroud and high diffusion, also associated with high leakage losses .
So In this case we have the possibility to use parabolic or spline defined distributions which help to control the local distribution of rVt and by reducing the shroud work we help to dissipate the tip leakage losses reducing the leakage flow and the impact on the following blade generally responsible for broadband noise. With inverse design we have direct control over this aerodynamic design intent.
Figure 6: Constant (free vortex), linear and parabolic spanwise distributions of work at the trailing edge of a fan. All of the examples do the same amount of work on the fluid, but place that work in different spanwise sections, which drives different aerodynamic behaviours
Can be shown that, for incompressible flows the pressure difference from one side of the blade to the other:
And similarly for compressible flow, where the constant density (rho) term is removed:
N = number of blades
Wmbl = mean meridional velocity at the blade surface
is the rate of rVtheta with meridional distance.
So we can control (𝑝+ − 𝑝−), or (ℎ+ −ℎ−) by varying
Figure 7: Example blade loading distribution between hub and should - in this example the blade is ‘fore-loaded’ with the majority of the work done in the fore section of the passage
By controlling the spanwise distribution of work and the meridional distribution of blade loading we take total control over the blade performance - we are specifying not only the total amount of work done/absorbed by the blade (that is Pressure Ratio, or Head, or Pressure Rise, or Expansion Ratio - depending on the machine in question) and how much work at each spanwise section, but also where that work is done in a meridional sense - either in the fore-part , aft-part, mid passage and/or any combination of those within the same blade passage. The remaining step is now to compute the 3D blade shape that will deliver this loading profile and overall performance.
Figure 8: Examples of mid- fore- and aft-loading on a fan blade, and the resultant changes to surface velocity distribution. Note how the loading correlates to the distance between the velocity curves at any given meridional location
Most designers will be familiar with the terminology of fore-loading and aft-loading the blade; the aerodynamic benefits of using such practices in specific turbomachinery applications is well known and documented in the literature.
However, the know-how associated with how to achieve such blade loading is kept as a most guarded secret by most companies and only developed over many years of design experience. With the 3D Inverse Design methodology, the blade loading is specified. As long as engineers understand the basic aerodynamic design principles of the blade they are designing, it is practically immediate to achieve a specific loading distribution.
The 3D Inverse solver is an iterative, inviscid potential flow solver which starts from a defined flow field, and then builds a geometry around that flow in order to perfectly sustain the field.
The blade shape is defined by a function f(r, z). The solver uses the Kinematic Boundary Condition, which dictates that the flow must be tangent to the blade surface.
The process iterates as follows:
Initial Guess: An initial blade shape is assumed represented by a sheet of discrete vortices, whose strength is directly related to the specified loading.
Velocity Calculation: The solver computes the 3D flow field based on the current shape and the prescribed 𝑟𝑉𝜃
Blade Evolution: If the calculated velocity is not tangent to the surface, the blade coordinates are updated using the constraint 𝑉.∇𝑓=0
This ensures that the blade "wraps" itself around the prescribed flow field.
Convergence: This continues until the blade shape stabilizes, meaning the geometry perfectly sustains the requested swirl distribution.
You can see the iterative process in real time below. Note how the 3D blade shape is adapting during the iterative process to fit the denied swirl distribution
To produce a 3D blade with specific thickness the inverse solver uses two types of sources in the potential flow field. Vortex Sheets, which represent the mean camber line and the loading (the pressure difference between the pressure and suction sides), and source sheets, which give the blade volume with a distribution of sources and sinks placed along the camber surface.
The power of 3D Inverse design can be summed up in 2 points:
Performance targeting and low-dimensionality are the keys to unlocking efficient optimisation and Machine Learning methods.
One of the most critical aspects of a robust turbomachinery design environment is to allow optimisers to explore the design space without losing control over the blade specifications. A recurring problem with direct design is that changing blade angle distribution affects the impeller head/pressure ratio or pressure rise. In the Inverse Design approach, the distribution of streamwise loading is input as a shape function, so as the distribution changes, the area underneath the loading curve (drvt/dm) is automatically scaled to match the specified spanwise rVtheta*. Therefore every design generated will automatically satisfy the required Euler Head / Pressure Ratio / Pressure Rise of the blade row.
We have generated numerous examples of custom Machine Learning systems making significant improvements to existing designs or creating new ‘clean-sheet’ turbomachine designs.
Francis Hydraulic Runner
Figure 9: Machine Learning designs a Francis hydraulic runner - resulting in significant increase in cavitation margin across the operating range
Read about how a Francis hydraulic runner was designed by Machine Learning
Centrifugal Compressor
Figure 10: Machine Learning designs a Centrifugal compressor - resulting in significant increase in performance across the operating range
Read about how a Centrifugal compressor was designed by Machine Learning
Axial Fan
Figure 11: Machine Learning designs an Axial Fan - resulting in significant increase in performance across the operating range
Read about how a Axial Fan was designed by Machine Learning
In all cases we show how the blade loading and meridional outline input parameters create a rich design space which is then efficiently explored by ADT’s Reactive Response Surface optimisation technology. The blade loading parameters are adapted, the 3D inverse design solution creates a new blade shape and the optimiser assesses its quality in a multi-objective, multi-point performance space. Because the 3D Inverse Design description of a blade geometry relies on so few parameters, the optimizer can fully explore the design space with a much lower computational overhead than direct design methods.
This also means that Machine Learning systems based on ADT’s 3D Inverse Design technology are truly rooted in the fundamental principles of turbomachinery fluid dynamics.
For more information on the operation of the Reactive Response Surface (RRS), see the blog article below.
This list of papers provide more detail on the 3D Inverse Design method:
Zangeneh, M.. "On 3D Inverse Design of Centrifugal Compressor Impellers with Splitter Blades." (ASME Paper No. 98-GT-507).
Zangeneh, M., Goto, A., and Takemura, T. "Suppression of Secondary Flows in a Mixed-Flow Pump Impeller by Application of 3D Inverse Design Method: Part 1—Design and Numerical Validation." (ASME Journal of Fluids Engineering).
Boselli, P., and Zangeneh, M. "An Inverse Design Based Methodology for Rapid 3D Multi-Objective/Multi-Disciplinary Optimization of Axial Turbines." (ASME Turbo Expo Paper).