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Title:
METHODS AND APPARATUS FOR SIMULATING LIQUID COLLECTION ON AERODYNAMIC COMPONENTS
Document Type and Number:
WIPO Patent Application WO/2019/186151
Kind Code:
A1
Abstract:
A computer-implemented method for simulating liquid collection on an aerodynamic component impacting a plurality of droplets comprises: defining initial simulation parameters including a droplet distribution, size, and velocity relative to the aerodynamic component, and a geometry of the aerodynamic component; using a computational fluid dynamics algorithm, determining an incident angle and velocity of each one of a plurality of droplets relative to a surface of the aerodynamic component at a distance from the surface of the aerodynamic component, based on the defined initial simulation parameters; for each one of the plurality of droplets, querying a lookup table to determine a pre-calculated liquid collection efficiency corresponding to the determined incident angle and velocity and the size of said one of the plurality of droplets; and combining the determined liquid collection efficiencies for the plurality of droplets to determine an overall liquid collection efficiency for the aerodynamic component. A method of generating the lookup table, and apparatus for performing the methods, are also disclosed.

Inventors:
CIMPEANU, Radu (Technology and MedicineSouth Kensington Campus,Faculty Building, Exhibition Road, London SW7 2AZ, SW7 2AZ, GB)
PAPAGEORGIOU, Demetrios (Technology and MedicineSouth Kensington Campus,Faculty Building, Exhibition Road, London SW7 2AZ, SW7 2AZ, GB)
Application Number:
GB2019/050870
Publication Date:
October 03, 2019
Filing Date:
March 27, 2019
Export Citation:
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Assignee:
IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE (South Kensington Campus, Faculty BuildingExhibition Road, London SW7 2AZ, SW7 2AZ, GB)
International Classes:
G06F17/50; B64F5/60; B64D15/20
Foreign References:
US20040155151A12004-08-12
GB2345772A2000-07-19
US20140257771A12014-09-11
Other References:
None
Attorney, Agent or Firm:
CORK, Robert et al. (Venner Shipley LLP, 5 Stirling House Stirling Road,The Surrey Research Park, Guildford Surrey GU2 7RF, GU2 7RF, GB)
Download PDF:
Claims:
Claims

1. A computer-implemented method for simulating liquid collection on an aerodynamic component impacting a plurality of droplets, the method comprising: defining initial simulation parameters including a droplet distribution, size, and velocity relative to the aerodynamic component, and a geometry of the aerodynamic component;

using a computational fluid dynamics algorithm, determining an incident angle and velocity of each one of a plurality of droplets relative to a surface of the

aerodynamic component at a distance from the surface of the aerodynamic component, based on the defined initial simulation parameters;

for each one of the plurality of droplets, querying a lookup table to determine a pre-calculated liquid collection efficiency corresponding to the determined incident angle and velocity and the size of said one of the plurality of droplets; and

combining the determined liquid collection efficiencies for the plurality of droplets to determine an overall liquid collection efficiency for the aerodynamic component.

2. The method of claim 1, wherein the lookup table stores pre-calculated liquid collection efficiencies for a plurality of discrete values of each of the incident angle, velocity and droplet size, and

wherein in response to a determined value of one of the parameters lying between two of said discrete values, the liquid collection efficiency is determined by selecting the liquid collection efficiency associated with the closest one of said two discrete values.

3. The method of claim 1, wherein the lookup table stores pre-calculated liquid collection efficiencies for a plurality of discrete values of each of the incident angle, velocity and droplet size, and

wherein in response to a determined value of one of the parameters lying between two of said discrete values, the liquid collection efficiency is determined by interpolating between respective liquid collection efficiencies associated with said two discrete values of the parameter.

4. The method of claim 1, 2 or 3, wherein said distance from the surface of the aerodynamic component at which the incident angle and velocity of the droplet is determined is a distance defined relative to the size of the droplet. 5. The method of claim 4, wherein the distance is about 20 d, where d is a diameter of the droplet.

6. The method of any one of the preceding claims, wherein the initial simulation parameters include one or more of:

an air temperature;

a surface temperature of the aerodynamic component; and

a material property related to a hydrophobicity of the surface of the

aerodynamic component. 7. The method of any one of the preceding claims, further comprising:

generating the lookup table and storing the generated lookup table in memory.

8. The method of any one of the preceding claims, wherein the droplet distribution defines a distance between neighbouring droplets which impact upon adjacent impact sites on the surface of the aerodynamic component.

9. A method of validating a design for an aerodynamic component, the method comprising:

using a computer-implemented method according to any one of the preceding claims to simulate liquid collection on the aerodynamic component, by defining the initial simulation parameters including the geometry of the aerodynamic component based on a design for the aerodynamic component;

comparing the determined overall liquid collection efficiency for the

aerodynamic component to an acceptable limit; and

validating the design in response to the determined overall liquid collection efficiency being within the acceptable limit.

10. The method of claim 9, further comprising:

in response to the determined liquid collection efficiency exceeding the acceptable limit, redesigning the aerodynamic component and repeating the method of claim 8 to validate the redesigned component. li. The method of claim 9 or 10, further comprising:

manufacturing the aerodynamic component according to the validated design. 12. A computer-implemented method of generating a lookup table for determining a pre-calculated liquid collection efficiency for a droplet impacting on a surface, the method comprising:

defining a computational domain for modelling a flow of air approaching a solid surface at an incident angle;

using a computational fluid dynamics algorithm to determine a steady state airflow within the computational domain;

initialising a liquid droplet at a boundary of the computational domain, the droplet having an initial size and an initial velocity;

determining a liquid collection efficiency on the surface when the droplet impacts the surface, by modelling a behaviour of the droplet as it approaches and impacts the surface taking into account interactions between the droplet and the steady state airflow; and

storing the determined liquid collection efficiency in a lookup table, wherein the liquid collection efficiency is associated with the incident angle, initial velocity and initial size of the droplet.

13. The method of claim 12, wherein a local velocity of the steady state airflow at a location of the initialised droplet is assigned as an initial velocity of the droplet. 14. The method of claim 12 or 13, wherein the interactions between the droplet and the steady state airflow include one or more of:

droplet deformation due to surrounding airflow;

droplet rupture and/or coalescence; and

motion of one or more ejected secondary droplets after the droplet impacts on the surface.

15. The method of claim 14, wherein the motion of each of said one or more ejected secondary droplets after the droplet impacts on the surface is modelled by:

determining a velocity and trajectory of the secondary droplet as it exits the computational domain; extrapolating a path of travel of the secondary droplet based on the determined velocity and trajectory to determine whether the secondary droplet would impact on another part of the surface outside of the computational domain; and

counting a volume of liquid contained in the secondary droplet in the determined liquid collection efficiency in response to a determination that the secondary droplet would impact on another part of the surface outside of the computational domain.

16. The method of any one of claims 12 to 15, comprising:

initialising a plurality of liquid droplets at the boundary of the computational domain, wherein the plurality of liquid droplets include said liquid droplet and the plurality of liquid droplets impact upon adjacent impact sites on the surface.

17. The method of claim 16, wherein the behaviour of the liquid droplet is modelled taking into account the effect of impact events at adjacent ones of the plurality of impact sites on the liquid droplet as it approaches and impacts the surface.

18. A non-transitory computer-readable storage medium arranged to store computer program instructions which, when executed, perform a method according to any one of the preceding claims.

19. Apparatus for simulating liquid collection on an aerodynamic component impacting a plurality of droplets, the apparatus comprising:

one or more processors for executing computer program instructions; and memory arranged to store computer program instructions which, when executed by the one or more processors, perform a method according to any one of claims 1 to 8.

20. Apparatus for generating a lookup table for determining a pre-calculated liquid collection efficiency, the apparatus comprising:

one or more processors for executing computer program instructions; and memory arranged to store computer program instructions which, when executed by the one or more processors, perform a method according to any one of claims 12 to 17.

21. A system comprising: a first apparatus according to claim 20, configured to generate a lookup table; and

a second apparatus according to claim 19, configured to query the lookup table generated by the first apparatus to determine a pre-calculated liquid collection efficiency corresponding to the determined incident angle and velocity and the size of said one of the plurality of droplets.

Description:
Methods and Apparatus for Simulating Liquid Collection on

Aerodynamic Components

Technical Field

The present invention relates to methods and apparatus for simulating liquid collection on aerodynamic components.

Background

Liquid collection on aeronautical components during flight can have significant consequences for both the component itself, and the aircraft as a whole. For example, supercooled droplets, which are found in the atmosphere at temperatures below o° C, can significantly affect aircraft performance and have been attributed as a factor in numerous incidents in the last few decades. Even in seemingly dry conditions, aircraft may encounter high liquid water content clouds during ascent or descent through the cloud layer, potentially resulting in a large number of droplets impacting on the fuselage and wings in a matter of seconds. Each droplet splashes onto the surface at velocities of approximately 50-250 m/s, and may leave behind a quantity of fluid. As a result a thin film of water forms on the surface, which can produce a significant alteration to the ideal flow properties for which the specific parts were designed.

As well as disrupting airflow over the surface of the component, sufficiently long periods of exposure to below freezing temperatures can cause the film of water to solidify. The layer of ice continues to grow until a critical mass is formed, at which point large pieces may break off and potentially cause serious damage on impact with other components, such as the aircraft engine or propellers.

Accordingly, when designing new aeronautical components it is important to understand the liquid collection behaviour of the proposed design. Software programs have been developed which attempt to model the so-called“retention rate” or “collection efficiency” of liquid droplets on a body, such as an aerofoil or nacelle. Such simulations rely on particle-based models in which the background air flow is first computed in the absence of water droplets. Virtual droplets are then initialised at a distance far in advance of the body, and a standard ordinary differential equation solver is employed to calculate the path of the fluid particle, accounting for its mass and density. However, such models typically rely on a number of crude approximations. Firstly, droplets are assumed to be spherical and non-deformable, even as they approach and hit solid surfaces. Secondly, interactions between droplets, such as collisions and/or coalescence, are either ignored or modelled in highly simplified empirical forms.

Thirdly, splashing (i.e. ejection of droplets upon impact with the surface) is not taken into account. Fourthly, all droplet and liquid film related effects on the surrounding air flow are neglected, since the background air flow is computed in the absence of water droplets. As a consequence of these approximations, such models effectively ignore all fluid dynamical related physics in the close vicinity of the body.

Summary of the Invention

According to a first aspect of the present invention, there is provided a computer- implemented method for simulating liquid collection on an aerodynamic component impacting a plurality of droplets, the method comprising defining initial simulation parameters including a droplet distribution, size, and velocity relative to the

aerodynamic component, and a geometry of the aerodynamic component, using a computational fluid dynamics algorithm, determining an incident angle and velocity of each one of a plurality of droplets relative to a surface of the aerodynamic component at a distance from the surface of the aerodynamic component, based on the defined initial simulation parameters, for each one of the plurality of droplets, querying a lookup table to determine a pre-calculated liquid collection efficiency corresponding to the determined incident angle and velocity and the size of said one of the plurality of droplets, and combining the determined liquid collection efficiencies for the plurality of droplets to determine an overall liquid collection efficiency for the aerodynamic component.

In some embodiments according to the first aspect, the lookup table stores pre- calculated liquid collection efficiencies for a plurality of discrete values of each of the incident angle, velocity and droplet size, and in response to a determined value of one of the parameters lying between two of said discrete values, the liquid collection efficiency is determined by selecting the liquid collection efficiency associated with the closest one of said two discrete values. In some embodiments according to the first aspect, the lookup table stores pre- calculated liquid collection efficiencies for a plurality of discrete values of each of the incident angle, velocity and droplet size, and in response to a determined value of one of the parameters lying between two of said discrete values, the liquid collection efficiency is determined by interpolating between respective liquid collection efficiencies associated with said two discrete values of the parameter.

In some embodiments according to the first aspect, said distance from the surface of the aerodynamic component at which the incident angle and velocity of the droplet is determined is a distance defined relative to the size of the droplet. For example, in some embodiments the distance is about 2od, where d is a diameter of the droplet.

In some embodiments according to the first aspect, the initial simulation parameters include one or more of: an air temperature; a surface temperature of the aerodynamic component; and a material property related to a hydrophobicity of the surface of the aerodynamic component.

In some embodiments according to the first aspect, the method further comprises generating the lookup table and storing the generated lookup table in memory.

In some embodiments according to the first aspect, the droplet distribution defines a distance between neighbouring droplets which impact upon adjacent impact sites on the surface of the aerodynamic component.

According to a second aspect of the present invention, there is provided a method of validating a design for an aerodynamic component, the method comprising using a method according to the first aspect to simulate liquid collection on the aerodynamic component, by defining the initial simulation parameters including the geometry of the aerodynamic component based on a design for the aerodynamic component, comparing the determined overall liquid collection efficiency for the aerodynamic component to an acceptable limit, and validating the design in response to the determined overall liquid collection efficiency being within the acceptable limit.

In some embodiments according to the second aspect, the method further comprises in response to the determined liquid collection efficiency exceeding the acceptable limit, redesigning the aerodynamic component and repeating the method to validate the redesigned component. In some embodiments according to the second aspect, the method further comprises manufacturing the aerodynamic component according to the validated design.

According to a third aspect of the present invention, there is provided a computer- implemented method of generating a lookup table for determining a pre-calculated liquid collection efficiency for a droplet impacting on a surface, the method comprising defining a computational domain for modelling a flow of air approaching a solid surface at an incident angle, using a computational fluid dynamics algorithm to determine a steady state airflow within the computational domain, initialising a liquid droplet at a boundary of the computational domain, the droplet having an initial size and an initial velocity, determining a liquid collection efficiency on the surface when the droplet impacts the surface, by modelling a behaviour of the droplet as it approaches and impacts the surface taking into account interactions between the droplet and the steady state airflow, and storing the determined liquid collection efficiency in a lookup table, wherein the liquid collection efficiency is associated with the incident angle, initial velocity and initial size of the droplet.

In some embodiments according to the third aspect, a local velocity of the steady state airflow at a location of the initialised droplet is assigned as an initial velocity of the droplet.

In some embodiments according to the third aspect, the interactions between the droplet and the steady state airflow include one or more of: droplet deformation due to surrounding airflow; droplet rupture and/or coalescence; and motion of one or more ejected secondary droplets after the droplet impacts on the surface.

In some embodiments according to the third aspect, the motion of each of said one or more ejected secondary droplets after the droplet impacts on the surface is modelled by: determining a velocity and trajectory of the secondary droplet as it exits the computational domain; extrapolating a path of travel of the secondary droplet based on the determined velocity and trajectory to determine whether the secondary droplet would impact on another part of the surface outside of the computational domain; and counting a volume of liquid contained in the secondary droplet in the determined liquid collection efficiency in response to a determination that the secondary droplet would impact on another part of the surface outside of the computational domain. In some embodiments according to the third aspect, the method comprises initialising a plurality of liquid droplets at the boundary of the computational domain, wherein the plurality of liquid droplets include said liquid droplet and the plurality of liquid droplets impact upon adjacent impact sites on the surface.

In some embodiments according to the third aspect, the behaviour of the liquid droplet is modelled taking into account the effect of impact events at adjacent ones of the plurality of impact sites on the liquid droplet as it approaches and impacts the surface. According to a fourth aspect of the present invention, there is provided a non-transitory computer-readable storage medium arranged to store computer program instructions which, when executed, perform a method according to the first, second or third aspect.

According to a fourth aspect of the present invention, there is provided apparatus for simulating liquid collection on an aerodynamic component impacting a plurality of droplets, the apparatus comprising one or more processors for executing computer program instructions, and memory arranged to store computer program instructions which, when executed by the one or more processors, perform a method according to the first aspect.

According to a fifth aspect of the present invention, there is provided apparatus for generating a lookup table for determining a pre-calculated liquid collection efficiency, the apparatus comprising one or more processors for executing computer program instructions, and memory arranged to store computer program instructions which, when executed by the one or more processors, perform a method according to the third aspect.

According to a sixth aspect of the present invention, there is provided a system comprising a first apparatus according to the fifth aspect, configured to generate a lookup table, and a second apparatus according to the fourth aspect, configured to query the lookup table generated by the first apparatus to determine a pre-calculated liquid collection efficiency corresponding to the determined incident angle and velocity and the size of said one of the plurality of droplets. Brief Description of the Drawings Embodiments of the present invention will now be described, byway of example only, with reference to the accompanying drawings, in which:

Figure l illustrates a computer model of an aerodynamic component impacting a plurality of droplets, according to an embodiment of the present invention;

Figure 2 is a flowchart illustrating a computer-implemented method for simulating liquid collection on an aerodynamic component impacting a plurality of droplets, according to an embodiment of the present invention;

Figure 3 illustrates a computer model of a droplet impacting on a flat surface, according to an embodiment of the present invention;

Figure 4 illustrates a computer model of a plurality of droplets impacting on a flat surface in sequence, according to an embodiment of the present invention;

Figure 5 illustrates a computer model of a plurality of droplets impacting at

neighbouring impact sites on a flat surface, according to an embodiment of the present invention;

Figure 6 is a graph plotting the collection efficiency of the intermediate droplets collapsed onto a single time frame, according to an embodiment of the present invention;

Figure 7 is a graph plotting the splashing profiles of the intermediate droplets collapsed onto a single time frame, according to an embodiment of the present invention;

Figure 8 is a flowchart illustrating a method of validating a design for an aerodynamic component, according to an embodiment of the present invention;

Figure 9 is a flowchart illustrating a method of generating a lookup table for determining a pre-calculated liquid collection efficiency, according to an embodiment of the present invention; and

Figure 10 schematically illustrates a system comprising apparatus for generating a lookup table for determining a pre-calculated liquid collection efficiency and apparatus for simulating liquid collection on an aerodynamic component using the lookup table, according to an embodiment of the present invention. Detailed Description

In the following detailed description, only certain exemplary embodiments of the present invention have been shown and described, simply by way of illustration. As those skilled in the art would realise, the described embodiments may be modified in different ways, all without departing from the scope of the present invention.

Accordingly, the drawings and description are to be regarded as illustrative in nature and not restrictive. Like reference numerals designate like elements throughout the specification.

As used herein,“aerodynamic” is employed in the technical sense to refer generally to any component that moves through, and interacts with, the air, and should be interpreted accordingly. As such, references to“aerodynamic” should be interpreted accordingly, as encompassing a body of any shape, and should not be taken to imply that the component necessarily has a streamlined form. Examples of types of aerodynamic components that maybe modelled in embodiments of the present invention include, but are not limited to, aeronautical industry components such as wings, aerofoils and nacelles, automotive industry components such as car wings, splitters and diffusers, and engineering industry components such as wind turbine blades. Furthermore, embodiments of the present invention are not limited to modelling the impact of water droplets suspended in the air. In some embodiments impact of heavier droplets can be modelled, such as the impact of seawater droplets on off-shore drilling structures (both static and dynamic) in strong winds and low-temperature conditions. Referring now to Fig. 1, a computer model of an aerodynamic component impacting a plurality of droplets is illustrated, according to an embodiment of the present invention. In the present embodiment an example of a nacelle is illustrated, however it will be appreciated that the principles discussed herein can readily be applied to modelling the impact of droplets on other types of aerodynamic body.

In the present embodiment, the position of the solid body 100 is fixed and the boundary conditions are constructed to accommodate an oncoming flow from left to right in Fig. 1, that is, in direction towards the nacelle 100. The background velocity of the flow is denoted by U ¥ , and in the present embodiment a representative value of lh ¥ ~ 130 m/ s (roughly 0.4 Mach) is selected to describe the desired flight conditions. In some embodiments, an angle of attack may also be taken into account when setting the background velocity.

The multi-fluid system is described by a group of liquid droplets 110 which travel towards and eventually impinge onto the solid surface of the body 100. For a total number of droplets, n, the diameters of the individual liquid droplets 110 can be expressed as , where i = i ,. .,h. The droplet diameters encountered in typical clouds range from a few microns (pm) up to the order of several hundred pm for the largest supercooled droplets. Each droplet is modelled as initially spherical, although in other embodiments a different initial droplet shape maybe used. In the present

embodiment, a maximum droplet size of D mSK = 236 pm is used since this is the largest droplet diameter for which experimental data is currently available. However, in general embodiments of the present invention can be used to model the impact of droplets of any size. From the perspective of splashing dynamics and water collection efficiency, the most significant interaction volume is the region 120 close to the surface of the aerodynamic component 100. This region 120 is hereinafter referred to as the‘region of interest for collection efficiency’, or simply the‘region of interest’. As will be described in more detail later, studies by the inventors have revealed that, surprisingly, the collection efficiency for any given droplet 110 entering the region of interest 120 is largely independent of droplets impacting the surface before and after the current droplet 110. Accordingly, a collection efficiency that is computed for a given droplet 110 after it has entered the region of interest 120 can be applied to other similar droplets, irrespective of their position in a sequence of droplets impacting upon the surface of the body 100.

Also, the inventors have discovered that the collection efficiency for any given droplet 110 entering the region of interest 120 can be described by a relatively small number of parameters, where the parameters relate to the state of a droplet 110 as it enters the region of interest 120. In the present embodiment, the parameters include: the incident angle of the air flow at the location at which the droplet 110 enters the region of interest 120, relative to the surface of the body 100; the velocity of the droplet 110 as it enters the region of interest 120; and the size of the droplet 110. Accordingly, it is feasible to compute liquid collection efficiencies for different combinations of parameters and store the results in a lookup table. In the present embodiment the droplet 110 is initialised to have the same velocity as the surrounding air flow when it enters the region of interest 120, and therefore the incident angle of the air flow at the location where the droplet 110 enters the region of interest 120 is equivalent to the incident angle of the droplet 110 itself. Referring now to Fig. 2, a flowchart showing a computer-implemented method for simulating liquid collection on an aerodynamic component impacting a plurality of droplets is illustrated, according to an embodiment of the present invention. First, in step S201 a computer model such as the one shown in Fig. 1 is initialised by defining initial simulation parameters. In the present embodiment the initial simulation parameters include a droplet distribution, droplet size, and droplet velocity relative to the aerodynamic component 100. The droplet distribution, size and velocity can be chosen according to specific cloud/humid environment characteristics. The initial simulation parameters also include the geometry of the aerodynamic component 100.

In some embodiments additional simulation parameters maybe defined in step S201, including but not limited to an air temperature, a surface temperature of the aerodynamic component 100, and a material property related to a hydrophobicity of the surface of the aerodynamic component.

Once the model has been initialised, in step S202 a computational fluid dynamics algorithm is used to determine the incident angle and velocity of each droplet 110 at the point at which the droplet 110 enters the region of interest 120, based on the initial simulation parameters that were defined in step S201. The incident angle is defined relative to the surface of the aerodynamic component 100. The point at which to determine the incident angle and velocity for querying the lookup table depends upon the width of the region of interest 120. The width of the region of interest 120 determines the distance from the surface of the aerodynamic component at which the values of the incident angle and velocity of the droplet are taken, for querying the lookup table.

In some embodiments this distance from the surface, which is the width of the region of interest 120, is defined relative to the size of the droplet. In the present embodiment a distance of 20 d is used, where d is the diameter of the droplet. The inventors have found that beyond distances of about 20 d from the surface of the aerodynamic component 100, interactions between the droplet 110, the background airflow and neighbouring droplets have negligible effect on the droplet 110 and can be ignored. In other embodiments the region of interest 120 may be set to have a fixed width, such that the values of the incident angle and velocity of a droplet for use in queiying the lookup table are always taken at the same distance from the surface of the aerodynamic component 100, irrespective of the diameter of the droplet. Next, in step S203 a lookup table is queried, for each one of the plurality of droplets, to determine a pre-calculated liquid collection efficiency. The liquid collection efficiency describes the proportion of liquid in the droplet which remains on the surface of the aerodynamic component 100 after impact. For example, a liquid collection efficiency of 50% indicates that 50% of the liquid contained in the droplet will be collected on the surface. The lookup table is queried using the droplet size, which in the present embodiment is defined in terms of the droplet diameter, and using the incident angle and velocity that were determined in step S202.

By using a lookup table in step S203, it is possible to obtain the result of a calculation of water retention on the real geometry of the aerodynamic component, in the specific flow and with the required conditions, within a very short time (e.g. a few seconds). At the same time, the calculation result can be based on a highly detailed simulation which may have taken years of CPU time to compute. For example, the liquid collection efficiency values stored in the lookup table can be derived using highly accurate direct numerical simulations, which are computationally intensive but provide the highest fidelity option currently available in the field of computational fluid mechanics.

Then, in step S204 the determined liquid collection efficiencies for the plurality of droplets are combined to determine an overall liquid collection efficiency for the aerodynamic component 100. The overall liquid collection efficiency describes the rate at which liquid builds up on the surface of the aerodynamic component 100 over time, and can be determined by taking into account the droplet distribution, droplet velocities, sizes, and liquid collection efficiencies. The overall liquid collection efficiency can be outputted in a suitable format for use in additional modules of an otherwise conventional design pipeline, for example to perform thin film formation and later icing calculations.

The lookup table may store pre-calculated liquid collection efficiencies for a plurality of discrete values of each of the incident angle, velocity and droplet size. In some embodiments, in response to a determined value of one of the parameters lying between two of said discrete values, the liquid collection efficiency can be determined by selecting the liquid collection efficiency associated with the closest one of said two discrete values. In other embodiments, in response to a determined value of one of the parameters lying between two of said discrete values, the liquid collection efficiency can be determined by interpolating between respective liquid collection efficiencies associated with said two discrete values of the parameter. Interpolation may be preferred when the discrete values are relatively far apart from each other in absolute terms. On the other hand, when the discrete values are finely spaced, the interpolation step may be omitted since the closest available value may provide a sufficiently accurate approximation of the liquid collection efficiency for the current droplet. Referring now to Fig. 3, a computer model of a droplet impacting on a flat surface is illustrated, according to an embodiment of the present invention. By modelling the droplet as impacting a flat surface, as in the present embodiment, the resulting liquid collection efficiency can be applied generally to any arbitraiy geometry of an

aerodynamic component on the basis that at the droplet scale, any surface curvature of the aerodynamic component is negligible and so the geometry can be approximated to a flat surface.

In the embodiments shown in Fig. 3, a computer model of a droplet 310 impacting on a flat surface 300 can take into account parameters such as the incident angle Q with which the droplet 310 enters the computational domain, the velocity of the droplet 310 as it enters the computational domain, U ¥ , and the droplet diameter D. The

computational model can be configured to simulate the break-up of the droplet 310 as it impacts the surface 300, resulting in a certain volume of liquid 311 remaining on the surface 311 and a number of secondary droplets 312 being ejected from the impact site. The trajectories of the secondary droplets 312 can be modelled to determine which of the droplets 312 will leave the computational domain, and which will remain within the domain and ultimately be collected on the surface 300.

In some embodiments, the motion of each secondary droplet 312 after the droplet 310 impacts on the surface 300 can be modelled by determining a velocity and trajectory of the secondary droplet 312 as it exits the computational domain. The velocity and trajectory of the secondary droplet 312 can then be used to extrapolate a path of travel of the secondary droplet 312 to determine whether it would impact on another part of the surface outside of the computational domain. In response to a determination that the secondary droplet would impact on another part of the surface outside of the computational domain, the volume of liquid contained in the secondary droplet can then be counted when determining the overall liquid collection efficiency. This approach can take into account whether liquid contained within a secondary droplet which exits the computational domain would ultimately re-impact upon the surface and add to the total amount of liquid collected on the surface, resulting in a higher liquid collection efficiency than would otherwise be obtained. For example, different generic component geometries may be modelled to determine whether a secondary droplet with a certain velocity and trajectory at the edge of the computational domain would re-impact on another part of the component, and the result for different geometries can be stored in the lookup table. When using the lookup table to determine a liquid collection efficiency for a particular component geometry, it can be checked which of the various generic geometries is the closest match for the component currently being simulated, and the appropriate value can be retrieved from the lookup table.

By summing the volume of liquid in the portion 311 of the droplet 310 which remains on the surface 300 after impact with the volumes of any secondary droplets 312 which ultimately return to the surface 300, the total volume of liquid from the droplet 310 which is collected on the surface 300 can be determined. This can then be compared to the volume of the original droplet 310 to determine the collection efficiency for the given incident angle Q, droplet velocity U , and droplet diameter D. In the present embodiment, the liquid collection efficiency for a droplet is defined in terms of a proportion of the initial volume of the droplet, and provides a measure of how much of the liquid inside the droplet adheres to the surface.

The simulation of the droplet 310 impacting the surface 300 can be performed using a direct numerical simulation method. The computer model for performing the simulation may be configured to take into account the interaction between the droplet 310 and the background airflow, such as deformation of an initially spherical droplet 310. Suitable computational algorithms for modelling the impact of a droplet 310 on a surface 300 are known in the art, and a detailed description will not be provided here so as to avoid obscuring the inventive concept.

Referring now to Fig. 4, a computer model of a plurality of droplets impacting on a flat surface in sequence is illustrated, according to an embodiment of the present invention. The model includes a total of n droplets 410a, 410b, 410c, labelled 1 to n, separated by distances S to S n - The diameters of the first droplet 410c, second droplet 410b, and n* droplet 410a are defined respectively as A, A,— , A. The droplets 410a, 410b, 410c are modelled as they approach and impact upon the surface 400 at an incident angle of Q, with an initial velocity of U ¥ , leaving a portion of liquid 411 on the surface 400 and creating a plurality of secondary droplets 412. As the simulation progresses, the liquid collection efficiency for each droplet 410a,

410b, 410c can be determined at each step in time through the simulation, as the proportion of the initial volume that remains in the computational domain. At the moment of impact the liquid collection efficiency for the droplet is 100%. The liquid collection efficiency then decreases over time, as secondary droplets 412 break off and leave the domain. The liquid collection efficiency for a droplet will eventually settle at a final value, which gives a measure of the total proportion of the liquid in the initial droplet which collects on the surface 400 of the aerodynamic component.

Qualitatively, it would be expected that the behaviour of each droplet 410a, 410b, 410c would be influenced by other nearby droplets in a chaotic manner, for example due to the surrounding airflow being affected by the presence of nearby droplets, and the droplet being impacted by and coalescing with secondary droplets from other nearby impact sites as it approaches the surface 400. Accordingly, it has not previously been attempted to carry out direct numerical simulations of the near-surface behaviour of droplets as they approach and impact upon the surface, for real component geometries, as such simulations are highly computationally intensive and may take several years of CPU time to compute for a realistic number of droplets. As explained above, for these reasons conventional modelling approaches rely on approximations which effectively ignore all fluid dynamical related physics in the close vicinity of the body.

However, as noted above, investigations carried out by the inventors have

demonstrated that the collection efficiency for any given droplet is largely independent of droplets impacting the surface before and after the current droplet. To put it another way, a sequence of droplets impacting the surface in turn are shown to exhibit self- similar behaviour, as will now be described in more detail with reference to Figs. 5 and 6. As a result, a given size and impingement angle will produce a splashing pattern which can be interpreted quantitatively, irrespective of when the respective drop impacts the surface.

Referring now to Fig. 5, a computer model of a plurality of droplets impacting at neighbouring impact sites on a flat surface is illustrated, according to an embodiment of the present invention. As in the model of Fig. 4, the model of the present

embodiment includes a plurality of droplets 510a, 510b, 510c and a flat surface 500 on which the droplets impact 510a, 510b, 510c. The model shown in Fig. 5 differs from the one in Fig. 4 in that the plurality of droplets 510a, 510b, 510c impact on a plurality of impact sites 511a, 511b, 511c, as opposed to a single impact site. In Fig. 5, each droplet D is identified using the notation Dy, where i denotes the impact site (1, 2 or 3 in Fig. 5), and j denotes the position of the droplet in a sequence of droplets approaching the i th impact site.

In the present embodiment three impact sites are 511a, 511b, 511c are illustrated, but in other embodiments any number of impact sites maybe modelled. Three impact sites 511a, 511b, 511c are used in the present embodiment since the dynamics of the droplets impacting the middle impact site 511b can replicate a typical droplet in a large scale system, where additional impingement events are likely to happen at nearby points. Droplets impacting the middle impact site 511b are less likely to be affected by events occurring at other impact sites beyond the neighbouring two impact sites, and so for computational efficiency only three impact sites 511a, 511b, 511c are modelled in the present embodiment.

In the model shown in Fig. 5, the distance Sy between the droplets for each impact site is set to be the same, and is set equal to 25 drop diameters. Any two consecutive droplets for any given impact site in the system, Dy and D ¾-+p , are therefore separated by the same distance S = 25 D, as do neighbouring droplets for adjacent impact sites, for example the pairs (D n , D ,) or (D 23 , D 32 ) in Fig. 5. The distance between neighbouring droplets, Sy, maybe explicitly defined in the initial simulation parameters or maybe estimated from a more general measure of liquid density (e.g. gmfy). Accordingly, droplets in the middle region ( D 2I , D 22 , D 23 ... Do,,) are shifted vertically in comparison to the positions of droplets in the neighbouring regions to maintain a uniform separation between droplets. Simple trigonometry shows that in the present embodiment the distance between adjacent impact sites 511a, 511b, 511c is therefore equal to 21.65D. It should be appreciated that the geometric arrangement of droplets 510a, 510b, 510c illustrated in Fig. 5 is used in order to simplify the calculations, and in other embodiments a different droplet distribution may be used when modelling a plurality of impact sites.

In the present embodiment, the model can take into account the effect of impact events at neighbouring impact sites on a droplet as it approaches and impacts the surface. For example, secondary droplets 512 from neighbouring impact sites with splashing dynamics likely to be affected by microdrop interaction and coalescence from the different sources in the computational domain. A model such as the one shown in Fig.

5 may therefore provide a more accurate simulation of the liquid collection efficiency on a solid surface, for a given set of initial simulation parameters. In the models shown in Figs. 4 and 5, the initial distribution of droplets in the computational domain can be set based on the liquid water content of a typical cloud, which can be used to approximate the mean distance between droplets in the cloud. Studies by the inventors have shown that changing the separation distance between droplets in a model such as the one shown in Fig. 4 or 5 within a certain margin does not produce a significant difference in the final splashing results and the calculated collection efficiency. The initial distribution of droplets can therefore be set to have a separation between droplets which is similar to the approximate mean distance between droplets in a cloud, thereby providing a more efficient simulation. Figure 6 is a graph plotting the collection efficiency profile of the intermediate

(excluding the first and last) drops on the middle impact site 511b in Fig. 5 collapsed onto a single time frame, with t = 80:0 as the time of initialisation of each drop. Figure 7 is a graph plotting the splashing efficiency profile of the intermediate (excluding the first and last) drops collapsed onto a single time frame, with t = 80:0 as the time of initialisation of each drop.

As in the case of the single impact site analysis shown in Fig. 3, the simulation of multiple drop dynamics taking into account neighbouring impact sites, as shown in Fig. 5, reveals the similarity in the retention efficiency and splashing profiles between intermediate droplets in the sequence when comparing the individual evolutions, as demonstrated by the graphs plotted in Figs. 6 and 7. The self-similar structure in the flow exists for all drops except the first and the last in the sequence, which represent special cases due to the first drop being the only one impacting a completely dry surface and the final drop no longer being affected by subsequent impingement events. As shown in Fig. 6, each retention profile also captures the effects of the subsequent impact event in the form of a sharp drop roughly 25 time units after the initial impact, coupled with a corresponding increase in the amount of liquid splashing away from the surface, as shown in Fig. 7. Accordingly, a collection efficiency that is computed for a given droplet in the computational domains shown in Figs. 4 or 5 can be applied to other similar droplets which enter the computational domain, irrespective of their position in the sequence of droplets 410a, 410b, 410c impacting upon the surface of the body 400. Hence it is feasible to compute liquid collection efficiencies for different combinations of parameters in advance and store the results in a lookup table.

Referring now to Fig. 8, a flowchart showing a method of validating a design for an aerodynamic component is illustrated, according to an embodiment of the present invention. First, in step S801 a candidate design to be evaluated is received. Then, in steps S802 to S805 a computer-implemented method such as the one described above with reference to Fig. 2 is used to simulate liquid collection on the aerodynamic component. Initial simulation parameters including the geometry of the aerodynamic component are defined in step S802 based on the candidate design for the aerodynamic component. Steps S803, S804 and S805 can be performed in a similar manner to steps S802, S803 and S804 respectively, and for the sake of brevity a detailed description will not be repeated here.

Next, in step S806 the determined overall liquid collection efficiency for the

aerodynamic component is compared to an acceptable design limit. In response to the determined liquid collection efficiency exceeding the acceptable limit, the aerodynamic component may be redesigned, and steps S802 to S806 can be repeated to validate the redesigned component. On the other hand, in response to the determined overall liquid collection efficiency being within the acceptable limit in step S806, then in step S807 the design can be validated, meaning that the design is deemed to be acceptable from a perspective of the overall liquid collection efficiency. Subject to the design meeting any other applicable design criteria, a physical embodiment of the designed component may be manufactured in step S808, according to the design that was validated in step S807.

Referring now to Fig. 9, a flowchart showing a method of generating a lookup table for determining a pre-calculated liquid collection efficiency is illustrated, according to an embodiment of the present invention. The method can be implemented using a computer model such as the one shown in Fig. 4 or Fig. 5.

First, in step S901 a computational domain for modelling a flow of air approaching a solid surface at an incident angle is defined. Then, in step S902 a computational fluid dynamics algorithm is used to determine a steady-state airflow within the

computational domain.

Next, once the background air flow has reached its steady state, liquid droplets are prescribed to enter the computational domain at a desired location in step S903. Here, the liquid droplets are initialised at the boundary of the computational domain, the droplets having an initial size and an initial velocity. In the present embodiment any given droplet is considered to be initially circular in the two-dimensional environment, with an origin located at ( i; yd. The initialised droplet inherits the local velocity field of the background air flow.

Then, in step S904, the behaviour of the droplet as it approaches and impacts the surface is simulated using the model, taking into account interactions between the droplet and the steady state airflow, to determine a liquid collection efficiency on the surface when the droplet impacts the surface. The simulation is progressed through time as the initialised droplet is advected towards the surface of the solid body. During the simulation, the droplet shape is subject to physical deformations up to the time of its impact. Full hydrodynamic coupling maybe used to determine the mean trajectory and shape of the droplet at each step in time through the simulation. In the present embodiment, a direct numerical simulation is performed without relying on any assumptions.

When calculating the liquid collection efficiency, the direction of a secondary droplet as it exits the computational domain may be taken into account to determine whether the secondary droplet would later re-impinge on the surface, or would permanently exit the domain. The volume of the secondary droplet is only counted in response to a determination that the secondary droplet would not re-impinge on the surface.

During step S904, in the present embodiment the liquid droplets fully interact with the flow and no assumptions are made in regard to particle trajectories, which in the near vicinity of the solid surface are computed as part of the time-dependent solution of the fluid dynamics problem itself. Consequently, the simulation of droplet behaviour within the region of interest can take into account phenomena such as droplet deformation and splashing as a result of impingement onto solid or liquid covered surfaces. In this way, embodiments of the present invention can provide a more accurate estimate of the liquid collection efficiency for any given droplet, while simultaneously reducing the processing burden during modelling of a specific aerodynamic component through the use of a lookup table. In contrast, conventional methods of simulating impacts of droplets onto a solid surface involve de-coupling the droplet dynamics from the background air flow, and therefore provide an estimate that is inherently less accurate than in embodiments of the present invention.

Finally, in step S905 the determined liquid collection efficiency is stored in a lookup table, wherein the liquid collection efficiency is associated with the incident angle, initial velocity and initial size of the droplet.

Using a method such as the one shown in Fig 9, a lookup table maybe generated which contains data produced using more than 10,000 CPU hours and several Tb (terabytes) of data, as an example. In spite of the enormous computational effort involved, the resulting lookup table may only measure several Kb (kilobytes). Accordingly, the lookup table can be easily stored and quickly accessed.

Referring now to Fig. 10, a system comprising apparatus for generating a lookup table for determining a pre-calculated liquid collection efficiency and apparatus for simulating liquid collection on an aerodynamic component using the lookup table is illustrated, according to an embodiment of the present invention.

The system comprises a first apparatus 1010 configured to generate a lookup table, and a second apparatus 1020 configured to query the lookup table generated by the first apparatus 1010 to determine a pre-calculated liquid collection efficiency corresponding to the determined incident angle and velocity and the size of one of a plurality of droplets. The first apparatus 1010 comprises one or more processors 1011 for executing computer program instructions, and memory 1012 in the form of a suitable non- transitory computer-readable storage medium. The memory 1012 is arranged to store computer program instructions which, when executed by the one or more processors 1011, perform a method such as the one described above with reference to Fig. 9, to generate a lookup table. In the present embodiment the first apparatus 910 stores the generated lookup table in a local storage unit 1013. However, in another embodiment the first apparatus 1010 may upload the generated lookup table to a server accessible by the second apparatus 1020, or may transmit the generated lookup table directly to the second apparatus 1020.

The second apparatus 1020 also comprises one or more processors 1021 for executing computer program instructions, and memory 1022 in the form of a suitable non- transitory computer-readable storage medium. The computer program instructions stored in the memory 1022 of the second apparatus 1020 are configured to perform a method such as the one described above with reference to Fig. 2, to determine an overall liquid collection efficiency for an aerodynamic component. While performing the method, the computer program instructions are configured to query the lookup table generated by the first apparatus 1010.

As described above, in the present embodiment the second apparatus 1020

communicates with the first apparatus 1010 to query a copy of the lookup table stored locally at the second apparatus 1020. However, as described above, in other embodiments the second apparatus 1020 may query a copy of the lookup table stored at a server, or may query a copy in local storage within the second apparatus 1020.

Embodiments of the invention can provide accurate predictions for the splashing dynamics near the surface of an aerodynamic component, and in particular for the rate at which water remains on the surface as a result of the impact process. In contrast to conventional modelling approaches which rely on approximations that neglect all fluid dynamical related physics in the close vicinity of the body, embodiments of the present invention can take into account the detailed physics involved, such as drop

deformation, topological changes such as drop coalescence and/ or rupture and the interaction of interaction of drops with the surface and other drops impacting neighbouring surface points. The results of the detailed direct numerical simulation can then be stored in a lookup table, for later retrieval and use in simulations of real component geometries. In this way, a simulation to produce an updated, physically motivated water collection efficiency estimate for any given aerodynamic component would only take a few seconds in addition to the present methodologies, while making use of extremely accurate calculations that required an immense computational effort to complete. Whilst certain embodiments of the invention have been described herein with reference to the drawings, it will be understood that many variations and modifications will be possible without departing from the scope of the invention as defined in the accompanying claims.