Brandon James Milne Robertson^[1], Faculty of Environment and Technology, University of the West of England

Abstract

This paper aims to investigate the effects of changing the shape of the aerofoil of a wing by changing its camber, thickness and maximum camber location to identify the design trends where the ground effect has the largest impact. This investigation conducted its primary research using Computational Fluid Dynamics (CFD), validated against wind-tunnel data and several full factorial design of experiments, and found that the only significant variables that affected the lift-to-drag ratio of a wing utilising the ground effect were the maximum camber and angle of attack of the wing against the freestream. This investigation also found a potential area to optimise a wing utilising the ground effect between zero and six degrees angle of attack.

Keywords: Ground effect, aerodynamics, CFD, wind tunnel, NACA aerofoil, optimisation

Nomenclature

• Precision error
• Bias error
• Drag coefficient
• Lift coefficient
• Standard deviation
• Uncertainty interval
• Mean
• Data point i of n
• Drag force
• Height above ground to chord length ratio
• Coverage factor
• Lift force
• Lift-to-drag ratio
• Number of data points
• Dynamic pressure
• Wing planform
• Velocity
• Density
• Positive value of the inverse of the standard normal cumulative distribution of the data

Introduction

The ground effect is a phenomenon that occurs when cambered bodies used to generate lift come into close proximity with the ground. This interaction between the lifting surface and the ground causes the lift-to-drag ratio of an aircraft to increase; this is often described by pilots as a ‘floating sensation’. However, considering all aircraft are influenced by the ground effect during take-off and landing, research into the area has been very limited as the use of ground effect was thought to be restricted to areas where the ground was relatively flat and the chord and span of the aircraft were both relatively large. Due to the more developed understanding of the nature of pollutants from aviation and its effect on the environment, the ground effect is now being investigated as a means to improve flight aerodynamics, thus reducing fuel consumption.

Wing in Ground Effect (WIG) vehicles are experimental aircraft that utilise the ground effect to cover large expanses of water and flat ground more efficiently than their freestream counterparts due to increases in lift-to-drag ratio. As long as the water is calm, the aircraft can fly at a low level, of around one chord length, so that vortices from the wing are prevented from growing, and lift to drag increases. This is a field that has great potential for very efficient aircraft that may be able, for example, to fly transatlantic, transpacific or transindian which would greatly reduce the impact of aircraft on the environment as they would require less fuel to remain in the air over the majority of the flight. The other benefit of this technology to society is the vastly improved speeds attainable by a WIG vehicle compared to a boat. This could be important in situations such as a rescue vehicle requiring extreme mobility and speed.

The investigation aimed to further understand how variations in maximum camber, maximum camber position, maximum thickness and angle of attack of a wing affect the influence of the ground effect in flight. The focus is on identifying the trends of these four variables and their effects on lift-to-drag ratio. This is so that further research is able to optimise the areas displaying the most effect to find a range of aerofoils that are more suited for use in WIG vehicles. This research has also attempted to investigate the use of NACA 4 series aerofoils for WIG vehicles which is an area of little research at the time of writing as often aerofoil shapes more suited to transonic flight are considered. NACA 4 series aerofoils are used in this investigation due to their numbering system which enables a parametric study. In the past, studies have been undertaken to assess uses of the ground effect on a number of different series of aerofoil and have found significant variations to freestream conditions suggesting that this theory could be implemented with significant impact on aviation as a whole.

Research questions

The proposed research questions are as follows:

1. What are the trends associated with changing the maximum camber position, maximum camber and maximum thickness and angle of attack of a NACA 4 series aerofoil with its lift-to-drag ratio when the wing is utilising the ground effect at an h/c of 1?
2. Are there any areas of further research for a NACA 4 series aerofoil wing using the ground effect?

Aims

To summarise, this project aimed to investigate how a NACA 4 series wing utilising the ground effect could be optimised for its lift-to-drag ratio if the maximum camber, maximum camber position and maximum thickness are changed. It also attempted to identify how changes in angle of attack affect the influence displayed across the three independent variables. The knowledge gained from this research project can potentially be used in further efforts to fully optimise aerofoil design in WIG vehicles.

Objectives

1. To investigate experimental data regarding the NACA 4412 aerofoil in the ground effect.
2. To conduct a numerical analysis of this aerofoil using computational fluid dynamics (ANSYS).
3. To conduct a series of CFD runs with variations of angle of attack, maximum camber, maximum camber position and maximum thickness to model the trends associated using a design of experiments method.
4. To discuss the results of these simulations using statistical analysis and to identify an area of local optimisation for future research.

Literature review

The ground effect is a phenomenon associated with lift-generating surfaces moving close to a solid surface, in that their lift-to-drag ratios are increased compared to a freestream model. This occurs because the solid surface prevents wing-tip vortices from becoming as large as they may for a freestream wing, as wing-tip vortices dissipate when colliding with the ground. Wing-tip vortices are caused when the pressure on the top and bottom of a wing’s surfaces equalise at the end or side of the wing as it uses these pressure differences to generate lift. Since wing-tip vortices are the direct cause of downwash, reduction of the size of wing-tip vortices also decreases downwash for a wing. The coincidence of a wing so close to the ground also causes ram pressure, whereby the flow is compressed between the wing and ground, thus providing an extra force in the form of lift (Scott, 2003). Figures 1 and 2 (Udris, 2014) show that by limiting the downwash from a wing, and considering that lift always acts perpendicular to the freestream and that downwash forces the air flow towards the ground, the direction of the normal force to the wing has been moved closer to the vertical. This then means that the vertical component, lift, has increased and that the horizontal component, drag, has decreased. This plus the reduction of induced drag caused by the reduction of the size of wing-tip vortices is the current theory behind the ground effect.

Figure 1: Effects of ground effect on wing-tip vortices

Figure 2: Freestream wing-tip vortices

Research into the ground effect has so far resulted in several WIG vehicles being developed that utilise this phenomenon. These are more efficient than similar-sized aircraft, due to their increased lift-to-drag ratios, and are much faster than marine vehicles (Halloran and O’Meara, 1999: 3). The increase in lift-to-drag ratio makes WIG aircraft an attractive prospect for applications that may require take-off on a short runway or over a long periods of flight over a relatively calm body of water. The Royal Australian Navy has conducted a review (Halloran and O’Meara, 1999: 3) to identify the possibility of using WIG vehicles in a military setting and found that there are issues with using a WIG vehicle close to a body of water such as increased corrosion due to the salt water, the possibility of impact with a wave at high speed and the likelihood of water ingestion into a turbofan engine. Halloran and O’Meara (1999: 3) also identified the possibility of salt build-up on the blades of a turbofan, altering the flow through the compressor and eventually causing compressor stall. This means that solutions need to be identified for these problems, or the maintenance costs for a WIG aircraft could remain relatively high. They did, however, identify that there are no technological barriers to the introduction of the WIG vehicle, meaning that should research deem it appropriate they would be unrestricted in waiting for future technology for conception.

A study was published (Malti, Hebow and Imine, 2016) that aimed to validate the literature surrounding the use of the ground effect using CFD. This study considered two-dimensional flow theories and CFD runs for a NACA 0015 aerofoil at varying angles of attack. Although this would be a good estimate as to the aerodynamic properties of the wing in three-dimensional flow, this estimate is limited in that it assumes that a wing has an infinite span and so does not take into account any of the aerodynamic effects around the wing tips. Since the concept of downwash is so fundamental to the ground effect, considering the flow around a two-dimensional wing may be much more limited in solving calculations close to the ground than its application to a freestream condition.

Mosaad et al. (1997) conducted a short CFD investigation into investigating the influences of the ground effect when only the camber of an aerofoil is changed. Their research influenced the methodology of this investigation as they utilised the properties of NACA 4 series aerofoils to investigate this effect. Their investigation determined that an increase in camber increased the overall aerodynamic efficiency of the aircraft and suggested that the optimum aerodynamic conditions for an aerofoil occur at around 4 degrees. This investigation suggests that should the primary results of this investigation agree with the results of the one undertaken by Mosaad et al. then the results could be considered replicable.

Furthermore Hsiun and Chen (1995) solved the Navier Stokes equations using finite volume method to identify the characteristics of WIG aerofoils. This is especially important to this investigation as it included the use of numerical data for NACA 4 series aerofoils and identified that the most appropriate aerofoil for a WIG vehicle has a large camber and small thickness. This paper will therefore be validated through a separate numerical theory if the conclusions reflect on Hsiun and Chen’s (1995).

Methodology

This investigation utilised ANSYS CFX to investigate the ground effect. It was therefore necessary to validate the results for one of the investigated aerofoils (NACA 4412) against previous experimental data (Ahmed, Takasaki and Kohama, 2007) for the ground effect to identify if numerical analysis was appropriate for interactions regarding the ground effect. This literature would be compared to a series of two-dimensional flow CFD runs which would at the very least identify the accuracy of CFD at modelling the pressure effects of the moving boundary underneath the wing.

This method of validation did not analyse the effectiveness of the three-dimensional flow CFD runs; however, as they would be attempting to simulate wing-tip vortex propagation, and so wind-tunnel experimentations were undertaken to validate both the three-dimensional runs conducted using numerical analysis and the results of the investigation.

It should be noted that a mesh independence study was conducted for both two-dimensional and three-dimensional flow CFD runs in ANSYS. The use of a mesh independence study before attempting to find numerical results in this way means that a rough optimum can be adopted for the investigation; the mesh can be refined so that its influences on the variables obtained are negligible without using a large amount of time and processing power. These can be graphically shown in Figures 3 and 4.

Figure 3: 2D Mesh Independence Study for a NACA 4412 in Ground Effect

Figure 4: 3D Mesh Independence Study for a NACA 4412 in Ground Effect

Computational fluid dynamics

The important non-dimensional value to compare the results of Ahmed, Takasaki and Kohama (2007: 46) to CFD was the Reynolds number of the investigation as this determined the properties of the fluid flow and would be necessary to match for dynamic similarity. Throughout this study, a velocity of 90 knots (46.3 m/s) was used, as research into modern aircrafts that utilise the ground effect indicated that this may be a suitable cruise speed (Wigetworks, 2017) which meant the chord of the aerofoil had to be 0.095 m in order to keep a Reynolds number of assuming International Standard Atmosphere (ISA) conditions. It is also important to note that all of the CFD runs simulated the ground effect by including a moving bottom plate at the same velocity as the inlet’s. The turbulence model used in this paper is k-epsilon as it is able to reliably simulate turbulent flow conditions.

This data was also compared to Abbot and Von Doenhoff’s (1959) research into aerofoil characteristics as this is a widely used publication and contains data over several Reynolds numbers. Considering that the comparison of several data sets may present outliers, quantitative analysis was undertaken using Chauvenet’s Criterion (Glen, 2018) to investigate whether any part of the CFD data does not correlate with the experimental data constituting in a failed validation.

A design of experiments approach was determined to be used, in that a number of CFD runs would be created. The variables changed were: maximum camber with respect to chord length at 2, 4 and 6%, maximum camber position through 30, 40 and 50% of the chord and maximum thickness of 8, 12 and 16% of the chord. These were conducted over angles of attack of 0, 3 and 6 degrees and at a constant h/c of 1. This was chosen because the literature review identified that the ground effect began having little effect past this point and because it matched the h/c used in the 2D comparison data. Therefore the aerofoils used in this investigation are summarised in Table 1.

NACA denomination	Camber (% of chord)	Camber position (% of chord)	Thickness (% of chord)
2308	2	30	8
2316	2	30	16
2508	2	50	8
2516	2	50	16
4412	4	40	12
6308	6	30	8
6316	6	30	16
6508	6	50	8
6516	6	50	16

Table 1: Summary of the aerofoils used in the investigation

Throughout the study, a Reynolds number of was used as it matched the Reynolds number of the experimental data found in literature (Ahmed, Takasaki and Kohama, 2007: 46). This meant that should the two-dimensional flow comparison prove CFD a viable method of investigation at this Reynolds number, then the resulting 3D runs would also be valid, subject to the results of the experimental validation conducted. Therefore using the velocity as mentioned and taking the kinematic viscosity of air to be at sea level, the chord length is set at 0.095 m and the span of the wing is 0.505 m; which also kept the aspect ratio the same as the experimental data at .

Statistical analysis using Minitab 17

Minitab 17 was used to construct and aid in the analysis of the design of experiments due to its capabilities with analysis of variance (ANOVA) tests. Within Minitab 17, a two-level full factorial design of experiments was chosen which included a third mid-point for each of the factors. Although this meant that more CFD runs would have to be conducted, it was considered key to the experiment in that it would greatly improve the reliability of the conclusions to this paper. This method would also automatically generate a factorial ANOVA model which would determine whether each factor was statistically significant in affecting L/D by comparing mean data points. This was crucial to this investigation as although a graph demonstrating the trends of a variable alone may suggest that there is causality, this statistical analysis method will determine whether the results found are more likely due to chance errors within the mesh or other areas of the numerical analysis. For this investigation, 0.05 was determined to be a suitable significance level as this infers that for every 200 sets of numerical data only one would have data sets outside of this trend.

Wind-tunnel runs post-numerical investigation

Since the investigation later found that the only significant variables that affect the lift-to-drag ratio were the maximum camber and angle of attack, a NACA 2316 and a NACA 6316 were created and tested in the wind tunnel using a non-moving flat bottom plate. Importantly, the runs would be conducted using dynamic similarity to the CFD runs, in that the Reynolds number of the set-up would be equal to . Geometric similarity could not be obtained and so the results of the wind-tunnel runs would be compared to the CFD using non-dimensional values of lift and drag coefficient. This, however, was not thought to be the greatest issue in the experimentation, as the non-moving bottom plate was thought to be likely to cause viscous interactions in the flow. Therefore the values found from this method would be used to validate the CFD data with some error assumptions.

In this investigation, error assumptions were considered as the product of bias and precision errors within the testing environment. In this instance the precision errors of the equipment could be simply modelled using a Gaussian distribution shown in Equation 1.

Equation 1: Precision error calculation

In this equation, K is given a constant of two to represent a 95% confidence limit and represents the standard deviation of the data. The bias error, however, would need to be calculated using the propagation of errors method, which would identify each of the different error sources and base a cumulative error on this data. This was chosen as it was suitable for single reading experimentation, which this data was assumed to be considering the failure to identify the variation with changes with time.

Results and discussion

Comparison of the NACA 4412

The two-dimensional CFD runs could be compared to the experimental data conducted by Ahmed, Takasaki and Kohama (2007: 46) and Abbot and Von Doenhoff (1959) by converting the data to dimensionless coefficients using Equation 2 and Equation 3.

Equation 2: Rearranging the lift equation

Equation 3: Rearranging the drag equation

This then meant that the numerical analysis had produced directly comparable data shown in Figures 5 and 6.

Figure 5: Comparison of lift coefficient data to literature

Figure 6: Comparison of drag coefficient data to literature

Figure 5 shows a strong correlation between the coefficients of lift found from the CFD and previous experimental data suggesting that the use of numerical analysis in modelling the effects of the ground effect when regarding the lift of a body is viable. Interestingly, Figure 6 shows a greater deviation in drag coefficient between experimental data sets than between the CFD runs and either data set. To understand whether there was any outlier data and to numerically consider the correlation seen in the coefficient of lift and coefficient of drag graph, Chauvenet’s Criterion was implemented at both 0 degrees and 6 degrees. First the means, standard deviations and tau value of the data sets were found. From this, the maximum and minimum values accepted before the data was considered to be an outlier were identified using Equations 4 and 5.

Equation 4: Calculating the maximum non-anomalous value

Equation 5: Calculating the minimum non-anomalous value

None of the data found from the CFD runs were found to be outlying to the rest of the data. Therefore the two-dimensional CFD runs were deemed to be close enough to the experimental data that the project could continue with reasonable certainty in its results. This validation, however, is only suitable for the range of angle of attacks used in this experiment and at a h/c of 1 as these were the only values tested.

CFD results

A large quantity of data was collected regarding the lift and drag values for each wing which compiled into a number of lift-to-drag ratio values under each specific given condition. This could then be inputted directly into the design of experiments. However, the data was also analysed graphically to identify whether the results found thus far were supported by the literature review conducted and to identify possible areas of optimisation. An example of this is Figure 7, which graphically shows that an optimum angle of attack falls within the bounds of 0 degrees and 6 degrees from the freestream for this wing. This is important as further research regarding the optimisation of a wing with this aerofoil can now be focused onto this smaller range of data points. It is also important as it correlates with the literature review (Mosaad et al., 1997). However this may be a local optimum as there may be another region of high lift-to-drag ratio after 6 degrees angle of attack.

Figure 7: Graph showing optimum lift-to-drag conditions for the NACA 4412

The CFD runs conducted may have affected the conclusions of this investigation. For instance, the results found from all of the CFD runs are entirely based on the assumption that any validation that can be accrued will show that the numerical models used in the software are accurate at modelling the effects on the flow when a wing is experiencing the ground effect. This may not be the case, however, as although computational fluid dynamic software is often used in modelling the flow of a freestream model, the model may not solve accurately for the time dependent wing-tip vortex’s interaction with the ground.

Design of experiments

All CFD runs were tabulated in and inputted into Minitab 17 as a series of two-level full factorial design of experiments; one for each angle of attack tested and one considering angle of attack as a fourth variable. The design of experiments considering the angle of attack as a variable yielded the most interesting results. A main effects plot was generated using Minitab 17 which plotted the minimum and maximum values for each independent variable against their mean lift-to-drag ratio considering the other factors. This is detailed in Figure 8, which shows that of the four variables considered in this investigation, increasing the maximum camber and increasing the angle of attack of the aerofoil both increase the effects of the ground effect whereas the camber position and thickness values almost have a negligible impact. This would suggest that this investigation is supported by Hsiun and Chen’s (1995) numerical investigation into wings operating in the ground effect.

Figure 8: Main effects plot from four variable design of experiments

The same design of experiments also resulted in the interaction plot, Figure 9. The interaction plot between maximum camber and angle of attack (bottom left corner) shows an interesting behaviour. At low angle of attack (AOA = 0°), increasing maximum camber may result in slight reduction in L/D. However at high angle of attack (AOA = 6°), increasing maximum camber may actually help to significantly increase L/D.

Figure 9: Interaction plot from four variable design of experiments

Statistical analysis

The design of experiments runs as a whole could also be used to identify the statistical significance of each variable using an ANOVA test. The normal plot of standardised effects returned from running an ANOVA test on the four-factorial design of experiments is shown in Figure 10. This graph shows that the only significant variables in this particular investigation were the maximum camber, angle of attack and the interaction between these two variables. Therefore only the maximum camber position, angle of attack and their interaction can reject the null hypothesis, meaning that the graphs observed from the four-factorial design of experiments for each of these factors can be taken as significant and should be considered in further research.

Figure 10: ANOVA test for four-factorial design of experiments

Wind-tunnel validation for CFD runs

The wind-tunnel runs conducted were important to validate the use of CFD in three-dimensional flow for a vehicle using the ground effect. Two data points were taken at each angle of attack and were taken up to 10 degrees angle of attack in two-degree increments. The coefficient of lift values found are represented by Figures 11 and 12, which demonstrates a close relationship between the coefficient of lift values found for both the NACA 2316 and the NACA 6316 CFD runs.

Figure 11: NACA2316 coefficient of lift comparison between CFD and wind-tunnel runs

Figure 12: NACA6316 coefficient of drag comparison between CFD and wind-tunnel runs

However, the results for drag coefficient deviate greatly for the wind-tunnel runs compared to the CFD runs, as shown in Figures 13 and 14. This seems to show that the coefficient of drag is not accurately modelled in one of the two sets of data.

Figure 13: NACA2316 coefficient of drag comparison between CFD and wind-tunnel runs

Figure 14: NACA6316 coefficient of drag comparison between CFD and wind-tunnel runs

It should be noted that this experimental method had its own errors associated with it. For example, the use of the non-moving bottom plate within the wind-tunnel chamber would have unrealistically modelled the ground effect, as it would not have assisted in the deviation of the wing-tip vortices due to viscous boundary layer interactions. Instead it would have simply caused an increase in ram pressure below the wing, similar to the two-dimensional flow model but without the interaction of the moving bottom layer and flow. This is likely to have increased the drag values found in the data and could explain the significant gap between wind-tunnel and moving-plate CFD data.

To assess the data’s bias, the propagation of errors method would need to be utilised for both lift and drag coefficient. Therefore considering the lift coefficient equation, Equation 6, the propagation of errors can be considered to be Equation 7.

Equation 6: Rearranging the lift equation

Equation 7: Ascertaining errors in lift coefficient

The errors assumed for each of the variables were 0.5 for the lift force, 0.05 for the planform area due to imperfections and 2 for the dynamic pressure calculation (Tuling and Kanaa, 2017). The lift value was assumed to be constant at its respective greatest value of 17 N and variables that held constant value such as speed, planform and density were simply constant. Thus the bias error in the coefficient of lift from experimental data was calculated to be . The precision errors found were all greater, though, and fell between 0.009 and 0.089.

The coefficient of drag was estimated using exactly the same methodology but assuming that the error in the drag measurement is 0.17 and that the drag force used in the method was equal to 4.2N as this was the greatest drag value (Tuling and Kanaa, 2017). This led to a drag coefficient bias error of . Similarly to the lift values, the drag precision error was much greater and found to be between 0.0034 and 0.078. Considering the set-up used, both of these bias errors found are reasonable and should be expected. Therefore the greatest uncertainty interval for the wind-tunnel investigation is calculated using Equation 8 and found to be for lift coefficient and for drag coefficient.

Equation 8: Calculating uncertainty interval

Although there is not a close correlation in lift coefficient, it can be observed that there are similarities between the moving and non-moving-plate results. This is likely to be due to the h/c chosen for this paper in that the effects of the non-moving plate on the aerodynamic characteristics have been mitigated by having a relatively high h/c for a ground effect vehicle.

Usability of this research

This investigation has proved that CFD is a valid method at determining the lift characteristics of a wing in the ground effect in both two-dimensional and three-dimensional flow. The results of this investigation are also supported by Mosaad et al. (1997) as the investigation found that an increase in camber increases aerodynamic efficiency and that an area of optimisation lies between 0 and 6 degrees angle of attack. Therefore the conclusions to this investigation should be reproducible, assuming that CFD is a valid method of investigating three-dimensional flow around a wing in the ground effect. The usability of this research will rely on further validation through experimental testing in facilities that can run a moving bottom section and for the exact same wing as determined in the numerical analysis.

Conclusions

To conclude, this investigation originally aimed to identify areas of optimisation regarding the lift-to-drag ratio of an aircraft utilising the ground effect considering the aerofoil’s geometric features and considering the angle of attack of the wing. Through a series of CFD and design of experiments runs, it found that the significant variables that affect the ground effect are the maximum camber and angle of attack of an aerofoil. The primary research conducted also outlined the angle of attack range, from 0 degrees to 6 degrees, to potentially be an area of local optimisation as the highest value was found in this range for all bar one of the aerofoils. This is supported by Mosaad et al. (1997). However the maximum camber value needed for a given lift-to-drag ratio should also consider the angle of attack of the wing to the freestream as the main effects plots for maximum camber displayed significant gradient changes between 0 and 6 degrees.

Further research should be conducted into the viability of using the ground effect for a full scale aircraft as no research was conducted in this investigation for parameters such as wing fuselage boundary layer interaction or propwash effects on the boundary layer. Furthermore although the primary research conducted during this investigation found the trends associated with changing each variable, it did not actually specify an optimum wing in an optimum condition. Therefore research into the ground effect should also investigate the properties of a number of different aerofoils to identify whether they follow the same trends as this data, and then efforts should be focused on creating a method that allows aerofoils to be easily identified as the optimum aerofoil for a given scenario in the ground effect using empirical data.

Similarly, a larger range of experimental data into the ground effect concerning three-dimensional flow should be conducted, as that will validate the conclusions discussed in this investigation. This is especially true of the aerofoils used in this investigation which could then be used as a comparison to determine whether CFD analysis is sufficient in analysing bodies utilising the ground effect. Other future research efforts should also be focused towards identifying technologies that reduce the effects of salt water spray from flying close to a body of water into engines or on to the fuselage, as this is a criticism that previous research into the ground effect has not taken into consideration but would be key to the development of an economically viable aircraft in future.

---

Notes

[1] Brandon Robertson is currently completing his MEng in Aerospace Engineering at the University of the West of England and hopes to begin a PhD role in September of 2019.

Acknowledgements

The author would like to thank Dr Budi Chandra for his invaluable advice, expertise and optimism when supervising this paper; Dr Chris Toomer for her efforts with computational fluid dynamics queries and Mr Zac Kanaa for his help developing the experimental results needed for this investigation.

List of figures

Figure 1: Effects of ground effect on wingtip vortices.

Figure 2: Freestream wing-tip vortices.

Figure 3: 2D Mesh Independence Study for a NACA 4412 in Ground Effect.

Figure 4: 3D Mesh Independence Study for a NACA 4412 in Ground Effect.

Figure 5: Comparison of lift coefficient data to literature.

Figure 6: Comparison of drag coefficient data to literature.

Figure 7: Graph showing optimum lift-to-drag conditions for the NACA 4412.

Figure 8: Main effects plot from four variable design of experiments.

Figure 9: Interaction plot from four variable design of experiments.

Figure 10: ANOVA test for four factorial design of experiments.

Figure 11: NACA2316 coefficient of lift comparison between CFD and wind-tunnel runs.

Figure 12: NACA6316 coefficient of drag comparison between CFD and wind-tunnel runs.

Figure 13: NACA2316 coefficient of drag comparison between CFD and wind-tunnel runs.

Figure 14: NACA6316 coefficient of drag comparison between CFD and wind-tunnel runs.

List of tables

Table 1: Summary of the aerofoils used in the investigation.

List of equations

Equation 1: Precision error calculation.

Equation 2: Rearranging the lift equation.

Equation 3: Rearranging the drag equation.

Equation 4: Calculating the maximum non-anomalous value.

Equation 5: Calculating the minimum non-anomalous value.

Equation 6: Rearranging the lift equation.

Equation 7: Ascertaining errors in lift coefficient.

Equation 8: Calculating uncertainty interval.

References

 

To cite this paper please use the following details: Robertson B.J.M. (2018), 'Optimisation of Lift-to-Drag Ratio for a Wing in Ground Effect', Reinvention: an International Journal of Undergraduate Research, Volume 11, Issue 2, http://www.warwick.ac.uk/reinventionjournal/archive/volume11issue2/robertson. Date accessed [insert date]. If you cite this article or use it in any teaching or other related activities please let us know by e-mailing us at .