MECH 203P Intermediate Fluid Mechanics Lab
MECH 203P Intermediate Fluid Mechanics Lab Group 18 Name
Student Number
Contribution %
Alper Bakici
14007530
16.67% Flat plate: Ansys Setup and meshing, and Data Analysis Aerofoil: Geometry design, aerofoil meshing, Data Analysis, and Graphical Representation Lab Report: Aerofoil explanation and theory, general report organisation
Youssef Hegazi
14035158
16.67% Flat plate : Ansys Setup, Data and Sensitivity Analysis, and Graphical Representation (Excel) Aerofoil : Data Analysis and Graphical Representation (Excel) Lab Report: Discussion for Flat Plate and Aerofoil. Graphs for Flat Plate and Aerofoil
Zijian Zhao
14001718
16.67% Theoretical: Flat plate calculations and analysis Error calculations: Superimposing and analysing theoretical and experimental results Lab Report: Formulas, Mathematical Theory, Boundary Layer Thickness, Velocity Profile
Karan Pinto
13055036
16.67% Flat plate: Boundary layer thickness, behaviour and calculation, Theoretical reasoning and explanation Aerofoil: Design and Lift Discussion, Theory Diagrams. Lab Report: Coalition and Organisation, results verification, analysis and discussions.
Kirill Markov
14041801
16.67% Flat Plate: Ansys Setup, Data Analysis, graphical representation (Excel). Meshing: fine and coarse mesh Lab Report: Boundary Layer explanation
George Lundy
14017121
16.67% Flat plate: Boundary layer calculation and analysis. Lab Report: Aerofoil discussion and editing.
Aim: To understand and familiarise ourselves with the fundamental concepts of fluid mechanics using ANSYS CFX and theory.
Part 1 Analysed the flow over a flat plate through computations (ANSYS/CFX)
Theoretical Calculation of Reynolds Number and Boundary Layer Thickness :
Reynolds Number A dimensional that helped determine the nature of flow of the fluid in the duct over the body. It is the ratio of the pressure forces to viscous forces. Larger the Reynolds number smaller the boundary layer. Smaller the Reynolds Number, the greater the dominance of the viscous forces (Inviscid Theory).
Figure 1: The development of boundary layer along a flat plate (not actual representation) Using the analytical solution for the configuration of a flat plate from the theory notes the following terms are defined with respect to the graph: Distance to the leading edge: x Inlet (maximum) velocity: U ∞ Velocity of flow at a given position: u Vertical distance to the plate: y Boundary layer thickness: δ
Reynolds number of a flat plate can be calculated as follows using data from table 3 in the appendices:
ρU ∞x 6 1.184×20.21x μ = 1.844 ×10−5 = 1.2976 × 10 x To find the position of the plate and thus the flow type over it, the maximum length of the plate was used in calculating Reynolds number, which was then compared to the critical value:
Rex =
Rex = 1.2976 × 106x = 1.2976 × 106 × 0.202 = 2.621 × 105 Rex, critical = 3 × 105 to 5 × 105 and since, Rex
≺Rex, critical ,
It can be confirmed that the flow is laminar even at the end of the plate. There will be no transitional or turbulent flow over the plate.
Figure 2: Actual representation of boundary layer along the flat plate Boundary Layer Velocity Variation: In Blasius solution for a laminar boundary layer, the thickness y varies along the velocity of 99% the free stream velocity.
u = 0.99u ∞ Boundary Layer Thickness: The fluid's velocity increases from zero at the surface due to the no slip condition to the free stream velocity (U ∞) at a distance from the surface which is equal to the boundary layer thickness (δ). It was observed that theoretically, boundary layer thickness increases across the plate as shear stress decreases. The value of x used is strictly the length of the flat plate which is equal to 0.202m. δ = 4.91x = 4.912×0.2025 = 1.937 × 10−3 m Re √ x √2.621×10
−3 So the theoretical boundary layer thickness at the end of the given flat plate is 1.937×10 m .
Thickness of Boundary Layer & Comparison with Blasius Solution Process: To obtain a visual of the boundary layer, a crosssectional plane is placed in the YZ direction, along the length of the plate and cut halfway through the width. Along this plane, the boundary layer thickness was defined by any points along the flat plate surface (0.199 → 0.401m) (where 1 the velocity was equal to 20.0079ms ). This gives us a 2dimensional visual in the 3dimensional simulation model. The position of this plane was chosen such that the boundary layer thickness is least affected by the width (resembles most closely to a 2D flow).
Figure 3 : The variation in boundary layer thickness across the length of the flat plate
The data provided by Ansys for the boundary thickness is displayed in the table below. Horizontal position, x (Flat Plate Length)
Vertical position, y (Boundary Layer Thickness)
0.20214
0.00198
0.20352
0.00204
...
...
0.35000
0.003684
...
...
0.40083
0.00271
0.40100
0.00271
Table 1: Data values for boundary layer thickness Preanalysis Observations: Ansys made calculations accounting for the horizontal distance along the entire duct, while the flat plate only starts at 0.199m into the duct. The data used from Ansys starts from 0.20214m onwards because the data between 0.1990.20214m was ignored due to the fluids inability to adjust at the sudden contact with the flat plate, demonstrated by a steeply decreasing boundary layer line (caused by boundary layer conditions).
Results: The boundary layer thickness is offset by 0.001m due to the thickness of the flat plate itself. As a result, the vertical position used for the data was adjusted by subtracting this offset in order to only represent the actual boundary layer thickness. Both table 1 and figure 3 show a maximum boundary layer thickness of 0.002684m(adjusted) at 0.151m into the flat plate (or 0.35000m into the duct).
Analysis: The boundary layer thickness begins to decrease towards the end of the flat plate, which is largely due to a lack of drag created by the walls. As a result, fluid begins to occupy the empty space behind the plate. Furthermore, the boundary layer is not developed from 0, but rather, from 0.001m. The presence of the flat plate thickness is believed to contribute to this offset. Using theoretical calculations, the flat plate is assumed to be thin, thus no additional fluid can affect boundary layer development. However, in this model, the stream of fluid comes into contact with a 0.002m thick flat plate (0.001m in the model due to symmetry) at the leading face. These fluids need to travel forward and cannot be ignored (can easily be explained by control volume or control mass). The clashing fluid is assumed to consume 0.001m of space on top of the near face of the flat plate, resulting in another offset as demonstrated by the graph.
Therefore, during the comparison with theoretical boundary layer below, the boundary layer thickness results from Ansys were adjusted by a 0.002m offset (relative to theoretical) in total. Theoretical Approach: With flow determined as laminar, Blasius solution states that boundary layer thickness at u=0.99umax can be expressed as
y = 4.91x . Substituting the given values (density, √Rex
maximum speed, and viscosity) for Reynolds number, the Blasius solution is thus given as:
y=
4.91x 1.184×20.21x 1.844×10−5
√
Figure 4 representing the Blasius solution falls between the xvalues of 0.199m to 0.401m, as this defines where the flat plate starts and ends. As previously explained, due the presence of the boundary layer (0.001m) and the fluid layer (0.001m) caused by the flat plate leading surface, an offset of 0.002m was added to Blasius’ solution. This allows for a more meaningful comparison between the curves and an analysis of the differences between theoretical and experimental results.
Figure 4: The variation in boundary layer thickness in comparison with Blasius Solution
Additionally, in order to calculate the error between the two sets of boundary thickness, a set of theoretical boundary thickness ‘y’ were calculated using the horizontal positions used by Ansys to indicate the experimental boundary layer thickness: Horizontal position
Theoretical Boundary Layer Thickness
Adjusted Boundary Layer Position
0.20214
0.00024
0.00224
0.20352
0.00029
0.00229
...
...
...
0.35000
0.00167
0.003675
...
...
...
0.40083
0.00194
0.00394
0.40100
0.00194
0.00394
Table 2: Blasius data adjustments to match numerical solution, shortened data set. Preanalysis Observations: The experimental boundary layer curve resembles that of the theoretical Blasius solution. However, the experimental data demonstrates a steep initial increase in the thickness, as well as a decrease towards the end of the plate. (The trend seen by the experimental boundary layer was previously discussed). Although the starting and ending trend of the simulation vary from that of the theoretical relationship, the overall boundary layer thickness demonstrates the correct behaviour. This shows that the simulation of the fluid's behaviour resembles the theoretical model considerably.
Results: Blasius Solution predicts boundary layer thickness of 0.003675m at 0.151m into the plate, compared to the experimental value of 0.003684m at the point (where maximum boundary layer thickness experimentally), or 0.001675m and 0.001684m relative to the top of the flat plate and after adjustment.
Error Calculation Between Theoretical and Numerical The error was calculated in two ways: 1. Percentage root mean square error:
The term [(experimental boundary layer position – theoretical boundary layer 2 position)/theoretical boundary layer position] was used to calculate the percentage error squared for each horizontal position x (provided by Ansys). Only data up to the position of maximum theoretical boundary thickness (0.00035m) was used in order to minimize the influence of the decrease in boundary layer due to the flat plate end on our analysis and increase the accuracy of prediction. The average value for these terms were calculated and the root of the result then computed, giving the percentage root mean square error: (experimental theoretical)/theoretical
2 ((experimental theoretical)/theoretical)
1.07974
1.16583
0.84660
0.71674
...
...
0.01531
0.000234
0.00539
0.000029
Table 3: Table of percentage error and squared percentage error at a horizontal position x By taking the averaging of the results in the second column of table 3 and obtaining the root value of the average, the percentage root mean square error was calculated to be: P ercentage Root Mean Square Error = 41.9%
2. Maximum boundary layer thickness error: The error of the maximum boundary layer thickness between experimental simulation and Blasius solutions:
P ercentage Error =
|Maxmimum experimental−Maximum theoretical| Maximum theoretical
P ercentage Error =
|0.001684−0.001937| 0.001937
× 100%
× 100% = 13.06%
It must be noted that the presence of the end of the plate is one of the major reasons of this difference. The maximum theoretical boundary layer thickness was calculated at the end of the plate where length is 0.202m, as compared to at 0.151m into the plate for the experimental results. The maximum experimental boundary layer results at the respective horizontal position (0.151m) was actually very accurately predicted (0.001675m versus 0.001684m).
Dimensionless Velocity Profile & Comparison with Blasius Solution
Process: In order to obtain a dimensionless velocity profile with ANSYS, the following steps in the Results section of CFX extension were taken. Firstly, a velocity profile isoclip was created. It’s visibility parameters were set as velocity w ≤ 20.0079 (which is 99% of the U ), m Fixed value of z (horizontal position) to analyse the velocity profile at a single horizontal position. In this case it is chosen to be the position where maximum boundary layer velocity occurs (most representative and informative).
Figure 5: The variation of velocity along the flat plate against Blasius Solution.
A set of velocity at different vertical position y was then taken as follows:
Vertical Position, y (m)
Vertical Position Offset, y (m) off
1 Velocity, u (ms )
0.00000
0.00100
0.00081
0.00003
0.00103
0.36239
...
...
...
0.00220
0.00320
19.57952
0.00266
0.00366
20.00790
Table 4: Velocity along height y at the point of greatest boundary layer thickness, shortened
Again, the vertical position y was offset by 0.001m to give a more informative comparison.
= yx × √Rex , the Reynolds number at the given position was calculated to be u Re(0.151) = 442.66. The term an also be calculated with the provided data, ultimately um c
Given that η
resulting in the table below: u/u m
η
0.00004
0.00000
0.01793
0.07467
...
...
0.96880
6.46216
0.99000
7.78965
Table 5: Data values for dimensionless velocity profile, shortened
The experimental velocity profile at z=0.35000m (0.151m into the plate) was then graphed as above. When u=0.99u , η has a value of 7.790. max
Blasius solution comparison: In order to obtain the theoretical Blasius solution velocity profile curve, the following set of data was obtained from the textbook: u/u m
η
0
0
0.0664
0.2
...
...
0.9878
4.8
0.9916
5
Table 6: Blasius data values for velocity profile, shortened
Again, the shape of the two curves resemble each other, indicating that our results are logically and correctly collected. Both curves show a positive correlation between u/u and η. Also, both m curves converge slightly upwards when u/u gets close to 1. On the other hand, the deviation m shows that there are still errors occurred. The similarity in trend signifies that the simulation, despite the numerical differences, modelled the correct relationship of a dimensionless velocity profile.
Error Calculation Between Theoretical and Numerical In order to analyse the error, a 6th order (maximum allowed on Excel) polynomial regression was performed to the Blasius solution curve. This is because the data points for the specific u/u values that matches the experimental outputs cannot be acquired from other resources. m The regression helped to achieve a set of η values at any u/u values that are needed, although m with a compensation of slight inaccuracy. The function achieved is as follows:
η = 154.31x6 − 412.68x5 + 415.14x4 + 191.49x3 + 39.665x2 + 0.1077x + 0.02 u where x = Um .
The following sets of data was then calculated using this equation, to match the experimental results: u/u m
η (theoretical)
0.00004
0.020004358
0.01793
0.033622991
...
...
0.96880
4.327219708
0.99000
4.806789892
Table 7: Data obtained from Excel polynomial as required, shortened The percentage root mean square error was then calculated as follows:
|experimental η theoretical η|/theoretical η
(|experimental η theoretical η|/theoretical 2 η)
0.84034
0.706175
0.57195
0.327130
...
...
0.49337
0.243418
0.62055
0.385083
Table 8: Table of root mean square error
Similar to the boundary layer error calculation, the following error was achieved:
P ercentage Root M ean Square Error = 48.4% However, there are sources of inaccuracy of this percentage error. Firstly, the regression model was used instead of the accurate Blasius solution values for the calculation. This only gives estimated values of η and therefore can create some dispense from the actual error. Also, not many sets of data were taken from Ansys (due to the limitations of meshing size and other parameters), which means that the actual shape of the experimental velocity profile curve might not be fully represented in this dimensionless graph. Since the error was calculated based on the differences between each set of theoretical/experimental data and there is an obvious convergence on the graph which is difficult to be fully described by the data acquired, it is further proved that this error calculation is subject to inaccuracy. On the other hand, the percentage error for the η at u=0.99umax was also calculated:
P ercentage Error = |Experimental−Theoretical| × 100% Theoretical
P ercentage Error =
|7.7897−4.91| 4.91
× 100% = 58.6%
Notice that 4.91 is the accurate Blasius solution for η at u = u . max
Skin Friction Coefficient Theoretical Local Skin Friction Coefficient for a flat plate with a laminar boundary layer aligned parallel to a stream is:
C f =
0.664
√Rex
=
0.664
√2.91×105
= 0.00119
Process: The skin friction coefficient was determined through the use of Ansys CFXpost and excel. First, a line was created at the top of the flat plate in the zdirection. This was used in order to define the direction of shear stress. A chart on Ansys was then plotted comparing the location along the line against the wall shear stress. This data was exported into excel and then used in the equation C f = τ . The following graph was then created:
√Rex
Skin Friction Coefficient Across the Plate
Figure 6: A variation of the value of the skin friction coefficient along the flat plate.
From the simulation in Ansys and using excel, the largest skin friction coefficient along the flat plate is determined as:
C f = 0.00121 In comparison to the theoretical value of 0.00119, this demonstrates a 1.68% error , demonstrating high accuracy between the simulation and theoretical value. This can be expected as the flow is better developed over the flat plate. From figure 6 we can see the strange data before the flat plate, due to the contact between the fluid and the edge of the flat plate in the model.
Numerical and Analytical Variation Analysis (Discussion):
The similarity in behaviour between theoretical relationships and simulated relationships demonstrates an accurate model. However, the numerical differences signify errors in the process.
Throughout the simulation of the flat plate, the models displayed significant similarity to the theoretical models. This is largely due to correct setup of the geometry and appropriate use of meshing. Had any of these variables been input incorrectly,the difference between theoretical and experimental would be significant, both visually and numerically. Visually, both the simulated boundary layer and the dimensionless velocity profile represented similar characteristics and trends to their respective theoretical models. These similarities are due to accuracies in the variables used in the simulation, as well as the implementation of correct procedure in obtaining and utilizing the results. However, numerically, the simulated models varied to the theoretical ones.
The simulated boundary layer, although similar in behaviour, provided different results to those of the Blasius boundary layer. The error between the theoretical and experimental maximum boundary layer thickness is 13.06%, which can largely be attributed to the presence of the end of the plate. Similarly, the dimensionless velocity profile recorded was visually comparable, but provided a numerical error of 58.6%. Moreover, the percentage root mean squared error were calculated to be 41.9% and 48.4% respectively, indicating a relatively notable error between the model and experiment. This reflects the presence of assumptions and limitations in the experiment.
A significant difference between the simulations and the theory is the presence of a duct in the model. The duct influences several variables, which in turn affect the simulations and thus the final results. Theoretically, the fluid flowing over the flat plate is uniform, however, with a duct during simulation, the fluid flow is affected, resulting in a nonuniform flow. The irregularities of flow influence the values of velocity, which consequently impact both the boundary layer thickness and the dimensionless velocity profile. This contributes to the difference in numerical value. Similarly, the walls present in the simulation influence the numerical results, creating an error, as the imperfection of the defined noslip boundary conditions create nonuniform flow.
Another contributing factor is the presence of the flat plate thickness and width. Theoretically, the thickness is considered negligible and flow continues over uniformly, creating the Blasius boundary layer. However, with the simulation, the thickness cannot be ignored, resulting in the disturbance of fluid flow. This affects velocity throughout, resulting in a difference between theoretical and experimental values. The simplification of the 3D model to a 2D graph due to the presence of the width may have attributed to the error too.
Further sources of error include limitations in Ansys itself. A decrease in accuracy was experienced due to the limits in mesh sizing. The limited mesh refinement means that numerical values, specifically the boundary layer thickness, can only be done to a certain accuracy. In other words, refinement is a direct source of error, contributing largely to the numerical differences observed between the analytical and numerical values.
Part 2 Modify the top of the flat plate to create an aerofoil
Flow Past the Aerofoil and Static Plot of Pressure In the duct, the air velocity increases as it flows over the aerofoil and therefore pressure reduces above the aerofoil inducing a transverse lift force.
Figure 7: Theoretical behaviour of pressure along the top surface of the aerofoil This theory can be validated by the velocity contours in figure 8. The flow coming in from the right hand side is interrupted by the aerofoil, resulting in the dispersion of the fluid. This creates local acceleration and results in the contours.
Figure 8 : Visual representation of velocity gradient around the aerofoil, high velocities in orange/red and lower velocities in blue and green will result in pressure difference From the above, the development of the pressure gradient around the aerofoil can be observed. By creating an isoclip to the aerofoil, the static pressure plot can be graphed (figure 10), where the blue line represents the negative static pressure along the upper surface of the aerofoil. For reference, the pressure below the aerofoil has been provided as well.
Figure 10: The plot of the static pressure over the aerofoil.
Coefficient of Pressure and Plot Along Blade Length
The coefficient of pressure represents a nondimensional pressure distribution on a body. Since the flow around a body accelerates and decelerates, there is a difference in the local static pressure and that of the mainstream.
Where,
p is the pressure at a particular point, p∞ is the pressure of the freestream, ρ is the freestream fluid density, u is the velocity at a particular point, U ∞ is the freestream velocity.
Figure 11: Theoretical variation of the pressure coefficient along the chord of the aerofoil.
Figure 12: The plot of the pressure coefficient over the blade length. The following can be inferred from the pressure coefficient plot in figure 12: ● Theoretically, the highest pressure achievable for an incompressible fluid occurs is the stagnation pressure at a stagnation point where Cp = 1. In the plot, this exists at the bottom part of the trailing edge where Cp ≈ 1. ● For a value of Cp > 1 at the bottom of the leading edge, it is indicative of the freestream flow being compressible, which does not seem to be the case with our fluid and must be accounted for as an error. ● The pressure is equal to the freestream pressure at points on the aerofoil where Cp = 0. These points are at the end of the trailing edge where Cp ≈ 0 (due to the formation of a minor wake).
Analysis of Lift Capability of Aerofoil (Discussion) In the scenario where an aerofoil generates lift, it creates a situation where the pressure above the aerofoil is lower than the pressure below the aerofoil. This pressure difference results in an upwards force, generating lift. Multiple simulations were run against the aerofoil from figure 7 to determine if it is capable of achieving lift. The primary concern to achieving lift was ensuring the relationship between the pressure difference was similar to the theoretical relationship. Figure 10 illustrates that the lower surface of the aerofoil experiences significantly larger pressures than the upper surface, representing the correct relationship. Furthermore, all the pressures above the aerofoil are negative, while those below are positive, signifying a large difference capable of generating lift. The coefficient of pressure along the aerofoil was also used to validate whether the aerofoil can generate lift. Figure 12 demonstrates that the upper surface is characterized by a smaller coefficient of pressure than the lower surface (excluding the front and tail of the aerofoil). This relationship enhances the statement that the aerofoil is capable of generating lift. Figures 7 and 8 match the criteria required for lift, meaning that this aerofoil design will work. With regards to the aerofoil stalling due to the presence of too much drag, the angle of attack and the risk of separation are crucially to determining left. Since the angle of attack is 5˚, the fluid flow will remain close to the aerofoil surface, ultimately redirecting it downwards. The pressure recovery area, which is defined as the space between the upper and lower surface in Figure 12. If this region is too severe, separation may occur, however this is not the case with this specific airfoil. As a result, there is no reason to believe that the aerofoil is not capable of lift. Even though, the simulations imply lift, there are limitations in the process that may cause error and misguided analysis. Again, error is caused by the boundary conditions of the walls due to the presence of a duct. Since they are defined as no slip, a boundary layer is created, which signifies nonuniform flow. Additionally, the data presented by figure 12 suggests the flow may be compressible, as the pressure coefficient is greater than 1. This could be due the excess pressure being exerted on the bottom of the aerofoil as a result of it being place in a duct. With these two differences in simulation acting on the model, the results calculated and presented by Ansys will vary from theoretical behaviour. Since these errors have not been calculated for the aerofoil, the degree to which they influence the calculated data cannot be determined. However, this suggests that the expected performance of lift by the aerofoil may be inaccurate.
Sensitivity Analysis & Mesh Refinement Simulations were run with two meshes coarse to increase the accuracy by including more integration or calculation points. The sensitivity analysis only changes 1 parameter (meshing grid refinement) in order to see the result on a simulated accuracy e.g. Boundary Layer Thickness. Fine: 578000 element mesh Coarse : 2640000 element mesh The maximum boundary layer thickness will be used as a reference value to compare the fine mesh and the coarse mesh in order to determine how good the refinement is. With the coarse mesh, the maximum boundary layer thickness is calculated as 0.004650.0015 = 0.00315m (refer to figure 13). The fine mesh creates a boundary layer thickness of 0.003670.002= 0.00167m . This demonstrates a good refinement, and a good sensitivity analysis.
Figure 13: Boundary Layer Thickness for Coarse Meshing The space was further broken down or organised using inflation layers and edge sizing with a view to minimize errors stemming from them. In order, to capture changes accordingly we needed a few thin layers within the boundary layer, which was probably to account for the large variation in the boundary layer thickness.
Appendix: Relevant Working and Project Information Flat Plate Setup A simple flat plate structured was created as geometry in Ansys based off the file provided and adjusted according to the geometrical details given in Table 1: Thickness [m]
Width [m]
0.0002
0.30
Appendix Table 1: The dimensions of the flat plate The flat plate was then placed in the middle of a rectangular wind tunnel section with dimensions as in Table 2. Length [m]
Width [m]
Height [m]
0.60
0.30
0.60
Appendix Table 2: The dimensions of the rectangular wind tunnel In order to simplify the simulation, the duct was sliced in half to take advantage of its symmetrical nature. A material called Labmaterial was created and the fluid properties were modified in CFXpre according to the given flow conditions and dimensions in Table 3. Also, the workbench and CFX files provided were then created and altered as per specifications mentioned in Table 3 to effectively set up a simulation. It should be noted that the inlet velocity is U ∞ which is parallel to the flat plate. Temp ( °C)
Air Density (ρ)[kg/m3]
Velocity (U ∞) [m/s]
Length (L)[m]
Viscosity ( μ ) [Pa.s]
25
1.84
20.210
0.202
1.844E05
Appendix 3: Given specification of simulation and flow characteristics
To run the simulation, after the setup was complete with the correct domains, we decided to run double precision. This was because in the mesh file, there were several elements which varied in size greatly and so double precision allowed more accurate results, regardless of the slower simulation time. Aerofoil Design and Setup An aerofoil was designed roughly to the NACA s7055il specification as shown below. The o diagram shows the tilt of 5 . This design was chosen due to its flat bottom and hence similarity to the flat plate.
Appendix Figure 1: A sketch of the aerofoil design with the labelled components and the flat bottomed aerofoil with chord line of 202mm at an angle of 5° to the fluid flow
To model the flow around the aerofoil, a new file was made to represent the whole duct. This was to allow ease in placement of the aerofoil in the centre of the duct, as shown in figure 2.
Appendix Figure 2: Image of the full duct represented in ANSYS geometry, aerofoil to locate in the centre to allow ease of positioning
Aerofoil Meshing After the sensitivity analysis for the flat plate, it is clear that the quality of meshing had an impact of the quality of results. This was a centre of discussion when trying to mesh the aerofoil. Therefore, it was considered to mesh certain regions within the duct with greater precision so as not to significantly increase the number of mesh elements or possibly to analyse the region with greater details. However, these options have not been used currently as a direct switch between the flat plate and the aerofoil was intended, without much change to the duct.
Figure 14: The possible meshing options for the aerofoil if the trail was analysed. What we have learnt: ● Explored the basic fluid mechanics concepts of external flow, boundary layer, pressure distributions around external objects like the flat plate and aerofoil. ● Furthered our understanding of the topics presented in class, cemented the principles taught and showed us how to adjust lift/drag accordingly. ● The importance of computational work in guiding our design and informing decisions to optimise lab work when building a prototype. References: [1] “Boundary Layer over a Flat Plate” P.P. Puttkammer [2] "Analytical approximations to the solution of the Blasius equation". Acta Mech. Parlange, J. Y.; Braddock, R. D.; Sander, G. (1981). [3] “ Buoyancy Effects of Steady Laminar Boundary Layer Flow and Heat Transfer over a Permeable Flat Plate Immersed in a Uniform Free Stream with Convective Boundary Condition” O lanrewaju P. O.1 , Titiloye E. O.2 , Adewale S. O.3 , Ajadi D. A.4 [4] "Coastal Inlets Research Program (CIRP) Wiki Statistics" . Retrieved 4 February 2015.
Student Number
Contribution %
Alper Bakici
14007530
16.67% Flat plate: Ansys Setup and meshing, and Data Analysis Aerofoil: Geometry design, aerofoil meshing, Data Analysis, and Graphical Representation Lab Report: Aerofoil explanation and theory, general report organisation
Youssef Hegazi
14035158
16.67% Flat plate : Ansys Setup, Data and Sensitivity Analysis, and Graphical Representation (Excel) Aerofoil : Data Analysis and Graphical Representation (Excel) Lab Report: Discussion for Flat Plate and Aerofoil. Graphs for Flat Plate and Aerofoil
Zijian Zhao
14001718
16.67% Theoretical: Flat plate calculations and analysis Error calculations: Superimposing and analysing theoretical and experimental results Lab Report: Formulas, Mathematical Theory, Boundary Layer Thickness, Velocity Profile
Karan Pinto
13055036
16.67% Flat plate: Boundary layer thickness, behaviour and calculation, Theoretical reasoning and explanation Aerofoil: Design and Lift Discussion, Theory Diagrams. Lab Report: Coalition and Organisation, results verification, analysis and discussions.
Kirill Markov
14041801
16.67% Flat Plate: Ansys Setup, Data Analysis, graphical representation (Excel). Meshing: fine and coarse mesh Lab Report: Boundary Layer explanation
George Lundy
14017121
16.67% Flat plate: Boundary layer calculation and analysis. Lab Report: Aerofoil discussion and editing.
Aim: To understand and familiarise ourselves with the fundamental concepts of fluid mechanics using ANSYS CFX and theory.
Part 1 Analysed the flow over a flat plate through computations (ANSYS/CFX)
Theoretical Calculation of Reynolds Number and Boundary Layer Thickness :
Reynolds Number A dimensional that helped determine the nature of flow of the fluid in the duct over the body. It is the ratio of the pressure forces to viscous forces. Larger the Reynolds number smaller the boundary layer. Smaller the Reynolds Number, the greater the dominance of the viscous forces (Inviscid Theory).
Figure 1: The development of boundary layer along a flat plate (not actual representation) Using the analytical solution for the configuration of a flat plate from the theory notes the following terms are defined with respect to the graph: Distance to the leading edge: x Inlet (maximum) velocity: U ∞ Velocity of flow at a given position: u Vertical distance to the plate: y Boundary layer thickness: δ
Reynolds number of a flat plate can be calculated as follows using data from table 3 in the appendices:
ρU ∞x 6 1.184×20.21x μ = 1.844 ×10−5 = 1.2976 × 10 x To find the position of the plate and thus the flow type over it, the maximum length of the plate was used in calculating Reynolds number, which was then compared to the critical value:
Rex =
Rex = 1.2976 × 106x = 1.2976 × 106 × 0.202 = 2.621 × 105 Rex, critical = 3 × 105 to 5 × 105 and since, Rex
≺Rex, critical ,
It can be confirmed that the flow is laminar even at the end of the plate. There will be no transitional or turbulent flow over the plate.
Figure 2: Actual representation of boundary layer along the flat plate Boundary Layer Velocity Variation: In Blasius solution for a laminar boundary layer, the thickness y varies along the velocity of 99% the free stream velocity.
u = 0.99u ∞ Boundary Layer Thickness: The fluid's velocity increases from zero at the surface due to the no slip condition to the free stream velocity (U ∞) at a distance from the surface which is equal to the boundary layer thickness (δ). It was observed that theoretically, boundary layer thickness increases across the plate as shear stress decreases. The value of x used is strictly the length of the flat plate which is equal to 0.202m. δ = 4.91x = 4.912×0.2025 = 1.937 × 10−3 m Re √ x √2.621×10
−3 So the theoretical boundary layer thickness at the end of the given flat plate is 1.937×10 m .
Thickness of Boundary Layer & Comparison with Blasius Solution Process: To obtain a visual of the boundary layer, a crosssectional plane is placed in the YZ direction, along the length of the plate and cut halfway through the width. Along this plane, the boundary layer thickness was defined by any points along the flat plate surface (0.199 → 0.401m) (where 1 the velocity was equal to 20.0079ms ). This gives us a 2dimensional visual in the 3dimensional simulation model. The position of this plane was chosen such that the boundary layer thickness is least affected by the width (resembles most closely to a 2D flow).
Figure 3 : The variation in boundary layer thickness across the length of the flat plate
The data provided by Ansys for the boundary thickness is displayed in the table below. Horizontal position, x (Flat Plate Length)
Vertical position, y (Boundary Layer Thickness)
0.20214
0.00198
0.20352
0.00204
...
...
0.35000
0.003684
...
...
0.40083
0.00271
0.40100
0.00271
Table 1: Data values for boundary layer thickness Preanalysis Observations: Ansys made calculations accounting for the horizontal distance along the entire duct, while the flat plate only starts at 0.199m into the duct. The data used from Ansys starts from 0.20214m onwards because the data between 0.1990.20214m was ignored due to the fluids inability to adjust at the sudden contact with the flat plate, demonstrated by a steeply decreasing boundary layer line (caused by boundary layer conditions).
Results: The boundary layer thickness is offset by 0.001m due to the thickness of the flat plate itself. As a result, the vertical position used for the data was adjusted by subtracting this offset in order to only represent the actual boundary layer thickness. Both table 1 and figure 3 show a maximum boundary layer thickness of 0.002684m(adjusted) at 0.151m into the flat plate (or 0.35000m into the duct).
Analysis: The boundary layer thickness begins to decrease towards the end of the flat plate, which is largely due to a lack of drag created by the walls. As a result, fluid begins to occupy the empty space behind the plate. Furthermore, the boundary layer is not developed from 0, but rather, from 0.001m. The presence of the flat plate thickness is believed to contribute to this offset. Using theoretical calculations, the flat plate is assumed to be thin, thus no additional fluid can affect boundary layer development. However, in this model, the stream of fluid comes into contact with a 0.002m thick flat plate (0.001m in the model due to symmetry) at the leading face. These fluids need to travel forward and cannot be ignored (can easily be explained by control volume or control mass). The clashing fluid is assumed to consume 0.001m of space on top of the near face of the flat plate, resulting in another offset as demonstrated by the graph.
Therefore, during the comparison with theoretical boundary layer below, the boundary layer thickness results from Ansys were adjusted by a 0.002m offset (relative to theoretical) in total. Theoretical Approach: With flow determined as laminar, Blasius solution states that boundary layer thickness at u=0.99umax can be expressed as
y = 4.91x . Substituting the given values (density, √Rex
maximum speed, and viscosity) for Reynolds number, the Blasius solution is thus given as:
y=
4.91x 1.184×20.21x 1.844×10−5
√
Figure 4 representing the Blasius solution falls between the xvalues of 0.199m to 0.401m, as this defines where the flat plate starts and ends. As previously explained, due the presence of the boundary layer (0.001m) and the fluid layer (0.001m) caused by the flat plate leading surface, an offset of 0.002m was added to Blasius’ solution. This allows for a more meaningful comparison between the curves and an analysis of the differences between theoretical and experimental results.
Figure 4: The variation in boundary layer thickness in comparison with Blasius Solution
Additionally, in order to calculate the error between the two sets of boundary thickness, a set of theoretical boundary thickness ‘y’ were calculated using the horizontal positions used by Ansys to indicate the experimental boundary layer thickness: Horizontal position
Theoretical Boundary Layer Thickness
Adjusted Boundary Layer Position
0.20214
0.00024
0.00224
0.20352
0.00029
0.00229
...
...
...
0.35000
0.00167
0.003675
...
...
...
0.40083
0.00194
0.00394
0.40100
0.00194
0.00394
Table 2: Blasius data adjustments to match numerical solution, shortened data set. Preanalysis Observations: The experimental boundary layer curve resembles that of the theoretical Blasius solution. However, the experimental data demonstrates a steep initial increase in the thickness, as well as a decrease towards the end of the plate. (The trend seen by the experimental boundary layer was previously discussed). Although the starting and ending trend of the simulation vary from that of the theoretical relationship, the overall boundary layer thickness demonstrates the correct behaviour. This shows that the simulation of the fluid's behaviour resembles the theoretical model considerably.
Results: Blasius Solution predicts boundary layer thickness of 0.003675m at 0.151m into the plate, compared to the experimental value of 0.003684m at the point (where maximum boundary layer thickness experimentally), or 0.001675m and 0.001684m relative to the top of the flat plate and after adjustment.
Error Calculation Between Theoretical and Numerical The error was calculated in two ways: 1. Percentage root mean square error:
The term [(experimental boundary layer position – theoretical boundary layer 2 position)/theoretical boundary layer position] was used to calculate the percentage error squared for each horizontal position x (provided by Ansys). Only data up to the position of maximum theoretical boundary thickness (0.00035m) was used in order to minimize the influence of the decrease in boundary layer due to the flat plate end on our analysis and increase the accuracy of prediction. The average value for these terms were calculated and the root of the result then computed, giving the percentage root mean square error: (experimental theoretical)/theoretical
2 ((experimental theoretical)/theoretical)
1.07974
1.16583
0.84660
0.71674
...
...
0.01531
0.000234
0.00539
0.000029
Table 3: Table of percentage error and squared percentage error at a horizontal position x By taking the averaging of the results in the second column of table 3 and obtaining the root value of the average, the percentage root mean square error was calculated to be: P ercentage Root Mean Square Error = 41.9%
2. Maximum boundary layer thickness error: The error of the maximum boundary layer thickness between experimental simulation and Blasius solutions:
P ercentage Error =
|Maxmimum experimental−Maximum theoretical| Maximum theoretical
P ercentage Error =
|0.001684−0.001937| 0.001937
× 100%
× 100% = 13.06%
It must be noted that the presence of the end of the plate is one of the major reasons of this difference. The maximum theoretical boundary layer thickness was calculated at the end of the plate where length is 0.202m, as compared to at 0.151m into the plate for the experimental results. The maximum experimental boundary layer results at the respective horizontal position (0.151m) was actually very accurately predicted (0.001675m versus 0.001684m).
Dimensionless Velocity Profile & Comparison with Blasius Solution
Process: In order to obtain a dimensionless velocity profile with ANSYS, the following steps in the Results section of CFX extension were taken. Firstly, a velocity profile isoclip was created. It’s visibility parameters were set as velocity w ≤ 20.0079 (which is 99% of the U ), m Fixed value of z (horizontal position) to analyse the velocity profile at a single horizontal position. In this case it is chosen to be the position where maximum boundary layer velocity occurs (most representative and informative).
Figure 5: The variation of velocity along the flat plate against Blasius Solution.
A set of velocity at different vertical position y was then taken as follows:
Vertical Position, y (m)
Vertical Position Offset, y (m) off
1 Velocity, u (ms )
0.00000
0.00100
0.00081
0.00003
0.00103
0.36239
...
...
...
0.00220
0.00320
19.57952
0.00266
0.00366
20.00790
Table 4: Velocity along height y at the point of greatest boundary layer thickness, shortened
Again, the vertical position y was offset by 0.001m to give a more informative comparison.
= yx × √Rex , the Reynolds number at the given position was calculated to be u Re(0.151) = 442.66. The term an also be calculated with the provided data, ultimately um c
Given that η
resulting in the table below: u/u m
η
0.00004
0.00000
0.01793
0.07467
...
...
0.96880
6.46216
0.99000
7.78965
Table 5: Data values for dimensionless velocity profile, shortened
The experimental velocity profile at z=0.35000m (0.151m into the plate) was then graphed as above. When u=0.99u , η has a value of 7.790. max
Blasius solution comparison: In order to obtain the theoretical Blasius solution velocity profile curve, the following set of data was obtained from the textbook: u/u m
η
0
0
0.0664
0.2
...
...
0.9878
4.8
0.9916
5
Table 6: Blasius data values for velocity profile, shortened
Again, the shape of the two curves resemble each other, indicating that our results are logically and correctly collected. Both curves show a positive correlation between u/u and η. Also, both m curves converge slightly upwards when u/u gets close to 1. On the other hand, the deviation m shows that there are still errors occurred. The similarity in trend signifies that the simulation, despite the numerical differences, modelled the correct relationship of a dimensionless velocity profile.
Error Calculation Between Theoretical and Numerical In order to analyse the error, a 6th order (maximum allowed on Excel) polynomial regression was performed to the Blasius solution curve. This is because the data points for the specific u/u values that matches the experimental outputs cannot be acquired from other resources. m The regression helped to achieve a set of η values at any u/u values that are needed, although m with a compensation of slight inaccuracy. The function achieved is as follows:
η = 154.31x6 − 412.68x5 + 415.14x4 + 191.49x3 + 39.665x2 + 0.1077x + 0.02 u where x = Um .
The following sets of data was then calculated using this equation, to match the experimental results: u/u m
η (theoretical)
0.00004
0.020004358
0.01793
0.033622991
...
...
0.96880
4.327219708
0.99000
4.806789892
Table 7: Data obtained from Excel polynomial as required, shortened The percentage root mean square error was then calculated as follows:
|experimental η theoretical η|/theoretical η
(|experimental η theoretical η|/theoretical 2 η)
0.84034
0.706175
0.57195
0.327130
...
...
0.49337
0.243418
0.62055
0.385083
Table 8: Table of root mean square error
Similar to the boundary layer error calculation, the following error was achieved:
P ercentage Root M ean Square Error = 48.4% However, there are sources of inaccuracy of this percentage error. Firstly, the regression model was used instead of the accurate Blasius solution values for the calculation. This only gives estimated values of η and therefore can create some dispense from the actual error. Also, not many sets of data were taken from Ansys (due to the limitations of meshing size and other parameters), which means that the actual shape of the experimental velocity profile curve might not be fully represented in this dimensionless graph. Since the error was calculated based on the differences between each set of theoretical/experimental data and there is an obvious convergence on the graph which is difficult to be fully described by the data acquired, it is further proved that this error calculation is subject to inaccuracy. On the other hand, the percentage error for the η at u=0.99umax was also calculated:
P ercentage Error = |Experimental−Theoretical| × 100% Theoretical
P ercentage Error =
|7.7897−4.91| 4.91
× 100% = 58.6%
Notice that 4.91 is the accurate Blasius solution for η at u = u . max
Skin Friction Coefficient Theoretical Local Skin Friction Coefficient for a flat plate with a laminar boundary layer aligned parallel to a stream is:
C f =
0.664
√Rex
=
0.664
√2.91×105
= 0.00119
Process: The skin friction coefficient was determined through the use of Ansys CFXpost and excel. First, a line was created at the top of the flat plate in the zdirection. This was used in order to define the direction of shear stress. A chart on Ansys was then plotted comparing the location along the line against the wall shear stress. This data was exported into excel and then used in the equation C f = τ . The following graph was then created:
√Rex
Skin Friction Coefficient Across the Plate
Figure 6: A variation of the value of the skin friction coefficient along the flat plate.
From the simulation in Ansys and using excel, the largest skin friction coefficient along the flat plate is determined as:
C f = 0.00121 In comparison to the theoretical value of 0.00119, this demonstrates a 1.68% error , demonstrating high accuracy between the simulation and theoretical value. This can be expected as the flow is better developed over the flat plate. From figure 6 we can see the strange data before the flat plate, due to the contact between the fluid and the edge of the flat plate in the model.
Numerical and Analytical Variation Analysis (Discussion):
The similarity in behaviour between theoretical relationships and simulated relationships demonstrates an accurate model. However, the numerical differences signify errors in the process.
Throughout the simulation of the flat plate, the models displayed significant similarity to the theoretical models. This is largely due to correct setup of the geometry and appropriate use of meshing. Had any of these variables been input incorrectly,the difference between theoretical and experimental would be significant, both visually and numerically. Visually, both the simulated boundary layer and the dimensionless velocity profile represented similar characteristics and trends to their respective theoretical models. These similarities are due to accuracies in the variables used in the simulation, as well as the implementation of correct procedure in obtaining and utilizing the results. However, numerically, the simulated models varied to the theoretical ones.
The simulated boundary layer, although similar in behaviour, provided different results to those of the Blasius boundary layer. The error between the theoretical and experimental maximum boundary layer thickness is 13.06%, which can largely be attributed to the presence of the end of the plate. Similarly, the dimensionless velocity profile recorded was visually comparable, but provided a numerical error of 58.6%. Moreover, the percentage root mean squared error were calculated to be 41.9% and 48.4% respectively, indicating a relatively notable error between the model and experiment. This reflects the presence of assumptions and limitations in the experiment.
A significant difference between the simulations and the theory is the presence of a duct in the model. The duct influences several variables, which in turn affect the simulations and thus the final results. Theoretically, the fluid flowing over the flat plate is uniform, however, with a duct during simulation, the fluid flow is affected, resulting in a nonuniform flow. The irregularities of flow influence the values of velocity, which consequently impact both the boundary layer thickness and the dimensionless velocity profile. This contributes to the difference in numerical value. Similarly, the walls present in the simulation influence the numerical results, creating an error, as the imperfection of the defined noslip boundary conditions create nonuniform flow.
Another contributing factor is the presence of the flat plate thickness and width. Theoretically, the thickness is considered negligible and flow continues over uniformly, creating the Blasius boundary layer. However, with the simulation, the thickness cannot be ignored, resulting in the disturbance of fluid flow. This affects velocity throughout, resulting in a difference between theoretical and experimental values. The simplification of the 3D model to a 2D graph due to the presence of the width may have attributed to the error too.
Further sources of error include limitations in Ansys itself. A decrease in accuracy was experienced due to the limits in mesh sizing. The limited mesh refinement means that numerical values, specifically the boundary layer thickness, can only be done to a certain accuracy. In other words, refinement is a direct source of error, contributing largely to the numerical differences observed between the analytical and numerical values.
Part 2 Modify the top of the flat plate to create an aerofoil
Flow Past the Aerofoil and Static Plot of Pressure In the duct, the air velocity increases as it flows over the aerofoil and therefore pressure reduces above the aerofoil inducing a transverse lift force.
Figure 7: Theoretical behaviour of pressure along the top surface of the aerofoil This theory can be validated by the velocity contours in figure 8. The flow coming in from the right hand side is interrupted by the aerofoil, resulting in the dispersion of the fluid. This creates local acceleration and results in the contours.
Figure 8 : Visual representation of velocity gradient around the aerofoil, high velocities in orange/red and lower velocities in blue and green will result in pressure difference From the above, the development of the pressure gradient around the aerofoil can be observed. By creating an isoclip to the aerofoil, the static pressure plot can be graphed (figure 10), where the blue line represents the negative static pressure along the upper surface of the aerofoil. For reference, the pressure below the aerofoil has been provided as well.
Figure 10: The plot of the static pressure over the aerofoil.
Coefficient of Pressure and Plot Along Blade Length
The coefficient of pressure represents a nondimensional pressure distribution on a body. Since the flow around a body accelerates and decelerates, there is a difference in the local static pressure and that of the mainstream.
Where,
p is the pressure at a particular point, p∞ is the pressure of the freestream, ρ is the freestream fluid density, u is the velocity at a particular point, U ∞ is the freestream velocity.
Figure 11: Theoretical variation of the pressure coefficient along the chord of the aerofoil.
Figure 12: The plot of the pressure coefficient over the blade length. The following can be inferred from the pressure coefficient plot in figure 12: ● Theoretically, the highest pressure achievable for an incompressible fluid occurs is the stagnation pressure at a stagnation point where Cp = 1. In the plot, this exists at the bottom part of the trailing edge where Cp ≈ 1. ● For a value of Cp > 1 at the bottom of the leading edge, it is indicative of the freestream flow being compressible, which does not seem to be the case with our fluid and must be accounted for as an error. ● The pressure is equal to the freestream pressure at points on the aerofoil where Cp = 0. These points are at the end of the trailing edge where Cp ≈ 0 (due to the formation of a minor wake).
Analysis of Lift Capability of Aerofoil (Discussion) In the scenario where an aerofoil generates lift, it creates a situation where the pressure above the aerofoil is lower than the pressure below the aerofoil. This pressure difference results in an upwards force, generating lift. Multiple simulations were run against the aerofoil from figure 7 to determine if it is capable of achieving lift. The primary concern to achieving lift was ensuring the relationship between the pressure difference was similar to the theoretical relationship. Figure 10 illustrates that the lower surface of the aerofoil experiences significantly larger pressures than the upper surface, representing the correct relationship. Furthermore, all the pressures above the aerofoil are negative, while those below are positive, signifying a large difference capable of generating lift. The coefficient of pressure along the aerofoil was also used to validate whether the aerofoil can generate lift. Figure 12 demonstrates that the upper surface is characterized by a smaller coefficient of pressure than the lower surface (excluding the front and tail of the aerofoil). This relationship enhances the statement that the aerofoil is capable of generating lift. Figures 7 and 8 match the criteria required for lift, meaning that this aerofoil design will work. With regards to the aerofoil stalling due to the presence of too much drag, the angle of attack and the risk of separation are crucially to determining left. Since the angle of attack is 5˚, the fluid flow will remain close to the aerofoil surface, ultimately redirecting it downwards. The pressure recovery area, which is defined as the space between the upper and lower surface in Figure 12. If this region is too severe, separation may occur, however this is not the case with this specific airfoil. As a result, there is no reason to believe that the aerofoil is not capable of lift. Even though, the simulations imply lift, there are limitations in the process that may cause error and misguided analysis. Again, error is caused by the boundary conditions of the walls due to the presence of a duct. Since they are defined as no slip, a boundary layer is created, which signifies nonuniform flow. Additionally, the data presented by figure 12 suggests the flow may be compressible, as the pressure coefficient is greater than 1. This could be due the excess pressure being exerted on the bottom of the aerofoil as a result of it being place in a duct. With these two differences in simulation acting on the model, the results calculated and presented by Ansys will vary from theoretical behaviour. Since these errors have not been calculated for the aerofoil, the degree to which they influence the calculated data cannot be determined. However, this suggests that the expected performance of lift by the aerofoil may be inaccurate.
Sensitivity Analysis & Mesh Refinement Simulations were run with two meshes coarse to increase the accuracy by including more integration or calculation points. The sensitivity analysis only changes 1 parameter (meshing grid refinement) in order to see the result on a simulated accuracy e.g. Boundary Layer Thickness. Fine: 578000 element mesh Coarse : 2640000 element mesh The maximum boundary layer thickness will be used as a reference value to compare the fine mesh and the coarse mesh in order to determine how good the refinement is. With the coarse mesh, the maximum boundary layer thickness is calculated as 0.004650.0015 = 0.00315m (refer to figure 13). The fine mesh creates a boundary layer thickness of 0.003670.002= 0.00167m . This demonstrates a good refinement, and a good sensitivity analysis.
Figure 13: Boundary Layer Thickness for Coarse Meshing The space was further broken down or organised using inflation layers and edge sizing with a view to minimize errors stemming from them. In order, to capture changes accordingly we needed a few thin layers within the boundary layer, which was probably to account for the large variation in the boundary layer thickness.
Appendix: Relevant Working and Project Information Flat Plate Setup A simple flat plate structured was created as geometry in Ansys based off the file provided and adjusted according to the geometrical details given in Table 1: Thickness [m]
Width [m]
0.0002
0.30
Appendix Table 1: The dimensions of the flat plate The flat plate was then placed in the middle of a rectangular wind tunnel section with dimensions as in Table 2. Length [m]
Width [m]
Height [m]
0.60
0.30
0.60
Appendix Table 2: The dimensions of the rectangular wind tunnel In order to simplify the simulation, the duct was sliced in half to take advantage of its symmetrical nature. A material called Labmaterial was created and the fluid properties were modified in CFXpre according to the given flow conditions and dimensions in Table 3. Also, the workbench and CFX files provided were then created and altered as per specifications mentioned in Table 3 to effectively set up a simulation. It should be noted that the inlet velocity is U ∞ which is parallel to the flat plate. Temp ( °C)
Air Density (ρ)[kg/m3]
Velocity (U ∞) [m/s]
Length (L)[m]
Viscosity ( μ ) [Pa.s]
25
1.84
20.210
0.202
1.844E05
Appendix 3: Given specification of simulation and flow characteristics
To run the simulation, after the setup was complete with the correct domains, we decided to run double precision. This was because in the mesh file, there were several elements which varied in size greatly and so double precision allowed more accurate results, regardless of the slower simulation time. Aerofoil Design and Setup An aerofoil was designed roughly to the NACA s7055il specification as shown below. The o diagram shows the tilt of 5 . This design was chosen due to its flat bottom and hence similarity to the flat plate.
Appendix Figure 1: A sketch of the aerofoil design with the labelled components and the flat bottomed aerofoil with chord line of 202mm at an angle of 5° to the fluid flow
To model the flow around the aerofoil, a new file was made to represent the whole duct. This was to allow ease in placement of the aerofoil in the centre of the duct, as shown in figure 2.
Appendix Figure 2: Image of the full duct represented in ANSYS geometry, aerofoil to locate in the centre to allow ease of positioning
Aerofoil Meshing After the sensitivity analysis for the flat plate, it is clear that the quality of meshing had an impact of the quality of results. This was a centre of discussion when trying to mesh the aerofoil. Therefore, it was considered to mesh certain regions within the duct with greater precision so as not to significantly increase the number of mesh elements or possibly to analyse the region with greater details. However, these options have not been used currently as a direct switch between the flat plate and the aerofoil was intended, without much change to the duct.
Figure 14: The possible meshing options for the aerofoil if the trail was analysed. What we have learnt: ● Explored the basic fluid mechanics concepts of external flow, boundary layer, pressure distributions around external objects like the flat plate and aerofoil. ● Furthered our understanding of the topics presented in class, cemented the principles taught and showed us how to adjust lift/drag accordingly. ● The importance of computational work in guiding our design and informing decisions to optimise lab work when building a prototype. References: [1] “Boundary Layer over a Flat Plate” P.P. Puttkammer [2] "Analytical approximations to the solution of the Blasius equation". Acta Mech. Parlange, J. Y.; Braddock, R. D.; Sander, G. (1981). [3] “ Buoyancy Effects of Steady Laminar Boundary Layer Flow and Heat Transfer over a Permeable Flat Plate Immersed in a Uniform Free Stream with Convective Boundary Condition” O lanrewaju P. O.1 , Titiloye E. O.2 , Adewale S. O.3 , Ajadi D. A.4 [4] "Coastal Inlets Research Program (CIRP) Wiki Statistics" . Retrieved 4 February 2015.
