IV. ANÁLISIS DE LOS PERFILESMIGRATORIOS DE LOS INMIGRANTES AFRICANOS Y
3. Estructura del hogar y la nacionalización
This section will discuss what happens when the HPV fairing ground clearance changes, and its effect on the drag, lift, and handling. Tables 5-4 and 5-5 illustrate the percentage difference in drag and lift coefficients for the HPV fairing with various ground clearances, compared to the freestream cases.
Table 5-6:
129
According to “Race Cars Aerodynamics,” by Katz, the handling of the vehicle with similar body shape increases with a lower center of gravity and closer ground proximity. However, the drag and lift coefficient increase as well. [14, 15]
In order improve the stability of the HPV, it needs to be closer to the ground; however, it will consequently have higher aerodynamic drag which limits top speed. Ultimately, an HPV design has to balance these effects.
In order to estimate what is the best height for the HPV a few trade-off studies were done. The first trade-off study estimates the power required as a function of velocity for each ground clearance and compares it to the average human output of 221𝑓𝑡−𝑙𝑏
𝑠𝑒𝑐 used
by the 2010 HPV team. In order to find the required power the following equations are used:
𝑃 = 𝐹𝑉 = 𝐹𝐷𝑉 =12𝐶𝐷𝐴𝜌𝑉2𝑉 = 12𝐶𝐷𝐴𝜌𝑉3 (5-10)
Here 𝐴 = 2.84𝑓𝑡2 and 𝜌 = 2.37𝐸 − 3𝑠𝑙𝑢𝑔
𝑓𝑡3 are constants and CD is used from the Table 5-4 for different ground clearances .required power values are calculated as a function of velocity for each ground clearance, and compared with the average human output, as shown in Figure 5-32.
130 0 25 50 75 100 125 150 175 200 225 250 0 10 20 30 40 50 60 70 80 Po we r [ (lb -ft )/ se c] Velocity (ft/sec)
Power as a function of velocity
Clearance=3" Clearance=6" Clearance=9" Clearance=12" Clearance=15" Clearance=18" Clearance=30" Clearance=free" Average human power
Figure 5-32:
131
The second trade-off study is to find the rollover speed for a corner with a 15ft radius as a function of ground clearance. It is assumed that the ground clearance and center of gravity location differ by a constant. Using an Excel spreadsheet for rollover calculation provided by Dr. Robert Ryan and the CSUN HPV design team the rollover speeds for ground clearances of 3, 6, 9, 12, 15, 18, and 30 inches were found and shown in Figure 5-33 and Table 5-7.
65 85 105 125 145 165 185 205 225 245 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 Po we r [ (lb -ft )/ se c] Velocity (ft/sec) Clearance=3" Clearance=6" Clearance=9" Clearance=12" Clearance=15" Clearance=18" Clearance=30" Clearance=free" Average human power
Figure 5-33:
132 Ground clearance to the
bottom of the HPV fairing in inches
ground clearance and seat height total in
inches Rollover velocity in ft/sec V in ft/sec for max power 3 10 30.23 77 6 13 27.50 77.5 9 16 25.50 78 12 19 23.83 78.5 15 22 22.45 79.5 18 25 21.30 81 30 37 17.97 88.4 17 19 21 23 25 27 29 31 3 6 9 12 15 18 21 24 27 30 Ro llo ve r V el oc ity (ft /s se c)
Groound Clearance (inches)
Ground Clearance vs. rollover speed
3 inches 6 inches 9 inches 12 inches 15 inches 18 inches 30 inches Figure 5-34:
Rollover speed for various ground clearances of the HPV
Table 5-7:
Rollover speed and maximum velocity to achieve average human power of 221.40 for various ground clearances of the HPV
133
Chapter 6: Conclusion
The flow fields around the HPV fairing at different ground proximities and oblate ellipsoid baseline models have been computed using the ANSYS FLUENT CFD package. This software solves the Reynolds Averaged Navier–Stokes equations for 3-D flow with incompressible flow assumption using the one equation Spalart-Allmaras (SA) turbulence model. The surfaces of the HPV fairing and the oblate ellipsoid were assumed to be smooth. The computational domain was selected to be large enough, and the mesh was fine enough, so that the flow calculated field is independent of these parameters, which results in solution convergence.
The drag and lift coefficients for the HPV fairing at different ground clearances and the drag coefficient of oblate ellipsoids were calculated from the CFD computations. The computed values for the oblate ellipsoids were compared to experimental values from “Fluid Dynamics of Drag” by Hoerner to verify that FLUENT was providing correct values and that the mesh was sufficiently fine. Then the computed values for the HPV fairing with different ground clearances ranging from 3 to 18 inches were compared to the HPV fairing in freestream and 30 inches away from the ground.
Based on the results for the HPV fairing with different ground clearance, the following can be concluded.
1. The streamlined shape of the HPV fairing does in fact help to reduce drag coefficient; however, due to its shape when the ground proximity increases, a lot of the flow is blocked near the ground and it creates a higher coefficient of drag is created. Also, in addition to the increase in drag there is also an increase in lift coefficient. This is because due to the ground proximity, the flow is no longer
134
symmetric and it will have an increased speed near the top of the HPV fairing and low pressure region beneath the HPV fairing. This results in a more positive lift coefficient.
2. The pressure coefficient (CP) on the fairing bottom surface decreases as ground clearance decreases. This is because as air tries to flow around the HPV fairing as the ground becomes closer, less airflow is permitted to flow beneath the vehicle. This then causes a low pressure region. Additionally, the pressure above the HPV fairing also decreased due to the effect on velocity on this region.
3. As discussed in Chapter 5, the HPV fairing’s drag coefficient (CD) and lift coefficient (CL) increase as ground clearance decreases. For example, the HPV fairing with a ground clearance of 15 inches has a remarkable increase in CD from 0.0923 to 0.131. This value represents a nearly 34.13% increase in drag from the benchmark simulation. In addition, the CL increases from 0.0652 to 0.0880. This value is nearly 29.68 % higher than the CL in freestream.
4. To take advantage of the drag decrease afforded by greater ground clearance the vehicle height is such that cornering stability is severely compromised. It may be possible that changes to fairing geometry can be used as an alternative approach to minimizing ground effect. This is recommended as a topic for further study.
135 References
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2. Anderson, J.D. Jr., “Fundamentals of Aerodynamics,” McGraw-Hill, Inc, 3rd edition, February 2001.
3. Barlow, J.B., Rae, W.H. Jr., and Pope, A., “Low-Speed Wind Tunnel Testing,” New York, NY: John Wiley& Sons Inc, 3rd edition, 1999.
4. Celik, I.B., “Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications,” ASME Journal of Fluids Engineering, Vol. 130, July 2008.
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6. DeMoss, J.A., “Drag Measurements on an Ellipsoidal Body,” Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, August 2007.
7. Diasinos, S., Barber, T.J., Leonardi, E., and Hall. S.D.,” A Two-Dimensional Analysis of the Effect of a Rotating Cylinder on an Inverted Aerofoil in Ground Effect,” 15th Australasian Fluid Mechanics Conference, The University of Sydney, Sydney, Australia, December 13-17, 2004.
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New York, NY: John Wiley& Sons Inc, 6th edition, July 2003.
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12. Hoerner, S., and Borst, H.V., “Fluid Dynamic Lift,” Mrs Liselotte A. Hoerner, 1985. 13. Hunco, WH., “Aerodynamics of Road Vehicles,” Warrendale, PA: SAE Int, 4th
edition, 1999.
14. Katz, J., “Aerodynamics of Race Cars,” Annual Reviews Fluid Mechanics Journal, 38:27-63, January 2006.
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16. Fidkowski, F.K.., and Darmofal, L.D., “Review of Output-Based Estimation and Mesh Adaptation in Computation Fluid dynamics,” AIAA Journal , Vol. 49, No. 4, April 2011.
17. Krishnami, P.N., “CFD Study of Drag reduction of a Generic Sport Utility Vehicle,” Master Thesis, California State University, Sacramento, CA, Fall 2009.
18. Little, R.P., “Flight Simulator Database Population from Wind Tunnel and CFD Analysis of a Homebuilt Aircraft,” Master Thesis, California Polytechnic State University, San Luis Obispo, CA, May 2006.
19. Mafi, M., “Investigation of Turbulence Created by Formula One™ Cars with the Aid of Numerical Fluid Dynamics and Optimization of Overtaking Potential” ANSYS Conference & 25th CADFEM Users’ Meeting 2007, Congress Center Dresden , Germany, November 21-23, 2007.
20. Milliken, W.F., and Milliken, D.L., “Race Car Vehicle Dynamics,” Warrendale, PA: SAE Inc, August 1995.
21. Mohammadi, A., “Computation of Flow Over a High Performance,” Master Thesis, California State University, Northridge, CA, May 2009.
22. Monsch, S.C., “A Study of Induced Drag and Spanwise Lift Distribution for Three- dimensional Inviscid Flow Over a Wing,” Master Thesis, Clemson University, SC, May 2007.
23. Munk, M.M., “Fundamentals of Fluid Dynamics for Aircraft Designers,” New York, NY, The Ronald Press Company, 1929.
24. Munson, B.R., Young, D.F., and Okiishi T.H., “Fundamentals of Fluid mechanics,” New York, NY: John Wiley& Sons Inc, 5th edition, 2006.
25. Panton,R.L., “Incompressible Flow,” New York, NY: John Wiley& Sons Inc, 3rd edition, 2005.
26. Paparone, L., Tognaccini, R., “A Method for Drag Decomposition from CFD Calculations”, ICAS 2002 Congress, pp 1113.1-1113.9, 2002.
27. Roy, C.J., Raju, A., and Hopkins, M.M., “Estimation of Discretization Errors Using the Method of Nearby Problems,” AIAA Journal, Vol. 45, No. 6, June 2007.
28. Spalart, P.R., and Allmaras, S.R., “A One-equation Turbulence Model for Aerodynamic Flows,” AIAA paper No.92-0439, January 1992.
137
29. Scibor-Rylski, A.J., “Road Vehicle Aerodynamics,” London, UK, Pentech Press, 1975.
30. Schlichting, H., “Boundary-Layer Theory,” McGraw-Hill, Inc, 7th edition, 1979. 31. Steenbergen, C.K., “Vortices in favorable pressure gradients,” Master Thesis, Delft
University of Technology, Delft, South Holland, Netherlands, July 2004.
32. Thompson, J.F., Soni, B.K.., and Weatherill, N.P., “Handbook of Grid Generation,” CRC-Press, 1st edition, September 1998.
33. Versteeg, H.K.., and Malalasekera, W.,”An Introduction to Computation Fluid Dynamics The Finite Volume Method,” Pearson Education Ltd, 2nd edition, February 2007.
34. “Tutorial on CFD verification and Validation,” NPARC Alliance CFD Verification and Validation Web Site, 20 December 2010,
http://www.grc.nasa.gov/WWW/wind/valid/tutorial/tutorial.html
35. “The Spalart Allmaras Turbulence Model,” Langley Research Center, 15 April 2011,
http://turbmodels.larc.nasa.gov/spalart.html
36. “History of Theoretical fluid Dynamics,” Centrum Wiskunde & Informatica, 17 September 2010, https://www.cwi.nl/fluiddynamicshistory
37. ANSYS Tutorials: 12 August2011, https://www1.ansys.com/customer/default.asp 38. ANSYS FLUENT 12.0 User Guide, 2009.
39. ANSYS ICEM CFD 12.1 User Guide, 2009. 40. ANSYS WORKBENCH 12.1 User Guide, 2009.
41. California State University, Northridge, ME692 Course notes
42. Javaherchi, T., “Review of Spalart-Allmaras Turbulence Model and its Modifications,” University of Washington, ME Department, March 2010.
43. Roache, P. J. , “Quantification of Uncertainty in Computational Fluid Dynamics,” Annual Reviews Fluid Mechanics Journal, 29:123-60, 1997.
44. Kusunose, K.., “Development of a Universal Wake Survey Data Analysis Code.” AIAA-1997-2294, pp. 617-626, 1997.
138 Appendix A
Boundary layer calculation using the integral approach: • The integral Equations:
Figure A illustrates an infinitesimal controlled volume of a thickness dx. The continuity equation supplies the mass Flux ṁtop that crosses into the controlled volume from the top.
ṁtop = ṁout − ṁin =∂x∂ �∫ ρu dy0δ � dx (1)
The X-component momentum equation is:
∑ 𝐹𝑥 = 𝑚𝑜𝑚̇ 𝑜𝑢𝑡 − 𝑚𝑜𝑚̇ 𝚤𝑛− 𝑚𝑜𝑚̇ 𝑡𝑜𝑝 (2)
Then it will become the following equation: Figure A-1:
The infinitesimal controlled volume of thickness dx for the boundary layer (adopted from Schaum’s Outline of Fluid Mechanics )
139
−𝜏𝑤𝑑𝑥 − 𝛿𝑑𝑝 =∂x∂ �∫ ρu0δ 2dy� dx − U(x)∂x∂ �∫ ρu dy0δ � dx (3)
In the above equation the term pdδ and dpdδ are neglected because they are a smaller order then the entire term because pdδ is very small since the δ assumed is to be very small and dδ is then an order smaller ; also the term for the momentum 𝑚𝑜𝑚̇ 𝑡𝑜𝑝 = 𝑈(𝑥)𝑚̇𝑡𝑜𝑝 . In addition divide the entire equation (3) by (-dx) and the new equation
becomes the von Karman integral equation:
𝜏𝑤 +𝛿𝑑𝑝𝑑𝑥 = 𝜌𝑈(𝑥)dxd ∫ u dy0δ − ρdxd ∫ u0δ 2 dy (4)
After using ordinary derivatives on equation (4), where the density is assumed to be constant over the entire boundary layer and the δ is a function of x. as a result for a flow over a flat plate with a zero pressure gradient , such as U(x)=U∞ and 𝜕𝑝
𝜕𝑥 = 0 this can
be simplified and put into the following form:
𝜏𝑤 = 𝜌𝑑𝑥𝑑 ∫ 𝑢(𝑈∞0𝛿 − 𝑢)𝑑𝑦 (5)
For the velocity profile of u(x, y) for the specific flow, equation (5) along with 𝜏𝑤 = 𝜇𝜕𝑢𝜕𝑦 �
𝑦=0lets both δ(x) and τ0(x) be determined. Where δ=y and u=0.99U∞
𝛿𝑑 = ∫ (1 −0𝛿 𝑈𝑢∞) 𝑑𝑦 = 0 (6)
𝜃 = ∫ 𝑈𝑢∞(1 −𝑈𝑢 ∞)
𝛿
0 𝑑𝑦 = 0 (7)
δd is the displacement thickness and it is the distance the streamline outside the boundary layer is displaced due the slow moving fluid inside the boundary layer. Θ is the momentum thickness and it is the thickness of the fluid layer with the velocity U∞ that
140
possesses the momentum lost because of the viscous effect. It is frequently used as the characteristic length for the boundary layer.
𝜏𝑤 = 𝜌𝑈∞2 𝑑𝜃𝑑𝑥 (8)
• Laminar and Turbulent Boundary Layer:
The main boundary conditions that need to be meet for the velocity profile in the laminar boundary layer for the flat plate with a zero pressure gradient are:
u=0 at y=0 u=0.99U∞ at y=δ
𝜕𝑢
𝜕𝑦= 0 at y= δ
As figure B illustrates a general flow over a flat plate with uniform velocity and the development of the boundary layers along the flat plate.
• Laminar boundary layers:
Figure A-2:
Boundary layer development along a flat plate. (Adopted from Fluid Mechanics
141
According Prandtl/Blasius boundary layer solution that can be solved for by the governing Navier-Stokes equations with negligible gravitational effects. They become the following two equations:
𝑢𝜕𝑢𝜕𝑥+ 𝑣𝜕𝑢𝜕𝑦= −𝜌1𝜕𝑢𝜕𝑥 + 𝜈( 𝜕𝜕𝑥2𝑢2+𝜕 2𝑢 𝜕𝑦2 ) (9) 𝑢𝜕𝑣𝜕𝑥+ 𝑣𝜕𝑦𝜕𝑣 = −1𝜌𝜕𝑣𝜕𝑦+ 𝜈( 𝜕𝜕𝑥2𝑣2+𝜕 2𝑣 𝜕𝑦2 ) (10)
Using conservation of mass equation for incompressible flow becomes. In addition Where v<<u and 𝜕
𝜕𝑥≪ 𝜕 𝜕𝑦 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦= 0 (11) 𝑢𝜕𝑢𝜕𝑥+ 𝑣𝜕𝑢𝜕𝑦= 𝑣𝜕𝜕𝑦2𝑢2 (12) Figure A-3:
Typical characteristics of boundary layer thickness (δ) and wall shear wall stress (τw) for laminar and turbulent boundary layer. (Adopted from Fundamentals of
142
Although both the boundary equations (11) and (12) and Navier-Stokes equations (9) and (10) are non-linear partial differential equations, there a considerable difference between them. For one, the y momentum equation has been eliminated and only leaves the x momentum equation. The pressure variable has been eliminated and only leaving the x and y velocity components as the only unknowns. In addition for the boundary layer flow over the flat plate the pressure is assumed to be constant and the flow represents a balance between viscous and inertial effects with the pressure playing no role.
For the laminar boundary layer, a parabolic velocity profile is assumed due to the fact that the boundary layer is very thin.
𝑢
𝑈∞ = 𝐴 + 𝐵𝑦 + 𝐶𝑦
2 (13)
The above boundary conditions are used to find the values for A, B and C 0=A
1=A+Bδ+Cδ2 0=B+2Cδ
The solution of the problem then:
𝐴 = 0 𝐵 =2𝛿 𝐶 = −𝛿12 This results in the laminar flow velocity profile:
𝑢 𝑈∞ = 2 𝑦 𝛿− 𝑦2 𝛿2 (14) Then substitute equation (14) in to equation (5) and the result of it is:
𝜏𝑤 = 𝜌𝑈∞2 𝑑𝑑𝑥 ∫ �2𝑦𝛿−𝑦 2 𝛿2� �1 − 2𝑦𝛿−𝑦 2 𝛿2� 𝛿 0 𝑑𝑦 = 152 𝜌𝑈∞2 𝑑𝛿𝑑𝑥 (15)
143
The wall shear stress is given by the following expression: 𝜏𝑤 = 𝜇𝜕𝑢𝜕𝑦 �
𝑦=0= 𝜇𝑈∞ 2
𝛿 (16)
Then equate the equation (15) and (16) to obtain 𝛿𝑑𝛿 =𝑈15𝜇
∞𝜌𝑑𝑥 (17) Then integrate equation (17) with δ=0 at x=0 to find the expression for δ(x) in the laminar boundary layer, where𝜇
𝜌= 𝜈:
𝛿(𝑥) = 5.48𝑥�𝜈𝑥𝑈∞ (18) To find the shear stress (τw(x)), local skin friction coefficient (cf) and
dimensionless drag force that is the skin friction coefficient (Cf) substitute equation (18)
into equation (16). 𝜏𝑤(𝑥) = 0.365𝜌𝑈∞2�𝑥𝑈𝜈∞ =0.365𝜌𝑈�𝑅 ∞2 𝑒𝑥 (19) 𝒄𝒇(𝒙) =𝟎.𝟓𝝆𝑼𝝉𝒘 ∞𝟐 = 𝟎. 𝟕𝟑�𝒙𝑼𝝂∞ =�𝑹𝟎.𝟕𝟑 𝒆𝒙 (20) 𝑪𝒇 = 𝟎.𝟓 𝝆𝑼𝑭𝒅∞𝟐𝑳= ∫ 𝝉𝟎𝑳 𝒘𝒅𝒙 𝟎.𝟓 𝝆𝑼∞𝟐𝑳= 𝟏. 𝟒𝟔� 𝝂 𝑳𝑼∞ = 𝟏.𝟒𝟔 �𝑹𝒆𝑳 (21)
• Buffer boundary Layer:
Buffer boundary layer is the layer in between the laminar and turbulent boundary layers; it is sometimes call the transition layer this is shown in figure B. The boundary layer thickness increases in proportion to √𝒙 where x denotes the distance from the leading edge according to Boundary-Layer theory by Schlichting. Near the leading edge of the flat plate the flow is always laminar and it is becoming turbulent further
144
downstream on the plate. The transition takes place at distances x from the leading edge and it determined by:
𝑹𝒙,𝒄𝒓𝒊𝒕 = �𝑼∞𝝂𝒙� = 𝟓 × 𝟏𝟎𝟓 𝒕𝒐 𝟏. 𝟖𝟏𝟒 × 𝟏𝟎𝟔 (22)
And 𝜹𝒄𝒓𝒊𝒕 ≡ 𝜹|𝒙=𝒙𝒄𝒓𝒊𝒕 determined by:
𝜹𝒄𝒓𝒊𝒕 = 𝟓. 𝟒𝟖𝒙 �𝑼𝝂∞𝒙𝒄𝒓𝒊𝒕�
𝟏 𝟐
(23) In addition the process of transition also involves a large decrease in shape factor H12=δ1/δ2 where δ2=θ and δ2= δd
Using Prandtl-Schlichting local skin friction equation for a Smooth Plates that is given by:
𝒄𝒇 =(𝐥𝐨𝐠𝟏𝟎𝟎.𝟒𝟓𝟓𝑹
𝒆𝒙)𝟐.𝟓𝟖−
𝑨
𝑹𝒆𝒙 (24)
Where the A is the function of only the transition Reynolds number and it is given by: 𝑨 = � 𝟎.𝟒𝟓𝟓 �𝐥𝐨𝐠𝟏𝟎𝑹𝒆𝒙�𝟐.𝟓𝟖− 𝟏.𝟑𝟐𝟖 �𝑹𝒆𝒙� 𝑹𝒆𝒙 (25) Figure A-4:
Changes in the shape factor (adopted from Boundary-Layer Theory 7th edition by H. Schlichting)
145
Where local shear stress is a function of the local skin friction and is given by: 𝒄𝒇 =𝝆𝑼𝟐𝝉𝒘∞𝟐 ; 𝝉𝒘= 𝟎.𝟒𝟓𝟓 �𝐥𝐨𝐠𝟏𝟎 𝑹𝒆𝒙�𝟐.𝟓𝟖− 𝑨 𝑹𝒆𝒙 𝟐 𝝆𝑼∞𝟐 (26)
• Turbulent boundary layer:
In the turbulent boundary layer it is often assume a power law velocity profile for this study the maximum Re number was found to be 2.95x106 so the n=7
𝒖� 𝑼∞ = � 𝒚 𝜹� 𝟏 𝒏 (27) Then substitute this velocity profile equation (27) into equation (5) and integrate to get the following expression for τw:
𝝉𝒘 = 𝟕𝟐𝟕 𝝆𝑼∞𝟐 𝒅𝜹𝒅𝒙 (28)
This τw from the velocity profiles yields a 𝝉𝒘 = 𝝏𝒖�
𝝏𝒚= ∞ 𝒂𝒕 𝒚 = 𝟎 so this
expression cannot be used at the wall. A new expression is needed to be derived for τw; the Blasius formula is selected for local skin friction coefficient for turbulent model and given by:
𝒄𝒇 = 𝟎. 𝟎𝟒𝟓 �𝑼𝝂∞𝜹�
𝟏 𝟒
(29) In addition to the local skin friction coefficient an experimentally determined formula for shear stress is given by:
𝝉𝒘 = 𝟎. 𝟎𝟐𝟐𝟓𝝆𝑼∞𝟐�𝑼𝝂∞𝜹�
𝟏 𝟒
(30) Combine equation (28) with equation (30) and we get:
𝛿14𝑑𝛿 = 0.231 �𝜈
𝑈∞� 1
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Equation (31) can be integrated from δ=0 and x=0 to obtain ∫ 𝛿𝛿 14 0 𝑑𝛿 = ∫ 0.231 �𝑈𝜈∞� 1 4 𝑑𝑥 𝑥 0 ; 𝛿 = 0.37𝑥 �𝑈𝜈∞𝑥� 1 5 𝑅𝑒𝑥 < 107 (32)
Equation (29) can substitute into equation (32) to get the local skin friction coefficient to be: 𝑐𝑓(𝑥) = 0.0577 �𝑈𝜈∞𝑥� 1 5 𝑅 𝑒𝑥 < 107 (33)
Equation (33) can be substituted into equation 30 and divided by 2 and right side multiplied by 𝜌𝑈∞2 to get the shear stress
𝜏𝑤(𝑥) = 0.02885𝜌𝑈∞2�𝑈𝜈∞𝑥�
1 5 𝑅
𝑒𝑥 < 107 (34) In addition the dimensionless drag force that is the skin friction coefficient (Cf)
becomes:
𝑐𝑓(𝑥) = 0.072 �𝑈𝜈∞𝐿�
1 5 𝑅
𝑒𝐿 < 107 (35)
• Law of the Wall:
Law of the wall is the average velocity of the flow at a specific point. It is proportional to the logarithm of the distance a point to the wall (flat plate surface). The law was first published in the early 1930’s by von Karman; it only applies to the flow that is close to the wall. The general logarithmic equations are:
𝑢+ = 𝑢 𝑢𝑡 𝑤𝑖𝑡ℎ 𝑦 + =𝑦𝑢𝑡 𝜐 𝑎𝑛𝑑 𝑢𝑡 = � 𝜏𝑤 𝜌 (36) Where:
147
y+ is the wall coordinate, the dimensionless distance y to the wall u+ is the dimensionless velocity
τw is the wall shear stress ρ is the fluid density ut is the friction velocity
For laminar study the y+ is assumed to be 5, for the buffer layer the y+ =11 and, turbulent the y+=30
148 Appendix B
In Appendix B we can find all of the figures for velocity, pressure and vectors contours from ANSYS FLUENT for HPV fairing at ground clearance of 3,6,9,12,18 inches.
Figure B-1:
Velocity Vectors about the HPV fairing with ground clearance of 3 inches
Figure B-2:
149 Figure B-3:
Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 3 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
Figure B-4:
Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 3 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
A B
C D
A B
150 Figure B-5:
Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 3 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
A B
C D
Figure B-6:
151 Figure B-7:
Velocity Vectors about the HPV fairing with ground clearance of 6 inches
Figure B-8:
152 Figure B-9:
Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 6 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
Figure B-10:
Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 6 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
A B
C D
A B
153 Figure B-11:
Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 6 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
Figure B-12:
Vorticity contours of HPV fairing with ground clearance of 6 inches
A B
154 Figure B-13:
Velocity Vectors about the HPV fairing with ground clearance of 9 inches
Figure B-14:
155 Figure B-15:
Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 9 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
Figure B-16:
Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 9 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
A B
C D
A B
156 Figure B-17:
Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 9 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).
Figure B-18:
Vorticity contours of HPV fairing with ground clearance of 9 inches
A B
157 Figure B-19:
Velocity Vectors about the HPV fairing with ground clearance of 12 inches
Figure B-20:
158 Figure B-21:
Static pressure contours of the benchmark simulation at symmetry plane with ground