# Computational Analysis

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NACE airfoil profileTRANSCRIPT

<p>Evan Kontras, Kyle Gould, Davide Maffeo MAE 4440/7440 Aerodynamics University of Missouri Department of Mechanical and Aerospace Engineering NACA Airfoil Evaluation Abstract Aerodynamic analysis has been conducted for the wingtip airfoil of the A10 Thunderbolt. The NACA 4212 airfoil was analyzed using three separate methods. Results from each method are compared to gain an understanding of the capabilities and limitations of each method. To begin, analytic tools and hand calculations were used to obtain a simple solution for the lift coefficient over a range of different angles of attack. Next, a 2D panel method was implemented using Matlab to compute the lift and drag coefficients over a range of angles of attack. Finally, the Ansys Workbench commercial computational fluid dynamics (CFD) program, Fluent, was used to similarly obtain and plot lift and drag coefficients. Both analytic tools and the panel method assume inviscid flow and therefore detailed drag computations, which are largely based on viscous effects are not accounted for. Using Fluent, three separate cases of fluid flow were analyzed. Inviscid flow, laminar flow, and turbulent flow were all simulated and the results compared. The general procedure for each of the three methods used is presented, and the results compared and discussed. INTRODUCTION Developed in the early 1970s by Fairchild Republic, the A10 Thunderbolt is a strait wing close air support aircraft used by the United States Air Force to combat ground vehicles such as tanks and armored vehicles. Designed around a 30 mm automatic cannon (GAU8 Avenger), the A10 has an unrivaled capability to quickly and effectively destroy armored targets on the ground. The A10 s strait wing design limits its aerial characteristics, and as such it is not known as a fighter plane. The wingtip cross section is designated by the NACA 4212 airfoil, as shown below in Fig. 1.</p>
<p>NACA 4212 Airfoil Profile0.3 0.25 0.2 0.15 0.1 0.05 3E-16 -0.05 0 Distance (m)</p>
<p>Y Upper Y Lower Camber 0.2 0.4 0.6 0.8 1 1.2</p>
<p>-0.2</p>
<p>Distance (m)</p>
<p>Figure 1. Profile of the NACA 4212 Airfoil.</p>
<p>Using three separate methods, the lift and drag coefficients of the NACA 4212 airfoil are computed over a range of angles of attack. Though the A10 is a well proven aircraft and its flight capabilities are not in question, the main goal of this analysis is to compare each method and determine how the aerodynamic properties differ between them. Using analytic tools and the assumption that the airflow over the airfoil is inviscid, hand calculations were first done to obtain a general knowledge and an expectation for the aerodynamic properties using the other methods. A 2D panel method was then used, also assuming inviscid flow. Central to the panel method is the segmenting of the airfoil into separate pieces connected by strait lines. The coefficient of pressure is calculated for each segment, and all segments then summed to obtain the overall properties of the airfoil. In the hopes of obtaining an even more accurate description of the aerodynamic properties, the Ansys Workbench CFD program, Fluent, was used as well. As both the panel method and analytic tools do not account for viscous effects, inviscid flow was first used in the Fluent simulation for comparison. To obtain a more complete description of the aerodynamic properties of the NACA 4212, both laminar and turbulent flows were then simulated.</p>
<p>PROCEDURE ANALYTIC TOOLS Central to the analytical approach to determining the aerodynamic properties of an airfoil is the assumption that the fluid flow is inviscid. The process used is a generalization of the method used for analyzing a symmetric airfoil. Unlike a symmetric airfoil, the NACA 4212 is a cambered airfoil. When the slope of the camber line for this airfoil is considered, the term (1) becomes non-zero. After obtaining the geometric information for the NACA 4212, x and y coordinates were plotted along with the camber line in Excel. A polynomial trend line was then fit to the camber line coordinates, to obtain the camber line as a function of the x coordinate alone, as shown below. x 12.3594x5 36.4845x4 41.052x3 21.8421x2 5.58x 0.5357 (2)</p>
<p>For our purpose, we needed an expression for the vortex strength 1 2 x d dx dz</p>
<p>in the following integral. (3)</p>
<p>0</p>
<p>where is the angle of attack, and is the distance from the leading edge of the airfoil, given below. Though this integral is difficult to solve, the assumptions corresponding to the following equations allow for simplification, using the substitution variable . c 1 cos 2 x d c 1 cos 2 c 2 An sin ni n 0</p>
<p>(4) (5) (6) (7)</p>
<p>2</p>
<p>1 cos A0 sin</p>
<p>For this study, coefficients from zero to two were needed. The first three Fouriers coefficients are then given by (8)</p>
<p>A1 A2</p>
<p>20</p>
<p>dx dy dx dy</p>
<p>0</p>
<p>cos</p>
<p>0</p>
<p>d</p>
<p>0</p>
<p>0.3439909</p>
<p>(9)</p>
<p>20</p>
<p>0</p>
<p>cos 2</p>
<p>0</p>
<p>d</p>
<p>0</p>
<p>0.184326</p>
<p>(10)</p>
<p>Once the first three Fouriers coefficients were found, the lift coefficient can be calculated using the following.2</p>
<p>c</p>
<p>A0 A1 2</p>
<p>2</p>
<p>A1</p>
<p>(11) (12)</p>
<p>cl</p>
<p>2</p>
<p>A0</p>
<p>Results were then computed over a range of angles of attack using Excel. Hand calculations can also be found in the appendix. 2D PANEL METHOD Panel methods are a technique to solve incompressible potential flow over thick 2D and 3D geometries. For a 2D analysis as was done for this report, the geometry of the body being analyzed is segmented into piecewise strait line segments. Each line represents a boundary element, and vortex sheets are placed along the segment to act as the boundary around the airfoil,</p>
<p>giving rise to circulation, and hence lift. For an airfoil generating lift, in general the upper surface is characterized by clockwise rotating vortices while the lower surface is characterized by counter-clockwise rotating vortices. If there are more clockwise rotating vortices than counter-clockwise rotating vortices, there is a net clockwise circulation around the airfoil, creating lift. For each line segment along the airfoil, there is a vortex sheet of strength0</p>
<p>ds0</p>
<p>(13)</p>
<p>where is the length of the line segment. Each line is defined by its end points, and by a control point located at the segments midpoint as shown in Fig. 2 below. At this control point, the boundary condition constant is applied. (14)</p>
<p>Figure 2. Schematic of Panel Approximation. To ensure that the flow velocity is tangential to the airfoil surface, it is treated as a streamline and assumed that no flow occurs through the surface. The stream function is an addition of the effects due the uniform free stream flow velocity and the effects due to the vortices on each panel. Using the definition of the velocity components based on the stream function, y u x v (15)</p>
<p>the free stream function is given by the following. u y v x The stream function of a counter-clockwise vortex of radius r and strength 2 ln r is given by (17) (16)</p>
<p>where the radial and tangential components of velocity are shown below, respectively. vr 1 r 0 v (18) r 2 r</p>
<p>For each individual line segment, the stream function can be written in terms of the differential line length and strength as0</p>
<p>ds0 ln r r0 2</p>
<p>(19)</p>
<p>where</p>
<p>is simply calculated using the following. (20)</p>
<p>By integrating over the entire airfoil surface, the stream function for all infinitesimal vortices at each control point can be obtained.0</p>
<p>ln r r0 ds0 2</p>
<p>(21)</p>
<p>Adding the free stream and vortex effects, the equation used to obtain the circulation for each line segment is given by, u y v x0</p>
<p>ln r r0 ds0 2</p>
<p>(22)</p>
<p>where C is a constant. This integral equation is subject to the constraint that the vortex strength on the upper and lower surface must be the same at the trailing edge, commonly called the Kutta Condition. The unknowns are the vortex strength on each panel and the value of the stream function, C. Once the vortex strength is obtained, the coefficient of pressure is calculated using cp where is the free stream velocity. 10 2 2</p>
<p>(23)</p>
<p>To implement these equations in Matlab to obtain a solution for the aerodynamic properties of the airfoil, Eq. (22) is written in terms of two indices. The airfoil is divided into N panels, of which each is numbered j, where j 1,2,N . On each panel it is assumed that is constant, therefore the vortex strength is indexed also, The control points for each line segment are also denoted by an index i, where i 1,2,N . The integral Eq. (22), is then written as follows.</p>
<p>N</p>
<p>u yi v xij 1</p>
<p>0,j</p>
<p>(24)j</p>
<p>2</p>
<p>ln ri r0 ds0</p>
<p>0</p>
<p>The index i refers to the control point at which the equation is applied, the index j refers to the line segment over which the line integral is evaluated. After obtaining the coefficient of pressure the coefficient of normal force is computed by integrating the difference between the upper and lower surface pressure coefficients as, cn 1T</p>
<p>cp,l cp,u ds</p>
<p>(25)</p>
<p>from which the coefficients of lift and drag are resolved using cl cn,y cos cd cn,y sin cn,x sin cn,x cos (26) (27)</p>
<p>where is the angle of attack of the airfoil. The geometry of the airfoil being analyzed is opened from a text file, containing x and y coordinates of each end point for all line segments. This geometry information was obtained for the NACA 4212 from the University of Illinois online database. CFD ANALYSIS USING FLUENT The Ansys Workbench is a powerful engineering simulation software suite. The computational fluid dynamics module named Fluent, utilizes a finite element method to solve the fundamental equations governing fluid flows. The first step in any Fluent analysis is to create the fluid domain, and the geometry of the object of study. For this report, the geometry of the NACA 4212 airfoil was imported from an online data base as a simple text file. The x and y coordinates were placed and merged, creating the 2D profile of the airfoil. A box was then created around the airfoil, to be the fluid volume, with a circular shape on the side corresponding to the airfoils leading edge. A Boolean subtraction was performed to fully designate the airfoil as a solid body, and the outer geometry as the fluid domain. After appropriately naming the edges of the geometry, edge sizing control was used to create a C shaped mesh around the airfoil, with a rectangular mesh from the trailing edge and behind, as shown in Fig. 3 and 4.</p>
<p>Figure 3. Fluent Airfoil Mesh.</p>
<p>Figure 4. NACA 4212 Meshed in Fluent.</p>
<p>With the mesh generated, the computation/solution module of Fluent was launched. Velocity, pressure, and wall boundary conditions were imposed. Critical to this analysis was to observe three different cases for the fluid simulation. To begin, inviscid flow was selected. The simulation was run with a free stream velocity of 10 over a range of different angles of attack. The angle of attack was controlled by modifying the x and y components of the free stream velocity appropriately. It was expected that the inviscid flow results be similar to those found using both analytic tools and the 2D panel method, as these were also only applied for inviscid flow. To obtain results that account for viscous effects, a laminar flow condition was then selected, and the simulation recalculated. Accounting for viscous effects, it was expected that stall would be observed beyond some angle of attack unlike the case for inviscid flow. However, simulating laminar flow did not account for turbulence phenomenon, therefore a third and final flow condition was selected. A k-epsilon k_ turbulent flow condition was specified, which assumes an isotropy of turbulence where by the normal stresses are equal. Upon defining the appropriate characteristic length, density, and flow velocity, Fluents built in force calculations were selected to calculate the lift and drag coefficients for all three flow conditions, over a range of angles of attack. Having the lift coefficients output to the display window, the values were recorded and plotted with the corresponding angles of attack using Excel. RESULTS ANALYTIC Using the equations outlined in the analytical procedure section along with Excel, a plot of the coefficient of lift over 13 separate angles of attack was created, as shown below in Fig. 3.</p>
<p>Analytical Method: Cl vs Angle of Attack5 Coefficient of Lift, Cl 4 3 2 1 0 0 5 10 15 20 25 30 35 y = 0.1097x + 0.8124</p>
<p>Angle of Attack (Deg.)</p>
<p>Figure 3. Coefficient of Lift vs. Angle of Attack Using Analytical Method.</p>
<p>2D PANEL METHOD A total of nine separate text files were created containing the airfoils x and y coordinates, as well as specifying the angle of attack to be used in the calculation. An example text file can be found in the appendix. The Matlab m-file Panel_Method.m was run using 9 separate files, corresponding to the 9 angles of attack that were analyzed. The coefficients of lift and drag, which were output by the m-file were recorded and plotted with the corresponding angles of attack using Excel. Plots of coefficient of lift and drag, as well as the lift/drag ratio are shown below as functions of the angle of attack.</p>
<p>Panel Method: Cl vs. Angle of Attack6 y = -4E-05x3 + 0.0022x2 + 0.1166x - 1.0915 5 4 3 2 1 0 -1 0 10 20 30 40 50 -2 Angle of Attack (Deg) Coefficient of lift, Cl</p>
<p>60</p>
<p>70</p>
<p>Figure 4. Coefficient of Lift vs. Angle of Attack for NACA 4212 Using Panel Method.</p>
<p>Panel Method: Cd vs. Angle of AttackCoefficient of drag, Cd 8 6 4 2 0 -2 0 10 20 30 40 50 Angle of Attack (Deg) 60 70 y = 0.0016x2 + 0.0144x - 0.1055</p>
<p>Figure 5. Coefficient of Drag vs. Angle of Attack for NACA 4212 Using Panel Method.</p>
<p>Lift to Drag Ratio10 5 0 -5 0 -10 -15 -20 -25 -30 -35 y = 5E-07x5 - 1E-04x4 + 0.0082x3 - 0.3124x2 + 5.4475x - 31.202 Lift to Drag Ratio, Cl/Cd</p>
<p>10</p>
<p>20</p>
<p>30</p>
<p>40</p>
<p>50</p>
<p>60</p>
<p>70</p>
<p>Angle of Attack (Deg.)</p>
<p>Figure 6. Lift to Drag Ratio (Cl/Cd) for NACA 4212 Using Panel Method.</p>
<p>FLUENT Printing the coefficients of lift and drag directly from Fluent made plotting these values vs. angle of attack strait forward. The capabilities of Fluent were also used to create contour plots of static pressure. To better display the characteristics of the fluid flow, a plot of velocity vectors was also created. For simplicity and conciseness, only one angle of attack, 15 degrees, was selected to create all the plots for display as follows. Coefficients of lift and drag for each type of flow are shown as well.</p>
<p>INVISCID FLOW</p>
<p>Figure 7. Pressure Contours at 15 Degree Angle of Attack, Inviscid Flow.</p>
<p>Figure 8. Velocity Vectors at 15 Degree Angle of Attack, Inviscid Flow.</p>
<p>Coefficient of Lift, Inviscid Flow2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 Coefficient of Lift, Cl y = -0.0004x2 + 0.0438x + 0.4854</p>
<p>10</p>
<p>20</p>
<p>30</p>
<p>40</p>
<p>50</p>
<p>60</p>
<p>Angle of Attack (Deg.)</p>
<p>Figure 9. Coefficient of Lift vs. Angle of Attack, Inviscid Flow.</p>
<p>Coefficient of Drag, Inviscid Flow0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 y = -5E-06x2 + 0.0025x + 0.0276</p>
<p>Coeffic...</p>

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