Pressure Coefficient as a Function of Angle of Attack

COMSATS Institute of Information Technology Sahiwal

Pressure Coefficient as a Function of Angle of Attack CFD Analysis of Change in Coefficient of Pressure along the Upper and the Lower Surfaces of NACA 2414 Airfoil as a Function of Angle of Attack

Awais Ahmad Shah SP13-BME-007

Muhammad Sufyan SP13-BME-011

Supervised by: Dr. Tareq Manzoor

Pressure Coefficient as a Function of Angle of Attack

CFD

Abstract The report is about the computational study of subsonic, inviscid, incompressible air flow over NACA 2414 airfoil. The study parameter is pressure coefficient whose distribution, along the upper and the lower surfaces of the airfoil, for different angles of attack is simulated. The analysis involves the 2D assessment and the results obtained are in agreement with the expected values.

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Contents 1. Introduction -------------------------------------------------------------------------------------------- 3 2. Literature Review ------------------------------------------------------------------------------------- 4 2.1. Pressure Coefficient ----------------------------------------------------------------------------- 4 2.2. Airfoil Nomenclature --------------------------------------------------------------------------- 4 2.3. Standard NACA Airfoils ----------------------------------------------------------------------- 5 2.3.1. NACA 4-Digit Airfoils (N2414) --------------------------------------------------------- 5 2.4. The CFD Process -------------------------------------------------------------------------------- 5 2.4.1. Discretization ------------------------------------------------------------------------------- 6 2.4.2. Grid Generation ---------------------------------------------------------------------------- 7 2.4.3. Boundary Conditions ---------------------------------------------------------------------- 9 3. Methodology------------------------------------------------------------------------------------------- 9 3.1. Preprocessing------------------------------------------------------------------------------------- 9 3.1.1. Geometry ------------------------------------------------------------------------------------ 9 3.1.2. Meshing ------------------------------------------------------------------------------------ 10 3.1.3. Applied Boundary Conditions ---------------------------------------------------------- 11 3.2. Solution ----------------------------------------------------------------------------------------- 11 3.3. Post-Processing -------------------------------------------------------------------------------- 11 3.3.1. Color Diagrams --------------------------------------------------------------------------- 11 3.3.2. Data Plots ---------------------------------------------------------------------------------- 13 4. Results Discussion ---------------------------------------------------------------------------------- 15

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1. Introduction Pressure coefficient is an important aerodynamic parameter which determines the relative pressure at a point in the flow field. It is a dimensionless number. Since it is related to the lift generation, therefore its significance in the aeronautics is no less than other dimensionless numbers such as lift coefficient and drag coefficient. An airfoil is a two-dimensional cross section of a streamlined body. It is usually not a regular shape. The purpose of such a shape is to create a pressure difference between the upper and the lower surfaces, so that a net upward or downward is generated. In case of lifting flow such as over the wings of an aircraft, the upper surface has lower pressure than the bottom surface, providing a net upward force. However, this phenomenon is reversed in case of automobiles where the reverse airfoil shapes are extensively used for creating the downforce. CFD provides a good approximation of the flow parameters such as velocity, pressure, turbulence, forces etc.; pressure coefficient being one of them. The Navier-Stokes equations are the governing equations of inviscid incompressible flow. In this study, we have simulated a low speed subsonic flow over a cambered airfoil NACA 2414 at three different angles of attack. We used Modified Petrov-Glerkin Method for our solution. The distribution of pressure coefficient at the upper and the lower surfaces of the airfoil was determined. The data obtained was plotted to understand the variation of pressure coeffi cient with the angle of attack. The initial section of this report give a theoretical background and a mathematical description of pressure coefficient. This literature review section also deals with the nomenclature and the theory of airfoils. The remaining sections of the report explain our computational methodology and the results obtained.

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2. Literature Review [1] [2] 2.1. Pressure Coefficient The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field. Mathematically, it is given as

 = 1⁄−∞ 2 Where,

 = Stagnation Pressure P = Free-stream pressure ∞

1⁄2   = Dynamic Pressure The above definition is used throughout aerodynamics, from incompressible to hypersonic flow. In the aerodynamic literature, it is very common to find pressures given in terms of



rather than the pressure itself. Indeed, the pressure coefficient is another similarity parameter. For incompressible flow,

 can be expressed in terms of velocity only. Using Bernoulli’s

equation, we get

   = (1 − ∞) Where, V = Local velocity V ∞ = Free stream velocity

2.2. Airfoil Nomenclature Airfoil: Any section of the wing cut by a plane perpendicular to the wingspan is called an airfoil. Fig. 1.

Fig. 1: Geometry of an Airfoil

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Mean Camber Line: The mean camber line is the locus of points halfway between the upper and lower surfaces as measured perpendicular to the mean camber line itself. Leading Edge and Trailing Edge: The most forward and rearward points of the mean camber line are the leading and trailing edges, respectively. Chord line and Chord: The straight line connecting the leading and trailing edges i s the chord line of the airfoil, and the precise distance from the leading to the trailing edge measured along the chord line is simply designated the chord c of the airf oil. Camber: The camber is the maximum distance between the mean camber line and the chord line, measured perpendicular to the chord line. Thickness: The thickness is the distance between the upper and lower surfaces, also measured perpendicular to the chord line.

2.3. Standard NACA Airfoils The shape of the airfoil at the leading edge is usually circular, with a leading-edge radius of approximately 0.02c. The shapes of all standard NACA airfoils are generated by specifying the shape of the mean camber line and then wrapping a specified symmetrical thickness distribution around the mean camber line. 2.3.1. NACA 4-Digit Airfoils (N2414)

The NACA identified different airfoil shapes with a logical numbering system. For example, the first family of NACA airfoils, developed in the 1930s, was the “four -digit” series, such as the NACA 2414 airfoil. Here, the first digit is the maximum camber in hundredths of chord, the second digit is the location of maximum camber along the chord from th e leading edge in tenths of chord, and the last two digits give the maximum thickness in hundredths of chord. For the NACA 2414 airfoil, the maximum camber is 0.02 c located at 0.4c from the leading edge, and the maximum thickness is 0.14 c. It is common practice to state these numbers in percent of chord, that is, 2 percent camber at 40 percent chord, with 14 percent thickness. An airfoil with no camber, that is, with the camber line and chord line coincident, is called a symmetric airfoil. Clearly, the shape of a symmetric airfoil is the same above and below the

chord line.

2.4. The CFD Process Computational fluid dynamics (CFD) involves replacing the partial differential equations with discretized algebraic equations that approximate the partial differential equations. These equations are then numerically solved to obtain flow field values at the discrete points in space and/or time. Since the Navier – Stokes equations are valid everywhere in the flow field of the fluid continuum, an analytical solution to these equations provides the solution for an 5


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infinite number of points in the flow. However, analytical solutions are available for only a limited number of simplified flow geometries. To overcome this limitation, the governing equations can be discretized and put in algebraic form for the computer to solve. The CFD simulation solves for the relevant flow variables only at the discrete points, which make up the grid or mesh of the solution (discussed in more detail below). Interpolation schemes are used to obtain values at non-grid point locations. 2.4.1. Discretization

The process of discretization involves developing a set of algebraic equations (based on discrete points in the flow domain) to be used in place of the partial differential equations. Of the various discretization techniques available for the numerical solution of the governing differential equations, the following three types are mos t common: 1) The finite difference method, 2) The finite element (or finite volume) method, and 3) The boundary element method. In each of these methods, the continuous flow field (i.e., velocity or pressure as a function of space and time) is described in terms of discrete (rather than continuous) values at prescribed locations. Through this technique the differential equations are replaced by a set of algebraic equations that can be solved on the computer. Finite Element Method: For the finite element (or finite volume) method, the flow field is broken into a set of small fluid elements (usually triangular areas if the flow is twodimensional, or small volume elements if the flow is three-dimensional). The conservation equations (i.e., conservation of mass, momentum, and energy) are written in an appropriate form for each element, and the set of resulting algebraic equations for the flow field is solved numerically. The number, size, and shape of elements are dictated in part by the particular flow geometry and flow conditions for the problem at hand. As the number of elements increases (as is necessary for flows with complex boundaries), the number of simultaneous algebraic equations that must be solved increases rapidly. Problems involving one million (or more) grid cells are not uncommon in today’s CFD community, particularly for complex three-dimensional geometries. Boundary Element Method: For the boundary element method, the boundary of the flow field (not the entire flow field as in the finite element method) is broken into discrete segments and appropriate singularities such as sources, sinks, doublets, and vortices are distributed on these boundary elements. The strengths and type of the singularities are chosen so that the appropriate boundary conditions of the flow are obtained on the boundary elements. For points in the flow field not on the boundary, the flow is calculated by adding the contributions from the various singularities on the boundary. Although the details of this method are rather 6


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mathematically sophisticated, it may (depending on the particular problem) require less computational time and space than the finite element method. Finite Difference Method: The finite difference method for computational fluid dynamics is perhaps the most easily understood and widely used of the three methods listed above. For this method the flow field is dissected into a set of grid points and the continuous functions (velocity, pressure, etc.) are approximated by discrete values of these functions calculated at the grid points. Derivatives of the functions are approximated by using the differences between the function values at local grid points divided by the grid spacing. The standard method for converting the partial differential equations to algebraic equations is through the use of Taylor series expansions. 2.4.2. Grid Generation

CFD computations using the finite difference method provide the flow field at discrete points in the flow domain. The arrangement of these discrete points is termed the grid or the mesh. The type of grid developed for a given problem can have a significant impact on the numerical simulation, including the accuracy of the solution. The grid must represent the geometry correctly and accurately, since an error in this representation can have a significant effect on the solution. The grid must also have sufficient grid resolution to capture the relevant flow physics, otherwise they will be lost. This particular requirement is problem dependent. For example, if a flow field has small-scale structures, the grid resolution must be sufficient to capture these structures. It is usually necessary to increase the number of grid points (i.e., use a finer mesh) where large gradients are to be expected, such as in the boundary layer near a solid surface. The same can also be said for the temporal resolution. The time step, ∆t, used for unsteady flows must be smaller than the smallest time scale of the flow features being investigated. Generally, the types of grids fall into two categories: structured and unstructured, depending on whether or not there exists a s ystematic pattern of connectivity of the grid points with their neighbors. Structured Grid: As the name implies, a structured grid has some type of regular, coherent structure to the mesh layout that can be defined mathematically. The simplest structured grid is a uniform rectangular grid, as shown in Fig. 2(a). However, structured grids are not restricted to rectangular geometries. Fig. 2(b) shows a structured grid wrapped around a parabolic surface. Notice that grid points are clustered near the surface (i.e., grid spacing in normal direction increases as one moves away from the surface) t o help capture the steep flow gradients found in the boundary layer region. This type of variable grid spacing is used wherever there is a need to increase grid resolution and is termed grid stretching.

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Fig. 2: Structured Grids (a) Rectangular Grid (b) Grid around a parabolic surface

Unstructured Grid: For the unstructured grid , the grid cell arrangement is irregular and has no systematic pattern. The grid cell geometry usually consists of various-sized triangles for two-dimensional problems and tetrahedrals for three-dimensional grids. An example of an unstructured grid is shown in Fig. 3. Unlike structured grids, for an unstructured grid each grid cell and the connection information to neighboring cells is defined separately. This produces an increase in the computer code complexity as well as a significant computer storage requirement. The advantage to an unstructured grid is that it can be applied to complex geometries, where structured grids would have severe difficulty. The finite difference method is restricted to structured grids whereas the finite volume (or finite element) method can use either structured or unstructured grids.

Fig. 3: Unstructured Grid over a NACA 0012 airfoil

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2.4.3. Boundary Conditions

The same governing equations, the Navier – Stokes equations, are valid for all incompressible Newtonian fluid flow problems. Thus, if the same equations are solved for all types of problems, how is it possible to achieve different solutions for different types of flows involving different flow geometries? The answer lies in the boundary conditions of the problem. The boundary conditions are what allow the governing equations to differentiate between different flow fields (for example, flow past an automobile and flow past a person running) and produce a solution unique to the given flow geometry. It is critical to specify the correct boundary conditions so that the CFD simulation is a well-posed problem and is an accurate representation of the physical problem. Poorly defined boundary conditions can ultimately affect the accurac y of the solution. One of the most common boundary conditions used for simulation of viscous flow is the no-slip condition. Thus, for example, for twodimensional external or internal flows, the x and y components of velocity ( u and v) are set to zero at the stationary wall to satisfy the no-slip condition. Other boundary conditions that must be appropriately specified involve inlets, outlets, far-field, wall gradients, etc. It is important to not only select the correct physical boundary condition for the problem, but also to correctly implement this boundary condition into the numerical simulation.

3. Methodology 3.1.

Preprocessing

3.1.1. Geometry

The first problem regarding the airfoil geometry was its non-trivial shape which is approximated by parametric interpolation formulas. We found it feasible to avoid using these formulas. Instead, an alternative approach was adopted. We got the data points

[3]

for

NACA2414 and imported them in the Autodesk ® Inventor. The chord was plotted in the +x direction and thickness above and below the y-axis. An interpolation spline was used to generate the upper and lower surfaces of the airfoil , Fig. 4. The chord length of the airfoil was taken as 1000 mm. A rectangular control surface was also sketched around the airfoil, Fig. 5. We developed three models of airfoils at three angles of attack of 0 o, 5o and 10 o.

Fig. 4: Sketch geometry of NACA 2414 airfoil at 0o AoA

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Fig. 5: Surface geometry showing control surface around the airfoil

3.1.2. Meshing

The surface model of the airfoil was imported in Autodesk ® CFD. The materials were assigned to the airfoil and the control surface. An automatic meshing was used as initial meshing approach. Then, we applied two refinement regions one around the airfoil to the boundary edge and the other just around the airfoil. Fig. 6. This mesh refinement increased the calculated time but it certainly improved the results as the model was better approximated. The structured mesh of the boundary layer region and the unstructured mesh of the remaining portion are shown in Fig. 7.

Fig. 6: Meshed surfaces showing refinement regions

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Fig. 7: Structured and Unstructured Meshing

3.1.3. Applied Boundary Conditions

We used velocity and pressure boundary conditions. The given Mach No. 0.15 corresponds to 52 m/s. This velocity was applied as inlet boundary condition and a gage pressure of 0 Pa was applied at the outlet. The top and bottom edges of the control surface were given slip/symmetry condition as there was no flow in the y-direction.

3.2. Solution The solution scheme used for all simulations was Modified Petrov-Glerkin Advection Method. The solution converged between 600-800 iterations for all three cases.

3.3. Post-Processing 3.3.1. Color Diagrams

A colored contour plot was generated for the pressure coefficient distribution. It showed a clearly distinction between the high pressure and the low pressure regions around the airfoil. Fig. 8, 9 and 10 show the coefficient of pressure distribution around the airfoil for 0 o, 5o and 10o angles of attack, respectively.

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Fig. 8 (a): Cp distribution around N2414 at 0o

Fig. 8 (b): Cp distribution around N2414 at 5o

Fig. 8 (c): Cp distribution around N2414 at 10o

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3.3.2. Data Plots

Our primary objective was to discuss the distribution of pressure coefficient along the upper surface and the lower surface of the airfoil. In our 2D analysis, these surfaces correspond to the edges. The extraction of coordinates for these edges caused some serious problems. It took a lot of time and effort to come to resolve the issue. Finally, we extracted the geometry coordinates from the Inventor model, performed some tedious manipulations and incorporated them into our analysis file. Pressure distribution plots along the upper and lower surfaces for three angles of attack were generated.

Fig. 9 (a): C p along the UPPER surface of N2414 at 0 o AoA

Fig. 9 (b): C p along the LOWER surface of N2414 at 0 o AoA

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4. Results Discussion Fig. 9 through 11 show the distribution of pressure coefficient at the upper and the lower surfaces of the airfoil. For all three cases, the value of C p is maximum at the leading edge. This is because of the stagnation point at the leading edge. However, its maximum possible value is 1 because the flow is inviscid. This approximation is very accurate for low speed subsonic flows because the compressibility effects are negligible. C p decreases rapidly as the air flows over the upper surface of the airfoil. It becomes zero in the interval 800 mm < x < 1000 mm for all three cases. At 0 o angle of attack, C p < 0 for the most part at the upper surface as well as lower surface. This shows a net downward pressure which causes downforce. As the angle of attack increases, values of C p rise on the lower surface and become positive on the upper surface. This shows a net positive (upward) force on the airfoil.

References 1. Anderson, J. D. (2011). Fundamentals of Aerodynamics. New York: McGraw-Hill Companies, Inc. 2. Munson, B. R. (2009). Fundamentals of Fluid Mechanics. John Wiley & Sons, Inc. 3. Airfoil Tools. Retrieved from http://airfoiltools.com/airfoil/details?airfoil=n2414-il

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Pressure Coefficient as a Function of Angle of Attack

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