Flow over an airfoil
Flow over an airfoil
AIM: Model the flow over a NACA0017 airfoil For the angle of attacks 0,2,4,6,8,10 compare the lift and drag forces.
ABSTRACT:
In this project we are going to perform transient state simulation of flow over a four digit NACA 2412 airfoil at various angle of attack. We are going to look at the lift and drag coefficients at various angle of attack for a particular Reynolds number. By performing simulations the stall condition for an airfoil is calculated. Graph of lift/drag coefficients vs angle of attack is plotted.
INTRODUCTION:
Airfoil is a structure with curved surfaces designed to give the most favourable ratio of lift to drag in flight, used as the basic form of the wings, fins, and horizontal stabilizer of most aircraft. An Airfoil shaped body moved through a fluid produces an aerodynamic force. The forward part of an airfoil is rounded and is called the leading edge. The aft part is narrow and tapered and is called the trailing edge. A reference line often used in discussing airfoils is the chord, an imaginary straight line joining the extremities of the leading and trailing edges. The angle between the chord line and the flight direction is called the angle of attack and has a large effect on the drag generated by the wing. The component of this force perpendicular to the direction of motion is called lift. The component parallel to direction of motion is called drag. The lift of an airfoil is primarily the result of angle of attack and shape (in particular its chamber). When either is positive, the resulting flow field about the airfoil has a higher average velocity on the upper surface than the lower surface. This velocity difference is necessarily accompanied by a pressure difference (via Bernoulli's principle) which in turn produces lift force. As the angle increases the drag quickly rises because of increased frontal area and increased boundary layer thickness. When the boundary layer separates, the wing is said to be stalled and both drag and lift become unsteady. In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment. The drag coefficient is always associated with a particular surface area. The drag coefficient is defined as.
`cd = (2Fd)/(ρ μ^2A)`
where:
Fd- is the drag force, which is by definition the force component in the direction of the flow velocity, p- is the mass density of the fluid, U is the flow speed of the object relative to the fluid, A- is the reference area.
The lift coefficient (CL) is a dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. The lift coefficient CL is defined by
`CL=L/(qS)=1/(1/2ρμ^2S)=2L/(ρμ^2S)`
Where:
L-is the lift force,
S- is the relevant surface area,
U is the flow speed of the object relative to the fluid,
p- is the mass density of the fluid.
Turbulence in a flow field is often characterized by the irregular,random fluctuation in the flow properties. Since the statistical quantities like mean,variance etc were reproducible, Sir Osborne Reynolds(1883) has brought up the clever idea of decomposing the randomly varying instantaneous flow property into mean plus fluctuating components.
Now, substituting mean and fluctuating properties in instantaneous Navier-Stokes equations and averaging results in the so-called RANS(Reynolds averaged Navier stokes equations). They are the time averaged equations of motion for fluid flow. Now there is a problem with these equations. They are not closed. There exists a "unknowns" tensor called Reynolds stress tensor. No matter how many manipulations we do,we shall always end up with more statistical unknowns than the equations relating them. This is the famous turbulence closure problem.
So Reynolds Stress terms needs to be modeled using functions containing empirical constants and information about the mean flow hence known as "Turbulence Modelling".
In this simulation we are going to use K-omega turbulence model. K omega (k-w) model is one of the most commonly used models. This include 2 additional transport equations to represent turbulent properties of flow - to account for history effects like convection and diffusion of turbulent energy. The transport variable k determines the energy in turbulence and w determines the scale of turbulence.
The basic k-w model can be used for boundary layer problems, where the formulation works from the inner part throught the viscous sublayer, till the walls - hence the k-w SST model can be used as a low reynolds flow applications without extra damping functions. SST stands for Shear Stress Transport. The SST formulation also switches to a k-e behaviour in the free-stream, which avoids that the k-w problem that the model is very sensitive to the inlet free-stream turbulence properties. The k-w SST model also accounts for its good behaviour in adverse pressure gradients and separating flow. The k-w SST model does produce some large turbulence levels in regions with large normal strain, like stagnation regions and regions with strong acceleration. This effect is much less pronounced than with a normal k-e model though. The SST model has the ability to account for the transport of the principal shear stress in adverse pressure gradient boundary-layers.
When using a low Reynolds turbulence model to solve the near-wall region, the first cell center must be placed in the viscous sublayer leading to the requirement of abundant mesh cells. Another way is to use the so called wall functions, which can model the near wall region. Wall functions are equations empirically derived and used to satisfy the physics in the near wall region. The first cell center needs to be placed in the log-law region to ensure the accuracy of the results. Wall functions are used to bridge the inner region between the wall and the turbulence fully developed region. When using the wall functions approach, there is no need to resolve the boundary layer causing a significant reduction of the mesh size and the computational domain. As long as we are in the non-separated flow region, the results will be reasonable regardless of how the wall stress is derived. But if we decide to compute the separated flow problem, we use a low Reynolds number model or a simple algebraic model where the whole flow field is covered. This is because it can be very confusing in the separation point region where the flow direction, the sign of the wall stress, and the validity of the wall function are all unknowns. So, for separated flow problems, we move on to the low Reynolds number model like k-w SST.
Y+ is a non-dimensional distance. It is often used to describe how coarse or fine a mesh is for a particular flow pattern. It is important in turbulence modeling to determine the proper size of the cells in domain walls. The turbulence model wall laws have restrictions on the Y+ value at the wall.
A non-dimensional wall distance for a wall-bounded flow can be defined in the following way Y+ = (U Y)/V
Where,
U-is the friction velocity at the nearest wall,
Y - is the distance to the nearest wall,
V-is the local kinematic viscosity of the fluid.
If we are using cell-centered scheme for finite volume method then the value of Y is multiplied by 2 since the equations are solved at the cell center. After getting the value of Y we set the grid size at wall equal to 2Y.
To get the data accurate near the wall the value of Y+ should be as low as possible such as Y+ = 1. It is important to choose right turbulence model for the given Y+ to resolve the flow near the wall.
SIMULATION SETUP:
To perform the external aerodynamic simulation we need to create a virtual wind tunnel. The wind tunnel is of cuboid shape. The size of the wind tunnel is very important in the simulation since in reality their is no such wind tunnel, the object is in open environment. The size of wind tunnel should be such that the boundaries of wind tunnel does not affect the flow near the object under test. By performing some simulation we came to a conclusion that the wind tunnel length should be minimum 50 times the length of the object in our case 50 times the chord length).
The height of wind tunnel should be minimum 30 times the length of object (length in the direction of flow). The airfoil is kept at a distance of 25 times chord length from the inlet and 15 times the chord length from top wall. The airfoil under test is NACA 2412 having a chord length of 0.1m.
Boundary Conditions: We are going to perform 2-D simulation for the Reynold's number of 200,000. By using formula Re (density x velocity x characteristic length)/ dynamic viscosity. Density of air at 300K is 1.1839 Kg/m^3, characteristic length here is chord length equal to 0.1m, dynamic viscosity of air at 300 K is 1.846 x 10^-5 Kg/m-sec. Using above formula we get velocity equal to 31.18 m/sec. The wall in front of airfoil has assigned inflow boundary having inflow velocity of 31.18 m/sec. The wall at the back of airfoil in the direction of flow is assigned outflow boundary condition having static pressure of 101325 Pa. The top and bottom wall are assigned to symmetry boundary condition. We can't use very big wind tunnel due to more computational time hence we tell solver that the condition on either of the wall is same. The front and back wall are assigned to 2D-Wall boundary condition. The material used is air having 02(0.23 mass fraction) and N2(0.77 mass fraction). The region is assigned with initial velocity of 31.18 m/sec.
The base grid size is set to 0.01m. Fixed boundary embedding is used for airfoil having scale of 4 and embedded layer equal to two. Box grid is created around the airfoil having scale of 2. These values are calculated using the formula [ scaled grid = (dx_base)/(2*grid scale)]. The scaled grid is calculated using formula for Y+ and then the grid scale value is calculated. The grid scale value 4 correspond to a Y+ = approx.30.
The following cases with different angle of attack are simulated
Case 1: Angle of attack = 0 Degree
Case 2: Angle of attack = 2 Degree
Case 3: Angle of attack = 4 Degree
Case 4: Angle of attack = 6 Degree
Case 5: Angle of attack = 8 Degree
Case 6: Angle of attack = 10 Degree
COMPUTATIONAL METHOD:
The governing equations Reynolds-averaged Navier-Stokes with k-w SST turbulence model having automatic wall function is solved to simulate the flow. Transient simulation is performed. PISO algorithm is used for pressure-velocity coupling. Initial time step is 1e-9, minimum time step is 1e-9 and maximum time step is 1. The discretization is based on cell centered finite volume scheme. To solve equations SOR solver is used. Simulation is run for 0.3 sec.
COMPUTATIONAL METHOD:
The governing equations Reynolds-averaged Navier-Stokes with k-w SST turbulence model having automatic wall function is solved to simulate the flow. Transient simulation is performed. PISO algorithm is used for pressure-velocity coupling. Initial time step is 1e-9, minimum time step is 1e-9 and maximum time step is 1. The discretization is based on cell centered finite volume scheme. To solve equations SOR solver is used. Simulation is run for 0.3 sec.
Case 1: Angle of attacks 0 degree .
I run the simulation only for 150 iteration.
Drag force 0.544 N
Lift force 2.274e-04 N
CASE 2: Angle of attacks 2 degree .
I run the simulation only for 150 iteration.
Drag force 0.584 N
Lift force 1.129 N
CASE 3: Angle of attacks 4 degree .
I run the simulation only for 150 iteration.
Drag force 0.681 N
Lift force 2.287 N
CASE 4: Angle of attacks 6 degree .
I run the simulation only for 150 iteration.
Drag force 0.889N
Lift force 5.115 N
CASE 5: Angle of attacks 8 degree .
I run the simulation only for 150 iteration.
Drag force 1.199 N
Lift force 5.115 N
CASE 6: Angle of attacks 10 degree .
I run the simulation only for 150 iteration.
Drag force 1.577 N
Lift force 6.369 N
RESULTS:
* For 2D simulation CONVERGE calculates the pressure force in N/m. So, we need to first multiply by characteristics length to the pressure force and then apply the formula.
The average pressure force is calculated by taking average of quantities for last 0.1 sec. Using above mentioned formula in introduction section the co-efficient of drag and lift are calculated. The characteristics length of airfoil at 0 degree is 0.1m (chord length) and height is 0.01211m these values changes as the angle of attack increases hence the effective area also changes.
CONCLUSION:
From the above results it is observed that at the location, where the flow is splitting up, the flow velocity is reduced to zero and high pressure is generated. This point is called stagnation point. The flow over the top of airfoil has high velocity and low pressure. At the bottom of airfoil their is low velocity and high pressure hence lift force is generated on the airfoil.
As we increase the angle of attack the drag force increases since the front area increases and the lift force also increases. But there is a certain angle after which drag force increases but lift force decreases this angle is known as critical angle and this condition is known as stall condition. Stalling is caused by flow separation which in turn, is caused by the air flowing against a rising pressure. Due to this adverse pressure gradient the boundary layer separate from the airfoil. This separation produces vortex behind the airfoil. The calculation of drag and lift co efficient in such situation is very difficult because of its transient nature. Because of this transient nature we should perform transient simulation since the steady state simulation may give inaccurate results. The solution in steady state may blow up after some iteration.