In the Pre-Analysis step, we'll review the following:

**Mathematical model:**We'll look at the governing equations + boundary conditions and the assumptions contained within the mathematical model.**Numerical solution procedure in Ansys:**We'll briefly overview the solution strategy used by Ansys and contrast it to the hand calculation approach.

Our simulation is governed by the continuity, Navier-Stokes (momentum conservation), and energy equations, but since this is turbulent flow, we will be using the Reynolds Averaged version of these equations. We will also be using the Spalart-Allmaras turbulence model to close the Reynolds Averaged equation set. The Spalart-Allmaras turbulence model has been developed for aerospace applications that are wall-bounded and subject to adverse pressure gradients. The Spalart-Allmaras model has only one equation which solves for the kinematic eddy viscosity. We need to solve for the variables of the problem in all cell centers of our mesh. In total, we have six variables to solve for: 3 components of velocity, pressure, temperature, and the kinematic eddy viscosity. The equations are given below.

The Continuity Equation is given by equation 1

The Reynolds Averaged Navier-Stokes equation is given by equation 2

The conservation of energy equation is given by equation 3

The Spalart-Allmaras turbulence model is given by equation 4

The equations are converted to algebraic equations. It then solves for our six variables at each of the cell centers of our mesh. This means that if we have 300,000 cells, Fluent is going to solve 1.8 million equations to solve the problem.

In this simulation, we expect to see many flow features in three dimensions. We will be especially interested in the suction peak (low-pressure zone) that forms on the wing and how the size of the suction peak changes spanwise. We will also be looking for a shock along the wing surface as this is transonic flow. We also expect to see trailing edge vortices forming downstream of the wing, due to the interaction of the high and low-pressure zones since this is a finite length wing.

We will also try to predict the lift coefficient of the wing. This calculation will assume that you have seen some of the basics of aerodynamics when it comes to finite wings.

We need to get the lift curve slope for our airfoil assuming that there it is part of an infinite wing. We know that our airfoil is symmetrical, hence at 0 angle of attack, it produces no lift. We find our airfoil (or one that can be closely approximated as our airfoil since it exhibits no lift at a 0 angle of attack). The airfoil that we used is detailed here: ONERA OA206 Airfoil

We calculate the slope for an infinite wing using its characteristics and find a0 = .0884

Using this, we then can calculate what our lift curve slope for the finite wing will be using the correction for a swept wing. We need to use this correction for a swept wing since the free stream Mach Number is not seen by the entire wing and instead the wing sees a lesser Mach Number, delaying the onset of a shock and increasing the critical Mach Number.

To use this equation, we must first know the aspect ratio of the wing.

where b is the span and S is the planform area of the wing. These two values are either known or easily calculated. We have an aspect ratio of 3.8 for our model.

After we use all of our values, we get a corrected slope a = .0760. For finite wings, we know that the lift curve will be lower than for infinite wings due to 3D effects and this is reflected in our calculation here. Once we know the slope for our finite wing, we can calculate the lift coefficient for the wing.

Since the airfoil is symmetric, we will say that αL = 0. Then, we get that our lift coefficient is .2328. However, this is for the entire wing (remember that we use only half for our simulation) and the data that we used to calculate this is for low-speed incompressible flows. This means that we need to use a correction for the compressibility at M = .8395.

Using this correction factor, we get that the lift coefficient for the entire wing will be 0.4284.

This would be for the entire wing and for our half wing, we simply divide by 2 to predict a lift coefficient of .2141.

This is a ballpark estimate of our lift coefficient and we expect that Fluent will give us something comparable but less than what we predict with our hand calculations, due to the presence of the shock on the wing surface.

We will be setting this to the type wall. Basically, a wall boundary condition is setting the velocity there to be 0.

We will be setting this to type symmetry. This basically means the solution is symmetrical with respect to this plane.

We will be setting these to type pressure far-field. At these boundaries, we need to specify the pressure, Mach number, temperature, and components of the velocity. This allows for the calculation of the speed of sound, and the velocity direction.