Task 3

Theoretical Analysis vs. Experiment

Jae Hwan Choi, Shane Hills, Matthew Berk, Michael Vegh

Task 3 focuses on modeling the Bixler 2 aerodynamics and performance and assessing the model as compared to experimental data obtained from real flights outlined in Task 2. In order to model the aerodynamic characteristics of the Bixler 2, we used a variety of measurements of the main airfoils and incorporated them into a model in AVL. Measurements of current interest are as follows:

The full suite of Bixler measurements can be found in a spreadsheet file attached at the bottom of this page. This is purely for reference purposes, as many measurements were used in the aerodynamic simulation both to define the airfoil geometries and to calculate wetted areas for drag calculations.

Model Analysis:

In order to model the aerodynamic performance we were required to estimate the shape of the Bixler 2 airfoils. It is clear that the main wing is not a symmetric airfoil, and as a result we chose to model the wing as a cambered airfoil, specifically the NACA 2411. By measuring the thickness to chord ratio for the bixler it was clear that the NACA 2411 was a good match. The profile this the main wing is shown below.

In contrast, the tail utilizes symmetric airfoil profiles, and as a result we chose to model the rear wing (both horizontal and vertical) as NACA 0011 airfoils for similar reasons outlined above, which has a profile shown below.

Using these 2 airfoil geometries and our measurements for the Bixler 2 we were able to input the geometry of the lifting members into AVL, which is shown below.

The key assumptions that went into our model were the following:

<AVL>

- assumes inviscid flow

<Main Wing>

- The angle of incidence is 5.71 degrees throughout the wing (no wash-out or wash-in)

- NACA 2411

<Horizontal Tail>

- The angle of incidence is 1.61 degrees throughout the wing (no wash-out or wash-in)

- NACA 0011

<Vertical Tail>

- NACA 0011

<Fuselage>

- Currently not including model of fuselage geometry, as is common in AVL

Furthermore, since we are operating near ground level the atmospheric conditions were taken to be at sea level, in particular μ = 1.812 x 10^-5 kg/ms, ρ = 1.225 kg/m³ L = 0.2032 m. We also assume that the Reynolds number is between 5 x 10⁴ < Re < 10⁵.

The Trefftz plane plots for select AOA are shown below.

Let c_l be the lift coefficient for a given differential span-wise airfoil element, i.e. a 2D cross section. Using Xfoil we can analyze the 2D wing section for the NACA 2411 airfoil to determine lift coefficient as a function of angle of attack. Let c_l1 be the 2D section lift coefficient of the NACA 2411 air foil at stall. The pressure coefficient as a function of angle of attack is shown below, as well as the pressure profile across the 2D section.

We can see from the tabulated data that the lift coefficient falls off at an angle of attack of approximately α = 10.5 degrees, which corresponds to c_l1 = 1.2036. Furthermore, tabulated data exists for the NACA 2411 airfoil. Given our flight velocity and atmospheric conditions, we consider a Reynolds number of approximately Re = 100 000, which falls within our range given above. For this Reynolds number and the NACA 2411 tabulated data on airfoiltools.com, we have the following data for lift coefficient as a function of angle of attack.

c_l vs α

Notice that the tabulated data agrees well with our maximum section lift coefficient from Xfoil. Clearly then we can be confident that the wing will stall when c_l in the Trefftz plane exceeds c_l1 any section. Since this approach is reasonably conservative, we will assume that stall occurs when 60% of the wing exceeds c_l1. Tabulated data from AVL is given below for different angles of attack (AOA).

Here maximum c_l is the lift coefficient from the section that produces the maximum lift, while C_L is the total lift coefficient across the model, obtained by including the tail, which is automatically performed by AVL. Notice from the above Trefftz plane plots that the UAV is most likely to experience stall on the main wing, so c_l in the above table is the maximum section lift coefficient on the main airfoil, while the total C_L is the result of the total lift across the main wings and tail.

In our most conservative view (that is where any section experiences stall by the above criteria) the aircraft experiences stall at an AOA of α = 5.7 degrees, which corresponds to a maximum lift coefficient of C_{L max} = 1.3407. In our less conservative approach, where approximately 60% of the airfoil stalls, the stall AOA is α = 7 degrees, corresponding to C_{L max} = 1.4568. To be conservative we will assume the minimum maximum lift coefficient for analysis.

From the tabulated data for the Bixler 2, including electronics and airframe, it has a total weight of m = 0.862 kg. Important parameters are determined from measurements and are as follows:

W = (0.862 kg)(9.81 m/s²) = 8.4562 N

S = .21145 m²

In order to determine L/D we require a measure of the drag coefficient, C_D. Recall that AVL employs a purely inviscid model for fluid flow. As a result the drag coefficient determined by AVL is purely a lift-induced drag, and does not take into account skin friction effects. In order to develop a usable model for drag we consider parasitic drag terms separately that are calculated using the Bixler 2 geometry and the tools available in the AA 241 notes. Note that while we did not model the fuselage geometry in AVL, it is now possible to include a fuselage drag effect by considering parasitic drag contributions to the overall drag. Thus, we consider drag from the main wings, horizontal and vertical tail components, and the fuselage. Obviously this is by no means an exhaustive list of drag sources, but it is simple to add more sources to our model in the future. Drag coefficients for the different components are tabulated below.

Based on these viscous drag terms, calculated with considering component geometries and the tools available in the notes as discussed, we can determine total C_D by including lift-induced drag at various AOA. This allows us to calculate L/D for various angles of attack, which are plotted below.

As can be seen from the plot, the max L/D calculated through theoretical methods was about 16. From simple actuator disk theory, a propulsive efficiency of about 70% at cruise was estimated. However, the propeller has significant interference, and using a conservative estimate of a 60% conversion efficiency from actuator disk theory yields a 42% propulsive efficiency ; using an estimated 80% conversion efficiency for the power train, this yields an overall conversion efficiency of 32%. Assuming a cruise velocity of 10 m/s (which is reasonable considering the experimental data from Task 2, as well as below), this yields an estimated powered cruise flight power consumption of 16.8 W.

NOTE: The above stall AOAs take into account the angle between the fuselage and the free stream. As a result, the true effective angle of attack is obtained by summing the AOA and the angle of incidences given above. In this case the stall AOA is similar to what we would expect for a thin airfoil type design.

Experimental data:

Using the experimental data it was possible to determine the times when the L/D glide tests took place, then measure the altitude loss against the ground track. The plot showing this comparison is below, where regions of no throttle indicate glide:

Note that the times in the table are absolute time from arming, whereas the plots are timed from the beginning of the flight.

It can be seen that the L/D ranges from about 8.2 to 15.5, which can be accounted for by light thermal activity and slight trim changes between each test. Averaging the values from the 5 tests we find an experimental L/D of 11.8 with a standard deviation of 3.33.

For the experimental stall determination and CLmax, a number of stalls were performed during the flight testing. The post-analysis of the data looked for a stall in which the acceleration was near 9.81 m/s^2 (which indicates a smooth stall) at the point when the airplane pitched over. Upon choosing the candidate stall, the GPS and pitot airspeeds were compared at the pitch-over point. Since the two indicators differed by more than 1 m/s, the readings were averaged. From this average, the stall occurred at about 6.693 m/s. Using a density obtained from the ideal gas relation and the sensor data for temperature and absolute pressure CLmax was found:

Finally, the level flight power consumption was determined from a ~30 s test attempting to maintain minimum power level flight with relatively constant throttle. This seems to show that about 30 W is sufficient for level powered flight. Note that the non-normalized barometric altitude was used for this section so the y-axis only represents relative altitude, staying within a range of ~5 m.

Note that this value for powered cruise flight is nearly 2X the theoretical predicted value. There are a number of potential reasons for this; firstly, the profile drag correlations used in the analysis section are primarily scaled for aircraft with higher Reynolds numbers, so some discrepancy is to be expected. Additionally, the power electronics, avionics, servos, etc. all consume battery power, and are not directly connected to flight operation. Assuming a more conservative L/D of around 10, wherein the profile drag will have to re-calibrated for yields an estimated power consumption of 26 W, which is fairly close to the measured value.