The purpose of this investigation was to reduce the computational cost of vortex mapping for helicopter flight simulators, with the particular case of landing a helicopter on a moving ship. We pursued this goal by modeling the velocity using a series of Fourier transforms, linear fits, and Legendre polynomial fits. The end result was a matrix of 9 coefficient values which could be used to recreate the vortex behavior of the original data set with some accuracy. This technique will allow flight simulators to generate vortex data quickly to improve their effectiveness in training pilots

Peters, David

Tibbetts, Matthew

English

Washington University St. Louis: Open Scholarship

Washington University in St. Louis Washington University Open Scholarship Mechanical Engineering and Materials Science Independent Study Mechanical Engineering & Materials Science 12-21-2017 Reducing Computation Time of Vortex Position For Helicopter Flight Simulations Matthew Tibbetts Washington University in St. Louis David Peters Washington University in St. Louis Follow this and additional works at: https://openscholarship.wustl.edu/mems500 Recommended Citation Tibbetts, Matthew and Peters, David, "Reducing Computation Time of Vortex Position For Helicopter Flight Simulations" (2017). Mechanical Engineering and Materials Science Independent Study. 54. https://openscholarship.wustl.edu/mems500/54 This Final Report is brought to you for free and open access by the Mechanical Engineering & Materials Science at Washington University Open Scholarship. It has been accepted for inclusion in Mechanical Engineering and Materials Science Independent Study by an authorized administrator of Washington University Open Scholarship. For more information, please contact digital@wumail.wustl.edu. Reducing Computation Time of Vortex PositionFor Helicopter Flight SimulationsDavid Peters, Matthew TibbettsDecember 16, 2017AbstractThe purpose of this investigation was to reduce the computational cost of vortexmapping for helicopter flight simulators, with the particular case of landing a helicopteron a moving ship. We pursued this goal by modeling the velocity using a series ofFourier transforms, linear fits, and Legendre polynomial fits. The end result was amatrix of 9 coefficient values which could be used to recreate the vortex behavior ofthe original data set with some accuracy. This technique will allow flight simulatorsto generate vortex data quickly to improve their effectiveness in training pilots.1 IntroductionFlight simulators are an incredibly valuable tool for training pilots without riskingexpensive equipment or endangering the pilots. As computation power grows, sim-ulators become more able to recreate the experience of flying in various conditions,presenting pilots with challenges before they have to face them in the real world sothat they can be better prepared. As the fidelity of simulations rises, so too doesthe computing power required to calculate the hundreds of variables that affect anaircraft’s flight. Any property which can be successfully modeled mathematicallywith a small number of inputs frees up computing power for other vital systems.One of the most common challenges a pilot-in-training must prepare for is theeffect of air vortices on the aircraft’s flight. Helicopters in particular are vulnerableto flight instability when encountering vortices due to the large surface area andhigh speed of their rotors. Calculating the position and magnitude of vortices is,1therefore, a vital component of any realistic flight simulator. Conventional methodsfor calculating vortices are time-consuming and computationally expensive.Our goal with this investigation is to reduce the amount of inputs necessary togenerate a velocity field mathematically. We used a computational fluid dynamicsimulation as test data, and derived a method by which the velocity data can bemodeled using only a small array of coefficients.Our method of obtaining these coefficients consisted of five steps. First, we tooka Fourier transform of the velocities with respect to time for each position. We thenfit the resulting velocity data with a least squares fit line, and plotted the fittingparameters, U0 and λ, with respect to x position. We then took a Fourier transformof this data with respect to x, and collated the resultant frequency plot so thatthe parameters fell into two peaks. We created an array of the peak values for thesecond peak with respect to y and z position and fitted this surface with a seriesof Legendre polynomials. The coefficients of these polynomial products is our finalproduct.The ultimate goal of this project is to use these coefficients to recreate the originaldata set and determine how accurate our method is at modeling velocity data. Ifsuccessful, this would allow a flight simulator to model vorticity positions while onlystoring a small array of values, preserving fidelity while decreasing computation timeand space.1.1 GeometryFor this investigation we used as our original data set a computational fluid dynamics(CFD) simulation consisting of velocity in the x(U), y(V ), and z(W ) directions at43493 positions: 61 points in x, 31 points in y, and 23 points in z. The simulationtakes place over 125 time steps of 0.373 seconds, and the position values are in stepsof 2 feet. The field of interest is above the landing pad and in the wake of a SimpleFrigate Shape (SFS-2), as seen in Figure 1.2Values in x are positive behind the ship with the origin at the hangar/superstructurewall, ranging from -30 to -150. Values in y are positive starboard centered at themid line of the ship, ranging from -30 to +30. Values in z are positive down withthe origin at the water line, ranging from -15 to -59. The velocities are measuredwith respect to the ship.Figure 1: Simple Frigate Shape (SFS-2) used in the computational fluid dynamics simula-tion that produced the original data set3Description ValueLength 455Beam 45Waterline Z = 0Flight deck elevation Z = -15End of flight deck X = -120Flight deck extent Y = -22.5 → 22.5Superstructure elevation Z = -35Tower elevation Z = -55Hangar face X = -30Tower longitudinal extent X = 20 → 40Tower lateral extent Y = -5 → 5Table 1: Important values defining the SFS-2 geometry1.2 Mathematical TheoryThis investigation primarily uses two mathematical modeling techniques. Fouriertransforms are used to reduce the number of dependent variables by expressing valuesas a function of frequency rather than the variable of interest. These frequency plotsare often useful for determining a characteristic frequency which captures most ofthe behavior of the value. Legendre polynomial fitting is here used to further reducean array of values in two dimensions to a much smaller matrix of coefficients. TheLegendre polynomials are a set of orthogonal polynomials which are often used tomodel uneven surfaces.The Fourier transform takes the formfˆ(ω) =∫ ∞−∞f(x) e−2piixω dxWhere x is the variable of interest and ω is the frequency of the function. The4Fourier transform can be reversed with the inverse Fourier transform, of the formf(x) =∫ ∞−∞fˆ(ω) e2piixω dωIn this way a function can be transformed into its frequency form, analyzed, andthen returned to its original dependency. In our investigation this process is repeatedtwice: first on the velocity with respect to time, then on the fitting coefficients withrespect to x.The Legendre polynomials are a set of orthogonal polynomials of the formPn(x) =12nn!dndxn(x2 − 1)nWith the normalization of Pn(1) = 1. In our investigation, these polynomials areused as fitting functions for a surface dependent on both y and z. The fitting surfaceis of the formU0, λ =N∑i=1N∑j=1cijPi(z)Pj(y)Where N is the degree of the fit. So, for example, the 2nd order Legendre fit of theU0 parameter would take the formU0 = c11P1(z)P1(y) + c12P1(z)P2(y) + c21P2(z)P1(y) + c22P2(z)P2(y)With these two tools, we were able to take an array of more than 21 millionvalues and reduce it to an N ×N array of coefficients that a computer could use toreasonably recreate the original data set.2 ProcedureOur process begins with selecting a component of velocity to analyze. In this inves-tigation we chose U , the velocity in the x direction, because it is the most importantin determining vortex positions. Our original data is an array of velocity values in5x,y,z, and t, as seen in Figure 2. Our first goal is to reduce the number of dependentvariables we have to work with.We start by analyzing each position independently, plotting the velocity withrespect to time as seen in Figure 3. Our test point for these plots is at the frontend of the landing pad (x = −30), on the centerline of the ship (y = 0), midwaybetween the landing pad and the top of our area of interest (z = −39). Our processrepeats each step for the full range of positions.The first step of the process is to take a Fourier transform of the velocity datawith respect to time, producing Figure 3. In Figure 3, we plot the natural log ofthe velocity versus the frequency in Hertz. We then find the least squares fit linefor this data set and record the slope and intercept values. These will become ourfit parameters. The fit line is of the formU = U0 e−λfWhere U is the velocity, U0 is the exponential of the intercept of the fit line, and λis the slope of the fit line.We now have two data sets,U0 and λ, each with respect to only x,y, and z. Wewant to reduce this dependency further, so we plot the fit parameters with respectto x, as seen in Figure 5. We take the Fourier transform of these parameters withrespect to x, producing a plot of values with respect to the wave number, k. Thiscan be seen in Figure 6.We can see in Figure 6 that only two distinct peaks appear on this plot. Thefirst and largest peak is at k = 0, representing the constant base velocity. We areinterested in relative motion within the field, so we focus on the second peak, neark = 4. In order to capture the full behavior of the velocity field, we sum the valuesbetween k = 2.5 and k = 5.5 and place them at k = 4, as seen in Figure 7. Thevalues of U0 and λ at this point, then, are our values of interest.We plot these two arrays of peak values in y and z, as seen in the interpolation6plots of Figure 8. We then use the Legendre fitting polynomials to create fit surfacesof order 3 to the data, as seen in Figure 9. Each fit surface is characterized by a3× 3 array of coefficients. These coefficients can be found in Table 2 and Table 3.3 ResultsFigure 2: Contour plot of velocity in the x direction at the ship centerline in y at a giventime7Figure 3: Velocity in the x direction plotted as a function of time for a particular location(x,y,z)Figure 4: The natural log of the Fourier transform of velocity plotted with respect to f fora particular location (x,y,z)8Figure 5: Fitting parameters plotted with respect to x for a particular (y,z)Figure 6: Fourier transform of the fitting parameters plotted with respect to k for aparticular (y,z)9Figure 7: Summed Fourier transform of fitting parameters with respect to k for a particulary,zFigure 8: Interpolation plot of parameters U0 and λ for all y, z10Figure 9: Legendre fit of order 3 of parameters U0 and λ for all y, zP1(y) P2(y) P3(y)P1(z) 1.16e-01 2.95e-03 1.702e-05P2(z) 6.93e-04 2.754e-05 1.749e-07P3(z) -4.815e-05 -1.524e-06 -1.202e-08Table 2: Coefficients of U0 peaks for 3rd order Legendre polynomialsP1(y) P2(y) P3(y)P1(z) 3.62e-03 1.15e-04 8.517e-07P2(z) -1.598e-05 -5.744e-07 -3.751e-09P3(z) -2.112e-06 -1.057e-07 -6.026e-10Table 3: Coefficients of λ peaks for 3rd order Legendre polynomials4 AnalysisIn this investigation, we successfully reduce a data set dependent on four variablesto an array of coefficients through the use of Fourier transforms and fitting. Thisprocess makes several assumptions and introduces error at each reductive step. Ourfit line to the Fourier transform of velocity captures only a trend, and the wide scatterseen in Figure 4 certainly introduces error into our final product. We also collatethe range of values in the Fourier transform of the fitting parameters in Figure 7,11which removes some of the nuance of the actual shape. Finally, our Legendre fittingsurfaces do not line up with all the data, as seen in Figure 9.Our investigation could improve the accuracy of our results primarily by increas-ing the order of the Legendre fit. In our process, the 3rd order Legendre surface wasthe highest order which produced usable results, but ideally a 10th or higher orderfitting surface would capture the data better, and the increase in fidelity would onlyrequire a modest increase in data, from 9 values to 100. While that may seem like alarge increase, these arrays are still replacing the millions of cells required to recordthe CFD simulation.5 ConclusionOur analytical process reduced a 61 × 31 × 23 × 125 array of values to a 3 × 3array, requiring only the mathematical framework to restore the data set from thesecoefficients. We use Fourier transforms and fitting functions to decouple dependencyon two of the four dependent variables from the original data.Given our desired outcome, the error we introduced during the fitting process isacceptable. Vortex position and velocity are highly stochastic, and any mathemat-ical model must necessarily provide the broad strokes of the shape of the velocityfield, allowing the simulation to stochastically augment in order to provide a realis-tically challenging experience.While we achieved the results we wanted from this exploration, there is stillwork to be done improving and verifying this process. We were unable due to timeconstraints to implement a reversal of the reduction process, taking the coefficientsand generating a full velocity data set. We also only modeled a Legendre fit of oder3, which does not capture the true shape of the surface. Future work on this subjectcould increase fidelity and demonstrate the full functionality of this process.12

Reducing Computation Time of Vortex Position For Helicopter Flight Simulations

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Reducing Computation Time of Vortex Position For Helicopter Flight Simulations

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