Thin Airfoil Theory Lab

Transcription

1 Thin Airfoil Theory Lab AME 3333 University of Notre Dame Spring 26 Written by Chris Kelley and Grady Crahan Deember, 28 Updated by Brian Neiswander and Ryan Kelly February 6, 24 Updated by Kyle Heintz February 8, 26 Abstrat The purpose of this report is ) to measure and analyze pressure distribution around an airfoil, 2) to alulate lift, drag, and moment oeffiients for the entire airfoil from disrete, loal pressure oeffiient measurements, 3) and to ompare experimental results with thin airfoil theory. To start the lab, a NACA 5 symmetri airfoil will be plaed in the wind tunnel with pressure taps loated at known distanes from the leading edge. Then pressure measurements will be taken and reorded to the omputer with a pressure transduer onneted to a Sanivalve. The loal pressure oeffiient will be alulated at eah pressure tap on both the upper and lower airfoil surfae. Next, the normal and axial fore oeffiients will be tabulated for eah pressure tap loation whih will then be transformed to a Riemann sum to find the total lift, drag, and quarter hord moment oeffiients for the entire airfoil. By repeating these measurements at different angles of attak, the lift urve slope an be alulated and ompared to the expeted value from thin airfoil theory. Similarly, the oeffiient of pressure an be used to alulate the moment oeffiients at the leading edge and the quarter hord loations.

2 Introdution 2. Thin Airfoil Theory Thin Airfoil Theory is the work of Max Munk, a olleague of Ludwig Prandtl. For the interested reader his original report was first translated into English in 923 is NACA report 42, General Theory of Thin Wing Setions []. Some preliminary kinematis are neessary for thin airfoil theory. The first omponent is a vortex, defined mathematially as a point that produes purely tangential veloity inversely proportional to the distane from the vortex: V θ = Γ 2πr. ) V θ is the veloity indued at a point of interest distane r from the vortex with strength Γ, positive when produing lokwise veloity. Now assume that this vortex is a line that extends infinitely into and out of the page forming a vortex filament. A vortex sheet is formed by having a ontinuous funtion of vortex filaments. It is this vortex sheet that Prandtl defined whih allowed Munk to quantify thin airfoil theory as will be seen. The irulation, or strength, of the vortex filament per unit length is defined as γs). Using Eq., the infinitesimal veloity dv indued due to a vortex filament is desribed by dv θ = γds 2πr, 2) where ds is the infinitesimal ar length along the vortex sheet. The primary assumption of thin airfoil theory is to assume that a thin airfoil one where the thikness is on the order of magnitude smaller than the hord) an be replaed with a vortex sheet. This sheet orresponds to the path of the amber line of the airfoil whih is also a streamline so that the amber line is impermeable. Figure shows this amber line, zx), where the loal slope an be written as artan dx) dz. In addition to the vortex sheet there is a free-stream veloity V at some angle of attak α. Figure : Vortex Sheet [2] The veloity indued by the vortex sheet, ws), at any point along the amber line is equal and opposite to the free-stream veloity normal to the amber line, V,n. When these two veloities are equal and opposite the amber line lies along a streamline beause the flow is everywhere parallel Updated: :5

3 . Thin Airfoil Theory 3 to streamlines by definition. Next the assumption is made that the vortex sheet instead of being aligned along the amber line, is along the hord line. Therefore w s) wx) beause the distane between the amber line is very small, and for a symmetri airfoil the differene is zero. So for the normal free-stream to be anelled out by the indued veloity of the vortex sheet, V,n + wx) =. 3) Figure 2: Vortex Sheet Approximated along Chord Line [2] From trigonometry and Figure 2, the loal normal free-stream veloity V,n an be written as V,n = V sin α + artan dz )). 4) dx Assuming small angles of attak and a thin airfoil, a small angle approximation is used α sin α artan α), and so 4) beomes V,n = V sin α dz ). 5) dx Using 5) and integrating from the leading edge ) to the trailing edge TE) with the dummy variable along the hord line ξ, the total indued veloity due to the vortex sheet wx) an then be found as follows: γξ) wx) = dξ. 6) 2πx ξ) Plugging 6) and 4) into 3) with small rearrangement yields the fundamental equation of thin airfoil theory for an unambered airfoil: 2π γξ) x ξ) dξ = V α. 7) In this lab the NACA 5 symmetri airfoil is being used, so the unambered assumption that dz dx = is aurate, and remember that in thin airfoil theory α is always measured in radians. Next it is neessary to make a hange of variables from ξ and x to θ suh that ξ = 2 os θ), dξ = 2 sin θdθ, and x = 2 os θ ). Substituting into 7), and noting that the bounds of integration must hange suh that θ = at the leading edge and that θ = π at the trailing edge. Updated: :5

4 . Thin Airfoil Theory 4 The solution of the integral of 7) is then + os θ γθ) = 2αV. 8) sin θ Now it is neessary to find the total irulation due to the vortex sheet. Γ = γξ)dξ = 2 π γθ) sin θdθ = παv 9) Equation 9) is quite useful when ombined with the Kutta-Joukowski theorem: lift per unit span 2D lift) L is equal to ρ V Γ. Hene the lift oeffiient from thin airfoil theory is proportional to angle of attak: C l = L 2 ρ V 2 = ρ V παv 2 ρ V 2 = 2πα ) Equation ) implies that the onstant of proportionality between lift and angle of attak is the lift slope C lα : C lα = C l α = 2πα) = 2π. ) α Another oeffiient that an be obtained from thin airfoil theory is the moment oeffiient about both the leading edge and the quarter hord. Consider one vortex filament of strength γξ)dξ. Again from the Kutta-Joukowski theorem the differential lift from this vortex filament is dl = ρ V dγ. See Figure 3 for a diagram of this inrement of lift. Figure 3: Indued Lift Inrement [2] To find the total moment per unit span about the leading edge integrate aross the entire hord: M = ξdl) = ρ V ξγξ)dξ = 2 ρ V 2 2 πα 2. 2) By definition of the moment oeffiient per unit span 2D) and using the previous equation for the leading edge moment, the following is found: C m = M 2 ρ V = πα = C l 4. 3) More importantly the moment oeffiient at the quarter hord an be found from the leading Updated: :5

5 .2 Aerodynami Fores and Moments 5 edge sine the moment about the aerodynami enter is where all the lift ats: C m/4 = C m + C l 4 = 4) Thin airfoil theory predits that the moment oeffiient about the quarter hord is, whih means that the aerodynami enter oinides with the enter of pressure. No moment is required to hold a symmetri airfoil at onstant angle of attak about the quarter hord. Finally take note that drag has been impliitly defined as zero in the derivation of thin airfoil theory. The Kutta-Joukowski theorem relates irulation to lift for a potential flow. Using thin airfoil theory and the Kutta-Joukowski theorem, a real airfoil an be replaed by a streamline segment with a vortiity distribution in the inompressible flow..2 Aerodynami Fores and Moments.2. Integral Coeffiients The main idea of this lab is to alulate the lift, drag, and moment oeffiient from the pressure taps and ompare them to those predited by the above thin airfoil theory results. All fores and moments ating on a body are solely due to the pressure and shear distribution. For this lab it will be useful to deompose the drag and lift oeffiients into axial and normal fore oeffiients. The axial diretion, A, is parallel to the hord line, and the normal diretion, N, is parallel to the hord line Figure 4). Therefore, the axial and normal oeffiients only differ from drag and lift oeffiients by the angle of attak, α. C l = C n os α C a sin α 5) C d = C n sin α + C a os α 6) By definition the pressure oeffiient, fore oeffiients, lift oeffiient, drag oeffiient, and moment oeffiient are defined as follows: C p = p loal p q, C n = N q, C a = A q, C l = L q, C d = D q, C m = M q 2 7) where q is the dynami pressure and the indiates per unit span sine airfoil analysis is, by definition, 2-dimensional. By breaking the airfoil into a lower setion denoted by subsript l) and upper setion denoted by subsript u) the total normal fore oeffiient per unit span is found. N = p u os θ + τ u sin θ)ds u + p l os θ τ l sin θ)ds l 8) Here θ is the loal angle between a pressure normal vetor and the line normal to the hord line. θ is positive when measured lokwise from the normal, and negative when measured ounterlokwise. See Figure 4 for a diagram of this oordinate system. And similarly for the axial fore oeffiient per unit span: A = p u sin θ + τ u os θ)ds u + p l sin θ + τ l os θ)ds l 9) The assumption is made that the pressure terms are of greater signifiane than the shear stress Updated: :5

6 .2 Aerodynami Fores and Moments 6 Figure 4: Normal and Axial Coordinates terms. This redues equations 8) and 9) to simply N = A = p u os θ)ds u + p u sin θ)ds u + p l os θ)ds l 2) p l sin θ)ds l 2) The moment about the leading edge per unit span M ) is found by integrating the axial and normal differential fore omponents multiplied by eah loations moment arm. M = dm n + dm a = p u os θ x p u sin θ y)ds u + p l os θ x p l sin θ y)ds l 22) Positive pithing moment is defined as leading edge up or lokwise diretion about the leading edge. As will be seen in the next setion when these equations are disretized, it will be advantageous to have the axial and tangential fore oeffiients and the leading edge moment oeffiient in terms of the pressure oeffiient. First substitute equation 2) into equation 7) to get the following for the normal fore oeffiient: C n = p u su ) os θ d + q p l q os θ d sl ) Seeing that ds os θ = dx then the normal fore oeffiient in terms of the loal pressure is p u x ) p l x ) pl C n = d + d = p ) u x ) x ) d = C pl C pu )d. 24) q q q q By doing the same operations the axial fore and leading edge moment oeffiients in terms of the loal pressure oeffiients follow: C m = C a = C pu C pl ) x d x C pl C pu )d ) + 23) y ), 25) C pl C pu ) y d y ). 26) Updated: :5

7 7.2.2 Disretized Coeffiients In this lab the pressure is not known as a funtion of x and y, but the pressure is known at eah tap loation. Using the integral form of the equations for the normal, axial, and leading edge moment oeffiients, we an approximate them with a Riemann sum sine the pressure taps are quite lose together. Using the trapezoidal rule in defining the summations, equations 24), 25), and 26), beome the following: C n = #taps i= #taps C a = i= ) x i+ Cpi + C pi+ 2 ) y i+ Cpi + C pi+ 2 x ) i, 27) y ) i, 28) #taps x i C m = C pi 2 + C x ) i+ xi p i+ x ) #taps i+ y i C pi 2 + C y ) i+ yi p i+ y ) i+. i= i= 29) Splitting the upper and lower surfaes, the disretized equations beome C a = C m = [ [ C n = [ lower upper lower ) x i+ Cpi + C pi+ 2 ) y i+ Cpi + C pi+ 2 x ) i ) x i+ Cpi + C pi+ 2 upper y ) ] i [ upper ) y i+ Cpi + C pi+ 2 x ) i 3) y ) ] i 3) x i C pi 2 + C x ) i+ xi p i+ x ) i+ y i C pi 2 + C y ) i+ yi p i+ y ) ] i+ upper ) lower x i C pi 2 + C x ) i+ xi p i+ x i+ 2 Desription of Experiment 2. Pressure Transduer lower y i C pi 2 + C y ) i+ yi p i+ y ) ] i+ A pressure transduer is a devie that onverts a pressure into a quantity that an be measured. One quantity is a voltage that an be measured using an analog-to-digital onverter or a digital voltmeter, and another quantity would be a height differene in a U-Tube manometer. Two ommon types of transduers are strain gage and apaitane types. The strain gage pressure transduer, whih is the one used in this lab, onsists of a thin irular diaphragm on the bottom of whih are bonded tiny strain gages wired as a Wheatstone bridge. When the diaphragm experienes a pressure on its exposed upper surfae that is different from the pressure in a small avity under the diaphragm it deflets, and the resulting bridge imbalane is a measure of the 32) Updated: :5

8 2.2 Sanivalve 8 defletion. This defletion is usually very small and will need amplifiation after it is onverted to a voltage. The apaitane-based pressure transduer has a strethed membrane lamped between two insulating diss, whih also support apaitive eletrodes. A differene in pressure aross the diaphragm auses it to deflet, inreasing one apaitor and dereasing the other. These apaitors are onneted to an eletrial, alternating-urrent AC) bridge iruit, produing a high level of voltage output. Strain gage transduers an be made small, hene they an be internally mounted in a wind tunnel model. Also, they have reasonably good frequeny response beause of the small mass of the diaphragm and the short distane between the pressure tap and the diaphragm fae. Capaitane transduers usually are not well suited for internal mounting and suh systems do not have a fast response. In this experiment, the pressure transduer measures the stati pressure at eah tap on the airfoil ontrolled by Sanivalve) referened to the freestream stati pressure. The freestream stagnation total) pressure is measured after the Sanivalve has stepped through all ports on the airfoil. This final measurement is the very last olumn of data in the saved.mat files saved. 2.2 Sanivalve A Sanivalve will take multiple pressure taps and allow them to be measured using only one pressure transduer. The tubes are all onneted to one stationary disk that is on top of a moving disk. The moving disk has a hole in it and will rotate the stainless steel tubes oiniding with eah pressure tap) that will be measured by the pressure transduer. Pitured in Figure 5 is the 4SDS-24 Sanivalve as installed with a stepper motor and the pressure tap tubing assoiated with this lab. Updated: :5

9 2.3 Airfoil 9 Figure 5: Sanivalve 2.3 Airfoil A NACA 5 airfoil is being used for this experiment. It has a thikness of mm, and a hord of 52.4 mm. This is a symmetri airfoil that is relatively thin, so thin airfoil theory will apply reasonably well. Pressure taps have been plaed on this airfoil at the x and y loations in Table, with the origin at the leading edge. These values will be useful in alulations of Cn and Ca. Updated: :5

10 2.4 Pressure Tap Design Upper Lower Tap Number x mm) y mm) Tap Number x mm) y mm) Table : Tables of NACA 5 oordinates for the UPPER surfae on the left and the LOWER surfae on the right. 2.4 Pressure Tap Design The motivation for the seond portion of the lab is to study the effets of varying pressure tap geometry. Pressure taps are holes drilled into a surfae with whih the loal stati pressure an be measured. Tygon tubing is usually used to onnet these taps to a pressure transduer. The basi assumption being made is that the stati pressure within the tubing is the same as the stati pressure at the wall of the tap loation. Careful design of pressure taps will minimize error between measured stati pressure and the atual stati pressure. The diameter of the pressure tap orifie should be twie as great as the tubing diameter. Also the ratio of the depth of the orifie to orifie diameter should be between. and.75 to redue variation of pressure error. [3] However, as is the ase with many measurement devies, measuring the flow usually affets the flow. For example, if a burr was reated in the drilling of the pressure tap, it may ause the pressure in the tubing to be higher or lower than the true loal stati pressure beause the streamlines bend around the burr. As seen in Figure 6, an obstrution upstream of a pressure tap auses the streamlines to have urvature onave down over the pressure tap. In the seond portion of this lab, the presene of burrs near taps is emulated by either plaing tape upstream or downstream of a tap. Updated: :5

11 Figure 6: Upstream Burr Conversely, a downstream obstrution will ause onave up streamlines over the pressure tap. Figure 7: Downstream Burr By using the Euler-N equation one ould deide how to hange the geometry near the tap, and hene the streamlines, to ause a pressure reading not indiative of the loal stati pressure. Also if a pressure gradient exists, a larger pressure tap will give an average pressure over a larger area, thus reduing its preiseness. 3 Proedure 3. Part - Thin Airfoil Theory. Chek that the pitot tube is oriented parallel to the flow inside the tunnel. 2. Chek the pressure system. The pitot tube stati pressure port parallel to flow) is hooked up to the TOP referene) port on the transduer. The Sanivalve is hooked up to the BOTTOM measurement) port on the transduer. 3. Chek the Analog In system. The BNC able from the Sanivalve is onneted to the AI hannel. 4. Chek the Digital Out system. The Sanivalve s STEP BNC is onneted to DO and the HOME BNC is onneted to DO3. 5. Turn on the Sanivalve ontroller and send it to the home setting by pressing the Home button. Updated: :5

12 3.2 Part 2 - Effet of Modified Pressure Tap 2 6. Start the Aerodynamis DAQ Utility program in MATLAB. 7. Make sure Analog In is enabled. Set Channels to. Set Sampling [Hz] to Set No. Samples to 8. Make sure Digital Out is enabled. Set No. Steps to. Leave Diretion as Pos.. 9. Set Repitions to 3. Set Timeout [s] to. Turn tunnel fan on to 25 Hz.. Set Airfoil to -5 degrees angle of attak. 2. Clik the Run button. The program will automatially aquire the mean pressure transduer voltages for eah of the 29 pressure taps and the upstream stagnation pressure. The data olleted is displayed in Plot When the aquisition is omplete, lik Save.MAT in the Plot 2 - All Data panel. Name the file with the appropriate angle of attak aoa n5, aoa, aoa 5, aoa 7p5, aoa, aoa 2p5, aoa 5, aoa 7p5 ). 4. One the data has been saved, press Clear Data in the Plot 2 - All Data panel. Clik Yes and Yes to the popup dialog boxes. 5. Repeat steps -4 for angles of attak of, 5, 7.5,, 2.5, 5, and 7.5 degrees using appropriate the filenames for eah. 6. Turn off the wind tunnel. 3.2 Part 2 - Effet of Modified Pressure Tap. One the tunnel is off, plae three thin strips of eletrial tape between pressure taps 5 and 6. Be sure not to over up any of the taps. 2. Make sure that the data is leared in Plot 2 - All Data. If it is not, press Clear Data and lik Yes and Yes to the popup dialog boxes. 3. Set the airfoil to 5 degrees angle of attak. 4. Turn on the wind tunnel to 25 Hz. 5. When the aquisition is omplete, lik Save.MAT in the Plot 2 - All Data panel. Name the file aoa 5 tape. 6. Turn off the wind tunnel. Updated: :5

13 3 4 Data Analysis and Disussion The data proessing and plotting instrutions are given below.. Pressure Coeffiient Plots: a) Calulate the pressure oeffiient C P at eah pressure tap loation. The position data for eah pressure tap is given in Table. b) Plot the negative of the pressure oeffiients C p of both the sution and pressure sides versus the hord-wise loation in mm: C p versus x. Make a separate plot for eah angle of attak. 7 plots) 2. Lift, Drag, and Quarter-Chord Moment Plots: a) For eah angle of attak, alulate the axial fore oeffiients C a, normal fore oeffiients C n, and leading edge moment oeffiients C m, using the split disretized equations. b) Calulate the lift oeffiients C l, drag oeffiients C d, and the quarter hord moment oeffiients C m,/4. ) Plot the lift oeffiient versus angle of attak in degrees: C l versus α. In the same graph, plot the 2π/rad theoretial slope on your plot be areful with your units!). Provide brief omments on the data and how it ompares to theory. plot) d) Plot the drag oeffiient versus angle of attak in degrees: C d versus α. Provide brief omments on the data. plot) e) Plot quarter-hord moment versus angle of attak in degrees: C m,/4 vs. α. Provide brief omments on the results. plot) 3. Pressure Port Biasing Plot: a) For the data set with tape on the airfoil aoa 5 tape.mat ) alulate the pressure oeffiient C P at eah pressure tap loation. b) Using your C p data from step a) and 3a), plot the negative pressure oeffiients verus hord-wise position in mm for α = 5 degrees with and without tape: C p versus x. How does the tape affet the pressure upstream and downstream from it? How do you expet the tape to affet the pressure? If you do not see any effet, try to explain why this might be. plot) 4. Inlude your proessing ode, eg. MATLAB.m files). Updated: :5

14 REFERENCES 4 Summary of Report Requirements. C p versus hordwise loation showing both the upper and lower surfae. There should be one for eah angle of attak 8 total). 2. C l versus α in degrees. Overlay the 2π slope line that is predited by thin airfoil theory. Comment on what your data shows and how well it ompares with thin airfoil theory. 3. C d versus α in degrees. Comment on what your data shows. 4. C m,/4 versus α in degrees. Comment on what your data shows. 5. C p versus hordwise loation showing just the upper surfae for α =5 o with and without tape on the same plot. Comment on what your data shows inluding how the tape affets the pressure at the nearby ports. If you do not see any effet due to the tape, try to explain why this may be? How do you exet the tape to affet the pressure. 6. Inlude your MATLAB ode. IMPORTANT: Make sure plots are printed large enough to see everything learly. Make sure all plots have appropriate titles and axes labels. If there are multiple lines on a single plot, make sure they are labeled and distinguishable by line style, markers, and/or olors. Referenes [] Munk M.M., General Theory of Wing Setions, Teh. Rep. 42, NACA, 923. [2] Anderson, J. D., Fundamentals of Aerodynamis, MGraw-Hill, 4th ed., 27. [3] MKeon, B. and Engler, R., Pressure Measurement Systems, Springer Handbook of Experimental Fluid Mehanis, 27, pp Updated: :5