AAE490 - Design of a Contraction for a Transonic Ludwieg Tube

Transcription

1 AAE490 - Design of a Contraction for a Transonic Ludwieg Tube Report No. 3 November 30, 1998 Shin Matsumura School of Aeronautics and Astronautics Purdue University West Lafayette, IN ABSTRACT The Mach 4 Ludwieg tube at Purdue University is being converted to test airfoils at transonic speeds. The modification consists of a new contraction and a test section with contoured walls. The contraction was designed using matched cubics and a superellipse function. The boundary layer simulation had shown that the flow will not separate. However, in calculating the pressure gradient parameter, faulty values for the radius of the cross sections were used. This error was corrected by using the equivalent radius of the cross sections, and by running the data points through a spline function to smooth out the values. The pressure gradient parameter calculated from these values has shown that the first design of the contraction is unacceptable, due to some separation that occurs in the first quarter section of the contraction. The new contraction has been designed so that the curvature at the beginning of the contraction is much more relaxed to avoid separation. The boundary layer simulation has shown that no separation should occur anywhere in the new contraction.

2 INTRODUCTION It has been almost a century since the first manned aircraft designed and built by the Wright Brothers flew, and several advanced methods for designing airfoils and simulating flowfields using high-powered computers have been developed. However the final word on the aerodynamic performance of a vehicle and/or a vehicle component must still come from wind tunnel testing [6]. Today, with the severe budget cuts in the aerospace industry and research area, it is crucial that wind tunnels are capable of operating at low cost [4]. To further complicate matters, boundary layer transition tests done in noisy facilities with pressure fluctuations and noise in the freestream flow do not properly simulate the quiet flow of actual flight [5]. The answer to the above two problems is the Ludwieg tube, a wind tunnel design first proposed by Ludwieg of Germany [7]. Many Ludwieg tubes have been built around the world due to the low cost of construction and operation [9]. Ludwieg tubes have an advantage over a conventional blowdown type wind tunnel because of the high Reynolds number and high quality flow. According to the author of reference [8], The authors are unable to conceive of a fundamentally quieter process for providing the nozzle airflow. This is due to the shockless inlet design of the Ludwieg tube [8]. A more detailed explanation of the Ludwieg tube is available in references [5] and [8]. In the early 90 s, a Mach 4 Ludwieg tube was built at Purdue University to study high-speed boundary layer transition. A new Mach 6 Ludwieg tube with a larger test section and longer test time is currently under construction to replace the Mach 4 Ludwieg tube. A few possibilities for the future use of the older facility have been discussed, and one option would be to modify the tunnel so that airfoils at transonic speeds can be tested. This would require a new contraction to accelerate the flow to the desired Mach number. The Mach number range will be decided on and reported later. A contraction was designed and the boundary layer simulation had shown that no separation should occur [2]. However, the values for the pressure gradient parameter were calculated using erroneous values for the radius of the cross sections. This paper will focus on how the boundary layer calculations for the first design were corrected, and on the design of the new contraction.

3 PREVIOUS DESIGN Documentation on the design method, geometry, and boundary layer simulations can be found in reference [1] and [2]. The first contraction was designed using matched cubics and a fifth order polynomial for the exponents in the superellipse function. In the first design, the match point for the semi-major and semi-minor axes was placed at the downstream location of 4 inches. This made the curvature of the contraction very steep at the beginning, and relaxed at the end. The boundary layer simulation was then performed assuming that the contraction is axis-symmetric. The actual flow in the contraction is probably three dimensional, but because of the limited time frame, budget, and resources of this project, it was necessary to simplify the problem and assume a twodimensional flow. The boundary layer computations were performed using the E0D code obtained from Douglas Aircraft Company, and additional codes were written that used the Rott- Crabtree Method [5]. The inputs to the E0D code were the radius of the cross sections of the contraction. For these values, the values for the semi-major axes were originally used. This resulted in a very smooth λ curve, the pressure gradient parameter. The smallest value for λ was about 0.05, which suggested that the boundary layer should not separate anywhere in the contraction. However, this method assumes that the flow is driven by the radius of the contraction, and not by the cross sectional area of the contraction. Since the definition of one-dimensional flow is that the flow parameters are driven only by the cross sectional area of the contraction, this assumption was erroneous. To correct this error, the equivalent radius of the cross sections were calculated from the areas of the cross sections, and these values were used to simulate the boundary layer. The resulting λ s are plotted in Figures 1 and 2. Figure 1 uses a resolution of 0.01 inches, and Figure 2 uses a resolution of inches. Both plots show that the values jump around, and are not smooth. The values for λ drop as low as 0.6, which is a major problem. Judging from the fact that using the semi-major axis to calculate the pressure gradient parameter results in a much smoother and acceptable curve, it was concluded that Figures 1 and 2 are unreliable, and that an error occurred somewhere in the numerical process of the calculations. The two plots both do have a region where the plot is smooth. These regions are the regions upstream and downstream of the contraction, where the values for the radius are constant. This lead to the conclusion that there is an error somewhere in the matched cubic section of the contraction. First, the input files to e0d were checked for simple errors such as duplicate points, missing points, and erroneous points. None of these errors were found, so the next step was to vary the resolution. Numerical methods are very sensitive to the resolution, because of the round-off and truncation errors that can occur during the computational process. However, results have shown that changing the resolution does not change the magnitude of the λ s or the noise level significantly. Comparisons to the old λ plot were again made, and it was noted that the inputs to the e0d code in the previous computations were created with values that were calculated from a cubic function. The new values for the radius however, were calculated numerically from the cross sectional areas of the contraction. It was possible that the data

4 points in the new input file were not smooth enough. To smooth out the data points in the new input file, the data points were run through a spline function. The resulting λ plots can be found in Figures 3-5. The plots use a resolution of 0.02, 0.05, and 0.1 inches respectively. It can be seen that the noise level has significantly decreased, and that as the resolution is made coarser, the points seem to converge to a discrete curve. The curve in Figure 5 is very similar to the curve plotted in the previous calculations, which can be found in reference [2]. The problem here is that the solution did not seem to converge onto a solution as the resolution was varied. For example, according to Figure 5 there will be no separation, but according to Figure 4 there will be separation. As the resolution was made coarser, the plot became smoother, and the values increased. Similarly, as the resolution was made finer, the plot became noisier, and the values of λ decreased. If the resolution is too coarse, the plot will not be accurate, but if the resolution is too fine, the plot will be unreliable due to the higher noise level. The λ s seem to be independent of the resolution, so it was necessary to make a judgement call as to what resolution is acceptable. At a resolution of 0.05 inches, the plot seemed to contain some detail and accuracy without having too much noise. Figure 4 actually shows some of the values jumping around, especially around the downstream position of 2 and 3 inches, but the magnitude is so small that it will probably not be a problem. It was decided that a resolution of 0.05 gives the best results overall. Before Figure 5 could be chosen as the acceptable lambda plot, one other issue had to be resolved. The previous lambda plots were all generated assuming that the inlet to the contraction is two inches in length. It was possible for the solutions change, if the length was increased. Figures 6-8 shows the lambda plots using inlet lengths of 5 inches, 10 inches, and 20 inches. All three plots retain the same general shape as Figure 4, but the values at the minimum and maximum points are much greater. As the length is increased to 20 inches, lambda decreases to about 2. As with the issue of resolution, the inlet length to use was chosen to be 2 inches, based on the resulting lambdas and on judgement. Going back to Figure 4, this meant that a new contraction had to be designed, because of the separation that occurs at 2 inches downstream from the inlet of the contraction. NEW CONTRACTION DESIGN The semi-major and semi minor axes of the contraction were designed using the same method as before, matched cubics with the same boundary conditions. To relax the curvature at the beginning of the contraction, the match point was moved much further downstream, to a position of 7 inches. The overall length of the contraction was kept at the original length of 13.5 inches. The resulting plots of the axes can be found in Figure 9. The circled points represent the semi-major axis, and the cross points represent the semi-minor axis. The relaxation of the axes will prevent the boundary layer from separating, but caused a problem in finding a function for the exponents. Previously, the exponents were calculated using a single fifth order polynomial. This function however did not work for this new contraction, because the axes of the inlet area were too relaxed. In another words, the axes were decreasing too slowly, and the exponents were increasing too

5 rapidly. This resulted in the same problem encountered in the design of the first contraction, having a region where the 45 degrees radius and cross sectional area increases within the contraction. A new method to find the exponents was needed. The exponents for the new contraction was again calculated using a single fifth order polynomial, but the boundary conditions were modified somewhat to account for the extremely relaxed region in the axes. The boundary conditions used are as follows: 1. a(zs)=2 the cross section at the beginning of the superelliptical region is a circle 2. a (zs)=0 3. a (zs)=0 4. a(z2)=30 the cross section at the end of the contraction is a rectangle 5. a (z2)=0 6. a (zi)=0 Here, zs is the z position where the superelliptical region starts, z2 is the end of the contraction, and zi is the z position of the inflection of the exponent function. The first three boundary conditions allowed the contraction to have an axi-symmetric region from z equal to 0 to zs. This was done so that the exponents will not increase until the axes have started to decrease fast enough. The position of the inflection point was also set to be a variable, because in the previous design process, it was found that by varying the inflection point, the steepness of the exponent function could be controlled. Setting zs to 4 inches, z2 to 13.5 inches, and zi to inches led to the exponent function shown in Figure 10. It can be seen that the first few inches of the contraction is axi-symmetric, and even at the beginning of the superelliptical region, the exponents increase very slowly. Figures 11 and 12 shows the 45 degrees radius and the cross sectional areas respectively. Neither of the plots have a region where the values actually increase due to the exponents increasing too fast. Figure 13 shows the cross sections of the contraction in 1 inch intervals. The outer 4 plots are all circular, which is the result of keeping the first 4 inches an axis-symmetric region. BOUNDARY LAYER ANALYSIS The boundary layer was analyzed using the same method as before, and a resolution of 0.05 inches and inlet length of 2 inches. The resulting λ s are plotted in Figure 14. As with the old contraction, there is still some noise, but the magnitude is so small, that for the accuracy that this resolution gives, it is good enough. The minimum value for λ is about 0.025, which is more than good enough to avoid separation. This minimum value occurs twice, once at the inlet region of the contraction, and then at the outlet region of the contraction. The curvature at the end of the contraction was not a problem at all in the previous design, but because in this new design the curvature here was made much steeper, the λ s are much smaller. There are also two peaks in the plot, one around z equal to 5 inches, and then about 2 inches after. These areas are the favorable regions, where the flow is accelerating and the boundary layer is stable. The pressure gradient parameter is independent of the total pressure, but the boundary layer momentum thickness and the Reynolds number are dependent on the total pressure. Figure 15 shows the momentum thickness at three different total pressures.

6 The plot shows that as the total pressure is decreased, the boundary layer thickness increases. As with the previous design, this thickness is so small in comparison to the semi-major and semi-minor axes, that no boundary layer thickness correction was made to the values of the semi-major and semi-minor axes. Figure 16 shows the boundary layer Reynolds number based on momentum thickness at the three different total pressures. This plot is very similar with the plot of the previous design. The only difference is that the values are somewhat smaller than the previous design. 1-D FLOW THROUGH THE CONTRACTION The isentropic flow through the contraction was investigated using 1-D gas dynamics. For this analysis, a test section Mach number of 0.8 was chosen. Calculations showed that for the existing test section, a double wedge throat that will reduce the crosssectional area to sq. inches will be needed. The wedge will have a maximum cross-sectional area of inches and will be inches thick at that point. A plot of the Mach number, pressure, and temperature distribution can be found in Figures 17, 18, and 19 respectively. The Mach number starts below 0.1 in the charge tube and smoothly increases to the test section Mach number of 0.8. Both the pressure and temperature also start at the total condition and decrease smoothly. For the pressure plot, total pressures of 120 psi, 60 psi, and 15 psi were used. For each case, the pressure dropped about 33% from the total values at the test section. The total temperature was set at room temperature of 300 K, and was calculated that the temperature will drop to about 265 K in the test section. This decrease in temperature is small enough that some of the effects seen in higher Mach number wind tunnels such as the liquefication of air will not be a problem here. The deformation of the contraction due to temperature change will also not be a problem. Finally the test section Reynolds number based on the width of the test section was calculated to be 1.3 million at a total pressure of 120 psi. CURRENT STATUS AND FUTURE PLANS The error that was found in the transonic contraction was corrected, and a new contraction has been designed. It is now ready to be manufactured on the CNC mill at the Aerospace Sciences Laboratory. The manufacture of the contraction will take a few weeks due to the large block of aluminum that will have to be milled. The milling is to be done sometime during the winter of 1998, or in the spring of To test the transonic contraction, a new pitot-tube mount with a smaller cross sectional area will have to be built in order to avoid choking the flow in the test section. The flow quality this contraction will produce is planned to be measured during the summer of The test section is also under the process of being designed. The test section will have contoured walls to minimize the influence of the walls on the main flow around the airfoil. The date of completion of the design of the test section is still unknown. More information on the test section design can be found in reference [3].

7 REFERENCES [1] S. Matsumura. AAE-490 Design of a Contraction For a Transonic Ludwieg Tube. Report-1, 6/25/1998 [2] S. Matsumura. AAE-490 Design of a Contraction For a Transonic Ludwieg Tube. Report-2, 8/7/1998 [3] S. Matsumura. AAE-415 Design of a Test Section For a Transonic Ludwieg Tube. Final Report 12/17/98. [4] S.P. Schneider. A Quiet-Flow Ludwieg Tube for Experimental Study of High Speed Boundary Layer Transition. Paper , AIAA, [5] S.P. Schneider. A Quiet-Flow Ludwieg Tube for Experimental Study of High Speed Boundary Layer Transition. Paper , AIAA, [6] S.P. Schneider and J.P Sullivan. A&AE 334 L Aerodynamics Laboratory Manual. Purdue University. [7] J.D. Warmbrod. A Theoretical and Experimental Study of Unsteady Flow Processes in a Ludwieg Tube Wind Tunnel. NASA MSFC, 70N [8] J.D. Whitfield, C.J. Schueler, R.F. Starr. High Reynolds Number Transonic Wind Tunnels Blowdown or Ludwieg Tube? AGARD, CP-83-71, [9] D.R. Wilson and S.Y. Chou. Development of the UTA High Reynolds Number Transonic Wind Tunnel. Paper , AIAA, 1985

8 Figure 1 Diagram of Purdue Mach 4 Ludwieg Tube Figure 2 Sample Plots of the Superellipse

9 Figure 3 Sample Plots of x0 and y0 with Varying z1 Figure 4 Plot of x0 and y0 for the Designed Contraction

10 Figure 5 Plot of the Exponents Figure 6 Plot of the Cross-Sectional Area

11 Figure 7 Plot of the 45 Degree Radius Figure 8 Sample Plots of the Exponents with Varying Inflection Points

12 Figure 9 Plot of the Exponents for the Designed Contraction Figure 10 Plot of the 45 Degree Radius for the Designed Contraction

13 Figure 11 Plot of the Cross-Sectional Area for the Designed Contraction Figure 12 Plot of the Cross-Sections in 1 inch intervals

14 Figure 13 Plot of the Pressure Gradient Parameter Figure 14 Plot of the Boundary Layer Momentum Thickness

15 Figure 15 Boundary Layer Reynolds Number Figure 16 Plot of the Mach Number Distribution

16 Figure 17 Plot of the Pressure Distribution Figure 18 Plot of the Temperature Distribution