Aerodynamic Drag of B Altitude Designs of Varying Fineness Ratios

A Research and Development report
submitted at NARAM-40, August 1998,
by Wallace & Gromit Go To NARAM, T-471

Copyright © 1998 by Chuck Weiss (cbweiss at frontiernet dot net) and Jeff Vincent (jeffvincent at verizon dot net).
This report may be copied for non-commercial use, provided this notice remains intact.
Potential commercial users should contact the authors for further information.

Abstract - Aerodynamic Drag of B Altitude Designs of Varying Fineness Ratios

Wallace & Gromit Go To NARAM - NARAM-40

A long-held rule of thumb in model rocketry is that a minimum diameter model has the lowest drag and achieves the highest altitude. This report investigated the aerodynamic drag and altitude performance of three designs. The designs are similar, with equal body surface area, but have differing fineness ratios: a minimum diameter design (0.45 inch diameter, fineness ratio of 20.4) and two larger-diameter, boattailed designs (diameters of 0.525 and 0.64 inches and fineness ratios of 15.9 and 11.5, respectively).

A fleet of fifteen models were constructed, and several tools and techniques were developed to aid in that process. The models were then flown and tracked. Engine delay abnormalities of a batch of Apogee B2-9 motors resulted in little useful data. A second round of flying, using Czechoslovakian Delta A2-5 motors resulted in seven valid altitude data points for comparing airframe design performance. Drag coefficients and drag form factors were derived from the tracking data.

The data from the A2 flights indicated no significant altitude performance or drag difference between the minimum diameter and 0.525 inch diameter designs. The 0.64 inch diameter design had a significantly lower altitude performance and higher drag. For models of the performance class and surface area tested, these results indicated that the advantages of a boattailed design can be utilized without loss of altitude performance. However, there is a fineness ratio limit where performance will begin to be compromised.

DATCOM drag calculations utilizing an all-turbulent body boundary layer were found to be highly representative of the performance (within 5%) of the 0.45 and 0.525 inch designs, but less accurate (18% error) for the 0.64 inch design.

The tracking data was analyzed on several levels. Angular deviation from the vertical of the flight paths was calculated, presented, and considered in evaluating tracking data. A plot of model apogees in the horizontal plane was used to display model dispersion and was an aid in recovering models.

Chuck Weiss
Email: cbweiss at frontiernet dot net

Jeff Vincent
Box 523
Slingerlands, NY 12159
(518) 439-2055
Email: jeffvincent at verizon dot net

Table Of Contents

List of Tables, Figures, and Graphs


The traditional rule of thumb for minimum aerodynamic drag and maximum altitude performance has been minimum diameter (designing the model with the body tube the smallest diameter to fit the selected engine). William P. Bengen writes in Topics In Advanced Model Rocketry (referring to the practice of boattailing): "The technique is limited in usefulness to those models which, for some reason, must use a main body section significantly greater in diameter than the engine casing; there is, of course, no sense in enlarging the diameter of a rocket just to enable it to be built with a boattail!" Several developments have led us to question this conventional wisdom.

First, rule changes in FAI flying in the mid-1980s (18 millimeter minimum diameter over 50% of model length, later changed to 30 millimeter minimum diameter over 50% of model length) mandated flying non-minimum diameter, boattailed models in certain events. Internats flyers quickly adopted fiberglass techniques to airframe construction, making a wide variety of model shapes possible (compared to commercially manufactured tube and adapters). Also, these models demonstrated surprisingly good performance. An example is Matt Steele's small single-stage model which won the bronze medal at the 1987 Internats, an 18 millimeter diameter B-powered model flying to 700+ meters.

A second, more recent, development was the introduction of Apogee micro 1/4A - B engines in 1995-1996. The small diameter (10.5 millimeters) and the long length (3.5 inches) of the micro B engines has led to relatively long minimum-diameter models (to contain the engine, recovery device, and tracking powder), with a fineness ratio (length/diameter ratio) of 20 or more. The higher fineness ratio leads to higher drag coefficients than traditional altitude models of lower fineness ratios.

Our experience with such small diameter models has shown that they present unique problems of their own. The small diameter makes prepping models more difficult and time-consuming. The small diameter makes ejecting tracking powder more difficult (the high skin friction of the tracking powder on the inner walls of a long, slender tube can result in ejecting the engine instead of the tracking powder). A minimum-diameter design has limited options in securing the engine casing. A larger diameter, boattailed design offers solutions to some of these problems. Even if the altitude performance is not improved (so long as it is equivalent), the boattailed design may prove more usable and thus more desirable.

For these reasons, we decided to re-examine the choice of minimum-diameter design for the B Altitude event. This report examines the aerodynamic drag and altitude performance of three designs. The three designs are similar, with equal body surface area, but have differing fineness ratios: a minimum diameter design (0.45 inch diameter, fineness ratio of 20.4) and two larger-diameter, boattailed designs (diameters of 0.525 and 0.64 inches and fineness ratios of 15.9 and 11.5, respectively). The aerodynamic drag of the designs was derived from tracked altitudes of prototype flights.

Background and Theory

For analytical purposes, the aerodynamic drag of model rockets is often broken down into several components:

We are interested in the drag of the body component for our experiment. Every effort was made to equalize the drag (not related to body shape) of the three designs :

The drag of each design was calculated, using the implementation of the DATCOM method presented in Topics in Advanced Model Rocketry. A weighted average airspeed of 85 meters/second was used. Drag force is computed by the equation:

Drag = (air density) (drag coefficient) (reference area) (velocity)2.

Since the reference area (based on the nose diameter) of each design is different, drag coefficients cannot be directly compared. Instead, the Drag Form Factor (CDA, drag coefficient times reference area) is used for comparison. Below is a table of representative values.

Table 1 - Calculated CD and CDA Values of the Three Designs

All turbulent boundary layer (BL) on nose & tube / all laminar BL on fins
Diameter Area CD CDA Normalized CDA
0.45" 0.159 in2 0.528 0.0840 in2 1.000
0.525" 0.216 in2 0.390 0.0842 in2 1.002
0.64" 0.322 in2 0.270 0.0869 in2 1.035

All laminar boundary layer (BL) on nose & tube / all laminar BL on fins
Diameter Area CD CDA Normalized CDA
0.45" 0.159 in2 0.298 0.0474 in2 1.000
0.525" 0.216 in2 0.218 0.0475 in2 1.002
0.64" 0.322 in2 0.149 0.0480 in2 1.013

As you can see, the predicted drag form factors are quite close for the three designs. The slightly higher predicted CDA values for the boattailed designs is due to a pressure drag correction in the equations for the "fatness" (lower fineness ratio) of these bodies. The slightly smaller base diameter of the boattailed designs resulted in overall drag decreases of 0.3% in the all turbulent BL body case and 0.9% in the all laminar BL body case.

Further calculations revealed an interesting point. When calculating drag with the body boundary layer transitioning at a certain critical Reynolds Number, the boattailed designs gained an advantage. Due to their shorter length, these designs were fully laminar, while the minimum diameter design had developed turbulent flow. This resulted in overall drag reductions as great as 5.5% (for the 0.525 inch design) and 9.0% (for the 0.64 inch design). Such behavior would only occur at very high Reynolds Numbers (1.22E6 and 1.11E6, respectively), making it unlikely that we could maintain a laminar boundary layer to take advantage of this fleeting phenomenon, but it is interesting, nonetheless.

Figure 1 - The Test Designs (scanned at 100 dpi)

Download Figure 1 in .TIFF format for printing (scanned at 300 dpi, 81 Kbytes)

The Test Designs

Experimental Method

I. Approach

A total fleet of 15 models were initially constructed to test the drag relationships for each of the three airframe designs that have been discussed. Because of engine problems discussed later in the report, one additional B and C design airframe were constructed between day one and day two of flight tests to finish the flight tests. Figure 1 illustrates the airframe designs. For convention, the straight airframe with outside diameter 0.45 inches and the boattailed 0.525 inch and 0.64 inch diameter airframes are referred to as design categories A, B, and C, respectively, throughout this report. The diameters selected for each model were influenced by commercially available paper tubing for the straight design and the availability of mandrels from previous rocketry projects for construction of the boattailed designs.

Five models of each airframe design were constructed with the intentions of ideally obtaining five altitude data points for each design from which relative drag relationships could be calculated and performance compared. A considerable amount of labor, time, and effort is associated with the level of craftsmanship required to minimize variables that can influence performance other than airframe design. In consideration of the labor burden and typical problems such as inappropriate engine performance, unclosed or lost tracks, etc. that can be encountered, it was hoped that a minimum goal of three data points for each airframe design could be obtained and that these data points would be enough to demonstrate significant performance similarities or differences among the airframe designs.

II. Airframe Construction

The straight A design airframes were constructed from Apogee nominal 10.5 millimeter diameter paper tubing. The tubes were cut to length and lightly sanded prior to finishing.

The design B and C boattailed airframes were constructed from 0.5 ounce per square yard fiberglass cloth and PIC epoxy coating resin. A small amount of epoxy coloring agent was added to the resin mix to visually assist in the application of the resin. The airframes were laid up on aluminum mandrels which were machined in a local machine shop. Attention was paid to insure that techniques for preparing the mandrels that could influence airframe diameter were consistent. Each mandrel was coated with a thin coat of paraffin wax and three coats of Crown epoxy release agent prior to applying the fiberglass cloth and resin. Fiberglass cloth was measured and cut to wrap lengths that would generate a thickness of four cloth layers on the mandrel. Resin was applied and brushed into the cloth with a stiff bristled brush. After curing, each airframe was sanded to a consistent finish with 120, 320, 400, and 600 grit sandpaper successively prior to removal from the mandrel. The diameter of each airframe was checked after sanding and prior to removing from the mandrel and found to be within +/- 0.002 inches.

All airframes were painted with four coats of Plasti-Cote red oxide automotive sandable primer. Each coat was lightly sanded with 400 grit sandpaper . The final coat was sanded with 600 grit sandpaper and buffed out on an electric drill finishing lathe with 000 grade steel wool.

Preliminary airframe size calculations had been made prior to construction. Once the airframes had been completed, they were remeasured with dial calipers and final calculations were made to determine proper body lengths to insure equal surface area of the three designs.

Fins were cut from 0.010 inch waferglass, sanded to a rounded leading and trailing edge, and washed. A special fin jig was constructed to insure that the fins were precisely mounted on each airframe with minimal angle of attack. The precision and accuracy of the fin jig was beyond visual detectability. Using the fin jig as a guide, the paint was scraped from each airframe with an X-acto blade where the fin root attaches to the body. Three fins were glued to each airframe with thin cyanoacrylate (CA) glue 120 degrees apart and one body diameter from the base . A thin fillet of CA was applied to both sides of each fin root.

Balsa noses for the A paper tube design were purchased from Apogee Components. Noses for the B and C designs were turned out of eight pound per cubic foot balsa on a Unimat modeler's lathe. A special jig was constructed to insure uniformity in nose cone shape and size. The two-dimensional elliptical pattern for each nose design was transcribed onto a piece of aluminum. The aluminum was cut and filed to exact shape. After turning the balsa to the correct cylindrical diameter required, the aluminum guide was firmly mounted parallel to and the correct distance from the rotational axis of the lathe. A right angle sanding guide with the sandpaper mounted on the inside surface was used to cut the nose cone shape. The leg of the sanding guide perpendicular to the cutting edge laid on top of and in the same plane as the elliptical pattern. The cutting edge butted against and followed the elliptical pattern. This maintained the spatial orientation of the cutting edge so that the elliptical pattern could be cut from the cylindrical work piece to form the nose. Good reproducibility was obtained using this technique. This technique can be utilized with an electric drill or Dremel tool providing a way to firmly hold the nose shaping guide in correct orientation with the rotational axis is established. After the nose shape was correctly shaped, the appropriate diameter matching the inside diameter of the airframe was cut into the base of the nose for mounting the nose on the airframe. A tolerance of +/- 0.0015 inches was necessary to correctly fit the nose to each airframe. The nose was then fitted to the airframe and sanded to establish a smooth nose/body tube joint. All noses were finished with four coats of sandable primer similar to the airframes. A kevlar shock cord was glued into each finished nose with a small ball of cotton and a CA glue. A one inch by 24 inch red mylar streamer was attached to each shock cord.

Each completed airframe with the recovery device included was adjusted to within 4.6 to 4.85 grams final weight with modeling clay applied inside the nose section.

Total construction time of each individual model was estimated at three to four hours for the A design and five to six hours for the B and C designs.

III. Flight Testing

A. Weather Conditions

Flight tests were conducted over two consecutive days. Weather conditions on both days included temperatures in the low to mid-eighties and light winds out of the west at five to ten miles per hour. Sky conditions included minimum haze with patchy clouds. Tracking was occasionally delayed to allow cloud cover to clear and a few flight tracks may have been lost because of poor visibility due to cloud cover. We wish to thank Wolfram von Kiparski for his selfless assistance with the tracking of the models, which proved critical to completing the flight testing.

B. Tracking

Tracking was conducted using theodolites and standard visual techniques. A baseline of 515 meters was used. The geometry of the tracking set up is depicted in Figure 2. A Radio Shack hand held computer/calculator was used to reduce data on the field. The geodesic method of calculation was used for computations.

C. Model Preparation

Models were prepped collectively in two groups with the intentions of flying three and two round salvos for each body type. The kevlar shock cord was taped to the top of the engine and the engine was fitted to the model with tape. A snug friction fit was used for the A style airframes. In addition to friction, engines were locked into the B and C style airframes by the forward tape band holding the shock cord. A small pouch was made from MRC recovery wadding to hold the tracking powder and placed in the engine body tube. Past experience flying small diameter models for Internats competition indicated that the pouch is necessary to assure proper deployment of the tracking powder. Tracking powder was added to the pouch until the desired gross launch mass was obtained. Approximately five to six grams of tracking powder was used to obtain a gross launch mass of 22 grams and 14 grams for the B and A engine flights, respectively. (The use of different impulse engines is discussed below and in the Discussion of Results - Engine Performance and Flight Plan Modification section of the report.) The center of gravity of each model was determined and checked against Barrowman center of pressure calculations (by Gary Crowell's VCP program) to insure stable flight. Finally, the ignitor was installed.

D. Engines

The original test plan involved the purchase and use of 18 Apogee B2-9 engines. Engines were obtained from the same manufacturing batch. Because of inconsistent and excessive ejection delays encountered during the first day of flight testing, the need for a more accurate and precise engine was recognized. The remainder of flight tests on the second day of flight testing were conducted with Czechoslovakian-manufactured Delta A2-5 engines. The unanticipated excessively-poor performance of the B2-9 engines on day one resulted in a desperate need for a reliable engine in order to salvage the project and the exhaustive amount of effort and expense that had been expended until this point. The base diameter of the airframe design severely restricted the choice of engines available to complete the project. A supply of the Delta engines obtained for Internats flying seemed the best choice to complete the project under the circumstances. We felt that, due to the similarly low thrust levels, model velocities and drag characteristics of the Delta A2 would be similar to the Apogee B2. While not NAR certified, the safety and precision record of the Delta engines had been well documented through experience and FAI testing for World Championship competition. The poor performance of the B2-9 and the performance of the A2-5 engines are discussed further in the Results section of the report.

E. Flights

All tests were conducted from a three rail tower launcher with four foot long rails. Total model mass was verified on a triple beam balance prior to flight. The launcher was adjusted with a slight angle (5 degrees) in the windward direction to compensate for weather-cocking. The collectively prepped models were launched in two groups of three for each airframe design in order to minimize tower adjustment and to track as many flights as possible during optimum sky conditions. Launch order is designated by flight number in the Results section of the report. As previously indicated, all B2-9 flights were conducted on day one of flight testing and all A2-5 flights on day two of flight testing.


Results for flight tests using B2-9 engines on the first day of flying are shown in Table 2. Day two flight test results using Delta A2-5 engines are listed in Table 3. Both tables include information on model identification, flight number, stability margin in calibers, total flight time, tracking azimuth and elevation raw data, flight altitude, tracking percent closure, and the calculated angle that the flight deviated from vertical boost.

The A, B, or C suffix on the model ID number indicates the airframe design according the defined convention.

Flight time was determined by timing the flight from the launch area with a stopwatch from ignition to observed ejection. The measured engine delay time was calculated by subtracting the engine burn time from the flight time observed. The burn time for B2-9 engines is 2.5 seconds (source: NAR Standards & Testing). Delay times for the B2-9 engines ranged from 9.5 to 20.4 seconds with an average delay time of 15.2 seconds for eight timed flights. The standard deviation and percent relative standard deviation for B2-9 delays was 3.51 seconds and 23.1%, respectively. The delay time average error calculated as the absolute value of the measured delay time minus the claimed delay time divided by the claimed delay time was 69% for the B2-9 engines.

Claimed burn time for Delta A2-5 engines is 1.3 seconds (source: manufacturer's data sheet). Observed flight times ranged from 8.18 to 8.90 seconds. The average measured delay for the Delta A2-5 engines for six timed flights was 7.3 seconds. The standard deviation and relative standard deviation was 0.267 seconds and 3.7 %, respectively. The delay time average error was 46%. This determination excludes flight number 8 which experienced a burn-through catastrophic failure and flight number 10 which was actually a Delta A2-6 engine. The A2-6 engine was a last resort to try and glean one more data point. The track was lost for this flight.

The angular deviation from the vertical listed in the tables was determined by calculation. By knowing the length of the baseline and the azimuth angles to the launch site, the launch area could be plotted in a two-dimensional coordinate system. Likewise, the azimuth angles of each track could be used to determine the point of ejection in the horizontal plane. The horizontal distance between this ejection point and the launcher was calculated mathematically. This horizontal distance could be used, with the tracked altitude, to calculate the angular deviation from the vertical of the flight path.

A graphic representation of the tracking geometry and dispersion of flights in a plane around a point vertical to the launch point is shown in Figure 2. The model ID, altitude, and angular deviation from the vertical is given for each point. A, B, and C airframes and B2 and A2 engines are represented by specific symbols.

This information is also represented in a bar graph form in Graph 1. The tracked altitude is represented on the Y axis. The tilt of the bar accurately represents the angular deviation from the vertical. Flights are dispersed along the X axis with model ID, altitude, and angular deviation listed below each bar representation. The airframe type is represented by the thickness of the bar with thin, thicker, and thickest representing A, B, and C airframes, respectively.

Table 2 - Day One Flight Test Results, 8/1/98

Engines: Apogee B2-9, June 13 1998 batch

Gross Liftoff Mass: 22.0 grams +/-0.1 grams

Suffix Convention: A = 0.45 ", B = 0.525 ", C = 0.64" model diameter

Stability Margin is in calibers (maximum body diameter)

Tracking was done on a 515 meter baseline, geodesic data reduction method

Angular Deviation is the angular deviation in degrees from the vertical of the flight path, calculated from the azimuth data.

Az East El East Az West El West Altitude Closure Angular
1A 1 4.0 18s track lost
2A 9 3.5 19s track lost
3A 8 3.5 14s 19 70 14 68 699m 2.06% 10.0
1B 2 2.6 17s track lost
2B 3 3.2 18s 31 64.5 35.5 66.5 675m* 0.66% 3.6
3B 4 3.1 12s 44.5 62.25 46 63 706m 0.25% 10.0
1C 5 1.6 22.9s track lost
2C 6 1.7 No ejection seen or heard - track lost
3C 7 1.6 21s track lost

* - Tracking East values approximated (tracking powder not seen until several seconds after deployment).

Delay time = flight time - burn time (2.5 seconds)

Average delay time = 15.2 seconds

Delay time standard deviation = 3.51 seconds

Delay time % RSD = 23.1 %

Delay time average error = Absolute value of ( Measured Delay time - 9 sec.) / 9 sec. x 100% = 69%

Table 3 - Day Two Flight Test Results, 8/2/98

Engines: Czech Delta A2-5, 1994 batch, except for flight 10 which was an A2-6, 1990 batch

Gross Liftoff Mass: 14.0 grams +/-0.1 grams

Suffix Convention: A = 0.45 ", B = 0.525 ", C = 0.64" model diameter

Stability Margin is in calibers (maximum body diameter)

Tracking was done on a 515 meter baseline, geodesic data reduction method

Angular Deviation is the angular deviation in degrees from the vertical of the flight path, calculated from the azimuth data.

Az East El East Az West El West Altitude Closure Angular
3A 1 2.6 TL 32 52.5 41.5 58 460m 0.69% 5.4
4A 2 4.0 TL Unstable flight
5A 3 3.8 8.4s 25.5 50 43.5 61 444m 1.9% 4.0
4B 4 2.4 8.59s 61 58.5 31.5 44 437m 0.45% 25.2
5B 5 2.5 8.18s 29 57 33 59 480m 1.18% 4.6
6B 6 2.5 8.80s 17 50 34 68 453m 1.89% 6.9
4C 7 1.6 8.90s 28.5 54.5 29.5 56 424m 0.78% 7.2
5C 8 1.2 5s Engine cato
6C 9 1.8 8.7s 48 56 31 45 394m 1.23% 20.2
5C 10 - 10.7s* track lost

* - Engine A2-6, 1990 batch - chuffed after ignition, irregular boost. Flight time based on ejection sound.

Delay time = flight time - burn time (1.3 seconds)

Average delay time (A2-5 excluding cato, N = 6) = 7.3 seconds

Delay time standard deviation = 0.267 seconds

Delay time % RSD = 3.7 %

Delay time average error = Absolute value of ( Measured Delay time - 5 sec.) / 5 sec. x 100% = 46%

Table 4 - Average Altitude in Meters

(Number of data points)

Airframe B2 Engine A2 Engine
A 699 (1) 452 (2)
B 691 (2) 457 (3)
C no data 409 (2)

Table 5 - Calculated CD, CDA, and Flight Time for A2 Averages

Airframe CD CDA Flight Time
A 0.555 0.0883 in2 8.44 sec
B 0.400 0.0866 in2 8.50 sec
C 0.328 0.106 in2 7.94 sec

These values are based on the average A2 altitude for each design. They were generated using the wRASP 2.1- program's built-in DigiTrak backtracking.

Table 6 - Calculated CD, CDA, and Flight Time for B2 Flights

Model ID CD CDA Flight Time Observed Flight Time
3A 0.622 0.0989 in2 10.54 sec 14 sec
2B 0.486 0.105 in2 10.33 sec 18 sec
3B 0.449 0.0972 in2 10.61 sec 12 sec

Note that these values are for individual B2 flights. Note the discrepancy between the calculated and measured flight time for these flights.

Figure 2 - Overhead View of Tracking Geometry (scanned at 100 dpi)

Download Figure 2 in .TIFF format for printing (scanned at 300 dpi, 134 Kbytes)

Overhead View of Tracking Geometry

Graph 1 - Altitude Performance of the Models, Displaying Angular Deviation from the Vertical (scanned at 100 dpi)

Download Graph 1 in .TIFF format for printing (scanned at 300 dpi, 97 Kbytes)

Altitude Performance of the Models, Displaying Angular Deviation from the Vertical

Discussion of Results

I. Engine Performance and Flight Plan Modification

As previously indicated, results of the day one flight testing indicated that a serious delay problem existed with the B2-9 engines and that useable data could not be obtained using these engines. Six out of nine flights resulted in lost tracks and one set of tracking data approximated. Two usable data points were obtained. The two B2 flights that could be tracked had delay times closest to claimed delays. The excessive delay times for seven of the nine flights impacted the trackers' ability to track the models. The ejection clouds were probably more observable from the range head for determining ejection delays because the direction of flight of the model could be better determined from the onset of the flight. Also, since many of the models were rapidly descending at ejection (due to excessive delays), the tracking powder cloud was often dispersed over a greater vertical distance. Finally, the variation of the delays meant that trackers could not rely on a consistent placement of ejection in the elevation axis. Thus the person in the launch area could better guess where the ejection would occur in the sky. It is estimated that some B2 models lost 10 - 20% or more of their altitude from expected apogee to ejection. This is also a critical factor, as the experiment was designed to measure the maximum altitude of the models - ejecting significantly after apogee resulted in data of extremely poor quality.

This situation resulted in only two usable data points for B2-9 flights and three out of five of the B and C airframes and two of the A airframes lost and unreturnable at the end of day one. One A airframe was found that could be used for further testing. After a grim day, our ability to salvage the project with remaining airframes and available engine choices was evaluated. Working until dawn, one additional B and one additional C airframe were constructed to restore the fleet to three of each of the A, B and C airframes. An inventory of available engines produced nine Delta A2-5 engines from a 1994 batch, one Delta A2-5 from a 1990 batch and one Delta A2-6 from a 1990 batch. Also available were 24 Delta 3/4 B2-8s from a 1994 batch. An insufficient number of Apogee A2s with similar delays were available to complete the project and our faith in their performance was diminished by the B2-9 flight tests and past experience with other Apogee engines. Our initial decision to use Apogee B2-9s was based on some favorable results obtained with B2-9s at a recent regional. Unfortunately, these engines were of limited supply in our current inventory and were from another manufacturing batch. After considering options it was decided that even though more Delta 3/4 B2-8s were available, they would probably achieve altitudes equal to or higher than the B2-9s and the ability to track them in patchy skies could be a problem. The number of airframes was limited to three for each design and experience showed that the ability to return B2 flights for reflight was minimal. Based on these factors, the decision was made to use the Delta A2-5 engines.

Flight tests on day two were started by using the 1990 batch A2-5 as a test. Unfortunately, the engine cato'd with a blown nozzle and the flight testing had to resume without a tracking test flight. That engine was suspect from the start because the ejection cap had been taped over for reasons that could not be recalled (which is why it was considered expendable for preliminary testing). The remaining flight tests conducted on day two showed that the precision of the Delta A2-5s was remarkable and much better than the B2-9s. Even though the average observed delay accuracy was 46%, it was still much better than the B2-9s. In addition, results obtained from drag coefficient calculations indicated that the average measured delay of 7.3 seconds was actually ideal for the altitudes achieved by the models. Calculations showed that the models ejected very close to apogee with the Delta A2-5s.

From 10 attempted A2 flights, two, three, and two usable data points for A, B, and C airframes, respectively, were obtained. Even though one model was recovered for a tenth flight attempt with a single Delta A2-6, one additional engine cato, one unstable flight for unknown reasons, and one lost track with the Delta A2-6 limited the number of useable data points to the seven obtained. Given the situation, we felt out efforts to salvage a potential project disaster were rewarded with the data obtained.

II. Flight Test Data

A. Altitude Data

The average altitudes determined for each airframe design and engine combination are listed in Table 4. The data suggests that there is no difference in altitude performance between the A and B airframe designs and the performance of the C airframe design is slightly less than the performance of the A and B airframes. A one-way statistical analysis of variance of the Delta A2-5 flight test results indicate a significant difference between the performance of the three designs at the 89% confidence level. T-tests show that the C design (0.64 inch diameter) was outperformed by the B design (0.525 inch diameter) at the 93% confidence level and by the A design (0.45 inch diameter) at the 89% confidence level. There was no statistically significant performance difference between the A and B designs. Although the limited B2-9 flight data suggest some similar relationships, the difference in tracked altitude from predicted altitudes derived from drag coefficient calculations indicate that tracked altitudes were as much as 75 meters or 10% below apogee (and the untracked flights with the longest delays probably had a much more severe altitude loss). This difference significantly reduces the usefulness of the B2-9 altitude data for comparison.

An examination of the flight angles for the A2-5 flights also support the differences and similarities suggested by the average altitude data. Data showed that most flight paths were within 5 to 10 degrees of vertical. Comparison of the most vertical flights and similar angular deviation from vertical flights for each airframe indicate that the performance of the A and B airframes are similar and outperform the C airframe. The worst B airframe flight (25.2 degrees) still outperformed the best C airframe flight (7.2 degrees) by 13 meters. The flight angles also verified that the lowest altitudes achieved for the B and C airframe designs had the greatest flight angles from vertical (437 meters at 25.2 degrees and 394 degrees at 20 degrees, respectively). Though this relationship was reversed for the A airframes, the flight angles were very straight (< 6 degrees) and within 1.5 degrees of each other. In this case the angular difference was not the reason for altitude variation. The representation of flight angular deviation could prove to have a general usefulness in verifying questionable altitude data in other applications.

The representation of flight geometry relative to the launch area and tracking geometry in Figure 2 pictorially illustrates the dispersion of flight paths over the launch site. This view gives an excellent single representation of the flight paths relative to the launch site. It should be noted that this data was used to successfully locate the few models that were retrieved after flight testing. One of the models recovered was located without the streamer attached to assist with visibility. It also indicated that most of the models landed in very dense, impenetrable vegetation where recovery was next to impossible. This representation of data could be the basis for a future research and development project for assisting in the recovery of difficult to find models.

B. Drag Data

Drag data was calculated using the wRASP 2.1- program by Chuck Gibke et al. CD, CDA, and flight time values were obtained with the built-in DigiTrak backtracking by Larry Curcio. A time thrust curve for the Delta A2 engine was developed based on manufacturer's data sheets. This version of wRASP already had Apogee B2 data based on NAR S&T curve data.

The results presented in Table 5 are values based on the average A2 altitude for each design. It appears that (within the limit of significant digits and for this specific experiment) that a single CD obtained from an average altitude value is equivalent to an average CD obtained from multiple CDs from multiple altitudes. Statistical analysis shows that the significance of the CD values is similar to the significance of the A2 altitude data discussed above.

Calculation of drag coefficient values from the A2-5 altitude data indicates that the A and B airframes performed very close to the all-turbulent body boundary layer CD predictions shown in Table 1. These predicted CDs and CDs derived from tracking data were within 5% and 2.5% of each other for the A and B airframes, respectively. It is unlikely that the body is all-turbulent flow, but the high drag value suggests an early transition (ie: a short distance along the body) from laminar to turbulent flow. This lends credibility to the popular supposition that the nose/body joint may trigger this transition. Balancing the lower than observed drag of this partially-laminar flow is induced drag from the models flying at an angle of attack due to inevitable in-flight disturbances. However, while the specific components of drag may be indeterminate, the values calculated by the all-turbulent body method seem to have some real-world predictive value.

The difference between the all-turbulent predicted CD and derived CD for the C airframe was 18%. The explanation for this difference may lie in marginal dynamic stability (and resulting higher induced drag) associated with the lower fineness ratio. Dynamic stability parameters were not investigated for the designs in this study, but the C design's fineness ratio (11.5) flirts with the long-time rule of thumb of a minimum fineness ratio of 10 to 12 for sufficient dynamic stability.

One final difference in theoretically predicted CDs and derived CDs is that the theory predicts a slight advantage for the A airframe over the B airframe. This was not observed. If significant, the derived results suggest a slight advantage for the B airframe.

Drag coefficient data from the B2 flights was not subjected to further analysis, aside from the presentation in Table 6, due to the low quantity and quality of the data. The large discrepancy between observed and calculated flight times indicate that the model went higher than calculated (with a correspondingly lower CD), descended for some time, and ejected significantly after (and below) apogee. One interesting note is that, based on the CDs obtained from the A2 flight data, the calculated B2 performance should have been in the ballpark of 750 meters with a flight time of 11 seconds.


The purpose of this study was to compare relative drag form factors derived from altitude data and the flight performance of three different airframe designs of different fineness ratios. Every effort was made to equalize body surface area and other drag components, making the fineness ratio the sole variable. Background and theory was presented, along with general observations from flying experience, which question the accepted belief that a minimum diameter design will produce a lower drag form factor and greater altitude performance than boattailed designs of lower fineness ratio. Advantages which facilitate model prepping and improve reliability were discussed which would make a boattailed model with lower fineness ratio desirable.

The results of the study derived from altitude data indicated that there was no statistically significant difference in altitude performance between a boattailed design with a fineness ratio of 15.9 and a minimum diameter straight tube design with a fineness ratio of 20.4 and equal surface area. The drag coefficients derived from the altitude data for these designs were within 5% of the CDs predicted from all-turbulent body boundary layer calculations.

The altitude performance of a boattailed airframe of equal surface area and fineness ratio 11.5 was 9.5% lower than the straight minimum diameter airframe at the 87% level of statistical significance and 10.5% lower than the 15.9 fineness ratio boattailed design at the 93% level of statistical significance. The derived CD for the lowest fineness ratio boattailed design was 18% higher than the CD predicted from all-turbulent body boundary layer calculations.

For models of the performance class and surface area tested, these results indicated that the advantages of a boattailed design can be utilized without loss of altitude performance. However, there is a fineness ratio limit where performance will begin to be compromised.

A method of calculating and presenting data showing the angular deviation of flight path from vertical flight was developed which could prove useful in other applications for evaluating questionable altitude data. The flight path angular deviation data determined for this project assisted in interpreting altitude data and supported the conclusions derived.

A two-dimensional pictorial representation of flight path distribution around the launch point in relation to tracking geometry was also presented which assisted in the prediction of model location and recovery. This representation of flight path distribution could be further developed and prove useful in model recovery and related applications.

Last, a rather simple method for constructing reproducible nose cone shapes from balsa by rotating the nose with a modelers lathe, electric drill or Dremel tool and controlling the shaping tool was developed and presented.

Cost Estimate

18 Apogee B2-9 motors $54.00
5 Apogee noses $ 5.00
3 Apogee tubes $ 2.70
Finishing supplies (paint) $13.00
Computer and copying costs $20.00
Total Cost $94.70

Items "On Hand" (no additional cost)

Modeling lathe, drill lathe, & other workshop tools
Triple-beam balance
Dial calipers
Two mandrels & fiberglass supplies
10 Delta A2 motors


Crowell, Gary. VCP Rocket Stability Calculator V1.64 PC program. 1994, 1995, 1996.

Chuck Gibke et al. Rocket Altitude Simulation Program for Windows wRASP 2.1- PC program, including DigiTrak 5.4 by Larry Curcio. 1998.

Gregorek, Dr. Gerald. Technical Report TR-11: Aerodynamic Drag of Model Rockets. Penrose, CO: Estes Industries, 1970.

Mandell, Gordon; Caporaso, George; and Bengen, William. Topics In Advanced Model Rocketry. Cambridge, MA: The MIT Press, 1973.

Shapiro, Ascher H. Shape And Flow: The Fluid Dynamics of Drag. Garden City, NY: Doubleday & Company, Inc., 1961.

Stine, G. Harry. Handbook of Model Rocketry, Fourth Edition. Chicago: Follett Publishing Company, 1976.

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Last modified January 26, 2006

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