V = Flow velocity, ft/sec

Transcription

1 1 Drag Coefficient Preiction Chapter 1 The ieal force acting on a surface positione perpenicular to the airflow is equal to a ynamic pressure, enote by q, times the area of that surface. Dynamic pressure is one-half the square of the flow velocity times the ensity of the flui. In equation form this is: q = 1 ρ V ρ = Air ensity, lb.-sec / ft 4 V = Flow velocity, ft/sec Imagine putting your han or a flat plate out the winow of a car such that the flat surface is positione normal to the airflow. The total force require to hol it in position when meeting the oncoming airflow will be approximately equal to that efine above. Realistically, the actual force is epenent on the shape of the object, an the -imensional flow characteristics of the flui (i.e.; flow relief, turbulent an/or laminar flow, etc.. Often these influences are summe up in a single coefficient known as an aeroynamic coefficient. In our particular case, we are intereste in a rag force D, an likewise a rag coefficient C D. Depening on equation formulation an reference area, C D can take on ifferent values. For rockets, C D is typically base on the rocket s maximum cross-section area. Most rockets are circular in cross-section, therefore its cross-section area is escribe by the equation for an area of a circle. In general, the equation for rag D is given by: D = qc D A = qc πr D q = Dynamic pressure, lb. / ft CD = Drag coefficient A = Reference area, ft = πr π = Constant Pi R = Raius of maximum cross-section, ft Most of the equations presente in this chapter are empirically base (base on physical ata. The majority of the equations will be simply presente an not erive, particularly those escribing Friction Drag. Others, such as those escribing Base Drag an Wave Drag, will be erive as we go along. I believe that the source for the basis of most of these equations is the U.S Air Force methoology known as DATCOM (Reference 4. Many of these equations have been transcribe from han written notes that I ha use in my past career as an aircraft conceptual esigner with the U.S. Air Force, many moons ago. The graphs of these notes have been transforme to equations for ease of writing computer programs. I have eviate somewhat from the DATCOM methos for Base Drag estimation, an have evelope an approach that fits rocket ata more accurately. A constant K F equal to 1.04 has been aopte to estimate the Interference Drag contribution for rocket ata.

2 Before we get into the etails, it woul be wise to iscuss some terminology. Many public or commercially available rocket performance programs require the user to input a value of average rag coefficient C D. This is often a guess on the part of the user. However, more sophisticate programs attempt to preict the time history of rag, an hence, a more precise time history of the rocket s performance. These programs have their equations embee in the coe, an o not attempt to enlighten the user as to their egree of integrity or sophistication. For those of us who are of a nery an curious nature, this just oesn t meet our nees. In fact, some of you may wish to evelop your own custom software using the equations provie here. We will limit our iscussion to Zero Lift Drag or Parasite Drag, which is the total rocket rag inepenent of lift. This occurs at zero Angle of Attack for rockets. Angle of Attack refers to the angle of incience between any lifting component (wing, fin, boy, etc. an its velocity vector. Inuce Drag is that rag associate with the generation of lift. We will not be ealing with Inuce Drag. One of the largest contributors to Zero Lift Drag over Subsonic spees is Skin Friction Drag. Subsonic refers to flight spees well below the spee of soun. Skin Friction Drag is the rag resulting from viscous shearing stresses acting over the surface of the rocket. Form Drag or Pressure Drag is the rag on a boy resulting from the summation of the static pressure acting normal to the rocket s surfaces, resolve into the irection opposite of flight. Base Drag is a contributor to Pressure Drag, an is attribute to the blunt aft en of the rocket. Base Drag can be a significant contributor to the rocket s overall rag uring power-off flight (after engine burnout. Another contributor to Pressure Drag is Wave Drag. Wave Drag makes its ebut uring Transonic spees (about Mach 0.8 to Mach 1. an through Supersonic spees (above Mach 1.. Wave Drag is a Pressure Drag resulting from static pressure components locate to either sie of compression or shock waves that o not completely cancel each other. Finally, the last rag contributor we will consier is Interference Drag. Interference Drag results from two boies in close proximity, such as fin to boy junctures an launch lug to boy junctures. Specifically, we will account for the following rag contributors: Skin Friction Drag (Viscous Effects Base Drag (Pressure Drag increment ue to blunt boy uring power-off flight Wave Drag (Pressure Drag increment ue to compression or shock waves Interference Drag (Drag increment ue to boies in close proximity Below is a typical flight history of rag versus Mach number. Drag Coefficient, C Decelerate Flight Accelerate Flight Mach umber, M In the above graph the rocket accelerates from a stanstill on the launch pa to a maximum Mach umber of about 1.6 (118 mph!. This is epicte by the ashe curve. The rocket then begins to

3 ecelerate, still uner thrust flight, as epicte by the soli curve. Then at a Mach umber of about 1.5 (951 mph, the engine burn is complete an the rocket continues its eceleration uner zero thrust flight. At this point, the soli curve is isplace above the ashe curve, epicting a rag rise ue to Base Drag. This Base Drag is a result of transitioning from thrust flight to coast or zero thrust flight. The existence of thrust, or lack thereof, can have a significant effect on the rocket s rag coefficient for blunt aft boies, as in the case above. Both curves exhibit a jump in rag coefficient over the range of about Mach 0.9 to 1.. This is the characteristic transonic region where Wave Drag makes its ebut an may ominate. Again, the shape of the fin an aft boy region will have a significant effect on rag rise ue to Wave Drag. Wave Drag can be somewhat minimize by tailoring the shape of the fin an aft boy geometry using a technique calle Area Ruling. Although I have use Area Ruling in some of my transonic rocket esigns, its overall effect on the rocket s performance is minimal ue to the short time spent in transonic flight. For most projects, time spent in efforts such as Area Ruling woul yiel little return on investment. There exists a steep increase in rag coefficient as the Mach umber approaches zero. This rag rise is associate with small Reynols umbers. At very small Reynols umbers the momentum of the airflow about the rocket is insufficient to remain attache to its surface an maintain well-efine streamlines, hence the flow separates an becomes turbulent. The result is an increase in Friction Drag. The Reynols umber is the ratio of inertia forces to viscous forces as a vehicle penetrates flow. In mathematical form it is efine by, R e = ρv L µ ρ = Density of flui V = Free stream velocity of the flui about the vehicle L = Characteristic length of vehicle (rocket iameter µ = Absolute coefficient of viscosity Reynols umbers are commo nly use as scaling factors to approximate similar flow conitions in the laboratory (i.e., win tunnel testing for situations where it is ifficult to recreate actual flow about full scale boies. Mach umber is the ratio of the flui velocity to the spee-of-soun in that flui. The spee-ofsoun in a flui is the spee at which a pressure isturbance is propagate through the flui. In air, at Stanar Sea Level conitions, the spee of soun is about 76 mph or ft/s. If the flui velocity about an object is equal to the spee-of-soun of that flui, then the flui is sai to be travelling at Mach 1.0 relative to that object. Viscosity is a characteristic of a flui escribe by the flui s ability to resist shear. A flui having high viscosity will better resist eformation uner shear than a flui having low viscosity. Viscosity of a flui is often measure by applying a pure torque to the flui. If one were to integrate the prouct of shear stresses times the istance to the center of applie torque over the flui volume, the result woul be equal to the applie torque. Shear stress within a flui is proportional to the graient of the flui velocity acting normal to the shear plane. The constant of proportionality is known as the absolute coefficient of viscosity µ. For the case of the applie torque, we efine the shear stress as: u τ = µ r

4 4 µ = Flui coefficient of viscosity u = Differential of flui tangential velocity r = Differential of raial istance from torque center of application The Friction Drag previously mentione is irectly relate to the shear stresses in the flui. Without viscosity, there woul be no shear stres s an likewise no friction rag. 1.0 Friction Drag A given rocket s rag will not only be a function of Mach umber, but also altitue. As altitue changes, so oes the air viscosity, spee of soun an air ensity. Viscosity, ensity an spee of soun will play a role in the equations for rag as well as a strong epenence on Mach umber. 1.1 Boy Friction Drag The following equations can be solve in sequential orer to etermine the rocket s boy coefficient of rag ue to friction. a = Spee of soun, fps = 0.004h , if h 7000 ft = , if 7000 ft h ft = h , if h ft h = altitue, ft ν = Kinematic viscosity, ft /s ah b = e + a = an b = 0.0, For h feet a = an b = , For h 0000 feet a = an b = , For h 0000 feet Rn* = Compressible Reynols umber aml = ( M 0.04M M 1ν Cf * M = Mach umber L = total length of rocket, inches = Incompressible skin friction coefficient = Rn * M M 5 Cf = Compressible skin friction coefficient

5 5 4 = Cf *( M 0.181M M 0.009M M 5 Cf * ( term = Incompressible skin friction coefficient with roughness 1 =.5 L log 10 K K = 0.0, for smooth surface = to , for polishe metal or woo = , for natural sheet metal = , for smooth matte paint, carefully applie = to 0.001, for stanar camouflage paint Cf (term = Compressible skin friction coefficient with roughness Cf * ( term = ( M Cf ( final = Final skin friction coefficient = Cf, if Cf Cf (term = Cf (term, if Cf Cf (term C f (boy = Boy coefficient of rag ue to friction Equation = Cf ( final ( L / = maximum boy iameter L = total boy length S B = total wette surface area of boy 4S π B ( L / L 1. Fin Friction Drag The following equations can be solve in sequential orer to etermine the rocket s total fin coefficient of rag ue to friction. a = Spee of soun, as efine in Section 1.1- Boy Friction Drag

6 6 ν = Kinematic viscosity, as efine in Section 1.1 Boy Friction Drag Rn* = Compressible Reynols umber amc = r ( M 0.04M M 0.089M M 1ν Cf * M = Mach umber C r = Root chor of fin, inches = Incompressible skin friction coefficient = Rn * Cf Cf = Compressible skin friction coefficient 4 = Cf *( M 0.181M M 0.009M M * ( term = Incompressible skin friction coefficient with roughness 1 = C log 10 r K.5 5 K = 0.0, for smooth surface = to , for polishe metal or woo = , for natural sheet metal = , for smooth matte paint, carefully applie = to 0.001, for stanar camouflage paint Cf (term = Compressible skin friction coefficient with roughness Cf * ( term = ( M Cf ( final = Final skin friction coefficient = Cf, if Cf Cf (term = Cf (term, if Cf Cf (term Rn = Incompressible Reynols umber amc = r 1ν Ct λ = = Ratio of fin tip chor to root chor C r

7 7 Cfλ = Average flat plate skin friction coefficient for each fin panel = Cf ( final 1 + log (, if λ = Rn λ.6 [ log10( Rn ] log 10( Rnλ = Cf ( final ( λ log.6 [ ] [ log ( Rn ] λ [ ] [ ] ( Rnλ log 10( Rn C f ( fins = Coefficient of friction rag for all fins Equation 1. 1 = Cf λ t C r X t c t C r 4 f S π f t = Maximum thickness of each fin at root C r =Fin root chor X t c X t = c Cr = Distance from fin leaing ege to maximum thickness X t c = Fin tip chor f = umber of fins S f = Total wette area of each fin b ( C r + C t = Maximum iameter of rocket boy Ct C r t X t/c b/ C r C t

8 8 1. Protuberance Friction Drag Protuberances are components that are foun on the exterior of a vehicle. An example of a common protuberance for a rocket woul be a launch lug. The contribution of a protuberance to rag is often accounte for by its friction rag. Protuberances will also generate an incremental rag rise ue to the interaction of their pressure istributions an bounary layers with that of the host boy. This incremental rag rise is known as Interference Drag an is ifficult to preict. A etaile analysis of interference rag is beyon the scope of this text. The following equations can be solve in sequential orer to etermine the coefficient of rag ue to a protuberance. a = Spee of soun, as efine in Section 1.1 Boy Friction Drag ν = Kinematic viscosity, as efine in Section 1.1 Boy Friction Drag Rn* = Compressible Reynols umber aml = P ( M 0.04M M 0.089M M 1ν Cf * M = Mach umber L P = Length of protuberance, inches = Incompressible skin friction coefficient = Rn * Cf Cf = Compressible skin friction coefficient 4 = Cf *( M 0.181M M 0.009M M * ( term = Incompressible skin friction coefficient with roughness 1 = log 10 L P K.5 5 K = 0.0, for smooth surface = to , for polishe metal or woo = , for natural sheet metal = , for smooth matte paint, carefully applie = to 0.001, for stanar camouflage paint Cf (term = Compressible skin friction coefficient with roughness Cf * ( term = ( M

9 9 Cf ( final = Final skin friction coefficient = Cf, if Cf Cf (term = Cf (term, if Cf Cf (term Cf pro Cf pro = Friction coefficient of protuberance 0.14 a = Cf ( final Lp a = Distance from rocket nose to front ege of protuberance L p = Length of protuberance C = Drag coefficient of protuberance ue to friction Equation 1. pro A 4S pro = Cf pro L p π A = Maximum cross-section area of protuberance S pro = Wette surface area of protuberance = Maximum rocket iameter a L p 1. Drag ue to Excrescencies Excrescencies inclue features such as scratches, gouges, joints, rivets, cover plates, slots, an holes. These will be accounte for by assuming they are istribute over the wette surface of the rocket. The coefficient of rag for excrescencies is estimate with the equations below. Ce = Change is rag coefficient ue to excrescencies Equation 1.4 4Sr Ce = Ke π S r = Total wette surface area of rocket = Maximum iameter of rocket boy K e = Coefficient for excrescencies rag increment K = , For M < 0.78 e 4 = M M.106M M For 0.78 M = 0.000M 0.001M , For M >

10 Total Friction an Interference Drag Coefficient The total friction rag coefficient is assume proportional to the sum of the rag coefficients for the boy, fin, protuberances, an excrescencies. The total skin friction rag coefficient with consieration for interference effects is estimate with the following equation. F [ C boy + K C ( fins + K C C ] C = ( + Equation 1.5 f F f F pro e KF = Mutual interference factor of fins an launch lug with boy Base Drag Coefficient - Base rag can be escribe as a change in mass momentum. Imagine laminar airflow traveling over a smooth graually contoure boy at velocity when suenly it encounters a blunt aft en where the velocity rops to zero. The mass momentum (mass x velocity changes abruptly, generating a force that acts opposite to the irection of flight. Fortunately, an particularly in subsonic flow, Mother ature helps reuce the severity of this change in mass momentum through the generation of a bounary layer. Most likely, the bounary layer is not laminar but turbulent an the momentum thickness is well evelope. The change in mass momentum at the blunt en is less severe with the avent of a fully evelope bounary layer. The resulting form rag is less severe as well. Unfortunately, nothing is free when it comes to Mother ature. The bounary layer is evelope from the presence of viscosity. Recall that viscosity is the culprit that causes skin friction rag. Generally, as friction rag increases the tren is a reuction in base rag. Base rag is ifficult to preict. An attempt to fin an existing metho to estimate base rag that correlates well with rocket ata was unsuccessful. In lieu of continuing a search, an effort has been mae to formulate a metho that woul estimate base rag with reasonable correlation. The metho escribe below is ivie into two regimes, the first for Mach umber less than or equal to 0.6, an the secon for Mach umber greater than Base Drag Coefficient for M < 0.6 Recall that for coasting flight, as friction rag increases, the tenency is for a reuction in base rag. In fact, base rag is typically escribe as inversely proportional to the square root of the total skinfriction rag-coefficient. In the formulation given below, it is assume that the base rag is inversely proportional to the square root of the total skin-friction rag-coefficient, incluing interference effects. Base rag is also relate to the ratio of boy base iameter to maximum boy iameter. The following is a general form of the equation for base rag. C b n b ( M < 0.6 = Kb Equation 1.6 C F K b = constant of proportionality b = base iameter of rocket at aft en = rocket maximum iameter C F = total skin-friction rag-coefficient, incluing interference n = exponent

11 11 The values for K b an n are given as 0.09 an.0 in Reference 6. The above equation using these values oes a poor job in preicting base rag for the rocket type shapes of References 1 an 5. As escribe in Reference 5, base rag is strongly relate to the rocket s length to boy ratio, where the length is taken aft of the maximum boy-iameter position. Configuration #1 b L o Configuration # b L o For the above two configurations, reasonable values of K b an n are: K b = tan Lo n = Lo When using the above equations efining K b an n along with the general equation for C b (M<0.6, the correlation as epicte by the graph below is reasonable. Preicte Cb Cb for Mach umber ~ Actual Cb Line of Perfect Agreement

12 1. Base Drag Coefficient for M > 0.6 For Mach umbers greater than 0.6, the base rag coefficient is calculate relative to the base rag value at Mach = 0.6 (M = 0.6. Using the equation of Section.1 to calculate the base rag coefficient at M = 0.6, the base rag coefficient for higher values of Mach umber is etermine by multiplying the value at M = 0.6 by the function f b. b ( M 0.6 = C b ( M = 0. f b C 6 Equation 1.7 ( ( M 1.798( M ( f b = M, For 0.6 < M < 1.0 f b =.0881 M, For 1.0 < M <.0 f b = 0.97( M 0.797( M ( M , For M >.0 The function f b is base on the souning rocket ata of Reference 1. A plot of f b against this souning rocket ata is given below. Function fb Mach umber 'M'.0 Transonic Wave Drag Coefficient The approach taken was to start with a clean sheet of paper. The DATCOM methos, as recore in my han written notes, prove to preict the transonic rag rise reasonably well for the Hart missile, esignation L-6591, of Reference. However, the metho i not perform as well for a variety of other configurations of References 1 an 5, incluing the variations of the Hart missile esignation L of References an. The epth of the DATCOM methos is beyon the scope of this effort. The metho presente here constitutes a series of equations that characterize the rag rise over the transonic region. These equations are curve fits of actual tren ata taken from a variety of rocket configurations, an attempt to preict the rag rise with basic boy imensional ata only. Equations bas e on curve fits of tren ata can be angerous an lea to erroneous results if use outsie the range of

13 1 parameters use in their evelopment. Specifically, the equations presente below shoul only be use for rockets having a ratio of nose length to effective rocket length less than 0.6. It is guarantee that the use of these equations for rockets having a ratio greater than 0.6 will result in a truly ba answer. Equations evelope this way will ten to lack quantities that relate to the actual physics of the problem, so extrapolation outsie the atabase use in their evelopment is a ba iea. Other methos with more substance, such as DATCOM, may be capable of accounting for the effects of iniviual component characteristics, such as fin sweep angle. Drag rise over the transonic region can be preicte for a given Mach umber (M an basic boy imensions, using the following equations: Configuration #1 L L e L b L Configuration # L e = L b MD = Transonic rag ivergence Mach umber L L = = Length of rocket nose L = Maximum Boy Cross Section Diameter MF = Final Mach umber of Transonic Region = a Le b Le = Effective length of rocket L a =.4, For < 0. Le

14 14 C DMAX L L = Le L L b = 1.05, For < 0. Le L = L e L L = Maximum rag rise over transonic region g L L e = c e, For 6 c( 6 g L e =, For < 6 L c = L b L g =.58 L b e L L b L L e L 6.48, For 0. Le + L 1.744, For 0. Le b CT = Transonic rag rise for given Mach umber M Equation 1.8 = C DMAX F, If MD M M F = 0., If M < MD or M > M F 5 4 F = 8.474x x 4.946x x x x = ( M M ( D M F M D 4.0 Supersonic Wave Drag Coefficient For all Mach umbers greater than M F, the supersonic rag rise is assume to equal the transonic rag rise at Mach umber = M F. This greatly simplifies calculations, an the results compare well with actual test ata. C = Supersonic rag rise for given Mach umber M Equation 1.9 S = C DMAX, If M MF = 0., If M < MF

15 Total Drag Coefficient The rocket s total rag coefficient for any given Mach umber is the summation of the iniviual coefficients given by equations 1.5 to 1.9. C D [ C f ( boy + K FC f ( fins + K FC pro + Ce ] + Cb + CT + CS = Preictions of rag coefficient versus Mach umber were performe for the two free-flight rocket configurations of Reference an the souning rocket of Reference 1. In all cases, it was not clear as to the actual surface finish of the rocket. Therefore, the surface finish constant K was ajuste until the preicte rag coefficients approximate the measure values. Due to the level of sophistication of the equations presente here, excellent agreement between preiction an measure ata was not expecte. However, it was hope that preiction of rag magnitue with configuration an variation with Mach umber is reasonable; that is, the trens are correct. For accurate preictions more sophisticate methos such as Finite Difference or Finite Element base Computational Flui Dynamics shoul be employe. The first example is the rocket configuration L-6590 of ACA T 549, Reference. The analysis suggeste that the rocket surface finish must have been very smooth with very few scratches an/or imperfections. Drag Coefficient Preiction for L-6590 Preiction With Escrescenes Preiction Without Excrescences Measure Data C Mach o.

16 16 The Secon example is the rocket configuration L-6591, also of ACA T 549, Reference. Drag Coefficient Preiction for L6591 Preicte Measure Data C Mach o. The final example is the 10-mm iameter souning rocket of Reference 1, How to Make Amateur Rockets. The example compares preicte versus measure rag coefficient for both the thrust (burn an zero-thrust (coast phases. Here we can see that the rag rise over the transonic region is unerpreicte for both burn an coast flight. The Mach umber at maximum rag rise is uner-preicte as well. The geometric configuration of this rocket falls outsie the range of ata for which the preiction methos were evelope. The methos were base on rocket configurations with nose cone length to total

17 17 rocket length ratios of 0. to 0.6. The 10-mm iameter souning rocket has a ratio of The preicte jump from the burn curve to coast curve compares well with that of the measure ata over the range of Mach umber. C for the 10 mm Souning Rocket Burn - Preicte Coast - Measure Coast - Preicte Burn - Measure C Mach o.

18 18 REFERECES 1. Wickman, John H.: How to Make Amateur Rockets, CP Technologies, Hart, Roger G.: Flight Investigation at Mach umbers from 0.8 to 1.5 to Determine the Effects of ose Bluntness on the Total Drag of Two Fin -Stabilize Boies of Revolution, ACA Technical ote 549, Langley Aeronautical Laboratory, Langley Fiel, Virginia, June Waliskog, Harvey A. an Hart, Roger G.: Investigation of the Drag of Blunt-ose Boies of Revolution in Free Flight at Mach umbers form 0.6 to., ACA Research Memoranum L5D14a, Langley Aeronautical Laboratory, Langley Fiel, Virginia, June Ellison, D. E.: USAF Stability an Control Hanbook (DATCOM, AF Flight Dynamics Lab., AFFDL/FDCC, Wright-Patterson AFB, Ohio, August McCormick, Barnes W.: Aeroynamics, Aeronautics an Flight Mechanics, pp. 15-1, John Wiley an Sons Inc., ew York, icolai, Lelan M.: Funamentals of Aircraft Design, 1975.