The drag force opposes the true speed direction and appears due to several effects of the flow around the rocket. As seen in (3.33), the drag coefficient contributes to the aerodynamic force and it may change throughout the flight. It can be estimated by adding all the drag coefficients of the external components of the rocket and, for each one of them, the drag may come from different sources: pressure drag, skin friction drag, base drag and wave drag.
Pressure drag arises due to the distribution of normal forces on the components and the skin friction drag, on the other hand, is a tangential force generated by the viscosity of the flow that creates a boundary layer on the surface of the rocket [47]. Therefore, assuming a flight without angle of attack, the nose, connectors and fins generate these two kinds of drag and the body tubes of each stage only produce skin friction drag. For conical noses, the pressure drag coefficient variation is represented in Figure 4.4. Since the drag coefficient for cones describe, approximately, a linear variation with the half vertex angle (ε)
Figure 4.4: Pressure drag coefficient of wedges, cones and similar shapes as a function of the half-vertex angle (ε) [47].
from 0 to 90°, the pressure drag coefficient may be found from
CDp(ε) = 0.233 + 0.011ε . (4.52)
The total drag coefficient for the nose and connector sections is CDnose/connect=CDp
Sb
S +CDf
Sw
S , , (4.53)
whereCDf is the skin friction drag coefficient (determined in Section 4.5.1),Sb is the nose base area and Sw is the wetted area. Note that, since the connector sections describe a cone without the apex, Sb, in that case, must represent the difference between the aft and fore areas in order to discard the pressure drag already included in the nose coefficient.
The fin’s drag coefficient, is given, for the two surfaces, as [48]
CDf ins = 2CDf
1 + 2tf
¯ c
Sw
S , (4.54)
where ¯candtf are the fin’s average chord and thickness, respectively, and the wetted area,Sw, must take into account the two sides of the fin. Considering a flight with angle of attack, there is an extra source of drag created by the fins, the induced drag. It can be estimated by [48]
CDind = CL2 πΛe
Sf
S , (4.55)
where Λ is the fin aspect ratio,e≤1 is the Oswald efficiency factor (e= 1 for elliptical wings),Sf is the fin planform area and CL is the fin’s lift coefficient that, assuming the fins as flat plates at low angles of
attack, may be approximated by (3.37). Since the flow around the fuselage interacts with the flow passing through the fins, it is created an interference drag. This extra drag can be interpreted as coming from the air flowing on fins that are extended into the fuselage [48]. Then, the interference drag coefficient is measurable by
CDint= 2CDf
crrext
S n , (4.56)
where nis the number of fins andcris the fin’s root chord.
Due to the lower pressure at the rear of the rocket, the boundary layer separates from the surface and increases the drag. This fact originates the base drag which can be found by [47]
CDbase= 0.029 pCDbody
dbt
d 3
, (4.57)
where CDbody is the drag coefficient of the fore body (not counting with the fins) and db is the boat-tail aft diameter. Ifdbt=d, the rocket does not have a boat-tail and the base drag coefficient is maximum, as expected.
Model rockets are provided with launch lugs that guide the rocket through the rods. They are an extra source of drag estimated as [48]
CDll =1.2Sll+ 0.0045Swll
S , (4.58)
where Sll and Swll are, respectively, the sectional and wetted (inner and outer surfaces) areas of the launch lugs.
4.5.1 Skin friction drag
Skin friction drag changes according the regime of the flow. In turbulent flow, the skin friction drag is higher than in laminar flow (although turbulence, in some cases, is desirable in order to delay separation and decrease the pressure drag). As the air hits the rocket’s surface, the flow is laminar and rapidly turns to turbulent with increasing Reynolds number. Between this two regimes appears a laminar-turbulent transition when critical Reynolds is reached (Recr) and the flow becomes fully turbulent above transition Reynolds (Retr).
Reynolds number, representing the relative order of magnitude between inertial and viscous forces of the fluid, is defined as [49]
ReL= ρV L
µ , (4.59)
whereL is the surface’s length which must be selected according the respective part of the rocket andµ is the dynamic viscosity of the air found from the empirical Sutherland’s law as [49]
µ= 1.458×10−6 T3/2
T+ 110.4. (4.60)
Since transition is hard to predict, a way to find the skin friction drag coefficient in a flat plate (considering zero pressure gradient) is to assume Recr≈Retr, yielding [49]
CDf = CDturb
l−Retr
ReL h
CDturb
xtr− CDlam
xtr
i
= 0.074Re−0.2L −Retr
ReL
h
0.074Re−0.2tr −1.33Re−0.5tr i
, (4.61)
where Retr ≈3×105 to 106. Equation (4.61) determines the turbulent drag coefficient for the entire plate and replaces the turbulent contribution by the corresponding laminar value up to the point (xtr) where the transition occurs [49].
4.5.2 Compressibility effects
Mach number is the dimensionless parameter [49]
M = V
a , (4.62)
where ais the speed of sound given as
a=p
γRT , (4.63)
where γ≈1.4 for air is the ratio of specific heats at constant pressure and volume.
From M>0.3, the flow no longer can be assumed as incompressible since there are changes in density and temperature and shock waves begin to increase the drag. The Prandtl-Glauert rule is commonly used to relate incompressible and compressible coefficients (CD and CDc, respectively) for slender and planar bodies [50]. This rule is given as
CDc= CD
p|1−M2|, (4.64)
that must be applied between 0.3<M<0.7 and 1.2<M<5. In transonic and low supersonic regimes (0.7<M<1.2), an interpolated function should be used to estimate the compressible drag coefficient and avoid the singularity at M = 1. For a conical nose, the wave drag due to shock waves can be estimated by the empirical equation [51]
CDw =
0.083 +0.096 M2
ε 10
1.69
for M>1, (4.65)
where ε is again the half vertex angle in degrees. Equation (4.65) should match the subsonic drag coefficient at M = 0.7 which is when the transonic effects begin to appear.
There are also corrections for the skin friction coefficient. These relations are [47]:
CDlamc =CDlam, (4.66)
CDturbc =CDturb(1−0.12M2), (4.67)
for subsonic (M<1) and
CDlamc = CDlam
(1 + 0.045M2)0.25, (4.68)
CDturbc = CDturb
(1 + 0.15M2)0.58, (4.69)
for supersonic (M>1) regimes.
4.5.3 Recovery device
After the apogee, or when it is desired, the recovery system is deployed and the main source of drag comes from the parachute or streamer. Since the respective drag coefficient depends on the shape of the recovery device, Table 4.1 lists several ranges of values for common parachute designs. The tabulated
Shape dproj/dref CD Flat circular 0.67 – 0.7 0.75 – 0.80
Conical 0.7 0.75 – 0.90
Biconical 0.7 0.75 – 0.92 Triconical 0.7 0.80 – 0.96 Hemispherical 0.66 0.62 – 0.77
Annular 0.94 0.85 – 0.95
Cross 0.66 – 0.72 0.60 – 0.85
Table 4.1: Solid textile parachutes’ projected to reference diameter (dproj/dref) and respective drag coefficient [52].
coefficients are associated to the reference area which is given as the total area of the canopy including the vent or other openings. To find the projected area when the parachute is inflated, are also listed the values for the ratio between the projected and reference diameters.
If the recovery device is a streamer, its drag coefficient may be found from the Open Rocket docu- mentation that formulates the empirical function [15]
CDstreamer= 0.034
ρs+ 25 105
ls+ 1 ls
, (4.70)
where ρs is the streamer’s material density (in kg/m3) andlsis the streamer’s length (in m).
Chapter 5
Rocket trajectory simulator
The developed rocket trajectory simulator aims to:
determine the rocket trajectory and respective flight data given known/estimated inputs;
give the user flexibility to change the default trajectory and rocket’s models;
enable to simulate from micro rockets up to sounding rockets’ models;
forecast the landing region from a Monte Carlo simulation;
optimize the rocket characteristics.
This tool is developed inMathematica® [53] since it is a powerful language in functional programming and to manipulate lists. Due to these characteristics, this language gives flexibility to implement the features we desire: structure of lists for the rocket, listable databases and modular programming. Al- thoughMathematica®becomes slower as it is an interpreted language, the mentioned advantages surpass this handicap. Besides the following sections aim to explain the methods applied to develop the simu- lator, their goal is also to expose some of the advantagesMathematica® offers while programming and manipulating data.