Samara National Research University Fundamentals of the theory of aerodynamic calculations of rockets (practical lesson)

1.

Samara National Research University
Fundamentals of the theory of
aerodynamic calculations of
rockets (practical lesson)
Vladimir Frolov
E-mail: [email protected]
Winter School, 2024
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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2.

Outline
Forces acting on a rocket
Rocket motion equation
General Formula for Rocket Drag Coefficient
Area calculation
Total friction drag coefficient of the body
Conclusions
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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3.

Forces acting on a rocket
We will consider the flight of a rocket as the
movement of a material point located at the center
of mass of the rocket.
The position of the center of mass changes
depending on the burnup of the fuel.
The thrust force T changes over time and depends
on the engine operation.
The drag force changes with flight altitude and,
therefore, air density, as well as flight speed.
Figure 1 Forces acting on a rocket
The force of gravity changes as the flight altitude
changes (the acceleration of gravity changes) and
the mass of the fuel changes
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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4.

Rocket motion equation
W mg ;
T f ;
D CD
V 2
2
Sm.body ;
dV
V 2
m
T W D T mg CD
S m.body ;
d
2
So, the main task of
dV
V 2
m
T mg CD
Sm.body ;
CD ? aerodynamics is to calculate
d
2
the drag coefficient
We will assume, CD const, const, g const. But, CD=f(Re , M)
V Lref
V
Re
const ; M
const.
( h)
a h
We will assume also, (h) const; a(h) const and V const= Vmean
h is altitude of flight
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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5.

Geometry of rocket
Figure 2 Geometry of rocket
f
lf
d cylinder
lnose
lcylinder lnose
d cylinder
100
nose
1.389
d cylinder 72
900 100 1000
13.89 is fuselage fineness ratio
72
72
is nose path fineness ratio
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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6.

General Formula for Rocket Drag Coefficient
CD rocket CD body 3 CD wing
S wet wing
Sm. body
(C f body CD bottom ) 3 2c f plate
S wet wing
Sm.body
1
S wet body
S wet wing
Sbottom
(2c f plate ) body
c p bottom
.
3 2c f plate
Sm.body
Sm.body
Sm.body
2
?
?
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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7.

Area calculation
S wet. body
?
S m.body
S wet. body S wet. nose ogive S wet. cylinder ;
2
d cylinder d cylinder
2
S wet. nose ogive S wet. nose cone
l
nose
2
2
0.036 0.0362 0.12 0.036 0.1063 0.01202
S wet. cylinder d cylinder lcylinder 0.072 0.9 0.2036;
S wet. body S we. nose S wet. cylinder 0.01202 0.2036 0.2156;
S wet body
0.2156
52.952.
S m.body 0.004072
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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8.

We have roughly calculated the wetted area of the rocket's nose section.
Let's now calculate this area more precisely.
Let's divide the length of the nose of the rocket into N parts. The function of the
radius from the coordinate along the length of the nose can be determined from
the drawing of the rocket.
?
N
N
i 1
i 1
Sogive 2πri x Δx 2πΔx ri x
An analytical method can be used.
Frolov V.A. Practical on aerodynamics, August 20-21, Samara
University, Russia
8

9.

Let us assume that the surface of the nose is described by a second-degree
polynomial
r x a0 a1 x a2 x 2
1) x 0 r x 0;
2) x ln
r x
a0 0
D
;
2
D
a1 x a2 x 2 a1ln a2ln 2
2
dr
0 0 a1 2a2 x a1 2a2ln
dx
a1ln a2ln 2 D 2
2) 3)
2a2ln2 a2ln2 D 2
a1 2a2ln 0
D
D
a2ln2 D 2 a2 2 a1 2a2ln
2ln
ln
3) x ln
D
D 2
x
x2
r
x x
r x x 2 x
r ( x)
2
ln
2ln
λn 2 λn ln
( D 2)
ln ln
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
2
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10.

r
x x
r ( x)
2
( D 2)
ln ln
2
for our rocket D/2=36 мм; ln 100 мм
let N be equal to 10 then
N
N
N
i 1
i 1
i 1
Sogive 2πri x Δx 2 36πΔx ri x 72 π 10 ri x
N
2261,95 ri x
i 1
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University,
Russia
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11.

N x. мм x/ln
1
10
0.1
2
20
0.2
3
30
0.3
4
40
0.4
5
50
0.5
6
60
0.6
7
70
0.7
8
80
0.8
9
90
0.9
10 100 1.0
sum
(x/ln)2
0.01
0.04
0.09
0.16
0.25
0.36
0.49
0.64
0.81
1.00
2(x/ln)-(x/ln)2
0.19
0.36
0.51
0.74
0.75
0.84
0.91
0.96
0.99
1.00
7.25
N
Sogive 2261,95 ri x 2261,95 7.25 16399,1375
i 1
16399,1375 (мм мм)=0,0164 (м м)
(0,01640-0,01204)/0,01640 100%≈27% (!)
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University,
Russia
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12.

Area calculation
S wet. body
?
Sm.body
Sbottom
?
Sm.body
Sbottom Sm.body Snozzle
0.072
4
Sm.body
2
cylinder
0.038
0.0722
4
2
2
d
d
cylinder nozzle
2
nozzle
0.002934;
0.004071;
4
Sbottom 0.002934
0.7207.
Sm.body 0.004071
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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13.

Calculation Area of the Wing
ct cs
0.03 0.12
S wet.wing
s
0.08
2
2
0.075 0.08 0.006 m 2
S wet wing 3 0.006
3
4.4215
Sm. body 0.004071
1
S wet body
S wet wing
Sbottom
CD rocket (2c f plate ) body
c p bottom
3 2c f plate
Sm.body
Sm.body
S m.body
2
1
(2c f plate ) body 54.03 c p bottom 0.7207 2c f plate 4.4215.
2
2c f plate ?
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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14.

Total friction drag coefficient of the wings
V cmean 75 0.075
5
Re
3.85
10
h
1.46 10 5
For a laminar boundary layer at Reynolds number Re <4.85 105
2,656
2c f M 0
Re
2,656
2,656
2c f M 0
0.00428
5
Re
3.85 10
V
75
M
0.22
a 340,29
M 1 0,12M
2 0,125
1 0,12 0.22
2 0,125
0.999 1.0
t f t f 0.027 1.0
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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15.

Total friction drag coefficient of the rocket
1
CD rocket (2c f plate )body 52.952 c p bottom 0.7207 0.00428 4.4215
2
1
(2c f plate )body 52.952 c p bottom 0.7207 0.01892.
2
Wing
Body
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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16.

Total friction drag coefficient of the body
Re
V Lbody
h
75 1.0
6
5.14
10
1.46 10 5
For a mixed boundary layer at Reynolds number 4.85 105< Re< 3.0 107
40 x
2c f M 0
1 xt
2,58
Re
lgRe
0,91
0,625
t
0,375
0,8
xt
xt
?
Lbody
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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17.

Calculation of the relative coordinate of the transition
point of a laminar boundary layer into a turbulent
boundary layer
For a mixed boundary layer at Reynolds number 4.85 105< Re< 3.0 107 the relative
coordinate of the transition point calculated by the formula
10n
xt
,
Re
0,5
hr
lg
Re 1
Lref
2
n 5 1,3 0,6 M 1 0,25M 1
,
2
2,2 0,08M
1 0,312 M
2
where hr is roughness of the surface of the body, m;
Lref is reference size of the body (wing chord or length of the body), m;
M is number Mach.
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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18.

0,5
hr
lg
Re 1
Lref
2
,
n 5 1,3 0,6M 1 0,25M 1
2
2,2 0,08M
1 0,312M
Lref Lbody 1.0 m; hr 25 10 6 ; M 0.22
2
n ?
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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19.

Table 1.2: Characteristics of surface roughness
Surface nature
Cleanliness
class
Approximate height of
roughness tubercles, micron
Machined parts
4
5
6
7
8
9
40.0
20.0
10.0
6.3
3.2
1.6
Duralumin sheets,
anodized
The same, spray- painted
–
6…10
–
20…30
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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20.

0,5
2
hr
lg
Re 1
Lref
2
,
n 5 1,3 0,6M 1 0,25M 1
2
0,08M
2,2
1 0,312M
Lref Lbody 1.0 m; hr 25 10 6 ;
M 0.22; Re 5.14 106
2
25 10 6
6
lg
5.14 10 1
1.0
n 5 1,3 0,6 0.22 1 0,25 0.222 1
2
0,08
0.22
2,2
1 0,312 0.22
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
0,5
20

21.

25 10
6
lg
5.14 10 1
1.0
2
n 5 1.3 0.6 0.22 1 0.25 0.22 1
2
0.08
0.22
2,2
1 0.312 0.22
2
6
2
lg 25 5.14 1
5 1,4304 1
0.003872
2,2
1.06864
5 1,4304 1 0.9190
2 0,5
0,5
0,5
2.1055
5 1,4304 1
2,1964
2
5 1,4304 1 0.8449
0,5
0,5
5 1,4304 0.3938 5.563
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
21

22.

n
5.563
10
10
xt
0.071,
6
Re 5.14 10
LLBL 1000 0.071 71 mm
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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23.

40 x
2c f M 0
1 xt
2,58
Re
lgRe
0,91
0,625
t
0,375
0,8
40 0.071
1 0.071
2,58
6 0,375
6
(5.14
10
)
lg(5.14 10 )
0,625
0,91
7.6577
0.006698 0.929
328.557
0,8
0.8
0.00644
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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24.

1
CD rocket (2c f plate ) body 54.03 c p bottom 0.7207 0.00428 4.4215
2
1
0.00644 54.03 c p bottom 0.7207 0.01892
2
1
0.00644 54.03 c p bottom 0.7207 0.01892
2
0.00322 54.03 c p bottom 0.7207 0.01892
0.00322 54.03 c p bottom 0.7207 0.01892
0.1740 c p bottom 0.7207 0.01892 .
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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25.

Sb
Cd b c p b
,
Smax
0.0155
0.0155
0.0733
c p b C 1000
b D f body
0.00322
72
CD rocket 0.1705 0.0733 0.7207 0.01892
0.1705 0.0528 0.01892 0.2233 0.01892 0.2422.
CD rocket 0.24
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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26.

Thanks for attention!
I will be glad
to answer any questions.
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
26

27.

Calculation geometry parameters
2
2
Sb Smax Snozzle
d
4
cylinder
d nozzle ,
lrocked 1000
b
13.889;
d cylinder
72
Sb
4
2
2
0.072
0.038
cylinder
nozzle 0.002934;
Sb
0.0382nozzle
0.2785;
2
Smax 0.072cylinder
t
2
t
0.027.
ctip 30 120 / 2
ctip is tip cord; t is thickness of aerofoil.
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
27

28.

Total friction drag coefficient of the rocket
Cd C f body 3C f fin
Smax b
2
dcylinder
4
Sfin
,
Smax b
0.0722
0.004072 m 2 ;
4
Sfin 0.5 ctip conboard s 0.5 0.02 0.13 0.08 0.006 m 2
Sfin
0.006
1.473
Smax b 0.004072
Cd C f body 3C f fin 1.473 C f body 4.42 C f fin C f body 4.42 C f fin
C f body 4.42 0.00428 C f body 0.01892
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
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29.

Reference Areas of the Body
S wet body S wet nose ogive S wet cylinder ;
S wet nose ogive 0.0164
S wet cylinder d cylinder lcylinder 0.072 0.9 0.2036;
S wet body S wet nose S wet cylinder 0.0164 0.2036 0.22;
S wet body
Sb
0.22
54.03.
0.004072
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University,
Russia
29

30.

We will perform all the necessary calculations of the
drag coefficient in subsequent practical exercises.
Thanks for attention!
I will be glad
to answer any questions.
Frolov V.A. Practical on aerodynamics, August 20-21, Samara University, Russia
30