Download _Study of relevant parameters which influence the time of the “low

Study of relevant parameters which influence
the time of the “low-friction” state of a CD
hovercraft
Author: Zhang Chao-Ran
Advisor: Ding Hong-Juan
Dongying Shengli No.1 Middle School
1
Study of relevant parameters which influence the time of the
“low-friction” state of a CD hovercraft
Zhang Chao-Ran
Advisor: Ding Hong-Juan
Dongying Shengli No.1 Middle School, Dongying257100, China
Abstract
In this paper, CD hovercrafts have been built to investigate the relevant factors that
influence the time of the “low-friction” state of a CD hovercraft. Our experimental results
indicate that upon altering of the volume of air contained in the balloon, the weight of the
loaded hovercraft, the size of the aperture of the sports cap and the bottom area of the CD
hovercraft as well as the roughness of the surface, the time of the “low-friction” state of the
CD hovercrafts shows significant changes. Moreover, the time equation of the “low-friction”
state established from Navier-Stokes equation on the basis of the lubrication theory has been
applied to illustrate our experimental results.
Key words: low-friction state, volume of air, weight of the hovercraft, CD hovercraft
1. Introduction
Since 1959 the world’s first hovercraft was built, the hovercrafts have received more and
more people’s attention for their unique feature that they can fly just a few inches off the
ground on a cushion of air. This feature lets the hovercraft maintain the identity of being an
amphibious vehicle, not confined by roads or water. And it can travel over most smooth
surfaces including ice, dirt, snow, water, and asphalt. Thus, hovercrafts have been widely used
in military, recreation, rescue, and other fields.
Why hovercrafts can leave the water or sail on the ground? That relies on inflating a
cushion of air. Hovercrafts all use a system of high-power fans which push air into the bottom
of boats. Because of the water barrier, accumulation of air flow in the bottom forms the
cushion of air, and produces a powerful lift to lift the boat out of the water surface. Since there
is much smaller friction with the air than the body with water, so hovercraft can glide at high
speed on the water. And this state is called “low-friction” state.
2
In this paper, the toy CD hovercrafts have been built for the purpose of study the relevant
parameters which influence the time of the “low-friction” state of the CD hovercraft. Factors
that influence the time of the “low-friction” state such as the volume V of air contained in the
balloon, the weight of the loaded hovercraft, the size of the aperture of the sports cap, the
bottom area of the CD hovercraft and the roughness of the surface have been investigated
experimentally. Moreover, the Stokes equation on the basis of the lubrication theory [1,2]
enabling us to establish the time of the “low-friction” state applied on the toy hovercraft has
been used to illustrate our experimental results.
2. Experimental and theoretical method
2.1 Experimental method
Figure1 CD hovercraft made with a balloon, a CD, super glue and a sports cap.
Make a balloon hovercraft using the following household items: a balloon, a CD, super
glue and a sports cap from a water bottle (Figure 1). Place the tip of the balloon over the
sports cap. Glue the bottom of the cap onto the middle of the CD. Now twist open the sports
cap and blow into the balloon from the bottom of the CD. Once the balloon is inflated, twist
3
shut the sports cap. Now place the CD on a flat surface (such as the surface of the glass) and
release the sports cap. The CD will glide across the surface on a cushion of air, much like a
puck would in air hockey.
The total mass of this system is measured by a balance. The inner and outer radii R0 and
R1 of the CD need to be measured using a dividing rule. The air flow between the cylinder of
radius R0 in the central part and the cylinder of radius R1 can be considered as an air flow with
a cylindrical symmetry located in a thin film of height h.
The thickness h of the air film between the CD surface and the table is measured using
sheets of adhesive papers whose thicknesses are measured by a micrometer screw gauge,
successively placed on a sheet of paper put on the table. When the CD hovercraft is stopped
by the papers, we can consider that the thickness of the papers is slightly higher than h.
To measure the volume V of air contained in the balloon and the time t, we have applied
the following procedure that needs a chronometer, a flexible tape measure and two operators.
First, the spherical balloon is inflated, and its circumference C is measured using a flexible
tape measure. The circumference C of the balloon is utilized to calculate the volume of air V it
contains. The first operator places the hovercraft on the table and carefully opens the cap.
Simultaneously, the second operator starts a timer and stops when the balloon is empty to
measure the time t. The volumetric air flow is given by Q = V / t . Considering the
incompressible of the flow, we have Q = v0 S0 , where v0 is a mean air velocity value at the
radius R0 and S0 = 2π hR0 is the lateral surface of the cylinder of radius R0 and height h. The
same approach for the radius R1. For any value of the radius r, r being the radial coordinate,
the mean velocity is vr = Q / 2π hr , and a possible mean value U of the air velocity evaluated
over the total surface of the CD is given by
R1
Q
U π ( R12 − R02 ) =
∫R0 2π hr 2π rdr .
(1)
And that reduces to
U=
Q
.
π h( R0 + R1 )
(2)
This indicates that if the mean velocity does not change, the volumetric air flow Q is
4
proportional to h.
2.2 Theoretical method
Using the Navier–Stokes equation to solve for an incompressible fluid, we have
neglected the external forces such as the weight,


∂v
  

ρ + ρ (v ⋅∇)v = −∇P + µ∆v .
∂t
Where ρ is specific mass and µ is dynamic viscosity. For a permanent flow,
(3)

∂v 
= 0.
∂t
In the case of a system having two very different typical lengths R1 and h, with R1 >> h , the
reduced Reynolds number R* defined in [1] gives the ratio of inertia to viscous forces,
  
ρ (v ⋅∇)v
R* =
 .
µ∆v
(4)
Considering the typical length R1, the inertia force can be evaluated. While the viscous
force is defined by considering the typical length h. Using the averaged velocity U, we have,
where υ = µ / ρ is the kinematic viscosity,
ρU 2 / R1 UR1 h 2
=
( ) .
R* =
µU / h 2
υ R1
(5)
υ 1.5 ×10−5 m 2 / s , and the
In the case of air at ambient temperature and pressure, =
numerical value of the reduced Reynolds number R* is 0.72, tending to zero as h→0. We can
consider only the pressure terms and the viscous in the Navier–Stokes equation, which
reduces to the Stokes equation on the basis of lubrication theory [3].
The Stokes equation is

 
− ∇P + µ∆v = 0 .
and the continuity equation of an incompressible flow is
 
∇⋅v = 0.
(6)
(7)
Making use of the cylindrical coordinates (r ,ϕ , z ) , the velocity vector is



  
v =v r er +vϕ eϕ +v z k , where (er , eϕ , k ) is the local orthonormal basis of the cylindrical
5
coordinate system. In our case, it is clear that the velocity components vϕ and v z are null. The
radial component v r only depends on the variables r and z. Using the definition of the
Laplacian of a vector and the expression of the gradient in cylindrical coordinates [4],
equation (6) becomes
v 
∂P  1 ∂P  ∂P 
er +
eϕ +
k = µ (∆vr − r2 )er ,
∂r
∂z
r ∂ϕ
r
(8)
with
∆v=
r
∂ 2v
∂ 2 vr 1 ∂vr ∂ 2 vr
1 ∂ ∂vr
.
(r
) + 2r ⇒ ∆v=
+
+
r
r ∂r ∂r
∂z
∂r 2 r ∂r ∂z 2
(9)
From equation (7), we have
∂v
v
1 ∂ (rvr )
0⇒ r =
=
− r.
r ∂r
∂r
r
The operator
(10)
∂
applied to equation (10) gives
∂r
∂ 2 vr
2 ∂vr 2vr
=
−
=
.
2
∂r
r ∂r
r2
(11)
Taking the result given by equation (11), ∆vr (equation (9)) becomes
vr ∂ 2 vr
.
+
r 2 ∂z 2
∆vr =
(12)
Finally, equation (8) is reduced to
∂ 2v 
∂P  1 ∂P  ∂P 
µ 2r er .
er +
eϕ +
k=
∂r
r ∂ϕ
∂z
∂z
(13)
A quick analysis of equation (13) indicates that the pressure does not depend on the angle ϕ ,
and is constant over the direction z , i.e. on the thickness of the air film. Integration of the
radial component is straightforward:
=
vr
1 ∂P 2
z + Az + B
2 µ ∂r
(14)
with the integration constants A and B found with the limit conditions vr (z=0)=0 and
6
vr (z=h)=0. We obtain the parabolic shape
=
vr
1 ∂P 2
( z − hz ) .
2 µ ∂r
(15)
Thus, it is possible to compute the volumetric air flow Q ,
 
v
∫∫ r ⋅ dS .
=
Q
(16)

Where dS is the elementary lateral surface of a cylinder of radius r and height h. The
calculations give
Q= −
π r ∂P 3
h .
6 µ ∂r
(17)
Integrating the pressure gradient ∂P / ∂r and taking into account that the pressure P = PA
(the atmospheric pressure) for r = R1 , leads to the pressure distribution P ( r ) :
P ( r=
) PA +
6µQ
ln ( R1 / r ) .
π h3
(18)
We obtain the lifting force F by integrating the overpressure δ P=
( r ) P ( r ) − PA over all
the surface of the CD. Let us underline that the overpressure in the central area of radius R0 is
constant, δ P( R0 ) =
6µQ
ln ( R1 / R0 ) .We have
π h3
=
F
∫
R1
R0
δ P ( r ) 2π rdr + π R02δ P ( R0 ) .
(19)
Finally, we obtain the lifting force
F
=
3µ Q 2
R − R02 ) .
3 ( 1
h
(20)
Taking into account Q = V / t and Q= v0 ⋅ 2π R0 h , we finally obtain
t
=
1
F
⋅V .
24π µ v R S
2
3
0
3
0
(21)
Where=
S π ( R12 − R02 ) is the area of the CD surface.
3 Results and discussion
3.1 Study of volume V of air contained in the balloon
The control variable method is implemented to study the relationship between the time t
7
and volume V of air contained in the balloon. A small size of CD with R0 = 7.5mm and
R1 = 40mm was chosen in this part. The total mass of this system is ms −CD1 = 14.0 g . We
obtain h = 0.6mm by the method mentioned in part 2.1. In this experimental part, we have
measured the time t of the “low-friction” state with different volume V of air. It is noteworthy
that the sizes of the apertures of the sports caps are the same in each measurement by means
of drawing a mark. The experimental results are presented in Figure 2. It is clear that the time
t is proportional to the volume V. And this is consistent with our theoretical results (21). By
linear fitting of s-CD1, we have =
Q 6.83 ×10−4 m3 / s . Furthermore, taking into account
equation (20) the numerical evaluation of the force F with these experimental results and
=
µ 1.85 ×10−5 kgm −1s −1 is F = 0.270 N .This value is comparable to the weight of the CD
hovercraft (0.140N). It is obvious that the lifting force is of the same magnitude as the weight
of the system, and allows it to glide on a table. It also demonstrates that our theoretical
method is reasonable.
70
s-CD1
linear fit of s-CD1
60
t/s
50
40
30
20
0.035
0.040
0.045
0.050
0.055
0.060
0.065
3
V/m
Figure 2 Time t of the “low-friction” state with different volume V of air of a small size of CD hovercraft.
3.2 Study of weight of the loaded CD hovercraft
In order to study the weight of the loaded CD hovercraft, CD hovercrafts with one small
size of CD disc, two small size of CD discs and three small size of CD discs separately have
8
been made. They are donated as s-CD1, s-CD2 and s-CD3 respectively. Other factors are
unchanged. Then the time t of the “low-friction” state with different volume V of air of these
three CD hovercrafts is measured and the results are shown in Figure 3. From Figure 3, we
can see obviously that the time t is proportional to the volume V, which is the same with the
previous conclusion. Additionally, we can find that the hovercraft with more weight owns
more time t of the “low-friction” state at the constant volume V of air. For example, when the
volume V = 0.051m3 , the time t of s-CD1, s-CD2 and s-CD3 is 49.76s, 51.94s and 55.72s
respectively, the weights of them are 0.140N, 0.205N and 0.276N. This phenomenon is easy
to understand. When the hovercraft glides on the table, the lifting force F is equal to weight of
the loaded hovercraft mg. Since the size of the apertures of the sports caps are the same, the
mean velocities v0 are nearly unchanged. Thus taking into account the equation (21), we can
deduce that the time t of the “low-friction” state is proportional to the
F when the CD
hovercrafts possess the same v0 , S and V.
80
s-CD1
s-CD2
s-CD3
70
t/s
60
50
40
30
20
0.035
0.040
0.045
0.050
0.055
0.060
0.065
V/m3
Figure 3 Time t of the “low-friction” state with different volume V of air of the s-CD1, s-CD2 and s-CD3.
3.3 Study of size of the aperture of the sports cap
The CD hovercraft s-CD1 with V = 0.051m3 have been chosen for the study of the size
of the aperture of the sports cap. When the size of the aperture is gradually increased, the time
t of the “low-friction” state decreases rapidly (487.47s, 100.56s, 49.76s, 34.50s, 26.37s and
9
10.97s). The longest time t of the “low-friction” state reaches up to 487.47s. Similarly, this
phenomenon can be clarified with our theory above. As well known, the v0 increases with
the increase of the size of the aperture. As other factors unchanged, the time t of the
“low-friction” state is inversely proportional to the value of
v03 .
3.4 Study of bottom area of the CD hovercraft
To investigate the effects of the bottom area of the CD hovercraft, the toy CD hovercraft
have been made using large size of CD disc with R0 = 7.5mm and R1 = 60mm (donated as
l-CD1) for comparison. The total mass of this system is ml −CD1 = 22.8 g . The time t of the
“low-friction” state with altered volume V of air of l-CD1 is measured and the results are
shown in Figure 4. The sizes of the apertures of the sports caps are the same in each
measurement. In addition, the measurement results of s-CD2 are also submitted in Figure 4
for comparison. As is shown in Figure 4, we can see clearly that when the volume V values of
air are the same, the time ts-CD2 is larger than tl-CD1. This can also be explained by our theory.
As other factors unchanged, the time t of the “low-friction” state is proportional to the value
of
F / s . Since the F/S value of l-CD1 64.34 N / m 2 is smaller than that of
s-CD2 132.79 N / m 2 , the time tl-CD1 is smaller than ts-CD2.
80
l-CD1
s-CD2
70
t/s
60
50
40
30
20
0.035
0.040
0.045
0.050
V/m3
0.055
0.060
0.065
Figure 4 Time t of the “low-friction” state with different volume V of air of l-CD1 and s-CD2.
10
3.5 Study of the roughness of the surface
The influence of the roughness of the surface has also been taken into account. The time
t of the “low-friction” state of s-CD1, s-CD2, s-CD3, l-CD1 and l-CD2 have been measured
on glass and sandpaper with V = 0.051m3 . And the results are listed in Table 1. From Table 1,
we can find that the time t of the “low-friction” state of all the hovercraft measured on
sandpaper is shorter than that measured on glass. We also put the inflatable CD hovercraft on
sheet and carpet. Finally, the “low-friction” state was not appeared. These indicate that the
rough and breathable surface is not conducive to the formation of the “low-friction” state.
Table 1 Time t of the “low-friction” state with V = 0.051m3 on glass and sandpaper.
s-CD1
s-CD2
s-CD3
l-CD1
l-CD2
t on glass
49.76
51.94
55.72
45.54
50.12
t on sandpaper
43.50
46.50
47.98
40.23
46.06
4 Summary
In this paper, we have built CD hovercrafts to study the relevant parameters which
influence the time of the “low-friction” state of the CD hovercrafts. Our experimental results
demonstrate that increase of the volume of air contained in the balloon and the weight of the
loaded hovercraft as well as decrease of the size of the aperture of the sports cap and the
bottom area of the CD hovercraft is contribute to the enhancement of the time of the
“low-friction” state of the CD hovercrafts. While the rough and breathable surface is not of
benefit to the formation of the “low-friction” state. Moreover, the equation of the time of the
“low-friction” state deduced from Navier-Stokes equation in the case of incompressible flows
has been used to explain our experimental results. This paper contributes to the understanding
of the principle of the hovercrafts.
11
References
[1] H. Schlichting, Boundary-layer theory (McGraw-Hill, 1968).
[2] C. d. Izarra and G. d. Izarra, European Journal of Physics 32 (2011) 89.
[3] M. Sen, Applied Mathematical Modelling 17 (1993) 226.
[4] J. Huba, N. P. Formulary, (2009) 94-265.
12