Scaling Law for Exper i ment of Under w ater Explosion
with Sever a l Independent Geometr i cal Scales
ZHA NG Xiao- c i
(China S hip S cientific Rearch Center, Wuxi 214082, China)
Ab st r act: Within the limits of experimental conditions,where the law of similarity with one and only geometrical scale for experiment of underwater explosion will not be suitable,veral independent geometrical scales have to be lected in the experimental design. From the point of view of energy, this paper designed the scaling law between the prototype and the model to aim at the dynamic re- spon experiment of underwater explosion. The measurements for two kinds of sho ck environments,
<, the first sho ck wave and the cond pressure wave of bubble pulsing, can be obtained together.
Key wor ds: und er water explosion; shock envir onment; scaling law; geom e t r ical scale
CLC n um be r : O382.4Document code: A
The ex plosiv e pressure in an ex periment is only a very little percentag e of that in deep ocean,and hence it is necessary to inquire that if the measurements can be adopted to forecast the practical situation.The whole non- similarity will result in a scale effect obviously, and the whole similarity will lead to a full- scale test.Each ex perimental subject has own law of simi- larity. Within the limits of ex perimental conditions, the scaling law, w hich requirem ents are w eak, m ay be ud as a substitute f or the law of similarity in the practical operation. It is pos- sible that the law of similarity will hav e to be given up and a scaling law will be usable f or the underw ater ex plosive test in a w ater pond.
1 Conceptive differ ence between scaling law and law of similar ity梦见排队打饭
According to the law of similarity of ex perimental science, the dimensionless similar co- efficients are deriv ed by the dimensional analy sis and/or Пtheorem.The dimensional analysis is a m athem atical method.The m echanical quantities are all ex presd by three primary quan- tities (leng th, time and mass). The principle of dimensional analysis is that each tern in any mechanical equation must contain the sam e pow ers of the three primary quantities,respectiv e-ly.Buckingham’s Пtheorem states that if there are n v ariables in a phy sical problem,and the v ariables contain m primary dim ensions (f or a mechanical problem, m=3), then the equation re- lating the variables will contain (n- m) dimensionless g roups, which can be found by using the dimensionless m ethod.
The corresponding relationship of ex perimental design between the m odel and the proto-
Received date: 2008- 04- 21
Biog raphy: ZHANG Xiao- ci(1945- ), male, rearcher of CSS RC.
ty pe will be the simplest, only if there are g eometrical, kinem atical and dynamical similarities betw een the model and the prototy pe, respectively.
When the ratio of corresponding lengths,widths and heights betw een the m odel and the prototype is constant, the tw o sy stems are geom etrically similar.
When the ratios of corresponding times,v elocities and accelerations betw een the model and the prototy pe are constant respectively, the tw o sy stems are kinematically similar.
When the ratios of corresponding static f orces and dynamic forces acting on corresponding elem ents are constant respectively, the tw o sy stems are dynamically similar.
In g eneral, the f ormer similarity is the basis of the latter tw o similarities.
The derivation of law of similarity bas upon the one and only g eom etrical scale in the whole mechanical field. The similar param eters m ay conflict with each other, and hence - lecting the dimensionless similar param eters wholly/partially may be result in doing full - scale ex periments. Therefore, for a practical problem of hy drodynamics, v eral ev en one simi- lar param eter are chon in test design. The rearch of law of similarity for underwater ex plo- sion has been carried out [1].
Under the real ex perimental conditions,only using a g eometrical scale in the whole field may be im possible,or the practical similar param eters are not derived by the standard similar theorem. In the cas, the scaling law between the model and the prototy pe has to be derived by using some em pirical f ormulas or a lot of measurements.The scaling law of anti- shock test for submarine’s equipm ents had been reporte d[6,7].
This paper aims at the scaling law,which is suitable f or test design of underw ater ex plo- sion, and tries to solve the f ollowing tw o knotty problem s.
(1) Under conditions of ex periment, there are v eral g eometrical scales in the whole me- chanical field,and this geometrical distortion of whole field results in the non- similarity g eo- metrically in the field.
(2) Only doing one test of underw ater ex plosion can obtain the data of dynamic respons in the same time f or the shock w ave and the cond pressure pulsing w ave.
The strict law of similarity for underwater ex plosion describes that it is impossible to give consideration to the similarities of phenom ena both the shock w av e and bubble pulsing in one and only test design [1]. From the point of view of energy, this paper designed the scaling law betw een the prototy pe and the model to aim at obtaining the dynamic responsive data of tw o phenomena in one test.
In the ex periments of underw ater ex plosion,the m aterial of m odel is g enerally the same as that of prototy pe,and hence the derivation in this paper is bad on the same material. But, the m ethod of this paper can easily open up the ca of using different m aterials[1].
2 Envir onmental simulation of shock wave
The action of shock w ave is considered as being situated in non- gravitational field and its
dynamic respon should embody in middle- high frequency band.
In the outdoor w ater pond of the China Ship Scientific Rearch Center,a model ex peri- ment (5∶1 s
cale ratio of some ship) w as carried out in the shock environment of middle fre- quency by v eral independent geom etrical scales [2].
If the law of similarity with one and only g eometrical scale was lected,corresponding to the 1 000kg TNT w eight, which the ship may be f aced with, then 8kg TNT w eight must be u- tilized in the w ater pond. But, is is dang erous in the w ater pond. Another larger scaling factor f or TNT weight must be ud. So, in the mechanical field to be studied, the two geometric scales, w hich belong to tw o solids (ship model and TNT charge) respectiv ely, are chon each other independently. But the g eom etrical scale in w ater region is not chon arbitrarily ag ain, and must be determined by using ex perimental relationship of energy.
For ex am ple,the g eometrical scale of model is 0.2,and the g eometrical scale of T NT charg e is 0.171 (corresponding to 5kg TNT w eight), then the derived geometrical scale in w a- ter region is 0.158.
There are three different g eom etrical scales,and hence the g eometry of w hole mechanical field is distorted.
In m aking a model of a riv er or open channel,it may be necessary to distort the model by adopting
different v ertical and horizontal scales of leng th.Riv er flow is norm ally turbulent, but when the scale reduction for a model is larg e the value of Reynolds num bers in the m odel may be too low to ensure turbulent flow or, alternatively, there may not be sufficient operating head to allow the m odel to function.
This can be ov ercome by distorting the ,using different v ertical and horiz ontal scales to increa v ertical pressure head. As en, the method of distorting the whole mechan- ical field w as ud in designing tests.
[1]
In the gravitational field, !v =1 (the speed scale of rigid body) .
From
kl s="=E #
where k- structural stiffness,l- characteristic length of structure, s- area,"- structural stress, E- elastic m oment of structure and #- structural strain.
Time scale
3
!m
!m $ !m l !m $
k
!
!E !# !m l ! !
!E !# !t =
= = !m l ! where !m - m ass scale, !k - stiffness scale, !m $ - density scale, !m l - g eometrical scale, !E - elas-
tic mom ent scale, and !# - strain respon scale of ship m odel, respectively .
In order to meet
!m l
!v = ! =1, then
t
1/2 1/2
!m l !E !" =1, i.e.
! ! =! 1/2
!m l !m #
E " m
# The speed respon scale of ship model !m d !" !
m l !m v = = =!"
! ! t t
where !m d - displacem ent scale of ship model.
The ex perimental energy relationship in the environment of shock wav e is 2
!e =!sh !m s =!mm !m v
where !e - energy scale of ship m odel, !sh - energy density scale of shock load, !m s - area scale 2
3
of ship model (!m s =!m l ) and !mm - mass scale of ship model (!mm =!m l ).
Let !Rl be the geom etrical scale of ex plosiv e distance, due to the ex pression
2.03
2.03
!sh =!wm ! 1/3
" l !! "
! !
wm
1/3 wl
=! w
!Rl
Rl
(where !wl - g eometrical scale of TNT charg e and !wm - mass scale of TNT charge), then
2.03
1/2.03
! "
"
!Rl "
!
!wl
! wl Rl
Hence, the geom etrical scale in w ater region
- 1/2.03
!Rl =!wl
!! !" "
!m l 2
1.49 - 0.99
0.49
!m l
=! l !"
w wl
and the scale of peak pressure of shock w ave
1.13
1.13/
2.03
0.56
!p =
! " = ! "
" =
! "
!wl !m l !m l 2
1.11
"
! ! !Rl桂南会战
!wl !wl
max According to
!"=!p = !!
0.56 "
!m l
1.11
! "
wl
then
0.56
乒乓球横拍5.09
!" =
!! " , !"=
!! " !wl !wl
0.11
m l
m l
- 0.99×5.09
5.04
托福教育部
1.49
!!
" 1.49
!wl
!!
"
4.55
!m l
!wl !m l !wl !Rl =
0.49曹文轩的作品
= = 0.49
!
m l 3 .55 !wl wl 5.09×1.11 0.56
0.56 5.65
5.09
!p = ! " !! " = !! " !! " =
!! "高跟鞋搭配
!m l
!wl
!m l !wl !wl
!wl
max
m l
wl
m l
m l
As en, if !wl =!m l then !Rl =!m l , !"=1 and !p m a x =1. The scaling relationship is in agreement
with the similarity relationship with one and only geom etrical scale.In particular, we point out that just under the condition of one and only g eometrical scale,the scale of absolute stretch is equal to that of characteristic leng th, i.e., !" =1, and the scale of translational rigid- speed is equal to the scale of speed respon of body .
2.03
5.09×2
10.08
!wl
!
sh =!wl
!! " !
" "=! ! "
!wl
!m l !wl
2
2
=! ! ! =! = wl
m l " m l ! ! 9.18 !m l
Rl
wl m l
The scale of specific impul
0.89/2.03
& 0.89
0.89/2.03
5.09×2
!i =!wm
! w m =!wl
! 1/3
" " #! ! ! ! 2 ! 1/3
m l m l wl
=!wl $ 5.09
总胆红素偏高的原因×2 ’ " !wl
! !Rl
! % wl ( m l
9.18×0.89/2.03
4.02
5.02
!wl
=!wl
!! " =!wl
!! "
!wl
!
wl
= 4.02 !m l
m l
m l
The scale of shock f actor SF
3/2 3/2 3.55
!wl !wl 5.05
!wl
!wl
!SF = ! = = 4.55 !m l 4.55
!m l
Rl !m #
) E "
!
!v =1, !E !" =!m # " !t =!m l
=!m l , ! !
Hence, when choosing the same material to make model, in order to !" =!m # , adding /reducing counterw eights are necessary.
The acceleration respon scale
5.09
!m v !" !wl
!m a = = ! = 6.09
, and ! !
m l t m l 5.09
!wl
!m d =!" !m l = 4.09
!m l
3 Envir onment simulation of the cond pr essur e wave
The action of the cond pressure w ave due to bubble pulsing is situated at the gravita- tional field, and its dynamic respon should embody in low frequency band.
Neglecting the bubble energy after the first pulsing, the bubble energy is ex presd as
4! *4 + p 3 r m ax 0
3
where r m ax - the m aximum radius of bubble at the first pulsing [m ], and p 0- hy drostatic pressure
0.5 - 0.5
at the ex plosive depth [Pa]. Becau r m ax =0.546 6 p 0 #w T b (where #w - w ater density [kg/m ]
3
召有司案图翻译0.5 - 1.5 3
and T b - period of bubble pulsing [s]), the energy of bubble is equal to 0.684 p 0 #w T b . This
