JPRS ID: 9326 TRANSLATION WORLD OCEAN EXPLORATION AND ENGINEERING PROBLEMS ED. BY A.L. VOZNESENSKIY

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JPRS L/9643
- 3 Aprii 1981
= Translation
- PENETl~ATION
_ Cpenetration of Compressible
Cor~tinuous Media by Solid S~tates)
ay
A.Y. Sagomanyan
_ +
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~
JPRS L/9643
- 3 April 1981
' PE~JETRATION
- ~PENETRATiON OF COMPR~SSIBLE
CONTiNUOUS MEDIA BY SO~ID STATE5)
Moscow PRONIKANIYE (PRONIKANIYE TVEFcDYKN TEL V BZHIMAYEMYYE SPLOSHNYYE
SREDY) in Russian 1974 signed ~.o press 3 Dec 74 pn 1-164
~
jTrans~ation o~ Cfi~pter 1 from the book by A.Y. Sago~onyan, Izdatel'stvo
M~skovskogo Universitata, 3360 copies, 299 page~, UDC 534.26+539.374]
CONTENTS
Annotation 1
Introduction 2
Chapter 1. Penetration of an Ideal Liquid by Solid ~tates 1~
' 1. Equations of Seif-Similar Motion of a I,iquid 6
, Penetration of an Ideal Ir.compressible Liquid by Solid States....... 16
2. Penetration of an Tnccmpressible Liquid by a Wed~e 16
3. Penetration of an Incomprenssible I,iquid by a Cone 23
!t. Movement of a Th~n C~ne in a I,iquid of F`inite Depth 32
5. Ricochet of a Pl.ate From the Surface of an Ideal Incompressible
I,iquid l~l
Penetration of a Compressible Liquid by Thin Rodies l~9
6. Statement of the Froblem in ~quations of Motion ?~9
7. Vertical Submersion of the Thin Solid of Revolution Without
an Angle of Attack. Penetration by a Cone 51
~ a [I - USSR - L FOUO]
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' 8. Inclined Penetration by a Thin Body at, an Angle of Attack.
Inclin2d Penetration by Cone
9. Penetration of a Compressible Liquid by Thin Flat Bodies.
Penetration by a Wed~e b6
Penetration of a Compressible I,iquid by Blunt Bodies 7~
10. Penetration of a Compressible I,iquid by a Slightly
Distorted Outline 7~
X1. Penatration of a Compressible Liquid by a Blunt Wedge 75
12. Penetration of s Comgressible Liquid by a ~'l.at Plate 85 -
13. Conslderatior~ of the Lift of the Free Surface on Penetration
of a Compressible I,iquid by a Blunt Wedge ..........a,..... 92
~ 11~. Penetration of a Compre,~sible Liquid by a Blunt Wedge ,
in the Nonlinear Statement 97
- 15. Penetration of a Compressible Liquid. by Blunt Three- _
. I}imensional Bodies ...................................e...~ 103
16. Penetrat~on of a Compressible Liquid by a Blunt Coi~~
Considering the Lift of the F`ree Surface 108
~ 17. Penetration of a Compressible I,iquid by a Blunt ~one fcr
V > a .....o 121 .
18. Nonlinear Problem of Penetration ~f a Compressible Liquiri _
by a Blunt Cone 126
a "Ly. Imp~~: L oi' Ri~icl Cylinder With the Surface of a -
- Compc�essible Liquid 131 -
20. Impact of a Cylindrical Elastic Shell With a Liquid Fili_er
_ Against the Surface of a Compressible Liquid 137
21. Estimating the Effect of Viscosity on Penetration of a
- Liquid by Solid States .........................o...�..���� 148
Original Table of Contents 159
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ANNOTATION
[Text] Thia book discusaes the problems of the penetration of liquid and soils
by solid states and also penetration by eolid statas having high relative veloci-
ties on ~mpact. The penetrating bodies are asswned to be abeolutely solid, elas-
tic and in the form of elastic shella containing a liquid. When inveatigating the
_ problems of the penetration of soils by solid states, a model p,iastic compressible
continuous medium ia introduced, on the basis of which detailed solutiona of urgent
modern problema defining all of the dynamic parameters of the movement of the soil
and the penetrating body are presented. In addition, the solutiona of penetration
problems on impact of a solid deformable bodq of small dimensions with a deform-
able body of large dimensions are discussed. Various models fox analytical investi-
tat3on of the penetration problem are investigated as a function of the magnitude of
the reZative velocity at which the bodiea meet.
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r Vl~ vi� a.LVia'w u.Ja+ vLVLt .
N
INTRODUCTION
In thie book the results are preaented from studies of the modern problema of pene-
tration of a.llquid, soils and metallic obstacles by a solid ecate. The study of the
problems connected with tha penetration of various continuous.media by solid .
stat~sbegan long ago. However, until recently these atudies pertained to a limited
group of problems and baeically were of an empirical nature. They led to several
useful formulas which, however, did not give a repreaentation of the dynamics of the
_ penetration proceas itself. This is explained by the fact that for a long time the '
penetration problem was of interest for a narrow group of researchers, .snd, above
all, there were eignificant mathematical diff iculties in the theoretical study of '
the sub3ect, indeterminacy of the mechanical properties of the media, and absence of -
reliable measuring devices in the experiments.
In connection with the occurence of new technical problems in various branches of
modern practice in recent yeare the iuterest of a broad group of acientiats and
engineers in the problems connected with the penetration of various media by
soliu states has increased significantly. In addition, new posaibilitiea have ~ ,
come up which are promoting progress in the study of the penetration problem. These ,
fnclude the following: the availability of high-speed computera which ahorte~ the _
numerical calculation procedure, achievements in the deve].opment of the general
methods of studying the motion of continuoue media promoting the analytical investi-
- gation of the problem, achievements in improving the measuring equipment permitting
the reliability of the experimental results to be increased. -
This book takes up the analytical solution of penetration problems. It consists of
three chapters. The first chapter ia primarily on the study of the problems of the '
penetration of a compressible liquid by solid states. Among the results of the
' solutions of the problems of penetration of bodies into an incompreseible liquid
presented in thie chapter which are of independent intereat, many are uaed for
logical relations and comparison with the resulta of the solutions of these problems
considering the compressibility of the liquid. If the penetr~tion rate is high '
or the penetration of a sufficiently blunt body is considered, then the compressi- ~
bility of the liquid muat be considered to obtain reliable results~ These problems '
are of interest for entry of rockets and misailes into water, landing of spacecraft '
and seaplanes and other practical problems. It appears to us that this chapter con-
tains analytical solutions of the most intereating problems of the penetration of
~ a compressible liquid by solid states ~at the pre$ent time.
- On the basis of the plastic compresaible continuous medium simulating manv types of
soils, a method has been developed to atudy the penetration of doils by solid
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state~ permitting eubstantial expaneion of the claes of eolved problems. The
studies of the problems of p~netratioa of so~ls by aolid states are made in the
second chapter. The aolutiene obtaiaed here make it poseible to determine al' of
the dynamic parameters of mavement during the process nf penetration of the body
inLo the soil. In addition to acientific izitereste, the reaulte of the studiee tn
this chapter will be useful to engineer~ involved with the problems of impact and
penetration of bodise into soils.
In the last, third, chapter of the book the phenomenon of penetracion is conaidered
aa the result of colliaion of a metal hamm~rhead with a metal obstacle under super-
sonic relativ~ velocitp conditions. The urgency of the problem of the interaction
of aQlid states meeting at vesy high relative velocities is obvious. In this chag-
ter a discussian is presented of the exisring approximate analytical methods of
- sol+ution. As a rule, they are contained in journal articles. In all of the
chaptera where it is poasible, the reaults of the theoretical studies are compared
~with the resulta of experimental measurements. ~
= The first chap ter of the book was prepared jointly with I. P. Rhlyzov, the second
chapter, jointly with M. T~. Gartashteya and V. T. Noskov. V. I. Ploskov found the
_ aolutions to the problema of items 7, 9, 10 and 11 in Chapter 2. Beaides these
comrades, the following people assieted in preparing the manuscript: 0. N. Goman,
~ V. A. Yeroshin, A. N. Mar'yamov, V. V. Paruchikov. I should like to expreas my
appreciation and gratitude to all of the mentioned comradea. The author will be
grateful to everyone who wiehes to eend comments and suggeatione.
~ _ ~
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G'HAPTER 1. PENETRATION OF A,v I~~AL LIQUID BY SOLIil STATES
Thia chapter takes up the prablems of the penetration of soli~l states into an iueal ~
liquid. Thes~e problema are of interest with regard to the entry of missiles and I
rockets into water, wat~r landings of apacecraft, seaplanes, aad so on. Until com-
paratively recently the etudy of the penetration phenomenon was carried out on a ~
_ model of an incompresaible ideal liquid. It is poesible to familiarize oneaelf with '
the basic results of theae studies in the monographs [1, 2, 3].
- i
If the speed of the penetrating body is high or the head of the body is sufficiently
bluat, then in order to obtain reliable reaults it ie neceseary to coiasider the
compreasibilitp and wave nature of movement of a liquid.
In this chapter basically the re~ults are presented from studies of the problema of
peaetration of aolid states into initial],y atill compresaible ideal liquid occupying
the lower halfspace. xhe atudies are made considering the eff ecC of the movsment ~
of air over the surface of the liquid during the procesaes of apprc,aching the water !
and penetration of the body. In all problema the initial period of penetration pre- i
ceding the appearance of a cavity is coneidereduwhere the penetrating body has ;
beea incompletely submarged in the liquid. During this time period, the presaures !
acting on the nenetrating body reach maximum valuea by which the dyaamic str.ength ~
calculation is made. '
. i
Many iteme in thia chapter are devoted to the atudy of self-similar problems of '
peaetration iii the linear and nonlinear etatement. Therefore at the beginaing of the j
chapter the derivation of the basic equationa of aelf-similar motion of a compree- ~
aible liquid and the characterietics of these equations are preaented. Then comes ~
a secti,on on the probleme of penetratioa of en incompreasible liquid by solid -
- �tA~~s,. Many of the solutions of the problema of this section which have indepeadent '
significance are compared with the solutiona of analogous problems of penetration of a
~compressible liquid by solid s'tates. ~~e aection containe~an original solution to t~he prob-
lement of ricochet of a plate from the eurface of an incompreaeible liquid aad in- i
vestigation of the motion of a thin bodq from depth in the direction o� the free
surface. The laet proble~ does not directly pertain to the clase of pen~tration !
probleme, but it is close to them with respect to mathematical atatement and method -
of solution. ~
The problema of penetration of eolid statee into a compreesible liquid are ~
the sub~ects of two sectione of the first chapter. In the firat of them the prob- !
: of the p~netration of thin axisymmetric snd flat bodies are investigated. ~
Subsonic and superaonic penetration and also penetration at a rate equal to the ~
speed of sound in the liquid are investigated. ~
4 '
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The etudies of this section demonetrate that up io high aubsonic penetration
- velocities of ttiin sharp bodies, the effect of compreesibility on the resistance 1s
inaignif icant. In the next section on pen~tration of a compreasible liquid, blunt
- solid states are considered. T:Ze correct deter~ination of the reaistance to pene-
- tYAtion of blunt bodies, as a rule, requires conaideration of the compresaibility
of the liquid. For example, when a Gylindrical body with a flat front tip hits the
stirface of an incon~pressible liquid with any initial velocity, an inetantaneous
- change in momentum of the liquid by a finite amoun* takea placer The force acting
on tt~e body at the time of impact ia of a pulse nature in this case.
In reality, the disturbances in a medium are propagated with f ini*_e velocity, as a
result of which the change in momentum of the liquid and velocity of the penetrat-
_ ing body i.s continuous, and the force acting on the body at the time of beginning of
penetration is finite. When the diaturbance wavea travel far from the body, the
disturbances in the medium asymptotically approach the disturbances at the corre-� -
spondi.ng point in time arising after impact on the surface of an incompreasible
liquid. Thus, the theory of an incompressible liqutd replaces ~he eff ect of the
finite force in the initial period of penetration by a pulse force which can be
determined by the integral characteristic the loea of momentum at the time of
impact. Consideration of the compresaibility of the liquid permita determination of
the f inite pressur.es and force acting ~n the body in the initial penetration period.
On impact penetration of a compressible liquid by blunt bodies with arbitrary sub-
sonic velocity, the displacement rate of the generatrix of the body with respect to
the free surface can turn out to be cloae to and greater than the apeed of sound in a
liquid. In this case it is necessary to consider the compressibility of the medium.
The terms used above "impact agaix~st the aurface of tha liquid�' and "impact entry" -
are also encouatered when discussing the aolution of varioua problems in the firat
chapter. They are frequently used by many authora obviously to emphasize the aharp -
nature of the change in the parametera of motion during the initial penetration -
period. Howev~r, chese terms do not introduce any vaguenesL into the definition of
penetration as the process of submersion of the body in the liqufd through its free
surface.
Al1 of the cases of penetration of a liquid by solid states investigated in this
chapter are formulated as mathematical problems with apecif ic initial and boundary
conditions. In view of the extraordinary complexity of the analytical investigttii~n
~ of the essentially nonatationary problem of penetration into a compresaible i~iquid,
the penetrating bodiea investigated in thia chapter have a simple geometric shape.
This explains the small number of penetration problems in which deformation of the ~
penetrating body and the wave nature of its streased state are conaidered. `
In the required cases, the atudy is performed conaidering the effect of the lift of
the free s�urface of the cumpressible liquid on the dynamic penetration process.
~ The studies of the penetration int~ a compresaible liquid in the first chapter, as
a rule, are performed on the basis of the linearized equations of motion. =
- For a weakly compressible liquid such as water, when investigating a broad class of -
penetration problems, this approximation ia entirely ~ustif ied. I~Iowever, in modern
practice there are cases where a blunt solid state meets a free liquid surface with
high relative velocity. In such problems, during the initial period of penetration,
a shock wave will be propagated from the poin*_ of impact into the depths of the
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.v_. vir,~L
liquid after which the movement will be described by noniinear equatior~s. This
difficult problem has not as yet been iuvestigated.
In this chapter the problems of penetration of a compressible liquid by a blunt
wedge and a blunt cone are investigated in the nonlinear statement. tt is proposed
that tne edge of the wedge (and the generatrix of the cone, respectively) is -
shifted along the free surface of the liquid at supersonic velocity. The study of
th~ae probleme ia based on the analytical apparatua presented in the f irst itein. _
The results obtained here are of definite practical interest, and the statement of
the problems themaelves can attract the attention of researc'~ers to this difficult,
but proapective modern problem. This chapter also contains a section on the study
of the effect of viscosity on the penetration process. ~
~ 1. Equations of Self-Similar Motion of a Liquid
Let the adiabatic motion ~f a nonviscous compressible liquid with axial symmetry be _
considered in the abser_ce of mass forces. Let us direc~ the Ox axis of the station-
ary orthogonal coordinates along the axis of symmetry, and let us place the Oy axis
in the plane of the meridian. In these coordinates the equations of motion, con-
- tinuity and coi:g~~vation of entropy of a particle are written as follows: -
dUX dLx aUy _ ~ dl'
~r + U` va -~'y ~y a v., ,
d'U (JU C7V y I ~D
X ~ vx _.IC ; v
- dt dx Y~y P dy
`P -r p ~X - ~ v,~ ~ p~ ~ 0, -
ut { dx dy ) dx dy ~
at ur ~S -r vy ~y = ~1.1)
- where v, v are the velocity components ot the particles, p is the density of the _
x y
= liquid, p is the pressure, S is the entropy, t is time. ~iereaf ter it is propose~
- that the speed, pressure and all other parameters of motion are uniform 0-order
functions with respect to x, y and the time t. This means that the indicated pa-
rameters will be functions of the ratios
~ = zlr~ ~1 = ylt.
Tlie currents having this property wi11 be called self-similar. Usually by s~lf- _
siml.l~r we mean the motion of a liquid, ttie parametera oF which are functiona of -
tlie ratioe
x/ta, ylt�`, -
where a ie a constant.
Let us transform the equation (1.1), proceeding to the new coordinates t1� Using
the obvious expressions
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a ,r a y a, a_ i a a_ i a
2i i~ ' a;, - r= ' an ' ax ab ' ay t d,~ .
it is easy to find the form of the equations (1.1) in the new vflriables~
(U,~ - o~ ~ ~vy - ~l ) _ - p ~
. ~pX - ~ ~py - _ - P d~
(v: - o~ = (Ub - ~1) ~ P ( -I- ~ ) P�y =
~
, (vz - a~ ~ (vy- a~ = o. 2)
In the investigated case t1~e vorticity w(x, y, t) can be represented in the form
~~X, y~ t) _ ~ W~~, ~1), (l. 3)
where
W _ - i.uX
o~ ~ ' (1.4)
For convenience of further calculations let us introduce the notation:
U=v,~-~, V=vy-~I, W2=J2-~V~.
In this notation equations (1.2) and (1.4) are written as follows:
U -i- U 1- V ~ _ - ~ � ~S ,
V -f- U o~ -'r- V ~ _ - ~ � ,
_ d ( ) ; d (pV) + pV 3~~ _ 0~
aE ' ~n n f 15)
ll a~ V a~ =
~v _ cu =
og an
Let us introduce the expressi.on for the "vorticity" w(~, n) in the plane n) with
respect to formula (1.4) into the first two equations of system (1.5). As a result,
- we obtain
a tv~ t vo
U - V~ o~ ~ 2 ) _ - P ' ~
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~
V Uc~ -i- d~ ( ~ 1 = - P � .
~ /
- Multiplying the f irst of these relations by the differ ential d~,.the second by dt1
~ and adding the expressions obtained, wP have .
~ (Ud~ - i~d~) Ud~ Vd~ t d ~d `~p = ~ � (1. 6)
This relation along the "current line" defined by the equation
. ~ _ d~ (1.7) ~
U V -
assumes the ~orm
d ~ Ud~ Vd~ ~ P _ 0' (18) =
The next to the last equation of system (1.5) indicates that along the line ~1.7)
the entropy S is constant, and the pressure differential is represented by the
~
f ormula
dP = a'dP~
~
where a is the speed of sound in the liquid.
It is easy to check that the " current line" (1.7) is a characteristic of the system
of equatfons (1.5j with the relation (1.8) along it. The other characteristics of
the system (1.5) can be obtained, for example, from investigating the Cauchy prob-
lem. By excluding tr.e pressure, density and entropy from tY~e first four equations
of system (1.5) and the obvious relations
. aP _ 2 zP ap as a,~ z~P vP as
o~ ~ Cs ' a~ ' a,~ - Q a,~ r as ' a,~
we come to the following system of two equations:
- L'') ~ - ~l Y ~ (
~ 'r J~ I ~Q' ~l ~l ~ ' i'
I
3as - W2 = 0, _
dV _ dil _ cu.
d'~ (1.9)
- This syetem can be reduced to one equation of the type
~a' - ~~t - ?UV (az - y2) a~ _ ~UV - �2~ - Wz - 3az.
- Gb 0~ G~ ~ ~I.ZO~
8 -
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In the Cauchy prob~~m along the line L in the plane n) values of the desired
�unctions of the syatem of equations are given. Then in our case for determination
of the derivatives of U and V with respect to ~ and 11 along the line L, in addition
to the system (1.9) we shall have two expressions -
dU = ~ d~ a dr~,
(1.11)
_ dv _ dV ds oV d,~.
~db G~ _
~ Let us multiply the f irst of these expressions by the factor a and add it to the
second. As a result, by using the second equation of the system (1.9) we ~htain
- ~ "U - ~~dt~ -i- d5) ~U ~ d,n oV - dV ~ ~dU - ~d~. (1.12)
_ d~ dr~ dr~ _
As is k*~own, the characteristi~ expresaions are obtained f rom the condition of i~-
possibi~ity of a unique determination of the derivatives from the system (1.9) and
(1.11). This means that a linear relation must exist between the coeff icients of
the equationa (1.10) and (1.12), which leada to the equalitiea
Adb _ i~d~ d~ _ dn = dV +~1I - u~dE
az - C'= 2UV a~ - V' ~(IV - a~V -F l~t - 3~2 ~1.13)
~ -
From the equalities (1.13) two systems of characteristics are obtained in the plane
n), and two conditions along them, respectively:
d~ ~ -UV ta ~IG'-a' (1.14)
\ Q~a / - ~''2 4' - u~ '
- a ~ r a1V _ ~2 + 3Qa 1 = -
dU -T- ~iz.t ~ -I- ~ ~ ~
a Ws - a~
~
_ ~W a=-fl~
(1.15)
Ttie equation of the third char~cterietic of the system (1.5) and the corresponding
~ condition along it, according to the above-presented f acts, are written in the form
\ ~ /9 - ~3 = U ~
(1.16)
= d 2~ Ud~ Vdr~ -i- `~P = 0, dp = azdp.
P (1.17) -
From (1.14) and (1.15) it follows that the first two systemsof characteristics wi11 be
real under the condition
(v~ - ~)2 -1- ~U~ - ~I)' ~ az. (1.18)
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Using equation (1.6) wh~.ch is valid in any direction it is easy to obtain the follow-
ing expressions along the characteristics of the first and second families of ni.2'
respectively:
~d~ = f ~ W~ - a� d~ n' -(1~ d W' dP
a a(aV ~ UYW= - n') L ' P~ (1.19)
_ If we substitute this expression for the vorticity W i.n the right hand side of the
:.onditions (1.15), after some transformations along the characteristics ~1.2, we _
- obtain
(U V~t.2) ~ v dP
VdU - UdV f 21 t~Mz - 1-= 0,
y~M~ - 1 Tl / p (1. 20)
where
M = .
0
ln the case of plane-parallel f low the equatione of the characteristics (1.14) in
the plane n) retain their form, and instead of expressions (1.15) and (1.20), _
- we have ! -
_ aYW'-n~ I -
dU = ~s.i dV -I- a4 j2a' - ~z~ _ ~ c,~ a~ _ ~ d~,
' VdU - UdV � W~- vr~ ~,s~ d~ ~ V.M2 - 1 d~' = 0. -
,~'M'-1 P
The equation of the third characteristic (1.16) and the conditions along it (1.17) I
also retain their form in this case.
Hereaf ter, when solving the problems, the expression f or the displac~ement rate of
the brea~Cdown surface in the self-similar motion is needed. If Fl(x, y, t) is the
breakdown surface, then, as is known [4], the displacement rate D of this surface -
is expressed by the formula
D _ - aFi; ar
(dF~ldx)' -r (~F,l~d)' ~1.21)
In the case of self-similar motion the velocity D is a uniform function of zero or- _
der and, consequently, the equation of the breakdown surf acs will be a uniform
first-order f unction ~
Fl (X. y, t) = tF ~I)� (1.22) -
Using (1.22), formula (1.21) assumes thz form _
D_-�- G= (1.23) ;
c ~g ' G ~ ~~aE~s~~~~'
_ ~
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Obviously, the expressions
1 aF t aF _
G ~ c~ ~ G ~ ~
- are directional cosinea of the normal to the breakdown surface with the coordinate
axes 0~, 011 � -
The expresaions (1.20), (1.17) along the characteriatics (1.14), (1.16) togetU~er
with the b4undary cunditiona and the conditions on the breakdown surface, including
formula (1.23) permit "step-by-step" determination of the current field and the
sections of the breakdown surface in the regions where the condition (1.18) is
satisfied. The value of the "vorticity" u is determined by the formula (1.6), and
al.ong the c~iaracteriatics t~i.2, by the formulae (1.19) . Of course, for the calcula-
tions relations (1.20) can be replac~d by the expressions (1.15) equivalent to them.
The pracedure for determining the parameters using the presented characteristics is
analogous to the procedure for determining the pa-rameters in a supersonic ateady
state. eddy movement. Therefore we shall not describe the method of performing the
required ~pergtions for calculating these parameters. -
Let the flow be irrotational (w ~ 0). From the def inition itself, it follows that
for self-similar currents the velocity potentials ~(x, y, t) can be represented in
the f orm -
, ~ ~z, y~ t) = t~ '1)� (1. 24)
- Let us introduce the auxiliary function n) into the investigation;
' ~ ~I) _ ~ ~1) - 2 '~a).
It is easy to check that ~
v; o~ , vy = ~ , U = o~ ~ V = d~ .
According to (1.lOj, the function ~ satisfies the equation
(a2 - Ua) a~m - 2UV �y~ (a' - V Z) d ~ , "iV 3az - W 2 = 0 .
d~ �~`~t �'1' r~ (1.25~
- The characteristics of the equation (1.25) describing the potential movement in the -
plane t~) are defined by the equationa (1.14). The conditions along theBC charac-
teriatics in the case of potenCial motion will be obtained from the expreasions
- (1.15) setting w ~ 0 in the right-hand side:
dU -t- I s. ~ dV -f- dE ~"~v - W9 ~ 3u' ~ ~ 0.
a'-U' ~
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Let us return to the Lagrange integral~
~
~ d~ + ? + f ~p = f (v' _ ~'X + vy)� (1.26)
J P
Fox the self-sisnilar motion that on the basis of (1.24) thi~ equation ie trans~orm~d
into the form
~ ('s, - ~ a~ - ~ + 2 + ~ dp = f � -
P
= In this equation the expression on the left is the function of t1; on tr.e right
is the time function. It is impossible by any combination of variables n to
obtain the time dimenaion. Hence, it follows that for self-similar flows the time
function f(t) in the right-hand side of the Lagrange integral is equal to a constant:
- ~ ~1) - ~ ~ - ~1 ~1 2 ~ p � const.
Introducing the function p) into this equation, we obtain
~
- ~ (E, ~I) -4- 2 -I- J P = const. (1. 27)
It is clear that expression (1.8) is the differential of the integral (1.27), and
now it occurs in any direction and not only along the "current line" (1.7) as
ha.ppened in the general case of rotational motion.
Let us proceed to the derivation of the linearized equations of self-similar motion.
The linearization of the equations of motion of a compressible liquid is based on
the assumption of smallness of the diaturbed motion. H[ere the variations of the
parameters of motj.on and their derivatives are considered sma11 so that in the equa-
tions of motion and continuity only the linear terms are retained.
Then after linearization, the syatem (1.1) assumes the form (for convenience here-
after the coordinates are given the subscript "0"):
a~x i , ap ~ _ _ ~ , ,
- ~;t ~ P ax, ' ~ P aNa
~ (�Vx ~ ~ = 0, a~ = as ~
ar P\ axo T au, yo vr at '
(1. 28)
More strictly, the system (1.28) can be obtained as f ollows [5]. Let us propose that -
_ in the problem there is the small parameter d. Then if we distort the parameters of -
motion in the form of expansions with respect to powers of d, and substitute these
expansions in the equations of syetem (1.1) and equate the coeff icients for identi-
cal powers of 8, then in the first approximation these equations assum~�the form of
- (1. 28) .
_ There are three cases where it is possible to introduce the small parameter d and,
consequently, to construct the linear theory.
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1. The motion after a ~rave of weak intensity (a sound wave) when v/a = d(d)and the
bodp has an arbitrary shape. For an ideal liquid this condition is equivalent to
- ~he following:
(P-Po)lAo=o(8)
(here p~ ie the pre4sure in the medium at rest, v, p is the speed of the particle
and ttie prc:esure af cer the wave, respeerively) .
2. The wave intensity is arbitrary, but the body introduces a small disturbance in-
to the basic flow; in this case the small parameter is the relative thickness of the -
.body.
3. The wave of arbitrary intensity encounters almost a vertical wall with a slope
6= Tr/2 - o(8);in this case the effect a~ tae distorteu part of thewall can be considered
as a small reflected discontlnuity disturbance.
Thus, let any of these cases occur and we obtain the equations (1.28). In these
equatior.s with the accuracy adopted above, tt~^ density p entering in by a f actor is
assumed to be constant and equal to its initial value. Excluding the velocity com-
ponents and the density from the system, we obtain the pressure equation:
~~P a�P . 1 dp 1 d~p
d.r ~ dyp ~ 9o dyo a~ dts ~ (1.29)
It is necessary to note that the equation (1.29) is obtained wi~hout the assumption
of potentialness of the flAw, and the speed of sound a with the above-adopted
accuracy was assumed constant. For twa-dimensional motion, the pressure is deter-
mined from the equation
~sP ~ ~P _ 1 a~P
dxo ' d~ a~ ~ ots ' (1.30)
The Lagrange integral (1.26) for the potential motion a�ter linearization assumes
_ the f orm ~
P const. (1.31)
= Differentiating this equation with respect to t for a c~nstant value of the density
and excluding the derivative of the pressure, with the help of the two last equa- -
tions of system (1.28) we obtain the equation defining the velocity potential: -
a~ a~~ t aW t a=~
a.~ ~ ayo yo ~ aya � a~ ' ar~ ' (1.32)
In tl~e cuse of planar motion, ~he pc%.ential satisf ies the equation
~ a
o + ,~o - ~ (1.33) -
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From the equation (1.33) it follows that any velocity component of planar motion
also eatisfies the wave equation. For eacample, for vy we have
~vy d~v l , d~uv
aX~ - a~ vr~ � (i.34)
In the linear self-similar problems it is more convenient to introduce dimension-
less coordinates:
- x - ai ~ y - ar � (i.35)
In these coordinates the equations (1.32) and (1.33), correspondingly, assumz the
f orm
( ] - x') �9~ 2xt~ (1 - ya) 1 � a~ = 0,
axa vA�ay ay~ y :~y (1.36;
(1 .ts) ~ 2xy d--~- -i (1 - ya) _ p,
Jx dxdy Jy (1.37~
where ~(x, y) is a function related to the velocity potential ~ by the formuia
a2t~. If we introduce this function into the Lagrange integral (1.31), we obtain
I
~(X, y) - x a~ - y a~ p, = const, ~1. 38) i
ax ay ~
where a~ _ v~ d~ _ ~
a.r a' ay-~'
Correspondingly, equations (1.29) and (1.30) in the coordinates of (1.35) are written
- in the form
(1 - x' ) - 2xy ~P + (1- ya) _ - 2x az -
aX= axay ay
ap i ap (1.39)
= U,
- 2y ay -r y ay
( I- x~) - 2xy d~~' (1 - y~~ G' 2x a~' - 2 y ap = 1. 40)
aX= azay ay aX ay (
Equations (1.34) are also reduced to the form (1.40). ;
The characteristics of the equations (1.36), (1.37) and (1.39), (1.40) in the (x, y) '
- plane are the same. They are def ined by the equations
� _ XytYxY-y'- t
~J~.s - xs_'1 � (1.41)
Thus, the indicated equations are of .hyperbolic type outside a circle:
X2 ~ y~ _ Ci.42~ .
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By direct statement it is possihle to check that the integrals of equations (1.41)
will be straight lir.es;
G`i,Z (1 -F a') ( t - x)
yl. 2= .
~c~,~ (i.43)
where C~ 2= yo t V x+
X~o (xO, y~ are the coordinates of the point in the _
0
plane (x, y), through which a given padr of characteristics passes). It is easy to
- check that the characteristics of (1.43) are tangents to the circle (1.42).
In the case of axisymneCric motion for the equation (1.36) we obtain the system of
charncterieCics:
yi - Cj (1 -f- z) (t x) ~ dvs 1- C~ - ~ dv ~ .1' = 0, -
2Ci 2C= y+ 1-x' y
C2 (1 + x) + (1- X) , dv~ dv dz = 0.
" 2Cs 2C1 ~ l --x~ y (1.44)
The system of characteristics for the equation of two-dimensional potential motion
(1.37) will be ortained from (1.44) if we drop the laat term in the equations
of the characteristics in the plane (vX, v).
y
It is well known that in the region x2 + y2 < 1 the equation (1.40) using the
Chaplygin tranaformation
x= r cos A, y- r sin A, r- 2~
l es (1.45)
reducea to the Laplace equation in the region 1:
E a (e`~P a`p = o.
- as 1, a~ ~ ae= (1.46)
Thus, in the plane (e, 6) the pressure p is a harmonic function, and it can be
represented as the real part of an analytical function
_ ~ ~T) = P ~e~ e) ~f ~E~ e)~ z - eeie. (1.47)
_ In this statement the determination of the presaure reduces to a boundary problem
for a function of a complex variable. Any velocity component of the planar motion
in a liquid in the variables e, A will also satisfy the Laplace equation.
For example, f or vy we ehall have
E ae ~eGaa =o.
(1.48)
Consequently, analogously to (1.47) it is poasible to write
Z ~z) _ ~e~ 8) tf ~e~ e)� (1.49) -
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If ~(T) is de�ined, then another veloc~ty component v will be found by quadrature.
x
The Cauchy-Riemann conditions for the function (1.49) are written as follows:
.a~
a~ ' as s a~ ' (i ~ 50)
From the condition of absence of vorticity it follows that
_ dus _ dvy (1.51)
ay - aX '
Let us rewrite equation (1.37) in the form
2 ~h," dvy s~= 0.
(1- Y ) ax - 2zy ax + ~1 -y ~ dy (1.52)
Using expressions (1.50)-(1.52) the complete differential ~
dvx = aX dx ay dy
~ is easily reduced to the form
dvY - x~ (zd~~ ~ 1+ e' df
(1.53)
_ that is, it is expresaed in terme of the ctmuplete diff erentials of the real and
imaginary parta of the functions ~('t) .
Analogously, if the analytical function W(z) = vX + if is found, then
? I - E'
dv~ ~ y~ (xydv,~ - E~ df l. (1. 54)
~ I
Fenetration of an Ideal Incompressible Liquid by Solid States
~ 2. Penetration af an Incompreseible Liquid by a Wedge
Let a rigid wedge symmetrically penetrate an ideal liquid occupying all the
lower halfspace. The velocity of the wedge v~ ia constant, it is directed vertically
downward, normal to the horizontal surface of the liquid (see Figure 1.1). The
most general statement of this problem is presented in reference [1]. Let us take
the origin oE the cartesian coordinate ~ystem at the point of contact of the apex of
the wedge at the free surface at the time of beginning of the penetration. The Ox
axi~ will be directed along the horizontal eurface to the rig~nt, the Oy oxis will
be vertically downward in the direction of the penetration velocity. The problem
_ oUviously is aelf-aimilar, and the motion is potential. The velocity potential
c~(x, y, t), ~uat as the velocity of the liquid at infinity, ia equal to zero. On
the faces of the wedge obviously we have the boundary condition
z~
- vo cos
dn
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where B is the slope of the f ace o~ the wedge with the Ox axis, n is the outer nor-
mal to the face of the wedge. On the free aurface, during the entire time of move-
ment the preseure remains constant and equal to the atmosphere. The ehape of the
free aurface ia unknown in advance and is sub~ect to definition during the courae of
- solving the problem.
_ ~ Up U
~0
' -H
0 -
A A. X I X -
~ ~o
1 2~ 2~
~ ~ -
, s ~
~ H
X
_ y~ ya y5
Figure 1.1 Figure 1.2
The problem reducea to determining the velocity potential from the Laplace equation
for tlie above-indicated bouadary conditions. Af ter determining the velocity poten- -
tial, the preseure in the liquid is calculated uaing the Cauchy-Lagrange integral.
The exact eolution of the problem stated here has not been found. Mathematical
difficulties of t~e analytical solution of the problem are related to determining
the shape of the free surface. In the basic work [1], a procedure ia indicated for
solving the problem by the method of successive approximations, and the reault of a
specific calculation is presented. Many researchera have engaged in atudying this
problem. The results of all these studiea are diacussed in conaiderable detail in
the monograph [2], and they are not discussed heree
Let us proceed with the inveatigation of this problem in the two limiting cases where
the angle ~ ie close to zero or a right angle.
a) Penetration of an Incompressible L.iquid by a T.hin Wedge.
_ Let a wedge with a small apex angle 2Y penetrate a liquid which is initially at rest
and which occupies the lower halfspace. The initial penetration rate v0 is directed
vertically downward, perpendicular to the free surface. The Oy axia is directed
along the penetration rate into the liquid, the ox axis along the horizontal surface
to the right. The origin of tt?e coordinates is placed at ttie point of contact of
the apex of the wedge with the free surface of the liquid at the time t= 0. It ie -
possible to demonstrate [1J that for very small angles Y(Y-r0), with the exception of
insignif icant regiona adjacent to the pointa AA of the :ace of the wedge (Figure 1.2),
the sl.ope of the free surface is inf initely small; therefore in the investigated
_ linearized problem, ~his surface is aseumed to be horizontal after the penetration.
period. -
- Let us take the problem of determining the complex potential W(z) of the motion:
~(Z)=~-I-~~=f~z)~ z=y-f-Ix, (2.1)
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~ -
a VY ~ lU;~ ~x ~ d~Z ~ u~ ~ ~ ~2.2~ -
- The boundary conditions on the faces o~ the wedge are shif ted to the segment OH of
the Oy axis, where H(t) is the depth of penetration. In this segment the boundary _
conditions are written ae follows:
~
- Ux= fH~t)�Y� (2.3) -
The dot over the H denotes differentiation with respe~t to t. On the free surface,
the velocity potential ~ and the velocity component vX are equal to zero. By the
principle of symmetry, let us continue the function dW/dz of formula (2.2) to the
upper halfplane of the plane z= y+ ix (see Figure 1.2). Let us construct this
function in the entire plane x0y with a section of the segment (-H, H) of the Oy
axis. In this segment the velocity component vX, that is, the imaginary part of the -
function dW/dz has a discontinuity (the diacontinuity is denoted by bracketa): in
the OH segment -
I~,~1= 2HY,
in the HO segment
Ioz] _ - 2FIY� (2. 4)
Denoting the limiting values of the function dW/dz on approaching the axis from the
right and the left, respectively, by the signs and we have:
( dIp ~ (dp~ � 2t1Yi on - H0,
` az - 1, dz (2.5)
- 2Hyi on OH.
According to the Sokhotskiyformula [6:J, we have
H 0
dW l ` 2Flyidg 1 2!lyWy
dz 2r~i . 0 y- Z 2al f
N y- t .
Hence, af ter performing the integration
- Q~ _ ~ ln f 1 - ( Z l9~ . (2.6) -
l ~ 1
Here the branch of the logarithm is selected which in the segment 0< z< H of the
y axis on both sides gives
~ ~ = t Hy.
From equation (2.6), the complex potential is defined by quadrature. Calculating -
this integral, we obtain:
-W(t)- ~Y ~-2zlnz-r4z-zln(z'-N')-Hln z-H
z H
On the Oy axis the velocity potential ~ has the form;
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~Y ~2y~~y-~y-ytn~~-y~)THIn y+y
O~y~H. (2.7)
The pressure along the face of the wedge is defined by the linearized Cauchy-Lagrange
formula
aa _ a- no _ a~ ti~ ~r _
P P dt nY ~ ~ N~- y -
-~[2ylny-4y-yln(Ha-y2)-}-Hin H-y
n
H+y
. 4G!/~H. (2.8) -
At a constant velocity H~ v0~ Ii = 0
OP _ P- na �o Y 1 n vo~ - y
P p ~ n v~t y' '
~~y~~o~� -
The vertical force of resistance to penetration is
H
F- 2 f Opydy (2 ln 2� Hf/z 3, 4 Hztlj.
o n (2.9)
For a constant penetration rate H= v~, H= O,we obtain
F= PY'vot�21n2. (2.10)
Tl~e liiveHtigated problcm was solved for the first time in reference [1j, where Clie
followin5 formula was obtained for the force (in our notation)
Fl = 1,78pvoty=. ('L.11)
Thie force is twice the force defined by formula (2.10). The difference is explained
by the fact that in reference [1] when calculating the force F, the momentum of the
liquid particles on the free surface was not taken into account. Let us return to
this problem in the section devoted to the penetration of a campressible liquid
by solid states. Comparing formulas (2.10) and 2.11), we have _
41n 2
k = F = = 0,495 ~ 0,5.
Fi 1,78
b) The penetration of an incompressible liquid by a blunt wedge considering the lift
of the free surface.
Let a very blunt symmetric wedge (Bi0) penetrate an incompressible liquid. The pene-
tration rate v~ is perpendicular to the flat free boundary of the initially quiet
liquid (Figure 1.3,a). The problem is solved in the linear statement: the boundary
conditions are removed to the horizontal aurface of the liyuid and l~nearized by
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the Cuucliy-Lagraiic:u integral. Let us C~ke tfie oi-lgin of lhP carCesian ayatem of
coordinatex x0y at the point of contact of the apex of the wedge witli the liquid
surface, the Ox axis is directed al.ong the �ree surface to the right, and the Oy
a.Yis into the liquid, in the direction of the penetration rate. The velocity poten-
ti~ls satisfies the Laplace equations
~2~ - 0.
a.,.a ay~ (2.12)
~ uo
_ ~f
n p e
N(1) p R
y Q
C
y 6
Figure 1.3.
The pressure is determir.ed from the linearized Lagrange equation
d
_ ~p=P-Po=-p a~ � (2.13)
The problem is solved for the following boundary and initial conditions (Figure
1. 3,b) .
On the wetted surface of the wedge (-c, c) on the Ox axis at some point in time t
for -c < x< c we have
a~ _ aw _ vo� (2.14)
an - ay
On the rest of the Ox axis, the velocity potential is zero:
q~ = 0 for a�~-c, x~c. (2.15) -
At the initial point in time t= 0
4 _
_ a 0. (2.16)
~ it -
- Along with the velocity potential ~ in accordance with the continuity equation it
is possible to introduce the harmonic current function ~ conjugate to ~ and investi-
gate the analytical function W in the complex plane z= x+ iy, where
dW
_ - fJy. (2.17)
W = ~ ~ t~%+ dz x
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The problem stated in this way coincides in its mathematical formulation witli the
problem of impact of a plate 2c wide against the surface af the liquid. The solu-
tion of this problem is well known [2]. In our notation it has the form
- W = ~ ~ _ ~vo {1/2= - c- - z}. (2.18)
Hence, along the wetted part (-c, c) for the velocity potential we obtain -
~p = -vo1~c'-xz, I xl (2.19)
- The speed of the fluid on the f ree boundary according to (2.18) is defined by the -
formula
u;=0, -
- z~ ~
Uy ~ ~ X C.
1 - `-t ( (2.20)
.r ~
Let us note that in the solution obtained the velocity v~ can be both Gonstant and
. a time i~unction. The case of the constant penetration rate will be considered -
further. If we do not consider the lift of the free surface ana the increase in
wetted surface of the wedge connected with this when determining the wetted segment -
of the wedge, we have _
c - vot ctg
(2. 21)
_ In the iuvestigated self-similar problem the propagation rate of the ends of the
wetted part of the edge with respect to the free surface c considering the lift of -
this surPace is coiistant [1]. We shall limit ourselves to an investigation on the
riglit-hand side of the movement of the wedge during symmetric penetration (see _
Figure 1.3,a). Whz:l determining the point E(the edge of the wetted segment of the
face ot the wedge) the foam part on the free surface is neglected [2]. 'ihis part
lies above the deflectionpoint of the free surface and, therefore, it is considered
that the defleetion point coincides with the point E on the face of the wedge. The _
liquid particle e on the free surface which at the time t aftez the beginning of motion -
~s at the point E of the face of the wedge obviously has the coordinate xe = ct.
= From Figure 1.3,a it is clear that
r
h~H=c�t8~~ H=~~t~ h=~J v~(.rz, c)dtl, -
o (2. 22)
The last integral is calculated simply:
- r
h= I ~o ~ dt `'ot I.= vot n- vot.
~ V 1 - T9 2 _
t~
Substituting this solution in ttie first equality (2.22), we obtain ~y
21
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~ = 2 vo~~tg~)~� (2.23)
- Consequently, the velocity c is
2 vo ~tg (2. 24) :
'rhus, as a result of lif t of ttie free surface the wetted part of the wedge on the
x axis occupies the segment
2c = nuo ctg ~ t.
Without consid~ring the variation of the free surface from ;2.21) it follows that
_ 2c = 2z~a ctg ~ 1. -
~ If c is taken fron the formula (2.21), according to (2.13) and (2.19) the pressure
along the wetted part is determined from the expression ,
puo ctg ~
~P=P-Po= -
xz (2. 25)
_ ~ 1- vo ~i ~tg2 8
Here the total force acting on the wedge is _
F- npao t ctg2 (2. 26)
Corr~~K~~undingly, wlicn canei~lcriug the variation in Khape of the free surLace we
h3vc
for tlie pressure
- n Pvo ~tB ~
~P=P-Fo= 2 .Z ' (2.27) -
t) -
or
c `
~P=P-Po=pUo ~
- ~1 _ / c ~x
l
for tt~e force
F=~PUOC2~~ C= 2 vo Ct8 (2. 28) ~
liy nnFilyziii~ so~ne ot I~is results of numerical solutions, LJagner proposed the follow-
ing approximation formula for calculating the resistance force [1, 2) (for the angle
U < ~ < ~r/2) : _
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I
Fc~(t nl~r1 C fAL fJSE ON1.Y ~
F= nPUG H 1 lY -
2~ .1 (2.29) =
� 3. Penetration of an Incompressible Liquid by a Cone ~
Let a cone of arbitrary apex angle penetrate a liquid initially at rest, occupying
the entire lower halfspace. The speed of the cone v0 is constant and directed into
the depth of the liquid perpendicular to the plane of the free boundary of the
- liquid at rest. The period where the cone has not completely submerged in the liquid
is cont~idered. In this statement the problem is self-similar, and the motion of the
liquid is potential. The parameters of motion of the liquid depenc~. on the coordi-
_ nates -
. t - � 'C . t Z
b ~ ~ b j ~
where t is Che tiu~e , x, z are the cartesian coordinates in the meridional plane of
the investigated axisymmetric problAm. The origin of the coordinatee is placed at
the point of contact of the apex of the cone with the surface of the liquid at the
time of beginning of penetration. The Oz axis is directed vertically downward, the
Ox axis is alung the horizontal surface of the liquid to the right (Figure 1.4).
The pressure is constant on the free surface. On the generatrix of the cone we have ~
the boundary conditions
- vo cos ~
dn ~
where ~ is the velocity potential, n is the external normal to the generatrix of the -
cone, ~ is the angle of inclination of the generatrix to the Ox axis. Just as in
the case of penetration of a wedge of arbitrary apex angle with constant velocity,
_ the solution of the investigated problem here cannot be obtained atialytically. By
- I va
1
A -
x
1 n
H(r~j
~
z
Figure 1.4.
the method ~f successive approximationspresented in reference jl] the solution can
be constructed numerically. This method can be used successively to construct the
shape of the free surface and determine the force of resistance to penetration.
A detailed study of this problem, including some y.mportant theorems pertaining to -
the geometry and kinematics of self-similar flow and the properties of the f ree
surface connected with can ~e found in references [l, 2]. Here studies are made of
the cases of peneCration of a cone wiCh a sma'll apex angle and a blunt cone.
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a) Penetration of an Incompressihle Liquid by a Thin Cone. A study is made of the
- vertical penetration of an ideal liquid occupying the lower halfspace by a thin
cone with a small apex angle. 13efore the beginning of penetration the liquid is
at rest, the initial. penetxation velocity v0 is perpendicular to the plane of the
free surface of the liquid. The motion of Lhe liquid arising on pen~_~ation by the
cone will be small, and it is possible to demonstrate that with the exception of
small regions in the vicinity of the points of intersection of the cone, the fr~e -
- surface of.the liquid will differ little from the initi.al undisturbed surface [1,
7]; consequently, the problem can be considered in the linear statement and the _
boundary condition at the free surface taken down to its initial plane. The origin
of the cartesian coordinate system will be placed at the point of con~act of the
apex of the cone Witti a free surface at the tiine of beginning of pen~tration t= 0.
D ~
_ ~
~
x 2
d
z -
Figure 1.5.
The Oz axis will b~ directed vertically downward, and the Ox and Oy axes will be _
placed in the plane of the free surface of the liquid before beginning of penetra-
tion. The problem has axial symmetry. At some point in time t in the meridional
plane the picture of the motiox~ is illustrated in Figure 1.5. The velocity potential
~(x, y, z, t) satisPies the Laplace equation, that is, it is a harmonic function,
and on the basis of the linear statement its value on the free surface z= 0 is
equal to zero during the entire penetration time. Then the velocity potential, on
the basis of the principle of symmetry, can be continued unevenly to the upper half-
plane and represented in the form [8] -
t r q(El dE r2 = x' y2� (3.1)
~ ~ 47t H ~~5 - z)= ~ ~a ,
iiere H(t) denotes the depth of penetration. On the basis of smallness of the
angle y, the boundary condition on the cone is written as follows:
~ a~ - H~' ( 3 . 2 )
- where ti is the penetration rate. The problem reduces to determir.ing the unknown
function q(z) under the integral (3.1) from the boundary condition (3.2).
On the generatrix of a thin cone from formula (3.1) we ha.ve [7]
- d ~ ' ~ 1 9 (z)
(3.3)
- dr ) i-�0 2n r ~
24 "
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Thus, withiti the framework of, the linear approximation it is possible to write
2n.9~z~- { _~Y~ O~Z~N~ -
HY, -H ~i~0 (3.4)
Noting that the radius r nn the cone is related to the angle y by the formula r=
_ (H - z)'y, from the conditions (3.3) and (3.4) for force intensity q(z) we obtain
0, z ~ H,
2nHys (N - z) ~ z G H,
_ 9 (z) _
- 2ntly' (H a), - H< z< 0,
0, z < H.
On the basis of (3.1), the velocity potential in final form will be represented by
the formula [7] -
1 ~ � N (N - aE Y+~ ~ (N -I- F) d~
2 Y H' Y~_Z~,+~ 2 f Y~_1-~+~, � (3.5)
- The exceRS pressure ia determined by the linearized Cauchy-Lagran~e equationl:
p - - P o~ � (3. 6)
From �ormula (3.5) on the surface of a cone we obtain
- o~ = 2 y~lY 2(H - z) ln y-}- 2z -r (H - z) ln 4; -
-{-Hln N~Zt Z~ ~rln H+z } _
2 yyli2 (2(ln2-lny) 1 ln H=3 z~.
L ' ' (3.7)
wliere H is the acceleration ot the penetrating cone.
'1'he Hr
the velocity potential will satisfy the Laplace equation, and the boundary condi~ ~
tions will aseume the form:
0, O~z~ H-hl~,
- HY, H-H0 for n-~; therefore fn(x) also approaches zero
for n-?~, and proceeding to the limit in the inequality (5.7), we find that
~w(x)-wn(x) I->0 for n-~.
Now ]_et us proceed to the solution of the prob lem of incidence of a flat plate
~f n~asH m on tl~c Rurfac~ of an ide~l incompreasible liquid. The horizontal
velocity component of the plate at the time of contact with the free surface
will be set equal to VXp, and the vertical component, vp; the angle of the plate -
with the im diaturbed level of the free surface will b e considered small and
equal to S� The depth of submersion of the rear edge will be denoted by h(x).
Let us consider the plate quite long and its movement in the liquid will be con-
sidered until the upper edge is above the free surface. The problems of ricocl:et
of a plate f rom a free surface of the liquid and landing on the f ree surfacP will
~ be considered under the following assumptions: 1) horizontal component of the _
plate velocity is constant; 2) the angle of the plate with the undisturbed
- level of the free surface does not change; 3) the plate is acted on by gravita- _
tional force and the b uoyancy of the nonsteady planing (hydrostatic forces
are not considered, for their effect is insignificant).
In view of the constancy of the horizontal velocity component, as the independent
variable we shall take the path traveled by the plate f rom the time of contact
- with the free surface and replace the differentiation with respect to time by
differentiation with respect to path: d/dt+V~d/dx.
�
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Under these assumptions, the movement of the plate in the vertical plane can
be described by the follawing system of integral differential equations:
~ ~ 2c-}-x-a (5.$)
rnVxo = Y(x) - mg, - 2ncv2 w s) ds
x-s
0
with the initial conditions
x~ 0, h(0) - 0, h' (0) v;o ,
The magnitude of the wetted surface of the plate 2c is determined by integration
of tlie vertical velocity component of the liquid at the free surface in front of
the plate. From expression (5.1; let us determine the vertical velocity component
_ of the liquid on the ~ axis at some fixed point (0>-~M=-~~.~~ c~.~5~
M~-1 '
_ ~1= M,'_ 1 ~~a + ~or) + MYcoor + Z2~ - - i~.~~ , c~.16>
_ Here M=vp/a.
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The expression for the potential in this casz will be
~
2 R f~H~t~)-~lf'IH(t')-~l~-i-
_ o (~.1~>
0
+ 2 .1 R f ~N ~f ) -f- f~ ~H (t~) d~�
E~
If z2+r2>a2t2, then in the case of subaonic movement equations (7.13) and (7.14)
do not have real roots, and ~=0.
For v~>a equation (7.14) has no roots, and (7.13) can have two roots, one or none.
- For the roots of equation (7.13) we now have
~i,s = ~,~,1 1 [Msz - vot ~ M V(vot - z)z _ ~Mx _ 1~ rs~ ~ ( 7.18)
where both roots are positive and the larger one ~~corresponds to the sign.
Hence, it follows that for (vpt-z)2>(M2-1)r2 and z r2;a2tZ the potential ~
has the exp~ession: E
cp=- 2 ~ R fIy(t~)-~~f~[N(t)-~]~� (7.19) ~
The condition of multiplicity of roots of (7.18) has the form
(Mz - 1) r= - (vot - z)2.
1
This is the equation of the Mach cone with half-apex an~le a= arctg
On t i~~ M:~c.l~ cone fi~A.
As an example let us consider submersion with constant velocity of a thin cone
with half-apex angle y.
1. For subsonic movement v~Q -
Z
~
Z b -
Figure 1.19
_ In the two-dimensional case the potential in the vicinitv of y=0 is limited to
the opposite of th~ spatial case, where for r->0 it has the logaa~ithmic
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_ singularitiy (7.8), and there�ore the boundary conditions (9.2) can be taken
from the profile surface to the plane y=0.
A8 usual, we satisfy the condition on the free boundary by con tinuing ~ unevenly
- to the halfspace z 0, and they are related by the expressions (16.9), and 6 is determined
from (16.6) or (16.8). _
When solvi.}~g the investigated axisymmetric problem basically we shall follow the
procedure proposed in [32J when sulving similar elastic problems.
Differentiating the equations (16.7) for vZ and p with respect to t, for z= 0 we
obtain:
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FOR OF FICL~11. USl~: Otil.l'
n n ,
ap = 2Re ~ U~ ~e~ `~Z = 2Re `j m le) d S2. (16 .10 )
dt r cos i2 ot ~ r cos S2
u
In the equati~ns of (16.10), we introduce a new complex variable v by the formula
- 0= v1~2, performing the section in the plane v along the positive half.-axis [0,
Then the half-plane Im6 > 0 is mapped onto the plane v with the section [0, _
Considering equation (16.9) relating U'(8) and W'(6), from (16.10) we ob-
tain: _
F' (v) dv r d p Re f F~ ~v) ~a'' - y dv - rp dvZ
Re `~,-y - vo 2 ct ' J ~l v- va 2 dt ' (16 .11) -
J ~
!
where
~
t' U (6) - U (v ? ) � F (v)
v0 = 1 ,
/
and, consequently, F(v) must be regular in the p~~,~e v outside the section [0,
for the branch of the radical (a'2 - v)1~2, the section was taken along the inter-
val of the positive half-axis [a-2, -1~), and the radical is considered positive
for v= 0; for isolation of a single-valued branch of the radical (v - v0)1~2 the
section [v~, -F~) was taken, and for v= 0 the argument of the radical is consid-
ered equal to ~r/2.
The outline Q(Figure 1.42) was obtained as follows. In Figure 1.43 the outline
R,~ is shown in the plane 0 to which the integration path [0, ~r] crosses from the
formulas (16.7) for z> 0. For z= 0(z +0) the ends of the outline Qp (the
points L and K) will lie on the segment [-a-1, a-1] ~y~mnetrically with respect to
the origin 6= 0 and, consequently, on replacement of 8= v1~2, the outline Rp be-
comes the outline R whicti can be represented in the form of a circle of arb itrary
radius R(on the basis of the analytical nature of F(v) outside the section [0,
and two sections of identical length KT' and LT, passed, as is indicated in
Figure 1.42, along the lower and upper sides of the section [0, respectively,
where the points K and L hit the same point v= v~.
,
~O N p -
a'J~~~ T ro -
_ ~~x~ . r'
K L
A C
I (-a) (al
Figure 1.43.
Figure 1.42.
~
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For satisfaction of the initial conditions in equations (16.11) it is necessary
that it be possible for vp < a-2 to contract the integration loop R to one point,
that is, the function F'(v) must be analytical outside the section [~-2, -F~).
Since according to the boundary conditions, the expression for ap/at from (16.11)
must vanish for v~ < c-2, F'(v) must be regular outside the section [c-2, -t~), and
since avz/at from (16.11) must disappear for v0 > c-2, accordingly, the integral
function in this expression must be analytical for Re v> vp > c-2, and it de-
creases at infinity just as o(v-1) so that the integral with respect to the circle
disappears at R-~ Then it is possible to set
F' (v) _ ~ , -
~~-a ` ~~n
where n is an integer, A(v) is an integral analytical function which does not dis-
apgear at v= c-2. From the condition of integralness, the pressure on the edge
of the wetted surface of the cone must be n S 2, Hence, it is easy to see that
A(v) must be limited: A(v) = A= const and n= 2. Thus, we obtain
F~ (v) = A(c-'z - v)-2 and F(v) - rl v c-2 (c-' - v)-~ Ci.
- For z= 0 analogously tu the expressions (16.11), we obtain:
P = ~ Re ` F (v) dv ~
~ vy v-va
` (16.12)
� ~
_ L'~ = v v0 Re f 1 f~ F' V a-2 - � d� -i- C21 dv.
P ! vy~v-vo Lo J
In (16.12) the integration with respect to u is carried out by the loop lying to
the same side of the real axis as the point u= v.
From (16.12) it follows that for satisfaction of the initial conditions it is nec-
essar} th at the functions under the integral sign in (16.12) be analytical at the -
= point v= 0, that is C1 = C2 = 0 and, consequently,
F (v) = A vcz (c ' - v)-',
and the formulas (16.1'L) assume the form -
A dv
_ p _ c= l/,,o Re ~
~~3 - v~ Yv - yo
! (16.13)
v ~%'va Re ~ Adv ~/n-' _ � d�
: = p ` v ~v _ ~o ~ ~~_z _
Now from (16.13) it is obvious that p actually vanishes for vp < c-2 Let us de-
fine the constant A from the boundary condition vZ = v0 for vp > c-2, z= 0:
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, v
_ o
- 1/v I2e ~ Adv � � d� _ vo,
,'p .1 v Yv - vo O~~-~ -
where the integral with respect to u can be represented in the form
v -o~ v
c Va dN~ Ya 3-� d� -I- S ~a B`~ Fo~~)�
o ' - o 2 - � - F~)=
Here in the second integral the integration is carried out along the ray arg u=
arg v and B is given by the expression
> > i
B=- c j~y 2~- (1 - Y) 2 arccos (Y 2)]. Y= ~2 a-Z�
Since Fp(v) changes sign on going through the section [c-2, then
Fo (v) dv
S ~ _ ~ for vo ~ c-=,
v y v - vo
- :
and we obtain vp = 2nABp-1. Hence,
vapj~l-y (16.14)
A=- .
2nc (~~y (1 - y) arccos y~y ]
For the pressure distribution p on the wetted surface of the cone (v0 > c'2) we
� obtain
vop cst
p Y c=t -r= (l~Y (1-y) -}-arccos j/y 1~ (16.15)
' where p has the integral singularity at r=~t.
In order to obtain the pressure at any point of the half-space z? 0, we make the
substitution of variable �=(a'2 - 62)1~2 in formula (16.7) for p and, consider-
_ ing the branch of the radical (a-2 - 02)1~2, we obtain for aZt2 > r2 + z2:
P= 2Aca Re C (za'9 - t�) d�
~ Z- u-~ - l~r'a-a _ t~ 2tz (r~ IY~ (16 .16 )
!i
where the loop R1 in the region of Re u> 0 is illustrated in Figure 1.44 and A is
given by the formula (16.14). Here, at the points K and L we have: u= ul
u= u2, respectively, where ul and u2 are the roots of the quadratic trinomial un-
der the sign of the radical in (16.16)
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ntZ t ii Ya~1s -~i - 2~ (16 .17 )
N'l~2 = Q ~r~ +z1~ �
For isolation of the single-valued branch of the radical in (16.16), a section is
made in the plane u from the point L to K(the dotted line) and the branch of the
- radical is taken such that its argument will be equal to n/2 for real values of U
greater than a-1. Then considering that on different sides of the section LK the
expression under the integral sign in (16.16) assumes opposite values with respect
to sigr. but equal with respect to absolute magnitude and has two simple poles at
the points
i
� = t i (~2 - a-') ,
using the remainder theorem, from (16.16) we finally obtain:
z z
p - - 2nAc~ t + Bo - ~n -
2 ~Ao ~ Bo~ (16.18)
Z Y ~o + eo + A~
a= y' c-- - a-= I 2 cA0 Bo)
~ where
~ 2z ~ a2~2, Ao = z' - a-2) ~~2 ~Z - tY,
i
- Bo = 2t2 (c-2 - a-') 2 ,
and the radicals are considered arithmetic. In particular, for z} 0 and r> ct
the expression (16.18) vanishes, and for z-~ 0 and r< ct it coincides with the
expression (16,15). Let us note that the expression (16.18) disappears for r2 +
z2 a2t2, and for c-~ a from (16.14) we find A-? -vpp/4~ra, and expression (16.18)
gives at the limit:
vo pa2t a~l~ - r~ - z~ ~
P 2 's '
(a=t~ - r~) 2 (16 .19 )
Formula (16.19) coincides with the analogous limit formula obtained from solution
of the problem of penetration of a compressible liquid by a blunt cone at c=
vp ctg a, when vp ctg S~ a(see the following section).
Analogously, it is possible to calculate the velocity components vZ and vr in the
region r2 + z2 < a2t2. Let us note that for r2 + z2 > a2t2 (z < at) all the func-
tions vZ, vr and p vanish, for the points K and L of the ends of the integration
loop R.1 in the plane U are incident at the same point lying on a segment of the
real axis [0, a-1J in the plane u(since in the plane 6 they lie on the segment
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[-a-1, a-1] symmetrically with respect to the point 6= and, consequently, by
the Caucliy theorem the integrals over the closed loop Q1 vanish.
O ~ tg~
I C 'Q~ U
l~ 1,5~
_ A KT B N
1,?5
~ ~Q
L ~ ~
-l ~ ~'~0 0,25 0,5 0,75 1,0� t9~
Figure 1.44. Figure 1.45.
Now let us define the radius of the wetted surface c= ct, substituting the ex-
pression for vZ from (16.i3) in equation (16.2). Then after some transformations
equation (16.2) asstim?es the form
1 s �
- _ Ac d~ ~ d� ~ ti~v - y dv ; Uo - c tg (16 . 20)
P. J y �YT-� Y (1-v)
- where the radicals are considered to be arithmetic. After integration we finally
obtain the following expression relating v~a'1 ctg S and Y:
, ylY arccos 1~y 2vo ctg (16.21)
Y i ~ _
yl-y a
where Y = c2/a2.
The numerical solution of equation (16.21) is presented in Figure 1.45. It is ob-
vious that c depends on a for a constant value of v~ ctg S and L~ 4vp(ctg S)/n
for vp(ctg S)/a 0(the case of an incompressible liquid) and when v~(ctg s)/a 1,
then c-~ v0 ctg a, and the free surface of the liquid outside the cone remains
undisturbed, which corresponds to the physical picture of the flow.
- Returning to formula (16.15), we note that if we consider the parameter c given,
~ independent of a, then the pressure according to formula (16.15) depends on a in
contrast to the other limiting case--penetration of a compressible liquid by a thin
cone with subsonic velocity where the pressure of the cone does not depend on a
[23].
In the case of an incompressible liquid the formula for the pressure on the surface
of the blunt cone (�3) is obtained from (16.15) for a-}
i
( :
P= 2v:~P C[ 1- 1, et ~ J 2, (16.22)
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= where ct just as in (16.15), is the radius of the wetted surface of the cone.
Formula (16.15) differs from (16.22) by the presence of the factor -
i i ~
~-i jY ~ ~1 - y) " arccos (Y ` )J-'.
which varies within the limits from n/4 to 1 and, consequently, f or the same values
of v~, r, t and L the pressure in the compressible liquid is less than the p ressure
- in the incompressible liquid.
If in the formula (16.15) c is given by equation (16.21), for the pressure on the
surface of the blunt cone we obtain the formula:
i ~
- P_ P~2B~ (1- ( ~ IZl ' . (16.23)
I ~ I~
~ If we do not consider the lift of the free surface in the compressible liqu~id,
then c= v~ ctg S, and the formula (16.15) assumes the form
~
~
P = �oP ( 1 - ~ rtg~ '1z1- 2 x -
tg~ l ~ot / ~ (16.24)
t
x { atg~ + [1-- ( ate~ ~rJ ~ arccos at�
a~ 1
For the case of an incompressible liquid considering lift of the f ree surface,
, that is, according to (3.36) for c= 4vp(ctg S)/n, the pressure distribution on
the cone is given by the expre~sion
i
8~ p 1- ( :tr tg ~ a- ~
P= n~t~~ L 1 4ve ' C16.25)
0
_ and without considering the lift of the liquid, by formula (16.22), where c=
vp ctg
i
~z -
p n tB ~ L 1-\, ~o~~ 1) 2. C 16 . 26 )
Using formulas (16.23)-(16.26), the forces acting on the cone will be defined by
the following relations:
For a compressible liquid considering the lift of the surface
F~ = nP (tg ~`t"~ (16.27) -
For a compressible liquid without considering the surface lift
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- 2nuo pt~ -
FZ - v ,
v arcccs ~ a tg ~ I (16.28)
o /
t~g ~ a tg ~ T ~o ~ z
~1- ( atg~ /
For incompressible liquid considering the lift of the surface (see formula 3.38) _
� (16.29)
F3 tg rc tg ptY,
_ For incompressible liquid without considering the surface lift (see formula 3.29)
F4 - 4va pt2 ctg9 (16. 30)
Then, referring to F4, we have
Fi _ n r ~{o~ 14
1~ F - 4 ` ~ )
, ~
,
II) F: = 2{ a tg ~ T' L 1-\ a tB ~ 3tCCOS Q{g I
III) Fe 4~$, IV) F` = 1.
F~ ~ n F~ l
t
~ The graphs of the functions (I-IV) of v~(ctg S)/a are presented in Figure 1.46.
2,5
~
7,0
l
1,5
f,0 ~
I II 5(1, ) v~ _
QS �'t9 S -
0 0,25 0,5 0,75 f,0
Figure 1.46.
It is necessary to note that the difference in the hydrodynamic forces (I-IV) act-
ing on the surface of the penetrating cone in four different cases arises not only
from the difference in pressures, but also the difference in areas of wetted sur-
faces. From these graphs it is obvious that the behavior of the force is de-
scribed most exactly and physically correctly by curve (I), which for
v~(ctg S)/a 0 gives the case of an incoc~pressible liquid considering lift of the
free surface (III), and for vp(ctg S)/a 1, when c--~ a, it gives a result which
- coincides with the analogous limiting result obtained for the case c= v~ ctg S> a
118
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for vp ctg S-} a(the point S(1, n/4)). In the last, limiting cas~ the theory of
an incompressible liquid considering lift of the liquid (graph III) gives an in-
crease in the indicated limiting result by approximately 2.6 times. Consequently,
for a more exact description of the hydrodynamic forces for not very small valuea
of the parameter v~(ctg 6)1a by comparison with one, it is necessary to use the
curve (I) which is suitable for the entire range of variation of the parametar
vp(ctg ~)/a from zero to one.
Obviously, the expressions obtained for the pressure and the forces acting on the
penetrating cone according to forniulas (16.27)-(16.30) will also be valid for a
cone of finite height to the time t* when the free aurface of the liquid reaches
its base. Let us denote the radius of the base of the cone of finite height in
terms of rp. Then the time t= t* can be determined from the equality ct* = rp,
In this case for a cone of finite height penetrating a compressible liquid, consid-
ering the lift of the free surface, the magnitude of the force from (16.27) after
transformations will be
~
e ~ z
v
Fi ~ l t6 ~ P2~ ~ro, (16. 31)
0
For~ an incompressible liquid considering the lift of the free surface it is poasi-
b?e to write formula (3.36) of the present chapter:
cC ~ 4 vo t' ctg ro,
n
- then
- t~ - tg ~ r
4 vo o�
_ Thus, for t= t* from (16.29) the value of the force is -
32 P~o z (16.32
- tg Y 2 nro . ) ~n
- Let us introduce the resistance coefficients C1 and C3: for compressible liquid
C1-2 vo 1z~~
and incompressible liquid
~ = a2
_ � natg~ ~
- In Figure 1.47 values are presented for the parameter C1 tg S and Cg tg S as a ~
- function of the M number:
~o
, M_ atg~ ,
119 -
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6~ tg ~
4 , ~ ~
32 o I
~~1 ~~t
0 0,1 0,2 0,3 0,4 0,5
vo
_ ~
tg'~
Figure 1.47. -
For the number M= 0.5, the ratio C1/C3 = 0.8; for M= 1 the ratio C1/Cg = 0.61.
The dots on this curva denote the experimental data for the cone R= 10� for t~ t*.
The experimental results, as is obvious from the figure, are closer to the theo-
retical curve--the model of a compressible liquid. The deviation wil~ be up to 15
percent. _
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�17. Penetratian of a Compressible Liquid by a Blunt Cone for V> a
~ A study was made of the problem of the penetration of an ideal compressible liquid
at rest occupying the entire lower half-space by a very blunt cone of circular
cross section [7, 23]. The velocity of the cone v0 is assumed to be constant, less
than the speed of sound in the liquid vp < a and directed downward, per.pendicular
to the free horizontal surface. rt is proposed that the cone is Qo blunt that as
it is introduced the periphery of the intersection of the cone with the free sur-
face shifts over th e surface at a velocity greater t.han the speed of sound in the
liquid. Under this condition the shock wave arises which cuts off the region of
disturbed motion from ~he liquid at rest. It is proposed that the shock wave does
not depart from the line of intersection of the surface of the cone with the free
undisturbed surface of the liquid. The shock wave front will be an axisymmetr.ic
surface. Since the penetratian pattern has axial symmetry, her eafter we shall
_ limit ourselves to investigation of the motion in the meridional plane Ox~y~ (Fig-
ure 1.48).
a
- o a
8 y
_ o
- B2' IA1
B~ A~
xa
- Figure 1.48.
The origin of the cartesian coordi.nate system Oxpyp is taken at the point of con-
tact of the apex of the cone with the frQe svrface of the liquid at the time t= 0
_ (the beginning of pen~atration). In Figure 1.48 the line AA1B1B corresponds to the
front of the formed shock wave. The horizontal free surface of the liquid to the
- lef t of the point A and to the right of th~ point B for the given statement of the
problem has no influence cn the regi4u of disturbed motion of the liquid. The
points A and B are shifted along the free surface at a velocity V= v~ ctg S. The
angle of inclination of the shock wave front a tc horizontal at the point A is de-
~ termined from the law of formaeion of a plane shock wave. 2ndeed, the shock wave
_ front beginning with the points A and B wi11. not be rectilinear, for penetration
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of an axisymmetric configuration takes place. Here a study is made of the problem
i.n the linear statement. The penetraCion of the cone in the nonlinear statement
of the problem will be considered b elow. It is proposed that the penetration
takes place under the condition -
- vo �a~ (17.1)
V=voctg~7a� (17.2) -
The region of disturbed motion on penetration by this cone will be cut off from ~
the liquid at rest by the line of the weak wave consisting of the straight lines
- AA1 and BB1 tangent to the circle of radius at with its center at the origin of -
the coordinates and the arc A1B1 of this circle. L1 the plane of self-similar
- coordinates x= xp/at, y= yp/at, this region is fourid in Figure 1.49. The indi-
cated circle is the boundary of the region where diffraction from the apex of the
cone is felt. From the conditions (17.1) and (17.2) we find that S� 1 and, con-
- sequently, it is possible to carry over the boundary conditions to the plane
xp = 0. Then the solution of the problem by definition of the potential of the
disturbed motion of the liquid in the cylindrical coordinates xp, yp, A reduces to
solution of the equation
i
- a~~ a~~ , i a~ _~i a~~ (17.3)
- - - ~ ~
a~ ay~ yo ayo a ~
under the following boundary and initial conditions:
_ vn = d~ ~ d
W Is~ vo for o c yo ~ yo, 0~ 6 C 2n, (17 .4) .
vn - a~ I - o f o r yo > yo, o c e c 2:~,
axo x =o
a~ 0 for t= 0, (17.5)
~ - at - -
here y~(t) = V� t= v~t/S, for tg S= B�
_ ~ P Po
~ - po
_ 1,0
. 0, 8
Q6
0,4
~ O'2 -
_ p y _
A~ e~
x
Figure 1.49.
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The solution of the system (17.3)-(17.5) by the method of the delaying potential
- in the investigated case is written in the form
1 ~ ~'(f ) u~ (t', C) C dC
~~xo~ yo, t) - - dA ` ~ (17.6)
2'~ o O l/~ C= - 2yo C cos e
where ~
- r _ t - a ~xo yo + ~a - 2yo b ~os e . -
Since vn = v0, then
~ ~ dy0 dA
~7~ c� y0 dt (17 . 7)
or - 2n o ~/xo + yo ~yp)~ - 2yoJo cas 0 ~
and y~ is determined from the formula
. ~o ~ ~ ~ 2 . ~ .
~o t- Q y xo + yo (yo) - 2yoyo ~os e
: ~
Hence,
yo M1M 1[u~ - Myo cos 9 t
(17.8)
t ~(at - Myo cos 9)a (ast' - xo - yo) ~Ma - 1)~ ~
where M = (v0/aR) > 1.
- Differentiating (17.8) with respect to time, we obtain
dyo ,Na 1 t M(Mat - yo cos A) '
� 1. )
dt M- - 1 Y(al - Myo cas 9)' (a~t~ - xo - r,~~ (M~ - 1) ~ 7. 9
In the dimensionless variables x= x~/at and y= y~/at these formulas assume the -
f. orm
- y� jt-n-ry~oSetRJ, (i~.io~
oi - ,uZ -1
dyo M r 1~ M(M - y cos 9) 1~ (17 . 1.1) -
- a dt ,YI~ - L L R ~
where
~
R=V(1-Mycose)2 ; (1-x'-y~)(M'-1) .
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Let us introduce the following two integrals into the investigation:
n v
Q~ 1-,yR ~o~ e dg, Q9 = 2~ i- My ~os e de,
0 o R
where
y- arccos 1+ Y(1 - xd - y~) (1 - M')
My
A simple investigation of the solution of (17.6) shows that in the region of dis-
turbed flow the (excess) pressure p=-p8~/at is expressed by the formulas:
~
p n~M,av~l~ ~n - Qi) for xa -f- yz ~ 1,
(17.12)
P=- P n~M~~O~~ Qa f or x' -'r- y' ~ 1.
Ae the point B(x = 0, y= M) behind the reflected wave the pressure is defined as
Pe= yPM�M 1 .
Taking this into account, it is possible to represent formula (17.12) in the fol-
lowing form:
p M ~n - Qi) for zz ~ y2 C 1~
- . pa ~ ~M' -1
(17.13)
p M Q for x2 yz ~ 1.
P B n y~M= - 1 z
In Figure 1.49, the pressure distribution along the generatrix of the cone calcu-
lated by formulas (17.13) is presented for M= 3.73. According to �15, formula
(15.17), the force acting on the cone in the vertical direction is
F - npa va (ctg2 t2.
On the leading wave the equation of which has the form
y--xctga-r- 1 ; (sin a= ~ l, (17.14)
' sin a ~ M /
with the help of formulas (17.6) and (17.12) we obtain
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~ (cos a-- x) v~ = vx � t$ a,
vx - u� 2 y sin 2a '
(17.15)
P= pavo 2(cos a- x) .
cos a y sin 2a
Here let us point out that by analogy with the two-dimensional case (see ~11) from
the formula
sin a - 1 -
M vo
and from the condition (17.1) it follows directly that a� S. ~
Thus, in the given linear statement of the penetration problem the Mach angle a is
large by comparison with the angle S. In the special case a=~r/2 (M = 1), for-
mula (17.12) gives _
p- P��0 -'t~ 3 y'~ f or x~ y2 ` 1, p- 0 for xY rt- ya ~ 1. _
2 (1 --y') s
Let us note that formula (17.6) defines the velocity potential also in the case
where the penetrating axisymmetric body has a deformed line deviating little from
the Oyp axis as the generatrix. The penetration velocity here also can be a vari-
able that satisfies the condition (17.2).
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g18� Nonlinear Problem of Penetration of a Compressible Liquid by a Blunt Cone
Here a study is made of the vertical symmetric entry of a blunt cone with constant
- subsonic v~locity vQ into a compressible liquid half-space. The expansion rate of
the radius of the circle, which is the line of intersection of the cone with the
free surface, is greater than the speed of sound a0 in the undisturbed liquid.
Under these conditions, as was noted in �17, a shock wave is formed in the liquid
which cuts off the region of the disturbed motion from the liquid at rest. It is
Proposed that the shock wave does not depart from the line of intersection of the
cone with the free boundary of the liquid. Qn the basis of studying the corre-
sponding linearized problem, it is possible to propose that the shock wave in the
meridional cross section is a curve consisting of two lines AA1 and BB1 differing
little from straight lines and joined by the curve A1B1 under the apex of the cone
(Figure 1.48).
~ t~e meridional plane x~0y~ the origin of the coordinates is placed at the point
of contact of the apex of the cone with the free surface of the liquid at the be-
gl~ing of penetration. The Oxp axis is directed vertically downward in the direc-
_ tion of the velocity v~; the Oyp axis is directed along the surface of the liquid
at rest, to the right. -
- In the investigated axisymmetric problem the motion of the liquid is self-sim;lar.
In the plane of the self-similar coordinates xp/t, n= y~/t, the picture of
the motion is illustrated in Figure 1.50. The lines AlA2, B1B2 in this figure de-
_ note sections of the boundaries of the region of effect of the diffraction of the
- apex of the cone. The motion of the liquid is determined from the system of quasi-
linear equations (1.9) under the corresponding boundary conditions. This system _
of equations, just as the second-order partial differential equation equivalent to
_ it (1.10) is of the elliptical type inside the region where the following condition
is satisfied (vX, vy are the velocity components with respect to the xp, yp axes):
`�x (t1~, -1~)z ~ Q'. (18 .1)
This region coincides with the region of effect of the apex of the cone [14].
When the left-hand side of the inequality in formula (18.1) is larger than a2, the
indicated equations of the hyperbol.ic type and the corresponding characteristics
are real. These characteristics are represented by the formulas ('1.14) and (1.15).
The problem was investigated in this statement in [19, 14, 23]. The condition that
126
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the equations are hyperbolic is satisfied after the sections AA1 and BB1 of the
shock wave. As was noted above, these sections are almost rectilinear, and, con-
sequently, the flow behind th~.m can be considered potential. From the results of
the first section of this chapter it follows that in the case of potential motion
in the region where the condition
(U.~ - -1-- ~vy - r~)' ~ a~~ (18 . 2 )
is satisfied, the characteristics of the equation of motion of the liquid are rep-
resented in the form
l ~ / t.s - ~li.s w t Q~~ ~ - a~ ~ (18. 3)
dU ~ls,~ dV -f- ~ I a~~ -W2 3aYl = 0. (18.4)
a' - U' ~ ~l + ~
~ o ~
~ ~
B A 2
B A -
C
8~ q~
f
. Figure 1.50.
In these equations the following notation is introduced: -
U=Us-~, V=u~-r~, R~2=U~-}-V'. (18.5)
Let us note that the problem investigated here is analogous to the problem of -
_ shock wave diffraction at the apex of the cone [14, 19, 23]. On the basis of sym-
metry of the problem let us consider the re;ion of real characteristics to the
right of the 0~ a:cis (Figure 1.50). Let a be the angle of the shock wave AA1 with
the horizontal. This angle along the shock wave is a variable. In the region
AAlA2A where the effect of diffraction on the apex of the cone is not felt, using
the characteristics of (18.3) and the conditior.s on the shock wave it is possib le
to construct the flow parameters and the section AA1 of the shock wave. Let ixs
denote by D the velocity of the shock wave. From formula (1.23) of the first sec-
tion it follows that at any point of the shock wave in the plane n the follow-
ine equality exi~ts
D-~cosa-{-r~sina. ~18
The ec~uation of state of the liquid will be taken in the form already used previ-
ously in the sectio~ on the nonlinear problem of penetration by a wedge. The con-
ditions on the shock front at the point A are written as follows (Figure 1.50):
127 =
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' P~�0 fr Pi 1 n
Pi - Po = n L~ po 1, ' A~ - Po = Po~'~ ~ _
PoD = Pi (D-vl), (18. 7)
- cas ~ -
Jl - U� (a - ~
sin a
D-vo
. tg ~
Here S is the angle of the generatrix of the penetrating cone with the free sur-
face of the liquid at rest, pl, pl, vl are the pressure, density and velocity of
the liquid behind the shock wave at the point A; p0, p~, a~ are the pressure, den- -
- sity and speed of sound in the liquid at rest. The last equality in the system '
~ (18. 7) Follows from formula (18.6) . Five unknowns at the point A are defined from
the five equations in formula (18.7) (a is the least angle):
a, D~ Pi~ Pi~ vi.
Let us note that at the point A the parameters of the liquid, the velocity and an- ~
gle of the shock wave are determined fram the same equations from which these p a-
rameters are determined at the convergence point, on penetration of a compressib le
liquid by a blunt wedge with the same apex angle. Thus, for the wedge and the '
cone of identical apex angle at the point A the liquid parameters coincide. Fu r-
thermore, the flow is defined using the characteristics ~ust as for supersonic
steady-state motion of a liquid. The method of constructing the solution coin-
cides completely with the method of solving the problem of reflection of the sho ck
wave from the apex of the cone [14, 19, 23]. FrQm the point A the element of the
shock wave AM1 is plotted at an angle a defined from the system (18.7), and the
values of the parameters of the point A are carried over to the point M1. From _
_ the point M1, the characteristic (18.3) (its element) is drawn, which intersects
_ the generatrix of the cone closer to the point A. Let us denote this point by M 2
(Figure 1.51). The speed of the liquid at the point M2 is found using the bound- -
ary condition on the generatrix (the normal component of the velocity of the liq-
uid on the generatrix of the cone is equal to the normal velocity component of the
cone) and the corresponding equation of the characteristic (18.4) in which the
differentials are replaced by finite differences. The speed of sound at the point
M2 is determined from the Lagrange integral (1.27). Let us introduce the speed of
sound a:
a~ \ n .
dp Po o P (18. 8)
2
Q l
dP / P Po !
Then this integral in finite differences for any two close points 1 and 2 will b e
written as follows [14, 19~:
a2-a~ v2-v~ , (18.9)
_ 51 lU2z - viz~ ~ ~i ~UZ~ - Uly~.
- n-L 2
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~
3500 4000 4500 ~
M6 M~ d ' M ~
J
MS
M
Figure 1.51.
Formulas (18.8) and (18.9) define the speed of sound and density at the point M2 if
_ the parameters of motion at the point M1 are known. The pressure at the point M2
is determined from the equation of state of the liquid
P_ Po _ Poao r r~ I" - 1~. (18.10)
n L~PoI
From the point M2, an element of the characteristic (18.3) is drawn to the inter-
section with the continuation of the segment of the shock wave AM1 at the point M3.
The first three equations in formula (18.7), the equation (18.6) and the condition
along the characteristic M2M3 which we obtain from (18.4) replacing the differen-
tials by finite differences are va~id at the point M3 at the shock wave. The five
> obtained equations def ine f ive unknown parameters on the shock wave, including the
_ angle ag. For further solution of the problem from the point M3 the shock wave a
is plotted at an angle ag. During the course of constructing the solution, an-
other problem of determining the parameters at the intersection point of two char-
acteristics emerging from two close points at whi~h the liquid parameters are al-
ready known, is encountered. This problem is solved by the usual method. Thus,
the flow behind the shock wave is defined "step by step" in the region of real
characteristics.
As an example, the parameters of motion of the liquid are defined by the indicated
method for penetration by a blunt cone at eight points shown in Figure 1.51.
The calculations were performed for the follawing data. The penetration velocity
vQ = 397 m/sec, the angle R= 5�. -
n= 7.15, pp = 101.94 kG-sec2/m4, a~ = 1,515 m/sec.
The results of the calculation are presented in Table 1.2. From the table, the
slow variation of the parameters of motion on going away from the point A is obvi-
ous. The angle of inclination a of the shock wave varies insignificantly so that
the shock wave remains almost rectilinear. This is in agreement with the results
of the solution of the problem in the linear statement.
By the method of characteristics it is possible to calculate the parameters of the -
_ disturbed motion to some boundary close to the line which is defined by the equa-
tion
(v: - ~)Z -r (vb - ~l)2 = a2.
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Tab le 1. 2 '
I A I ,Nl I ~bl~ I ~~f~ I M~ I 111s I h(~ I hf~ I ,N~
~ m/sec 0 22,8 5,3 75,5 t7,3 229,6 ~19,i 139,6I 425,2
~l m/sec 4038 4500 4~77 4412 ~340 4155 39i0 ~063 ~3830
~x m/sec 3T7 377 377 375 377 371 345 358 I 357
vy m/sec 227 227 226 225 22=1 221 218 219 208
a m/sec 2Si7 2877 2873 2867 2866 2845 2816 2831 i27i6
D m/ s e c 2338 2338 - 2333 - 2322 - - I2285
p/po 2 1,232 1,232 1,231 1,230 1.230 1,22i 1,223 1,225 1,218
a, kg/cm 1~901 131�O1 10440 133005~ 10370 130�46 9830 I 9990 I 30�~5
In the diffraction region the parameters of motion of the liquid and the corre-
sponding section A1B1 of the shock wave remain unknown (Figure 1.50). The defini-
tion of the motion in this region encounters diff iculties of the same nature as
when determining the motion between a blunt body and the departing shock wave in
the supersonic steady-state flow. In the case of penetration investigated here
(just as in the problem of reflection of the shock wave from the apex of the cone,
the problem is still more complicated by the fact that the solution obtained in
the diffraction region must fit with the solution obtained by the method of char-
acteristics at a finite distance.
The effect of diffraction leads to the fact that beginning with the points A2 and
B2 along the gener3trices of the cone in the direction of its apex there will be
further, more intense continuous decrease in pressure. Its minimum is reached at
the apex of the cone.
In [27] the approximate solution was obtained for the problem of the diffraction
of a shock wave of constant intensity from the apex of a blunt cone. It is demon-
- strated that in the self-similar plane n in the diffraction region the pressure
on the cone in the section A2C (Figure 1.50) decreases by a parabolic law as a
function of the distance from the apex of the cone. In view of the complete anal-
ogy of the mathematical statement of the problem of reflection of a shock wave
f rom the apex of the cone and the problem of penetration of a compressible liquid
by a blunt cone investigated here, it is possible to expect that thE pressure will
vary similarly also iti this problem in the diffraction region along the generatrix.
130
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�19. Impact of a Rigid Cylinder With the Surface of a Compressible Liquid
_ Let an impact take place at the time t= 0 with a velocity vp � a between an abso-
lutely rigid circular cylinder of radius r0 with flat front tip and the free sur-
face of an ideal compressible liquid occupying the lower half-space z> 0(Figure -
1.52) [23, 33]. Here a is the speed of sound of the undisturbed liquid. Under
this condition, as is easy to demonstrate, for the initial time interval Ot - rp/a
- where the compressibility of the liquid is significant, the problem will be linear,
and it is described in the cylindrical system of dimensionless coordinates rizl by
the following equation and conditions:
+ a~ a2~ _ a~~ 0 for r~ ~ 1, z~ _
- d~~ ~l ~rl d ~ ~
- v(T) io for 0 C~i ~ 1, al = 0;
� vzl
(19.1)
d`~ =0 f_or T=O;
~ - dt -
at r z
T=-, r1= - Z1=-.
ro io rp
V~
OI r
I
Iz
Figure 1.52.
Here ~(rl, zl, T) is the potential of the disturbed motion, v(T) is the penetra-
tion velocity. Hereafter, the subscript 1 will be omitted on the dimensionless _
independent variables. This linear statement gives a proper solution at all points
except the small region at the edge of the disk where as a result of discontinuity
of the direction of the velocities, the latter must have a singularity.
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In the problem the force F(t) is found which acts on the disk at 0 5 T< 1.
The Laplace transformation [34] with respect to T is applied to the system (19.1):
a~~ , a~ -'r d=m = pz~ (Re p~ 0),
ar2 ~ , ar aZ9
(19.2)
~zao = p f or r~ 1~
~ ~m 1 -V r for OC~~1.
bz ~z=o (P) o
1
Then, applying the Hankel transformation to equation (19.2) [34], with respect to
r, it is easy to obtain for z= 0:
i
f~Z=o Jo (r x) r dr 1 X
o YP'+z' (19.3)
M ~
X I- J` a~ ~o ~r x) r dr T- V~P) ro s ~o ~r X) r dr l.
~
t o
- In (19.2) and (19.3) the following notation is adopted:
Jn(r x) is an n-th order Bessel function,
V ~P)� v ~ Z, P) ~ Z~ ti) �
The reverse Hankel transformation for (19.3) with replacement of r by p for p> 1
- gives
~ ~
S~ a~ 1 r dr f Jo (rx) Jo (px) .e dx _
a~ i Z=o .1 y' p~ ~ xa (19 . 4)
~
_ - V ~P) ro ~ ~i ~X) ~o ~XP) ~ .
Yn: + a'
0
It is easy to demonstrate that
0o b+i ~
r ~o ~rz~ ~o ~zp~ x dx _ 1 ~ a{s) s ds
1 ,
~ y-p. ~ x9 n~ b_~~ y p~ Sa
(19.5)
� a+~~
S J~ ~ Y~ Jo ~XP~ d.r _ 1 Ko ~SP~ f ~~S~ ~ p ds S.
0 1/ps + xa ni ~~oo a ~
Here 0< b< Rep; Kn(x) and In(x) are the MacDonald and Bessel functions of an
imaginary argument, respectively, of n-th order. The branch y p2 - s2 is selected
so that
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ReYp~-s~~0 for O~Res Ko ~SP) f o~Sr) f or P~ r.
- Substituting the expressions (19.5) in (19.4) and making the substitution of vari-
able sl = s/p, we obtain
~
( am r d~ a(ps) s ds
( a: )s~o l yl -s~
~ (19.6)
_ - r0 1~ ~P) `Ko ~SPP) ~i ~SP) ds �
- P ~ v, - ~
L
'l ~ p .
_ s
~ e -erg(p)
~1 ~
~ ~ ~ ~ ~ ~
Figure 1.53.
Here the subscript 1 on sl is omitted. The l,oop L is shpwn in Figure 1.53. Now
it is easy to note that the integration loop in (19.6) can be deformed along the
section of the branch ?1 - sZ, as shown in Figure 1.53. Then it is possible to
_ apply the asymptotic expansions of the cylindrical f unctions for large values of
the arguments, making one significant simplification.
_ Let us imagine
_ 1 n ~9) = 2 ~Hn ~ ~9~ ) N~'~ ~9~)1 ~9 = SP~ SP~~ SPP)� -
Here HA~~(qi) is the j-th type, n-th order Hankel function. It is easy to note
that the function I~1~(qi) gives the delay factor exp (-2q) by comparison with
1i~2~(qi). From the shape of the def ormed loop it is obvious that this factor sat-
i.sfies the inequality (exp (-2q)I exp (-2Rep). Con.:quently, the function
H~1~(qi) ma.kes a contribution to the solution of (19.6) in the form of secondary,
tertiary, and other diffraction waves.
For the inveYse Laplace transformation these terms appear only for T> 2, which is
excluded by the condition of the problem. Consequently, instead of In(q) it is
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necessary to substitute the function 1/2 (-i)nHn2~ (qi) in (19. 6) and use its _
asymptotic form. Then it is easy to see that for the first approximation with
- respect to 1/p, we obtain the following, if we deform the loop ta the former posi-
tiori after appZication of the asymptotic form) :
~
am 1 r_o V I d/ ` esw-SvP p V~P) ~ S es 1 S~ [~.S. (19 . 7)
~ \ l ~ ,~1-~ ~ Y
Let us assume that
cm
( dZ )Z_o~Mo~P- 1)~ for p-~.1, 1.
Then [35 J
`~~5, P)~SP.~..,Mls-i.-~ for s~.oo,
~
~ ~S~ P)=.~ ( ~ )Z~e'ayr dr.
i
Here Mp and M, are constants. Applying the Wiener-Hopf inethod to equation (19.7),
we obtain (F~gure 1.53)
~~S~ P) ~~p T V~P) ~o ds =
~ ~ y,-s as Y1-S s-S~ (19.s)
Using the fact that the first term in brackets of the expression under the inte-
gral sign (19.8) is an analytical function in the half-plane Res < 1, from (19.8)
it is easy to obtain
`Y (s, P) _ ~~~P) r" ~V 1-s - 1) eso. (19.9)
- P$
If we find the second approximation of 4'(s, p) with respect to 1/p, then instead
of (19.7) it is necessary to take
d~ esor-sv0
J( I j/r dr l ds =
~ ` az ~ ~o ~ ~ 1 - s9 (19 .10 )
- sP-sPP 3 1
_ - V ( p) ` e (1 - - ,J ds.
P ~ s 1- s' \ $PS BPSP /
It must be noted that in the expression under the integral sign of the left-hand -
side of (19.10) in the term 1+ Y, the value of Y has been omitted
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1 l
Y = 8ps~ - AP~ . 4
This is done for the following reason. In order to find the term of the second
approximation it is necessary to multiply both sides of the complete equation
(19.10) by exp (s'pp), integrate with respect to p from 1 to infinity and deter-
mine the second term in the asymptotic expansion of the left-hand side with re-
- spect to pownrs of 1/p. It turns out xhat the value of Y has no influence on this
term; it is related to the terms of high~r order. This ie easy to demonstrate if
- we substitute [a~/aZ~Z=o found from (19.9) in the left-hand side of the complete
equation (19.10). Then the result of integration of the terms related to Y wi11
give a value of the third order in the expansion with respect tu 1/p.
-r~
-
'p; Thus, after repetition of all the argumenta analogous to the case of the first ap-
proximation for (19.1Q), we obtain _
~S+ P) = e~'� ro ~ P(~ ~ ( ~ - 1 -r' 2sp (1- 1 - S) -
L (19.11)
- ~ j/1-sl.
4p ' ~
From (19.13.) it is easy to obt~in
_ ~
S ( a~ ~~rdr - p V ~P) 2 -I- 8p -f- o~P-~] � (19.12)
Then frem (19.3) and (19.12) we obtain:
~
_ ~ m~-p r dr = P y ~P) 2 2p - 8~~ -F ~ ~P-3) J ~
- 1
- F (T) _ - 2n ro pa ~ ( ~ 1 ~ r dr ~
~ /
0
s s
~ rtro Pa f v(~) - s o(z) dx 4( a(x) (z - x) dx 1.
_ ~ o o ~
If the cylinder has large mass, it is possible for 0 5 T< 1 to set v(t) = vp.
Then with accuracy to 0(T3) the following is obtained
F~'~) = nro Pavo ~ 1- T-~- 1 T' 1. _
~ 8 1
Analogously, considering the following terms of the asymptotic expansion, we ob-
- tain
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F(t) ~ nro pavo f 1- t-}- 8 T' 8 -�-0 (T�)1. ~19 .13) ~
l. .1
i
0,5
0
~ 0,?S QS 0,75 1 ~
Figure 1.54.
The function (19.13) is represented in Figure 1.54, where
- F (z)
F� n~ pavo .
In [36] a study was made ~f the problem of impact of a cylindrical body against
the surface of a compressible liqLid in the approximate statement considering the -
- nonlinear effect of the liquid medium and elaetic properties of the penetrating -
cylinder. As a result, the following relation was obtained for the maximum pres- '
sure at the time of impar.t:
P ~oD
A = Pp ' _
1
~ P~ E (1 - v)
(1 -F- Q) (1- 2v)
~ where D is the velocity of the shock wave in the liquid, pi, E, a are the density,
tb.e modulus of elasticity and the Poisson coefficient of the cylinder material.
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-d
�20. Impact of a Cylindrical Elas~ic She.11 With a Liquid Filler Against the Sur-
face of a Compressible Liquid
- A study is made of the impact of an elastic thin-walled shell of circular cross -
section, radius R, thickness h, with flat leading tip and liquid filler against
the horizontal surface of a compressible liquid at rest occupying the half-space.
The velocity of the impact is direc~ed along the axis of the sheZl perpendicularly -
to the surface of the liquid.
vo
~
- . _L.. .
a~�~
. / _
_ iP,
- - - -
- -P' ~ -
-
-
Figure 1.55.
At the time t= 0 normal impact of the leading tip of the shell, which is a rigid
disk of radius R and mass m against the surface of the liquid (Figure 1.55). It
is required that the stressed state of the shell be determined for small values of
the tim~ considering the filler--an ideal compressible liquid. The investigation
of ttie problem of impact of the cylinder against the surface of the compressihle
liouid was made in the ~receding section. If the depth of penetration of the disk -
U1(t) and its velocity Ui(t) are represented in the form ,
_ ~~t + ~ (t), (r~ _ - Uo � u (t~, (20.1~
for t the asymptotic value of the force of the resistance f of the iiquid can be
written as follows:
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t
- f (t) = nR' pa ~vo - U (f) - R (vot - f U dtil
- ~ o ~ (20.2)
~
a
a �0~ - f (I (t - dT 1 ~ .
4Ra 2 0 ~
In formulas (20.1), (20.2), vp is the initial velo~ity of the impact, p and a are
the density and the speed of sound in the liquid at rest into which the shell pene-
- trates. The additional mixing of U(t) occurring as a result of the forces of re-
- sistance to penetration, the elastic forces of the shell at the front tip and the
_ pressure of the liquid filler is subject to definition during the course of solu-
tion of the problem.
y y
e
t~r) U .c
_ ~
x
- - I
Figure 1.56.
Now it is possible to formulate the problem as follows. An external fnrce given
by formula (20.2) begins to act on the bottom of the shell moving with constant
velocity v~, from the point in time t= 0. It is required that the stressed state
of the shell be determined. In the meridional plane the origin of the cylindrical
coordinate axes x0y connected with the body will be taken at the center of the
disk; the Ox axis is directed along the axis of the shell, the Oy axis, perpendicu-
lar to the Ox axis (Figure 1.56). In agreement with formula (20.1), the displace- ~
ment of the goints of the shell in the axial direction ul(x, y, t) will be writ.ten _
in the form
ui ~x~ y, t) vot u(x, y, t). ~20. 3)
Let v(x, y, t) be the transverse shift o_` the shell walls. Since the bottom is
rigid, in the cross section x= 0 there is no transver~e shift, and on the basis
of formula (20.1) we have
X- o, u~o, y, t~ - u~c~, o. _
From the equality (20.3) we obtain tihe velocity and deformation of the shell along
its axis: _
dul _ du au, au
_ at - - + a~ ' ax - aX '
a~, au
ay - ay '
138 -
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The equations of motion of the shell in the adopted coordinated system have the _
form
dt,~y day ay - ve dav
~ y = P~ ~
(20.4)
daX 1 d d4u
a.~ + y . dy (yTxy) = P~ a~, .
where ~X, vy, oe, TXy are the stresses in the general.ly accepted notation, pl is
the density of the shell material. The stresses are related to the deformations
- linearly by Hook's law.
We shall consider that the longitudinal shift u of the shell does not depend on
the coordinate y and, as is usually assumed in shell theory, the normal transverse
stress Qy is negligibly small by comparison with the normal annular and axial -
stresses ve and QX. Let us integrate the equations of motion of the shell and the
relations expressing Hook's law with respect to small thickness of its wall
h([h/R] � 1). Then, introducing the stresses and shifts averaged with respect
to this thickness, using the theorem of the mean and the assumptions made above,
we obtain (the average values of the stresses and strains are denoted by the same
letters as de~:ioted the corresponding nonaveraged values)*: _
aT ve a(X, t) o~o aQX
aX - R'~" R = P~ a=~ ~ dX = P~ dt, ,
Q,~=P~ la' dx +k2 R ) (20.5)
~
a du ~ v 2 du
_ Qe = P~ (k a.Y y Q' R~. P~ a2 az, . _
a~ = 4� + _ 1 E , k2 - aai, as = (20 . 6)
P~ -F 2�) Pi ~1 - Qa) Pi
In these equations QX, Qg, T are the axial, annular and tangential transverse aver- -
_ aged stresses, u, v are the average longitudinal and transverse shifts, a, u are
the Lame coefficients, E, a are the Young's modulus and the Poisson coefficient,
_ p(x, t) is the difference between the liquid pressure and its initial pressure, _
pQ equal to tlie outside pressure. The system of equations (20.5) is reduced to two
equations in the shifts:
_ * If we do not consider the equality
r.
aTxy xxy _ a (y""y)
ay + y ay
and the averaging is carried out term by term, then instead of the second equation
of system (20.5), we obtain
dQx S ~'u -
dx + R - Pl dt' ~
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2 aav k' du 2 v p(x, t) d~u
- a2 axa R a.t - a, Ra - plh = at~ ,
(20.~>
2 ~u k2 du d~u
a~ c3xs + R dt Jt~ .
The initial and the boundary conditions of equations (20.7) will be
- du dv
0, x~ 0, u= v= ot at = 0, p(x, 0) = 0;
s (20.8) -
_ t~ x= m~Q = f-~ 2n.Rh QX (0, t) - nRzP -
v (0, t) - 0.
At infinity (x during the time of the motion the shell and the liquid in it
are at rest.
After the investigated small time interval, the disturbances arising at the front
end of the shell, as a rule, do not reach the other end of the shell. Therefore
a semi-i.nfinite shell is considered. The investigation of the shell of finite
length does not introduce any theoretical difficulties.
The solution of the systems (20.7) is represented in the form of the sum of the _
solutions of the equations
2 d~v k' du 2 u d'u
- a2 a.C2 R aX - Q~ az ar~ '
(20.9) .
2 d2u kY dv o~u
n~ dxa ~ R dx dt2
with the initial and boundary conditions
t-~, x~0, u-v- ac - ar (20.10)
t~ 0, x- 0, m a~u = j- 2nRh v; (0, t)- nR2p (0, t), v(0, t) - 0
~tZ
and the solutions of the equations
2 a2~ k= au z ~ p(X, r~ _ a=v -
a2 aaz R c3x - Q~ RZ r Plh - ar= ' (20. ~1)
2 d~u kz du a'u
a' ax~ + R ax~ ar= -
with the initial and boundary conditions
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au a~
t-0, x~0, u=v= ai - a~
(to.i2)
9
f~ 0, x= 0, m a~" - 2nRh a~ (0, t).
On the basis of the boundary conditions of (20.12) of equations (20.11) it is pos-
' sible to assume that the axial stress 6X arising out of the effect of the pressure
of the liquid on the sidewall of the shell is negligibly small by comparison with
the stresses Qg and T. That is, in this problem along with ay we assume crX equal
to zero. Then instead of the equations (20.11) with the conditions (20.12) from
the system (20.5) we obtain one equation -
a2 a2ti - c2 v+ P~-r, r~ _ a~, ~ ~a _ E (20.13) -
aX1 R~ ~~a~ a~y a~
with the initial and boundary conditions
t-0, z~0, v= a~ =0,
(20.14)
t~ 0, x= 0, v(0, t) = 0.
It is necessary to add the equations of motion of a liquid in the shell which are
considered in the one-dimensional acoustic approximation to the equations (20.9),
(20.13) :
a~, t ap
d t po d.r ~ _
(20.15)
aw i ap ~ ao 0,
_ aX + PoQO at ~ R ar -
where p0, w are the density and velocity of the liquid, a~ is the speed of sound
in it. For (20.15), we have the condition:
t-0, z~0, c~= a~ -0;
(20.16)
t>0 z_o r~_,~,o= auao,r~ =u(t)
The basic prYt of the transverse shift occurring as a result of the pressure of -
ttie liquid on the sidewall of the shell is taken into account further in the sec-
_ ond equation of system (20.15). Using equation (20.13), the system (20.15) is re-
duced to one fourth-order equation for determining the velocity of the liquid (or
the pressure in it): -
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a9 ( i a�w o~, a2 a9 ( i a~, _ a~,l +
- dta 1 a~ at~ dz~ ) 2 dxs l ap dt~ dx~ J
~ ~ i
c' 1 ~3'~ dzru _ 0 (20.17) _
+ R~ ( ~,o ar~ ax~ , - ,
/ p~ \-i
~u = I 2 Eh ~ '
` ao l
If we consider the inertia of the shell walls and the transverse tangential stress,
then the equation (20.17) becomes the known second-order equation describing the
propagation of the "hydraulic impact" with a velocity ao�
Equation (20.17) has two characteristic velocities _
dx dz (20.18)
dt - ~ a, d~ = t
The study of the asymptotic form of the solution of equation (20.17) for small
values of the time demonstrates that the wave with the velocity a2 does not in tro-
duce disturbances into the liquid. Accordingly (the initial period of motion is i
investigated) in equation (20.17) we drop the terms taking into account the trans- ;
verse stress. As a result, we obtain: ~
1 a2w d2u~ cz ' 1 d~~ d4~ 0.
_ ar~ ( ao at~ ax~ ) R= ~~,o ar2 aX= _ (20.19)
- The solution of equations (20.9), (Z0.13) and (20.19) and the first equation of
the system (20.15) will be found using the Laplace transformation. Let us intro-
duce the following correspondences between the originals and the transforms: -
� u(x, t) = U(x, s), v(x, t) =1% (x, s), ul (x, t} = V, (x, s), (20. 20)
r~ (x, t) = t~ (x, s), p(x, t) Q~z~ S)�
Here vl and V1 denote the solutions of equation (20.13) and its transform, s is
the complex parameter of the Laplace transformation. After solution of the corre-
_ sponding ordinary differential equations the transforms (20.20) are represerted in
. the following form:
U- Cl i I a~y~ - sz - Ri J exP IXY~) - Y, / a2y2 - s= - R= exP ~xYz)} ~
` ~ I
~ / Yz ~
~ _ ~~Y~ R {exP (XY~) - exP (XYz)}~ `
Y~._ - ~ ~~A + l~Az - B~ (20 . 21)
alas Y2
( 2 1 ~ ! Q2
A- a2s2 a~ I s= - RQ I- R~ . B= 4a ~a~2 I sz -r- R= l,
~ I \ l
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IV = Wo exp I- as V s= + v1 J~ Q= Poao ~S ~1 (20 . 22 )
l. o
n2
t~o = Wo = SU (0, s), cu' - Q hR v~, va - R ~
z
Vl = N~ k N2 {exp xN,) - exp xNl)},
t 2
k= ao PosU (0 , s) s~ + v' �
a2 pih Y s~ -F- ' (20 . 2 3)
z
2 s9 v~ s a' c~x
Ni = a~ ~ Nz = ~ ~ S~ + y~ .
z
The value of C1 in formulas (20.21) is determined from the boundary condition
(20.10) written in the transforms
x= 0, msaU (0, s) - F= 2nRhplai nR2poao V s~+ SU (p' S~' (20 . 24) -
Here F is the transform of the external force f. In the general case finding the
originals of the transforms (20.21)-(20.23) is a difficult problem. However, for
small values of the time it is possible to obtain an effective asymptotic repre-
sentation of the originals. It is easy to establish that for s-~ ~ in the rough
and more exact approximations the following asymptotic representations occur*:
s ~s'+ei _ s n~
- - 1
Y~ _ l , ~ - -
nl al al s=
2 2
n~ = e~ = R~ v=(1 -a),
s YS2+e2 s n, eZ ai (20.25)
Yx- i--, - ~--(1-; -l,n?=-=--n,,
Qq Q+ Qq \ / ~ 2R~
I
s s--~2 s s 1- k k- 1 c~ -v-
, - ( 2
U~ S' - V~ Qp ~ 4p ( S~ ) 2
According to formula (20.2) the transform F of the force f acting on the end of
the shell is defined in the form
_ F=;~R~puvo ( 1 1 t a1 1~- nR2~a ( SU - aU - a~U I (20. 26)
~ s R s= ~R1 s~ R 4R's J'
* For the averaging indicated in the first footnote of this section we have
_
n ~ R=2 ( l - 2a"-) .
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i~crlt U~~~~tCt;~~, t~ti~~, u~~.1'
From the first equation of formula (20.21) for the mapping of U and its derivative
with respect to x at x= 0 we obtain:
a?
U (0, s) _ ~l Y~ - Y~ a2y~Ys sz ~ , _
Y: R (20.27)
~lU
dx - C1Q2Yi ~Yi - Y2)�
Substituting these mappings and the mapping F from (20.26) in the boundary condi-
tion (20.2 4), we obtain the linear equation defining the integration constant
C1(s). Af ter determining this value it is easy to see that for the asymptotic
va lues of (20.25) the originals of the mappings (20.21)-(20.23) are represented
by quadratures [37, 38]. We shall limit ourselves to the investigation of the
simplest asymptotic representation where the terms of order 1/s2 and less by com-
_ parison with values on the order of one are neglected in the mappings. In this
case, we have (s _
s s 1/ s~ u~' s a 20 . 28
Y~ Ql ~ Yt a: ' V s' + v' 1' Nl a' NZ Q. ~ ) -
= o
He re the mappings introduced above are simplified and assume the form '
:
-
- U- nR'p~vo 1 s- a e ol
m sa s ~ ~
X a - z s (20.29)
nRapu~al 1 s- a Q, Q,
V = j e - e
~ - a2~ a~ S + ~
v Pa�o�z U(0, s) [e a, s- B a, S~ ~
i =
Plh ~ao - a~) S
The following notation has been introduced
a " 2Rh a + a ) (20.30)
- R, a= m( Pi i R'Po o�
y~Ihen the shell does not contain a liquid, the mapping V1 is equal to zero, and in
the formula (20.30) the last term in the expression for S is absent. As is obvious
- f rom the f irst equality (20.29) in the above-indicated approximation the mapping U
fo r longitudinal shift does not depend on the wave propagated at a velocity of a2.
When keeping the largest term considering this wave, the mapping U has the form
U~x~ S~ -:cR pavo I s- a e a, S_ i�~ 1 s- a e c, s
_ m ~ s� s ~ R2 (a~ - a~) s+ s -f- ~ ~ .
The mappings of the deformations and the shift rate of the shell are obtained by
differentiation of the corresponding mappings in (20.29):
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du dU du y SU dv _ dV dvl } dVl ~20. 31~
dz dx ~ dt dx dx ' dz dz ~
Let us introduce two functions into the investigation:
~ ~t) _ ~a T - ~ t9 ~ ~a -I- t - ~a (20. 32) -
f ~t) _ ~a -t- ~'s~ a~f - (a -i-
Then, going from the mappings of (20.29), (20.31) to their originals, for the ion-
gitudinal shifts, deformations and velocities we obtain:
u(x, t) H(t - l l~(t Q~ 1'
~ ~ ~ i
_ du :tR'pavo H(t - x\ t f t- x 1~ (20.33)
dx mal~~ ~ al ) l al J -
dt ~m~~av~ H(t-ai/f\t al/~
For the transverse deformations
dv nRpav~ ' H (t- rl ~ (t- xl -r
d.r m~a ~Q~ - a2) \ ai / \ ai /
2 2 ~ '
~'~a ~ ~ _ a a
~ ~RPuv~'aa H 1t ~ l 1t --1 / -
nR au (20.34)
dul ~P aPoao x x
00
H t--l ~`t--~ ~
dx mPihRa ~ap - a2) ao / ao
n~~~oQ~ X ~ ~ X
H~t--~~(t--~.
mPi~s~~ ~ao - a2) as . as ~
In the formulas (20.33) ,(20. 34) , H(z) denotes the unit Heviside function. The
transverse shifts v and vl are expressed by the formulas:
v(z~ t) _'~RP 2�k~a~ , H~ t- X 1 m r t- Xl
(a, - az) ai ! ~ al / -
-H(r-X~~(t- .
a: a.
(20.35)
ol (x, t) = ~ nRaP~~�P�~� h ,H ~t - x 1 ~ rt - x \ -
( p - 2) P~ �o / ~ ao /
-N(t- X lm(t--~~. -
as / , as ,
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i'OR nI~ FIC'L~l. t~SF: Oti I.1' -
In these formulas the function ~(t) has the form
~ ~r~ _ - ~a + e-~r - a68 te ~ ~2 ~a 2 tz _ ~ = t + (a + (ZO. 36>
= According to (20.5) the average stresses in the shell are defined as follows:
~ � du 2 v vl \ '
- ti= Piaz ( aX T~a
1), Qx = Pi (ai aX + k R l -
Qe = P~ ~k2 dr a~ v Rvl
The pr~ssure of the l~,iuid inside the shell is repr.esented in the form
P=- ~Z
~m~av0 PoaoH (t ao ) f\t ao 1
_ The penetration rate of the rigid bottom of the shell is defined by the formula
(20 .1) ~
~
1 I
JcR'p~vo f ~t) J � ~
~ �
ui = - Zo ~ m~:
Hence, for acceleration of the rigid bottom of mass m, we obtain
d2ic1 ~u rtR~pavo r \ 1 + ~ / ~~t ~ J -
_ dt~ = dt~ - m l
Now let the penetrating cylindrical body be a continuous elastic rod of semi-infi-
nite length of radius R. In the one-dimensional statement for the longitudinal
shi~t u(x, t) we obtain the equation
v~u _ 2 d'u (20.37)
-c-
dla dx~
Here the wave velocity is denoted by c
Cz = E . ~
p~
The initial and the boundary conditions of the problem will be
t 0' u- ar (20. 38) -
t~ p, x= 0, - nR~aX = f(1) ~
f(t) in the bour..dary condition (20.38) is defined by the formula (20.2). In the
investigated r~roblem the mapping U(x, s) of the shift u(x, t) has the form
146
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FpR UFf~CtAL US~E ONLY -
~
kv s4 R + 4R1
- U(x,s)--�
~
(20.39)
8 S~ - k( R S 4R' ,
k - ,~t M - Pavo .
M v� ~ Pi~' ~
c
The mappings of the deformation and the shift rate are defined by the correspon-
dences:
du } dU au _ SU, -
aX ~X ar
Formula (20.39) shows that these mappings are simple, and their originals are de-
fined without additional restrictions. As a result, for deformations and shift -
rate w~ have
N(t-Xl
ax = - ~'0 e ~ � ~ {sin Nw (t - ~ 1 c~ cos
Nw (t - ~ 1
~ / ~ /
~
4 N~~~l+~~ [sin N~ (t - ~ I-~cosNcu (t- ~ 71 -
~ 1 \ ~
- N sinN~ (t - c ~ ~ N (t _ x ` _ (20.40) `
\ ~ c~
ku0 bs ~ H (t- x ~u C au
c 4 N9 ( l-F- \ c)' c't ax '
b-R , w~= p~,,v= 2 ,
P
The atress csX and the total velocity ul of the particle.s of the elastic rod are -
defined as follows:
oX = E a`~ , a1= - uo ;~t ,
147
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- f
OR OF~[CIA1. t'SE: O;~iL1'
~21. Estimating the Effect of Viscosity on Penetration of a Liquid by Solid States
In order to es t imate the forces of viscosity on penetration of a liquid by solid
states, l.et us use the well-developed boundary layer theory [3�, 4~]. First let
us consider the problem of penetration by a semi-infinite plate witl~ constant ve-
locity vp.
Let us select the origin of the moving system of coordinates at the edge of the
plate, let us direct the ~ axis along the plate, the y axis along the normal to it
(Figure 1.57) .
~ {
I
2y -
y y
Q 6 _
Figure 1.57.
In the moving system of coordinates the equations of motion and continuity in the
boundary laye r of a barotropic liquid without pressure gradient are written in the
form
~u du du d~u (21.1) -
ac ag + v ay ay= ,
- (21.2)
au , a~
.~.-=0.
a~ ay
Here u, v ar_e components of th~ relative veloc~~y of the liquid with respect to
the ~ and y axes, respectivelY; v is the coefficient cf kinematic viscosity of
the liquid which is assumed to be constant. Let us proceed in equations (21.1)-
- (21.2) to the new variables: _
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_ u y, t) = vou (S, ~ y, t) _ ~ ~ ~ (S,
s- ~yt ,
where u and v are the dimensionless components of the velocity.
Omitting the bar over the dimen5i~nless variables, we obtain:
= aa ~ au a~ au ~su (21.3)
-s--- _ ,
ds 2~ u aa + v ~
au ~ av _0, O~sC 1, 0~~~~. (21.4)
c~s dr~
The boundary conditions for the equations (21.3)-(21.4) will be
u(s,0)-0 for t~~0,
u(s, oo) = 0 f or
For equations (21.3)-(21.4), i*_ is poss~.ble to introduce the current function ac-
cording to the equalities:
= u- v=- (21.5)
a~ ~ as '
Here the con~inuity equation is satisfied identically, and the substitution of ex-
pressions (21.5) in the equation of motion (21.3) leads to an equation for the
current functicn:
- s - ~ - a'~ _ (21. 6 )
asa,~ 2 a~~ a~ a,~as os dn~ - dti~a
with the boundary conditions
~ (s, 0) = 0, ~ - 0 for ~ = 0,
a'~ = l f"or t~ = oo.
a,~ ~
~ Let us represent the current function in the form
~ ~S, ~I) = j~s f \ V9 ~ � (21. 7 )
Then for the longitudinal component of the velocity we obtain the expression
d'V d~4 dy , - ~
u=-=---=f (~1, y - _ ~zi.8~
- d*~ ay ti~ s~
14 9
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Substituting the expression for 'Y from the expression (21.7) in equation (21.6),
for determination of the dimensionless current function f we obtain the Blasius
equation:
2 f,,, -F f~" = 0. ( 21. 9)
- The boundary conditions for the third-order ordinary differential equation (21.9)
- will be
- f- p~ f~ - p for y= 0,
_ (21.10)
f' = i for y = .
The equation (21.9) was integrated numerically, by the Runge-Kutta method, by many
authors, and its solution has been tabulated [39].
~ By the tangential frictional stress on the plate we obtain the known expression
au �va o~~ ~~o f" (0) (21.11)
_ ti=�(-l - ,
, dy /y~ ti/vt a~~~ )n~o Yvt ~ s
where u is the coefficient of dynan?ic viscosity of the liquid which is assumed to
be constant. _
_ The force of frictional resistance is
_ .
Fz (t) = 2f" Pvo Yvt S~S = 4~ Pvo V vt, (21.12)
0
f
(0) = 0,33206.
Let us define the force of the resistance on penetration of a viscous liquid by a
thin wedge. Neglecting the pressure gradient, for small half-apex angles of the
wedge, the force of the frictional resistance FT is close to the force of resis- _
tance to penetration of the plate, and it is possible to consider that it is de-
fined by the formula (21.12). In this approximation the total force of resistance _
to penetration of the wedge is
F ~t) = Fn ~t) + Ft ~t)� (21.13)
Here Fn is the force of resistance on penetration caused by the component pyy of
the viscous stress ter_sor. It is possible to show that the value of Fn(t) coin-
cides with high accuracy with the force of resistance to penetration of an ideal
liquid by a thin wedge. Actually, the expression for the component of the stress
tensor pyy has the form:
Pbd P~ (`s, t) - 2 � au rt 4 � ao (21.14)
s a~ a ay
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where pl(~, t) is the pressure in the boundary layer, u, v are the longitudinal
and the transverse components of the velocity, respectively. Let us write the
' continuity equation for the problem of penetration of a compressible liquid by a
, thin we~lge in the linear statement
i apl ~ p au + P av - ~ ( 21.15 )
a~ at aE ay
Substituting 8v/8y from equation (21.15) in equation (21.14), we obtain the ex-
= pression for the stress pyy fcr y= 0:
_ ~Pyy)u~o = - Pi t) - 4 v ap, (21.16 )
3 a~ ot
_ The values of pl in the stationary coordinate system for incompressible and com-
pressible liquids, respectively, are presented in 52 and �9. On substitution
of pl in expression (21.16) it is first necessary to proceed to the stationary co-
ordinate system.
The magnitude of the resistance force Fn(t) is calculated by the formul~.
~,t
F~ ~t) _ `l -r ~P~y) u-o~�
b
Using equality (21.16), after calculating the integral we shall have -
Fn ~t) = n~~M) PvoY2 (UOf -L- 3 ya l, t~ 0, ~
~
~P(M) = ln(1+Y1-M~~ ~~M~ 1~ (21.17)
1 - M~ '
~ ~M) = arct8 (l~,N' -1>, M ~ 1.
~Mz - i
As is obvious, the second term in the parentheses in the right-hand side of formula
(21.17), considering the effect of viscosity, is constant. For an incompressible
liquid (M = 0), the force Fn(t) does not depend on the viscosity. Introducing the
depth of penetration H= v~t, we rewrite formula ~21.17) in the form
Fn lt~ _~~P 1M~ P"pt'zH 1 1~ 3 aff ~21.18~ -
The depth H at which the second term in parentheses is 0.01 for water (v = 10-6
m2/sec, a= 1,500 m/sec, M~ 1), is on the order of 10-e m, that is, i.n reality
_ when determining the component Fn of the resistance force the viscosity effect can
be neglected. On penetration of an incompressible liquid by a wedge, from (21.17)
we hav~
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(21.19)
F~ ~t) = 4 ln 2p oy~t, ~p ln 2.
n
For th.is case let us compile the ratio
F'n l:t 2 2 t (21.20) -
~ ~t) = Fj = n~� ~0~ Y vo ~ y '
or, introducing ttie depth of penetration H= vpt,
F'� ln 2 , voH
~ ~t) = FS = ,~f� 1'" v ' -
The depth of ~ubmersion at which FT is 10 percent of Fn is defined by the formula
H _ toon~v jf" (0)1' .
y~vo ln' 2 ~
for a thin wedge with half-apex angle y= 10� penetrating water (v = 10-6 m2/sec)
with a velocity 100 m/sec, this depth is 2.~ � 10-3 m. Foz~ a depth of penetration
H= 2.5 � 10-1 m the frictional force FT will be 1 percent of the resistance force '
Fn.. Tlie estimate of the effect of the pressure gradient on the frictional force
_ on penetration by a thin wedge can be made by the Karman-Polhausen integral method _
[39, 40J.
Let us introduce the thickness of the boundary layer d into the investigation and
let us proceed to the dimensionless variables by the formulas:
u= vou (s, r~), P1= PavoP~ ~S), (21. 21)
, S - j/vt S (s), s = at ~ ~1 - yy~r '
In these equdlities the dimensionless parameters are noted by a bar at the top.
The velocities of the external flow with respe~t to the wedgp are Pxpressed in
terms of the absolute velocity of the disturbed motion of the liquid ul (see ~2, 9)
~ by the formuia
u� = vuu~ = vo - ui~ (21. 22 )
Let us select the distribution of the dimensionless velocity by the thickness of
the boundary layer in the form
2u (s) u~, (S) (21. 23)
u (s, ~I) _ ~ ~1- s, ~S~ ~12, ~ ~ ~1 C 8 (s),
b (s)
then for the tangential stress on the wedge we obtain the formula
2�~o u~, ~S) (21.24)
T - Yvt a f9~ .
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_ In formula (21.24) and below the bar o~rer the dimensionless parameters is omitted.
In accordance with the Karman-Polhauaen method for determination of the dimension-
- lese thickness of the boundary layer d(s), we obtain the equation
du u
( 3 5 Mu� 1 u� da 3 ds 6+
/
du 2u (21.25)
3 Mu�� ds rt 5 Mu�� da ~ b__ 8.
This ordinary first-order differential equation is solved under the condition
8 _ s = (21.26)
On integration of equation (21.25) usually the new desired function a= d2 is in-
troduced.
_ The force of the resistance F,*~ is defined by the formula -
~t
Fi = 2~ tid~ = 4pvo j/vt (Il !s)� (21. 27)
Here in the case of a compressible liquid I1, I2 denote the integrals:
M M
1 ds 1 ul (s) ds
- Il ~M' - M ~ d (s) ' 1 z (M, Y) _ /y g ~s~ , -
0 0 (21.28)
s- ~ ,
at
- In the case of penetration into an incompressible liquid these integrals assume
the form
t i
- 1 ~ ~Y) _ ~ 8
~s~ ~ ~z ~Y) = - ~ "'8 ~s~ s , s = ~ . ~ -
~ 21.29)
0 0
The dimensionless thickness of the boundary layer which enters into the term under
the integrals (21.29) is defined from the differential equation
s 2 dS , r s du� u_ + 2 u~ du� \ S=
( 3 1~ u`~) u~ ds 3 ds 6 5 ds ~
- 2u� ~ (21.30)
, s~ �
- 8 v~!
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. ,
fOR OPFlCIAL L!SE: ONI.I'
In the absence of a pressure gradient the integrals I2 are equal to zero. The
equations (21.25) and (21.30) were integrated numerically by the Runge-Kutta
method. In Table 1.3 with ar: apex angle of Y= 10� the values of I1 and I2 are
- presented for different Mach numbers (M = 0 corresponds to the case of the inCom- _
pressible liquid).
Table 1.3
M I !,(M,IO�I I l~(M~IO�) I Fi/Fs
- 0 0,3591 0,0711 1,2955
~ 0,01 0,40;4 0,0037 1,2408
0,3 0,3258 -0,0048 0,9666
0,6 0,3109 - 0,0176 0,8832
0,9 0,2867 -0,0241 0,7910
1,0 0,2401 -0,0248 0,6~84
- 1,2 0,2906 -0,0281 0,7905
1,5 0,2844 -0,0272 0,7745
In the last row of the table, the ratio of the force F* calculated by the formulas
(21.27) to the force of resistance of the plate FT calculated by the formula
(21.12) is presented. Table 1.3 shows that consideration of the pressure gradient _
leads to some variation of the force of frictional resi5tance by comparisan with
the plate. It is easy to show that this difference decreases with a decrease in
_ the apex angle of the wedge. However, the variation of the frictional force
_ caused by the pressure gradient does not change the order of the ratio of the
forces (21.20) .
The viscosity effect on penetration of a compressible liquid by blunt bodies can
be estimated using the generalized Newton's law. In �'2 the asymptotic solution
was obtained for the problem of impact entry of a rigid plate (half-band) of width
2c into a liquid for the initial period of ti.me (t < c/a, a is the speed of sound
in the liquid). In the initial stage of submersion, the component pyy of the
- stress tensor has primary influence on the resistance force, and the role of the _
tar.gential trictional forces is insignificant.
- The expression for pyy has the form [39]
2 du , 4 dn (21.31) _
P~b=-P~-3~`d~ ~�au~
it is assumed that pl(~, t) is the pressure known from the solution of the ex-
ternal problem (�12).
- The continuity equation has the forni
1 apl au av (21. 32 )
- n2 ~ f -F. p c~ ~ p ay = 0.
Substituting the value of av/8y from equation (21.32~ in expression (21.31), we
obtain
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4 v cPi (21.33)
~Pyy)u=o = - Pi - 3 a4 ot '
The magnitude of the resistance force for vertical penetration of a plate of width
2c with constant velocity v~ is
2~ as
F~ ~t) P~d~ 8 v~ aP' d~ = 2PuvoC (1 - ut 1~
3 a~ . at ~ 2~ ~ -
0 0 (21. 34)
- 3 �vQ, p C t C a
It is possible to assign the following form to the formula (21.34):
Fn ~t) = 2Pavo~ [ ~ 1 2c ) + 3 ac ~ C f ~ Q . ~21. 35) -
For t= c/a, from (21.35) w2 obtain:
- F~ (t~ - 2P~voC I 2'"~ 3 aC
~
~ior a plate with a width on the order of one (2c ~ 1) penetrating water, the value
in brackets in formula (21.35) has the order [(1/2) + 10'8], that is, it is possi-
ble to neglect the fo rces of viscosity and set Fn equal to
Fn (t) = 2pavoC I 1- 2c l, 0~ t C a. (21. 36 )
~ i
Let us note that in the problem of normal impact of a cylinder against a Iiquid
surface (�19) the relative effect of the viscosity on the re~istance force has the
= same order as in the two-dimensional problem of penetration of the plate, and it
can be neglected.
The solution of the problem of vertical penetration by a thin cone obtained in ~7
indicates that the cone introduces a weaker disturbance into the liquid than a
. thin wedge with the same apex angle and pen;~tration velocity. The estimates anal-
ogous to the estimates made in the two-dimensional case indicate that the magni_-
tude of the force Fn(t) coincides with great accuracy with the magnitude of the
force of resistance to penetration of an ideal liquid by a thin cone.
For determination of the tangential stress T and the frictional force FT it is
necessary to solve the problem of the boundary layer with the external flow veloc-
ity u~ equal to
I
u� = vo ~ 1- ul), (21. 37 )
- where ul is the dimensionless velocity of the disturbed motion of an ideal liquid
on the generatrix of a cone on the order of o(Y2) (~7). If we neglect the small
155 `
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_ addition in expression (21.37) by comparison with one, we obtain the problem of
the boundary layer with the veloc ity of the external flow equal to the penetra-
tion velocity of the cone v~. Iri the meridional plane the origin of the moving
r.oordinate system is placed at the apex of the cone, the ~ axis is d~rected altrtlg -
its generatrix, and the y axis is perpendicular to it (Figure 1.57b). Then for
the assumption made above, the b a sic equations of axisymmetric boundary layer in
the plane y0~ are written in the form
ou a� au a~� (21.38) -
ar a~ ay - ~ ay~ ' _
d (ru) , ~d (rv) _ (21. 39)
d~ ~ dy
Here u, v are components of the relative velocity of the liquid with respect to
the ~ and y axes, respectively; r(~) is the equation of the generatrix of the
cone.
Let us introduce dimensionless parameters and independent variables by the follow-
ing formulas into equations (21. 38)-(21.39) : -
~ v -
u(~, y, t) = vou (S, ~n), v(~, y, t) v(S, ~
s = ~ , ~l = ~
v~ , r = Y~ = y'~ots. _
Omitting the bar over the dimens ionless variables, we have
du r~ du vu , U v~~ _ d~u (21.40)
- s as 2 or~ * u os ' aYIZ '
d(su) + d(s~) - p, 0 5 s C 1, 0 C~ C oo� (21. 41)
ds d~l
The boundary conditions for the equations (21.40)-(21.41) will be
u(s, 0) = 0 for 0, ~
u(s, oo) = 1 for oo�
Let us introduce ttie current function according to the equality
a,~ aW ( 21. 42 ) -
su= a , os ~
~1
Substitution of expressions (21.42) in equatian (21.4U) leads to the equation for
_ the cur.rent function
,
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S~ a~? _ Sg v~~ a~ a
a~ a~ds 2 a,~= a?~ ~ 1
1 S diY . dlYr _ S dYi . d~iY _ SZ as~Y (21. 43)
' dr~ J~ds ds dt~ d~~
with the boundary conditions
ip= ~ =p for t~=0,
1 aY' = 1 for r~ _
s or~
_ Let us assume that the current function can be represented by the equality
~S~ ~1) = s ~ 3 f (Y3 ~s ) , (21. 44 )
then for the dimensionless current function f(z), z=~(n/~) we again have the
Blas~us equation (21.9) with the boundary conditions (21.10).
The tangential frictional stress on the surfa~e of a cone is
T _ � ~ a" \ _ �vo 1 ( ~~Y _ ~�vo f' (0) (21. 45 )
d y l ymo j/ vr s l dri~ ) n~ o vt y~s
For the force of frictional resistance on penetration of a viscous liquid by a
thin cone we obtain:
v.t
Fz ~t) = 2nY ~ T~ nf PvoYf 1~,
a (21.46)
j" (0) - 0,33206. ~ -
The ratio of the force of resistance Fn(t) during subsanic penetration of an ideal
liquid by a thin cone (formula (7.22)) to the force FT is (0 ~ M~ 1)
~ ~t~ _ Fn _ ~ Ywo (l n ~l1~ t ( 21. 7 )
F~ 4f" (0) \ 2Y / y .
For v~ = 100 m/sec, Y= 10�, the inverse of this ratio is 10 percent at a depth of
H= v~t = 1.9 � 10-2 m and 1 percent for H= 1.9 m. These es~imates are retained
also for supersonic penetration of the thin cone investigated in �7. Consideration
of the pressure gradient by the Karman-Polhausen method leads to the following
equation for the th~_ckness of the boundary layer:
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_ S dS ( s du~ _ I ~YIu~, ~p` -
3- i5 Mu�� 1 u�� ds 3 ds 6 ' 3 ds
~ (21. 48) -
~
2 .Vfu;, , lYIU du~ l S = - u~ ~ S = .
` 15 s T ~ ~ ds ~ b at
Here the dimensionless variables d, u~, pl are defined by the formulas (21.21).
- The equation (21.48) is solved under the condition 8(0) = 0. The frictional _
stress on the generatrix of the cone is defined by the formula (21.24), where d(s)
is the thickness of the boundary layer on the generatrix of the cone. For the
frictional resistance force we obtain the e~:pression
Fi = 4:cpv~ jl vt (Il l.:), (21. 49) -
_ where I1 and I2 denote the integrals
1 (N' sds 1(~l'~ ul (s) sds (21. 50)
I' - ,u J ~ 8 (s> ' IZ ~r1 J b (s) ~
0 0
For subsonic penetration of a compressi~le liquid by a cone (M < 1) as demanstrated
in �7, the external solution does not depend on the Mach number, and it coincides
with the corresponding solutian of the problem of penetration of an incompressible
liquid by a thin cone. In this case the thickness of the boundary layer is deter-
mined from the eqsation
- ( s 2 dS , ( s du.~ ? u;,
` 3 15 u`~ ) u`~ ds 3 ds ^ 6 15 s~ -
(21.51)
2 du� \ 2uo, ;
-I- 5 u� ~ ~ g = - s , s = p~ ,
- and the integrals I1 and I2 in formula (21.50) assume the form
1 sda 1 r~l (s) sds (21. 52)
Il ~ 6 ~S~ , 1~ _ - ~ a ~S~ .
0 0
The equation (21.51) was numerically integrated by the Runge-Kutta method. As a
- result of the calculations, the following values were obtained for the integrals
(21.52) and the ratio of the force F,*r to F.~ calculated by the formulas (21.44) and
(21.46), respectively:
� 2
I, - 0,1628, !Z 0,0038, F` = 0,8294. ( 1. 5 3
~ �
,
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~
ORIGINAL TABLE OF CONTENTS
Introduction 5
Chapter 1. Penetration of an Ideal Liquid by Solid States 7
~1. Equations of Se~.f-Similar Motion of a Liquid 9 -
- Penetration of an Ideal Incompresaible Liquid by Solid States 20
�2. Penetration of an Incompressible Liquid by a Wedge 20
�3. Penetration of an Incompressible Liquid by a Cone 27 _
_ g4. Movement of a Thin Cone in a Liquid of Finite Depth 37
~5. Ricochet of a Plate from the Surface of an Ideal
Incompressible Liquid 45
~ Penetration of a Compressible Liquid by Thin Bodies 55 ~
~ -
�6. Statement of the Problem in Equations of Motion 55
g7. Vertical Submersion of the Thin Solid of Revolution -
_ Without an Angle of Attack. Penetration by a Cone 57
- �8. Inclined Penetration by a Thin Body at an Angle of
Attack. Inclined Penetration by Cone 64
~9. Penetration of a Compressible Liquid by Thin Flat =
Bodies. Penetration by a Wedge 72
Penetration of a Compressible Liquid by Blunt Bodies 76
~10. Penetration of a Comgressible Liquid by a Slightly
_ Distorted Outline 76
�11. Penetration of a Compressible Liquid by a Blunt Wedge 81
_ �12. Penetration of a Compressible Liquid by a Flat Plate 91
~13. Consideration of the Lift of the Free Surface on
Penetration of a Compressible Liquid by a Blur~t Wedge 99
~14. Penetration of a Compressible Liquid by a Blunt Wedge
in the Nox~linear Statement 105
�15. Penetration of a Compressible L~quid by Blunt Three-
Dimensional Bodies 111
~16. Penetration of a Compressible Liquid by a Blunt Cone
Considering the Lift of the Free Surface 115 -
i
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�17. Penetration of a Compressible Liquid by a Blunt Cone
- for V > a 128
~18. Nonlinear Problem of Penetration of a Compressible
Liquid by a Blunt Cone 133
~19. Impact ofi. a Rigid Cylinder With the Surface of a
Compressible Liquid 138
g20. Impact of a Cylindrical Elastic Shell With a Liquid
Filler Against the Surface of a Compressible Liquid 143
~21. Estimating the Effect of Viscosity on Penetration of
a Liquid by Solid States 153
Chapter 2. Penetration of Soils by Solid States 165
~ 1. Experimental Methods of Studying the Process of Im-
_ bedding Bodies in an Obatacle 168
- ~ 2. Model Medium for Soils 171
~ 3. Approximations for Studying the Problem of the
Penetration of a Plastic Compre~sible Medium
by Bodies 173
~ 4. Penetration by Sharp Thin Solids of Revolution 178
~ 5. Penetration by a Thin Cone 185
~ 6. Numerical Results 190
~ 7. Penetration by a Cone with Finite Apex Angle 191
~ 8. Penetration of the Ground by a Solid Cylinder 204
~ with Flat Front Tip 204
~ 9. Penetration of the Ground by a Sphere Along the -
Normal to the Ground Surface 210 ~
~10. Penetration of the Ground by a Cone at an Angle
to the Ground Surface 22~ _
~11. Approximate Solution of the Problem of Ricochet
- from the Surface of the Gr~und 242
Chapter 3. Penetration on High-Speed Impact 25Q
~ 1. Penetration Process 251 -
~ 2. Hydrodynamic Model of Penetration on High-Speed
Impact 25~
g 3. Modified Hydrodynamic Theory of the Braking of a
Hammerhead 259
~ 4. Supersonic Penetration on High-Speed Impact 274 _
~ 5. Supersonic Penetration by Long Hammerheads 275
~ 6. Numerical Studies of Supersonic Penetration by
Long Hammerheads 280
~ 7. Results of Calculating the Impact of Viscoplastic =
- Bodies 287 -
g 8. Interaction of a Hammerhead with a Block of Finite
Thickneas 290
Bibliography 29~
* The remainder of the book was not translated. ,
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