On some problems of similarity flow of fluid with a free surface

1. Introduction

Hydrodynamic problems of flow with free surfaces _{(in particular, the} water-entry problems) are essentially non-linear. TI Ic difficulty in solving these problems is that of sat isfing the non-linear boundary_{conditions on the free surface, which} is not a stream-surface in unsteady motion.

The problems under consideration have been studied by many authors. The theoretical analysis of similarity flows of an incompressible fluid was first pre-sented by Wagner (1932) who obtained, in particular, an approximate solution of the wedge water-entry problem.

Thereafter this problem was investigated by Pierson (1950), Garabedian (1953), Borg (1957), Moiseev et al. (1959) and others. _{The similar problem of the impact} of a water wedge on a plate _{was studied by Cumberbatch (1960). But until no} HO exact solutions (analytical or numerical) of these problems have been

obtained.

The complete solution was obtained only for the linearized wedge water-entry problem. For the case of a compressible fluid it was given by Grigoryan (1956) and Sagomonyan (1956); the results for an incompressible fluid were first written explicitly by Mackie (1962).

In § 2 of the present paper the definition of Wagner's function and its main

J. i'1ujd jlferh.(19139), vol. 36, pare 4, pp. 805-529 ₈₀₅ Prirtecl in Great Britain

On some problems of similarity flow of

fluid with a free surface

By Z. N. DOBROVOL'SKAyA

Computing Centre of the Academy of Sciences of the USSR, Moscow (Received6 May 1968)

The paper presents the method of solving a class of two-dimensional problems of the similarity flow of an incompressible fluid with a free surface. The fluid is assumed to he non-viscous and wcightless. We consider two-dimensional irrota-tional similarity flows with dimciisionless hydrodynamic characteristics depend-ing only on the ratios x/0t, y/v0t, where _{x, y are Cartesian co-ordinates, t is time} and v0 is a constant of the velocity dimension.

The proposed method is based upon using the function introduced by Wagner (1932) and can he applied to the problems where_{the flow region is bounded by} free surfaces and uniformly moving (or fixed) rectilinear imfermeable boundaries. Introduction of Wagner's function makes it possible_{to reduce each of the} prob-lems under consideration to a non-linear singular integral equation for the real function.

The method is illustratcd by solving the _{classical problem of the uniform}

I?.

I

806 Z. N. Dobrovol'8kaya

properties are recalled. In § 3 we give the method of Wagner's function for

reducing the problems under examination to a singular integral equation. In § 4 the integral equation is derived for the problem of the unifirm symmetrical entry of a wedge into a half-plane of a fluid. In § 5 we show the applicability of the successive-iterations method for solving the integral equation obtained and give the scheme of numerical integration of this equation.

Results of the numerical solution of the wedge problem are given in § 6. Presented and analyzed here are the free surfaces for different wedge angles and the curves of the pressure distribution along the wedge; in particular, the free-surface behaviour near the wedge is investigated. The obtained numerical solu-tion is compared with the analytical solusolu-tion of the linearized problem.

2. Wagner's function

Let x, y be fixed Cartesian co-ordinates in the plane of flow (which we shall call the 'physical plane'), t, time and = x/(v0t), = y/(v0), the dimensionless similarity variables (v0 = const.). In the (, )-plane a stationary region corre-sponds to the physical flow region. varying with time. But the part of the boundary of the flow region, corresponding to the free surface, is unknown in advance in both planes.

The velocity potential (x, y, 1) and the stream-function (x, y, t) of the flows under consideration have the form

(x, y, t) = v(l)(, _), (x, y 1) = vtT(. j),

where (i)(,

) and T(,

) arc harmonic functions of and . U(z, t) = ç(x. y, t) ± ifr(x, j, t)

is the complex velocity potential in the z-plane (z = x±iy). Let us introduce

function

_{V() =}

)±i'F(1)

( =

The, function V'() is connected to the complex velocity U(z, t) by the obvious

relation _{U(z,t) = v0V'(L).}

(2.1) Therefore, it is natural to call the function V'() the complex velocity and V() the complex velocity potential in the c-plane.

Wagner's function h can be determined as followst

J()

f(dV))d

(2.2)

The purpose of introducing Wagner's function is that, in the plane of this function, the free surface of a fluid is always represented by a segment of a straight

t Wagner introduced the function h (for the wedge water-entry problem) in a slightly different way, namely

h=

where z = x + iy and x, y are the Cartesian co-ordinates moving uniformly with the wedge.

-

.-_=

-- a....

--.-.-'--- --.S...1. - _----. - - --- .. p.._

f:J(dU'(.. t))

--a----

--_-.._.a.. *a...lqna.*A.-. (2.4)

(2.7)

(2.9)

Similarity flow of fluid with a free surface R07

line (or h a broken line). We sliill demonstrate below that for the flows under COflS1(leFatiOn the flow region hounded by thin tree surfaces and the uniformly-moving (or fixed) rectilinear mpermeable boundaries, is always known in the plane of \Vagner's function. For this purpose we represeiit the integrand expres-sion of formula (2.2) in the form

(2.3) and investigate first of all the behaviour of function (2.3) on the free surface following Wagner's presentation.

In the similarity flow with the variables

gx

-,

Y

the following relation holds for the velocity vector ll(x,y, t)

11 (x,y.t) = v0 _(2.5)

After differentiating the left- and right-hand sides of (2.5) with respect _{to t,}

we obtain

d,y,t)

x/vot,:i//v0t)x _{t9J(/vot,y/v0f)}

?J

1 _{(° 6}

L (x/v0t) v012 ô(y/v0t) v012j

Let us consider (2.6) on the free surface when t is fixed. WTith t fixed, the variables x, y on the free surface are single-valued functions of the arc length .s measured along the free surface. In this case (2.6) reduces to the form

dll(x.ij.()

_{Jl.l(,1J,) (dsxd.sy}

dt s \dx t

The expression in the brackets in (2.7) is a scalar function, and hence the_vector hl(x. y, t)/s has the same direction as the acceleration vector dU(x, y. t)/dt at any point of the free surface. I'ressure on the free surface being constant, the pressure gradient on the free surface is normal to the latter at any of its points.

Therefore, according to the Euler equations, the acceleration dU/dt of fluid

Particles lying on the free surface is also normal to the free surface. Taking this into account., it follows from (2.7) that the increment of the velocity vector along the free-surface element is normal to the free surface at any of its points.

The same fundamental fact can be re-proved in a more formal way. In fact, the total differential of the function ll(x, y, t) has the form

clU(x,y,t) = _(2.8)

at a . ay

Introducing the similarity variables (2.4) and taking into account (2.5). the rela-tion (2.8) can be reduced without difficulty to the form

Fa3(, ?/)

_d+

b93(, ?/)

d dU(x, y. t) = v0

SOS _{Z. N. Dobrovol',s'Jcaya}

or, as might be expected,

(i1l(X, i/, 1) = v0d(, 'ij). _(2.10) Formula (2.10) shows that in the flows under consideration the differentil _of function 11(x,: t) with respect to variables x, y, t coin(ids (to within theconstant factor v0) with the di'erential of function _{?/) with respect. to variables E, ?/.} Let us fix variablet and consider (2.10) on the free surface. The vector dll(x, y,l), having the same direction as the acceleration vector, is normal to the free surface. Then, according to (2.10) ,tlie direction of the normal to the free _{surLice in the} (, )-plane is also the direction of the vector _{j) which is the increment, of} the velocity vector along the free-surface element. ''

This fact makes it possible to construct the free-surface image in the plane of Wagner's function.

Therefore, let he a point of the free surface

_{= () and let the contour of}

integration in formula (2.2) include the part of the free surface from _infinity to (the path of integration is chosen so that the flow region_{would be from the} left). Now let us investigate the behaviour of the _{argument of function}

V(cl V'() d')

on the free surface. As shown above, the increment d V'(C) of the velocity vector along the free-surface element is normal to it and _{can be directed along the} out-ward or inout-ward normal.

Let us first consider the case when (under the chosen path of' integration) the vector d V'() is in tire outward normal direction to the free surface. The argument of vector d at a point J11 on the fi'ee surface BC is denoted by 0 (figure 1). Then

argd V'() = 0 as d and d V'() are rnntriall- orthogonal.

_{and hence}

argrIV'(C) =

as vectors I V'() and (1 F) are conjugate. Taking this into account,_{we obtain}

arg \/(dF()d) = [argdV'(')+ar'gdfl =

(2.11)

Condition (2.11) is satisfied at any point of the free surface. Therefore the incre-ment of \Vagner's function h() has tile arguincre-ment _{7T on the considered part of} the free surface.

t The increment of the velocity vector is not orthogonal to the free surface in the case of flow with the similarity law of the form

U(x, y, 1) = ctY3(, ?1), where =

= In this case the relations

dU(x,y,t)

dU(x, y, t = ycgr_1 rfl clt+ctv d23(g, ₎

hold, instead of the conditions (2.7) and (2.10)

As will be seen from the following, the similarity of this form excludes the construction of the flow region in the plane of Wagner's function before solving the problem.

In the similarity problems under consideration, the free surface reaches the point at infinity of the 'planc.

imilarity flow of' fluid with _{a frec 8'arJuce 809} In a similar way it can be showii that the argument of increments of Wagner's function is equal to

-

,iT on those parts of the free surface where the

vector d V'() is in the inward normal direction to the free surhice.t

Thus, the free-surface image in the plane of Wagner's function, is in the general case a broken line, _{composed of the rectilinear segments inclined to the} axis of abscissas at the angle _{or - m. If the vector d V'() on the free surface} does not change its direction from outward normal to the inward one (or con-versely), then the free surface is represented in the plane of Wagner's function simply by a rectilinear segment making _{the angle jr or}

-

_{with the axis of} abscissas.

810 Z. N. Dobrovol'slcaya

arg \I(d.V()d) = ± m (or ± ii ± zn), the sign depending on the direction of

path along the boundary.

If the impermeable boundary AC (figure 1) is stationary in the (x, y)-plane, the behaviour of the argument of function V(c1J"()d) is the same as in the case just considered. Thus, the image of the uniformly moving or stationary recti-linear impermeable boundary in the plane of Wagner's function is a rectirecti-linear segment parallel or orthogonal to the axis of abscissas.

C C

x

A

FmuaE 2. The flow region in t in physical planc x, y for the unsymmetrical entry of a wedge into a fluid.

C

FIGURE 3. The flow region in the piano of Wagner's function for the symmetrical and

unsymmetrical entry of a wedge into a fluid, and for the impact of a water wedge on a wall. Below we consider only such problems in which the flow region is bounded by the free surfaces and uniformly moving (or stationary) impermeable rectilinear

boundaries. For the problems of this class, as seen from the above, the flow

region is always represented in the plane of Wagner's function by a polygon. For example, in the ease of symmetrical solid-wedge entry into a half-plane of a fluid (or into a fluid wedge) the flow region is represented in the plane of Wagner's function by a rectangular isosceles triangle. In the case of unsymmetrical wedge entry the flow region in the plane of Wagner's function is of the same form as in the symmetrical case (figures 2 and 3).

SnnIiarii!/ ow of fluid with a,free $uface 811

of the impact of a water wedge on a wall (figure 4), as in this case the flow legion in the plane of Wagner's function is known and represented by the same triangle us in the wedge water-entry problem.

In the problem of the uniform spreading of a constant pressure wave over the free surface being initially non-perturbed, the flow region in the plane of Wagner's function is rcpreseiited by a square.

B

/

/

/

/

/

B1 /

/

1/

1/

FIGURE 4. The flow region in the physical plane x, y

for the impact of a water wedge on a wall.

3. Method of Wagner's function for reducing the problems

under

consideration to a non-linear singular integral equation

The essence of the method consists in the iillowing. The problem of the similarity potential flow in the region bounded by the free surface and imperme-able rectilinear boundaries can be formulated as the problem of determination, in the flow region (in the similarity (, m,i)-plane), of the velocity potential _), a harmonic function, satisfying the constant-pressure condition and the kine-matic condition on the free surface ₌ and the impermeability conditions on the solid boundaries.

Let V(fl = I, ) + iW() be the complex velocity potential in the c-plane

( = +i). We introduce the auxiliary parametric variable w = u+iv and

con-sider the analytical function _{= (w) mapping conformally the upper half-plane} 1mw > 0 onto the flow region in the c-plane in such a way that the free surface of a fluid is represented by a segment L of the real axis of the u-plane and the rectilinear impermeable boundaries by the remaining part of the real axis which we denote by 1 (both L and 1 may contain a point at infinity).

Introduction of function (w) makes it possible to reduce the problem to the determination of two analytical functions V(w) and (w) in the upper half-plane 1mw> 0 using the following boundary conditions: functions V(u) and (u) have to satisfy the constant pressure and kinematic conditions on segment L, and the impermeability and obvious geometric conditions on 1; the latter follows from

812 Z. N. Dobrovol'skaya

the fact that the argument of the function '(u) on 1 is known and equal to the constant or piecewise constant function. Thus, both on L and I there are two conditions for the determination of two functions V(w) and (w) analytical in the upper ha If-plane.

Let us introduce Wagner's function h(). In the plane of this function the flow region is represented by a polygon. By moans of the Schwarz-Christoffel formula we can write the function h _{h(w) which maps conformally the upper half-1lane} 1mw > 0 onto the interior of the polygon in the h-plane. Elimination of _the

variable h from the expression h h(w) and relation (2.2) gives an explicit ex-pression for the complex velocity V'(w) in terms of the mapping function (w). The expression of function V'(w) in terms of (w) gives a possibility of excluding an unknown function V(w) from the obtained boundary-value problem for two functions V(w) and (w), and the problem is thus reduced to the determination of function (w), analytical in the upper half-plane Tm w > 0, which has to satisfy the kinematic condition on segment £ and the geometric condition on 1. The second condition on each of these intervals (the constant-pressure condition_{on £} and the impermeability condition on 1) has already been used for the determina-tion of the flow region in the plane of Wagner's funcdetermina-tion.

Introduce the real function f(u) (u a L) representing the argument of '(u) on L. Then, remembering that the argument of function '(u) on 1 is know-n, we can write (by means of the Schwarz integral) the representation of the mapping function (w) in terms of the real functionf(u). The substitution of(u) expressed in terms off(u) into the kinematic conclit ion on the interval L gives a non-linear singular integral equation for the determination of f(u). Function f(u) being

determined, the hydrodynamic problem can be considered _{as solved since the} mapping function (w) and the complex velocity V'(v) at any point of tiie flow region arc expressed in terms of the funetionf(u) by quadraturcs.

4. Symmetrical entry of a wedge

We consider uniform sninetrical entry ofa wedge into a half-plane of a fluid which is assumed to be incompressible, non-viscous and weightless, the wedge angle being arbitrary ( <

Let x, y be Cartesian co-ordinates with the origin at the point ofintersection of the unperturbed free surface with the axis of symmetry (the_{y-axis is directed} opposite to the wedge movement along the axis of symmetry). Because of the symmetry only the half x 0 of the flow region is considered.

As the flow under consideration is the similarity flow, the velocity potential (x,y,t) has the form _{(x,g,t) =}

(4.1)

where x/v0t, = y/v0t (v0 being the velocity of the wedge), and

_{i) is}

a harmonic function of and in the flow region CBA C (figure 5) .On the boundary of the flow region, the function _{) has to satisfy the following conditions: on} the free surface _{= 1(), the constant pressure condition}

a(I 1 /a(,E\2 1

fa(\2

C

Ficuu 5. The half of the flow region CBAC in the similrity plane = for the symmetrical entry of a wedge into a fluid.

The wodge-entryproblem can ho reduced, first, to the boundary-value problem for two functions analytical in the_{upper half-plane. Therefore, let}

J7() =

be the complex velocity potential in the

_{c-plane (= +i). We introduce}

w = u + iv and consider an analytical fimction _{= (w) which maps conform ally} the upper half-plane Im w > 0 onto the flow region in the -p1anc ii such_{a way}

that the points _{= (point of contact), = - i (wedge apex) and =} correspond tow = 0, w = 1 and w = co respectively (figure 6). In so doing the free

surface of a fluid _{1l be represented by the real negative semi-axis of the w-plane.}

Conditions (4.2) and (4.3) take the form

'(u)'(u)Re V(u)Re[(u)'(z1) V'(u)]+V'(u)

V'(u) = 0

_{(co < u}

0).

(4.6)

Re[iJ7'(u)] = Re[i'(u)(u)J (co

<U 0). (4.7)

On segment 0 u 1 tw-&conditions have to be satisfied: condition_(4.4)which can be reduced to the. form

Re[iV'(u)] =

_'(u)Isino

(0 u 1), _(4.8)

Similarity flow 0/fluid i.oith a free surface 813

and the kinematic condition

aq) 3'I)

'7

= 0; _(4.3) on the wedge and axis of symmetry, the impermeability _conditions

(I)

on AB, (4.4)

= 0 on AC. (4.5)

The free surface at infinity has to approach the unperturbed free surface

814 _{Z. N. Dobrovol'skaya}

and the obvious geometrical condition

arg'(u) = (iT+0)

(0 u 1).

When u _{1, we have condition (4.5) taking the form} Re[iV'(u)] = 0 (1 u

and geometrical condition

arg'(u) = -

(1 u < cc).

V

0

C _{B A C}

FIGuRE 6. The image of the flow region CBAC _{in the w-plane}

for the symmetrical entry ofa wedge.

The problem is thus reduced to the determination _{of two functions JT(w) and} (w), analytical in the upper half-plane, satisfying boundary conditions (4.6)-(4.11) on the real axis.

We reduce the obtained boundary-value problem to the determination of the function (w). To this end, we introduce Wagner's function

ft

/(dV'(C))

In the plane of this function the iiow region CBA C is represented by the interior of the isosceles right triangle (with the size unknown in advance) depicted _in figure 7. The function h(w), conformally_{mapping the upper half-plane onto the} interior of the triangle in the h-plane with the correspondence of the angular points indicated in figures 6 and 7, has the form

h = icof w(w_1)_dw (Imc0

= 0). _(4.13) Eliminating the variable h from expressions (4.12) and (4.13) and taking _into account that V'(11) _{= (see Dobrovol'skaya 1965), we obtain}

V'(w)

_{= D(W)}

_cç(w)f

_{w(w_ l)-1_L dw.}

(4.14)

0

Formula (4.14) gives an explicit_{representation of the complex velocity interms}

of the derivative of the function

_{(w) every-where in the upper half-plane} 1mw> 0. It should be noted that_{representation (4.14), obtained with the help} of Wagner's function, is the _{consequence of conditions (4.6), (4.8) and (4.10)}

-U

Similarity flow of fivid with a free sufccc 815

since these conditions have been used for the determination of the how region. iii the plane 0f \Vagner's function. Eliminating function V'(u) from the kinematic condition (4.7) with the help of (4.14), we obtain the following boundary con-dition for function (u) on the real negative semi-axis u 0

,, r

Re

{i'(u)J

'(u) + cui(u 1)1

1

1 du1 _{= 0.}

0 L '(u)j J

On the positive semi-axis, the function (u) has to satisfy condition (4.9) for

o _{u land (4.11) for!}

u < +.

FIGUnE 7. The flow region (JJ3AC in the plane of Wagner's function

for tlie symmetrical entry of a wedge.

The problem is thus reduced to the determination of the function (w), ana-lytical in the upper half-plane 1mw > 0 fron the boundary conditions (4.15), (4.9) and (4.11) on the real axis.

The boundary-value problem for (w) will be reduced below to the integral equation fbr a real function f(u) connected with '(u), when u 0, by the. relation

_{arg'(u) = [f(u)+ 1]}

_{<U 0).}

(4.16) The funet.ionf(v) has to satisfy the condition

f(u)-.0 when u-

,

(4.17)

as the free surface has to approach asymptotically the unperturbed free surfa.ce at long distances from the wedge. Using (4.16) and taking into account that the argument of function '(u) for 0

u < +

is known from the boundary con-ditions (4.9) and (4.11), we can write (with the help of the Sehwarz integral for the upper half-plane) the representation of mapping function (w) in terms of the real function f(u). This representation has the form

(4.15)

icw4+(w_1)expI

f '0 f(u) 1

-

I du (c >0,

= z/ir).

(4.18)

816 _{Z. N. Dobrovol'skaya}

Using the Sokhotsky-Plemelj formula for the limit values of the (Jauchy-type integral of expression (4.18), we can write the limit values of '(w) (determined by (4.18)) when variable w tends to the _{real value u from the upper half-plane} Im w > 0. The substitution of '(u) expressed in terms off(u) into the kinematic condition (4.15) gives the following non-linear_{singular integral equation for the} determination of the function

_f(u)t

It is seen from (4.14), (4.18) and (4.19)_{that the solution of the problemcontains} two real parameters (c and c0) and a complex parameter _{the real and imaginary} parts of the latter are connected to each other_{by the obvious dependence}

=

(l+B)tgoØ.

_(4.20)

Thus, the solution contains real _{parameters c, c0 and}

Bfor which determina-tion there are three condidetermina-tions:

(4.21) (4.22) (4.23)

Using these conditions, we find that the value_{c/c from integral equation (4.19)} is determined by the following functional

2

L

ul+(1 _u)exp[_f°

'd}d

-

I'I [ ('I)

[(it ) 1 ( . )

J0 u I - u)-l±1 exp_[J

-

_---

U1U

-

_j

For the numerical intcgrn.tion of (4.19), it is more convenient to introduce positive variables of integration t

_{(1-u)-1 and r}

_{= (1-u1)-1 which}

vary within 0 t 1 and 0 T 1. Then, (4.19) is transformed _to

f(t) = c%I

(1_t)1exp[tf1 f(r)

_dr1 OT(Tt)

_j

dtI(t) (4.25)

1 0L

_{t(1_t)-+exp[_tJ'}

o

f(T)d]d

(the notation 1(t) is introduced for brevity_{of the subsequent discussion).} Intro-ducing variable r = (1 + u) _{into expression (4.24), we obtain}

1 _f(T)dT

_f:r_1(1_+2r_

1)exp{_f

_{T[T{2_(l/r)}_l]}dr}

(4.26) J1 f(T)dT

c -

_{J(1_r)-1(2r_1)-1exp{}

T[T2_(1/r))_1]}

j- A copy of the author's derivation of_{this equation, which is taken from}

Dobrovot'skaya (1965), will be Bent to any interested reader on request to the Editor.

(1) =_i,

Im (u)

0

when u -

-V'()

= i at =

-i.

fj

jUj

_-

1c

-rro

I)-1+ _{exp I I} 'u' ' _rlu r ro _ci 1 '

Jo

I

iui+(u

-

l)exp

I

- I

1.

J_

'

"

_{du (lu} j

S'irnilarity of fluid flow with a free Rurfacc ₈₁₇

'I'hus. the wedge water-entry J)rOhle.m has been reducedto the determination of the solution of the non-linear singular integral equation (4.25) for the real lunction f(t).

Th( fund ionf(t) at any point I a [0, 1] is, by deflnition (to within the coftstant factor and sum), the angle of inelination of the free surface to the -axis at the corresponding point of the free surface, and because of this,f(t) is l)ounded on the whole segment [0, 1] including its ends (1 _{1 corresponds to the point of contact} of the free surface with the wedge face and_t= _{0 to the infinite point of the free} surface).

If the solution of (4.25) is found, the mapping function (w) can be determined in terms of fnnctionf by expression (4.18) and the complex_{velocity at any point} of the flow region by formula (4.14).

The free surface of a fluid is determined in terms of f(t) _{by the following}

parametric equations obtained from the formula (4.18):

1 N C1 f(r) ₁ (cos) [nf(t)]dt (4.27)

cJ ti(1_t)_±expL_t

I

-T(T -t) _j sin) (4.28)

where the constants c and have the fo m

1'' . F C' 1(T) dr ₁

c =

1eos0J rz(1 r)2 (2r- 1yexp1

Jo T[T{2_ (1/r)}- ijjdr ti (1

t)l

exp

[-if

((T)/)dr]

sin {f(ifl di);

(4.29)

= _{c f}

i -

exp

[-if

dT]sin [f(t)] (it. (4.30) It shou?d be noted that the asymptotic behaviour oft _{e free surface at infinity} can be obtained from equations (4.27), (4.28). Really, the analysis of (4.25) _near the point. t =0 shows thatf(i) = O(tl) when t 0. This estimate together with (4.27), (4.28) considered at I

_{- 0, gives an asptotic of function () at}

which, as in the linearized theory of a thin wedge (Mackie 1902), has the form

11 =

K1-,

(4.31)

where K,=

The pressure distribution on the wedge face can be obtained from the Cauchy-Lagrange integral

2 Re [17(r) - (r) V'() T)]+ [JT'()

V'()](r)

+p(r) = 0

(. (

r

1), (4.32)

where p(r) p/(pv) is the dimensionless pressureon the wedge face,

=

pp0

(p0 is the pressure on the free surface); point r = corresponding to the wedge apex = i and r 1 to the point of contact _{Correspondence between the} points of segment [ r 1] and the co-ordinates , on the wetted wedge face

is given by formula

= (r) + ii(r) ₌

_[n

c sin a0H0(r)] + [D c cos a0H0(r)], (4.33)

52 _{. Fluid Mech. 36}

818 Z. N. Dobrovol'skaya where

110(r)

=

f r1(,I - r)i

(2r- ] )

exp [-f'

rLr2-(1/r)}

dr. (4.34)

and Re V(r) in (4.32) have the form

Re V(r) _{= -(} +

) +J ri(1

-

(2r - 1)

x exp [5'

r[rf2

(i/)}fl

dT][c 00(r) c(L. Sifl0 + ?/ COS _{)] di.}

(4.38) Given the above formulas for the pressure on the wedge face it is possible to estimate the behaviour of the pressure curve in the vicinity of the wedgeapex. For this purpose it is sufficient to differentiate the Cauehv-Lagrange equation

(4.32) with respect to r, to take into account the singularities

_{at r = l of the}

improper integrals 110(r) and G0(r) and to use formula (4.33) connecting variable r with the similarity variables and on the wedge. As a result we have

P = Pi - J02!U-,

(4.39)

where p is the pressure at the wedge apex and 112_{= I12().}

5. Method of numerical integration of the basic integral

_{equation of}

the problem

For the numerical integration of (4.25) the method ofsuccessive iterations has been used. Below we dwell on some peculiarities of realization of this method for the given equation.

Point t = 1, as seen from the equation, is the singular point of the integrand of the outer integral 1(t). Integral 1(t) has to be convergent at this point since functionf(t) has to be bounded, by deffiuition, on segment [0, 1] including its ends. Let us analyze the behaviour of the integrand of integral 1(1) _{near the point t = 1.} We denote by fl0 the angle between the wedge face and free surface at the point of contact B. Then we have

f(l)=+-fl (fl=o).

(5.1)

-sa-.a.S4S

Functions

[V'()

Re K(r) V'()(r)I

kB _siiioGo(r)] +

1R

+2 cosO6o(?)],

(435)

[V'() V'()]()

= where

Cofr) =

fri

(1 -r)'

(2r - l)-'+ exp [f

dT] di; (4.3(i) T{T{2 _(i.)}

Re [(r) T"() [C

= - C Sifl (rfl [B - (cd/c) sin 0G0fr)]

Similarity of flow fluid with a free surface ₈₁₉ Now the Ciuchy-type integral in (4.25) has the following representationnear the point t = 1

JoTTt)

= (+-fl)In(1-t)+w(t).

The following estimates for the functions standing under the integrals in (4.25) near the point t = I (t < 1) can be obtained

(1 - t)-' exp [t

1

T2t)

dT] = (1 -

t)4fl

- t)4

_{exp [}

tfl

dr]

= t(1 - t)fie(°.

Thus we find from (4.25) that f'(t) behaves near t = 1 as (1 -t)4-2fl. Hence

integral 1(t) at the point t = 1 is an ordinary improper integral which converges if and only if the following condition is satisfied:

fi<1.

(5.2)

Thus, the magnitude fl0 of the angle between the free surface and the wedge face at the point of contact cannot exceed 4ir for any wedge angle.

From conditions (5.1) and (5.2) it follows in turn that, if we_{use the iterations} method, then for existence of every successive iteration f1+1 it is necessary that all the previous iterations would satisfy the condition

(k=O,1,...,n)

(53)

which is the condition of convergence of integral 1(t) at the point t = 1.

It can he shown that condition (5.3) gives a possibility of cdnstructing as many iterations as desired. Therefore, let us take as zero approximation a one-para-meter family of suitable (monr)tone, vanishing at I = 0 and providing con-vergence of 1(t) at t = 0) curvesf0(t) (0 t 1) witl1 parameter I f0( 1)

belong-ing to segment R

_{= [ +}

, - + ]. Substituting into (4.25)

the function

f0(t) a i with F0 = we obtain that F1 f1(1) = 0. 1fF0 = it follows from (4.25) that F1 = . Thus, choosing as zero approximations the curves with

F0 e R, we obtain the first iterations with 0 F1 < , and then there

neces-sarily exists a smaller segment R') (lying entirely in R) such that the values

F0 a R(') will correspond to F1 a R and 0

<) Carrying out the

same reasoning for the family of functions f(t) with .F a 1? and then continuing this

process further, we can obtain an unlimited sequence of segments R()

con-tained in each other such that, at F0 a the above process makes it possible

to construct at least (n+ 1) iterations; however, the following iterations,

be-ginning with a certain number, will not in general exist; in fact, it is sufficient that .F _{should be outside the segment R and under this condition the (m + 2)-th} iteration would not exist.

The sequence of continuously decreasing segments 1?), whenn - ,

deter-mines (at least) one point F', by approaching which we can obtain _{as many} iterations of (4.25) as desired.

820 _{Z. N. Dobrovol'.kaya.}

.[t is clear from the above that with this method of constrncting the _Successive iteratioii the process of iterations for (4.25) vill notconverge in the usual sense and cati prove to be only asymptotically _{convergent. The asymptotic}

con-vergence in this case is understood in the following sense: the number of 'Con-vergent' iterat ions of a given equation can he as great as desired but the following iterations generally- speaking, do not exist.

For tlie numerical integration of (4.25) we have taken as zero approximation

f(t) (0 t _{1) the family of monotone increasing functions having, near the}

ends of the integration interval, the following representation (arising from the elementary analysis of (4.25))

f(t) = O(t) when t 0, (5.4)

f(t) =

ly(1 _t)h2fi

when t 1 (0

<ft

_<

). (5.5)

The value F0=f(1) has been taken as free parameter.

All the integrals in (4.25) have been computed with the variable step of

integration decreasing near t = _{1, the number of steps having been chosen in}

such a way that the distance between the last point of integration and 1 is equal to some small value c which choice was determined by the necessary accuracy of calculation. The computation has shown that the value e is of the order of 1012_1O_18 (depending on the value of _{) decreasing with decreasing . The} refinement of the step of integration to such a small magnitude is due to the very special behaviour of the integrands of (4.25)near t = 1. The partial sums of the

integrals near the ends t = 0 and t= _{1 have been computed with the help of}

obtained estimations of the corresponding intcgrands near these points.

As would be expected. the process of iterations for (4.25) _{has proved to be} asymptotically convergent. The computation has shown that for obtaining the solutionf(1) to the practically required _{pi.ccision it is sufficient to construct only}

10-12 iterations, 4S mean iterations having proved_{to be almost coinciding.} The computation has confirmed to high accuracy the character of behaviour of the function f(t) near t = _{1, described by formula. (5.5). However, the curve}

f(t) can be represented in the form of (5.5) only in the extremely small neighbour-hood of point t = _{1. Such behaviour of f(t) has required the above mentioned} refinement of the step of integrationnear t= 1.

6. Numerical results of solving the wedge entry problem

The numerical computations have been done for the complete wedge angles = 0036°, 036°, 36°, 6°, 18°, 60° and 120° ₍ = 00001, 0001, 001. , 005,

11

6' 3

The curvesf(t), which are the solution of (4.25) for angles = , --and , are presented in figure 8. All these curves have_{very small ordinates at the large} interval adjoining zero and sharply increase immediately _{near the point t = 1} (the last points of the curves at t = _{1 are denoted in figure 8 by heavy points).} The curves f(t) for < _{practically coincide in the figure with the axis of} abscissas and the straight line parallel to the axis ofordinates and passing through the point t= 1.

-a - _{c%bc,am.f.P 4et.r. c}

Similarity flow of fluirl wifh a free surface R2 I With the help of obtained functions f(t) the free _{surface curves have been} computed froni the formulae (4.27)(4.3o). For the _{total wedge angles 2 x 30,} 2 x 9°, auicl 2 x 30° these curves are presented in figure 9. It is seen here that with increased wedge angle the free-surface disturbance, as would be expected, in-creases and the splash adjoining the wedge inin-creases lengthwise_{getting at the}

0 02 0-4 06 08

FIGURE 8. Curvef(t) for = 0O, and .

height of the splash and the region of the free _{surface disturbance decrease.} same time thinner and thinner by its apex. With decreased angle _{the maximum} The curve _{= "JB() of the maximum splash heightsversus is presented in} figure 10. Numerical values of magnitude _{Bfor different angles are presented} together with some other data of numerical computations in table 1. Figure II gives the dependence '//(mTcL) on log .From the analysis of this dependence _we

may conclude that with

- 05 the height

71Bseems to increase oniy up to some

limit value close to_9/B 3.

It is customary to assume that the linearized theory_{gives in general an infinite} splash height. However, Mackie (1962) has obtained_{from the linearized theory} on the basis of reasonable concepts the following approximate formula for the splash height

2 1

= ln.

- (6.1)

0

1

2

FIGURE 9. The free surfaces '

= i() for a

3, 9° i

30°

3

25 20 15 05

FICUBE 10. The maximum height of the splash

ijB versus a. 30

-:0 01 02 a 03 04 05

Smi1ar?ty flow of fluid with ( free .$ur face. 823 Careful treatment of our numerical data gives for the small angles the asymptotic dependence

1

(6.2)

the consttnt A practically being equal to 2/ 0637 and B O82. It is seen from here, that Mackie's formula, theoretically rightly describing the dependence

on when

- 0, gives a small relative error for the splash height jB on]y

log a

FIGuRE 11. The curve /(a) versus log a.

a _{Kilo p} tgia0/ 'tg2a0)3oP

/j' P \

I 9-4 _{00019 0100 102 100 100} 10-s 0014 0100 103 TOO 101 I 0- 0099 0004 105 104 110 0153 0089 111 1.00 117 039 01i72 131 12 15 113 0030 23 20 31 :1 20 001] 12 6

for the extraordinary

45'

40 35 30 25 20

30

0001

small wedge angles. a

0002 0006 002 006 02

0004 001 004 01 05

824 _{Z. N. Dobrovoi'skaya}

For comparing the obtained free surfaces with the data of linearized theory the free surface curves are presented in figure 12 in the form of the dependence

j/tg (i) on

. The free-surface curve obtained from the linearized theory and

described by the formula

= i!- [In (1+

+ 2 arc tg - 2

(U .3)

is represented on the same figure. Our numerical results, as it is seen from the figure, approach the latter when

- 0.

075 050 025 Linearized theory

-/

= /

/

/

/

/

/

/

4

At infinity the free-surface curves behave according to (4.31) as K1(c)

The treatment of our numerical results shows that at sufficiently large the equa-tion of the free surface can be represented in the _form

1 0028

= 9 r2{11()]

1TL 7T5

where e() and e2() are the functions vanishing when _{- 0. So, it follows from} the computations that, at small ,

K1() =

_{that coincides with the result of the}

linearized theory. Numerical values of the ratio K1/() are presented in table 1. The free-surface behaviour near the point of contact with the wedge can be characterized by the value ,8 fl0/7T,where is the angle between the free surface and the wedge at the point of contact. The values ,8 for the different wedge angles

3

0 ₂

Similarity flow of fluid with a free surface 82 havo beeii computed with the help of corresponding val ucsf( I) from the formula

(5.1). The curve ft = ft(a) is represented in figure 13 (numerical data are given in table 1). At small a this relationship can be expressed to high accuracy by the formula ft = 01

-010 008 006 ft 004 002 0 01 02 03 04 05

FIGURE 13. The curve of the angle of contact ft versus .

It is seen from the dependence ft = ft(a) that the angle of contact of free surface with the wedge increases with decreased wedge angle. However, with _{a - 0 the} angle ft0 does not practically exceed IS°. On the other hand, if a = 0 then from the statement of the problem it follows directly thatf(t) 0 and ft0 ?,-r. Thus,

the free surface inclination to the wedge face at the point of contact has a dis-continuity at a = 0 asafunction of a. This is similar to the problem of symmetrical impact of a water wedge on a solid wall. Really, Cumberbatch's (1960) approxi-mate computations have shown that with the decrease of half-angle of a fluid wedge from 225° to 1125° the tip angle of the streamon a wall increased from 3° to 48°. Therefore, it is reasonable to_{suppose that with the fluid wedge angle}

tending to zero the tip angle of the stream would approach _{some finite value}

rather than zero.

The dependence _{B = B(a) for the wedge entry (see figure 10) shows that} with a - 0 the maximum splash _Rtcnds to zero, and the region of disturbance vanishes, i.e. the picture of flow in the case of a wedge entry remains in the large continuous when a - 0 and at a 0.

The discontinuity of function ft = fl(a) at a 0 is the consequence of the model of ideal liquid and the similar result in the hydrodynamics ofan ideal liquid is not a single one. So, in the flow past the plate inclined to the flow direction at angle a, the fluid velocity at the edges of the plate is infinite at any a + 0, although

826 _{Z. N. Dohrovol'skaya}

at = 0 tho fluid velocity at the edges is equal to the velocity at inflnity. But the 1)icture of flow past the plate, as in the ease of a wedge entry, changes on the whole, continuously when _{-* (Jand at = 0.}

For estimating the accuracy of obtained numerical results we have checked two (OflditiO11S: the incompressibility of_{a fluid and the conservation of the are}

length along the free surfhce.

In order to verify the incompressibility condition we have computed the area H of the flow region above the axis of abscissas and then this area was compared with the area _{of that part of the wedge which was under the axis of abscissas.} For verifying the second condition mentioned _{above we have computed the} length 1 of the part of the free surface from the point of contact B to that point of the free surface which ordinate could be practically taken as zero, and then the value 1 was compared with the abscissa * of this point.

For the wedge angles up to 18° ( = 0.05) inclusive, the condition of

con-servation of the arc length (*

_{- 1 = 0) has P1oved to be satisfied to within}

(* 1): BI

= 0004 and the incompressibility condition to within

IsIs-lI = 0.01.

When _{> 18° the computations have been done with a smaller accuracy. So,}

for = _{the computations have been done to within 001 a.nd 003, respectively,}

and for = (the wedge angle = 120°) only an approximate estimation has been obtained (therefore the Curves for _{= 60° are drawn in t lie figures by dotted} lines). Increase of the accuracy of numerical results_{and their extension onto the} angles _{> 120° do not induce great difficulties and demand only more machine} time.

By means of obtained curves f(t) fir different c the distribution of hiydro-dynamic pressure on the wedge face has been computed from the formulas (4.32)-. (4.38). As would be expected, the computations have shown that _{the pressures} on the wedge decreased with the decrease of the wedge angle. The dimensionless ressiire p = ip/(pv) at the wedge apex tends with _{- 0 to a value close to 1,} and the pressures on the wedge face tend to zero. The value PA = I can be

obtained also from indirect theoretical considerations. For _{this purpose it is}

sufficient to use the estimate (4.39) for the behaviour of the pressure in the

neighbourhood of the wedge apex and to substitute there the values_{of pressure} at au' two points, calculated by the linearized theory (according to the linearized theory the pressure at the wedge apex itself is infinite). Numerical_{values of the} dimensionless pressure p.1 at the wedge apex are presented for different in table 1.

As the pressures on the wedge for the small angles _{are very small} (every-where except the neighbourhood of the apex), it is_{more con venient to compare} the pressure distribution urves for different wedge angles by considering the

dependence of p/tg0 on ?1 rather than p on i/. Figure 14 presents the relations As has been shown by Wagner (1932), the distance (measured along free surface) between two arbitrary fluid particles on the free surface remains constant in the similarity

flow under consideration.

-t.-.-

J -

...-.-,-_3-1' I

Similarity flow of fluid with a free surface 827 p/tga0 versus yfor different _{(y = - I is the wedge apex). Thepressure curve for} = t)0001 practically coincides w-ith the curve given by the linearized theory:

=

' iT 1+

Ior

> 3° the pressure distribution _{curves, as seen from the graphs, begin}

quickly to deviate from the pressure curve of the linearized theory and for close to 30° t hesu curves have nothing_{in common with the results of the linearized}

-3r

\

) _{/ N} Vertical scale 'I four Times :: diminished / 05 a Linearized theory

05

/ / /

/

/

T / / (6.5)

Fic;-n; 14. The curves of tIt distribution of the diinensionIos pressure p (livided by tg _{along t lie wedge b.cc for dif1reut angles} .

theory. The ordinates p/tg0_{near the wedge apex ( = - 1) become for the small}

, as would be expected, very large (but finite), and therefore the values

p/tg0at

_{= 1 arenotinelucledinthefirrure Asseenfromfigure l4thepressure}

at the upper Part of the splash_{proves to be practically equal to zero (the ends of} the splash jets are denotedoil the figure by heavy points).

The pressure distribution curve for = 60° shows that for the large angles a the maximum of the pressure displaces from the wedge apex to some point on the face of the wedge, this point being_{situated for the large above the level of the} initially undisturbed free surface. _{This qualitative result coincides with the} Borg (1959) approximate _{computations.}

With tile help of obtained_{pressure curves we have computed the va]ue of the} dimensionless total drag force

1'BLn

P=21

10dT .Jo PV

acting on the entering wedge (both its faces). The actual drag force P is connected with the value P by the obvious relation: P = pvtP. The dependence P/tg2

S28 _{Z. N. Dobrovol'skaya}

on a is represented in figure 15. Numerical values of_{the ratio of (I'/tg2a0) to the} magnitude (l'/tg2a0) _{obtained as the limit when}

_{a0 - 0, are prcsnted in}

table 1. The analysis of obtained data makes it possible to represent the foree P for small wedge angles (a < 0, 1) in the form

P

_{= l7GStg2a0+58tga0,}

where 1765 is the magnitude given by the linearized theory.

10 6 C 2 S

/

/

/

/

/

/

/

/

FIGU _{15. The dimensionless drag force P (dividedby tg2a0)} acting onto the entering wedge,_{versus a.}

The presented results of numerical _{calculations fr the wedge-water} entry problem demonstrate the efficiency and practical applicability of the general method when using modern digitia.l computers.

The theoretical part of this work has been done at the former Institute _of Mechanics of the USSR Academy of_{Sciences, in the Department of}

Prof. L. A. Galin, member of the Academy, _{to whom the author is grateful formany useful} discussions. The development of the numerical methods of solving the wedge

problem and all the calculations have been_{carried out by the author at the}

Corn-Juting Centre of the Academy.

The author is greatly indebted _{to Prof. G. K. Mikhailov for his constant}

attention to this work and many valuable _ideas.

&miiarity flow of fluad with a free surface. 829

RE FE fl EN CE S

Bow, 8. F. 1957 Some coniributiet1s to the wocigo-wator entry problem. Proc.A?n. $oc.

Ciril Eigrs, J. E'iqjng Mc.ch. Div. 83. no. EM2, Pap. 1214.

Bole, S. E". _{I USi) Tho maximum pressuros and total force on Straight-sided wedges with}

sndl (IOflCl-riSO. .1. Am. Soc. Naval Engrs. 71, 559-56].

CUMI3ERBATCH, E. 1960 The impact of a water wedge ona wall. J. Fluid lIla-h. 7, 353-374. DomovoL'sJ\y\, Z. N. 1965 Some non-linear problems of the similarity flow of in-compressible fluid with free surfaco (in Russian) .1n. Applications of the Theory of Fuiutons in Continuum Mechanics, vol. 2, pp. 150-170. Moscow.

DOaROVOLSRAYA, Z. N. 1966 Numerical solution of integral equation of _a

two-dimen-sional problem of the similarity flow with free surface (in Russian). J. Comput. Math. and Math. Phys. 6, 934-941.

CARABEDIAN, P. R. 1953 Oblique water entry of_{a wedge. Comm. Pure and Appl. Math.} 6, 157-165.

GRIGOTcYAN, S. S. _{1956 Some problems of the hydrodynamics of slender bodies (in} Russian). Thesis, Moscow Univ.

MAcTIE, A. G. 1962 A linearized theory of the water entry problem. Quart. J. Mech. and Appl. Math. 15, 137-151.

MOISEEV, N. N., BORISOvA, E. P. & KoRIAvov, P. P. 1959 Plane and axially_symmetrical automoclel (similarity) problems of stream impact. Appi. Math. and iltech. (PMM) 23, 490-507.

PIERsoN, J. D. 1950 The penetration of a fluid surface bya wedge. Stevens Inst. Technol.,

Expt. Towing Tank Rep. no. 387.

SACOMONYAN, A. Y. _{1956 Wedge entry into a compressible fluid (in Russian). Vestnik}

Moscow Univ. no. 2, 13-18.

WAGNER, H. 1932 crher Stoss- und Cleitvorgango an der Obcrflhche von Flussigkeiton.

Z. an yew. Math. und Mech. 12, 193-215.