Interaction of free surface waves with viscous wakes

Reprinted from THE PHYSICS OF FLUIDS Volume 11, Number 2, February 1968

Interaction of Free-Surface Waves with Viscous Wakes

JEROME R. LI7RYE

Control Data Corporation, TRG Division, Melville, New York (Received 18 August 1967)

A method of investigating the interaction of free-surface waves with viscous wakes is described. The method consists in constructing a viscous-wake solution to the Oseen equations that satisfies the three linearized free-surface conditions appropriate to a viscous fluid. The solution is characterized by a singularity which simulates approximately the effect of a body. More general flows of the same type can be formed by superposition. The solution obtained is believed to be the first one to represent explicitly a viscous wake in the presence of a free surface.

re0.0.itfre ,ffoi.esdiricti

1. IN TROD UCTION

A submerged or a surface-piercing body moving through a real incompressible fluid in a gravitational field creates, through its motion, both a system of waves and a viscous wake. An important problem connected with this phenomenon is to provide an

accurate quantitative description of the way in

which the wave system and the wake interact. The solution to this problem should help to answer such questions as: How shall the effect of the viscous wake on wave generation be accounted for when we

process wave-survey data in an attempt to obtain a

measured wave resistance?

The present paper offers a model of wavewake interaction which we believe to be both realistic and mathematically tractable. Its construction is based on two observations:

Sufficiently far downstream of a body,

viscous-wake flow is governed by the Oseen equations. There exist infinite-fluid flows that satisfy the Oseen equations throughout the fluid and which, in the downstream direction, become asymptotic to the viscous-wake flows that satisfy the exact Navier-Stokes equations.'

From the second observation, we can model vis-cous wakes in an infinite fluid with one of the in-finite-fluid Oseen flows. We then allow one of these

1 P. A. Lagerstrom, in Theory of Laminar Flows, F. K.

Moore, Ed. (Princeton University Press, Princeton, New

Jersey, 1964), p. 96.

261

latter flows to interact with a free surface by adding to it a "diffracted" flow which is a solution of the Oseen equations and regular throughout the fluid; the sum of the infinite-fluid and diffracted flows is to satisfy the free surface conditions appropriate to a viscous fluid. These free-surface conditions can be linearized since we take all fluid velocities to be small perturbations imposed on an assumed under-lying uniform stream.

This approach evidently possesses the advantage that by purely linear processes we can construct a flow which first Is a good model of a viscous wake,

and second displays a wave character (since it

satisfies the free-surface conditions). - In the next section, we exhibit such a flow. In this example, the infinite-fluid Oseen flow, i.e., the "incident" flow, is singular at one point and develops a viscous wake downstream. It is represented by Eq. (2.4). At the singularity, the fluid experiences a concentrated force opposite to the stream flow; thus this singular flow can be thought of as representing a body with

drag in a uniform stream. The associated "diffracted"

flow arising from the free Surface has been obtained

exactly, within the assumption of linearity. The

desired total disturbance flow is the

sum of the

incident and diffracted flows and is represented by Eq. (2.38).

The subject of infinitesimal free-surface waves in a

viscous fluid with gravity has been discussed

ex-tensively in the literature. The reader is referred to

Wehausen and Laitone2 for a concise summary of much of this work: More recently, there have ap-peared papers by Cherke,sov,3 Cumberbatch,4 and Nikitin5 on the infinitesimal waves generated by a moving pressure point o'n the surface of a viscous fluid. However, it appears that none of these papers consider flows with viscous wakes, so that the key topic of our interest, viz., wave-wake interaction, is absent.

The solution presented herein is a formal one in the

sense that expressions for the diffracted flow quanti-ties are left in the form of double Fourier integrals. The interpretation of these integrals by analytical

or numerical means remains to be accomplished.

2. A FLOW REPRESENTING A VISCOUS WAKE IN THE PRESENCE OF A FREE SURFACE We consider a -viscous incompressible fluid which,

in its undisturbed state, occupies the half-space

z <0 and is flowing with uniform velocity c. Here

i is a unit vector in the positive x direction; the

plane z = 0 represents the undisturbed free surface, in which the origin of a rectangular (x, y, z)

co-ordinate system is located. In the presenee of a

steady disturbance flow characterized by velocity V(x, y, z), the total fluid velocity becomes c.i V (x, y, z). As discussed in Sec. 1, we regard the

disturbance flow as the sum of an infinite-fluid

"incident" flow. and -a "diffracted" flow representing

the influence of the free surface. Thus we write

V(x, y, z) = V' (x, y, z) Vd(x, y, z), (2.1)

where V' (x, y, z) and Vi(x, y, z) are the velocities of the incident and diffracted flows, respectively. We now assume that IT I and IV' c, whence

IVI c, and so the vectors V, V', and V' satisfy

the Oseen approximation to the Navier-Stokes equa-tions, viz.,

av

1-c = Vp(x, y, z) + vV2V, (2.2)

where p and v are the fluid density and kinematic viscosity, respectively. The pressure p(x, y, z) is

such that (1/p)Vp is small to the same order as

V. p(x, y, z) does not include the hydrostatic pressure.

The hydrostatic pressure and the gravitational body force cancel one another, so that neither appearsin Eq. (2.2).

J. V. Wehausen, and E. V. Laitone, in Handbuch der

Physik, S. Fliigge, Ed. (Springer-Verlag, Berlin, 1960), Vol. 9,.p. 638.

3 L. V. Cherkesov, Dokl. Akad. Nauk SSSR 153, 1288 (1963). [Soy. Phys.---Doldady 8, 1166 (1964)].

4 E. Cumberbatch, J. Fluid Mech. 23, Pt. 3, 471 (1965).

6 A. K. Nikitin, J. Appl. Math. Mech. 29, 204 (1965).

Since V' and V also satisfy Eq. (2.2) with

ap-propriate pressure functions, and since Eq. (2.2)' is linear, it follows from Eq. (2.1) that

p(x, y, z) = p'(x, y, z) pa (x, y,!), (2.3)

where p1 and ill are the incident and diffracted flow pressures, respectively.

In accordance with the foregoing discussion, we

introduce as an incident flow the following

in-finite-fluid solution to the Oseen equation, Eq. (2.2),

viz., 1

vG)

+

1 v(e kocii-x)\ 47 pc 4-rpc R

)

V(x; y, z) e_k(R-) 47pp R (2.4) (x, y, z)

(x/ 4re)

(2.5)

In these equations, R = [e ± y2 ± (z

h)2] while k. = c/2v.

As discussed in Lagerstrom,1 Eqs. (2.4) and (2.5) represent a flow that may be pictured as induced in

a uniformly flowing infinite viscous fluid by a

"singu-lar needle" located at the point (0, 0, h). The

drag force on the needle is concentrated at that

point and is of unit strength in the positive x

direc-tion. The flow corresponding to a drag force of

strength D is obtained merely by multiplying the right sides .of Eqs. (2.4) and (2.5) by D. Note that by making c sufficiently large in Eq. (2.4), we can satisfy the inequality IV I << c everywhere except in an arbitrarily small neighborhood of the singular

-point (0, 0, h). Note also that the second and

third terms on the right-hand side of Eq. (2.4)

exhibit the character of a viscous wake in the down-stream region, x> 0.

The diffracted flow arising from the free surface in the presence of the incident flow defined by Eqs. (2.4) and (2.5) is calculated by imposing the free surface conditions on the total disturbance velocity

V

= V ± V

and on the total disturbance pressure

p = p'

pa, and requiring that vd(x, .y, z) and

(x,

y, z) be free of singularities in the region

z <0 and vanish as (x, y, z) ---> co within that region.

The free surface conditions state that the free

surface is a stream surface and that on it the normal and tangential components of fluid stress must van-ish. They may be linearized in view of the restriction

IVI c. Writing

V --= u(x, y, z)1 v(x, y, z)9 w(x, y,

can be shown that the linearized free surface

con-ditions area

In these equations, the subscripts denote partial derivatives. As usual, the equations apply on the undisturbed free surface, z = 0. In applying these conditions we are assuming that the depth h of the singular point in the incident flow is greater than 0.

In terms of ud (x, y, z), vd(x, y, z), and ?Da (x, y, z),'

i.e., the rectangular components of

yd, the free

surface equations become

ud.(x, y, 0) ± 14(x, y, 0) = r(x, y),

(2.9)

y, 0) +

y, 0) = G'(x, y),

(2.10) (c/ p)pdx(x, y, 0) 2vcwd,.(x, y, 0)

gwd(x, y, 0) = 1-1'(x, y), (2.11)

where the functions F'1, G`, and 1-1` are the result of

applying the free surface operators to V' and pi as defined by Eqs. (2.4) and (2.5).

The determination of the diffracted flow involves

four unknowns, viz., ud, vd, uyd, and pd. This number

may be reduced to three by invoking the

diver-genceless character of Vd. Specifically, we write, as we can for any vector field,

Vd(x,?1,z) = V0d(x, y, z) ± V x ad (x, y, z), (2.12) where Oa and ad are .scalar and vector potentials, respectively. Because fla is undetermined to within the gradient of an arbitrary Sealar, we choose that

scalar so that its y derivative equals minus the

y component of ad . Thus ad can always be made into

a two-component vector of the form

ad(r, y, z) = ad (x, y, z)i -yd(x, y, z)i. (2.13)

From the divergenceless character of Vd together with Eq. (2.12), it follows that Oa satisfies Laplace's equation, viz.,

7/20d

(2.14)

Imposing the Oseen equation, Eq. (2.2), on Vd and pd and utilizing Eq. (2.12), we find that the components of the vector potential, ad (x, y, z),

and -yd(2., y, z), satisfy the partial differential

equa-tions aad c = vV2a (2.15)d ayd C

ax

= 0\7274, (2.16)

while the pressure pd (x, y, z) is given by the relation

Pd y, z) = Pcibd(x, y, z). (2.17)

Finally, we can express the linearized free surface conditions in terms of Cild, ad, and 7". Combining Eqs. (2.9)-(2.11) with Eqs. (2.12) and (2.17), We get

2.(x, y,

± ,' (x,,, y7 0)

a(x, y

, 0) =

F'(x, y),

(2.18)

y, 0)

a (x, y, 0) ±

y, 0)

y, 0) = G'(x, y),

(2.19)

cY..x(x, y, 0) ± 2Pc[44`.(x, y, 0) + adY..(X 0)]

g[ctid.(x, y, 0) ± adz,(x, y, 0)] = 111(x, y). (2.20)

The boundary-value problem for the determina-tion of the diffracted flow may now be stated as

follows: find functions lkd y, z), ad(x, y, z), and

y, z) which are regular in the region z < 0,

satisfy the differential equations (2.14), (2.15), and (2.16) in that region, satisfy the linearized free sur-face conditions, Eqs. (2.18), (2.19), and (2.20) on

z = 0, and vanish as the point (x, y, z)

C°

z < 0. The velocity V" and pressure pd are then

given by Eqs. (2.12) and (2.17), respectively.

To solve the problem, we assume representations of the following form, valid in z < 0:

c6d , y,z) -=

f-

cf.d(x, A)

27

-exp [i(Xx AY) + 1114 dX (2.21)

ad(x, y, z) 217r fl Ad(X, A)

exp [i(Xx Ay) Nz] dX (2.22)

yd(x, y, z) =

.1:

ra(x,

exp [i(Xx Ay) Nz] dX (2.23)

where

m

(x2 . 1.12j

-(2.24)

N = (X2 +2 +

(2.25)

'ttith the real parts of the square roots positive. Here = c/2v. These expressions satisfy the differential equations (2.14), (2.15), and (2.16), respectively. When z = 0, they become double Fourier integrals. We determine the unknown transforms 4"(X,

Ad(X, A), and rd(x, Ai) from the three free surface %(x, y, 0) ± w.(x, y, 0) = 0, (2.6)

wz,(x, y, 0) ± v,(x, y, 0) = 0, (2.7)

(c/ p)pr(x, y, 0) 2vcw.(x, y, 0)

r(x, y) =

conditions, EqS. (2.18), (2.19), and (2.20), by

ex-pressing the inhomogeneous terms fr` (x, y) Gi (x, y),

and H'(x, y) as double Fourier integrals of the same type as those in Eqs. (2.21), (2.22), and (2.23). This is accomplished by first invoking the representations

exP [ko(R

x)]

1 1

= [i(Xx + Ay) h)] dX dA,

27r

_.N

h < z < 0

(2.26)

=f_

j: 13-

exp[i(xx + AY) 214. h)] dX dp

h < z < O.

(2.27)

(It will be recalled that R = [x2 ± y2 (z h)2]*.)

We now apply the gradient operations of Eq.

(2.4) to Eqs. (2.26) and (2.27), followed by the free surface operations as defined by Eqs. (2.6), (2.7), and (2.8). After some manipulation, we obtain the desired representations for , and H', viz.,

1 f

ei(Xx+Azi) 1 rixe-mh

2r 2ir pc'

(k0 _{iX)e-Nh] cix} ,

G'(x, y) =

_2r

f f ex'+"2)

'144. [e-mh e-w2] dX dA,

27 pc

111(x, y) =

f f

2r

ei0.+AY)

[(

x2e2

27 pc --2-1-1/1 iveXM g2)e-mh -7 (ivcXN Oe-N21dXclA. -(2.30)

We substitute the expressions for F', G1, and H' into the right sides of Eqs. (2.18), (2.19), and (2.20) and apply the free-surface operators in those equa-tions to the representaequa-tions defined by Eqs, (2.21), (2.22), and (2.23). Upon taking the double Fourier transform of the result, we finally obtain the follow-ing set of three linear algebraic equations fOr the three unknown functions, d3d (X, A), Ad (X, A), and

rd(x,

vxmitd XAAd

iora

Mh

=

[xXe (ko Negh},

air pc (2.31)

2413 (N2 --I- xL2)Ad ixNrd

27 pc

2ivaM2 gM)(Dd (2vcXAN igA)Aa

1(_x2c2 ipeim Oe_hfh

21-pc Lk 2M

(i.vaN 9-)e-g1]

2 (2,33)

The solution can be written at once. Denoting the right sides of these equations by Pi(X, Oi(x, A),

and g'(X, /./), we have

-43a(X, = (1/ D){ffi(X, u)[ iX2 iFLN(N2 + 12)] (2vcXAN ig A)[-iNXPi (X, A) ± iNACi(X, ,

(2.34)

Ad(X, A) = (1/D){iN(c2X2 2ipcXM2 gM)

- [XP(X, Gi(x,

+

i.i)MW), (2.35)

ra(x,

=/ D)( -1-r (X, A)[-2iXM(N2 12) ± 2iXA2M] Pi(X, /.4)[(c2X2 2ivcXM2 gM)(N2 AL2)

2iAM(2vcXAN ig A)]

-0(x, it) [(C2X2 2ipcX312 g31)XA

27:XM(2vcXAN ig A)] , (2.36)

where

D(X, A) = 2(2vcXAN igA)M2N

i1iN(M2 N2)(c2X2 2ipcXM2 gM). (2.37)

We may summarize our results as follows: the total disturbance velocity V(x, y, z) imposed upon a uniform stream by the sum of theincident viscous-wake flow, Eq. (2.4), and the diffracted flow due to the free surface is given by

V(x, y, z)

=

1

V

1 1

{exp [ko(R

xi}

Llir pc - R Lir pc R

1

exp [k0(R

_x)]

1 VcPd(x, y, z)

4rpv fl

+ V x [ad(x, y, 'Yd(x, y, (2.38)

where Oa, ad, and 7d are given by Eqs. (2.21), (2.22),

and (2.23), in which e, Ad, and rd have the forms shown in Eqs. (2.34), (2.35), and (2.36).

(c2 x2

(2.32)

(2.28)

By invoking Eqs. (2.3), (2.5), (2.17), and (2.21))

we can write for the total disturbance pressure:

p(x, y, z)

"d(X' P)

exp [i(Xx 'AO Mz] dX dj.L. (2.39)

Finally we note that in terms of quantities already

obtained, the waveheight t(x, y) takes the form

1

r(x, y) = p(x, y, 0) 2 w(x, y, 0). (2.40) Pg

This completes our solution. It should be remarked

that more general flows exhibiting wavewake inter-action can be constructed by superposing flows of the above type, the superposition being characterized by a suitable density function for the singularities.

ACKNOWLEDGMENT

This research was sponsored by the Bureau of Ships, General Hydromechanics Research Program administered by the Naval Ship Research and De-velopment Center under United States Navy Con-tract Number Nonr-4682(00).