Free-Surface Effects for Yawed Surface-Piercing Plates

Journal of Ship Research, Vol. 20, No. 3, Sept. 1976, pp. 125-136

J o u r n a l

tesearoh

n. B. Chapman"'

The problem of a yawed surface-piercing flat plate is solved by applying the slender-body approximation and solving the resulting equations by a finite-difference method. The solution is shown to depend on two parameters—the product of the length Froude number and the square root of the aspect ratio of the plate, and the ratio of the angle of attack to the aspect ratio. Numerical methods are developed with linear, sec-ond-order, and nonlinear free-surface conditions. Calculated side force and yawing moment coefficients show good agreement with experimental values near the limit of zero angle of attack. At finite angles of attack, the experimental data exhibit nonlinearities not contained In the present formulation.

Introduction

T H E S I D E F O R C E and yawing moment acting on a yawed slender body moving i n an unbounded f l u i d are essentially de-termined by the hydrodynamic masses of the cross sections i n the crossflow. The result that a f l a t plate and a circle have identical hydrodynamic mass suggests that the lateral stability characteristics of a body depend primarily on its outline. Thus, a flat plate may be a useful model of a maneuvering monohull, strut, or side-wall.

I f a free-surface is present, it may be regarded as a r i g i d plane at low Froude numbers, but, at higher speeds, the free-surface displacement cannot be ignored. The present paper presents a numerical method for calculating this free-surface displacement and its effects f o r a yawed surface-piercing flat plate. I f the ratio of depth to length is not too large, the problem can be simplified by the slender-body assumption that transverse variations of the flow f i e l d caused by the plate are an order of magnitude greater than the longitudinal variations. A n important result of this as-sumption is that the solution is independent of downstream con-ditions. This allows numerical solution by a single longitudinal pass starting at the leading edge. I f the longitudinal coordinate is replaced by a time-like variable, the steady three-dimensional problem assumes the f o r m of unsteady two-dimensional f l o w i n the transverse plane w i t h time advance corresponding to transla-tion of the solutransla-tion plane aft.

I n this way three-dimensional free-surface flows past slender bodies can be solved by combining the slender-body assumption with numerical methods f o r unsteady two-dimensional free-sur-face problems. The two most immediate limitations on this basic method are that it does not model f l o w details at a blunt bow and that i t does not generate transverse waves. Because of the zero thickness and because transverse waves should have small effect ^ David W. Taylor Naval Ship Research and Development Center, Bethesda, Maryland.

Manuscript received at SNAME headquarters August 14,1975; revised manuscript received January 5, 1976.

on the hydrodynamic coefficients, these problems are minimal for the case of the yawed flat plate. A slender-body f o r m u l a t i o n f o r the yawed flat plate w i l l be derived i n a manner analogous to that used by Ogilvie [1]^ for symmetric hulls at high Froude numbers. This formulation leads to a linear problem and a nonlinear prob-lem. Numerical methods f o r solving these two problems are de-scribed, and calculated side forces and yaw moments are compared w i t h experimental data.

Formulation

A flat plate is placed i n a u n i f o r m free-surface f l o w w i t h a freestream velocity [/ at a small angle of attack a. Flow geometry is shown i n Fig. L Cartesian coordinates are (x,y,z) w i t h z positive upward, x positive a f t i n the direction of U, and the origin at the intersection of the leading edge and the waterline. The plate extends to depth a below the waterline and has a length / and an aspect ratio e = a/1. Both angle of attack a and the aspect ratio e are considered small, but initially no assumption is made about the magnitude of their ratio.

I t is assumed that the f l o w relative to the plate is incompressible and irrotational so that i t m a y be described by a velocity poten-tial

i = Ux + <p{x,y,z)

The potential f u n c t i o n <p{x,y,z) must satisfy Laplace's equation throughout the f l o w , and on the free surface i t must satisfy the kinematic condition

-F

<5|

dip _ d<p

dy dy dz

and the dynamic condition

^ 4 .

i < ^

' dy dx'^ 2 \ dx

+

dip' (1)

(2)

2 Numbers in brackets designate References at end of paper.

Coordinate system

Both contiitions must be satisfied on z = ^ix,ij) where the variable

^ix,y) represents the free-surface elevation. The boundary

con-d i t i o n on the plate is

d<p dip\

= I t / -h — I tana aty = x tana, z > —a

dij \ dx/ (3)

I t is also assumed that (p and ^ vanish f a r upstream and (p ap-proaches zero as ij^ + z^ approach i n f i n i t y . I t is further assumed that the f l o w remains attached to the plate so that separation and ventilation are excluded f r o m this formulation.

A convenient method for applying the slender-body approxi-mation is to rewrite the d e f i n i n g equations i n nondimensional variables. Nondimensional coordinates are

X* = xl ^ y* = ya "1 and z* = za ^

₍₄₎

Operators d/dx*, d/dy*, and d/dz* are given magnitudes of order unity as a and e go to zero. Terms of order a and e are maintained and higher-order terms such as a^, ê, or ae are excluded. Thus, Laplace's equation reduces to two dimensions

(5) The nondimensional potential is established f r o m equation (3); the plate boundary condition after expanding tana is given b y

= all a = avQ at y*

dy*

(?)

> - 1 (6) A nontrivial solution exists only i f both sides of equation (6) have the same order of magnitude. Therefore, a nondimensional po-tential

= (pa ho ^ ( 7 )

is assigned order unity, and the plate boundary condition (6) be-comes

= 1 on y dy*

> - l

(8)

It may be shown f r o m the free-surface conditions that a nondi-mensional free-surface elevation defined by

(9)

is of order unity. A f t e r terms of order and ae are eliminated, the free-surface conditions (1) and (2) become

de dx*

+

and dip*

+

dx* 2 \ dy

(lOfc)

Both equations must be satisfied on z* = ( a / e ) f *. The governing equations are now equations (5), (8), (lOfl), and (lOb) ahd the in-f i n i t y condition. I in-f the longitudinal coordinate x* is considered as a nondimensional t i m e - l i ce variable, these equations immedi-ately assume the f o r m of unsteady two-dimensional free-surface f l o w i n the («/-z)-plane. I t follows that the f l o w i n any transverse plane is independent of the downstream solution and that nu-merical methods f o r two-dimensional free-surface f l o w are ap-plicable.

Only t w o parameters are contained i n the d e f i n i n g equations:

H = a/e and ri = U{t/glYl^. The first is the ratio of angle of attack

to aspect ratio; the second is the product of the Froude number and the square root of the aspect ratio. The order of magnitude of the acceleration of gravity is determined b y the order of r); i f ?j is of order unity, g must be of order e^/^ product T/M forms a Froude number based on a characteristic length equal to the plate depth and a characteristic speed equal to the cross velocity. The nondimensional f o r m of any variable such as force or moment may be expressed as a f u n c t i o n of 7/ and ii.

I f the f l o w past a f l a t plate is solved by an unsteady two-di-mensional numerical method, the result w i l l be a series of solutions

. E V I o m e n c l a t u r e . a = submerged d e p t h of plate C!,a = N/{y2pvoUaH) g = acceleration of gravity h = mesh length of f i n i t e d i f f e r -ence grid

I = chord length of plate N - yaw m o m e n t about plate

m i d c h o r d

p = hydrodynamic pressure on plate

s{x) = strength of dipole d i s t r i b u

-t i o n used -to represen-t plate outside of grid

(x,y,z) = dimensional spatial

coordi-nates

(x*,y*,z*) = [x/l, y/a, zja) =

nondi-mensional coordinates = l o n g i t u d i n a l coordinate of

nth station a f t of leading

edge

Ax = spacing between stations X;., 2;. = coordinates of

hydrody-namic center

Y(xn) = side force acting on plate

upstream of

U = freestream velocity Vo = all, equivalent t o cross

ve-locity under slender body approximation

Vy, Vz = y, 2 components of flow ve-locity

a = angle of attack i n radians;

positive when plate is yawed to starboard e = plate aspect ratio

T, = U{t/gl^'^) = length Froude

number x (aspect ra-t i o ) i/ 2

/ i = a/e = angle of attack/as-pect r a t i o p = f l u i d density ^ = free-surface elevation f * = nondimensional elevation = ^al $ = velocity p o t e n t i a l <p = p e r t u r b a t i o n p o t e n t i a l ip* = nondimensional potential

= ip/auo Subscripts R = conditions on plate at y = ax -1-0 L = conditions on plate at y = ax - 0 n = conditions at n t h longitudi-nal station

in transverse planes or stations starting at the leading edge. The f l o w at any one of these stations is independent of downstream conditions so that each station could be the trailing edge of a plate w i t h length equal to the local value of % and ?? = U{a/gx^y/'^. Since depth and cross velocity are the same at all stations, the product riix is fixed. Any nondimensional variable may be ob-tained as a f u n c t i o n of i} f o r constant IJM by nondimensionalizing at each station using the local value of x as the length scale.

The significance of the ratio = a / e for a typical hydrodynamic quantity is illustrated by Fig. 2, w h i c h shows some yaw moments measured by Van D e n B r u g [2] on a f l a t plate at f i x e d depth and speed and variable angle of attack. Since r] is fixed f o r all these cases, the behavior of the nondimensional f o r m of the yaw moment should depend only on fx. I n particular, the range over which the yaw moment is a linear function of a is directly proportional to the aspect ration e. Also, the slope i n the linear region is determined by the solution for the l i m i t i n g case ^ = 0.

Y A W M O M E N T

I , D A T A FROM ^ A N D E N B R U G , E T A L

Fig. 2 Yaw moment versus angle of attack

Linear problem

As / i = a / e approaches zero, the free-surface conditions (10a) and (lOfo) reduce to their Hnearized f o r m :

(11)

(12) | l ! = ^ a n d ^ = - r . - o n . * = 0

dx-^ dz-' dx*

The boundary condition (8) on the plate becomes ^ = 1 on?/* = 0 , z * > - l

dir

Laplace's equation and the i n f i n i t y conditions are unchanged. I t is assumed for the linear problem that the potential is antisym-metric about tj = 0. Then the condition on the y-axis below the plate is

<p*{x,0,z) = 0 o n z * < - I (13)

Numerical solution of linear problem

I t is convenient i n discussing the numerical solution to return to dimensional variables and transform the equations developed in the preceding section to their dimensional forms.

The numerical calculation is made w i t h a square f i n i t e d i f f e r -ence grid like the one in Fig. 3. This grid represents the (!/,z)-plane for successive values of Xn w i t h XQ corresponding to the leading edge and u n i f o r m spacing Ax between the planes or stations. Potentials are determined at grid line intersections under boundary conditions specified along the f o u r sides of the grid. Successive overrelaxation (SOR) is used to solve the standard second-order finite-difference f o r m of Laplace's equation. This method was motivated, i n part, b y the straightforward extension to the non-linear case. On the plane containing the plate, which corresponds to the left side of the grid, boundary conditions are finite-difference forms of equations (12) and (13). The f l o w extends to infinity, but a numerical calculation of this type is restricted to a f i n i t e grid. The lower and r i g h t sides of the g r i d f o r m the outer boundary where the potential is determined b y matching the interior n u -merical solution to an exterior analytic solution. The top of the grid is the static waterline, z* = 0. Here potentials and elevations are derived f r o m the upstream conditions b y the approximations to the dimensional forms of conditions (11):

f ( x „ + i/ 2, y ) = axn-i/2,y) + A x U - i ^ {xn,yfi) {Ua) and

V>{x„+i,y,0) = <p{x„,y,0) - gAxU-^^ixn + i/2,y) (Ub) so that the potential is specified on the top of the grid. The stability

of this type of algorithm is discussed i n the Appendix. The

ele-OUTER BOUNDARY

Fig. 3 Finite-difference grid

vation at %„ is not used i n the calculation but may be estimated f r o m

axn,y) = èix„-i/^,y) + Ax^ixn,y,0)/2U (14c) The i n f i n i t y conditions i m p l y that at a sufficient distance f r o m the plate, the f l o w may be represented by a source and a dipole and the resulting free-surface disturbance. Since the f l o w is an-tisymmetric, the source does not contribute. Therefore, on the lower and right sides of the g r i d and beyond, the solution is rep-resented by a line dipole distribution of strength s(x) on the line

(y,z) = (0, — a / 2 ) w i t h the dipole axis normal to the plate. Since

the governing equations are equivalent to those of an unsteady surface problem, analytic results derived f o r unsteady free-surface f l o w are directly applicable. L e t f{t,y,z) represent the two-dimensional analytic solution of the potential f o r a dipole oriented i n the y-direction under a linearized free surface w i t h unit strength at {y,z) = (0, - a/2) for 0 < / < AxlJ-'^ and zero strength otherwise. The analytic expression for this function may be f o u n d b y integration of a result of Havelock [3, equation (12)]:

f(t,y,z) = 2AxV-^ dke'''-^-''^^hinhj

• [cos lit - AxU-^) Vki\ - cos (t Vkg)] (15) A t x = Xn the potential on the outer boundaries may be matched to the analytic f o r m

0.10 0.0 -0.10 , O O " <J ü O O P L A T E D E P T H IS T E N M E S H L E N G T H S I N A L L C A S E S g S B ' ° ^ P O I N T S C A L C U L A T E D I O N 4 1 X 4 1 G R I D O P O I N T S C A L C U L A T E D O N 2 1 X 2 1 G R I D • P O I N T S C A L C U L A T E D O N 1 6 X 1 6 G R I D 0.0 1.0 zo 2 5

Fig. 4 Effect of grid size on free-surface elevations calculated at

0.0 -0.2 •0.4 z/a 0.6 -0.8 •n = 0.23 O NUMERICAL VALUES \ ) WITHOUT FINITE DIFFERENCE MODIFI-CATION AT z = 0.95a - V NUMERICAL VALUES WITH FINITE DIFFERENCE MODIFICATION QJ au

S —'—

1 1— 1 0.0 - 0.1 • 0.2 - 0.3 • 0.4 • 0.5

Fig. 5 Effect of finite-difference modification on plate potential calculated at leading edge

Fig. 8 Transverse free-surface profiles from linearized solution

,y/a = 0.05

Fig. 9 Longitudinal free-surface profiles from linearized solution

z/a -0.15 O NUMERICAL VALUE WITHOUTSURFACE PRESSURE A NUMERICAL VALUE WITH SURFACE PRESSURE -O.Ol -0.0 -0.12 -016 -0.20

Fig. 6 Effect of free-surface pressure on plate potential calculated at leading edge 060 0.50 0.40 z- 0.30 020 0.10 Ah/a = 0.05 • ti/a = 0.10 0.0 ₀₅ 1.0 1.5

Fig. 7 Effect of mesh size on free-surface elevations calculated at

7) = 0.714

<p{xr„y,z) = s{xn)e{y,z)

+ gixn,y,z), (v>z) on outer boundary (16)

where s(x„) is the constant dipole strength over

g{x„,y,z) = E sixj) + {Ax(n - j)U ',y,z} j = 0

w h i c h is the contribution f r o m all dipoles upstream, and

e{y,z) = y[{y' + {z + a)T' - \v' + (^ " a^-']

(16a)

is the potential f r o m a unit dipole.

Let (pA{y,z) represent a numerical solution w i t h homogeneous conditions on the free surface and the plane containing the plate. Then

fAiy,o) = 0 <PAiO,z) = 0, z < —a

dy

<pAiO,z) = 0, z> - a (17)

and on the outer boundaries.

PAiy.z) = e{y,z), {y,z) on outer boundary (18)

Also, let ipBixn,y,z) represent a numerical solution w i t h correct boundary conditions on the free surface and the plane of the plate and let (ps = g{xn,y,z) on the outer boundaries. Then, the f u l l solution m a y be expressed as

The analytic and numerical solutions have matching velocities normal to the outer boundary at a point on the boundary i f

dn dn

öe{y,z) dn

dgix,y,z) dn (20)

where d/dn represents differentiation normal to the boundary. Terms on the left side of (20) are f o u n d b y numerical d i f f e r e n -tiation, those on the right by evaluation of analytic expressions. The sum of the squares of the errors may be m i n i m i z e d by weighting this condition by the local value of (de/dn) — {d<pA/dn) at each point on the outer boundary. I n practice, de/dn was used as a weighting factor and s(3C„) taken to be

'^''"> - ^ dn\dn dn) /^~d^\-d^~Tn) (21)

where summations are over all points on the outer boundaries. This method has been tested for a plate ten mesh lengths deep using a 41 X 41 grid, a 21 X 21 grid, and a 16 X 16 g r i d corre-sponding to grid lengths of 4.0, 2.0, and 1.5 times the plate depth. Free-surface elevations at r; = 0.23 are shown i n F i g . 4. The profile f o r the 21 X 21 g r i d is quite close to that f o r the 4 1 X 4 1 , and even the 16 X 16 grid yields surprisingly good results, Po-tentials along the plate are essentially identical f o r all cases.

The velocity f i e l d for the yawed plate has singularities at the bottom edge of the plate and at the intersection w i t h the free surface. These are most easily illustrated by the special case of the leading edge {x = 0). Since the f l o w at the leading edge cannot be anticipated, f l o w there must be considered as inipulsive w i t h zero potential on the free surface. The condition on the plane of the plate is the dimensional f o r m of equations (12) and (13):

dip

= vo,z> —a and <p = Q,z< (22)

Let f{z) = <p (0,0,z) denote the potential on the plate. Sedov [4, p. 169] has derived the impulse potential on a surface-piercing ellipse. For a f l a t plate this reduces to

fiz) = 27r-h In [-z/{a + (a^ - z S j i / z ) ] (23)

The vertical velocity along the plate at the leading edge is d / / d z . I t has a square-root singularity at z = —a and a logarithmic sing-ularity at the intersection w i t h the free surface . These singular points are also grid points.. The potential at the lowest g r i d point on the plate is identically zero. The local solution of the first grid point above is improved b y replacing the quadratic t e r m i n the finite-difference expansion of Laplace's equation i n z b y a (z -h

a)^/^ term. Numerical solutions w i t h and without this m o d i f i e d

finite-difference f o r m near the lower portion of the plate are compared w i t h equation (23) i n F i g . 5.

The singularity at the free surface is not as strong as the one at the lower edge, but it is more awkward since free-surface velocities are needed to establish conditions at the next station.

This singularity may be removed by a pressure acting between the first and second grid points on the free surface such that.

dv^

dy = <Pyz{x,y,0)

vanishes over y \ <h where h is the mesh size of the grid. This condition applied to Laplace's equation produces the f i n i t e d i f -ference condition

^ j ; , 0 , 0 ) = ip{x,h,0) + <pix,0,h) - <({xXh) (24) The case of zero surface pressure is approached as h goes to zero since the magnitude and range of the surface pressure are both proportional to h. Computations indicate that the surface pressure required to maintain (24) is greatest at the first f e w stations near the leading edge and reduces to zero downstream. The effect on

Fig. 10 View of free surface for plate yawed to starboard

the leading-edge solution f o r h/a = 0.05 is illustrated by Fig. 6, w h i c h compares potentials calculated on the plate w i t h and without condition (24). The total impulse force applied to the free surface due to (24) is 1.0 X ICr^oUpa'^ i n this case, and the force acting on one side of the plate is 3.1 X lO~hoUpa^. The general effect of (24) on the downstream solution may be demonstrated by comparing profiles calculated w i t h h = 0 . 0 5 and h = 0 . 1 0 . Condition (24) should have a significantly greater effect on the latter case since i t imphes a surface pressure of twice the magnitude and extent. Profiles at T; = 0.714 are compared i n Fig. 7. T h e y d i f f e r only near the plate. Calculated side forces d i f f e r by about one percent.

Results of linear calculations

The linear problem was solved w i t h a 41 X 41 g r i d using t w o values of h/a (0.10 and 0.05) as a check. A step size of Ax = a / 1 0 was f o u n d to be adequate. Transverse-plane profiles at several stations are shown i n Fig. 8. Longitudinal plane profiles are shown i n F i g . 'è.^ These profiles are antisymmetric across the plate. A perspective plot of the free-surface is shown i n Fig. 10. Elevations i n these figures are normalized by the product of plate depth and angle of attack rather than according to equation (9), and are presented as a function of x/a rather than -q. This is done to keep profiles more directly related to the physical f l o w . T o properly define the f l o w , another parameter such as U^/ga has to be specified. Of course, the amplitudes shown go beyond the limits of linear theory. I t may be noted f r o m F i g . 9 that the elevation becomes zero over the a f t portion of the plate. This implies that at low values of j] the disturbance is confined to the forward region of the plate and the f l o w i n the aft section reduces to the lower half of the solution f o r i n f i n i t e f l o w over a plate of height 2a. This is a consequence of linearization. As ri becomes small, f l o w i n the

^ The author has recently been made aware of a paper by M. H. Hirato ("The Flow Near the Bow of a Steadily Turning Ship," journal of Fluid

Mechanics, Vol. 71, p. 283). The example solved by Hirato using a

Fourier transform method appears to be equivalent to the linear problem considered here. In particular, the free-surface elevation in Fig. 3 of Hirato's work should correspond directly to the curve near the plate, (y/a = 0.05) in Fig. 9. Interestingly, the two curves give similar heights and positions for the maximum elevation but are otherwise quite different.

'i.o Cya 3.0 2,0 1.0 0.0 O O -COMPUTtD LINE(f;>]=0) D A T A F R O M R E F E R E N C E [ 2 1 P L A T E A . A S P E C T R A T I O = 0 . 2 0 P L A T E D , A S P E C T R A T I O = 0 . 2 0 P L A T E A , A S P E C T R A T I O = 0 . 3 5 P L A T E A , A S P E C T R A T I O = 0 . 5 0 P L A T E D , A S P E C T R A T I O - 0 . 5 0 05 1.0 1.5 Fig. 11

n = FROUDE NUMBER x (ASPECT RATIO)

Theoretical and experimental side-force coefficients

S = 2.0 1.5 C N « 1.0 05 0.0 D A T A F R O M R E F E R E N C E [ 2 1 P L A T E A , A S P E C T R A T I O = 0 . 2 0 P L A T E D , A S P E C T R A T I O = 0 . 2 0 P L A T E A , A S P E C T R A T I O = 0 . 3 6 P L A T E A , A S P E C T R A T I O = 0 . 5 0 P L A T E D , A S P E C T R A T I O = 0 . 5 0 O O 05 1.0 1.5

n = FROUDE NUMBER x (ASPECT RATIO)''''

Fig. 12 Theoretical and experimental yaw moment coefficients

-02

O O

0.2

0.4

DATA FROM REFERENCE [2]

/ \

^ PLATE A. ASPECT RATIO = 0.20 ~ \ ^ PLATE D, ASPECT RATIO = 0.20

\ °

PLATE A, ASPECT RATIO = 0.35

\ ^ C O M P U T E D LINE ^ PLATE A, ASPECT RATIO = 0.50 PLATE 0, ASPECT RATIO = 0.5O

\ ° A —è-———•

— — COMPUTED LINE

1 1

0.0 0.5 1.0

r\ = FROUDE NUMBER x (ASPECT RATIO)'''' Fig. 13 Position of hydrodynamic center

, 15

transverse plane should correspond to the steady-state l i m i t of a plate moving w i t h constant speed «Q. Since side forces and wave elevations are proportional to VQ^, they do not appear i n the linear solution. Thus, the formulation of the linear three-dimensional problem includes only the transients of the equivalent two-di-mensional f l o w .

The linear solution may be used to predict forces and moments at small angles of attack. For the linearized case, the f l u i d pressure acting on one side of the plate is

(25)

side is equal and opposite to that on the other so that the side force on a differential element is

d^Y = -IpUip^dzdx (26) and the total force ahead of x = / is

Y{l) = -2pU j ° J z i p { l , z ) (27)

The moment acting about the m i d c h o r d is

« , o 4 m o - X > . ( ^ ) . (.SI Velocity squared terms are not included. The potential on one Nondimensional coefficients Cy^ = Y/S and C^a = N/Sl where

r

S - l/2pVoUa^ are functions of the parameter V- Calculated values f o r these coefficients f o r linear theory are shown i n Figs. 11 and 12. L i m i t i n g values are Cy„ = w, C^a = T / 2 for ?y = 0, and

Cya = 1-25, Cwa = 0.62 for infinite rj. Also shown i n these figures

are experimental values based on static hydrodynamic coefficients for a ^ 0 reported by Van Den Brug et al [2]. Data for an aspect ratio of 0.04 fall outside the general range of values predicted and are not shown. I t is perhaps surprising that agreement between theory and experiment is best for the larger aspect ratios since the f o r m u l a t i o n involves an assumption of small aspect ratio. The probable explanation is that f l o w separation along the lower edge of the plate is a greater factor for low-aspect-ratio data.

The hydrodynamic center is defined by Xf and Zf where

Xf = xY - 1 _{N ( / ) Y - (29)}

and Zf is similarly defined. Computed and experimental values are compared i n Fig. 13. One interesting feature of the theoretical prediction is a region of negative Xf at low values of r]. Data do not support this aspect of the prediction, b u t agreement is good at higher values of r}.

Nonlinear proMein

The side force and moment coefficients i n Figs. 11 and 12 are a measure of the rate of change of side force and moment w i t h respect to angle of attack near zero angle of attack. Data f o r a f i n i t e angle of attack may contain nonlinear effects. Figures 14 and 15 show data measured b y Van D e n Brug [2] on a plate w i t h

a = t a n - i (0.16) and aspect ratios of O50, 0.35, and 0.20

corre-sponding to ju = 0.32, 0.46, and 0.80. The forces are larger than predicted by linear theory, particularly at low Froude numbers. The greatest deviations occur for low aspect ratios corresponding to high values of the parameter n, w h i c h is a measure of the free-surface nonlinearity. However, calculations indicate that only a fraction of this deviation may be accounted f o r by nonlinear free-surface effects.

Numerical solution of the nonlinear problem was first attempted by a method which traced the exact free surface using an irregular grid. A t first this method was numerically unstable, and a second method was developed based on second-order expansion of the free-surface boundary conditions about the z = O plane. Exper-iments w i t h the second-order method suggested that the instability i n the f u l l y nonlinear solution was caused by the algorithm used to advance the free surface. A f t e r this portion of the nonlinear calculation was altered, the solution was successfully stabilized. A brief analysis of this numerical instability is given i n the A p -pendix. 7.0 fi.n 5.0 ' Y a 4.0 3.0 2,0 ' 1,0 0,0 1,6 0,8 0,0

DATA FROM REFERENCE (2) a PLATE A, ASPECT RATIO = 0,50 • PLATE A, ASPECT RATIO = 0,35 A I^LATE A, ASPECT RATIO = 0,20

LINEAR THEORY

0,5 1,0 >7 = FROUDE NUMBER X (ASPECT RATIO )'''' Fig. 14 Side-force coefficients for tana = 0.16

B PLATE A, ASPECT RATIO = ,50 • PLATE A, ASPECT RATIO = ,35 A PLATE A, ASPECT RATIO = .20

1,5

0,5 1,0 ri = FROUDE NUMBER X (ASPECT RATIO)'''" Fig. 15 Yaw nnoment coefficients for tana = 0.16

OUTER BOUNDARY

Fig. 16 Grid for second-order method

Second-order method

A second-order method was developed using a rectangular grid like the one i n Fig, 16. The upper boundary of this grid represents the free-surface, and the remaining three sides f o r m the outer boundary. The plate is i n the center of the grid, where there is a discontinuity i n the potential above the lower edge of the plate. The method of computation is generally similar to that described for the linear problem. The primary points of difference between the two methods are as follows:

(i) The free-surface boundary conditions are second-order expansions of the nonlinear equations about the plane z = 0.

(ii) Unlike the linear calculation, the second-order method must account f o r the lateral shift of the plate as x increases. To maintain the plate i n the center of the grid, the grid is shifted w i t h each advance of the calculation such that a g r i d point at {y,z) f o r Ï = x„ w i l l appear at (y + aAx,z) ioT x = Xn +

i-(iii) The solution on the outer boundary contains a source as well as a dipole since the potential is not assumed to be

antisym-metric. Also, the shift of the plate and the grid must be accounted for i n the analytic solution.

(iv) A second-order method is used to evaluate forces and moments on the plate.

The free-surface conditions consistent w i t h the slender-body assumption are represented by equations (lOa) and (lOb). I n the linear analysis, terms of order (a/e) were excluded. I n the present analysis these terms remain and terms of order (a/e)^ are excluded. The expansion

^(y*,z*) = <p*{y*,0) + (^) ^

dz' ' - +

o(^^y

(30)

is used to place these conditions on z* = 0. The resulting free-surface boundary conditions expressed i n dimensional f o r m are

dx ^ dy (31a)

and

where the subscripts R and L represent conditions on the right and left sides of the plate. This expression may be developed as

ly=ax

OUTER BOUNDARY Fig. 17 Grid for nonlinear mettiod

where

^ + A ) = i ( „ ^ 2 + , ^ 2 ) _ g ^ (3ifo)

The first t e r m represents a first-order quantity, and the remaining terms are of second order. The t h i r d and f o u r t h terms may be evaluated, correct to second order, by expanding about z = ^. For example, i f PR is approximated by

(31c) = P g ( & - ^ ) + P t/ ^ « z ( ^ = 0) (37) and all quantities are evaluated at z = 0.

W i t h each advance of the calculation, the free-surface conditions are applied i n the following steps:

1. The free-surface velocities are determined f r o m equation (31c). The elevation i n (31c) is |(j:„,!/) extrapolated f r o m

\{Xn-\l2,y)-2. The elevations are advanced f r o m Xn-1/2 to a:„+1/2:

? ( ^ n + i / 2 , f / + a A ^ )

(32a) where

t / = !/ + A ï j ü o - Vy{xn,y)}U-^ (32fo) Linear interpolation over the y-variable is used to evaluate

quantities at y'.

3. The dynamic free-surface condition is used to advance the potential at z = 0 :

(p{xn+i,y + aAx,0)

+ ê n - H / 2 ^iXn + l,y+ aAx,0} - ^ {Xn,y,0)

= <p{xn,y',0) + AxU - 1 • ( « • 2 _ , . - g f n + 1/2 (33) where

?n + I / 2 = ?(a:n+l/2,!/ + « A x ) ,

Vy = Vy{xn,y), and = v^ixn.v)

and y' is as defined i n equation (32fo).

This gives a mixed condition at z = 0 rather than the Dirichlet condition of the linear problem. Because the normal derivative term is of second order, i t is possible to apply condition (33) as a D i r i c h l e t condition i n two steps. First the problem is solved w i t h o u t the normal derivative t e r m i n (33). Then prior to the advancement of the calculation, i t is solved again using (33) w i t h normal derivatives calculated f r o m the first solution. This differs f r o m the conventional analytic method i n that second-order terms are returned to the calculation at each advancement.

The pressure on one side of the plate is

P = - f ^ p - f « + - vy)/2 - pgz (34) evaluated on y = ax ± 0. The net side force ahead of x = / is Y(0 = dx X ! ^''^ ~ ^ X '^'^^'^

- C^'CIZPL] (35) •JO

then

(38) Second-order results are discussed after the f u l l y nonlinear method is described.

Nonlinear method

The nonlinear calculation is made w i t h a grid of the type shown i n F i g . 17. The irregular upper boundary of this g r i d is the free surface. The free-surface potential is specified at all points where this boundary intersects either a horizontal or a vertical line of the grid. The potential and elevation of the free surface are assumed to be linear between these points. The three remaining sides of the g r i d f o r m its outer boundary where the potential is matched to an analytic solution by a method essentially identical to the one used i n the second-order problem. The grid shifts as the calcu-lation advances so as to maintain the plate i n its center.

Dimensional forms of equations (10a) and (lOb) are immediately applicable as free-surface boundary conditions. W i t h each ad-vance of the nonlinear calculation, these conditions are applied i n the f o l l o w i n g steps:

1. Free-surface velocities, Vy and Vz, are calculated at each point of intersection between the free-surface and the grid.

2. These points are displaced to positions ( X n + i,yn-i- i , 2 n + 1 ) determined b y

Xn+l =Xn + Ax yn+i = yn + VyAxXJ-^

Zn+i = z„ + VzAxU-K (39)

3. Potentials at the displaced points are determined by applying equation (lOb):

'p{Xn + l,yn + l,Zn + l ) = >p{Xn,yn,Zn)

n

(Vz^ + Vy^) + gyn + 1 AxU- (40)

4. The displaced points are joined by straight lines. Free-surface points for the next potential problem are established where these lines intersect the shifted grid. Potentials of these new free-surface points are determined by linear interpolation between the displaced points.

The f o r m u l a t i o n of the side force acting on the leading edge is identical for the linear, second-order, and nonlinear problems. For the nonlinear problem, the side force a f t of the leading edge may be determined f r o m the pressures at all grid points on the plate covered by the f l u i d at Xn + 1 / 2 , " = 0, 1, 2 . . . . I n most cases this

r

0.10 0.0 -0,10 A NONLIMEAR • SECOND ORDER , O LINEAR 1 1 2 = 0 O ^ Q 0 _

° n

0

" ° § D

z = 0

' ° ° " 0

0 0

g

§ i

o o I I I 1 -< 1 t 1 1

—i

1 1 1 1 I

1

I I -0.80 -0.60 -0.40 -0.20 0.0 0.20 0.40 0.60 0.80 y/a

Fig. 18 Free-surface elevations for -q = 0.667 and p. = 0.300

A NONLINEAR • SECOND ORDER • LINEAR • • Ü Ö n A • A O ^ n D 0 0 0 A

0

Ö ^ < A A D 0 z = 0 Ü 0 " ' O O ° I l l l

' • • •

I l l l _ l I I I i I 1 1 1 — -O80 -0.60 -0,40 -0.20 0,0 0,20 0.40 0.60 0.80 y/a

Fig. 19 Free-surface elevations for i] = 0.333 and p. = 0.600

pressure may be estimated by the f i n i t e difference approxima-tion

P(Xn+l/2.,yn + l/2,z) = -UpMXn+l,y„ + l>z}

- <piXn,y,„z)] + I ( « 0 ^ - vy{Xn,y,„z)) " PgZ (41)

where = aXm ± 0. Since the moving free surface may cover or uncover new g r i d points, this equation may not be apphcable for certain grid points close to the free surface. The pressure at these points may be estimated b y expansion about the free surface at Xn+i/2 as i n equation (38). L e t the grid points on the right and left sides of the plate covered b y the f l u i d at Xn + 1/2 be located at

zo^ >Zi^> ... ZM^ and zo^ > z i ^ > > . . . Z n ^ where Z Q' ^ and Z o ^ correspond to the free-surface height at x„+1/2. Then the net side force between x„ and x„+i may be estimated as

Y{Xn + i)-YixJ M- 1 1 = Ax

E

i ( p ( 2 m « ) +

p(z„+i«)Kz„«.

- Z „ + i « ) m = 0 2 -Ax'zhpiZn'^) + piZn+yMZn'^-Zn+y) (42) n = 0 2

where p (zn, ^ ) and p (Zn ^ ) are the pressures at the grid points and p(^o«) = p ( 2 o ^ ) = 0.

Nonlinear results

The second-order method and the f u l l y nonlinear method were used to solve the nonlinear problem f o r rjfi = 0.20 and ijn = 0,40. The case of T/JU = 0.01 was also solved to check against the linear solution. The second-order grid was 41 points wide and 21 points deep w i t h a mesh length7i = O.lOa. The nonlinear g r i d was also 41 points wide and extended 21 points below z = 0 w i t h h = 0.10a. Nonlinear, second-order, and linear solutions f o r ^a~^ = are compared i n Figs. 18 through 26.

Elevations f o r ryx = 0.20 i n transverse planes at 77 = 0.667 and

•n = 0.333 are shown i n Figs. 18 and 19. Figure 20 illustrates the

0 . 2 0 I / " 0,0 N O N L I N E A R S E C O N D O R D E R L I N E A R 0,8 1.6 2,4 3,2 4,0 4,8 5,6 6,4 Fig. 20 Free-surface profiles at y = O.lOa for t^r] = 0.200

free-surface profiles at (/ = i O . l O a . Linear and nonlinear ele-vations deviate significantly f o r TJH = 0.20, particularly at small values of ij. For r}p = 0.20, the second order gives a satisfactory approximation f o r the nonlinear effects on the free-surface ele-vations, as illustrated by Figs. 18 through 20, and on the side force and moment coefficients.

Free-surface elevations for rj/j, = 0.40 i n transverse planes at rj = 0667, 0.333, and 0.182 are shown i n Figs. 21 through 23. Figure , 24 shows the free-surface profiles at y = ± 0 . lOa. Deviations be-tween linear and nonlinear free-surface elevations are large, par-ticularly f o r small values of ?y. The second-order solution deviates f r o m the nonlinear solution f o r pT] = 0.40 between rj = 1.0 and rj = 0.50, where i t diverges violently. The nonlinear e f f e c t is very d i f f e r e n t f o r the elevations on the two sides of the plate. O n the suction side, the transients die out more rapidly as the free-surface nonlinearity increases, and the solution appears to approach a stationary solution w i t h the elevation rising to ^ = vo^/2g on the plate, as required by Bernoulli's equation. Of course, the elevation on the plate could be reduced i n the corresponding physical sit-uation i f the f l o w separates f r o m the plate. O n the pressure side.

S E P T E M B E R 1976 133

A NONLINEAR ^ • SECOND ORDER o LINEAR ( z= 0

1

G ^

- ° ^ 8

O A O 6 ° — ^0

* * I

1 z = 0 o o I l l l

: . .

1 1 1 1 — -0.80 -0.60 <l.40 -0.20 0.0 0.20 0.40 0.60 O.f Fig. 21 Free-surface elevations for T] = 0.667 and /t = 0.600

r

0.30 A NONLINEAR © LINEAR 0 A A 0.20 |/a 0.10 0.0 -0.10 - ^ ^ ^ ^ A A A A A ^ z= 0 ^ • • • n f ^ i

1

- - " ' O ° °

t

0.20 |/a 0.10 0.0 -0.10 O G O ° I l l i ' • • o ° " I l l l -0.80 -0.60 -0.40 -0.20 0.0 0.20 0.40 0.60 O80 V/a

Fig. 22 Free-surface elevations for ?; = 0.333 and IJ. = 1.200

" A A A A A A A A A , A NONLINEAR k • LINEAR , A A Z = 0 f I l l l • • ' • . • • ' ! • A A I l l l 0.20 0.10 0.0

-o.ioh

-0.80 -0.60 -0.40 -0.20 0.0 0.20 0.40 0.60 0.80 y/a

Fig. 23 Free-surface elevations for j) = 0.182 and n = 2.200

free-surface norilinearity appears to retard establishment of any stationary solution. A heuristic explanation can be based on the observation that f l o w tangential to the free surface is contractive

{{d/ds)vs < 0} on the pressure side and stretching f l o w {{d/ds)Vs

> 0} on the suction side. Dagan [5] has shown analytically that contraction destabilizes free-surface flows, while stretching has the opposite effect. This free-surface stability on the suction side is apparently confined to transverse f l o w since V a n D e n B r u g [2] observed breaking at the first crest on the suction side. Such breaking i n a longitudinal direction would not be modeled b y a nonlinear method w i t h the slender-body assumption.

Computed nonlinear effects on side force and yawing moment coefficients are small when compared w i t h the nonlinear effects on the elevations. Figure 25 shows nonlinear, second-order, and linear solutions f o r the side-force coefficients at pi} = 0.40, w h i c h is greater than (irj f o r most of the data points i n Fig, 14. The m a x i m u m side-force coefficient is 13.5 percent greater than the linear value (3.0 percent for iirj = 0.20), The moment coefficients for iiT) = 0.40 are shown i n Fig. 26, They exhibit even less effect f r o m free-surface nonhnearity.

Obviously, the nonlinearities i n the hydrodynamic coefficients measured by Van Den Brug [2] may not be explained by

free-surface nonlinearities i n a calculation based on the slender-body approximation. I n visualization tests. Van D e n Brug [2] observed f l o w separation, starting at a = 0.10, at high speeds, and ventila-tion. H e attributes the nonlinearities to separation and wave breaking at low speeds and ventilation at higher speeds. These considerations suggest that more attention be given to the f l o w near the sharp lower edge of the plate. W h e n viewed as an unsteady two-dimensional problem, even the application of a simple Kutta condition is unclear, since the plate must first shed vortices. There is presently available, however, a method described b y Kandil [6] f o r modeling separated f l o w f o r three-dimensional steady prob-lems. I t may be possible to combine such a method w i t h a slen-der-body calculation of the free-surface effects.

Conclusions

The present results indicate that a calculation based on the slender-body approximation and a linearized free surface may produce w o r t h w h i l e estimates of the side force and y a w i n g mo-ment f o r a surface-piercing f l a t plate at moderate to high values of ri, aspect ratios f r o m 0,20 to 0,50 or possibly higher, and yaw angle i n the linear regime. Nonlinear calculations show that

-0.40 L

0.8 2.4 3 . 2

x(g/u^a)

Fig. 24 Free-surface profiles at y = 0.10a for ni] = 0.400

free-surface nonlinearities can influence the f l o w and free-surface elevations greatly but do not significantly alter side force or yawing moment. Second-order calculations were satisfactory f o r mod-erate degrees on nonlinearity, but the second-order method was in several ways more d i f f i c u l t to implement than the f u l l y non-linear method. Experimental values of side force and yaw mo-ment coefficients contain nonlinearities not predicted b y the present formulation.

Acknowledgments

This study was conducted under the N u m e r i c a l Naval H y d r o -dynamics Program jointly funded by the O f f i c e of Naval Research, the Naval Ship Systems Command, and the D a v i d W . T a y l o r Naval Ship Research and Development Center.

References

1 Ogilvie, T. F., "Nonlinear High-Froude Number Free-Surfaee Problems," Journal of Engineering Mathematics, Vol. 1, No. 3, July 1967.

2 Van Den Brug, J. F., W. Beukelman, and G. J. Prins, "Hydrody-namic Forces on a Surfaee-Piercing Flat Plate," Report NR 325, Ship-building Laboratory, Delft University of Technology, Delft, The Neth-erlands, 1971.

3 Havelock, T., "Some Cases of Wave Motion Due to a Submerged Obstacle," Proceedings of the Royal Society, Series A, Vol. 93, 1917, p. 523.

4 Sedov, L. I . , Ttoo-Dimensional Problems in Hydrodynamics and Aerodynamics, Interscience Publishers, New York, 1965.

5 Dagan, G., "Taylor Instability of a Non-Uniform Free-Surface Flow," Journal of Fluid Mechanics, Vol. 67, Part 1, 1975.

6 Kandil, O. A., "Prediction of the Steady State Aerodynamic Loads on Lifting Surfaces Having Sharp Edge Separation," Dissertation, Virginia Polytechnic Institute, Dec. 1974.

A p p e n d i x

Numerical stability

The stability of the algorithm for advancing the free surface is most easily analyzed for the linear case. I n practice, linearly stable algorithms seem satisfactory f o r nonlinear problems, although nonhnear terms greatly exacerbate instabilities present i n the linear problem.

Consider a potential f u n c t i o n (p{x,y,z) d e f i n e d f o r z < 0 where it satisfies Laplace's equation. O n z = 0, the potential (p(x,y,z) satisfies the free-surface condition

dip _ Ó Z ' dx dx ' (43) 1.0 0.0 0.5 1.0 1.5

FROUDE NUMBER X (ASPECT RATIO)'

Fig. 25 Linear, second-order, and nonlinear solutions for side-force coefficient at /tij =. 0.40 1.6 1.2 08 0.4 0.0 NONLINEAR SOLUTION NONLINEAR SOLUTION I 0.5 1.0 1.5

r] = FROUDE NUMBER X (ASPECT RATIO)

Fig. 26 Linear, second-order, and nonlinear solutions for yaw moment coefficient at /xr] = 0.40

i n f i n i t y . I f (p{x,y,z) and ^{x,y) are specified f o r x = Xo, then they may be calculated for all x>Xohy methods similar to those already discussed i n the paper. For example, the free surface may be advanced i n x by the algorithm

^{x+ Ax) = ^{x) + ^-^(x)Ax

dz

dx + Ax) = <p(x) - ^ {^(x) + ^{x + Ax)}Ax (44)

where all quantities are defined at the same value of y. The re-lationship of algorithm (44) or similar algorithms w i t h equations (43) may be established by Taylor expansions about x' = x +

l/2Ax:

Hx + Ax) = èix') + I Ax^ix') + i A x ' ^ ^ + 0(Ax3)

. , : ) - ^ A x ^ { x ' ) + ^ A x ^ ^ ^ ^

dx^

+ 0(Acc3)

(45)

It is also assumed that <p{x,y,z) vanishes as y^ + z^ approaches w i t h similar expansions f o r (f{x) and {d(p/dz){x). A p p l i c a t i o n of

these expansions indicates that algorithm (44) is equivalent to the condition

dx

dx (*') = - i { x ' ) + 0( A x 2 ) (46) I n the l i m i t i n g case of zero Ax, equation (46) is equivalent to (43). I f As: is small but f i n i t e and terms of order (Axf are neglected, condition (46) reduces to

Consider a solution of the f o r m

<fix,y,z) = flejexp ( - kz + iky + ii^x)} (48)

This solution is unstable i f w has a negative imaginary part and stable i f co has a zero or positive imaginary part. The i n f i n i t y condition requires that k be positive, and condition (47) requires that

io' + ikAxco - /(2 = 0 ₍₄₉₎ This produces a negative imaginary component i n co and thus an unstable solution. By similar methods, i t may be shown that the algorithm used f o r the linear calculations:

<p{x + Ax) + <p{x) -^(^x + ^Axj Ax ₍₅₀₎

is stable, ancf that the linear l i m i t of the algorithm used f o r the nonhnear calculation is also stable.

Plotkin, Allen, " A Note on fhe Thin-Hydrofoil Theory of Keldysh and

Lavrentiev," JOURNAL OF SHIP RESEARCH, Vol. 2 0 , No. 2 , June 1 9 7 6 p p . 9 5 - 9 7 .

(7? should

reïd

"^^"''^'^ f o l l o w i n g correction to his paper. O n page 96, the first line of equatf CL = 27r« [1 - Koh-^ + (Ko- Ki)h-'