A comparison between the parabolic and partially-parabolic solution ...

A comparison between the parabolic and partiallyparabolic solution procedures for three-dimensional

turbulent flows around ships' hulls A. M. A b d e l m e g u i d

Department of Mechanical Engineering, lmperial College of Science and Technology, London SI¢7, UK

N. C. Markatos CHAM Ltd, Bakery House, 40 High Street, Wimbledon, London SW19 5A U, UK K. M u r a o k a 1HI Ltd, 1 Shinnakahara-Cho, Isogo-Ku, Yokohama, Japan D. B. Spalding

Department of Mechanical Engineering, lmperial College of Science and Technology, London SWT, UK [Received 1 July 1978J

This paper is concerned with the prediction of three-dimensional turbulent flows around bodies of arbitrary shape, with particular emphasis on a ship's hull. Two solution methods are compared employing a non-orthogonal coordinate system, in which the surface of the body is arranged to coincide with a coordinate surface. The velocity components are solved for the axial, radial and circumferential components in the cylindrical-polar system from which the non-orthogonal coordinates are derived. The partial-differential equations governing the flows under consideration are solved by two finite-difference methods for three-dimensional, parabolic I and partially-parabolic 2 flows. Turbulence is accounted for through a two-equation model of turbulence developed by Harlow and Nakayama 3 and modelled by Launder and Spalding. 4 Solutions are presented for flow around a ship's hull which demonstrate the potential of the present methods. Introduction

Classification of flows In order to classify the equations governing the flow of a fluid; it has been useful in numerical fluid mechanics to divide steady-flow problems into two classes, viz.: parabolic and elliptic. A flow is called parabolic if: (i), there exists a predominant direction of flow (i.e. there is no reverse flow in that direction); (ii), the diffusion of momentum, heat, mass, etc. is negligible in that direction; 0307-904X/79/030249-10/$02.00 O 1979 IPC Business Press

and (/ii), the downstream pressure field has negligible effect on the upstream flow conditions. Many duct flow,jet and boundary layer phenomena are of the parabolic kind; i.e. upstream conditions can determine the downstream flow properties, but not vice versa. A flow is called elliptic when at least tile last two of tile three prerequisites for parabolic flows are not satisfied. There is a class of flow situations, however, in which although the flow is strictly elliptic (because the downstream pressure field has a marked effect on the upstream

Appl. Math. Modelling, 1979, Vol 3, August

249

Turbulent flows around ships'hulls: A. M. Abdelmeguid et aL

flow conditions), the diffusion of the transport properties in the predominant direction is small enough to be neglected. This class of flows is called partially-parabolic (SpaldingS). Flow situations falling into the partiallyparabolic class include the flow in strongly-curved ducts, the flow in turbine and compressor cascades, and others.

Exchange laws In turbulent flows it is assumed that the shear stress, heat flux, etc. obey laws similar to those in laminar flows; thus, the following exchange laws are formed:

oil = -"eft" ~ bxl

The problem considered This paper compares two solution methods for flows around bodies such as the hull of a ship which are of basically cylindrical, but otherwise arbitrary in shape. The cross-sectional shape of the body may vary in an arbitrary, but smooth, way with axial distance. If the main flow is aligned at a small (or zero) angle to the axis of the body, and abrupt changes in body shape do not occur, the flow around the body may be presumed to be of the three-dimensional, parabolic (or boundary-layer) type. A solution method of this type has been reported by Markatos et aL 6 Under certain circumstances of great practical importance, however, the parabolic assumptions become invalid. For example, in regions where the body shape changes abruptly (such as at the stern of a ship, if the stern is a blunt one) or when the presence of a propeller causes a strong curvature of streamlines, it is necessary to take full account of lateral pressure variations. In these cases, it is necessary to employ the partially-parabolic method, whereby the governing partial differential equations are solved by the iterative marching integration scheme of Pratap and Spalding. 2 The remainder o f this paper is divided up as follows. The mathematical formulation and physical models employed in the present study are described, also the solution algorithms and the important aspects of the numerical solution, e.g. convergence and computer time requirements. The method is then applied to the predictions of flow around and behind the rear portion of a ship's hull. Results are presented and discussed, and conclusions drawn. M a t h e m a t i c a l basis

Partial-differential equations The partial-differential equations which govern steady, three-dimensional motion of fluids can be expressed in tensor notation as follows: Continuity: (1) Transport of fluid property ¢: __o ( p v , ¢ ) = Oxt

(2)

s0 - a ] ~ . , Oxi

where ¢ may stand for a velocity component, temperature, kinetic energy of turbulence or its dissipation rate, and Jo,i is the flux o f ¢ along the ith direction. For the transport of the velocity component U/, for example, equation (2) takes the form:

a

ax, (°u~ui) Fi

ap

oa~l

axi

axt

(3)

Fbeing a body force acting on the fluid, and aii the stress tensor.

250

Appl. Math. Modelling, 1979, Vol 3, August

'axt]

(4)

Jdp,i=--(/aeff-- / ~

w~rr,~: ~x--~

(5)

If the flow is laminar,/aeff becomes the laminar viscosity/at. For turbulent flows/aeff is calculated, in accordance with the k ~ e model of turbulence,4 as:

/,/err =/al + CDPkZ/e

(6)

where Co is a proportionality constant, k is the kinetic energy of turbulence and e its dissipation rate; both k and e being determined by solving separate partial-differential equations. The recommended values for oerr,~ to be used in equation (5), are shown in Table 1. In the present calculation it is assumed that Oeff,~b remains uniform in the flow domain. Co and other empirical constants are also given in Table 1.

Boundary-layer approximations The mathematical consequences of tile first two requirements, outlined in the introduction, and applicable to both parabolic and partially-parabolic flows, are the following:

(i) ox,i-~ 0, J~,x, = 0

(7)

x I being the predominant flow direction. (ii) Terms involving (OU2lOxl) and (aU3laxl) are neglected in the viscous terms ol2 and 013 in the xl-momentum equation. To take advantage of the third requirement of parabolic flows, outlined in the introduction, the pressure in the momentum equation for the predominant flow direction is presumed to be decoupled from the pressure of the momentum equations in the other two directions. Thus, the pressure gradients appearing in the three momentum equations are: ap xl-direction: - (8) axl 0p x2-direction: - (9) 0x2 0p xydirection: - ax3 where p stands for the average static pressure over a cross section. This decoupling of pressures is a necessary part of the parabolic procedure3 It permits a marching-integration teclmique to be used; and only two-dimensional computer Table 1 Constants of turbulence model Ct

C2

CD

aeff,k

Oeff,e

1 A4

1.92

0.09

1.0

1.23

Turbulent flows around ships" hulls: A. M. Abdelmeguid et aL

storage for all variables is needed. It is worth noting that the omission of this practice does not result in an increase in the accuracy for flows which are truly parabolic. For partially-parabolic flows, the above stated pressuredecoupling practice cannot be applied. Therefore, the pressure field which is common to all the three-momentum equations, requires a three-dimensional computer storage. Owvilinear coordhmte system In the current investigation, a non-orthogonal coordinate system is employed. The coordinate system has one axis normal to the axis of the body. The elements of the present system (~, ~, ~') are defined in terms of orthogonal, cylindrical-polar coordinates (r, 0, z), and in accordance with Figure I as follows:

General transport equation The general differential equation for the transport of the flow property 4, expressed in the r/, ~, ~"coordinates takes the form: 1

0

1

3

[Ar/Nttq5] + - - - - [r~b v -- ttG - w E ] rA~EA~ N O~ rAr/N 3r/

÷

1

0

a-----~[w~rAr/N]

rAr/N

+-- 1 0-[ PO r ( I + G 2 ) ~ + S ¢ ~ rAr/N 3r/LAr/N or// (12)

r-r S

In the above equations, five new geometrical quantities are present, and are defined as follows:

rN--r s O - Ow

(I0)

Ar/N = rN -- rs

OE - Ow

ff

~'=z

A t e = OE -- Ow = - -

where r s is the hull radius, and rN is the radius of the outer boundary. The governing partial-differential equations are transformed in'to the 77, ~, ~"system by using the following transformations: 0

1

3

Or

(rN -- r s )

or/

0 ao

I

3~

(r~-rs)

_1 (r~v

3z

+ 7?

3Ar/N I

(13)

--L-z J

1

[3r s

az

[3rs

Full expressions for S ~ and 1-'¢,,for various dependent variables (viz.: tt, v, w, k, e), are given in Table 2.

x/-E+r/ 3_3

l lOrs OAr/NI d~ G =+r/ K= -r (30 "--~1 ' dz F = /

3

(oE - ow) a~

2

3(rNzrs) I 3 3o ,

+

rs) i 3z

(11)

3 3z

Turbulence model The effective viscosity//eft is calculated via equation (6). The governing differential equations for k and e are given in equation (12) and Table 2 where GE stands for the generation rate o f k and is given by:

I ar/

The resultant governing equations in the r/, ~, ~"coordinate system are similar to the original equations with additional corrections for convection, shear stresses and pressure gradients due to the non-orthogonality.

GE=Iz t 2

Ar/N -~01

+ {~

(

37, N 3r/ 1 0w

+~rz~ E 0~j

S.hip'sbody .Hull

rAtE 3~

u + 1 av r r a t e 3~ G 3,q2+(

Ar/N 3~ l"

c

3vl2

Ar/N "~-~l 1 0wl2]

At/N Or/] kA-~N~ 1 J

(14)

The length-scale of turbulence, in this model, is given by: 1 = CD k312]e

o W

I."

I

7/ Flowdomain~ "

Figure I

"

~

11, ~ and[" coordinatesystem

/

Boundary conditions In addition to the equations governing the flow the complete specification of the mathematical problem requires an adequate prescription of boundary conditions. This means that the flow conditions must be specified at the inlet plane and at the lateral boundaries of the flow domain of interest. In the case of partially-parabolic flows, the exit conditions also need to be prescribed. Numerical analysis Discretization procedure The first step in the development of a numerical scheme for solving the equations given in the previous section, is to obtain discretized equivalents of the partial-differential

Appl. Math. Modelling,1979, Vol 3, August 251

Turbulent flows around ships'hulls: A. M. Abdelmeguid et aL Table 2 Diffusion coefficients and source terms of equation (12)

Equation

~b

r

Continuity

I

-

u-Momenturn

u

//eft

SQ 0 -

_

_

1

_

_

ap+

FA~E at

1_ L ( / J ~ , rA~E a t ~rA~E

rAil N a~

r2

/Jeff au

/Jeff

1

AllN all 2/jefv - r - 2

-

aAll N - -

G

w

{

/Jeff

1

Kinetic energy

k

-G

i

+

(/jeffU)

All N all ~rA~E

----

I~N art ~AllN art

a [.efa-~+.o,,

rA~£ at ~AllN arl) +

1

a.

+

2G \

2G

-)

r

a

rAil N art

2~.a-

rAilN all

,,ll-~~ v-E~ ~) + T

+

1

r2A~E at

,,ll,,, a,,

(Pefv)

a(/Jeff

AllN

All N all

a_~)}

~

.

rAGE a t rA~EAll N

/Jeff a w

.

1

O~,u)

alltr J - -

r

a

r2A~E at

G

~) + 2

a~_~)

~

--\r--~E ( I -~+aG

+ rA~E a t G

a~aG__~)r k

/Jeff

Oef.e

of '

a (_r. a.

E at +

A~N

~)

,

AllN

equations (i.e. finite-difference equations). In the present section it is shown how the partial-differential equations are discretized by integration over the control volumes which surround the nodes of the grid system, and how they are arranged to be solved by a numerical algorithm. Grid system. The arrangement of the finite-difference grid system used is indicated in Figures 2 and 3. It consists of a set of orthogonal intersecting grid lines in die/j-r/ plane corresponding to a constant value of~'. The intersections of the grid lines are termed grid nodes and are used as reference locations for identifying the flow variables. Since the choice of the grid affects accuracy and economy, the grid spacings need not necessarily be uniform over the flow domain, but can vary so that there is a high concentration of grid nodes in region~ of steep variations of flow properties. The same freedom applies in choosing the increment in the ~'-direction (or forward step). The only limits placed on this choice are those of accuracy, stability and economy. The finit&difference grid is chosen after experimentation with finer and coarser grids so as to make the results of the computation substantially independent of the grid employed. All the scalar variables (p, k, e, T, etc.) are stored at nodal points while velocities tt and v are stored at points midway between grid nodes in the/j and 7/direction respectively. The longitudinal velocity, w, is stored at

252 Appl. Math. Modelling, 1979, Vol 3, August

ak

AllN all

AllN

•

rAil N all

at \AllN

2 aG

a

at

+

- -

a --

(~.u)

rArt N art

a~ll N

(K--1l-at~ . . + AllN art

/jeff Oef,k

Dissipation rate

1

All N art

- - - al~ + + rAGE

-- l-~rt N

a

--(/jeffV)

r 2 AGE at

rAGE at

at~ ~ A l l N

a(Pef

1

- G- - ap

+

at

"llN T~ iv-~ -~ + -All N w-Momentum

2

~-~)+----

art~llN

- - 1- - a (.efa~) AllN art t~-N~N

- - _ _I_a_ p+

-

All N all krA~ E

All N art

rAGE at rA~EAllN

a,

--

/Jeff au(_

All N all

v

~-~)+

r

1 a(/jeffa~)}

v-Momentum

1 i[/jef

au

rA~EAll N G 2 rk ak r AllN art

a ( ~. % art

rAGE at

c

1

r Ar~N

at

a~ll. ~. a.

all + ClGE k

-- c2 --k

different places depending on whether the flow under consideration is parabolic or partially-parabolic. In parabolic flows, w is stored at the nodal points, along with the other scalar variables. In partially-parabolic flows, however, w is stored halfway between the node P and the downstream nodal point. The advantages of such a grid are that the velocity components u and v are stored at those points where they are I= .

"l .

.rS

South

West boundary

~

boundary

I

Figure 2

Grid (cross-stream)

Turbulent flows around ships'hulls: A. M. Abdelmeguid et aL North Initial section

Hull

Figure 3

boundary

Axis

Grid(longitudinaldirection)

required when either mass balances are made over the control volume surrounding each grid node, or the convective contributions to the balance of ¢ are calculated. The storage of pressures at nodal points simplifies the calculation of pressure-gradients that affect u and v momentum equations. As regards the storage location of w-velocity, in parabolic flows one is not interested in what is happening downstream as one is moving in the longitudinal direction, and consequently the w-velocity can be stored at the pressure location; in partially-parabolic flows by sto/ing w ahead of pressure one allows the downstream pressure to have its effect on w. Other characteristics of the present grid are: (i), the north boundary is of constant radius R2v; (ii), the south boundary has an axially varying radius equal to that of the hull, at the axial station under consideration, and (iii), the east and west boundaries are normal to each other. It is obvious that because of the shape of the hull which forms the solid south boundary of the domain, the grid is non-orthogonal. The hull shape normally varies with the longitudinal distance according to given tabulated values.

(ii), For the calculation of the longitudinal (~) direction convection of ~, and the source terms that may depend on ~, the variation of ~ in the cross-stream (~, r/) plane is also taken to be stepwise. Thus, in the G-r/plane the value o f ¢ is assumed to remain uniform and equal to ¢? over the control volume surrounding the point P, and to change sharply to ~N, ~S, ~E, ~N or ~U at the edges of tile control volume. (iii), For the cross-stream convection from the ~-]" and rl-~" faces of the control volume, an upwind differencing scheme is employed. According to this, the value of convected across the face is the value at the neighbouring node from which the flow is approaching towards the face concerned. (iv), For the cross-stream diffusion from the ~-g" and r/-~" faces of the control volume, the value of ¢ is assumed to vary linearly between grid nodes.

hztegration eqttation for ~. Tile result of the discretization process is an algebraic equation for each grid location, for each variable, of the form:

Control vohtmes. The finite-difference equations are

-

obtained by integrating the partial-differential equations over control volumes, which surround the nodes of the grid system. The control volume surrounding a grid node, P, is indicated in Figure 4, and is termed the main control volume. Control volumes appropriate to the u and v velocity components are 'staggered' from such main control volumes. The interfaces in the ~'-direction for all variables are located at the upstream and current cross-sectional planes, for parabolic flows.

=

~ = ~'o.

In both cases implicit schemes are employed which have the advantage that one is free to choose a reasonably long forward step, without numerical 'instability' as the result.

T~(~

-

~ )

-

T~(~2,- ~s) + N

Discretized equations. The general transport equation (12) is integrated for each variable, over the appropriate control volume that encloses the specific variable. These integrations are performed'after making presumptions about the manner in which the variable varies between grid nodes. 1 (i), In the longitudinal (~') direction, ¢ varies in a stepwise manner. In parabolic flows the current plane (~"= ~'c) values of ~ are supposed to prevail over the interval from the upstream station (~'u) to the current plane station (~'c) except at the upstream station itself (Figure 4). In partially-parabolic flows, the current plane (at ~"= ~'c) values of ~ prevail over th~ interval ~c to ~'o, except at

e , u ~ , u + L ' . ( ~ + ~ ) - L'.(~p + ~s)

T~(~E

-

~,.)

orth

/ direction ~..,~'

rj'qo,~ ~ z , Figure 4

~

"%W

l

.

~

South S

Maincontrolvolume Appl. Math. Modelling, 1979, Vol 3, August

253

Turbulent flows around ships" hulls: A. M. Abdelmeguid et aL

The coefficients of equation (15) are defined as follows:

F~e,u = (p,v)l,,uA~pA~?p r

xa

L~

= puiAoeA~[2

F~

=F~,u-2L~e+2L~w-2Lnn +2Lns

r?

e,"q/2

(16)

(I + c 7)

=

where the subscripts E, IV,N, S, P and U, and the subscripts e, n, s, w refer to the corresponding nodes and locations as shown in Figure 4. The expression for F,o may be obtained by integrating the continuity equation; and is as follows:

Fp = F u - 2Lnn+ 2rns + 2L~e+ 2L~

(17)

S o is the integrated form of the source (and/or sink) of ¢; although sources are nonlinear functions of ¢, in general, they are always expressed in the following linearized form for reasons of stability:

(18) Rearranging terms in equation (15), one obtains: c~p = A N ~ V + A s ¢ s + AE(PE +AwCw +B

(19)

whe re:

&v = ( r . - L .n)/Ae

A w =

(20)

+

=

+Sg)lA +L~w +F~,u:-S ~

The finite-difference equations for the velocity components are expressed in the same form as equation (19) but contain an additional source term representing the pressure-gradient (ap]O~ for-parabolic flows; bp[O~ for par tially-parabolic). The discretization of the continuity equation is quite simple. It merely states the requirement that the inflows and outflows of mass are locally in balance for each control volume in the flow domain.

Sohttion procedure The developed computer program embodies the parabolic procedure I and the partially-parabolic procedure3 The main features of these lbrocedures and the corresponding solution steps are: Parabolic procedure: (I) It is a general, finite-difference, non-iterative, forward

254 Appl. Math. Modelling, 1979, Vol 3, August

marching-integration procedure that takes full advantage of the boundary-layer character of the flow. (2) The solution operation moves plane by plane, ffoln upstream to downstream, in the predominant flow direction. All the dependent variables and associated coefficients, etc. require only two-dimensional computer storage (one plane being attended to at a time). (3) In each cross-stream plane, the finite-difference equations are solved by employing the TDMA (Tri-Diagonal Matrix Algorithm), along lines in the/j and r/directions. The equations are linearized by evaluating the finitedifference coefficients from the flow properties prevailing at the upstream plane (one section upstream). If local iterations at each axial plane are performed, the coefficients are evaluated from the flow properties of the preceding iteration. The sequence of calculation steps is as follows: (i), The pressure field which is stored in a two-dimensional array and the mean pressure, p, are assigned guessed values at the inlet plane of the calculation domain. At tile following stations, the general practice is to employ the upstream pressures as the guessed values, or the upstream pressure values in conjunction with the mean pressure-gradient if the latter is known. (it), Using the guessed values of the pressure field the three momentum equations are solved to get a first approximation to the velocity field at the current station. The coefficients in the equations are evaluated on the basis of the flow properties at the upstream station. Equations are solved by employing TDMA along lines in the ~ and 77 directions successively. The sequence of the calculations is to solve first for the tt and v velocities and then for the w velocity. (iii), Tile resulting velocity field is used in conjunction with the continuity equation to correct the guessed pressure field; the velocity fields are thereafter corrected, accordingly. (iv), Steps (it) and (iii) are repeated until the continuity errors reduce almost to zero. (v), The equations for the remaining variables (e.g. k, e, T, etc.) are then solved. (vi), The operations at the current station are now completed. The next downstream station is chosen and the process is repeated until the region of interest is covered. Once the flow domain has been covered the calculation has come to its end. Partially-parabolic procedure: (1) It requires several sweeps of marching integration, from the upstream to the downstream end of the calculation domain. Thus, it is an iterative procedure. (2) It requires three-dimensional computer storage for pressures so that in each integration sweep the corrected (improved) pressure field can be stored for its later use, in the subsequent integration sweep. (3) Full account is taken of the effects of pressure on axial velocity; that is, the source term for w is Op/8~, rather than ap/~ as is the case for parabolic flows. This allows pressure effects to be transmitted in both upstream and downstream directions. The sequence of calculation steps in partially-parabolic flows is exactly similar to that in parabolic flows except for the following differences: First, the pressure field is stored in a three-dimensional

Turbulent flows around ships" hulls: A. M. Abdelmeguid et aL

array and is assigned guessed values at the start of calculations. Second, in the calculation of axial velocity (w), the local pressure gradient, ap[8~, rather than ap[a~, is employed. Similarly at the correction stage the central coefficient of the pressure correction equation is increased by adding the axial-link coefficient. And third, after completing a sweep (i.e. plane by plane march up to the last downstream section), the flow domain is swept once again, each time employing an improved pressure field (stored from the preceding sweep). The procedure is terminated when the corrections to the pressure field become smaller than a preassigned value.

Boundary and &itial conditions In this section, the appropriate boundary and initial conditions needed to close the problem are described; they are, wall conditions, symmetry-plane conditions, freeboundary conditions, initial conditions and, for partiallyparabolic flows, exit conditions.

Wall conditions (httll boundary). The velocities at the hull are zero from the no-slip condition. Turbulent flows near a wall are distinguished in two ways from the flow away from the wall; the effect of molecular viscosity becomes prominent because of the damping effect of the wall on turbulence and the flow properties often show steep variations in the vicinity of the wall. The rigorous incorporation of the effects of the vicinity of a wall to turbulence proves expensive in computer time. One economical method of accounting for these effects is by way of'wall functions'. 7 These functions are embodied in algebraic expressions which force the numerical solution to behave in a prespecified manner, such as the log-law variation for velocity, in the vicinity of the wall. For k, a zero diffusive flux at the wall is used. For e, the empirical evidence that a typical length scale of turbulence such as l varies linearly with the distance from the wall, is used to calculate e itself at the near wall point.

Symmetry-plane conditions. The west boundary of the calculation domain is a symmetry plane. Downstream of the stern, the boundary which was previously the hull (the South boundary) also becomes a symmetry-plane. The East boundary of the calculation domain (the free-surface) has also been treated as a symmetry plane (i.e. this is the case of zero Froude number flow). Treatment of the water surface as a plane of symmetry is an approximation justified by the ignorance of thg true conditions there. The symmetry planes are of course zerogradient boundaries, since there is no flux of any quantity through them. Free-boundary conditions. At the free boundary of the calculation domain (North boundary), a special treatment is made. The flow variables (u, w and p) at this boundary are given the values calculated from the potential flow solution around the hull; the pressure at the near boundary node is set equal to the pressure at the boundary; the mass flow rate across this boundaiy is calculated from local continuity; and a zero-gradient condition is applied for k and e. lnlet conditions. The calculation starts at about threequarters of the ship's length from the bow. At that plane, the following conditions are imposed:

For tt and v: Equal to zero, everywhere. For w: It is assumed that, at the inlet section, the boundary layer is uniform around the girth of the hull and corresponds to a growth along a flat plate of length x I. Then w is calculated from:

w-=(Yl'17 w.

(21)

ka l

wherey denotes distance from the hull, • = 6(z) denotes the boundary layer thickness at distance Xlen from the plate's leading edge and w.o is the free-stream velocity. The flat plate l/7th power law prot~de is then scaled to produce a velocity at the free boundary which is the same as that calculated from the potential flow solution. 6 is calculated from the following relationS:

( w ~ x t P ) 'Is 6(z)=0.37xt ,

ta

-

(22)

For k: Uniform and equal to 0.4% of the free-stream velocity squared. For e: The Escudier formula 9 is used to calculate the distribution of the length scale, lm . For:

y co < 6 K Y > CD 6 K

l,n = Ky (23)

lm = CD~

where K = 0.42 is the von Karman constant and 1m is equal to ICD II4. The starting values of e are now calculated from:

cgl4k312 e-

lm

(24)

Ex# boundary condition for partiaTly-parabolic flows. At the exit of the calculation domain (nearly l]5th of a ship's length downstream of the stern), an additional boundary condition is needed for the pressure, for partiallyparabolic flows. It is assumed that the exit pressure is uniform, at the value given by the potential flow solution.

Comptttational details Compttter program. The differential equations described in the section on partial-differential equations above, are solved by a computer program called FLASH (for FLow Around Ships' Hulls).

The grid. In the present investigation, the distribution of grid lines in the ~- and rydirections was arranged so as to make the main control volumes evenly distributed; 12 grid nodes were used in the circumferential and 12 in the radial directions. In the longitudinal direction, more grid nodes were concentrated in regions of steep hull curvature; a total of 25 grid nodes was taken in the ~"direction. To obtain the hull radii at stations where no tabulated values were given, the following procedure was followed: (1), Transform the tabulated values of the hull radii into cylindrical-polar coordinates if they are initially specified in Cartesian coordinates. (2), Interpolate in the circumferential direction to obtain

Appl. Math. Modelling, 1979, Vol 3, August

255

Turbulent flows around ships'hulls: A. M. Abdelmeguid et aL

the hull radii at the required ~ stations (a linear interpolation with respect to 0 was used). (3), Interpolate in the longitudinal direction to obtain the hull radii at the required z stations (a linear interpolation with respect to z was used).

1.3

o

J

o

o

°

O : 25 °

J

1.0 . V

Convergence o/results obtained. The convergence of the partially-parabolic method is solely determined by the initial estimate for pressure; if this initial guess is correct, the momentum equations become uncoupled and the solution is obtained in one sweep over the flow domain; on the other hand, if the initial guess for pressure is quite far from the correct one, several sweeps are required to obtain the correct solution. In the present investigation, the potential flow solution pressures prevailing at the free boundary were taken to be the initial estimate of the pressure field. The sweeps were terminated when further sweeps caused a change of the flow variables of less than 1%. Figure 5 shows typical variations of the value of the longitudinal velocity, at different grid locations, with the number of sweeps; the figures show an oscillatory character which decays with an increasing number of sweeps; 40 sweeps were found necessary to dampen out such oscillatory behaviour.

0=60 o

-o

I O["

£

ln-

r OI

O = 25 °

e = 60 ° 0

Computational time. The computational time on a CDC6500 computer was of the order of 45 sec for one sweep over the flow domain, using a 12 x 12 x 25 grid.

I

I

5O IOO . Distonce from hull,(mm) Figure 6 Comparisons of longitudinal velocity profiles. Ordinate B. ( - - ) , partially parabolic; (------), parabolic; (o), experimental

Results and discussion

I-3

. ~ e : 23 °

Figures 6 - 9 present comparisons between experimentall° data and the present calculations. These figures present the longitudinal velocity distributions with distance from the hull at different computational stations characterized by their ordinate number, and at different angular locations, specified on the curves. The length ordinate position is calculated at (Lp - z)/O.1Lpp. The following may be observed from the figures: The present predictions are in good agreement with the experimental data.

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12 16 20 24 28 32 3~6 4'0 44 48 52 ' ' ' ' ' --~' ' ' 5'6 S.~¢ps Figure 5 Variation of longitudinal velocity with sweeps. A , 6 x 6 grid location; B, 3 x 3 grid location; C, 10 x 3 grid location

256

"8

Appl. Math. Modelling, 1979, Vol 3, August

o

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Distance from hull,[mm) Figure 7 Comparisons of longitudinal velocity profiles. Ordinate 2. (- ), partially parabolic; (------), parabolic; (o), experimental

Turbulent flows around ships'hulls: A. M. Abdelmeguid et aL

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50 I~)O Distance from hull,(mm} Figure 8 Comparisons of longitudinal velocity profiles. Ordinate 1. ( ), partlatly parabolic; (------), parabolic; (o}, experimental

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Distance from hull, (rnm) Figure 9 Comparisons of longitudinal velocity profiles. Ordinate ½. ( ), partially parabolic; (------), parabolic

Notation The partially-parabolic solution is, in general, far superior to the parabolic one because of the downstream pressure effects propagating upstream from the stem region (which are not taken into account by the parabolic procedure), At ordinate 3 (i.e. far away from the stern), the partiallyparabolic and the parabolic solutions are quite close to each other, indicating the weak influence of the stern region so far upstream. Nearer to the stern, however, large discrepancies between partially-parabolic and parabolic results are observed. It is concluded that the much cheaper parabolic solution may be used to produce sufficiently accurate results only in cases where the influence of the downstream pressure effects upstream is negligible, e.g. for bodies without abrupt changes in shape, Conclusions The present methods have been shown to produce converged plausible results. Further work is required to: (i) Make improvements to the code in respect of economy and ease of use. (ii) Investigate the numerical accuracy of the solutions by performing calculations for finer finitedifference'grids. (iii) Test the physical realism of the solutions by making comparisons with more experimental measurements. (iv) Assess the effects of the imposed boundary conditions and the locations of the boundaries on the solutions, and (v) Perform calculations for conditions of interest to ship designers and manufacturers.

Finite-difference coefficients After-perpendicular Cl, C2, CD Constants of turbulence model Generation of turbulence energy GE Turbulence kinetic energy k Length from initial section of calculation Lt, domain to Ap Length between the two perpendiculars Lpp Length scale of turbulence l Static pressure P Radius of outer boundary rN Hull radius rs Source (and/or sink) terms for variable q~ SC, Axial, radial and circumferential velocity comw, v, u ponents, in a polar-cylindrical coordinate or Ui system (z, r, 0), respectively Free-stream velocity ~v Distance from front perpendicular to initial x! section of calculation domain Distance from hull Y Exchange coefficient for variable PC, Boundary layer thickness 6 Dissipation rate of turbulence e Components of a non-orthogonal coordinate rT, ~, ~" system Van Karman constant (= 0A2) K Laminar viscosity It Effective viscosity Iterf Turbulent viscosity Itt Ai Ap

Appl. Math. Modelling, 1979, Vol 3, August

257

Turbulent flows around ships'hulls: A. M. Abdelmeguid et aL

t9 Oeff,~, oil

Density Effective Prandtl number for variable Stress tensor

4 5 6

References 1 2 3

Patankar, S. V. and Spalding, D. B. Int. J. tteat Mass Transfer, 1972, 15 (8), 1787 Pratap, V. S. and Spalding, D. B. The Aeronautical Quart. 1975, XXVl ttarlow, F. tt. and Nakayama, P. I. 'Transport of Turbulence Energy Decay Rate', Los Alamos Sci. Lab. Univ. California, REP LA-3854, 1968

7 8 9 10

Launder, B. E. and Spalding, D. B. 'Mathematical Models of Turbulence', Academic Press, London and New York, 1972 Spalding, D. B. Imperial College, Mech. Eng. Dept., Rep. No. HTS/75/5, 1975 Markatos, N. C. et al. Proc. First Int. Conf. Appl. Numer. Modelling, University of Southampton, July 1977, Comp. 3leth. Appl. ,~Iech. Eng. 1978, 15, 161 Patankar, S. V. and Spalding, D. B. 'fleat and Mass Transfer in Boundary Layers', 2nd Ed. Intertext Books, London, 1970 Schlichting, tt. 'Boundary Layer Theory', McGraw-llill Book Company Inc., 1960 Escudier, M. P. Imperial College, London, lleat Transfer Section, Rep. No. TWF/TN/1, 1966 Namimatsu, M. and Muraoka, K. lIIIEng. Rev. 1974, 7 (3)