The vortex of appendage body junction flow control using suitable fillets

CHINA SHIP SCIENTIFIC RESEARCH CENTER

The Vortex of Appendagebody Junction

Flow Control Using Suitable Fillets

Li Ding

Zhou Liandi

CSSRC Report

May 1995

English version 95001

Presented at the International Conference

on Hydrodynamics, 30th Oct. 3rd Nov. , 1994,

Wuxi, China.

Contents

Page

Abstract

i

Introduction

i

Numerical method

2 2. 1 Governing equations 2

2. 2 Bodyfitted coordinate systems

2

2. 3 Transformation of the equations

2

2. 4 Discretization of equations

3

2.5 Wall

functions

3

2. 6 Solution procedure 4

CFD visualization

4

Results and discussion

4

Conclusions

5

THE VORTEX OF APPENDAGE-BODY JUNCTiON

FLOW CONTROL USING SUITABLE FILLETS

Li Ding, Zhou Lian-di

China Ship Scientific Research Center, P. 0. Box 116 , Wuxi Jiangsu 214082, China

ABSTRACT

The possibility of using a suitable fillet form to

control the horseshoe vortex flow caused by the

turbulent shear flow around wing-body junction has been investigated numerically. A numerical method

for the solution of three dimensional incompressible,

Reynolds-averaged Navier-Stokes equations with the two-equation (k,e) turbulence model has been developed to evaluate the appendage-body junction vortex flow control devices. The wing investigated is NACAOO2O. The Reynolds number based ori a

chordlength is 1,0x1 Five configurations

including a baseline, a triangle fillet fonti, a constant

radius convex arc fillet fornì, an elliptic concave fillet form ánd an elliptic convex fillet forni along

the entire wing/flat-plate junction are presented. It is shown that the convex fillet form can significantly improve the stability of junction horseshoe vortex

and reduce the strength of vortex and the non-uniformity in the wake velocity profile. lt is also

demonstrated the capability of the numerical

approach n the design of vortex flow control

devices.

1. INTRODUCTION

When an laminar or turbulent boundary layer

on a surface encounters a wing or other

protuberances projecting from that surface, a

complex and highly three-dimensional flow results.

1'lie most significant feature of this flow is the

generation of a horseshoe vortex or a set of

horseshoe vortices. This horseshoe vortex forms

around the nose of the wing and its legs trail

downstream into the wake and forms streamwise

vortices in the wake. This phenomena occurs at

many places including wing/fuselage intersections in

aerodynamics, appendage/hull junctions in

hydrodynamics, and blade/hub junctions in

propellers and turbomachinery, etc.. As a

consequence of (lie generation of the horseshoe

vortices, the drag is in general increased arìd the

wake velocity profile becomes significantly non-uniform. This can be a great nuisance in marine

applications where a propeller operating in a

non-uniform wake results in significant unsteady forces.

As well known, when a boundary layer

encounters a protruded bluff body, the strearnwise

vortex in the form of the horse?hoe vortex cannot be

prevented from being generated. However, it is

possible to minimize the strength of the horseshoe vortex by sortie flow control devices such that tIre

resulting wake will be less non-uniform arid the

unsteady forces will be reduced. Experimental and numerical investigations on the use of fillet forms to reduce the interference effects have been reported

before . In these papers, the type of fillet forms are all concave. According to observation arid

analysis of our experimental research on the effect

of fillet formt61, (lie strength and the unsteadiness of.

the horseshoe vortex and non-uniform of wake may

be effectively reduced by designing a suitable

convex fillet form. Here, a numerical investigation was carried to evaluate the effectiveness of the

convex forms and to demonstrate the capability of the trumerical approach ¡n the design of vortex flow

control devices.

A general purpose computer code for (lie

solution of the complete three-dimensional

Reynolds-averaged Navier-Stokes equation has bten developed t. _{In this numerical piócedure, (lie 31)}

Navier-Stokes equation is solved by finite-volume

scheme in a nearly-orthogonal body-fitted

coordinate system. Ihe pressure-velocity coupling is

treated with SIMPLEC algorithmtt using a

non-staggered grid. The Rhie-Chow algorithrntl is

chosen to avoid the well known problems due to

chequerboard oscillations in the pressure and

velocity fields which are traditionally associated

with (lie naive use of non-staggered grids. The

system of algebraic equations formed by tIre

assembly of the convection-diffusion and pressure

equations are solved by the strongly implicit

conjugate gradients (PCG) algorithmí7ì. The two-equation k-e model with wafl functions is used for the turbulent flow involving sepaation and vortices.

This paper continues the previous workP'1 on the numerical simulation of the effect of fillet forms on appendage-body junction flow. Emphasis in this paper is made on two new configurations of fillet forms: a concave quarter-elliptic form and a convex quarter-lliptic forni. The numerical solutions were used tö evaluate the relative effectiveiessofthe live configurations. Particular emphasis is placed on the discussion of effectiveness of convex fillet forms on the stabilityofthe horseshoe vortex.

2.1 Governing Equations

We consider

the governing equations in

Cartesian coordinates (x,t) = (x,y,z,t) for unsteady, three-dimensional, Incompressible flow. The complete three-dimensional Reynolds-averaged equations in the general form are

(I)

wherè 4)

represents any one

of the convective

transport quantities (1, U, V, W, k, e). The scalar diffusivity r4 and source functions S for U, k and e

are, respectively, + Iii a 2

(p+pk) +

S+ =

-r' =

t.LI/ak rc

2.2 Body-Fitted Coordinate Systems

For the present application to wing-body

junction flow with

fillet forms, the body-fitted

numerical grids were generated by a system of

elliptic partial differential (Possion) equations ofthe form or 72 _{= f (i= 1,2,3) (3)}

au

TI,, 8x' (2b) S" = p(G-c) 4) k (2c) S = p(C,G-C2e) 4)= e (2d)

2-Here, V2 is the Laplacian operator in

Cartesian

coordinates x1. The nonhomogeneous so'irce

functions f may he assigned appropriate values to

yield desirable grid distributions. In practical applications, the inverse transformation of equation(3) is used to obtain the coordinate transformation relations x' = x'(), i.e.,

V2x _{= O}

_(i=l,2,3)

(4)

where V2 is the Laplacian operator in the

transformed plane.(). Using the

transfonnation

relations, the equation(l I) can be rewritten as,

where gik is the inverse metric tensort7t. The grid-control, functions f were determined by the specified boundary-node distribution. In present study, a corrective methodl7) is used to generate nearly orthogonal grids.

2.3 Transformation of the Equations

lt is convenient to write equation(i), in the steady state, in the equivalent form:

i=S

(6) 2. NUMERICAL METHOD g

ôx

(5) I' = pU'4) -ax (7)

When the Cardesian coordinate is transformed

to

the non-orthogonal coordinate system(), we

obtain immediately:

--(.jjl') = S

''

= .JjS

(8)

i.e. exactly the sanie form as equation(6), with the effective total flux given in contravariant and

normal components by:

ji

,Jjt.i

₌

_pÛ'4)ru

₍₉₎

r'=o,

s=o

(2a) where I is called the total(i.e. convective +

where Ûi the normal flux components are the scalar products of velocity vector U with the area vectors

Ûi=gUi U.A(1)

(IO)

_{c,,, = (pû'),, =(pU.A))0}

and

rii=-JgiiF

(Il)

Note that the effective diffusion tensor

Pi is a

symmetric tensor, by virtue of the fact that gi is symmetric. Also, it is diagonal if and only if gl is diagonal, - so the effective diffusion tensor is

orthotropic if and only if the- coordinate transformation is orthogonal, and it is fully anisotropic

if

and only if the coordinate transformation is strictly non-orthogonal. Due to its importance in the subsequent discretization process,

we give the tensor multiplier

in

(Il) a special

symbol:

Gui=.[ggu1 (12)

and we call G'i the geometric diffusion coefficients.

2.4 Discretization of Equations

The discretisation of the advection-diffusion equation is now straightforward. Integrating (8) over a control volume in computational space, we obtain, since computational space control volumes are unit

cubes:

t I]

+ E j2 J

+ [I

I = u n e

-s

w

< .jjs >,

(13) where nn(nearest neighboring face) = u, d, n, s, e, or w as shown iñ Figure 1. Using (9), wehave:

nn or =

( pÛ'4 -

ri.)

(a

nnaj )

1, nn=u,d

i= 2,nn=n,s

3, nn=e,w

(14) S'=FD2 + D'3 L Oq

+ÍD'

+ D)2

[

a,

where C,,,, D, are the convection and ari isotropic diffusion coefficients defined by:

D0 =

=

The Rhie-Chow algorithmI' is used for the interpolation of velocity components to control volume faces required for

the computation of

convection coefficients. We employ hybrid differences, i.e. central differences when mesh Peclet number is less than 2 and upwind differences when mesh Peclet number is greater than 2, for the values of 4 on control volume faces appearing in the

convection term of equation (15). We employ

central differences for

the normal and

'cross-derivatives' of 4 appearing in the diffusion term of

(15) (See Figure 2). Explicitly, we obtain,

substituting (15) into the discretisation equation

(13):

+ s_{m) p}

-

afl,NN - S' = <JS> (18)

where 4NN are the values of 4 at nearest neighboring points, are the standard matrix coefficients obtained using hybrid differencing normal to control

volume faces, and s is a mass source term, i.e.

au =

max(±Cj, D0)

-

-C,,

a0 =

max(-1jCdI, Dd) + _!Cd etc. (19)

S'is the extra term arising from the cross-derivatives due to the .non-orthogonality of the grid:

2.5 Wall Functions

Consider a solid boundary of general normal vector n, the velocity vector U may be split into

(15) _{parts perpendicular to and parallel to}the wall:

U = U + u = (U.n)n + (U - (U.n)n).

perp r

+ÍD2I + D235

L

The wall function approximation states that the velocity normal to the wall is identically zero in a neighborhood of the wall, and the velocity parallel to the wall is related linearly to the wall shear stress

t

as follow:

-

u

_perp o

=

t = T(U

_i-Up()

where T4 is

a multiplier dependent upon th6

distance y from the wall. For laminar flow, it is

given by TM = iiy and for turbulent flow (k-E

model ) it is given by:

lily'

y y

TM= _pitik

ln(Ey)

y y

where

t=

C2pk, y

and k, E,

C are constants associated with the turbulence

ni ode I.

2.6 Solution Procedure

The system of algebraic equations is formed by the assembly

of the

convection-diffusion and velocity-pressure correctfon equations (28) and (44). This approach ignores the non-linearity of the underlying differential equations. Therefore iteration is used at two levels; an inner iteration to solve for the spatial coupling for each variable and an outer iteration to solve for the coupling between variables. Thus each variables is taken in sequence, regarding all other variables as fixed. By always reforming

coefficients using

the most recently

calculated values of the variables.

In present paper, the U, V, W equations are

solved by the SIP algorithm with 5 internal

iterations; the P pressure equation is solved by the PCG algorithm with. 30 internal iterations; the k, e equations

are solved by the

A[)l (Alternating-Direction-Implicit) algorithm with 4 internal iterations. different iteration methods. The mass source residual, the error in continuity, is chosen as stopping criteria for the outer iteration.

3. CFD VISUALIZATION

One of the great problems with three-dimensional flow calculations is that of interpreting

4

the sheer amount of outputs of the numerical

method. Thus an

integrated CFD visualization

system VISPLOTIIO) is developed to process and analysis the numerical results of this paper. The VISPLOT has a friend user-interface which converts the results into a database and allows the user to dialogue with computer and database. Three forms of data processing are carried out: on one-dimensional grid lines, two-one-dimensional grid planes, or full three-dimensional graphics. The facilities of VISPLOT include:

Color contours of scalar quantities in grid slices and planes.

Profiles of the variables on arbitrary lines through the computational domain.

Velocity vector plots. Particle tracks.

Full exploitation of color to bring out the features of the flow, for example, to overlay velocity vector plots, with contours of the

pressure.

3D plot of grid distribution, geometry of

problems, or velocity fields etc..

4. RESULTS AND DISCUSSION

The baseline configuration consists of the wing mounted on the flat-plate(See Fig. 3(a)). The wing is 25.9 cm chord length, consisting of a 3:2

semielliptic leading edge and a NACA 0020 aft section. The triangle fillet form is a triangle form of side length 0.IC along entire wing/flat-plate junction(See Fig. 3(b)). The Convex arc fillet form is a circular arc form of radius 0.1C along entire wing/flat-plate junction(See Fig. 3(c)). The elliptic convex fillet form is a 2:1 quarter-elliptic arc form along entire wing/flat-plate junction(See Fig. 3(d)). The elliptic concave fillet form is a 3:1

quarter-elliptic arc form along entire wing/flat-plate

junction(See Fig. 3(e)). Ilere, C is chord length of the wing. All computations of turbulent junction

flows with and without fillet forms were made on the 50x30x30 nearly-orthogonal grids(See Fig. 3). The solution domain consisted of a semicircular cylinder, of radius R/C = 1.5 attached to a rectangular region O <X/C <2.5, 0 <Y/C < 1.5, 0 < Z/C < 1.5. The first spacing normal to the flat-plate and wing is on the order of 0.00IC. The mass source residual less than l0 was chosen to judge whether convergence has been achieved. In

generally the solution became convergence at about 200 cycle outer iterations using about 8 CPU hours on a PC-486/50.

As mentioned earlier, the main objective of using a fillet form is to weaken th horseshoe vortex generated. The pressure

_S

profiles on the flat plate are shown in Figs. 4a - 4e. lt can be seen that the strong kinks exist in the C profile of the baseline. Sharper kinks in the C, profiles imply that a stronger vortex has been generated.

The velocity vector fields in the symmetry plane in front of the leading edge of the baseline, the triangle fillet form, the convex arc fillet form, the elliptic convex fillet form and the elliptic concave fillet form are shown in Figures 5(a), 5(b), 5(c), 5(d) and 5(e). It can be seen that the vortex generated by the baseline

is an elliptic form but the vortices

generated by the fillet fonns are circular forms. lt means that the vortex generated by the baseline is unstable and will become low-frequently oscillating vortex but the vortex generated by the fillet forms, specially by the elliptic convex fillet form, may be stable and adhere the junctions. lt is very similar

with the observation of our experiments161. lt implies

that the convex fillet form can control the unstable vortex generated by the junction configuration.

Figures 6(a), 6(b), 6(c),

6(d) and 6(e) of

velocity vectors in transverse section xIC 2.14

clearly show the streamwise vortices in the wake formed by ahead horseshoe vortices. These vortices cause the wake velocity profiles becoming significantly nonuniform.

5. CONCLUSIONS

An improved numerical method for the solution of the complete three-dimensional incompressible, Reynolds-averaged Navier-Stokes equation based on finite-volume scheme in a

non-staggered grid has been applied to evaluate (he

effect of fillet forms on wing-body junction flow. Five configurations including a baseline, a triangle

fillet form, a convex arc

fillet form, an elliptic

convex fillet forni and ai elliptic concave fillet form were considered. The computed pressures on the entire flat-plate are first used to discuss the

effectiveness of the

fillet forms in terms of the

ability .of each

to reduce the strength

of the

horseshoe vortex. The velocity vector fields in the symmetry plane ahead leading edge of the wing are used to analysis the ability of each to improve the

stability

of the horseshoe vortex.

lt has been

demonstrated that the numerical approach is a

valuable tool for evaluation of the effectiveness of the fillet forms and the design of vortex flow control devices. It is the main purpose of this paper.

REFERENCES

I. Sung, CIL, Michael, 3.0., and Rodcrick, MC.. "Nwncricai Evaluation of Vortex Flow Control Devices," AJ.çpej 2..L..L2.5..AlAA22nd Fluid Dynamjçs,jsma ,yaanlics

Las.crs Conference, June 24-26, 199!, llawaii.

Kubendraii, L.R., Bar-sever, A. and larvey, WO..

Juncture Flow Control Using Leading-Edge Fillets," AJ'A t.per 85-4097, 1985.

Kubendran, L.R., Bar-sever, A. and Harvey, WO., "Flow Control in a Wing/Fuselage-type Juncture," ¿JAA Ppçi 88-0614. 1988.

Pierce, F3., Frangistas, GA. and Nelson, Di.. "Geometry Modification Effects on Junction Vortex Flow,"

for Marins..A.p1icatiou. Newport, Rl, November, 1988.

Sung. CH., and Yang, C.!., "Control of Horseshoe 'ortex Juncture Flow Using A Fillet," .Lr1pjgL,...Qn

flydrodynap,iç J'et[cjinnce Enhançcm.enj...,for Mscjne

Aoo1isition, Newport, Rl, November. 1988.

Li, D.,and Zhou, L.D., "The Effect of Juncture Form on

Appendage-Body juncture Flow," Osa...Ço!loqtthJm '9E

Japan.

Li; D., "Viscous Flow Around Body-Wing Junctures," Tlv-si China Ship Scientific Research Center, 1992. Von Doormat. J.P. and Raithby, GD., "Enhancements of the

SIMPLE Method for Predicting Incomprcssible Fluid

Flows," Numjical FleatTranfct, Vol 7, ppl47-163, 1934.

Rhie, C.M. and Chow, W.L., "Numerical Study oI'Turbulent Flow Past an Airfoil with Trailing Edge Separation," ¿1M Journal, Vol 21,No. lI,pp 1525-1532, 1983.

Li. D. and Zhou, LO., "Computational Fluid Dynamic Visualization System," Journal of l-1ydrudvnamJc. Vol.

1993

Il. Li, D. and Zhou, LO., "Numerical Simulation of Thc Effect of Fillet Forms on Appendage-Body Junction Flow,"

The Sixth International Conference on Numerical Ship

Hydrodynamics, Iowa, USA, 1994

= Mass control volume centers

o Mass control volume face centers

CDS = Central Oifíerence Schcm. lIDS = hybrid [)ifference Scheme..

DS

Fig. 2. Differcncc Schemes for Normal and CrossDerivatives.

Fig. 3. Geometric Configuration ofJuncion and Computational Grids

L) >, L) 0.5 -0.5 0.5 o -0.5 -0.5 -0.5

x/C

(a) Unfilleted Case

I.

I..

0 0.5 1

x/C

(h) Triangle Filleted Case

0.5 15 1

.-ç, 0.5 0 0.5 1 15

x/C

(c) Convex Arc Filleted Case

Fig. 4. Pressure Coefficient C, Profiles on TheFlat-Plate

x/C

(d) Elliptic convex FìIletd Case

/

b J.

I,.

\ j O ,." x/C

(e) Elliptic concave Fifloted Case

15 -0.5

\\\

f

--_-'

15 0.5 o -0.5 >-o

°:;

::"

.

u

N

u

N L) N 0.1 0.05 o 01 0.05 n 0.1 0.05

x/C

(a) Unfilleted Case

o

-02 -0.15 -0.1 -005

x/C

(c) Convex arc filleted case

Fig. 5. Velocity Vectors in The Symmetry Plane in Front of Wings

8-o N al 0.0 -0.15 xIC

(d) Efliptic Convex Filleted Case

--e-.,-- -- '-

.--

,-- .-.

-

---.---

-..

-'-- -..-.,---.--

,---,--. -:--.

_{Z - \}

«1, -0.1 -OE05 xIC

(e) Eliptic Concave Fulleted Case -(.2 -0.15 -01 -0.05 o

x/C

b) Triangle Filteted Case

-0.15

-4)2 -0.05 o

o

0.4 0.3 0.4 0.3 N 0.2 0.1 02 t t t t ' t t t _{t t} t II ,_s\ t I I t t 0 1 /11

.,,,,, ,

i i i i

__,,,,

0.2 0.4 0.6

v/C

(u) Unfilieted Case

Y/C

(b) Triangle Filleted Case

...\\\ lit

..\\\ \

I t

i,- -

Iii

i i . -I I I i i I ¡ t

,II'__,ii /

/ / I

\\

r i ¡ i . ¡. o -S 0.4 0.3 N 0.2 0.1

Il / /

-Fig. 6. Velocity Vectors in Transversc Section XIC 2.14

t I

/1/I'

" .'

'

t

Y/C

(d) Elptic Convex Edleted Case

-

- 's

'

'

s

11''

It

- -

-

r , t. \ \ t v/C

(e)Elliptic ConcaveFIetedCose

,. .

0 0.2 0.4 0.6 0.8 I

Y/C

(c) Convex Arc FilIeted Case

0.1 0.2 0.3 0.4 0.8 I 0.2 0.4 0.6 0.8

i

0.4 0.3 L)

Ñ02

0.1 o