A panel method for the analysis of unsteady flow around an arbitrary three dimensional body

CHINA SHIP SCIENTIFIC RESEARCH CENTER

A Panel Method

for

an

Arbitrary

Dong Shitang Feng Jinzhang CSSRC Report June 1986

English version 86008

P. 0 . BOX 116, WUXI, JIANGSU

CHINA TECHNISCHE UN_VERSITEff abcatorjum _Voor ftrCh let 2, 2628 _{CI) DeIft} TeL

A PANEL METHOD FOR THE ANALYSIS OF UNSTEADY FLOW AROUND AN ARBITRARY THREE DIMENSIONAL BODY

INTRODUCTION

It is often required to predict the hydrodynamic performance

of a 3-D body in unsteady onset flows which may either be caused

by unsteady motion of the body or be due to interference from environment the body sustains.

Our concern is mainly on periodical unsteady flow problems, aimed at its application to unsteady interaction between

propeller and other bodies such as in the case of ducted

propeller. _{However, the following method to be introduced can be}

easily applied to other more general unsteady flow problems. To improve numerical accuracy, surface panel method of

combined singularity of dublet and source are adopted in the

paper. _{For nonlifting flows, quasi-steady solution is equivalent}

to unsteady solution if only disturbed velocity is considered. While for lifting problem, at any time, the strength of shed vorticity on slip stream behind the body depends on the time history of the solution before the specified time, i.e. the solution has "memory" which makes the problem more difficult to

be handled. _{Time step iteration procedure is used in the paper}

to tackle such problems.

THEORY

Our theory is based on potential flow. _{There is a}

disturbed velocity potential which satisfies: and

Since nonlifting flow can be regarded _{as a special case of}

lifting flow (with zero lifting force), their solutions can be

expressed in an unified form. _{From Green's identity, it is}

wellknown that i i

3(q,t)

an

r(p,q)

- f

_[(q,t)

_-

_q,t)]

a

r(p,q)

V2(p,t)

C Te

,t)

V(p,t)!p+infinite 0

where Sb is the body surface, Sw is a surface often called slip stream surface which extends from body trailing edge to far down

stream. Usually, it is assumed, has a jump across Sw but the

normal partial derivative of is continuous. For nonlifting

flow,

[4(q,t) -

and the last term in the expression vanishes.

For convenience, let [4(q,t)-4(q,t)] be denoted by Uw(q,t)

thereafter. Sw is an infinite thin fluid layer which can not

stand pressure jump across it; i.e.

1+

-'..p -

'sw

From Bonoulli's integration, we have

.a.T _{+ (_Vb +} 1-+_{v )'v}++ 4- 4--- (_vb + v )'v = O or i -*+ -*- -+-+ 4--+ [_Vb 4--+ (v

+ y )](v -v

V represents the average value of disturbed velocities on

the tw sides of Sw. Since is continuous on S by

assumption, the above equation can be rewritten as

1w +

9-+ (-y + y ).V

j

t b m

sww

Due to the presence of

_mVsUw

differential equation. Either iteration or shed singularity trace

technique should be used to determine both the geometry location

of S and the streugth of i.i. However, such calculations could

be time consuming and expensive. In order to reduce numerical

work, some simplifications were made in the paper. First, Sb.. is

formulated with a certain model in which several parameters are

open to be evaluated. For example, for problems of flow around a

3-D wing or a circular nozzle, a parabolic curve is used in the near region of the body trailing edge as an approximation for S

in longitudinal direction, as shown in Fig. 1. The curve is

determined by:

y = (tga2 - tgai) .2+ tgaìx

OxL

y = x

tgc2

In most cases, L is taken as the local section chord length.

-2-on Sw

i _{is directional angle of local stream line at the}

trail-ing edge and is computed by iteration since the local streamline is not known in prior.

Now let (,n) be curvelinear coordinates on Sw and be in

the stream line direction on Sw. Define

,. = - + b m Then

ìI:

-o

u (,

If

_{=o denotes body trailing edge, u(, n,t) can be}

rearranged as

u(, n,

f[0 -

-U L',O,

The above equation means that the shed doublet strength at a

certain point equals to that of n corresponding point at body

trailing edge in an earliar time (tL1t). Where txt is the time

the shed doublet needs to travel _{from trailing edge to the}

specified point at the speed of V. _{This concept is the basic}

idea in our time step iteration approach. For periodic unsteady

flow, u is also a periodical function,

,

t)

where T is the period.

From eq. (8), it is possible to describe _{U,,, distribution on}

S.,, at any specified time with its history at the body _trailing

edge. Furtherly, eq.(9) tells us that such a time history is

corn-peletelydetermined by its variation in one time period. In lìfting flow problems, Kutta condition should be satisfied, which requires that the limits of pressure on the

upper and lower sides at body trailing edge be equal. _This

requirement can be fulfilled _{if doublet distribution and its}

derivatives are continuous _{across trailing edge from Sb to Sw.}

Substituting eq.(5) into (2), governing equation is established:

where a ds +

f

_u(q,t)

(q,t)

Vbfl

r(p,q)

ds means that the integral takes its principal value. ormal discretising procedure of surface panel method is used to change integral equation (10) into a N-linear-equation

system corresponding to N surface panels on Sb. Time step

iteration is applied to solve the N-linear equations. Suppose

the present time is t and past solutions have been known. Now

let time forward a step Lt. In this small time duration, the

shed doublet on S _{would be washed down and there is a small}

seg-ment next to the body trailing edge on which present l-1w is not

known. According to eq. (9),U(,fl,t+tt-T)which has been known by

assumption is used as a first approximation. _{The unknowns}

re-mained in the discretised governing equation _{are therefore u only}

which can be solved from the linear system. _{After U on Sb being}

solved, interpolation between U _{ori Sb and Uw on Sw is performed}

to obtain Uw(t+'t) on the small segment in the vincity of body

trailing edge. Then the newly interpolated Uw are used to

replace its first approximation. _{With time stepping on, Uw}

updated, U _{on S gradually converges to a periodical function of}

time. Within certain creteria, finally u(q,t), uw(q,t)can be regarded as the solutions of the problem since during the

iteration process, both boundary condition and Kutta condition are numerically satisfied.

NUMERICAL EXAMPLES

Since at the time being experimental results _{of lifting}

body in non-uniform onset flow are not available to the author,

numerical calculation were performed on two lifting bodies

moving with a uniform speed.

The first example was carried _{on a rectangular wing with}

chord-span-ratio 6. The angle of incidence is =6.75 . The

calculated pressure distribution together with corresponding

experimental values from [2] are plotted in Fig. 1.

The second example is on an axisymmetrical nozzel. In

Fig. 2, the angle of incidence O=0 . In Fig. 3.., c=8

Experimental results in both of the figures are from [3].

Comparisons show that theoretical results calculated with

the present approach and the meassured values from experiments are

in good agreements.

Further numerical examples of lifting bodies undergoing

non-uniform onset flows are being planned.

REFERENCES

B.R. Bristow, G.G. Grose "Modification of Douglas Neumann

Programme to Improve the Efficiency of Predicting Component

Interference and Higher Lift Characteristics." NASA Contract

Report 3020 1978.

Dong Shitang, Feng Jinzhang "Comparisons between Panel

Methods with Different Singularity Distributions" CSSRC

Technical Report. July, 1985.

Morgan W.B. "Prediction of the Aerodynamic Characteristics of Annular Airfoils." AD611018. Jan, 1965.

Cp -1.8 -1.2 0.6 1.2 Cp Calculation o o o Experiment a 6.75° Middle span - Calculation O O_°Eperiment = 6.75° End

Fig.1 Pressure Distribution on the Rectagular Wng(Aspect ratio =6) NACA 0012

op cp -0.4 -0.2 0.2 0.4

a-p.

Fig.2 Pressure Distribution on A Duct in Axisymmetric Flow

Fig.3 _{A. Pressure Distribution with an Angle of Attack a= 8°}

Calculation oO o Experiment

cp

-0.4

-0.2

0.2

0.4

Fig. 3 B. Pressure Distribution with an Angle of Attack (1=8°

Bottom Section

8

Calculation