Appendix 1: Upwinding of Convection Terms
7 4 Appendix 1: Upwinding of Convection Terms CFD m e thods r e quir e smoothing or di ff usion f or stability and cons e qu e ntly to converg e on a solution. This dif f usion is commonly a o de d e xplicitly into th e e quations being solved. An alt e rnativ e m e thod o f achieving th e same result is upwindingo f th e conv e ctive t e rms in the discrete f ormulation.
While the upwind method applies to multi-di m ensi c n a l flows solv e d using finite- el e ment and finit e -volume m e thods, the analysis o f this m e thod is most cl e arly shown for one dir a ension using a finite-dif f e r e nce approach. The model equation us e d 1 ,c- r eis the simple lin e ar conv e ctionequation, known as the wave equation: Ou Ou --= - c -- (A 1 -1) O t Ox I f a r tifi c i a l v i sc o si ty wa s e x plicitly app l ied, _.he equ at ion so l v ud wou ld ta k e the form, & + Ou O _ u --_-= - C _ x + aTx 2 (A1 - 2) wher e _ is th e, coefficient of artificial viscosity, o _ < < 1. Not e that o _must b e a function of A x in order f o r Eq. (A1-2) to be consistent with Eq. (A1-1) as Axe0.
Usir , g upwind diff e rences, the conv e ctiv e terms of Eq. (A1-1) ar e discretized as: I Ou f - c U _ - U "-_ ' c > O + -c -_ x , =t u ,_ . l&__ u . (A 1 -3)
i,-c ,c < o
Applying trunc a tion e rror a nalysis to th e differ e nc e d t e rm s in E q. (A1-3), w e e xp a nd the t e rms th a t a re not a t time lev e l , n, a nd a t s p ace i nd e x, i, u s ing T a ylor s eri es e xp a n s ions. For th e ca s e wh e n c • 0, w e h a v e :
A_I " _ _ '" 1 °
.," -,= . F - a x E + _ _-i x 21 , - O (z_'). (A1-4)
In a complete truncation error analysis, the discretiz a tion of th e time-dependent derivativJ would also be evaluated, using Taylor series e xpansions of those terms w hich are not at time level, n. However, the purpose of this discussion is t o show t he rela t ionship be t ' , veen differen t discre t e forms of t he space derivative, so _t is assumed that the same time discr e J zation is used in both cases, and an a nalysis of the discrete time terms is not shown her e .
Substituting Eq. (A1-4) into Eq . (A1-3), for c > 0 yields:
,, : _ .+ _ o,, F _' o _. I
a . ' Oxl, 2 Ox_] , +O(a x_)
- c _- x = - c A _ (A1-5) which reduces to:
Ou r ( _-_ a'u l"
+t, c TJ_l , + o _A< ' ). (A_ - 6 )
As might be e xpected from this one - sided difference, the truncation error o_the discrete f o rmulation is first order . Ign o ring the terms of second order and higher, the nu merical error, ¢ i n the upwind difference method shown here is:
( , _'_ o 2.1" a 2 . 1"
_=L < T J T, 'I , :_1 , (,,_-7)
Thus, we see the artificial viscosity, _ is evident in the solution, a s if it wer e , Jxplici t l y appli e d as in E q. ( A1 - 2 ). F o r th e cas e w h en c < 0, th e l e a d ing e rror , _ t e r m is: '!, 2 I n
( _'_ a u l
:t- < T ) _l. (A_ -8 )
W e confirm th at u p win d i n g is a s pe cial cas e o f a d d in g arti fi c i al vi scosity to a central di ff erence sc h e m e by obs e rving th e discr e t e f orm of the right hand sid e o f E q . (A1- 2! : P r l
- c _+ aTx T= - c"'+' _ " + a _, '-' (A1-9)
A x Substituting a - c --z- yields: Z - cu T+, - u ,"__ ( A,_'_ , ,"+_ - 2 u" + " " " 2 A x + _, c m2 ) A x 2 ui-' -- c U' _- ui-I , (AI - 10) which is identical to tne upwind difference in Eq . (A1-3) for c > 0 . Similarly, A x substituting a = - c m into Eq . (A1-9) yields the upwinddiff e renc e for c < 0 .
Appendix 2: Loca l Sweep Angles
Appendix 2: Loca l Sweep Angles
For a tapered wi ng, the angle o f s weep, A , varies along the c hord.
Although one might assume a linear variation i n A fr om A LE'.0 A TE, it is a c tuall y tan P . that varies linearly along the chord, as shown h e re . This geometric derivation is for a single - element wing. In order to find l o c al values of A on a multi-element wing, A LE and A TE must be known for ea c h ele me_, t.
Figure A2.1 shows a gene r ic swept, tapered wing . The root chord length is C r,so for a taper ratio, ; L , the tip c hord length is c t = _ .C r. The wing semispan is b, and a is the distan c e in the x - dire c tion to the wing tip leading edge due * o the leading - edge s w eep. From these dimensions, we fi n d relations for the leading - edge and trailing - edgesweep angles:
a (A2- 1 )
t a n A L _ = " / _ ( a + _,Cr) -- C, -- a + c ,(_ - 1) (A 2 -2) tan A rE - b b Solvin g E q . (A2-2) f or Cran d substitutin g Eq. (A2- 1 ) f or a yiel ds : b tan ArE - btan At.E ( A2 - 3)
c , = (Z- 1)
The c ho r dwise locat i on on t he air fo i l , non - dimensionaliz e d bet w ee n t he l ea d i ng e dg e and the t railing edge, is represented by _. For an arbitrary chord .4ation, _, t h e ioc a ! s weep angle, At ; , is giv e n by: t a aA{ = ( a + _j_,c ,) - _ c , _ , + _jcr(Z - 1). . (A2-4) b b S u b stitu t i ng Eq . (A2- 1) for a a nd Eq . (A2 -3 ) fo r Cry ield s : b tan A_ 4-_ b(tan Ar E - tan A t e) (, , 1 , _ 1)
(;t -l ) (A2 - 5)
tan A ¢ = --- b ' and simplifying, we find: tanA_ = (1- 4)t a n A L r + _ t anATE. (A 2 .-6) Thus, the tangent of the local sweep varies linearly betwe e n the tangent of the leading-edge swe e p and the tangent of the trailing-edge sweep.
Y
l
x b Figure A2.1 - Schematic of a swept, tapered wing.
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