On the theory of ventilated wind tunnels

I n t . J . M e c h . ,'~ci. Pergamon Press Ltd. 196(I. Vol. 1, I)P. 313-321. Printed in Great Britain

ON THE

THEORY

OF VENTILATED L.

WIND

TUNNELS

C. WooDs

U n i v e r s i t y of N e w S o u t h Wales, N.S.W., A u s t r a l i a

(Received 9 July 1959) S u m m a r y - - T h i s p a p e r gives a n e w a n d m o r e a c c u r a t e t r e a t m e n t of t h e t h e o r y of r e c t a n g u l a r w i n d t u n n e l s t h a t are v e n t i l a t e d b y m e a n s of l o n g i t u d i n a l slots o n one p a i r of o p p o s i t e walls. A n aerofoil is p l a c e d a t zero i n c i d e n c e m i d w a y b e t w e e n t h e s l o t t e d walls. T h e cross-flow, w h i c h is i n d u c e d b y t h e slots o n t h e basic t w o - d i m e n s i o n a l flow, is calculated accurately. This permits an averaged boundary condition to be obtained w h i c h is m o r e a c c u r a t e t h a n t h o s e p r e v i o u s l y p u b l i s h e d . I n t h e final s e c t i o n of t h e p a p e r t h e w a k e a n d solid b l o c k a g e c a u s e d b y t h e s l o t t e d walls is c a l c u l a t e d . L e t t h e slots h a v e a w i d t h 2 a a a n d b e s p a c e d 2a a p a r t a n d let t h e t u n n e l h e i g h t be H . T h e n t h e t h e o r y yields t h e a, 2a/H r e l a t i o n for zero b l o c k a g e w i t h o u t a n y l i m i t on t h e r a t i o 2a/H, w h i c h in earlier t r e a t m e n t s is r e q u i r e d t o b e small. 1. I N T R O D U C T I O N

T~E exact mathematical t r e a t m e n t of the flow past aerofoils in " t w o dimensional" wind tunnels, ventilated by means of longitudinal slots in the walls, is rather difficult. This is because the gaps or slots render the flow three-dimensional by imposing a periodic circulating "cross-flow" on the basic two-dimensional flow. This is illustrated in Fig. l, which shows a symmetrical /)

D ,:,,~H u

~'-I Ty

i ,"

v

( x ÷ i v ) - plane

~y-X-~

T-[ ',

/~

,

A

;,,j. i

,,t" =(z ÷,'~v) - plane

FIG.

1.

aerofoil at zero incidence on the axis of a slotted tunnel. The slots are uniformly spaced on the roof and floor of the tunnel, and this induces a flow pat t ern which is periodic in the z-direction. Efforts have been made to find exact solutions of this three-dimensional problem, 1 but even for the simple case shown in Fig. 1 the result is a very complicated set of equations which require numerical methods for their exact, solution. A much simpler t r e a t m e n t of the problem can be obtained if it is assumed t h a t the flow is quasi-plane, i.e. t h a t the cross-flow produced by the slots is only a small perturbation of the basic two-dimensional flow. These flows are assumed to be independent, except for a linking through the boundary conditions at the wall. The method thus has much in common with slender-body 21

313

314

L.C.

WOODS

theory. W h e n the cross-flow problem has been solved an average b o u n d a r y condition for the mean or basic two-dimensional flow can be deduced. Several authors h a v e given various derivations of this averaged b o u n d a r y condition (see Maeder and Wood, ~ who give references to the work of Guderley, Goethert, Davis and Moore on this topic ; also see Baldwin et al.a), b u t these derivations are not v e r y satisfactory for the following reason. The averaged b o u n d a r y condition is calculated on the assumption t h a t 2 a / H ~ O , where 2a is the distance from the centre of one slot to the next, and H is the tunnel height (this assumption is wrongly accepted b y some authors as being essential to the validity of using averaged b o u n d a r y conditions), b u t later this b o u n d a r y condition is used to derive a relation involving 2 a / H and a, the ratio of open to t o t a l b o u n d a r y (see Fig. 1). We shall give a new derivation, which overcomes this defect. 2. M A T H E M A T I C A L F O R M U L A T I O N L e t the axes Ox, Oy, Oz be chosen as shown in Fig. l, and let (u, v, w) be the velocity c o m p o n e n t s in these directions. The velocity u p s t r e a m at infinity will be U, and t h a t d o w n s t r e a m at infinity, V. We shall a d o p t linear p e r t u r b a tion t h e o r y and ignore second-order terms in u - U , v and w. In s t e a d y irrotational three-dimensional flow" we have

I ( ~p

~u ~v ~w

~?p

~ x + ~ + ~z +~ ~u~x+v~?J+w~. ~v ~x=

~u ~y'

~w ~u ~,x = ~ z '

~p~ ..

= o,

(l)

~u" ~v ~:y = ~/z'

(.2)

U=~x, v=~y, w=~z. and

(3) (4)

dp = a2dp = - q d q , P P

where p, p, q and a are the density, pressure, fluid speed and sound speed respectively. I n linear p e r t u r b a t i o n t h e o r y the last two terms of (1) can be ignored, while b y (4) the f o u r t h can be written u~p

u

~q2

u

~ x = - ~a~ ~x ~- - e ~

~u 2 ~

U 2 i,'u ~- - ~

~

_

~u

= - ~

~x'

M a being the Mach n u m b e r u p s t r e a m at infinity. Thereibre, with fl~ = 1

( 1) becomes

~u

~v

~a~+~+~

~w

M~,,

(~)

= 0.

As we are assuming the slots to be all of equal width, and to be u n i f o r m l y distributed across the tunnel wall, the flow p a t t e r n must be periodic in the 0z-direction. L e t (z~,~, z~) denote average values over one period - a ~ z < a, i.e. =

2a]f_audz,

~=

'2alf a, v d z ,

.~=

2alf_awdz

(6)

t h e n these quantities will depend only on x and y. I t is clear from Fig. 1 t h a t if we chose the Oz-axis to lie on a line of s y m m e t r y passing t h r o u g h the centre

On the theory of ventilated wind tunnels

315

of a slot, t h e n w will be an odd function of z. Thus e

= 0.

(7)

I f the first of (2) and (5) are now averaged over one period the result is 8x Let

8y

fla~x-~-. ~y = 0.

~- = ~ + i O = fla 1 -

+~,

(8) (9)

t h e n as ~-~ q, ~ and 0 are the speed p a r a m e t e r and the flow direction of the m e a n two-dimensional flow, a n d it follows from (8) and (9) t h a t for this flow

v = ~-(Z),

(Z = x + i f l a y ).

(10)

This equation, plus the corresponding averaged b o u n d a r y conditions, determine the m e a n flow. H o w e v e r , before these b o u n d a r y conditions can be o b t a i n e d it is necessary to calculate the cross-flow. To find a suitable equation for the cross-flow, let Substituting these into (5), and taking (8) into account, we get

a~xx-~-~-~ ~Z- ~-~ 0.

(12)

We now introduce the a p p r o x i m a t i o n t h a t ~t/~x is small c o m p a r e d with ~ / ~ y and ~(v/~z. This is the usual a p p r o x i m a t i o n of slender-body theory, a n d is valid if a typical d o w n s t r e a m distance is m u c h greater t h a n a typical crosss t r e a m distance, e.g. if 2a/H is small. In the present case, however, this restriction is not really essential; this can be shown as follows. The velocity u has two distinct components, viz. u a due to the aerofoit alone, i.e. with the walls at infinity, and u~ the i n c r e m e n t to u caused b y the walls. This increm e n t u w can be w r i t t e n u~. = ~ + ~ as u a is obviously i n d e p e n d e n t of z. W i t h walls t h a t are either completely solid or completely open, ~u,~,/~x is a v e r y small q u a n t i t y over the aerofoil surface. In fact, if Cv is the aerofoil drag coefficient and c is the chord length, this gradient can be expressed as k(TrcUCD/fl3aH2), where I: = ]½ with solid walls and ~ with open " w a l l s " Thus with slotted walls the variations in ~uw/~x with z will certainly not exceed (rrcUCD/16fi3aH2), and will, in fact, be a good deal less. This means t h a t ~t/~x is quite small, regardless of the m a g n i t u d e of the ratio 2a/H, and as the variations we h a v e described are greater at the aerofoil t h a n elsewhere in the tunnel, this r e m a r k applies generally. W e are now able to replace (12) b y

which b y (3) and the last of (12) is equivalent to = 0.

This is the e q u a t i o n governing the cross-flow.

316

L . C . WOODS

T h e b o u n d a r y c o n d i t i o n s for t h e cross-flow a r e : (i) On t h e solid sections o f t h e walls, 0 = v = ~¢/~y = ~(¢ + ¢)/~y, oi'

~-~ =

- ~y.

(14)

(ii) I n t h e g a p s t h e pressure is constant., h e n c e q = U, or i g n o r i n g s e c o n d - o r d e r t e r m s , u = ~(q~+ ~ ) / ~ x = U. L e t t h e origins o f ¢ a n d x coincide, t h e n this b o u n d a r y c o n d i t i o n c a n be w r i t t e n =

(15)

(iii) On z = + a t h e p e r i o d i c i t y o f t h e flow (see Fig. 1) a n d r e s u l t i n g s y m m e t r y requires t h a t w = ~, = 0, so

-~z. = 0.

(16)

(iv) On t h e aerofoil surface v = UO~, w h e r e 0s is t h e slope o f t h e surface. as 0~ is i n d e p e n d e n t o f z, e = UOs, w h e n c e ~ = 0, or

~$ i~y =

0.

But

(~7)

This c o m p l e t e s t h e m a t h e m a t i c a l f o r m u l a t i o n o f t h e p r o b l e m . 3. T H E

MEAN

BOUNDARY

CONDITIONS

T h e first step in d e t e r m i n i n g t h e m e a n b o u n d a r y c o n d i t i o n s is to c a l c u l a t e t h e cross-flow. Fig. 2 shows t h e A( = z + iy) plane, a n d t h e b o u n d a r y c o n d i t i o n s in this plane. I t will be n o t i c e d t h a t t h e n o r m a l d e r i v a t i v e is specified e v e r y where, e x c e p t on t h e g a p E ' E . T h e A - p l a n e is m a p p e d on to t h e u p p e r h a l f of t h e t-plane s h o w n b y t = sn(hA, k),

(18)

w h e r e t h e m o d u l u s / c a n d p a r a m e t e r A are d e t e r m i n e d b y K

K'

= ~ =iiT

(19)

A t E, t = t o a n d A = aa + ( i / 2 ) H = aa + iK'/A, so 1

1

t = sin (Aaa + i K ' ) = k ns (Aaa) = k ns (Ke),

(20)

T h e t-plane is m a p p e d into t h e [ - p l a n e s h o w n in Fig. 2 b y t = t0sin½~ , as is r e a d i l y verified f r o m the S c h w a r z Christ offel m a p p i n g t h e o r e m . ing (18) t o (21) we n o w h a v e sin ½[ = k s n ( K a ) s n ( K A / a ) .

(21) Combin(22)

T h e p o i n t s B a n d A at A = a a n d A = a + i K ' a / K m a p on to [ = Y0 a n d [ = 71, and hence sin 17o = k s n (Ka) (23) and

sin 171 = sn (Kc 0.

(24)

On the theory of ventilated wind tunnels L e t ~ be the harmonic conjugate of ¢, t h e n

317 (25)

cos½~ ,~-{['sin¢(7)d ), F (oo~ ¢+(~7) + d~-(~7) ] 2~r ½7-sin½~ 3o \eosh-l~Zs-sin½~ c o s h ½ v + s i n ½ ~ / d r , (see WoodsY). Now on ? = + s b o t h ¢+(V) and ¢-(V) have the value - ¢ + Ux, which is independent of 7. Substituting this value in (25), and integrating the first t e r m by parts, we get sin¼@-7*) ~ _, ,:. In c o s ¼ ( ~ + 7 , ) l a ~ O + u x , at a p o i n t y = y * o n ~ / =

(26)

0. ¢=-¢+u~ E'

DiE

~-"-'~¢-o

t

t~

~

! ~y I

E' -o

I

,4

o -*~

,.

0

'

iB ~--

z

o

.,Y-plane I

- ~ . . . . -(o

Do~

-~

I

-I

E'

0

,

~

to

B

~

E

z

k

co

O~

t-plane

E'~'

'

~-plane Fm.

2.

B y a C a u c h y - R i e m a n n equation, on V = 0, we have ~7

~

~n ~

'

where n is measured along the normal to the b o u n d a r y in the x-plane. I t is apparent from Fig. 2 t h a t ~¢/~y vanishes except in - ~r < 7 < - Yl, 71 < ~ < ~, where 8¢ ~¢ ~y ~

--

~y

×

~"

Differentiation of (22) gives dA

d$

a n s (Ka) cos ½~ 2Kk cn (KA/a) dn (KA/a)

and on ~ = ~,, A = x + ½ill, this becomes ~y ~

a n s (Ka) cos ½~ 2K ds (Kx/a) cs (Kx/a)'

(27)

318

L.C. WooDs

where b y (22) and (24) ns (Kx/a) = sin }F/sin 171. Hence

~¢

~ ~ ns (K~)

~Y

~y

2K

cos } 7

t/sin}7~2-k2} ½f(sin}7~2- I " ([sin }71] [[sin }71] 1J ½ .

Substituting this value in (26), we obtain ¢(7") =-¢+Ux+A(7*

A(y, )

ans(Ka) ~-

c°s }Y In cos}y + cos 17 * i / s i n ~7~2_ k2t½ f{ sin }y~ 2 }~d 7 . v, (~sin½7,] / [[~] - 1

2IrK

(2s)

co_s)7 :- cos }7*

~.

where

,

(29)

F r o m the definition of ~ its average value over - a < z < a is zero, so the average of (28) with respect to z is A ;y - ~ + Ux = 0, where

(30)

A~ = 1 (7o A(V,)d~_,d7, 2a j_~,°

A(7*)c°slT*dY*

- ns(~K) f_° {

4Kk -

70

(31)

1 _ /sin ½7"~2/½ / _ k"~/sin ~v*\21½' ~l a, / 1 ~sin}70] t [1 ~sin}70] i

by (23), (24) and (27). The required averaged b o u n d a r y conditions now follow by differentiating (3) with respect to x d ~~vx + U - 5 = O, or b y (9) where

h ~~x' O - a = 0,

(32)

~ = - fla zt.

(33)

The published work cited in the second paragraph of the introduction applies only to the limiting case 2a/H = 0. B y (19) this is the limit K/K' = O, K = w/2, k = 0 and s n a K = sin@a/2). It now follows from (23), (24), (31) and (29) t h a t in this limit

= - f l . d = -fiaA(O) - fla whence

a s i n 171 F" ln{(1 + coslT)/(1 - cos ly)} ~ | cot 17 ~si~2~- - sinl½7,} ~ - dT, A = 2a fla In sin ~ 77

(34)

Z

A more useful special case is obtained b y assuming a to be small, for it is shown in the next section that the values of a which eliminate tunnel blockage are always less than 0.1 regardless of the value of 2a/H. If ~ is small it follows from (13) and (14) that 70 and 7~ are small. Then b y (29) and (31), _A = A(0),

On the theory of ventilated wind tunnels

319

which reduces to the same integral as obtained in the derivation of (34), except t h a t K # 7r/2 and ½~1-Ka. Thus in this case = ~aft, 1 ln~K[. 4. B L O C K A G E

(35)

IN A S L O T T E D

TUNNEL

The problem of determining the flow past an aerofoil in a slotted tunnel is now reduced to t h a t of finding r(Z) in the strip - ~ < x < ~ , 0 < y < ½H, subject to the b o u n d a r y conditions

0 = Oo(x) on y = 0,

\~x] h - ~(2h = 0 on y = ½H,

(36)

where the subscripts 0 and h denote values on the tunnel axis and wall respectively. This problem is readily solved b y the Laplace transform method. We shall p u t q = U at x = - ~ , so t h a t ~_~ = 0. N o w (see Woods 4, 5) 1

~

zr

,

g2o(X*) = ~ f'-oo (O°(x)c°sech~ ( x - x

Oh(X*) =

and

~-

)+~h(x)seeh~(x--x*)}

Oo(X)sech~(x-x*)+g~h(x)cosech~(x-x*

dx

(37)

) dx

(38)

o3

where

b ~ ½ft, H.

Let ~f denote the two-sided Laplace transform introduced b y van der Pol and Bremmer, 6 i.e.

.~f{f(x);p} =_p / ]

e -ptf(t) dt,

then from results given b y these authors it follows t h a t

1

~o s e c h ~ , and

p = 2pseepb

~ ° { e o s e e h ~ ; p} = p ~ { f x

c o s e e h ~ d x ; p}

= 2 p b ~ { l n tanhTr~ ; p } = - 2pb tanpb. W i t h these and with the aid of the convolution theorem we can transform (37) into ~f{O0(x) ; P} = tan pb~{Oo(x) ; P} + see pb~f{F~h(x) ; p}, ~f{0h(x); P} = sec pb~{t~o(X) ; P} + tan pb~Lf{~2h(x) ; p}. The second b o u n d a r y condition in (36) transforms into

p}-

p } = 0.

On eliminating ~h and 0h we get

[ p~,+tanpb I

~{f~°(x); P} = ~{0°(x) ; P} [1 - p A t a n p b )

-

Ipl~f{Oo(x);p

p } + p ~ { 0 o ( x ) ; p}~f{F(x); p},

(39 /

320 where

L.C.

WooDs

l ff+i~ ,,xJJPl p2+tanpb ] F(x) = 2m= Jc-~ e (-P- + i ZpXtanpbl alp,

(40)

c being a real c o n s t a n t chosen so t h a t the integral converges. V a n der Pol a n d B r e m m e r 6 give ~f{1/x; p} = 7rip I; c o n s e q u e n t l y on i n v e r t i n g (39) we o b t a i n

~o(X) = l f ~ ~ xOo(x*)dx* ~ 'x* t j_~Oo(x*)F(x-x*)dx*.

(41)

As the first t e r m on t h e r i g h t - h a n d side is the well k n o w n result for a n aerofoil in an infinite s t r e a m , the second t e r m m u s t r e p r e s e n t the i n c r e m e n t to ~20 d u e to the slotted wall. E q u a t i o n (40) for F(x) can be expressed in real f o r m as follows. T h e real n u m b e r s c a n d ~2p m u s t be chosen to secure the convergence of the integral, otherwise t h e y are a r b i t r a r y . L e t c = 0, a n d p = - i w / b , where ~o is real, t h e n (40) can be t r a n s f o r m e d into

F(x)

1 fff (b - oJ;~)e ~ sin wx/b _ = ~r~b~ t F b eosh~o+~Ji~sinhww d~°"

(42)

T h e " b l o c k a g e f a c t o r " E a t a p o i n t x on the aerofoil surface is defined b y where ~* is t h e v a l u e ~ would h a v e a t t h e s a m e p o i n t x in an infinite stream, i.e. w i t h H = ~ . H e n c e f r o m (9) a n d (40),

e = (~-~*)/U,

1 (c/2 on replacing 00 b y

(dy/dx*)o. =

I n t e g r a t i o n b y p a r t s gives

c

)

yo(X*)F'(x*-x)dx*,

(43)

as y = 0 at x = - ½ a n d Y0 = 8/2 a t x = c/2 where ~ is t h e w a k e d i s p l a c e m e n t thickness c/2 Cn (see Fig. 1). A useful a p p r o x i m a t i o n for e(x) can be found b y e x p a n d i n g the i n t e g r a n d in (43) in a series in (x-x*)/H a n d ignoring t e r m s 0(H-a). The result is

GA

8 F[C__ x) 2f3 _' where f r o m (38) a n d (42) G is a function of )l/b given b y

(44)

=

G-

fie"HeR'(O)= 2

2 fo

~v

°a(~/b-1)e-°~d~°

(45)

cosh oJ + co()l/b) sinh w"

B y (33) a n d (38) the ratio )t/b is 2 ( - A ) / H , a n d is therefore i n d e p e n d e n t of the compressibility factor fla" W i t h small slots it follows f r o m (35) t h a t bA = ( H ) ' l n K ~K' W i t h c o m p l e t e l y solid walls ~ = 0, so b y (23), (24) a n d (31) Yo = and -/3 aA=2=~. Hence = ~ J0 sinh~

d ~ = 6"

(46) ~/1

=

0,

O n t h e t h e o r y of v e n t i l a t e d w i n d t u n n e l s

While with completely open " w a l l s " , ~ 0 = Y l = T r , and - f l a - ~ = A = 0 . Thus G=

321

~ = l, so (23), (24) and (29) give

_ 2 (~° oJe ~ 7r J0 cosecodc° -

12"

T h e corresponding forms of (44) are the well k n o w n s t a n d a r d results. Zero average blockage can be o b t a i n e d b y choosing ~/b so as to m a k e E(0) vanish. Suppose CD is negligible, t h e n the wake blockage t e r m in (44) can be ignored, and the condition for zero blockage reduces to G = 0. This e q u a t i o n can be solved numerically; let its solution be ~/b = r. I f a is small, this value can be s u b s t i t u t e d in (46) to give cr -~ ~1

e_rK(H/2a) -~ K e - r K ' ,

(47)

b y (19). F r o m a graph given b y Baldwin et al. 3 it appears t h a t r z 1.2. W i t h this value of r in (47) we arrive at the a, 2a/H relationship outlined in the following table: 2a/H

~

¼

~

1

2

4

a a*

0.0000 0.0000

0.0003 0.0003

0.0143 0-0147

0.084 0.097

0.047 0.255

0.024 0-423

The value a* shown in the table is derived similarly from e q u a t i o n (34) which is valid only if 2a/H is small. T h e r e are three conclusions to be d r a w n from this table: firstly, t h a t the a p p r o x i m a t e t h e o r y of previous a u t h o r s (which gives a*) fails for 2a/H> ½; secondly, t h a t a r e m a r k a b l y small value of a is required to eliminate the blockage effect regardless of the value of 2a/H; and finally t h a t a has a m a x i m u m value at 2 a / H ~ 1. REFERENCES 1. R . C. TOMLINSON, T h e Theoretical Interference Velocity on the A x i s of a T w o - d i m e n s i o n a l W i n d T u n n e l with Slotted Walls. C u r r e n t P a p e r No. 181, A e r o n a u t i c a l R e s e a r c h Council, L o n d o n (1954). 2. P. F. MAEDER a n d A. D. WOOD, Z. angew. M a t h . P h y s . 7, 177 (1956). 3. B. S. BALDWIN, J . B. TURNER a n d F. D. KNECHTEL, Tech. Notes N a t . A d v . Comm. Aero., Wash., No. 3176 (1954). 4. L. C. WOODS, A p p l . Sci. Res., Hague, A 6, 351 (1957). 5. L. C. WOODS, Prec. Roy. Soc. A 233, 74 (1955). 6. B. VAN DER POL a n d H . BREMMER, Operational Calculus Based ou the Two-sided Laplace I~tegral. C a m b r i d g e U n i v e r s i t y P r e s s (1950).