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A99-33432

AIAA 99-3106 Design of Low-Drag Airfoils in Transonic Flow of BZT Gases Zvi Rusak Rensselaer Polytechnic Institute Troy, NY

Chun-Wei Wang Chung-Ch-eng Institute of Technol,ogy Taiwan

17th Applied Aerodynamics Conference 28 June - 1 July, 1999 / Norfolk, VA

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(c)l999 American Institute of Aeronautics & Astronautics

Design of Low-Drag Airfoils in Transonic Flow of BZT Gases *

By Zvi Rusak’ Department of Mechanical Engineering, Aeronautical Engineering and Mechanics,

Rensselaer Polytechnic Institute, Troy, New York 12180-3590

and Chun-Wei Wang2 Department of System Engineering,

Chung-Cheng Institute of Technology, Ta-Shi, Taoyuan 33509, Taiwan, R.O.C.

Abstract We design low-drag airfoils for transonic

flow of BZT gases using the nonlinear small- disturbance theory on this topic. This transonic flow is characterized by the high nonlinearity of the fluid thermodynamic behavior that is closely coupled with its compressible flow dynamics. Utilizing BZT gases may result in low drag exerted on airfoils operating at high transonic speeds. This advantage may be further improved by a proper design of the airfoils’ shape. The modified airfoils are characterized by arcs along which the flow is sonic and that connect to a sharp tail. These airfoils may give the highest free stream Mach number for which the flow is nowhere supersonic. Using an analysis in the hodograph (velocity) plane, an analytical formula is derived to describe the sonic arc. Numerical computations using the Euler equations demonstrate that the modified airfoils have significantly lower pressure drag against standard airfoils and their critical Mach number is increased to higher values.

1 Associate Professor 2 Associate Professor

Copyright @ 1999 by Zvi Rusak and Chun- Wei Wang. Published by the American In- stitute of Aeronautics and Astronautics, Inc. with permission.

*Dedicated in memory of Julian D1 Cole, Margaret Darrin Distinguished Professor of Applied Mathematics.

1. Introduction The nonclassical behavior of organic dense

gases such as fluorocarbons (C,F&+sN or CnF3+4, with n normally greater than 10) or hydrocarbons (CnHzn+2, with n normally greater than 7) are of increasing interest because they may result in low drag exerted on airfoils operating at high transonic flows. This advantage, together with their large heat ca- pacity, makes them excellent heat transfer fluids in Rankine cycle turbomachinery (see De- votta and Holland’). It may be further improved by a proper design of the airfoil shape.

Specific interest has recently been developed in dense gases of retrograde type, i.e., they vaporize as being compressed and condense as being expanded in a certain range of pressures and temperatures. Such gases are known as the Bethe-Zel’dovich-Thompson (BZT) fluids (see Bethe, 2 Zel’dovich,3 Thompson4). At these conditions the perfect gas law can not adequately predict the thermodynamic and dynamic properties of the gas flow. Improved equations of state, such as the van der Waals (Moran and Shapiro,5), Redlich-Kwong, or Martin-Hou models, should be used for a more accurate description of BZT gas flows.

The main parameter commonly used to describe the nature of the BZT fluids is the thermodynamic property:

r,1+P d” ( > a % Se

Here p, a and s are the density, speed of sound and specific entropy, respectively. Thompson4 defined this parameter as the fundamental derivative of gasdynamics (see also Duhem’)

1


and it reflects the intrinsic gasdynamics nonlinearity. For a perfect gas model, I’ = (y + 1)/2 is a constant and greater than 1 (y is the ratio of specific heats > 1). For dense gases, I’ is no longer a constant and may become less than 1 or even negative in a certain range of temperatures and pressures (see Cramer7vs). The BZT fluids are characterized by I < 0 in a range of densities and pressures near their critical values.

In addition to I’, the second and the third nonlinear parameters,

strongly affect the BZT gases compressible flow behavior (see Tarkenton and Cramerg). In the range of pressures and densities where turbines in a Rankine cycle usually operate, I’, A and C may change from positive to negative values.

Transonic flows of BZT fluids around airfoils have recently been studied by Tarken- ton and Cramer9 and Rusak and Wang”. This kind of transonic flow is characterized by the high nonlinearity of the fluid thermodynamic behavior that is closely coupled with its compressible flow dynamics. References 9 and 10 presented an extended transonic small- disturbance theory for the flow of a BZT gas around a thin airfoil with a thickness ratio 0 < E C< 1. The oncoming flow is near sonic, with Mach number M, = U-/a, N 1 (Uoo is the oncoming flow speed and aoo is the up stream speed of sound). It is also characterized by small values of I and A, IO0 N 0 and A, N 0, whereas C, N O(1). The theory9 showed a significant increase of the critical Mach number in flows of the BZT fluids over airfoils. The numerical solutions by Morren,” using the Euler equations and the van der Waals equation of state, also revealed substantial reductions in the strength of com- pression shock waves. A further benefit is the decrease of the airfoil’s pressure drag in a flow of BZT gas compared to that in a flow of a perfect gas with the same Mach number.

One of the ideas to improve the perfor-

mances of airfoils operating in transonic flows is to increase the critical Mach number at which shock waves appear for the first time. This may be achieved by constructing airfoils along which the flow is sonic on some portion of the airfoil surface. Gilbarg and Shiffmanr2 showed that for a given thickness ratio the symmetric airfoil with the highest critical Mach number consists of an arc on which the velocity is exactly sonic. Schwende;- man et a1.13 constructed numerical solutions of such airfoils. They also derived an exact analytical solution of the airfoils in the framework of the transonic small-disturbance theory for a perfect gas. They demonstrated significant increase in the critical Mach number compared to traditional airfoils.

Low-drag airfoils for transonic flow of BZT gases are constructed in this paper using the nonlinear small-disturbance theory of Refer- ences 9 and 10. The modified airfoils are characterized by arcs along which the flow is sonic and that connect to a sharp tail. These airfoils may give the highest free stream Mach number for which the flow is nowhere supersonic. An analytical formula of the sonic arc shape is derived by an analysis in the hodograph (velocity) plane. This exact analytical solution extends the work of Schwendeman et all3 to BZT gases. Numerical computations using the Euler equations demonstrate that the modified airfoils have significantly lower pressure drag against standard airfoils and their critical Mach number is increased to higher values.

2. The construction of a sonic arc A small disturbance theory of transonic

flows of a BZT gas around a thin airfoil has been recently developed (see Ref. 10). It shows that the velocity potential 0 of a two- dimensional near sonic uniform flow around an airfoil of unit chord can be given by 9 = Uoo(z + e2i5d(z, @) + . ..). The parameter e is the airfoil’s thickness ratio, 0 < e << 1, and @ = e3j5y is the resealed vertical coordinate. The velocity potential perturbation, 4, is de scribed by a modified K&man-Guderley equa-

2

.I . (c)l999 American Institute of Aeronautics & Astronautics

Y

tion,

This equation was first derived by Kluwick14 for studying transonic flows of dense gases in a nozzle. The solution of (1) must satisfy the linearized tangency boundary condition on the airfoil’s surface,

&(x, Ok) = F:,,(x) for every -l/2 2 x 2 l/2, (2)

and the far-field conditions,

& + 0 and ~$6 + 0 as 1x1 or 161 tend to 00. (3)

Here K, Kr, KA, and C, are the similarity parameters of the flow problem; K = w, Kr = %, KA = %. The airfoil’s shape function is given by y = cFtc,l(x) where the indices u, 1 refer to the upper and lower surfaces of the airfoil, respectively. The theory also shows that the local Mach number, Ml of the flow is given by

M; = 1-e6f5 K - 2Kr& + K,& - +c#: . ( >

We look for the shape, F,,l (x), of a symmetric two-dimensional body for which the flow is sonic, Ml = 1, over the entire surface and nowhere supersonic around the body, Ml 5 1. This can be found in the limit as e approaches zero or Mm tends to one.

For a variety of transonic aerodynamic ap plications of dense gases, we are specifically interested in the case where M, = 1, rrn = 0 and A, = 0. In this case, (1) becomes:

J%&& - &j = 0, (4

where E = X,/3. Let:

w = E”3~, and v = E1/3+g, (5)

then equation (4) becomes:

w3w, - vy = 0 and WC = v,.

Notice that in this case the local Mach number , Mf = 1 + c6i5w3. The small-disturbance

flow is subsonic (elliptic region), Ml < 1, when w < 0 and sonic, Ml = 1, when w = 0.

The transformation from the transonic plane, (w(x,y), v(x,@)), to the hodograph plane, (x(w, v),G(w, v)), results in a linear equation known as the modified Tricomi equation:

w3jjvv - GwuJ = 0. (6) In the elliptic region (where w < 0) we define the variable:

7 s 2 -w)i. 5( (7)

Substituting (7) into (6) results in an equation for solving G,

LJ + ?L + $7 = 0, (8)

and the relation between x and y,

57 3/5 _ xv = - ( > y- Yr* e-9

In the search for a twodimensional body that is symmetric about both the x and g axes and that is also a sonic arc, we look for a solution of (8) where --oo < x 5 0, 0 5 g < co, and which satisfies the following boundary conditions (see Figure 1):

e(r, 0) = 0 when x < - f and 0 < r 5 00,

G(O,v)=Owhen -~<x~OandO<v<oo,

x(7,0) = 0 when 0 5 @ < 00 and r 2 0, x + -oo or y -+ 00 as v and 7 tend to 0. (10)

From this solution, we can derive the entire sonic body shape and the flow field around it.

Fundamental separation of variables solutions of (8) are e -xur1~5Jl,5(Xr) where X 1 0 is a constant of integration and J is the Bessel function of the first kind (see Ref. 15). Using these fundamental solutions, the solution of jj for v 2 0 (see Figure 1) can be written in the following integral form:

ST, v> = J- ;,47*7/10

X J 0

O” e-‘~~1~5J1,5(Xr)X1’2J7/100dX. (11)

3


y to zero (see formula (9.1.7) in Ref. 15). There

t fore, (13) becomes:

From (14) and formula (11.4.16) in Ref. 15 we

1 2 3/5 Figure 1: Small disturbance problem in the 0 5 (15) domain

A=G I(7/10)’ I?(1/5) -cc < x 2 0 and 0 5 y < 00.

Using (15) and formulas (11.4.34) and (15.1.1)

The constants A and r* in (11) will be de- in Ref. 15, it can be verified now that the third

termined in the following paragraphs from the boundary condition in (10) is also satisfied.

boundary conditions of the problem. Notice The sonic arc formula is now constructed.

that when v = 0 the first boundary condition Along the sonic arc r = w = 0 and according

in (10) is satisfied since to (10):

J 0 O” T’/~J~,~(XT)X’/~J~,~~(XT*)~X = 0

according to formula (11.4.41) in Ref. 15. X

Also, notice that since T~/~J~/~(XT) N Jrn

e 0

-XvX-3~‘0J7,10(X~*)d~. (16)

w4 / ‘j5r2i5 I’(6/5) approaches zero as 7 Therefore, along the sonic arc: tends to zero (see formula (9.1.7) in Ref. 15), 1 the second boundary condition in (10) is also satisfied.

xv(07 ‘) = -27/10T*r(7/10)

J

al From (9) and (11)) and using formula X e --Xv/T* i7/10 J7,10(~)&. (17)

(9.1.30) in Ref. 15, we find that: 0

Xv = - (;)3’5 fp*7/10

Here x = XT*. Using Laplace transforms formula (29.3.57) from Ref. 15 we find that

X J

1 I(6/5) Om Xe-Xu~4’5J_4,5(Xr)X’/2J?/100dX.(12) zv(O,V) = --

1 r*fiI(7/10) (1+ (v/7’)2)6’5’

Integrating (12) results in: (18) From boundary condition (2) we have

x(7-, v) = -; + 4x, W/E l/3 = F:,(x) for every -l/2 5 x I 0 which results in: ’

I

oc X

X e V(Zl, O+n -0

-Xu~4’5J-4,5(X~)X1’2J7,~o(A~*)dX (13) Q(x) = & J 0 1 v

for every v 2 0. =gpoo I wv(O, v&h

From the boundary condition that x = 0 (middle of the airfoil) is the point of the maxi- = -L- 1 r(6/5)

J 2, v1dv1

mum sonic arc thickness we have v = T = 0 at J?w3 T’J?; r(w) ccl (1 + (vl/7-*)y5

this point. Also, notice that r415 Jm4j5(X7) - 5 1 I(6/5) * 24/5X-4/5/I’(1/5) approaches zero as r tends = z Eq/x(7/10) (1 + &*)2)1/5

G F(v).(19)

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According to the boundary condition at x = 0, F(0) = l/2 and v = 0 at this point. Therefore, from (19) we can find the scaling parameter

7t = E’/3J?;ir(7/10)

5 W/5)

and so,

F(v) =; (I, (;)‘)-‘“- (21)

To determine the related x(v) we integrate (17):

J

V

x(v) = XL@, Vl)dVl = 00

1 r(6/5) v/r* dVi -- J J;Frwo) o (1 + vq)s’5*

(22)

Equations (21) and (22) where 0 5 v I 00 formulate a parametric representation of the sonic arc shape in the region x I 0 and g > o+. From the symmetry considerations about the &axis, we can extend the sonic arc to the region x 2 0 by using the parameter v in equations (21) and (22) in the range -oo < v 5 0. Same symmetry considerations about the x-axis are also made. The shape of the sonic arc y = EF(x) when E = 0.12 is described in Figure 2. According to the

0.0 0.1 0.2 0.3 0.4 05 0.6 0.7 0.6 0.9 1.0

TT

Figure 2: The body with sonic arcs in TSD flow of dense gases and the modified airfoil.

small-disturbance theory the pressure coefh- cient is zero, C, = 0, along the entire surface. It can be shown from (21) and (22) that near the nose of the airfoil (when x tends to -l/2+ or when v tends to +oo) the sonic arc shape is given by F - g(x + 1/2)2’7 -t- . . . where g = i[w]“li = 0.357828. This

nose shape matches with the x2’7 optimal nose shape that has a minimum pressure drag that was predicted by Wang and Rusak.“j

Notice that the solution (11) and (13) pro vides an exact solution of the transonic small- disturbance problem (l)-(3) where IO0 = A, = 0. It describes, in the limit as E tends to zero, a sonic flow around the airfoil shape given by (21) and (22) at zero angle of attack. Such a solution can be used for validating numerical codes for solving the problem (l)-(3).

An extended discussion on the flow field around the sonic arc is given in the Appendix.

3. Numerical studies of modified airfoils

In order to demonstrate the effect of the sonic arc on airfoil’s pressure drag, we construct a special, modified airfoil. This airfoil, Y = cFm(x) where 0 5 x < 1, has a nose shape given by (21) and (22) and a tail angle of the NACA 4-digit family of airfoilsi (we use a sharp tail for practical aerodynamic reasons to minimize the boundary layer separation near the trailing edge). The maximum thickness of the airfoil is located at x = 0.4. The modified airfoil shape is given by the formula:

y = E (2gxf + blx + b2x2 + b3x3 + b4x4)

(23) where g = 0.3578288, bl = -0.1919425337, b2 = 1.242479388, b3 = -3.330595693 and b4 = 1.575013529. The modified shape with e = 0.12 is also presented in Figure 2. It can be seen that it almost matches the sonic arc shape up to about 40 percent of the chord. The modified airfoil is compared with the NACAO012 airfoil in Figure 3. It has the same tail as the standard airfoil. Notice that the modified airfoil has a blunter nose.

The transonic flows with IO0 = I&, = 0 and C, = 16.05, for which pm/p, = 1.0696 and v,/vuc = 1.3605, around the modified airfoils with E = 0.12 and e = 0.06 were studied. Results were compared with those of NACA0012 and NACA0006 airfoils, respectively, at same flow conditions. The Euler solver of Morren” was used in these studies.


0.06 , 1

~ij’,~,~ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0

x

Figure 3: The geometry of the modified airfoil and the NACA0012 airfoil.

This code uses the van der Waals equation of state with R/c, = 0.02. Notice that the error in computing the pressure drag coefficient using this code is within &0.0005 (see References 11 and 18 which the code of Ref. 11 is based on).

Figure 4 shows the pressure coefficient and Mach number distributions over the surfaces of the modified airfoil with c = 0.12 and NACA0012 at Moo = 0.925. It can be seen that there is a shock wave over the NACA0012 airfoil but not on the modified airfoil. The Mach number on the modified airfoil increases quickly along the blunter nose and keeps an almost sonic level until about 40 percent of the airfoil’s chord. It reflects the flow behavior of a sonic arc geometry. The pressure drag coefficient for the flow around the modified airfoil is 0.0004 and is less than 0.0014 of the NACA0012. (notice that a drag coefficient of 0.0004 is very small and is within the accu- racy of the computations, i.e. it is expected that the drag of the modified airfoil actually vanishes at these flow conditions). These results show that the modified airfoil creates at Mm = 0.925 an essentially shock-free transonic flow and achieves less drag. Therefore, a higher critical Mach number for a transonic flow of BZT gases is attained for the modified airfoil. A similar behavior is found in the case of the modified airfoil with E = 0.06 and NACA0006 at M, = 0.96 (see Figure 5).

It is interesting to compare the critical Mach number of a flow of a perfect gas around a NACA0012 airfoil with that of a flow of a BZT gas with R/h = 0.02 at IO0 = Aoo = 0

M.=O925. A&O 1.51. I.,.,.,.,.

1.0 .

0.5 .

0.0 .

0.5 .

-1.0 .

-1.5 .

- Mcdi66dairloacp -MOOWaitilM&hllJd~ -----' NACAM)lPairblCp ~---~N4CAWl2aitioilMachmmtm

Figure 4: The, pressure coefficient and Mach number distributions along the modified and the NACA0012 airfoils in transonic flow of dense gases.

-1.0 * -1.0 * U - Wi airfoil Cp

- kdai aml Mach nrmbec - kdai aml Mach nrmbec ---NACACQX~~IIO~IC~ ---NACACQX~~IIO~IC~ *-*NACMMGairiadUachnunbec *-*NACMMGairiadUachnunbec

J

4.5' B ' ' ' ' 0.1 0.1 0.3 0.5 0.7 0.9 1.1

X/C

Figure 5: The pressure coefficient and Mach number distributions along the modified and the NACA0006 airfoils in transonic flow of dense gases.

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.

008 o---0 NACAOOl2 airfoil 0-0 Modified airfoil

0.08 .

cD

0.82 OS3 0.04 0.85 0.86 0.97 0.98

MO3

Figure 6: The drag coefficient distributions with various Mach number of the modified airfoil with e = 0.12 and the NACA0012 airfoil.

around a NACAO012 and around the modified airfoil with E = 0.12. The comparison shows an increase from Mm = 0.74 (see Ref. 19) to about Mm = 0.88, just because of the use of a BZT gas, and a further increase to about Mm = 0.92 as a result of the use of a modified shape. In the case of a E = 0.06 the comparison shows an increase from Mm = 0.79 (see Ref. 19) to about Mm = 0.92, just because of the use of a BZT gas, and a further increase to about Mm = 0.94 as a result of the use of a modified shape.

Moreover, comparing the pressure drag coefficient, CD, of the modified and the NACA airfoils at higher Mach numbers shows (see Fig- ures 6 and 7) that the modified airfoils result in less drag in transonic flows. Specifically, for practical low-drag applications, where CD has to be less than 0.02, we can see that the modified airfoils provide a reduction of about 40% in the pressure drag coefficient.

4. Summary The nonlinear small-disturbance theory can

be used to construct low-drag airfoils for transonic flow of BZT gases. This kind of transonic flow is characterized by the high nonlinearity of the fluid thermodynamic behavior that is closely coupled with its compress-

0.06 - ) ,‘,’ ,,,f !

CD O.M . ,,j

0.02 * /

,’ P ,’ ,’

._- d’ 0.00 I

0.920 0.040 0.960 0.980 1.m M

Figure 7: The drag coefficient distributions with various Mach number of the modified airfoil with E = 0.06 and the NACA0006 airfoil.

ible flow dynamics. The modified airfoils are characterized by arcs along which the flow is sonic and that connect to a sharp tail. An analytical formula of the sonic arc shape is derived by an analysis in the hodograph (velocity) plane. The numerical computations using the Euler equations demonstrate that the modified airfoils have significantly lower pressure drag against standard airfoils and their critical Mach number is increased to higher values.

The present results are limited to the case where in the unperturbed flow the fundamental derivative of gas dynamics and its first derivative with respect to the density van- ish (I’, = Aoo = 0). Our experience shows that similar results can be found at other flow states where the absolute value of these derivatives is small, lIool < 0.1, IAml < 0.15. However, away from this range of parameters, specifically when I becomes significantly negative, the nonlinear effects of the terms in (1) related with rm and Aoo play a more dominate role, and, therefore, the present results do not always apply.

The present work is also limited to an invis- cid two-dimensional study. From the experience with perfect gas flows, it is expected that the addition of small viscosity will only add a

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nearly constant viscous drag to the pressure drag of the airfoils, atleast up to the critical Mach number of each airfoil. Therefore, the present results may apply to high Reynolds number flows of BZT gases.

It should be emphasized that the previous research on compressible flows of BZT gases focuses primarily on theoretical and numerical studies. Experiments to demonstrate the im- portant phenomena that occur in BZT flows are not yet available, including drag charac- teristics of airfoils. The present paper may provide additional insight and guidelines for performing such future experiments.

Acknowledgements .The authors would like to thank Professor

Mark S. Cramer for providing them with the computer code of Morren (1990) and for many insightful discussions on the topic. The authors also thank Dr. Grey M. Tarkenton for helpful discussions. The second author (C.- W. Wang) wishes to thank the Chung-Cheng Institute of Technology (CCIT) in Taiwan for supporting his graduate studies.

Appendix. The sonic flow around the sonic arc

Using Wang and Rusak,16 it can also be shown that the small-disturbance nose singularity of a near sonic flow around the sonic arc is given by the asymptotic self-similar solution

(24

where

and -7r/2 5 Q 5 1r/2. The variation of (f/~:‘~) as function of (</cy”‘) is described in Figure 4 in Ref. 16.

It can be shown that according to this nose singularity, the TSD solution is valid only when both 121 and IyI are greater than c7i5. In order to account for the correct behavior of the flow near the nose of the profile, where a stagnation point exists, an inner region analysis can be performed. In the inner region a resealing in radial direction is applied where x* = x/c7j5 and y* = y/e7i5. Using similar methods to those described in Rusak and Wang,lO it, is found that the flow in the inner region can be described by the solution of a sonic flow (M, = 1) with Foe = A, = 0 around a y* = hx*2/7 surface (see Figure 8). The Euler solution of this problem provides

Y*t

!p""

/ y* = t g (x*)2/7

X*

Figure 8: The inner problem.

the distribution of the pressure coefficient, C;, the density ratio, p*/p,, and the axial velocity, u*, in the nose region. The matching between the TSD (outer) solution and the inner region solution results in a uniformly valid composite solution for the pressure distribution along the entire sonic arc, CO(x; E) = cg-) - 2(p*,p&*(l - jp)lI3,&3 + 1. .*

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(c)l999 American Institute of Aeronautics & Astronautics !

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