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The ray theory of supersonic propeller acoustics
Journal
of Sound and Vibration (1988) 127(l), 145-153
THE
RAY THEORY
OF SUPERSONIC
PROPELLER
ACOUSTICS C.
J. CHAPMAN?
Institute of Sound and Vibration Research, University of Southampton, England
Southampton SO9 SNH,
(Received 25 August 1987, and in revised form 31 May 1988) The wavefront surface produced by a supersonic propeller is constructed in the standard way as an envelope of expanding spheres. Focusing of rays is shown to occur, determined by the curvature of this surface and consequent variation of ray tube areas during propagation. Two important factors, the amount of sweep and the length of edge with supersonic normal component of velocity, are investigated analytically by means of a family of “quadratically swept” edges; the length of the supersonic part may be varied from zero to infinity, to give different waveforms. A straight radial edge, which emerges as a special case, is further analyzed. These results are believed to be a helpful preliminary to applying sonic boom methods, such as those based on Whitham’s weakly non-linear theory or the transonic equation. 1. INTRODUCTION A rapidly rotating aeroplane propeller produces shock surfaces in the air through which it passes. On linear acoustic theory, these emanate from the supersonic edges of the propeller (i.e., those parts of the leading and trailing edges whose normal component of velocity is supersonic) and spiral around the axis of motion in a complicated way; if the edges are sharp, linear theory predicts a pressure field which is discontinuous on one sheet of such a surface and logarithmically singular on the other, the two sheets being separated by a cusp line. The whole of a moving “shock and singularity” surface will be denoted SS; its equation may be obtained by mathematical analysis of the integral for thickness noise, and aspects of the pressure field in its vicinity determined [l]. An alternative method of obtaining SS is to exploit the properties of envelopes and rays, as in studies of sonic boom [2-71. Extremities of a body, such as edges or tips, are assumed to be continuously producing spheres, which start from zero radius and expand at the speed of sound: their envelope constitutes SS, and the family of normals gives the ray system. (Here the simplest case is being considered, in which the ambient air is at rest and of uniform properties.) One type of problem which arises in practice may be called the “moving point” variety, where the envelope comes from a one-parameter family of spheres formed, for example, by the pointed nose of a supersonic projectile, or the tip of a propeller. The “moving line” type, on the other hand, requires a two-parameter family, such as would be produced by the leading or trailing edge of a wing or propeller; the parameters labelling the spheres are time of emission and position on the edge when this took place. The moving point problem for a supersonic propeller has been solved [2,8], and the structure of SS elucidated. Thus the wavefront produced by a propeller tip is known. The t Present address: University of Cambridge, Department Silver Street, Cambridge, England CB3 9EW.
of Applied
Mathematics
and Theoretical
Physics,
145 0022-460X/88/220145
+ 09 %03.00/O
@ 1988 Academic
Press Limited
146
C.
J. CHAPMAN
present paper is concerned with the corresponding line problem, to determine the effect of a supersonic edge. One method would be to take the envelope of the previous solution, by varying the point on the edge; but the easier approach of working directly with the spheres will be followed. Note that a point P on an edge may be moving supersonically, even though the edge is subsonic. In this case, the two-parameter family does not have an envelope near P, but the one-parameter family does; therefore the edge does not produce a ray from P, despite the fact that there would be one from P taken in isolation. The paper is arranged as follows. In section 2, the equation of SS is obtained for a non-translating propeller with arbitrarily positioned leading or trailing edges, and the rays found. (An extension to include axial velocity is straightforward, but produces more complicated formulae.) The results are then applied to the special case of a “quadratically swept” edge, represented by the equation 0 = -&r2 in polar co-ordinates, with (Ya positive constant. This example is useful because the sweep can be varied: large (Y makes the whole edge subsonic, LY= 0 gives an infinitely long supersonic part, and intermediate values make any desired length supersonic. For CY= 0, a straight edge, the rays are sketched. In section 3, the curvature of SS is analysed, with particular reference to focusing. The results are discussed in section 4, and related to more complete theories of propeller acoustics. 2. ENVELOPES
2.1.
GENERAL
AND
RAYS
EQUATIONS
In a system (r, 0’, z) of cylindrical co-ordinates, fixed in space, the propeller is assumed to lie across the plane z = 0 and rotate with angular velocity 0 about the z-axis. An edge whose equation is 19’= h(r) at time t = 0 is regarded as continuously emitting spheres which expand at the speed of sound c. A sphere produced at time T from the point (r, , h ( r2) + Lh, 0) has radius c( t - T) at time t a T, and equation
C*(t-~)2=r2+r:-2rr2COS(e’-h(r2)-RT)+z2.
(1)
This gives a family labelled by the parameters r2, T. It is convenient to non-dimensionalize lengths by the sonic radius c/Q and use the parameter e2 = a( t - T) instead of T; then the “blade-fixed” co-ordinate 0 = 8’- at and the variable
R={r2+r~-2rr2cos(e-h(r2)+8,)+z2}”2 enable equation (1) to be written as R = e2. To obtain the envelope, one differentiates with respect
aR/ar, = 0,
R=e2, may then be re-arranged r=
to give SS parametrically
[id - W2Wr2)12+ W/* 9
(2) to r2 and e2; the equations
aR/ae* = 1
(3)
by means of r,, f3*. A little algebra yields
e=-e,+h(r,)+tan-’
r2
z=*{r:-l-r$‘*(r2)}“*e2/r2.
e2 (495) r: e,r,h’ ( r2)I ’ { (6)
Sections of SS by co-ordinate surfaces may easily be drawn. For example, if r is fixed, then the values of e2 obtained from equation (4) may be substituted into equations (5) and (6) to determine 0, z in terms of r, r,; variation of r, gives the required cylindrical section. The surface SS has the geometrical property that rotation about the z-axis is equivalent to translation along the normals by fixed amounts: for SS both rotates with the propeller and propagates at right angles to itself with the speed of sound. In fact, the argument can be made quantitative by noting that e2 is the distance travelled by a ray in the
RAY
THEORY
OF SUPERSONIC
147
PROPELLERS
space-fixed frame, and that rays correspond to holding r2 constant while varying e2. These remarks may be justified by returning to the spheres (1) from which SS was constructed; a ray has travelled a distance equal to the radius of an expanding sphere, namely c( t - r), which after scaling by c/0 is just e2. Moreover, each ray always remains on the surface of one expanding sphere, identified by r,, the radial co-ordinate of the propeller edge at the point where the sphere came into being; hence fixing r2 in equations (4)-(6) determines a ray. In what follows, the term ray will be used to refer either to the straight line in the space-fixed frame, or to the corresponding curve on the stationary surface SS as viewed from the propeller. One may obtain the latter from the former by rotating SS, and noting the locus of intersections with the straight line. From each point on a supersonic edge, two rays emerge, one into z > 0, the other into z < 0. They lie in the plane normal to the edge, and their directions are such that the normal velocity of the edge, resolved along a ray, is exactly sonic at the time of emission. (For a moving point problem a cone of rays, not just two, is emitted at each instant.) Consideration of a triangle with sides dr, and r2h’(r2) dr, reveals that the normal component of velocity at a point r2 on the edge has Mach number r2/{1 + r:hf2( r2)}“2. Thus the criterion for ray production is r22 3 1 + rzh’2( r2), as might have been expected from the square-root term in equation (6). This condition will be assumed to hold for the analysis below. In general, SS contains a cusp line, caused by the focusing of rays. The most direct method of detecting it is to determine where the tangent vectors (8r/8r2, r ae/tw,, az/ar,) and (ar/aO,, r af3/M2, i3z/ao2) have zero cross-product; a relation between r2 and e2 is obtained, leading to the equation of the cusp line on SS. Alternatively, the Hessian of a certain function may be equated to zero [l]; and in a third method, given in section 3, curvature and ray tube areas are used. Since r,, O2are parameters, one is free to transform them. The algebra may conveniently be simplified by defining r, = rZ and tan 0, = e,/{r: - B,r,h’(r,)}, to give SS the representation r=
r, set 8,
rf tan 8,
e=e,-
1+ r,h’(r,) tan 8,’
1 + r,h’( r,) tan
8,
+ Wr,),
(738)
z = f r,{r: - 1 - r:hr2( r,)},‘2 (tan e,)/{ 1 + r, h’( r,) tan 0,).
(9)
Rays are still obtained by fixing r, , but 8, does not measure distance along them. 2.2.
QUADRATICALLY
SWEPT
EDGE
The above results will now be applied to the special case h( r,) = -$ar:, By equation (9), the condition for r, to produce a ray is
a2r:-rf+1S0,
where a L 0. (10)
i.e., _
+
r, S r, S r,,
(11)
where
try,r:) =
1 -(I
(1
_4a2)1/2
2ff2
l/2
I
’
1+(1-44aZ)“2 “2 2cX2 1 >*
1
(12)
Note that if (Y exceeds $, no rays are formed, while if it takes a value slightly below $ only a small length of edge is supersonic; as (Yapproaches zero, this length increases, and at zero the whole edge beyond r, = 1 produces rays. In what follows, the assumption OC a G; will be made. (If forward sweep were of interest, it would be useful to allow negative a.)
148
C. .I. CHAPMAN
Since the variables that 0s 8, s T and
R, r, r,, rz are non-negative, (tan f?,)/(l-curttan
it may be deduced 8,)sO.
from equation
(3)
(13)
The required domain of r, , 8,) drawn in Figure 1, is bounded by 8, = 0, cot-’ (ar:) and by any of the methods given in section 2.1, reveals that r, = r;, r:. A routine calculation, the cusp line has equation r~(2~2r~-1)tan26,+~r~(~2r~-1)tan8,+(r~-1-a2r~)=0, 7r/4). and extends from (r;, 0) to (l/A, Sections of SS by cylinders of fixed r correspond tan 13~={ar’r,
-(cy*r*r:+
(14)
to
r*- rf)“‘}/r(a*r*rf-
l),
(19
where r, must be varied between rl and min {r, r:}. The family of curves (15) is drawn in Figure 1 as P,P2P3, Q1Q2Q3, etc. Numerical computations at various I and CYshow that sections always take the form of Figure 2: the supersonic edge AB generates “arrowheads”, which shrink in size to a point as A is approached, and simultaneously flatten. The semi-angle at the front, namely the Mach angle, tends to 7r/2 as either A or B is approached. Beyond B, the arrowhead has detached itself from the edge which produced it, and become blunt-nosed; its extent in the z-direction increases without limit as r increases, whatever the length of AB. That part of SS swept out by PIP2 and QIQZ as r varies will be called the front sheet; P2P3 and Q2Q3 sweep out the rear sheet. Lines of constant r, in Figure 1 become rays on SS. The position of the cusp line on the edge will pass through a focus, implies that rays from between r ; and l/G
Figure 1. For a quadratically swept edge, the parameter region inside ABCD generates the portion of SS occupying z > 0, in such a way that the lettering corresponds to that of Figure 2. Thus one may imagine folding the region along APsQsE and stretching it in three-dimensional space so that it is located as follows: AP,B is attached to the supersonic part of the propeller edge; BQ,C is the leading edge of that part of SS detached from the propeller, with C at infinity; CED is the cylindrical section at infinity; DQ,P,A is the centreline of the rear sheet of SS. Note that A is the vertex of the cusped cone, and APsQsE the cusp line. The’ family of lines joining ABC to AD (such as P,P,P, and Q,Q2Q3) generates the cylindrical sections, obtained by fixing r.
RAY
THEORY
OF
SUPERSONIC
PROPELLERS
149
(a)
(b)
!cl
Figure 2. (a) A rotating blade, with a quadratically swept leading edge containing a supersonic segment AB; the lines AP,BQ, . . and APsQ, . . mark the intersection of the piane z = 0 with the front and rear sheets of the cone. (b) Cylindrical section of SS at a radius less than that of B, demonstrating the sharp edge at P, and cusp at P2. (c) Section at radius r greater than that of B; note the blunt nose of SS for such r. All lettering corresponds to that of Figure 1: thus the front and rear sheets of SS are generated by the regions inside ABCE and ADE respectively.
whereas those from between I/& and r: will not; i.e., the former cross the line APzQz - * a, while the latter remain always on the front sheet. 2.3.
STRAIGHT
EDGE
The choice (Y= 0 gives a straight radial edge [ 11, on which the points A, B‘of Figure 2(a) are at radii 1, co, respectively; every section of SS has the form shown in Figure 2(b). Ray paths are readily drawn in this case, and details will be presented. The variables (u, u) = (I-:, 6,) enable equations (4)-(6) to be written
150
C. J. CHAPMAN
I
0
B’
0
I
;p’ IS’
A',C' ,
I
Figure 3. Points A, B, in (a) correspond to A’, B’, in (b) under the mapping from II, o to r, z. Typical rays P’Q’R’ and S’T’U’ are shown; they move over from the front to the rear sheet on crossing the cusp line B’Q’T’D’.
It is convenient to project the rays cylindrically onto a meridional plane, by ignoring the &equation and using the other two for a mapping from (u, V) to (r, z). Rays are obtained by holding u constant and varying V; equivalently, u may be eliminated to give the equation of the ray for a particular u, namely (r2/u)-z2/u(u-l)=l.
(17)
Figure 3 shows that the domain u 2 1, u 3 0 is mapped in a “two-to-one” way onto the regionOaz~(r2-l)/2,andthefoldlinev2=u(u-1)senttothecusplinez=(r2-1)/2, as represented by BQTD-, B’Q’T’D’. The areas bounded by CBD and ABD map to the front and rear sheets of SS. Typical rays are indicated by the lines P’Q’R’and S’T’U’; if P corresponds to the value u, then P’ and Q’ have (r, z) co-ordinates (4, 0) and (m, u - 1). Neighbouring rays cross on the cusp line; the apparent intersection in Figure 3(b) is due only to the projection adopted: P’Q’and S’T’are situated on the front sheet of SS, while Q’R’ and T’U’ are on the rear. 3.
CURVATURE
AND
FOCUSING
The surface SS, and its propagation through space, lend themselves to analysis by the simpler parts of differential geometry [3,9]. If the principal radii of curvature at a point are R, , R2, then ray tube areas vary in proportion to (R, + s)( R2 + s), where s is the distance travelled along a ray. When the area passes through zero, focusing occurs, and
RAY
THEORY
OF
SUPERSONIC
151
PROPELLERS
geometrical acoustics ceases to give a valid approximation to the pressure field. Recall that motion along a straight ray in space is equivalent to remaining on SS but varying & in equations (4)-(6), or v in equation (16). It is therefore sufficient to know, as a function of position on SS, the value of the product R, R2, which is the reciprocal of the Gaussian curvature. The requisite geometrical theory is standard [9], and the following results are quoted to define the notation. For a surface represented parametrically by r(u, v), the square of the distance between points r( u, v) and r( u + du, v + do) will be written ds2=Edu2+2Fdudv+Gdv2,
(18)
where (E, F, G) = (lrU/*,rU* TV, bul*), subscripts denoting differentiation. magnitude
(19)
The vector ru x rv, parallel to the unit normal N, has
Ir, x rvl = (EG - F*)“* = H, so that N = (r, x r,)/H.
Hence the second fundamental Ldu2+2M
(20)
form,
du dv+ N dv*,
(21)
where (L, M N) = (N - ruur N - ruv, N - rJ, may be obtained by further differentiation. K=
LN-M* HZ
’
(22)
The Gaussian and mean curvatures, namely tJ=
EN+GL-2FM 2H2
(23)
’
the principal curvatures K,, K* through the relations K = K,K* and p = then R, , R2 are the reciprocals of K~, K*. For a general surface, the formulae lead to complicated expressions. But they simplify in the problem being considered here, because B2, or v, gives the distance s along a ray (as shown in section 2.1), and l/K varies quadratically with s, in accordance with the remarks at the beginning of this section. As an application of these results, the expressions will be evaluated for the straight edge specified in section 2.3. Equation (16) gives determine
(K, + ~*)/2;
(66 N=
(
G)=
u(u-l)+v* 4u2(u-1)
(u(u2+vL12)]l,29
T
2, u(u-l)+v* u
(24)
>’
(25)
(-g-J*,(y*),
V (L,MN)=
-4u2c;_l)’
-&,
-;),
(K,)=
-l
(
(
(R,,R,)=(-v+{u(u-l)}“*,-v-{u(u-l)}”*).
u(u-l)-v2’u(u-1)-v*
>’ W,27)
(28)
As expected, the principal radii of curvature are linear in v. Since u 2 1 and o > 0, the product RI R2 can vanish only when v = {u( u - l)}“‘, consistent with the location of the cusp line found earlier.
152
C.
J. CHAPMAN
4. CONCLUDING
REMARKS
The main use of ray analysis in propeller acoustics is that it leads directly to the weakly nonlinear theory based on Whitham’s F-function [3,10-131. In particular, it is possible to determine when this theory will not apply. Near SS, except close to the cusp lines, the pressure obtained from the linear equations s , w h ere A(s)represents the ray tube area at a distance s takes the form F(s - ct)/m along a ray. On the front sheet of the surface SS coming off a sharp edge, F has a simple discontinuity, while on the rear sheet it is logarithmically singular. If the ray crosses the cusp line at s = sO, where A(q)= 0, linear acoustics [lo] introduces a complicated correction to the singularity involving 1s - s01-“6. Whitham’s theory will give a non-linear pressure near the front sheet of SS, but will not eliminate the singularity at the cusp; indeed, it is known [lo] that if the formalism is carried through as a weak shock approaches sO, then a pressure variation proportional to (s - s,,) -I’* is obtained, violating the assumptions of the theory. What in fact happens near the cusp line is that transonic effects predominate, and permanently affect the pressure field on the rays passing through. Thus it seems likely that the pressure near the rear sheet cannot be calculated by any simple modification of the logarithmically singular F-function. These ideas are familiar from research into sonic boom [ 14,151 and the focusing of weak shocks [16-181. They may well provide a useful adjunct to the methods of propeller acoustics developed over the years by Farassat [19] and Hanson [20].
ACKNOWLEDGMENTS
Thanks are due to M. J. Fisher, G. M. Lilley and C. L. Morfey for many helpful comments while the work of this paper was carried out. Funding was provided by the Royal Aircraft Establishment, and benefit was gained from a visit by the author to NASA Lewis Research Center as a Case/NASA Research Fellow.
REFERENCES 1. C. J. CHAPMAN 1988 Journal of Fluid Mechanics 192, 1-16. Shocks and singularities in the pressure field of a supersonically rotating propeller. 2. G. M. LILLEY, R. WESTLEY, A. H. YATES and J. R. BUSING 1953 Journal of the Royal Aeronautical Society 57, 396-414. Some aspects of noise from supersonic aircraft. 3. G. B. WHITHAM 1956 Journal of Fluid Mechanics 1, 290-318. On the propagation of weak shock waves. 4. P. S. RAO 1956 Aeronautical Quarterly 7, 21-44, 135-155. Supersonic bangs, parts I and II. 5. F. WALKDEN 1958 Aeronautical Quarterly 9, 164-194. The shock pattern of a wing-body combination, far from the flight path. 6. C. H. E. WARREN 1968 in Noise and Acoustic Fatigue in Aeronautics (Eds E. J. Richards and D. J. Mead), 266-284. Sonic bangs. New York: John Wiley. 7. G. B. WHITHAM 1974 Linear and Nonlinear Waves. New York: John Wiley. 8. M. V. LOWSON and R. J. JUPE 1974 Journal of Sound and Vibration 37,475-489. Wave forms for a supersonic rotor. 9. T. J. WILLMORE 1959 An Introduction to Diflerential Geometry. Oxford: Clarendon Press. 10. J.-P. GUIRAUD 1965 Journal de Mkcanique 4,215-267. Acoustique geometrique, bruit balistique des avions supersoniques et focalisation. 11. D. L. HAWKINGS and M. V. LOWSON 1974 Journal of Sound and Vibration 36, l-20. Theory of open supersonic rotor noise. 12. C. K. W. TAM 1983 Journal of Sound and Vibration 89, 119-134. On linear acoustic solutions of high speed helicopter impulsive noise problems. 13. C. K. W. TAM and M. SALIKUDDIN 1986 Journal of Fluid Mechanics 164, 128-154. Weakly nonlinear acoustic and shock-wave theory of the noise of advanced high-speed turbopropellers. 14. I. R. SCHWARTZ (Ed.) 1971 NASA SP-255. Third Conference on Sonic Boom Research.
RAY THEORY OF SUPERSONIC PROPELLERS
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15. H. H. HUBBARD, D. J. MAGLIERI and D. G. STEPHENS 1986 NASA TM 87685. Sonic-boom research-selected bibliography with annotation. 16. B. STURTEVANT and V. A. KULKARNY 1976 Journal of Fluid
17. 18. 19.
20.
Mechanics 73, 651-671. The focusing of weak shock waves. M. S. CRAMER and A. R. SEEBASS 1978 Journal of Fluid Mechanics 8?3,209-222. Focusing of weak shock waves at an a&e. F. OBERMEIER 1983 Journal of Fluid Mechanics 129, 123-136. On the propagation of weak and moderately strong, curved shock waves. F. FARASSAT 1984 AIAA/ NASA 9th Aeroacoustics Conference, Williamsburg, Virginia, AIAA Paper 84-2303. The unified aerodynamic and acoustic prediction theory of advanced propellers in the time domain. D. B. HANSON 1985 American Institute of Aeronautics and Astronautics Journal 23, 499-504. Near-field frequency-domain theory for propeller noise.
of Sound and Vibration (1988) 127(l), 145-153
THE
RAY THEORY
OF SUPERSONIC
PROPELLER
ACOUSTICS C.
J. CHAPMAN?
Institute of Sound and Vibration Research, University of Southampton, England
Southampton SO9 SNH,
(Received 25 August 1987, and in revised form 31 May 1988) The wavefront surface produced by a supersonic propeller is constructed in the standard way as an envelope of expanding spheres. Focusing of rays is shown to occur, determined by the curvature of this surface and consequent variation of ray tube areas during propagation. Two important factors, the amount of sweep and the length of edge with supersonic normal component of velocity, are investigated analytically by means of a family of “quadratically swept” edges; the length of the supersonic part may be varied from zero to infinity, to give different waveforms. A straight radial edge, which emerges as a special case, is further analyzed. These results are believed to be a helpful preliminary to applying sonic boom methods, such as those based on Whitham’s weakly non-linear theory or the transonic equation. 1. INTRODUCTION A rapidly rotating aeroplane propeller produces shock surfaces in the air through which it passes. On linear acoustic theory, these emanate from the supersonic edges of the propeller (i.e., those parts of the leading and trailing edges whose normal component of velocity is supersonic) and spiral around the axis of motion in a complicated way; if the edges are sharp, linear theory predicts a pressure field which is discontinuous on one sheet of such a surface and logarithmically singular on the other, the two sheets being separated by a cusp line. The whole of a moving “shock and singularity” surface will be denoted SS; its equation may be obtained by mathematical analysis of the integral for thickness noise, and aspects of the pressure field in its vicinity determined [l]. An alternative method of obtaining SS is to exploit the properties of envelopes and rays, as in studies of sonic boom [2-71. Extremities of a body, such as edges or tips, are assumed to be continuously producing spheres, which start from zero radius and expand at the speed of sound: their envelope constitutes SS, and the family of normals gives the ray system. (Here the simplest case is being considered, in which the ambient air is at rest and of uniform properties.) One type of problem which arises in practice may be called the “moving point” variety, where the envelope comes from a one-parameter family of spheres formed, for example, by the pointed nose of a supersonic projectile, or the tip of a propeller. The “moving line” type, on the other hand, requires a two-parameter family, such as would be produced by the leading or trailing edge of a wing or propeller; the parameters labelling the spheres are time of emission and position on the edge when this took place. The moving point problem for a supersonic propeller has been solved [2,8], and the structure of SS elucidated. Thus the wavefront produced by a propeller tip is known. The t Present address: University of Cambridge, Department Silver Street, Cambridge, England CB3 9EW.
of Applied
Mathematics
and Theoretical
Physics,
145 0022-460X/88/220145
+ 09 %03.00/O
@ 1988 Academic
Press Limited
146
C.
J. CHAPMAN
present paper is concerned with the corresponding line problem, to determine the effect of a supersonic edge. One method would be to take the envelope of the previous solution, by varying the point on the edge; but the easier approach of working directly with the spheres will be followed. Note that a point P on an edge may be moving supersonically, even though the edge is subsonic. In this case, the two-parameter family does not have an envelope near P, but the one-parameter family does; therefore the edge does not produce a ray from P, despite the fact that there would be one from P taken in isolation. The paper is arranged as follows. In section 2, the equation of SS is obtained for a non-translating propeller with arbitrarily positioned leading or trailing edges, and the rays found. (An extension to include axial velocity is straightforward, but produces more complicated formulae.) The results are then applied to the special case of a “quadratically swept” edge, represented by the equation 0 = -&r2 in polar co-ordinates, with (Ya positive constant. This example is useful because the sweep can be varied: large (Y makes the whole edge subsonic, LY= 0 gives an infinitely long supersonic part, and intermediate values make any desired length supersonic. For CY= 0, a straight edge, the rays are sketched. In section 3, the curvature of SS is analysed, with particular reference to focusing. The results are discussed in section 4, and related to more complete theories of propeller acoustics. 2. ENVELOPES
2.1.
GENERAL
AND
RAYS
EQUATIONS
In a system (r, 0’, z) of cylindrical co-ordinates, fixed in space, the propeller is assumed to lie across the plane z = 0 and rotate with angular velocity 0 about the z-axis. An edge whose equation is 19’= h(r) at time t = 0 is regarded as continuously emitting spheres which expand at the speed of sound c. A sphere produced at time T from the point (r, , h ( r2) + Lh, 0) has radius c( t - T) at time t a T, and equation
C*(t-~)2=r2+r:-2rr2COS(e’-h(r2)-RT)+z2.
(1)
This gives a family labelled by the parameters r2, T. It is convenient to non-dimensionalize lengths by the sonic radius c/Q and use the parameter e2 = a( t - T) instead of T; then the “blade-fixed” co-ordinate 0 = 8’- at and the variable
R={r2+r~-2rr2cos(e-h(r2)+8,)+z2}”2 enable equation (1) to be written as R = e2. To obtain the envelope, one differentiates with respect
aR/ar, = 0,
R=e2, may then be re-arranged r=
to give SS parametrically
[id - W2Wr2)12+ W/* 9
(2) to r2 and e2; the equations
aR/ae* = 1
(3)
by means of r,, f3*. A little algebra yields
e=-e,+h(r,)+tan-’
r2
z=*{r:-l-r$‘*(r2)}“*e2/r2.
e2 (495) r: e,r,h’ ( r2)I ’ { (6)
Sections of SS by co-ordinate surfaces may easily be drawn. For example, if r is fixed, then the values of e2 obtained from equation (4) may be substituted into equations (5) and (6) to determine 0, z in terms of r, r,; variation of r, gives the required cylindrical section. The surface SS has the geometrical property that rotation about the z-axis is equivalent to translation along the normals by fixed amounts: for SS both rotates with the propeller and propagates at right angles to itself with the speed of sound. In fact, the argument can be made quantitative by noting that e2 is the distance travelled by a ray in the
RAY
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space-fixed frame, and that rays correspond to holding r2 constant while varying e2. These remarks may be justified by returning to the spheres (1) from which SS was constructed; a ray has travelled a distance equal to the radius of an expanding sphere, namely c( t - r), which after scaling by c/0 is just e2. Moreover, each ray always remains on the surface of one expanding sphere, identified by r,, the radial co-ordinate of the propeller edge at the point where the sphere came into being; hence fixing r2 in equations (4)-(6) determines a ray. In what follows, the term ray will be used to refer either to the straight line in the space-fixed frame, or to the corresponding curve on the stationary surface SS as viewed from the propeller. One may obtain the latter from the former by rotating SS, and noting the locus of intersections with the straight line. From each point on a supersonic edge, two rays emerge, one into z > 0, the other into z < 0. They lie in the plane normal to the edge, and their directions are such that the normal velocity of the edge, resolved along a ray, is exactly sonic at the time of emission. (For a moving point problem a cone of rays, not just two, is emitted at each instant.) Consideration of a triangle with sides dr, and r2h’(r2) dr, reveals that the normal component of velocity at a point r2 on the edge has Mach number r2/{1 + r:hf2( r2)}“2. Thus the criterion for ray production is r22 3 1 + rzh’2( r2), as might have been expected from the square-root term in equation (6). This condition will be assumed to hold for the analysis below. In general, SS contains a cusp line, caused by the focusing of rays. The most direct method of detecting it is to determine where the tangent vectors (8r/8r2, r ae/tw,, az/ar,) and (ar/aO,, r af3/M2, i3z/ao2) have zero cross-product; a relation between r2 and e2 is obtained, leading to the equation of the cusp line on SS. Alternatively, the Hessian of a certain function may be equated to zero [l]; and in a third method, given in section 3, curvature and ray tube areas are used. Since r,, O2are parameters, one is free to transform them. The algebra may conveniently be simplified by defining r, = rZ and tan 0, = e,/{r: - B,r,h’(r,)}, to give SS the representation r=
r, set 8,
rf tan 8,
e=e,-
1+ r,h’(r,) tan 8,’
1 + r,h’( r,) tan
8,
+ Wr,),
(738)
z = f r,{r: - 1 - r:hr2( r,)},‘2 (tan e,)/{ 1 + r, h’( r,) tan 0,).
(9)
Rays are still obtained by fixing r, , but 8, does not measure distance along them. 2.2.
QUADRATICALLY
SWEPT
EDGE
The above results will now be applied to the special case h( r,) = -$ar:, By equation (9), the condition for r, to produce a ray is
a2r:-rf+1S0,
where a L 0. (10)
i.e., _
+
r, S r, S r,,
(11)
where
try,r:) =
1 -(I
(1
_4a2)1/2
2ff2
l/2
I
’
1+(1-44aZ)“2 “2 2cX2 1 >*
1
(12)
Note that if (Y exceeds $, no rays are formed, while if it takes a value slightly below $ only a small length of edge is supersonic; as (Yapproaches zero, this length increases, and at zero the whole edge beyond r, = 1 produces rays. In what follows, the assumption OC a G; will be made. (If forward sweep were of interest, it would be useful to allow negative a.)
148
C. .I. CHAPMAN
Since the variables that 0s 8, s T and
R, r, r,, rz are non-negative, (tan f?,)/(l-curttan
it may be deduced 8,)sO.
from equation
(3)
(13)
The required domain of r, , 8,) drawn in Figure 1, is bounded by 8, = 0, cot-’ (ar:) and by any of the methods given in section 2.1, reveals that r, = r;, r:. A routine calculation, the cusp line has equation r~(2~2r~-1)tan26,+~r~(~2r~-1)tan8,+(r~-1-a2r~)=0, 7r/4). and extends from (r;, 0) to (l/A, Sections of SS by cylinders of fixed r correspond tan 13~={ar’r,
-(cy*r*r:+
(14)
to
r*- rf)“‘}/r(a*r*rf-
l),
(19
where r, must be varied between rl and min {r, r:}. The family of curves (15) is drawn in Figure 1 as P,P2P3, Q1Q2Q3, etc. Numerical computations at various I and CYshow that sections always take the form of Figure 2: the supersonic edge AB generates “arrowheads”, which shrink in size to a point as A is approached, and simultaneously flatten. The semi-angle at the front, namely the Mach angle, tends to 7r/2 as either A or B is approached. Beyond B, the arrowhead has detached itself from the edge which produced it, and become blunt-nosed; its extent in the z-direction increases without limit as r increases, whatever the length of AB. That part of SS swept out by PIP2 and QIQZ as r varies will be called the front sheet; P2P3 and Q2Q3 sweep out the rear sheet. Lines of constant r, in Figure 1 become rays on SS. The position of the cusp line on the edge will pass through a focus, implies that rays from between r ; and l/G
Figure 1. For a quadratically swept edge, the parameter region inside ABCD generates the portion of SS occupying z > 0, in such a way that the lettering corresponds to that of Figure 2. Thus one may imagine folding the region along APsQsE and stretching it in three-dimensional space so that it is located as follows: AP,B is attached to the supersonic part of the propeller edge; BQ,C is the leading edge of that part of SS detached from the propeller, with C at infinity; CED is the cylindrical section at infinity; DQ,P,A is the centreline of the rear sheet of SS. Note that A is the vertex of the cusped cone, and APsQsE the cusp line. The’ family of lines joining ABC to AD (such as P,P,P, and Q,Q2Q3) generates the cylindrical sections, obtained by fixing r.
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149
(a)
(b)
!cl
Figure 2. (a) A rotating blade, with a quadratically swept leading edge containing a supersonic segment AB; the lines AP,BQ, . . and APsQ, . . mark the intersection of the piane z = 0 with the front and rear sheets of the cone. (b) Cylindrical section of SS at a radius less than that of B, demonstrating the sharp edge at P, and cusp at P2. (c) Section at radius r greater than that of B; note the blunt nose of SS for such r. All lettering corresponds to that of Figure 1: thus the front and rear sheets of SS are generated by the regions inside ABCE and ADE respectively.
whereas those from between I/& and r: will not; i.e., the former cross the line APzQz - * a, while the latter remain always on the front sheet. 2.3.
STRAIGHT
EDGE
The choice (Y= 0 gives a straight radial edge [ 11, on which the points A, B‘of Figure 2(a) are at radii 1, co, respectively; every section of SS has the form shown in Figure 2(b). Ray paths are readily drawn in this case, and details will be presented. The variables (u, u) = (I-:, 6,) enable equations (4)-(6) to be written
150
C. J. CHAPMAN
I
0
B’
0
I
;p’ IS’
A',C' ,
I
Figure 3. Points A, B, in (a) correspond to A’, B’, in (b) under the mapping from II, o to r, z. Typical rays P’Q’R’ and S’T’U’ are shown; they move over from the front to the rear sheet on crossing the cusp line B’Q’T’D’.
It is convenient to project the rays cylindrically onto a meridional plane, by ignoring the &equation and using the other two for a mapping from (u, V) to (r, z). Rays are obtained by holding u constant and varying V; equivalently, u may be eliminated to give the equation of the ray for a particular u, namely (r2/u)-z2/u(u-l)=l.
(17)
Figure 3 shows that the domain u 2 1, u 3 0 is mapped in a “two-to-one” way onto the regionOaz~(r2-l)/2,andthefoldlinev2=u(u-1)senttothecusplinez=(r2-1)/2, as represented by BQTD-, B’Q’T’D’. The areas bounded by CBD and ABD map to the front and rear sheets of SS. Typical rays are indicated by the lines P’Q’R’and S’T’U’; if P corresponds to the value u, then P’ and Q’ have (r, z) co-ordinates (4, 0) and (m, u - 1). Neighbouring rays cross on the cusp line; the apparent intersection in Figure 3(b) is due only to the projection adopted: P’Q’and S’T’are situated on the front sheet of SS, while Q’R’ and T’U’ are on the rear. 3.
CURVATURE
AND
FOCUSING
The surface SS, and its propagation through space, lend themselves to analysis by the simpler parts of differential geometry [3,9]. If the principal radii of curvature at a point are R, , R2, then ray tube areas vary in proportion to (R, + s)( R2 + s), where s is the distance travelled along a ray. When the area passes through zero, focusing occurs, and
RAY
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151
PROPELLERS
geometrical acoustics ceases to give a valid approximation to the pressure field. Recall that motion along a straight ray in space is equivalent to remaining on SS but varying & in equations (4)-(6), or v in equation (16). It is therefore sufficient to know, as a function of position on SS, the value of the product R, R2, which is the reciprocal of the Gaussian curvature. The requisite geometrical theory is standard [9], and the following results are quoted to define the notation. For a surface represented parametrically by r(u, v), the square of the distance between points r( u, v) and r( u + du, v + do) will be written ds2=Edu2+2Fdudv+Gdv2,
(18)
where (E, F, G) = (lrU/*,rU* TV, bul*), subscripts denoting differentiation. magnitude
(19)
The vector ru x rv, parallel to the unit normal N, has
Ir, x rvl = (EG - F*)“* = H, so that N = (r, x r,)/H.
Hence the second fundamental Ldu2+2M
(20)
form,
du dv+ N dv*,
(21)
where (L, M N) = (N - ruur N - ruv, N - rJ, may be obtained by further differentiation. K=
LN-M* HZ
’
(22)
The Gaussian and mean curvatures, namely tJ=
EN+GL-2FM 2H2
(23)
’
the principal curvatures K,, K* through the relations K = K,K* and p = then R, , R2 are the reciprocals of K~, K*. For a general surface, the formulae lead to complicated expressions. But they simplify in the problem being considered here, because B2, or v, gives the distance s along a ray (as shown in section 2.1), and l/K varies quadratically with s, in accordance with the remarks at the beginning of this section. As an application of these results, the expressions will be evaluated for the straight edge specified in section 2.3. Equation (16) gives determine
(K, + ~*)/2;
(66 N=
(
G)=
u(u-l)+v* 4u2(u-1)
(u(u2+vL12)]l,29
T
2, u(u-l)+v* u
(24)
>’
(25)
(-g-J*,(y*),
V (L,MN)=
-4u2c;_l)’
-&,
-;),
(K,)=
-l
(
(
(R,,R,)=(-v+{u(u-l)}“*,-v-{u(u-l)}”*).
u(u-l)-v2’u(u-1)-v*
>’ W,27)
(28)
As expected, the principal radii of curvature are linear in v. Since u 2 1 and o > 0, the product RI R2 can vanish only when v = {u( u - l)}“‘, consistent with the location of the cusp line found earlier.
152
C.
J. CHAPMAN
4. CONCLUDING
REMARKS
The main use of ray analysis in propeller acoustics is that it leads directly to the weakly nonlinear theory based on Whitham’s F-function [3,10-131. In particular, it is possible to determine when this theory will not apply. Near SS, except close to the cusp lines, the pressure obtained from the linear equations s , w h ere A(s)represents the ray tube area at a distance s takes the form F(s - ct)/m along a ray. On the front sheet of the surface SS coming off a sharp edge, F has a simple discontinuity, while on the rear sheet it is logarithmically singular. If the ray crosses the cusp line at s = sO, where A(q)= 0, linear acoustics [lo] introduces a complicated correction to the singularity involving 1s - s01-“6. Whitham’s theory will give a non-linear pressure near the front sheet of SS, but will not eliminate the singularity at the cusp; indeed, it is known [lo] that if the formalism is carried through as a weak shock approaches sO, then a pressure variation proportional to (s - s,,) -I’* is obtained, violating the assumptions of the theory. What in fact happens near the cusp line is that transonic effects predominate, and permanently affect the pressure field on the rays passing through. Thus it seems likely that the pressure near the rear sheet cannot be calculated by any simple modification of the logarithmically singular F-function. These ideas are familiar from research into sonic boom [ 14,151 and the focusing of weak shocks [16-181. They may well provide a useful adjunct to the methods of propeller acoustics developed over the years by Farassat [19] and Hanson [20].
ACKNOWLEDGMENTS
Thanks are due to M. J. Fisher, G. M. Lilley and C. L. Morfey for many helpful comments while the work of this paper was carried out. Funding was provided by the Royal Aircraft Establishment, and benefit was gained from a visit by the author to NASA Lewis Research Center as a Case/NASA Research Fellow.
REFERENCES 1. C. J. CHAPMAN 1988 Journal of Fluid Mechanics 192, 1-16. Shocks and singularities in the pressure field of a supersonically rotating propeller. 2. G. M. LILLEY, R. WESTLEY, A. H. YATES and J. R. BUSING 1953 Journal of the Royal Aeronautical Society 57, 396-414. Some aspects of noise from supersonic aircraft. 3. G. B. WHITHAM 1956 Journal of Fluid Mechanics 1, 290-318. On the propagation of weak shock waves. 4. P. S. RAO 1956 Aeronautical Quarterly 7, 21-44, 135-155. Supersonic bangs, parts I and II. 5. F. WALKDEN 1958 Aeronautical Quarterly 9, 164-194. The shock pattern of a wing-body combination, far from the flight path. 6. C. H. E. WARREN 1968 in Noise and Acoustic Fatigue in Aeronautics (Eds E. J. Richards and D. J. Mead), 266-284. Sonic bangs. New York: John Wiley. 7. G. B. WHITHAM 1974 Linear and Nonlinear Waves. New York: John Wiley. 8. M. V. LOWSON and R. J. JUPE 1974 Journal of Sound and Vibration 37,475-489. Wave forms for a supersonic rotor. 9. T. J. WILLMORE 1959 An Introduction to Diflerential Geometry. Oxford: Clarendon Press. 10. J.-P. GUIRAUD 1965 Journal de Mkcanique 4,215-267. Acoustique geometrique, bruit balistique des avions supersoniques et focalisation. 11. D. L. HAWKINGS and M. V. LOWSON 1974 Journal of Sound and Vibration 36, l-20. Theory of open supersonic rotor noise. 12. C. K. W. TAM 1983 Journal of Sound and Vibration 89, 119-134. On linear acoustic solutions of high speed helicopter impulsive noise problems. 13. C. K. W. TAM and M. SALIKUDDIN 1986 Journal of Fluid Mechanics 164, 128-154. Weakly nonlinear acoustic and shock-wave theory of the noise of advanced high-speed turbopropellers. 14. I. R. SCHWARTZ (Ed.) 1971 NASA SP-255. Third Conference on Sonic Boom Research.
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15. H. H. HUBBARD, D. J. MAGLIERI and D. G. STEPHENS 1986 NASA TM 87685. Sonic-boom research-selected bibliography with annotation. 16. B. STURTEVANT and V. A. KULKARNY 1976 Journal of Fluid
17. 18. 19.
20.
Mechanics 73, 651-671. The focusing of weak shock waves. M. S. CRAMER and A. R. SEEBASS 1978 Journal of Fluid Mechanics 8?3,209-222. Focusing of weak shock waves at an a&e. F. OBERMEIER 1983 Journal of Fluid Mechanics 129, 123-136. On the propagation of weak and moderately strong, curved shock waves. F. FARASSAT 1984 AIAA/ NASA 9th Aeroacoustics Conference, Williamsburg, Virginia, AIAA Paper 84-2303. The unified aerodynamic and acoustic prediction theory of advanced propellers in the time domain. D. B. HANSON 1985 American Institute of Aeronautics and Astronautics Journal 23, 499-504. Near-field frequency-domain theory for propeller noise.