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The effect of forward flight on the diffraction radiation of a high speed jet
Journal of Sound and Vibration (1977) 50(2), 183-l 93
THE EFFECT
OF FORWARD RADIATION
FLIGHT
ON THE DIFFRACTION
OF A HIGH SPEED JET M. S. HOWE
Engineering Department, University of Cambri&e, Cambri&e CB2 IPZ, England (Received 9 July 1976) This paper describes a theoretical modelling of the effect of aircraft tlight on the diffractional generation of sound which occurs when shear layer turbulence convects at high speed past a trailing edge. This is relevant to the study of noise problems associated with blown flaps, powered lift and aerodynamic shielding devices. The analysis is conducted for a two-dimensional configuration at arbitrary subsonic flight velocity. It is concluded that in the absence of a Kutta condition at the trailing edge, the effect of flight results in a forward arc amplification of the diffraction radiation through a single Doppler factor on the linear acoustic pressure field. The forward arc lift in the field shape disappears when a Kutta condition is imposed. In all cases the intensity of the diffraction radiation at 90” to the flight path is diminished by forward motion of the aircraft.
1. INTRODUCTION
This paper discusses the effect of forward flight of an aircraft at arbitrary subsonic speed on the field shape of the diffraction radiation arising from the convection of turbulence past a trailing edge. The analysis is intended to model the situation depicted in Figure 1, in which a
Nozzle
Rigid
trailing
edge
uu
Figure 1. The practical configuration modelled by the analysis. The jet nozzle is fixed relative to the trailing edge. Roth are in motion at velocity U in the negative q-direction. U, is the nominal centreline velocity of the jet relative to the nozzle.
high speed jet exhausts from a nozzle which is fixed relative to a sharp trailing edge, such as that of an aircraft wing. It is particularly relevant to the study of noise generation by externally blown flaps [l], by analogous devices in which a jet is deflected onto the upper surface of a wing to produce additional lift [2], and in estimating the effect that forward flight has on the shielding of aerodynamic sound when an engine is located above a wing [3]. The present study contributes to the theory of the interaction of aerodynamic sources and moving bodies initiated by Crighton [4]. Crighton considered the generation of sound by source distributions fixed relative to a trailing edge, and demonstrated the possibility of an amplification of the diffracted sound which is radiated into the forward arc (i.e., in directions lying ahead of the aircraft at the time of emission of the sound) at low, subsonic forward flight velocities, as well as a variation with flight speed of the field radiated at right angles to the flight path. In the particular case of a line source aligned parallel to a trailing edge it was 183
184
M. S. HOWE
shown that, provided the source is located well within an acoustic wavelength of the edge, the effect of the forward motion is to amplify the linear pressure field by two powers of the Doppler factor l/[l - M,], M, being the Mach number of the velocity at which the aircraft is approaching the observer at the time of emission of the sound. The flow induced at the trailing edge by the source is singular unless the infinite velocity is alleviated by the shedding of vorticity to form a wake. The application of such a Kutta condition at the trailing edge had no appreciable effect on the radiated sound levels unless the source was located well within a hydrodynamic wavelength U/w of the edge, U being the flight speed and w the characteristic radian frequency. In this case the field was dominated by that generated by vortex shedding from the edge, and was amplified by a single Doppler factor (see also reference [5]). In this paper a model of the situation illustrated in Figure 1 is examined. The jet is assumed tb be two-dimensional, and the shear layer turbulence frozen during the interval in which a significant interaction occurs with the trailing edge. The analysis thus takes account only of the diffraction radiation [6] of the jet, and at low jet velocities the aerodynamic sources are therefore always located within a hydrodynamic wavelength of the edge. The principal conclusions of the investigation are : (a) the effect of source motion relative to the edge reduces the Doppler amplification of the diffraction field from the two powers obtained by Crighton for stationary sources to at most one power; (b) the application of a Kutta condition at the trailing edge eliminates completely the Doppler amplification in the forward arc; (c) at 90” to the flight path, the effect of forward motion is to reduce the intensity of the radiated sound. 2. SPECIFICATION OF THE MODEL PROBLEM We consider first the diffraction radiation generated when an ideal two-dimensional eddy in the form of a line vortex of strength r translates at a constant velocity U, relative to a semi-
h
vu 0 ------j,iJ
Woke
Figure 2. A line vortex of strength r translatesat a velocity U, parallel to the xl-axis and at a distance h from a rigid plate. The plate is semi-infinite and lies in x1 -z 0, x2 = 0, the origin of the co-ordinates being at the trailing edge 0. A m&n flow of velocity U exists on both sides of the plate, and vorticity is shed into the wake when a Kutta condition is imposed at the edge. infinite plate (see Figure 2) parallel to a mean flow of velocity U on both sides of the plate. The plate occupies x1 < 0, X, = 0, the origin of co-ordinates being at the edge, and the vortex is located at x1 = U,t, x2 = h at time t. The back-reaction of the plate on the eddy-which close to the edge would deflect it from a rectilinear path-is neglected, and this is equivalent (cf. [7]) to assuming that the modification of the basic flow near the plate is sufficiently small that conditions there and in its wake are governed by the linearized equations of motion. When a Kutta condition is imposed at the edge, so that the pressure and velocity remain finite and the disturbed flow leaves the edge tangentially, there is an unsteady distribution of vorticity shed into the wake and convected downstream at the mean flow velocity U. The linearized approximation requires that the vorticity be conlined to the region x2 = 0, x1 > 0,
EFFECT OF FORWARD FLIGHT
185
and will be represented by the singular distribution x(x1 - Uf)6(x,)l where x1 > 0 and 1is a unit vector directed out of the paper. The flow is assumed to be isentropic and dissipation is ignored. The sound generated as the vortex r passes the edge will be determined by means of the formulation of the acoustic analogy theory of aerodynamic sound discussed in detail by the author in reference [8], in which unsteady distributions of vorticity o drive the inhomogeneous wave equation
=div(aanv)-$g.(onv),
(1)
in which v denotes the fluid velocity, c the speed of sound, D/Dt the material derivative, and B is the stagnation enthalpy defined in the present case by B=
s5 +
iv”.
Here p and p represent, respectively, the pressure and density, and in irrotational regions of the flow B is equal to -@/at, C#J being the perturbation velocity potential. The wave operator on the left of equation (1) is implicitly dependent on the flow, a feature which results in the scattering and refraction of the sound. This, however, is additional to the very much stronger diffraction field which is expected to arise from the trailing edge interaction, and is ignored in the present discussion. Also, the second term on the right of equation (1) may be neglected since the vortexr has a constant velocity of translation USand conditions in the wake have been linearized. When these approximations are introduced equation (1) becomes
variation in the sound speed c now being neglected. The terms on the right of this equation account respectively for the contributions of the vortex r and the wake vorticity x(x, - Ur) to the “dipole” source div(o A v) of equation (1). A formulation analogous to equation (3) was used in reference [7] in connection with the analysis of a sequence of model problems involving vortex shedding in the asymptotic limit as the mean flow Mach number M = U/c is small. 3. THE DIFFRACI’ION RADIATION Equation (3) determines the stagnation enthalpy B in terms of r and the wake vorticity. In the vicinity of the upper and lower surfaces of the rigid plate B = -a&at, and it follows that the vanishing of the normal velocity on the plate implies that aB/ax, = 0 on x2 = 0, x1 < 0. Hence B may be calculated by first determining the Green function G(x, y, r, T), from the equation
($,+
u~)i-02]G=s(x-y)a(r-,),
(4)
where x = (x1,x2) and G vanishes for c < z and satisfies aG/ax, = 0 on the plate. IfJ(x, t)
186
M. S. HOWE
denotes the terms on the right of equation (3), B is then given by the convolution product
{‘3x,Y, t,MY,
B(x,1)=
4 d2y dz.
(5)
The functional form of G may be deduced from standard results given by Noble ([9], page 57) in the manner described, for example, in reference [5]. If the Fourier time transform 9(x, y, W) of G is introduced by means of
G(x,Y, f, 7) =
&
Y(x,
y,w) e-ico(r-r)
dw,
where the integration is along a path parallel to the real axis and lying above all the singularities of $J in the o-plane, one has
s1
m+1e
sgn(x2) -8n24=
m
dk
-m+ie
da
exp W
- KM) x1 - (a - KM) Y1+ ~(4 Ix2I +
WY211, c7j
(k-a)--
-a0
e being a small positive quantity which is subsequently allowed to tend to zero, and it being assumed that y2 2 0. Other quantities appearing in this result are defined as follows :
K=
d&z,
x, = y(k)=
;TtK)(K2
Y1=
&p
_ k2/1/2,
according as K2 - k2 p 0. These expressions may now be applied to determine the radiation field B,, say, due to the incident vortex and represented by the first term on the right of equation (3). First set
(10)
and consider the contribution Br(i2) from a single harmonic component: i.e., take
(11) in equation (5); the complete result is obtained by integration with respect to a. Thus one has
187
EFFECT OF FORWARD FLIGHT
Upon substituting for Q from equation (7), it follows easily that for x2 2 0
WQ = ; exp[iQ(z-r)]
(exp [i(xz + h) ~Udl -
+-$&p
K
+
k
w (-G- 4 ev IiIx2- h IYNJII+
exp[i{(k - KM) xl + ~2 y(k) + hy(k,) - WI dk, (13)
where, upon writin; $2for o in the first of equations (8),
iU, = US/c, and the path of integration lies below the pole at k = k,. When the result (13) is integrated over all Sz the contribution from the first term on the right-hand side gives the field due to a line vortex and its image in an infinite rigid plane x2 = 0, so that the corresponding x,-component of the velocity perturbation vanishes identically on x2 = 0. Accordingly this term makes no contribution to the radiated sound unless the vortex r translates at a supersonic velocity relative to the mean flow: i.e., il4, - M > 1. The mathematical expression of this statement is obtained by noting that, by equation (14),
y(k) =
(1%
according as M, - M ><1, so that the first term on the right of expression (13) decays exponentially with increasing x2 unless A4, - M > 1. The integral in equation (13) gives the contribution to the sound field from the interaction at the edge. It is convenient to consider the two cases MS - it4 ><1 separately. In the case of subsonic relative velocity the whole of the acoustic radiation is furnished by the integral and this case is examined first. At large distances from the edge of the plate the integral may be approximated by deforming the path of integration on to one of steepest descents [lo]. Define (R, 0) by
XI E
d&2=
RCOS8,
x2
=
R&e;
(16)
then the new path of integration in the k-plane is the usual one associated with half-plane diffraction problems : (l - Kcos e) (C cos 8 - K) (17) ‘= sine1/(KrcosQ2 + r2sin28’ where k = t + iv. The pole at k = k, gives an exponentially small contribution in the far field provided that MS - M < 1, and the leading contribution is from the stationary point located at k = KcosO, giving &(a)
r 2: -
4x&i?
sin (e/2)4(Kcos 8 - k,)
exp{i[KR(l - Mcos 0) + hy(kS) - at + x/4]}. (18)
The acoustic pressure in the far field is obtained by noting that
p__i:+44”-((l_1M2)-&4f$ PO
(19)
M. S. HOWE
188
where p. is the mean density, and use has been made of the asymptotic functional dependence of the field variables expressed by the argument of the exponential in equation (18). Recalling that B = -6’q5/&, one has p/p0 2: ((1 - Mcos 0)/(1 - M2)} B. (20) Now results (18) and (20) express the field in terms of a co-ordinate system fixed relative to the trailing edge: i.e., the field experienced by an observer travelling with the aircraft at velocity U. For an observer fixed relative to the ground, and therefore translating in the positive x,-direction at velocity U relative to the edge of the plate, the appropriate representation is in terms of polar co-ordinates (r, 0) of the observer at the time of emission of the sound. By inspection of Figure 3 the following transformation formulae are seen to apply: x* = r(M + cos O),
x,=rsin@.
(21) Upon using these together with equations (18) and (20) it follows that the diffraction radiationp,(SZ),,say, due to a single harmonic component of the line vortexf is given by Pm
_
PO
us
-r
J
47rfi
(1 + MS - A4)1’2sin (B/2)
n(1+McosQ)(1-[MS-M]cos6)X
hPl x exp -i[Q(t - r/c) - n/4] - u dl - (MS - M)’ . (22) s 1 ( The complete space-time representation may now be obtained by integrating over all values of Q. Before doing this and discussing the practical implications of equation (22), the additional contribution to the scattered sound which arises when the Kutta condition is imposed at the trailing edge and which is given by the second source term on the right of equation (3) will be determined.
‘1
Rigid
/(x
x2
plotc
Figure 3. Illustrating the relationship between the observer positizh (r, S) at the time of emission of the sound and the co-ordinates (xl,xs) fixed relative to the mean flow.
Introduce the representation
[’
x(x, - W = _,1 xo(Wexp 161($--l)]dR
(23)
of the wake vorticity distribution function, and consider the sound generated by the harmonic component of frequency 8. The corresponding contribution B,(Q), say, to the wake generated sound is given by the convolution product B,(Q)=
r G(~,Y,CZ)~ -C0
8Y2
UXo(S2)6(y2)H(y,)exp [I‘62(Us 2 --z )])dY+‘Y2d7.
Proceeding as before one finds that for x2 2 0
the path of integration passing below the pole at k = K/M.
(24)
EFFECT OF FORWARD
189
FLIGHT
The Fourier amplitude ~~(62)of the wake vorticity is determined by requiring the perturbed flow induced by the line vortex r and the wake vorticity, and given respectively by equations (13) and (25), to leave the edge of the plate tangentially. This implies that aB/ax, should vanish asx, ++Oonx,=O.Nowonx,=O
elhy(kd + +
&m~K(l
+ l/W
I
mexp {i[k - KM] X1}dk. (26) (k - KIM) As x1 + +0, the behaviour of the integral is determined by the asymptotic form of the nonexponential terms of the integrand as k + 03 (see, e.g., reference [lo]). It is clear from near the edge expression (26) that the latter is of order I/&, implying that aB/ax2 k: l/G unless Xc@) = &
,Jzi
eihyfk*).
(27)
With-this choice of x0(Q), aB/ax2 vanishes at the edge of the plate. One may also note that equations (26) and (27) imply further that when k, = K/M the normal derivative actually vanishes at.all points of the plane x2 = 0, in which case no diffraction radiation is generated by the shed vorticity and the line vortex f. This occurs when the line vortex convects at the mean flow velocity US= U, and extends the corresponding low Mach number result obtained in reference [7] to arbitrary subsonic values of U. Equations (25) and (27) may now be used to write down expressions for the far field sound generated by the wake, leading to an expression analogous to equation (22) for the corresponding pressure perturbation p,,,(Q), say. Explicit forms for the space-time dependence of the sound field are then obtained by integration with respect to 8. These integrals reduce to. standard forms and it is not necessary to detail their evaluation. The results of these calculations are conveniently presented separately in terms of (i) the diffraction radiation pr of the line vortex, i.e., the field in the absence of Kutta condition; (ii) the wake generated sound pW and (iii) the total diffraction radiation p =pr +p,,, when the Kutta condition is imposed. Thus one has (i) diffraction radiation, no Kutta condition: -l-u, PrlPo = -2*fi
(1 + MS - MY2
sin (O/2)
cos (WQ ; (28)
(1+Mc0s8)(1-[M,-M]c0s@)
2/%
(ii) wake generated sound: (29)
(iii) diffraction radiation, Kutta condition imposed: PIP0 =
w-
27cfi
UJ
(1 + MS - MY2
sin (Q/2) (1 -[M,--M]cos@)
(33s(W)
4G
’
(30)
In these formulae (&, 0,) are defined by Roe’% = Us(t - r/c) + ihfl
- (MS - M)‘,
(31)
and when (MS - M)2 4 1 represent the polar co-ordinates relative to the edge of the plate of the retarded position of the line vortex r.
M. S. HOWE
190
The three cases (i), (ii), (iii), all exhibit the characteristic sin(8/2) half-plane dependence on the polar angle 0. The motion of the aerodynamic sources relative to the plate eliminates one of the Doppler factors I/( 1 + Mcos 0) in cases(i), (ii) which Crighton [4] obtained for sources at relative rest. In particular in case (i) (no Kutta condition) the intensity of the radiation is alleviated further in the forward arc by the presence of an additional Doppler factor associated with the relative velocity between the eddy and the mean flow. In case (iii), the diffraction radiation when the Kutta condition is imposed, the forward arc lift is eliminated entirely apart from the weak relative velocity effect. Before considering the consequences of these predictions for the diffraction radiation produced by a high speed jet, one can quote the results analogous to (i)-(iii) which obtain when the line vortexr translates supersonically relative to the mean flow: i.e., for A4, - M > 1. The radiation B,(Q) from the incident line vortex given by equation (13) now has contributions from the first term on the right-hand side corresponding to Mach wave radiation in a direction making an angle GM = cos-‘{ l/(M, - M)} with the positive x,-axis. Similarly the pole of the integrand at k = k, gives a Mach wave contribution for radiation directions 0 < OM, and arises when (for K > 0) Kcostl > k,, this being the condition that the pole is crossed when the integration contour is deformed onto the steepest descents path (17). It
Figure 4. Illustrating the form of the Mach wave radiation and the edge-diffracted field when h4, - A4 > 1 and for two positions a, /3 of the line vortex r on its trajectory. The plate is shown at the time of emission of the diffraction radiation.
corresponds to the cut-off in the Mach wave reflected from the plate as the vortex passes the trailing edge (see Figure 4). Let the Mach wave radiation propagating into x2 > 0 be denoted by pM; then these considerations imply that PM _ WG - u
PO-
2
(H(O,-0)6[x,+(~,+h)z/(M,-M)~-l-U,t]-
- 6[x, + (X2- h)1/(M, - M)2 - 1 - us t I},
(32)
the first term in the curly brackets being the Mach wave A of Figure 4 reflected from the plate and the second term being that labelled B. The corresponding expressions for the diffraction fields pr, p,,, and p when M, - M > 1 are as follows : (i) diffraction radiation, no Kutta condition: -ru, Pr - = y---& (1 + ius - A4)“Z PO
sin (812) (1 + Mcos@)(l
-
[i&f,- M]cos@) x (33)
EFFECT OF FORWARD
191
FLIGHT
(ii) wake generated sound: PW - = PO
&$
(1 + M, - M)“2 (1 ?(@I)@)
x (34)
(iii) diflraction radiation, Kutta condition imposed: P -= PO
r(U2nfi
Us)
(l + Ms -
M)“2
sin (O/2) (1 - [MS - M]c(-Js 0) x
x W’)/m
(35)
where T = t - r/c + (h/U,)&4, - II~)~- 1. Thus the effect of supersonic motion of the line vortex r relative to the mean flow changes only the form of the diffracted wave profile from the rather gentle pulse of equations (28)-(30), characterized by the non-singular function of the retarded time COS(~~/~)/~~, to the sharp fronted, mildly singular waves described by the last factor in each of equations (33)-(35). The respective variations of the field shape (i.e., the dependence on 0) with forward flight are unchanged. The geometry of the singular wave fronts is illustrated in Figure 4 for two positions of the line vortex r along its trajectory past the trailing edge. The explicit forms given above in equations (32) (33) and (35) actually cease to be valid at the points marked C in the figure, where the Mach waves merge with the diffracted field along rays at angles 0 = fro, emanating from the retarded position of the trailing edge. 4. DIFFRACTION RADIATION OF A HIGH SPEED JET In order to apply the predictions of the above analysis to the more practical problem of Figure 1, the line vortex r must be identified with a typical eddy in the shear layer of the jet. Naturally such an eddy must have a finite diameter which is neglected in the model, and this will tend to alleviate the intense Mach wave and diffraction fields which arise in the analysis at supersonic relative velocities. This rather artificial feature of the results is clearly the only significant difference between corresponding components (i)-(iii) of the diffraction radiation at subsonic and supersonic relative velocities, the variations in the field shapes with forward flight velocity U being unchanged. In considering the relevance of these field shape variations it should also be noted that in general the characteristic eddy strength r and the eddy convection velocity U, change with the flight velocity U if the centreline jet velocity U,, say, relative to the plate is constant. A reasonable representation of these variations would not be expected to depart substantially from r = ;Z(U, - U) and Us = (U, + U)/2, where 1 is a length determined by the geometry of the jet nozzle. On this basis one can make the following general statements regarding the diffraction radiation of the high speed jet of Figure 1. Referring to equations (28)-(30), or (33)-(35), it is apparent that the effect of forward flight depends crucially on whether or not it is appropriate to apply a Kutta condition at the trailing edge. In particular when such a condition is imposed it is obvious that, since Jl4, must always exceed M, the intensity of the diffracted sound experiences no forward arc (0 > 90”) lift. It is questionable, of course, whether at the acoustic frequencies of interest conditions at the trailing edge can adjust rapidly enough to justify the application of a Kutta condition, and indeed it appears that this possibility must be specifically excluded if diffraction radiation
192
M. S. HOWE
is to account for the field shape variations in the forward arc in the manner proposed by Crighton [4]. Consider, therefore, the case of no Kutta condition. Now forward arc lift is provided by a single Doppler factor on the linear field (equations (28) and (33)). In addition the variation with Mach number of the coefficients of the diffracted field indicate that at constant frequency its amplitude decreases with increasing flight velocity U, which in turn implies that forward flight reduces the intensity of the sound radiated at right angles to the flight path. These statements apply strictly to the modelling of the two-dimensional jet of Figure 1. An estimate of the modification required for a jet of finite diameter may be made by formally multiplying the predictions for the sound pressure by a factor of the order of as/(1 + Mcos @)“2, noted by Crighton [4] to be necessary to account for the spherical spreading of the diffracted sound in the “flyover” plane. In this case it is interesting to note that equations (28) and (33) would predict an increase of about 4 dB in the sound pressure level at 0 = 140” for Mj = 2 and when the flight Mach number is increased from 0.25 to 0.5, a result consistent with flight measurements of the Rolls-Royce Olympus on a Vulcan flying test bed [ll], although the issue is complicated here because of the presence of shock-cell noise [ 121. In conclusion one can note that the present theory predicts a significant difference between the diffraction radiationp, occurring in the absence of vortex shedding at the trailing edge and p, that which arises when the Kutta condition is imposed. If d dB denotes the difference in the sound pressure levels the above discussion implies that A = SPL(p,) - SPL(p) N 2Olog,,{[(M,
+ M)/(M, - M)J/(l + McosO)}.
(36)
Thus at a flight Mach number M = 0.5 and for Mj = 1.5, A increases from 2.9 dB at 0 = 30” in the rear arc through 6.02 dB at 90” to 10.95 dB in the forward arc at 0 = 150”. When the jet Mach number Mj = 2 the corresponding values of A are 1*31,4&l and 9.37 dB, respectively. These figures indicate the need for a more thorough appreciation of the precise boundary condition which must be imposed at the trailing edge. It seems likely that in practice a condition intermediate between the “full-Kutta” and “no-Kutta” conditions must actually be applied, so that the expressions forp andp, would then represent, respectively, the lower and upper bounds for the intensity of the diffraction radiation.
ACKNOWLEDGMENT
This work documents a study undertaken as part of the Rolls-Royce (1971) Limited programme of research into high speed jet noise. REFERENCES 1. D. J. MCKINZIE and R. J. BURNS 1975 NASA TM X-3171. Analysis of noise produced by jet
impingement near the trailing-edge of a flat and a curved plate. 2. M. RESHATKO,J. H. GOODYKOONTZ and R. G. DORSCH1973 AIAA 6th FluidandPlasma Dynamics Conference, AIAA paper 73-61. Engine-over-the-wing noise research. 3. E. G. BROADBENT 1976 R.A.E. Technical Report 76002. Noise shielding by aircraft. 4. D. G. CRIGHTON1975 Journal of Fluid Mechanics 72, 209-227. Scattering and diffraction of
sound by moving bodies. 5. D. S. JONES1972 Journal of the Institute of Mathematics and Its Applications 9, 114-122. Aerodynamic sound due to a source near a half-plane. 6. H. LEVINE1975 Philips Research Reports 30, 240-276. Acoustical diffraction radiation. 7. M. S. HOWE1976 Journal of Fluid Mechanics 76,711-741. The influence of vortex shedding on the generation of sound by convected turbulence. 8. M. S. HOWE1975 Journal of Fluid Mechanics 71, 625-673. Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute.
EFFECT OF FORWARD
FLIGHT
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9. B. NOBLE 1958 Methods Based on the Wiener-Hopf Technique. Oxford: Pergamon. 10. D. S. JONES1964 The Theory of Electromagnetism. Oxford: Pergamon. 11. R. HOCH and R. HAWKINS 1973 AGARD Conference Proceedings No. 131, Paper 19. Recent studies into Concorde noise reduction. 12. M. HARPER-B• URNE and M. J. FISHER 1973 AGARD Conference Proceedings No. 131, Paper 11. The noise from shock waves in supersonic jets.
THE EFFECT
OF FORWARD RADIATION
FLIGHT
ON THE DIFFRACTION
OF A HIGH SPEED JET M. S. HOWE
Engineering Department, University of Cambri&e, Cambri&e CB2 IPZ, England (Received 9 July 1976) This paper describes a theoretical modelling of the effect of aircraft tlight on the diffractional generation of sound which occurs when shear layer turbulence convects at high speed past a trailing edge. This is relevant to the study of noise problems associated with blown flaps, powered lift and aerodynamic shielding devices. The analysis is conducted for a two-dimensional configuration at arbitrary subsonic flight velocity. It is concluded that in the absence of a Kutta condition at the trailing edge, the effect of flight results in a forward arc amplification of the diffraction radiation through a single Doppler factor on the linear acoustic pressure field. The forward arc lift in the field shape disappears when a Kutta condition is imposed. In all cases the intensity of the diffraction radiation at 90” to the flight path is diminished by forward motion of the aircraft.
1. INTRODUCTION
This paper discusses the effect of forward flight of an aircraft at arbitrary subsonic speed on the field shape of the diffraction radiation arising from the convection of turbulence past a trailing edge. The analysis is intended to model the situation depicted in Figure 1, in which a
Nozzle
Rigid
trailing
edge
uu
Figure 1. The practical configuration modelled by the analysis. The jet nozzle is fixed relative to the trailing edge. Roth are in motion at velocity U in the negative q-direction. U, is the nominal centreline velocity of the jet relative to the nozzle.
high speed jet exhausts from a nozzle which is fixed relative to a sharp trailing edge, such as that of an aircraft wing. It is particularly relevant to the study of noise generation by externally blown flaps [l], by analogous devices in which a jet is deflected onto the upper surface of a wing to produce additional lift [2], and in estimating the effect that forward flight has on the shielding of aerodynamic sound when an engine is located above a wing [3]. The present study contributes to the theory of the interaction of aerodynamic sources and moving bodies initiated by Crighton [4]. Crighton considered the generation of sound by source distributions fixed relative to a trailing edge, and demonstrated the possibility of an amplification of the diffracted sound which is radiated into the forward arc (i.e., in directions lying ahead of the aircraft at the time of emission of the sound) at low, subsonic forward flight velocities, as well as a variation with flight speed of the field radiated at right angles to the flight path. In the particular case of a line source aligned parallel to a trailing edge it was 183
184
M. S. HOWE
shown that, provided the source is located well within an acoustic wavelength of the edge, the effect of the forward motion is to amplify the linear pressure field by two powers of the Doppler factor l/[l - M,], M, being the Mach number of the velocity at which the aircraft is approaching the observer at the time of emission of the sound. The flow induced at the trailing edge by the source is singular unless the infinite velocity is alleviated by the shedding of vorticity to form a wake. The application of such a Kutta condition at the trailing edge had no appreciable effect on the radiated sound levels unless the source was located well within a hydrodynamic wavelength U/w of the edge, U being the flight speed and w the characteristic radian frequency. In this case the field was dominated by that generated by vortex shedding from the edge, and was amplified by a single Doppler factor (see also reference [5]). In this paper a model of the situation illustrated in Figure 1 is examined. The jet is assumed tb be two-dimensional, and the shear layer turbulence frozen during the interval in which a significant interaction occurs with the trailing edge. The analysis thus takes account only of the diffraction radiation [6] of the jet, and at low jet velocities the aerodynamic sources are therefore always located within a hydrodynamic wavelength of the edge. The principal conclusions of the investigation are : (a) the effect of source motion relative to the edge reduces the Doppler amplification of the diffraction field from the two powers obtained by Crighton for stationary sources to at most one power; (b) the application of a Kutta condition at the trailing edge eliminates completely the Doppler amplification in the forward arc; (c) at 90” to the flight path, the effect of forward motion is to reduce the intensity of the radiated sound. 2. SPECIFICATION OF THE MODEL PROBLEM We consider first the diffraction radiation generated when an ideal two-dimensional eddy in the form of a line vortex of strength r translates at a constant velocity U, relative to a semi-
h
vu 0 ------j,iJ
Woke
Figure 2. A line vortex of strength r translatesat a velocity U, parallel to the xl-axis and at a distance h from a rigid plate. The plate is semi-infinite and lies in x1 -z 0, x2 = 0, the origin of the co-ordinates being at the trailing edge 0. A m&n flow of velocity U exists on both sides of the plate, and vorticity is shed into the wake when a Kutta condition is imposed at the edge. infinite plate (see Figure 2) parallel to a mean flow of velocity U on both sides of the plate. The plate occupies x1 < 0, X, = 0, the origin of co-ordinates being at the edge, and the vortex is located at x1 = U,t, x2 = h at time t. The back-reaction of the plate on the eddy-which close to the edge would deflect it from a rectilinear path-is neglected, and this is equivalent (cf. [7]) to assuming that the modification of the basic flow near the plate is sufficiently small that conditions there and in its wake are governed by the linearized equations of motion. When a Kutta condition is imposed at the edge, so that the pressure and velocity remain finite and the disturbed flow leaves the edge tangentially, there is an unsteady distribution of vorticity shed into the wake and convected downstream at the mean flow velocity U. The linearized approximation requires that the vorticity be conlined to the region x2 = 0, x1 > 0,
EFFECT OF FORWARD FLIGHT
185
and will be represented by the singular distribution x(x1 - Uf)6(x,)l where x1 > 0 and 1is a unit vector directed out of the paper. The flow is assumed to be isentropic and dissipation is ignored. The sound generated as the vortex r passes the edge will be determined by means of the formulation of the acoustic analogy theory of aerodynamic sound discussed in detail by the author in reference [8], in which unsteady distributions of vorticity o drive the inhomogeneous wave equation
=div(aanv)-$g.(onv),
(1)
in which v denotes the fluid velocity, c the speed of sound, D/Dt the material derivative, and B is the stagnation enthalpy defined in the present case by B=
s5 +
iv”.
Here p and p represent, respectively, the pressure and density, and in irrotational regions of the flow B is equal to -@/at, C#J being the perturbation velocity potential. The wave operator on the left of equation (1) is implicitly dependent on the flow, a feature which results in the scattering and refraction of the sound. This, however, is additional to the very much stronger diffraction field which is expected to arise from the trailing edge interaction, and is ignored in the present discussion. Also, the second term on the right of equation (1) may be neglected since the vortexr has a constant velocity of translation USand conditions in the wake have been linearized. When these approximations are introduced equation (1) becomes
variation in the sound speed c now being neglected. The terms on the right of this equation account respectively for the contributions of the vortex r and the wake vorticity x(x, - Ur) to the “dipole” source div(o A v) of equation (1). A formulation analogous to equation (3) was used in reference [7] in connection with the analysis of a sequence of model problems involving vortex shedding in the asymptotic limit as the mean flow Mach number M = U/c is small. 3. THE DIFFRACI’ION RADIATION Equation (3) determines the stagnation enthalpy B in terms of r and the wake vorticity. In the vicinity of the upper and lower surfaces of the rigid plate B = -a&at, and it follows that the vanishing of the normal velocity on the plate implies that aB/ax, = 0 on x2 = 0, x1 < 0. Hence B may be calculated by first determining the Green function G(x, y, r, T), from the equation
($,+
u~)i-02]G=s(x-y)a(r-,),
(4)
where x = (x1,x2) and G vanishes for c < z and satisfies aG/ax, = 0 on the plate. IfJ(x, t)
186
M. S. HOWE
denotes the terms on the right of equation (3), B is then given by the convolution product
{‘3x,Y, t,MY,
B(x,1)=
4 d2y dz.
(5)
The functional form of G may be deduced from standard results given by Noble ([9], page 57) in the manner described, for example, in reference [5]. If the Fourier time transform 9(x, y, W) of G is introduced by means of
G(x,Y, f, 7) =
&
Y(x,
y,w) e-ico(r-r)
dw,
where the integration is along a path parallel to the real axis and lying above all the singularities of $J in the o-plane, one has
s1
m+1e
sgn(x2) -8n24=
m
dk
-m+ie
da
exp W
- KM) x1 - (a - KM) Y1+ ~(4 Ix2I +
WY211, c7j
(k-a)--
-a0
e being a small positive quantity which is subsequently allowed to tend to zero, and it being assumed that y2 2 0. Other quantities appearing in this result are defined as follows :
K=
d&z,
x, = y(k)=
;TtK)(K2
Y1=
&p
_ k2/1/2,
according as K2 - k2 p 0. These expressions may now be applied to determine the radiation field B,, say, due to the incident vortex and represented by the first term on the right of equation (3). First set
(10)
and consider the contribution Br(i2) from a single harmonic component: i.e., take
(11) in equation (5); the complete result is obtained by integration with respect to a. Thus one has
187
EFFECT OF FORWARD FLIGHT
Upon substituting for Q from equation (7), it follows easily that for x2 2 0
WQ = ; exp[iQ(z-r)]
(exp [i(xz + h) ~Udl -
+-$&p
K
+
k
w (-G- 4 ev IiIx2- h IYNJII+
exp[i{(k - KM) xl + ~2 y(k) + hy(k,) - WI dk, (13)
where, upon writin; $2for o in the first of equations (8),
iU, = US/c, and the path of integration lies below the pole at k = k,. When the result (13) is integrated over all Sz the contribution from the first term on the right-hand side gives the field due to a line vortex and its image in an infinite rigid plane x2 = 0, so that the corresponding x,-component of the velocity perturbation vanishes identically on x2 = 0. Accordingly this term makes no contribution to the radiated sound unless the vortex r translates at a supersonic velocity relative to the mean flow: i.e., il4, - M > 1. The mathematical expression of this statement is obtained by noting that, by equation (14),
y(k) =
(1%
according as M, - M ><1, so that the first term on the right of expression (13) decays exponentially with increasing x2 unless A4, - M > 1. The integral in equation (13) gives the contribution to the sound field from the interaction at the edge. It is convenient to consider the two cases MS - it4 ><1 separately. In the case of subsonic relative velocity the whole of the acoustic radiation is furnished by the integral and this case is examined first. At large distances from the edge of the plate the integral may be approximated by deforming the path of integration on to one of steepest descents [lo]. Define (R, 0) by
XI E
d&2=
RCOS8,
x2
=
R&e;
(16)
then the new path of integration in the k-plane is the usual one associated with half-plane diffraction problems : (l - Kcos e) (C cos 8 - K) (17) ‘= sine1/(KrcosQ2 + r2sin28’ where k = t + iv. The pole at k = k, gives an exponentially small contribution in the far field provided that MS - M < 1, and the leading contribution is from the stationary point located at k = KcosO, giving &(a)
r 2: -
4x&i?
sin (e/2)4(Kcos 8 - k,)
exp{i[KR(l - Mcos 0) + hy(kS) - at + x/4]}. (18)
The acoustic pressure in the far field is obtained by noting that
p__i:+44”-((l_1M2)-&4f$ PO
(19)
M. S. HOWE
188
where p. is the mean density, and use has been made of the asymptotic functional dependence of the field variables expressed by the argument of the exponential in equation (18). Recalling that B = -6’q5/&, one has p/p0 2: ((1 - Mcos 0)/(1 - M2)} B. (20) Now results (18) and (20) express the field in terms of a co-ordinate system fixed relative to the trailing edge: i.e., the field experienced by an observer travelling with the aircraft at velocity U. For an observer fixed relative to the ground, and therefore translating in the positive x,-direction at velocity U relative to the edge of the plate, the appropriate representation is in terms of polar co-ordinates (r, 0) of the observer at the time of emission of the sound. By inspection of Figure 3 the following transformation formulae are seen to apply: x* = r(M + cos O),
x,=rsin@.
(21) Upon using these together with equations (18) and (20) it follows that the diffraction radiationp,(SZ),,say, due to a single harmonic component of the line vortexf is given by Pm
_
PO
us
-r
J
47rfi
(1 + MS - A4)1’2sin (B/2)
n(1+McosQ)(1-[MS-M]cos6)X
hPl x exp -i[Q(t - r/c) - n/4] - u dl - (MS - M)’ . (22) s 1 ( The complete space-time representation may now be obtained by integrating over all values of Q. Before doing this and discussing the practical implications of equation (22), the additional contribution to the scattered sound which arises when the Kutta condition is imposed at the trailing edge and which is given by the second source term on the right of equation (3) will be determined.
‘1
Rigid
/(x
x2
plotc
Figure 3. Illustrating the relationship between the observer positizh (r, S) at the time of emission of the sound and the co-ordinates (xl,xs) fixed relative to the mean flow.
Introduce the representation
[’
x(x, - W = _,1 xo(Wexp 161($--l)]dR
(23)
of the wake vorticity distribution function, and consider the sound generated by the harmonic component of frequency 8. The corresponding contribution B,(Q), say, to the wake generated sound is given by the convolution product B,(Q)=
r G(~,Y,CZ)~ -C0
8Y2
UXo(S2)6(y2)H(y,)exp [I‘62(Us 2 --z )])dY+‘Y2d7.
Proceeding as before one finds that for x2 2 0
the path of integration passing below the pole at k = K/M.
(24)
EFFECT OF FORWARD
189
FLIGHT
The Fourier amplitude ~~(62)of the wake vorticity is determined by requiring the perturbed flow induced by the line vortex r and the wake vorticity, and given respectively by equations (13) and (25), to leave the edge of the plate tangentially. This implies that aB/ax, should vanish asx, ++Oonx,=O.Nowonx,=O
elhy(kd + +
&m~K(l
+ l/W
I
mexp {i[k - KM] X1}dk. (26) (k - KIM) As x1 + +0, the behaviour of the integral is determined by the asymptotic form of the nonexponential terms of the integrand as k + 03 (see, e.g., reference [lo]). It is clear from near the edge expression (26) that the latter is of order I/&, implying that aB/ax2 k: l/G unless Xc@) = &
,Jzi
eihyfk*).
(27)
With-this choice of x0(Q), aB/ax2 vanishes at the edge of the plate. One may also note that equations (26) and (27) imply further that when k, = K/M the normal derivative actually vanishes at.all points of the plane x2 = 0, in which case no diffraction radiation is generated by the shed vorticity and the line vortex f. This occurs when the line vortex convects at the mean flow velocity US= U, and extends the corresponding low Mach number result obtained in reference [7] to arbitrary subsonic values of U. Equations (25) and (27) may now be used to write down expressions for the far field sound generated by the wake, leading to an expression analogous to equation (22) for the corresponding pressure perturbation p,,,(Q), say. Explicit forms for the space-time dependence of the sound field are then obtained by integration with respect to 8. These integrals reduce to. standard forms and it is not necessary to detail their evaluation. The results of these calculations are conveniently presented separately in terms of (i) the diffraction radiation pr of the line vortex, i.e., the field in the absence of Kutta condition; (ii) the wake generated sound pW and (iii) the total diffraction radiation p =pr +p,,, when the Kutta condition is imposed. Thus one has (i) diffraction radiation, no Kutta condition: -l-u, PrlPo = -2*fi
(1 + MS - MY2
sin (O/2)
cos (WQ ; (28)
(1+Mc0s8)(1-[M,-M]c0s@)
2/%
(ii) wake generated sound: (29)
(iii) diffraction radiation, Kutta condition imposed: PIP0 =
w-
27cfi
UJ
(1 + MS - MY2
sin (Q/2) (1 -[M,--M]cos@)
(33s(W)
4G
’
(30)
In these formulae (&, 0,) are defined by Roe’% = Us(t - r/c) + ihfl
- (MS - M)‘,
(31)
and when (MS - M)2 4 1 represent the polar co-ordinates relative to the edge of the plate of the retarded position of the line vortex r.
M. S. HOWE
190
The three cases (i), (ii), (iii), all exhibit the characteristic sin(8/2) half-plane dependence on the polar angle 0. The motion of the aerodynamic sources relative to the plate eliminates one of the Doppler factors I/( 1 + Mcos 0) in cases(i), (ii) which Crighton [4] obtained for sources at relative rest. In particular in case (i) (no Kutta condition) the intensity of the radiation is alleviated further in the forward arc by the presence of an additional Doppler factor associated with the relative velocity between the eddy and the mean flow. In case (iii), the diffraction radiation when the Kutta condition is imposed, the forward arc lift is eliminated entirely apart from the weak relative velocity effect. Before considering the consequences of these predictions for the diffraction radiation produced by a high speed jet, one can quote the results analogous to (i)-(iii) which obtain when the line vortexr translates supersonically relative to the mean flow: i.e., for A4, - M > 1. The radiation B,(Q) from the incident line vortex given by equation (13) now has contributions from the first term on the right-hand side corresponding to Mach wave radiation in a direction making an angle GM = cos-‘{ l/(M, - M)} with the positive x,-axis. Similarly the pole of the integrand at k = k, gives a Mach wave contribution for radiation directions 0 < OM, and arises when (for K > 0) Kcostl > k,, this being the condition that the pole is crossed when the integration contour is deformed onto the steepest descents path (17). It
Figure 4. Illustrating the form of the Mach wave radiation and the edge-diffracted field when h4, - A4 > 1 and for two positions a, /3 of the line vortex r on its trajectory. The plate is shown at the time of emission of the diffraction radiation.
corresponds to the cut-off in the Mach wave reflected from the plate as the vortex passes the trailing edge (see Figure 4). Let the Mach wave radiation propagating into x2 > 0 be denoted by pM; then these considerations imply that PM _ WG - u
PO-
2
(H(O,-0)6[x,+(~,+h)z/(M,-M)~-l-U,t]-
- 6[x, + (X2- h)1/(M, - M)2 - 1 - us t I},
(32)
the first term in the curly brackets being the Mach wave A of Figure 4 reflected from the plate and the second term being that labelled B. The corresponding expressions for the diffraction fields pr, p,,, and p when M, - M > 1 are as follows : (i) diffraction radiation, no Kutta condition: -ru, Pr - = y---& (1 + ius - A4)“Z PO
sin (812) (1 + Mcos@)(l
-
[i&f,- M]cos@) x (33)
EFFECT OF FORWARD
191
FLIGHT
(ii) wake generated sound: PW - = PO
&$
(1 + M, - M)“2 (1 ?(@I)@)
x (34)
(iii) diflraction radiation, Kutta condition imposed: P -= PO
r(U2nfi
Us)
(l + Ms -
M)“2
sin (O/2) (1 - [MS - M]c(-Js 0) x
x W’)/m
(35)
where T = t - r/c + (h/U,)&4, - II~)~- 1. Thus the effect of supersonic motion of the line vortex r relative to the mean flow changes only the form of the diffracted wave profile from the rather gentle pulse of equations (28)-(30), characterized by the non-singular function of the retarded time COS(~~/~)/~~, to the sharp fronted, mildly singular waves described by the last factor in each of equations (33)-(35). The respective variations of the field shape (i.e., the dependence on 0) with forward flight are unchanged. The geometry of the singular wave fronts is illustrated in Figure 4 for two positions of the line vortex r along its trajectory past the trailing edge. The explicit forms given above in equations (32) (33) and (35) actually cease to be valid at the points marked C in the figure, where the Mach waves merge with the diffracted field along rays at angles 0 = fro, emanating from the retarded position of the trailing edge. 4. DIFFRACTION RADIATION OF A HIGH SPEED JET In order to apply the predictions of the above analysis to the more practical problem of Figure 1, the line vortex r must be identified with a typical eddy in the shear layer of the jet. Naturally such an eddy must have a finite diameter which is neglected in the model, and this will tend to alleviate the intense Mach wave and diffraction fields which arise in the analysis at supersonic relative velocities. This rather artificial feature of the results is clearly the only significant difference between corresponding components (i)-(iii) of the diffraction radiation at subsonic and supersonic relative velocities, the variations in the field shapes with forward flight velocity U being unchanged. In considering the relevance of these field shape variations it should also be noted that in general the characteristic eddy strength r and the eddy convection velocity U, change with the flight velocity U if the centreline jet velocity U,, say, relative to the plate is constant. A reasonable representation of these variations would not be expected to depart substantially from r = ;Z(U, - U) and Us = (U, + U)/2, where 1 is a length determined by the geometry of the jet nozzle. On this basis one can make the following general statements regarding the diffraction radiation of the high speed jet of Figure 1. Referring to equations (28)-(30), or (33)-(35), it is apparent that the effect of forward flight depends crucially on whether or not it is appropriate to apply a Kutta condition at the trailing edge. In particular when such a condition is imposed it is obvious that, since Jl4, must always exceed M, the intensity of the diffracted sound experiences no forward arc (0 > 90”) lift. It is questionable, of course, whether at the acoustic frequencies of interest conditions at the trailing edge can adjust rapidly enough to justify the application of a Kutta condition, and indeed it appears that this possibility must be specifically excluded if diffraction radiation
192
M. S. HOWE
is to account for the field shape variations in the forward arc in the manner proposed by Crighton [4]. Consider, therefore, the case of no Kutta condition. Now forward arc lift is provided by a single Doppler factor on the linear field (equations (28) and (33)). In addition the variation with Mach number of the coefficients of the diffracted field indicate that at constant frequency its amplitude decreases with increasing flight velocity U, which in turn implies that forward flight reduces the intensity of the sound radiated at right angles to the flight path. These statements apply strictly to the modelling of the two-dimensional jet of Figure 1. An estimate of the modification required for a jet of finite diameter may be made by formally multiplying the predictions for the sound pressure by a factor of the order of as/(1 + Mcos @)“2, noted by Crighton [4] to be necessary to account for the spherical spreading of the diffracted sound in the “flyover” plane. In this case it is interesting to note that equations (28) and (33) would predict an increase of about 4 dB in the sound pressure level at 0 = 140” for Mj = 2 and when the flight Mach number is increased from 0.25 to 0.5, a result consistent with flight measurements of the Rolls-Royce Olympus on a Vulcan flying test bed [ll], although the issue is complicated here because of the presence of shock-cell noise [ 121. In conclusion one can note that the present theory predicts a significant difference between the diffraction radiationp, occurring in the absence of vortex shedding at the trailing edge and p, that which arises when the Kutta condition is imposed. If d dB denotes the difference in the sound pressure levels the above discussion implies that A = SPL(p,) - SPL(p) N 2Olog,,{[(M,
+ M)/(M, - M)J/(l + McosO)}.
(36)
Thus at a flight Mach number M = 0.5 and for Mj = 1.5, A increases from 2.9 dB at 0 = 30” in the rear arc through 6.02 dB at 90” to 10.95 dB in the forward arc at 0 = 150”. When the jet Mach number Mj = 2 the corresponding values of A are 1*31,4&l and 9.37 dB, respectively. These figures indicate the need for a more thorough appreciation of the precise boundary condition which must be imposed at the trailing edge. It seems likely that in practice a condition intermediate between the “full-Kutta” and “no-Kutta” conditions must actually be applied, so that the expressions forp andp, would then represent, respectively, the lower and upper bounds for the intensity of the diffraction radiation.
ACKNOWLEDGMENT
This work documents a study undertaken as part of the Rolls-Royce (1971) Limited programme of research into high speed jet noise. REFERENCES 1. D. J. MCKINZIE and R. J. BURNS 1975 NASA TM X-3171. Analysis of noise produced by jet
impingement near the trailing-edge of a flat and a curved plate. 2. M. RESHATKO,J. H. GOODYKOONTZ and R. G. DORSCH1973 AIAA 6th FluidandPlasma Dynamics Conference, AIAA paper 73-61. Engine-over-the-wing noise research. 3. E. G. BROADBENT 1976 R.A.E. Technical Report 76002. Noise shielding by aircraft. 4. D. G. CRIGHTON1975 Journal of Fluid Mechanics 72, 209-227. Scattering and diffraction of
sound by moving bodies. 5. D. S. JONES1972 Journal of the Institute of Mathematics and Its Applications 9, 114-122. Aerodynamic sound due to a source near a half-plane. 6. H. LEVINE1975 Philips Research Reports 30, 240-276. Acoustical diffraction radiation. 7. M. S. HOWE1976 Journal of Fluid Mechanics 76,711-741. The influence of vortex shedding on the generation of sound by convected turbulence. 8. M. S. HOWE1975 Journal of Fluid Mechanics 71, 625-673. Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute.
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FLIGHT
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9. B. NOBLE 1958 Methods Based on the Wiener-Hopf Technique. Oxford: Pergamon. 10. D. S. JONES1964 The Theory of Electromagnetism. Oxford: Pergamon. 11. R. HOCH and R. HAWKINS 1973 AGARD Conference Proceedings No. 131, Paper 19. Recent studies into Concorde noise reduction. 12. M. HARPER-B• URNE and M. J. FISHER 1973 AGARD Conference Proceedings No. 131, Paper 11. The noise from shock waves in supersonic jets.