# Compressible Flow Behind a Wing

Compressible Flow Behind a Wing 160 AIRCRAFT ENGINEERING June, 1942 By D. I. Husk H E changes in the relationship between Tha t compressibility may have an important Let influence on stability can be seen from the the downwash behind a wing and the M = Mach Number corresponding to flow following. The pitching moment coefficient Twing incidence in passing from flow at abou t aerofoil. for the whole aircraft may be written :— low values of Mach Number to flow at high CLW = lift coefficient of wing values of Mach Number came to the notice of a = dC/, slope of lift curve of wing corres- the author when investigating some high speed da ponding to low speed flow stressing cases. If 7] (elevator angle) be regarded as constant, a = wing incidence measured from the we have, on differentiating with respect to C/_, While the compressible flow round an aerofoil no-lift line or streamline body has been discussed by •n, = tail setting measured relative to the various authors, the effect of compressible flow no-lift line of th e wing on the flow behind an aerofoil or body appears TJ = elevator angle th e static stability is neutral and to have passed unnoticed. a = dCx , corresponding to low speed flow t ( The basic assumption underlying the follow­ da, so h=h , the rear neutral point. ing work is tha t the downwash behind a wing is V = tail volume dependent only upon the lift coefficient of the Therefore h = C.G. position on A.M.C. wing which produces it and is independent of h = Aerodynamic Centre on A.M.C. Now suppose the aircraft is flying at a Mach the incidence. An elementary proof of the C,„= pitching moment coefficient of aircraft. above has been given before and is reproduced number M, then. The suffix M on any coefficient indicates flow below for emphasis. a t Mach Number M. Coefficients without this Let suffix correspond to low speed flow. V = true airspeed Using the Glauert-Prandtl approximation we If h„ be the neutral point of the aircraft A = cross section area, normal to the flight have :— when M is small we have, remembering that path, of that part of the atmosphere affected by the wing v = downwards velocity imparted to the air and also in passing through A Then since th e downwards momentum/sec. is equal to the upload on the wing, therefore e oc C/, Data on the variation of the position of the aerodynamic centre with Mach Number appears The downwash at any point will vary with to be scanty, but since all tests up to date have the position of that point relative to the wing, been conducted on small chord aerofoils, such but clearly this variation will not invalidate the dat a as is available is probably not very above proof. reliable. I t is usual to consider the lift on a wing as the resultant force due to the airflow about the However th e change in stabilit y due to change wing. However, we may consider the system in may be investigated. In equation (10) in the inverse way ; i.e. imagine the wing exerts forces equal to the resultant lift on the air as referring only to that part which includes the it flows about the wing. It is then apparent term we see that all the coefficients on the tha t any variation in the manner in which these forces are applied to the air will produce but right hand side are essentially positive. But small alteration in the configuration of the flow since M is always less than unity then AA„ is outside the immediate neighbourhood of the always negative. Thus the change in neutral aerofoil profile. Since the compressible flow point due to the influence of compressible flow is confined to the immediate neighbourhood of on the downwash characteristics is de-stabilis­ th e aerofoil, it may be argued that, whether ing. An idea of the magnitude of the shift th e flow is compressible or not, provided the forward may be obtained by substituting resultant lift coefficient is unaltered, then the In the expression for a' given by (3a) and average values for the coefficients. value of the downwash in the neighbourhood (3b) one would not expect tha t any error in the of the tail will be unchanged. (This is analo­ calculation of a would occur since it is well gous to St. Vcnant's Principle in the Theory of known that the slope of the lift curve is in­ Elasticity). creased by compressibility. However, it is More briefly, we may state :—that the down- probable that the factor associated wash behind a wing is a function of the lift coefficient of that wing and is independent of with would be overlooked, and so the prob­ th e type of flow about the aerofoil profile. Now, generally speaking, CL and e are not able error in computing the angle of attack of This is a shift of some magnitude and unless bot h equal to zero at a given balue of a. this effect is mitigated by a progressive shift th e tail is given by Present wing theory indicates that the down- aft of the aerodynamic centre as M increases, it wash across the span is constant only if the would appear that high speed aircraft will wing plan form is elliptical. Since the tail span need a relatively larger tail volume than low is bu t a fraction of the wing span, and also be­ In the full load C.P. A. cases a,„ may be of the speed aircraft. cause of fuselage and nacelle interference order about -5, and so if M =8 the Furthe r to the above remarks on static effects, one would not expect to find the effec­ stability, it has been shown theoretically (and tive downwash at the tail plane zero when the error is approximately realized practically) that any increase in opera­ wing is in the no-lift attitude. This expecta­ tional altitude has a deteriorating- effect on tion is generally fulfilled, and if a is measured dynamic stability. Since dynamic stability from the no-lift line, the downwash-incidence is largely dependent on the degree of static Whilst this error will generally make but little relationship, deduced from tunnel tests, may stability, and since the tendency is for aircraft difference to the tail load, since ther e is a corres­ be expressed. to develop high speeds at high altitude, it ponding error in the value of n (elevator angle), e = A + Da (1) appears tha t considerable care an d thought must th e C.P. of the tail load will, in general, shift Before proceeding further it is convenient to be given to problems of stability and control forward and so increase the torsion on the introduce a list of symbols. arising on high speed high altitude air-craft. structure. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

# Compressible Flow Behind a Wing

, Volume 14 (6): 1 – Jun 1, 1942
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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb030911
Publisher site
See Article on Publisher Site

### Abstract

160 AIRCRAFT ENGINEERING June, 1942 By D. I. Husk H E changes in the relationship between Tha t compressibility may have an important Let influence on stability can be seen from the the downwash behind a wing and the M = Mach Number corresponding to flow following. The pitching moment coefficient Twing incidence in passing from flow at abou t aerofoil. for the whole aircraft may be written :— low values of Mach Number to flow at high CLW = lift coefficient of wing values of Mach Number came to the notice of a = dC/, slope of lift curve of wing corres- the author when investigating some high speed da ponding to low speed flow stressing cases. If 7] (elevator angle) be regarded as constant, a = wing incidence measured from the we have, on differentiating with respect to C/_, While the compressible flow round an aerofoil no-lift line or streamline body has been discussed by •n, = tail setting measured relative to the various authors, the effect of compressible flow no-lift line of th e wing on the flow behind an aerofoil or body appears TJ = elevator angle th e static stability is neutral and to have passed unnoticed. a = dCx , corresponding to low speed flow t ( The basic assumption underlying the follow­ da, so h=h , the rear neutral point. ing work is tha t the downwash behind a wing is V = tail volume dependent only upon the lift coefficient of the Therefore h = C.G. position on A.M.C. wing which produces it and is independent of h = Aerodynamic Centre on A.M.C. Now suppose the aircraft is flying at a Mach the incidence. An elementary proof of the C,„= pitching moment coefficient of aircraft. above has been given before and is reproduced number M, then. The suffix M on any coefficient indicates flow below for emphasis. a t Mach Number M. Coefficients without this Let suffix correspond to low speed flow. V = true airspeed Using the Glauert-Prandtl approximation we If h„ be the neutral point of the aircraft A = cross section area, normal to the flight have :— when M is small we have, remembering that path, of that part of the atmosphere affected by the wing v = downwards velocity imparted to the air and also in passing through A Then since th e downwards momentum/sec. is equal to the upload on the wing, therefore e oc C/, Data on the variation of the position of the aerodynamic centre with Mach Number appears The downwash at any point will vary with to be scanty, but since all tests up to date have the position of that point relative to the wing, been conducted on small chord aerofoils, such but clearly this variation will not invalidate the dat a as is available is probably not very above proof. reliable. I t is usual to consider the lift on a wing as the resultant force due to the airflow about the However th e change in stabilit y due to change wing. However, we may consider the system in may be investigated. In equation (10) in the inverse way ; i.e. imagine the wing exerts forces equal to the resultant lift on the air as referring only to that part which includes the it flows about the wing. It is then apparent term we see that all the coefficients on the tha t any variation in the manner in which these forces are applied to the air will produce but right hand side are essentially positive. But small alteration in the configuration of the flow since M is always less than unity then AA„ is outside the immediate neighbourhood of the always negative. Thus the change in neutral aerofoil profile. Since the compressible flow point due to the influence of compressible flow is confined to the immediate neighbourhood of on the downwash characteristics is de-stabilis­ th e aerofoil, it may be argued that, whether ing. An idea of the magnitude of the shift th e flow is compressible or not, provided the forward may be obtained by substituting resultant lift coefficient is unaltered, then the In the expression for a' given by (3a) and average values for the coefficients. value of the downwash in the neighbourhood (3b) one would not expect tha t any error in the of the tail will be unchanged. (This is analo­ calculation of a would occur since it is well gous to St. Vcnant's Principle in the Theory of known that the slope of the lift curve is in­ Elasticity). creased by compressibility. However, it is More briefly, we may state :—that the down- probable that the factor associated wash behind a wing is a function of the lift coefficient of that wing and is independent of with would be overlooked, and so the prob­ th e type of flow about the aerofoil profile. Now, generally speaking, CL and e are not able error in computing the angle of attack of This is a shift of some magnitude and unless bot h equal to zero at a given balue of a. this effect is mitigated by a progressive shift th e tail is given by Present wing theory indicates that the down- aft of the aerodynamic centre as M increases, it wash across the span is constant only if the would appear that high speed aircraft will wing plan form is elliptical. Since the tail span need a relatively larger tail volume than low is bu t a fraction of the wing span, and also be­ In the full load C.P. A. cases a,„ may be of the speed aircraft. cause of fuselage and nacelle interference order about -5, and so if M =8 the Furthe r to the above remarks on static effects, one would not expect to find the effec­ stability, it has been shown theoretically (and tive downwash at the tail plane zero when the error is approximately realized practically) that any increase in opera­ wing is in the no-lift attitude. This expecta­ tional altitude has a deteriorating- effect on tion is generally fulfilled, and if a is measured dynamic stability. Since dynamic stability from the no-lift line, the downwash-incidence is largely dependent on the degree of static Whilst this error will generally make but little relationship, deduced from tunnel tests, may stability, and since the tendency is for aircraft difference to the tail load, since ther e is a corres­ be expressed. to develop high speeds at high altitude, it ponding error in the value of n (elevator angle), e = A + Da (1) appears tha t considerable care an d thought must th e C.P. of the tail load will, in general, shift Before proceeding further it is convenient to be given to problems of stability and control forward and so increase the torsion on the introduce a list of symbols. arising on high speed high altitude air-craft. structure.

### Journal

Aircraft Engineering and Aerospace TechnologyEmerald Publishing

Published: Jun 1, 1942