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ConstantVelocity Aerofoils

ConstantVelocity Aerofoils December , 1939 AIRCRAF T ENGINEERING Theoretically-derive d Profiles of Thin Sections Suitabl e for High Speeds B y J. Lockwood Taylor, D.SC. OR aerofoils intended for use at high a t a point on the back is equal to the defect for the first few coefficients can be found by speeds, it is evident that the maximum of velocity at the corresponding point on the successive approximation, and these suggest lift on the back, consistent with a given face, which is of course the same point in for the true values a fairly simple expression, maximum intensity of suction, will be attained space, since the thickness is ignored. This which proves to be correct since it gives sum- when the suction, and hence the velocity, over excess velocity, which is to be made constant, mable series, enabling the fulfilment of the th e back is uniform ; in other words, for a can be expressed as the sum of a series required conditions to be directly verified. given mean lift on the back, the greatest suction Σ An sin n0 and the usual method of evaluating The value of the profile slope thus found is will be a minimum, and so will the greatest th e coefficients of a Fourier series gives dy/dx =Σ A sin n0+Σ A cos n+10 n n velocity. On the hypothesis that compressi­ A —1/n, odd values of n only occurring. The bility stalling and compressibility effects value of the slope of the camber line is then where A =1 .1/2.3/ 4 . . . . n-1/n generally depend on the amount of the greatest given by the corresponding scries Σ A cos n 0. n having all even values, including zero velocity at any point, such an aerofoil will This series has a known sum which may be (A = 1). have the ideal shape for high speed work. written log (1+cos 0/1—cos 0). Integration Summation of the series is effected by noting Interest, therefore, attaches to the theoretical of th e slope dy/dx gives the ordinate y in terms tha t the sine series is the imaginary part of solutions which are possible giving the form of 0 an d hence in terms of x as x . log x+(c— x) the expression (1—e2f0) − ½, and the cosine of an aerofoil having this property, particularly log (c— x), omitting as already mentioned the series of the real part of the expression in view of the success of the theory in predicting constant factor which specifies the scale of ei0 (i −e2f0)−½ each having the value velocity and pressure distributions for the the curve. This then is the required solution, (cos 0/2 —sin 0/2) /2 √ si n 0, when 0 is positive, ordinary range of aerofoils. When the thick­ and gives th e shape of a camber curve develop­ so that the required condition is satisfied. ness is finite somewhat involved problems in ing constant suction at zero nose-tail incidence. Similarly the corresponding series for the conformal representation require to be solved, excess velocity on the back of the aerofoil, viz. : although there is no fundamental difficulty in Thin Flat-faced Aerofoil Σ A cos n0 —Σ A sin n + 10 is equal to the n n working out the solution. For thin aerofoils Other constant-suction aerofoils can be difference of the real part of the first expression of small camber, such as those considered by derived by " clothing " the camber curve just above, and the imaginary part of the second Glauert in his classical theory, the problem is specified. If, however, the face ordinate is expression, less the term in A in the cosine much simpler and it is known that the results given, the logarithmic camber must be de­ series, there being no corresponding sine term. of this theory are actually applicable to aero­ parted from in general, in order to satisfy the Since the real part of the first expression and foils such as are used in practice. Glauert condition of constant velocity. The case of a the imaginary part of the second expression, himself considered only the case of infinitesimal flat face is of particular interest both practically both have the value (cos 0/2+si n 0/2)/2√sin 0, thickness ; this gives one solution of the and theoretically, since it happens to yield the net value of the excess velocity is —A , problem here considered. Others are obtained ο an exact solution to a problem which at first which is constant, as required. by extending the theory to thicknesses of the sight seems somewhat intractable. same order as the camber, the order of ap­ The actual form of the back of the aerofoil is In order to bring in the effect of thickness, proximation remaining the same. A particular found by integration of the expression already it is necessary to add to the " odd " sine series case which will be treated here is that of an given for the slope to be y = √sin 0 (1 + sin 0) previously used for the excess velocity an aerofoil having a flat face, i.e., camber equal —1/2 . cosh − 1 (1+ 2 sin 0) which is the solution " even " cosine series, Σ A cos n 0, n being an to half the thickness. even number. The corresponding sine scries required. appears in the expression for the slope of the Both the curves thus determined prove to Thi n Cambered Aerofoil profile, and it is readily seen that if the sums be of the same general type, being somewhat With the usual notation the ordinate and of the two series are equal at a point on the fuller at the ends than a parabola. This is abscissa of a point on the aerofoil will be back (each being equal to half the slope) where as might have been anticipated, since the expressed in terms of a parameter 0, given 0 is positive, they will be equal in magnitude parabolic aerofoil or camber line has its maxi­ by : abscissa x=c/ 2 (1—cos 0), c being the but opposite in sign at a point on the face, mu m velocity at mid-chord. Independent chord. There will be no loss of generality in so tha t the total slope will be zero as required. verification of the form of the constant-pressure However, there is no straightforward method considering the case of unit velocity and leaving camber line has been obtained, using Theodor- of determining the values of the coefficients A the scale of the ordinate arbitrary, since all sen's well-known graphical method, but re­ to give this result and at the same time satisfy similar camber curves which are sufficiently versing the usual procedure, i.e., calculating th e condition for constant velocity on the back. flat will satisfy the conditions. the form for a given pressure distribution With considerable labour approximate values instead of vice versa. Fo r the case considered, the excess velocity http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

ConstantVelocity Aerofoils

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Emerald Publishing
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10.1108/eb030578
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Abstract

December , 1939 AIRCRAF T ENGINEERING Theoretically-derive d Profiles of Thin Sections Suitabl e for High Speeds B y J. Lockwood Taylor, D.SC. OR aerofoils intended for use at high a t a point on the back is equal to the defect for the first few coefficients can be found by speeds, it is evident that the maximum of velocity at the corresponding point on the successive approximation, and these suggest lift on the back, consistent with a given face, which is of course the same point in for the true values a fairly simple expression, maximum intensity of suction, will be attained space, since the thickness is ignored. This which proves to be correct since it gives sum- when the suction, and hence the velocity, over excess velocity, which is to be made constant, mable series, enabling the fulfilment of the th e back is uniform ; in other words, for a can be expressed as the sum of a series required conditions to be directly verified. given mean lift on the back, the greatest suction Σ An sin n0 and the usual method of evaluating The value of the profile slope thus found is will be a minimum, and so will the greatest th e coefficients of a Fourier series gives dy/dx =Σ A sin n0+Σ A cos n+10 n n velocity. On the hypothesis that compressi­ A —1/n, odd values of n only occurring. The bility stalling and compressibility effects value of the slope of the camber line is then where A =1 .1/2.3/ 4 . . . . n-1/n generally depend on the amount of the greatest given by the corresponding scries Σ A cos n 0. n having all even values, including zero velocity at any point, such an aerofoil will This series has a known sum which may be (A = 1). have the ideal shape for high speed work. written log (1+cos 0/1—cos 0). Integration Summation of the series is effected by noting Interest, therefore, attaches to the theoretical of th e slope dy/dx gives the ordinate y in terms tha t the sine series is the imaginary part of solutions which are possible giving the form of 0 an d hence in terms of x as x . log x+(c— x) the expression (1—e2f0) − ½, and the cosine of an aerofoil having this property, particularly log (c— x), omitting as already mentioned the series of the real part of the expression in view of the success of the theory in predicting constant factor which specifies the scale of ei0 (i −e2f0)−½ each having the value velocity and pressure distributions for the the curve. This then is the required solution, (cos 0/2 —sin 0/2) /2 √ si n 0, when 0 is positive, ordinary range of aerofoils. When the thick­ and gives th e shape of a camber curve develop­ so that the required condition is satisfied. ness is finite somewhat involved problems in ing constant suction at zero nose-tail incidence. Similarly the corresponding series for the conformal representation require to be solved, excess velocity on the back of the aerofoil, viz. : although there is no fundamental difficulty in Thin Flat-faced Aerofoil Σ A cos n0 —Σ A sin n + 10 is equal to the n n working out the solution. For thin aerofoils Other constant-suction aerofoils can be difference of the real part of the first expression of small camber, such as those considered by derived by " clothing " the camber curve just above, and the imaginary part of the second Glauert in his classical theory, the problem is specified. If, however, the face ordinate is expression, less the term in A in the cosine much simpler and it is known that the results given, the logarithmic camber must be de­ series, there being no corresponding sine term. of this theory are actually applicable to aero­ parted from in general, in order to satisfy the Since the real part of the first expression and foils such as are used in practice. Glauert condition of constant velocity. The case of a the imaginary part of the second expression, himself considered only the case of infinitesimal flat face is of particular interest both practically both have the value (cos 0/2+si n 0/2)/2√sin 0, thickness ; this gives one solution of the and theoretically, since it happens to yield the net value of the excess velocity is —A , problem here considered. Others are obtained ο an exact solution to a problem which at first which is constant, as required. by extending the theory to thicknesses of the sight seems somewhat intractable. same order as the camber, the order of ap­ The actual form of the back of the aerofoil is In order to bring in the effect of thickness, proximation remaining the same. A particular found by integration of the expression already it is necessary to add to the " odd " sine series case which will be treated here is that of an given for the slope to be y = √sin 0 (1 + sin 0) previously used for the excess velocity an aerofoil having a flat face, i.e., camber equal —1/2 . cosh − 1 (1+ 2 sin 0) which is the solution " even " cosine series, Σ A cos n 0, n being an to half the thickness. even number. The corresponding sine scries required. appears in the expression for the slope of the Both the curves thus determined prove to Thi n Cambered Aerofoil profile, and it is readily seen that if the sums be of the same general type, being somewhat With the usual notation the ordinate and of the two series are equal at a point on the fuller at the ends than a parabola. This is abscissa of a point on the aerofoil will be back (each being equal to half the slope) where as might have been anticipated, since the expressed in terms of a parameter 0, given 0 is positive, they will be equal in magnitude parabolic aerofoil or camber line has its maxi­ by : abscissa x=c/ 2 (1—cos 0), c being the but opposite in sign at a point on the face, mu m velocity at mid-chord. Independent chord. There will be no loss of generality in so tha t the total slope will be zero as required. verification of the form of the constant-pressure However, there is no straightforward method considering the case of unit velocity and leaving camber line has been obtained, using Theodor- of determining the values of the coefficients A the scale of the ordinate arbitrary, since all sen's well-known graphical method, but re­ to give this result and at the same time satisfy similar camber curves which are sufficiently versing the usual procedure, i.e., calculating th e condition for constant velocity on the back. flat will satisfy the conditions. the form for a given pressure distribution With considerable labour approximate values instead of vice versa. Fo r the case considered, the excess velocity

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Aircraft Engineering and Aerospace Technology – Emerald Publishing

Published: Dec 1, 1939

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