Introduction .sx When considering the design of a jet-flapped aircraft from a stability and control aspect , it is necessary to have fairly accurate information concerning the downwash field behind the jet-flapped wing , particularly in those regions where it is practicable to locate the tailplane .sx The evaluation of the downwash at the tailplane is dependent upon a knowledge of the strength and position of the vorticity distributions which represent the wing and the jet .sx In his treatment of the flow past a wing with a jet-flap , of infinite span , Spence assumes that the incidence of the wing and the deflection of the jet are small , and hence the usual assumptions of thin aerofoil theory , in which the wing and jet are replaced by vortex sheets in the direction of the free stream , apply .sx The results so obtained for the vorticity distributions on the wing and jet are used in Part =1 to give the downwash at any position relative to the plane vortex sheet in the form [FORMULA] , where 5e = downwash angle , 5t = jet deflection angle , and 5a = wing incidence .sx However , in the calculation of the downwash induced at a point ( P ) in the field , it is necessary to allow for its location relative to the actual wing and jet .sx To the order of accuracy consistent with the previous assumptions , this implies calculating the downwash at a point whose ordinate relative to the plane vortex sheets is equal to the distance of the tailplane from the jet ( as shown in Figs .sx 1a and 1b) .sx The functions 15de/ dt and 15de/ da depend upon the jet momentum coefficient C;J ; , and on the relative position of the tailplane ; charts for these functions , and for the position of the jet , are given for various specific C;J ; values .sx The downwash has been evaluated for ranges of the tailplane position , wing incidence , jet deflection and jet momentum coefficient .sx For the unswept wing of finite span , with a full-span jet-flap , considered in Part =2 , Maskell has introduced the concept of an effective wing and jet flap of infinite span , in order to obtain the strength of the bound vorticity , elliptic spanwise loading being assumed .sx This solution may be used to give the contribution to the downwash from the bound vorticity , in a similar way to that described in Part =1 , but it does not account for the effect of the trailing vortices arising from the pressure gradients along the wing and jet spans .sx In the case of a wing without a jet-flap , it has been found that the downwash is very sensitive to the relative distance between the tailplane and the wake , and that the spanwise loading has more effect on the downwash than the chordwise loading , and so the wing and its wake are replaced by a lifting line and its trailing vortices , the latter being displaced in order to keep the tailplane at the correct height above the wake .sx The effect of the rolling-up of the wake has also been investigated for a wing without a jet-flap , and it is shown that rolling-up is not important for normal tailplane positions behind wings of large aspect ratio .sx The distance e behind the wing at which rolling-up may be assumed to be complete is given by e/ c = k@7/ C;L ; for a wing without a jet-flap , where k@7 depends upon the plan-form and spanwise loading of the wing .sx For the jet-flapped wing , the C;L ; will be greater than for the normal wing , but k@7 may now be a function of C;J ; , and will probably increase with increasing C;J ; ( since the bound vorticity on the jet will tend to resist rolling-up ) , so that e/ c will not decrease so quickly with increasing C;L ; and C;J ; , as might have been expected from first considerations .sx Thus , in order to evaluate the contribution to the downwash behind a jet-flapped wing from the trailing vorticity , it is assumed that the majority of the load is carried on the wing , so that the trailing vortices may be considered to arise from one chordwise position on the wing with no rolling-up taking place .sx The displacement of the jet and trailing vortices is accounted for by taking the position of the tailplane relative to the wake , and a chart is given for the downwash due to the trailing vorticity .sx Calculated values of the downwash are in good agreement with the few experimental results available , especially if the difference between the experimental and theoretical lift coefficients is taken into account .sx Theoretical results for the downwash on the centre-line are also given for a wing of aspect ratio 6.0 , showing variation with tailplane position , wing incidence , and jet parameters .sx PART =1 .sx 1 .sx Vortex Representation of the Wing and Jet-Flap of Infinite Span .sx The wing and jet-flap of infinite span may be represented in two dimensions by vorticity distributed on the chordal plane of the wing and the median line of the jet ( assumed to be thin) .sx The downwash relations have been solved by Spence , using the assumptions of thin-aerofoil theory , so that the aerofoil incidence and jet deflection are considered to be small .sx The vorticity distributions and the position of the jet are given in Fourier-series forms , with coefficients as functions of the jet momentum coefficient C;J; .sx Let U;0;f(x ) be the vorticity distribution on the aerofoil ( at incidence 5a to the mainstream ) and 5g(ch ) the vorticity distribution on the jet ( emerging at deflection 5t to the extended chord-line of the aerofoil ) , as shown in Fig. 1a .sx The x axis is taken parallel to the main stream , and the z axis vertically downwards , with the origin at the leading edge of the aerofoil .sx The chord of the aerofoil is taken to be unity , so that x and z are non-dimensional .sx Thus the vortex representation of the flow which is in accordance with the assumptions of thin aerofoil theory is as shown in Fig. 1b , with U;0;f(x ) located on the x axis , between 0 and 1 , and 5g(ch ) also on the x axis , between 1 and @25 .sx Then the expressions for f(x ) , 5g(ch ) and z;J;(x ) , the jet displacement , as obtained from Ref .sx 1 , are :sx For [FORMULA] .sx For [FORMULA] .sx .sx The Downwash .sx The downwash induced by the vortex distributions U;0;f(x ) and 5g(ch ) at the point ( X , Z ) is given by to the first order in 5a and 5t ( see Fig. 1b) .sx In order to apply the results calculated for the simplified configuration ( Fig. 1b ) to the actual configuration ( Fig. 1a ) , where the jet is displaced a distance z;J;(X ) below the x axis , it is assumed that the downwash w(X , z ) calculated for the point P@7(X , z ) in Fig. 1b is equal to the downwash at the point P(X , z + z;J; ) in Fig. 1a .sx A similar procedure is followed in Ref .sx 3 , where the displacement of the wake of a finite wing has to be considered .sx In general , the tailplane will be located a distance H above the jet , as indicated in Fig. 1a , so that to evaluate the downwash at the tailplane , i.e. , at the point ( X , z;J ; - H ) in Fig. 1a , we must evaluate the downwash at the point ( X , - H ) in Fig. 1b .sx The position of the tailplane is usually given as the distance along and height above the extended chordline .sx If l is the distance of the aerodynamic centre of the tailplane behind the wing leading edge , measured along the extended wing chord-line , and h the height above the chord-line , when the chord is of length c , as shown in Fig. 1a , then the non-dimensional co-ordinates ( X , Z ) at which the downwash is to be evaluated are given by [FORMULA] , where z;J ; may be obtained from Fig. 3 ( or equation ( 4)) .sx For the numerical evaluation of the two integrals in equation ( 6 ) , it is necessary to change the variables of integration , in the first integral using equation ( 1 ) in order to avoid the infinite value of f(x ) at the leading and trailing edges , and in the second integral using equation ( 3 ) to make the range of integration finite .sx Thus , if we write [FORMULA] , then the downwash at the tailplane is given by [FORMULA] , where f;1;(x ) sin 5th and f;2;(x ) sin 5th remain finite as x and 5th tend to zero , and as [FORMULA] , [FORMULA] .sx Equation ( 10 ) may be rewritten in the form [FORMULA] , where 15de/ dt and 15de/ da are functions of C;j ; , X and Z. These have been evaluated for C;j ; = 0.5 , 1.0 , 2.0 and 4.0 , with [FORMULA] and [FORMULA] , the results being shown as charts in Figs .sx 4a to 4d .sx Thus the procedure for the evaluation of the downwash at a given tailplane position , h/ c and l/ c , and given 5a , C;J ; and 5t , is to calculate the functions in the following order :sx ( =1 ) X from equation ( 8a ) ( =2 ) z;J ; from Fig. 3 ( =3 ) Z from equation ( 8b ) ( =4 ) 15de/ dt , 15de/ da from Figs .sx 4a to 4d ( =5 ) 5e from equation ( 11) .sx Interpolation will be necessary for C;J ; values other than 0.5 , 1.0 , 2.0 and 4.0 , and it seems better to evaluate 5e for a range of C;J ; , and then to interpolate the final result , rather than to interpolate for z;J ; , 15de/ dt and 15de/ da separately .sx For large X , the downwash is given by [FORMULA] , ( see Ref .sx 1 ) so that [FORMULA] and [FORMULA] .sx It may be noted that the value of C;L;/ ( 415pX ) for the downwash far behind the aerofoil is also obtained when the aerofoil is without a jet-flap .sx .sx Results .sx The results for the downwash behind an infinite wing and jet-flap are shown in Figs .sx 7 to 11 .sx It should be remembered that the theory is only strictly valid for small 5a and 5t , so that the use of the method to obtain the downwash for the larger values of 5a and 5t must wait to be justified or otherwise until experimental data are available .sx However , the results should indicate the trends in the variation of downwash with the various parameters .sx In Figs .sx 7 and 8 , the variation of the downwash with tailplane position is shown for two values of jet deflection angle , 5t , and two values of wing incidence , 5a , for C;J ; = 2.0. Fig. 7 shows that on the extended chord-line , h/ c = 0 , the downwash decreases quite sharply with increasing distance behind the wing , l/ c , but when h = 2c , the downwash is practically constant in each case for [FORMULA] .sx The results have been replotted in Fig. 8 to show the downwash field ( i.e. , contours of equal downwash ) , in the tailplane region .sx A comparison between the fields for the various 5t and 5a shows that the downwash is more sensitive to tailplane position for the higher 5t and 5a values , as might be expected .sx The results for the variation of 5e with C;J ; , 5t and 5a are given in Figs .sx 9 and 10 for a representative tailplane position , l/ c = 3.5 , h/ c = 1.5 , and also for a position on the extended chord-line , l/ c = 3.5 , h/ c = 0 .sx It will be noticed in Fig. 9a that 5e does not increase linearly with 5t for a given C;J ; value ( as might be implied by a glance at equation ( 11) ) due to the correction made to the downwash field for the displacement of the jet relative to the tailplane position .sx Fig. 9b indicates that 15de/ dC;J ; decreases with increasing C;J; .sx The variation of downwash with wing incidence is more important for stability and control considerations and the results are shown in Figs .sx 10a to 10d for 5t = 30 and 60 deg , and for various C;J ; values .sx Ranges of values of [FORMULA] are also indicated on the diagrams , and are seen to be the same for the two different 5t values over the same range of C;J ; for a given value of h/ c. Since [FORMULA] increases with C;J ; , it is not possible to assess a maximum , but for C;J ; = 4.0 , [FORMULA] is well below 1.0 at the tailplane and on the extended chord-line , being 0.20 and 0.35 respectively .sx It also appears that 15de/ da increases as 5a increases , but this is only noticeable at the higher values of C;J ; , and for C;J ; = 4.0 , 5a = 20 deg , 15de/ da is still less than 0.4 at the extended chord-line position .sx