1、A METHOD OF CALCULATING WIND TUNNEL INTERFERENCE FACTORS FOR TUNNELS OF ARBITRARY CROSS-SECTION by Robert G. Joppd Prepared by UNIVERSITY OF WASHINGTON Seattle, Wash. f OT i NATIONAL AERONAUTICS AND SPACE ADMINISTRATION . WASHINGTON, Oh. . JULY 1967 Provided by IHSNot for ResaleNo reproduction or ne
2、tworking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM NASA CR-845 A METHOD OF CALCULATING WIND TUNNEL INTERFERENCE FACTORS FOR TUNNELS OF ARBITRARY CROSS-SECTION By Robert G. Joppa Distribution of this report is provided in the interest of information exchange. Responsibility for the
3、 contents resides in the author or organization that prepared it. Prepared under Grant No. NGR-48-002-010 by COLLEGE OF ENGINEERING UNIVERSITY OF WASHINGTON Seattle, Wash. for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information
4、 Springfield, Virginia 22151 - CFSTI price $3.00 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-A METHOD OF CALCULATING WIND TUNNEL INTERFERENCE FACTORS FOR TUNNELS OF ARBITRARY CROSS-SECTION By Robert G. Joppa SUMMARY A new method of.calculating th
5、e wind tunnel wall induced in- terference factors has been developed. The tunnel walls are rep- resented by a vortex lattice of strength sufficient to satisfy the boundary conditions at the wall. The vortex lattice is then used to calculate the interference velocities at any point in the wind tunnel
6、. The resulting interference factors agree with the classical results that are available for square and circular tun- nels. Calculations are also presented for a rectangular tunnel, and they can be made to closely approximate a tunnel of any cross- section. INTRODUCTION Current interest in V/STOL ai
7、rcraft has resulted in a renewed interest in the problems of the wind tunnel measurement of their characteristics. Among the problems of critical importance is that of calculating the interference velocities due to the presence of the tunnel walls, particularly the longitudinal distribution of the P
8、rovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-interference because of its large effect on the measured pitching moment. Classical methods of computing these interference veloci- ties are inadequate for V/STOL models that characteristically produce l
9、arge downwash, and so a new method is required to handle this case. In this report a new method of representing the tunnel walls is developed which should be applicable to the large down- wash case, and it is tested by being applied to the limiting case of small downwash in order to compare results
10、with classical theory. The classical solutions have depended upon the assumption of a proper set of images outside the tunnel, of the vortex flow inside the tunnel, such that the walls become streamlines. Unfor- tunately, no proper image system has been found for any tunnel except the rectangular cr
11、oss-sections. Prandti (Ref. 1) has pre- sented a solution for the circular wind tunnel with an undeflected wake which gives correct values of upwash at the wing. Glauert (Ref. 2) has solved the rectangular wind tunnel problem, including the effects downstream, i.e., at the tail location. Others (Ref
12、s. 3 to 9) have extended it to include other tunnel shapes. Lot2 (Ref. 10) has offered a solution for the upwash interference for circular and elliptical tunnels which will yield results at down- stream locations as well as at the wing. In Lotzs solution, an image system is used which is valid at th
13、e wing and far downstream, and an additional potential function is assumed in infinite series form which is required to cancel the remaining normal velocities Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-at the wall, also expressed in series form.
14、 To the degree that these series do not completely converge before being truncated, this solution is an approximation. The object of this work is to present an alternate method of representing the tunnel walls which would be applicable to a tun- nel of any arbitrary cross-section and which could be
15、extended to handle the large downwash case. It is hypothesized that the tunnel walls might be represented by a network of vortex lines whose mag- .nitude and direction are just sufficient to prevent flow through a set of control pointson the walls. The approach is similar to that of approximate lift
16、ing surface theory. This paper presents the mathematical development of the theory. The results of sample calculations for the interference factors and the distribution of interference over the longitudinal and lateral axes are presented for uniformly loaded wings of various spans in a variety of wi
17、nd tunnels. The wind tunnel configurations include square, rectangular and circular cross-sections. These results are then compared, where possible, with prior work which have obtained corresponding values by other theoretical treatments. SYMBOLS b C cL Wing vortex span Wind tunnel cross-section are
18、a Wing lift coefficient 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-h( 1 h( I( 1 H - - - i, j,k L,G ii R( 1 R( )( 1 S s 7 1 6 v Velocity induced at a point W W w X,Y,Z B r 6 Normal distance to a point p from a line containing a vortex segment i
19、dentified by subscript Normal distance to a point p from a plane containing vortex segments identified by subscript Height of wind tunnel Unit vectors in the directions X, Y, Z Dimensions of rectangular vortex ring (Fig. 3) Unit vector normal to vortex ring Vector from point (X,Y,Z) to end of a vort
20、ex vector s indicated by subscript Magnitude of component of vector R ( 1 indicated by second subscript Wing area Vector representing a vortex segment of strength r and length S component of s indicated by subscript Unit vector in the direction of the total velocity vector at a point Vertical compon
21、ent of wall-induced interference velocity Width of wind tunnel Vector representing a wing bound vortex of strength rw Cartesian coordinate of a point (see Fig. 1) Angles defining direction to a point from the end of a vortex segment (Fig. 2) Circulation strength of a vortex Tunnel-wall-induced inter
22、ference factor 4 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-STATEMENT 6F PROBLEM The problem is to find that distribution of VOrtiCity lying in the tunnel walls which will prevent any flow through the wall due to the action of a lifting system i
23、n the wind tunnel. The lifting surface is assumed to be uniformly loaded and is re?re- sented by a simple horseshoe vortex with.the trailing Pair un- deflected. In principle, any desired distribution of lift could be built up of such simple elements. The walls are represented by a tubular vortex she
24、et of finite length composed of a network of circumferential and longitudinal vortices having equal spacing (Fig. 1). Helmholtz theorem that a vortex filament can neither end nor begin in the flow is satisfied most readily by constructing the network of square vortex rings lying wholly within the pl
25、ane of the walls. Each such square has a vortex strength lYi, and each side is coincident with the side of the neighboring square. Thus, the strength of any segment is the sum of the strengths of the two adjoining squares. The boundary condition that the wall must be impervious to flow is satisfied
26、at a control point in the center of each square. This results in a set of simultaneous equations, one written for each control point, in which the unknowns are the lYi. A large number of equations results if the tube is very long, thus some judgment is required in choosing the geometric arrange- men
27、t. The use of square vortex rings requires a tunnel of constant cross-section. One notes that for a wing mounted in the center 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-of the tunnel, lateral symmetry always exists: and if the wake is undefle
28、cted, vertical symmetry also exists, thus reducing the number of unknowns. The trailing edge of the finite length tube which represents the long tunnel requires a slightly different treatment. At a far downstream section only longitudinal vorticity should exist. This is represented by elongating the
29、 last ring of squares by a large amount, while keeping the control point at the same location with respect to the last circumferential station. Figure 1 shows the arrangement for a rectangular tunnel with filleted corners. SETUP OF THE EQUATIONS A right-hand axis system is established with the X-axi
30、s on the longitudinal centerline of the tunnel, positive downstream. The Y-axis is taken positive upward and the Z-axis positive to the right side of the tube facing downstream. Since the surface of the tunnel is to be made of square ele- ments, its cross-section is a polygon of equal segments arran
31、ged to approximate any desired shape. In this development the cross- section will be assumed to be symmetrical about the X,Y plane. In general, the velocity induced at any point p (Fig. 2) due to a vortex segment may be written: v=r 4rrh (cos P 1 + cos 8,) v (1) 6 Provided by IHSNot for ResaleNo rep
32、roduction or networking permitted without license from IHS-,-,-The terms required are written as follows: cos Bl + cos 8, = - (Rl-R2) 2 1 := ii1 x 5 Iii, = x 4 - i j k Rlx Rl RIZ Y sx sy % RIS Sin 8, h But Sin p, = - Rl I y Rl SZ-R S Lz y RlxSZ RlxSY -Rl sx k ;= y i Sh Sh Finally, the velocity induc
33、ed at a point due to a vortex segment is: v Rl+R2 r/4rrh = 2 2RlR2S h s2-(l-2)2 li RlySZ (2) + Rl;x-Rlx%) 3 i +(RlxSy-RlySX k ) 1 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-One could then add the contributions of all four sides of a vortex squar
34、e, but it is more convenient to take advantage of the lateral symmetry and sum the effects due to a pair of symmetrically located vortex squares of the same strength. The arrange- ment is shown in Figure 3 and the following equation results: v r/m = hAB f cos GB 1 % A+RNB 2 Hal RNAR + cos 4, RMA+RMB
35、 2 hMl RMA%B L2- %A-%B 2 1 L2- Qg. RIDRMC - (RMc-Rm) 3 SD+% + $21 %DRNC L2- k-RRm%A 3 i QD+RMD 1 sCSRMC + h; sCRMC 1 L2- (GfR %BRm - (sB-R + Rw1+%J2 b2 - (-s) s r 1 2b - %y h; The boundary condition is expressed at each control point by writing v.n = 0 where 6 is the unit outer normal to the surface
36、 at that point. - - EC i x (R-R) II x (Ii,-Ii,)1 10 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Thus there is a set of N equations, one for each control point. Because of the right and left symmetry and vertical symmetry, there are N/4 unknown lY
37、i. An electronic computer is used to solve the matrix for the I? 1 Once the Ti are known, the induced velocity due to the walls can be calculated at any point in the tunnel by the use of eq. (3) summed over all the vortex rings in the tunnel walls. The inter- ference is expressed as an angle whose t
38、angent is the vertical component of interference velocity divided by the tunnel wind speed. Results are expressed in terms of the classical interference fac- tor 6, defined by the equation: The factor is computed in terms of wing circulation and vortex span Results are presented graphically to show
39、the longitudinal varia- tion of the factor 6 for different wing spans in a variety of tunnels. 11 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-RESULTS OF CALCULATIONS Results of calculations made for three representative tunnel shapes are presente
40、d in the form of graphs of the wall interference factor 6. Values of 6 were calculated at points along the tunnel centerline from the wing location downstream for several values of wing vortex span. These are presented for a circular, a square, and a 3:5 rectangular tunnel in Figures 5, 6, and 7. Th
41、e average value of this interference factor over the vortex span of the uniformly loaded wing was also calculated and is shown as a function of vortex span for each of these tunnels along with the centerline values in Figure 8. COMPARISON OF RESULTS WITH CLASSICAL WORK Square Tunnel Very little prev
42、ious work exists which can be used for a check on the accuracy or convergence of the present method. Prandtls concept of an infinite array of images of the wing located outside the tunnel is applicable only to rectangular (including square) tunnels and has been applied by Silverstein l tunnels are n
43、ot common. The number of line segments, each corresponding to the side of a vortex square, to be used to adequately represent the square Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-tunnel cross-section was determined by making a series of calcu-
44、lations with Increasing numbers of segments. Figure 9 shows the results of using 12, 16, and 20 segments to make up the periphery of the square cross-section. The results for 1.6 and 20 segments differ only slightly and correspond very closely to the data taken from,Reference 9. The excellent agreem
45、ent shown indicates that 16 segments are enough to represent satisfactorily the square cross- section tunnel. Circular Tunnel In the case of the circular tunnel, no exact solution is avall- able for the downstream interference factors, so two approximate results are compared with the new calculation
46、s in Figure 10. The treatment presented by Lotz (Ref. 10) would be exact except that the numerical values depend upon the point at which an infinite series in truncated. Reference 10 gives. no indication of the accuracy expected in its numerical values. The result taken from Silverstein each was rot
47、ated so that either points or flats of the polygon were at the top and side centerline. All four calculations yielded the same curve, with 13 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-values within one-tenth of one percent. Thus, it is conclude
48、d that a 12-sided polygon is adequate to represent the circular tunnel. Length Effect The effect of length of the tunnel to be used in calculations was explored for the circular tunnel. A twelve-sided polygon was used in the calculation, with the model vortex span equal to 0.4 of the tunnel diameter. It is evident from Figure 11 that a length-to- diameter ratio of 3 or 4 is ample for convergence. The reason for this may be seen in an examination of the distribution of the wall vorti
