1、NASA Technical Paper 1435 NASA ! TP I 1435 c. 1 Similitude Requirements and Scaling Relationships as Applied to Model Testing Chester H. Wolowicz, James S. Bowman, Jr., and William P. Gilbert AUGUST 1979 I Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHST
2、ECH LIBRARY KAFB, NM NASA Technical Paper 14-35 Similitude Requirements and Scaling Relationships as Applied to Model Testing Chester H. Wolowicz, Drydeiz Flight Research Center Edwards, Califoriiia James S. Bowman, Jr., and William P. Gilbert Langley Research Ceizter Humpton, Virginia National Aero
3、nautics and Space Administration Scientific and Technical Information Branch 1979 Provided by IHSNot for Resale-,-,-r SIMILITUDE REQUIREMENTS AND SCALING RELATIONSHIPS AS APPLIED TO MODEL TESTING Chester H. Wolowicz Dryden Flight Research Center and James S . Bowman, Jr . and William P . Gilbert Lan
4、gley Research Center INTRODUCTION Experimental data for scale-model aircraft are used to define the aerodynamic characteristics of full-scale aircraft, verify theoretically predicted aerodynamic characteristics, and provide data where theory is deficient. To apply the data to a full-scale aircraft o
5、r its components with maximum validity, certain similitude conditions must be met. The similitude of the geometric configurations is a fundamental requirement, as is the similitude of the angles of attack. Reynolds number and Froude number, as well as Mach number in the case of compressible flow con
6、ditions, are pertinent parameters for steady-state (static) or dynamic test conditions. A number of other similitude parameters may be important, depending on the test objectives and aircraft elasticity. In general, any one experimental technique will not satisfy all the similitude requirements for
7、correlation of wind-tunnel data with free-flight data or for correlation of free-flight data obtained from models of different scale. Most tests are designed for certain similitude conditions at the expense of other parameters. For example, an elastic, rigidly mounted wind-tunnel model tested at ful
8、l-scale Mach number and dynamic pressure through an angle of attack range does not properly account for the effects of mass on elastic deformation except at one angle of attack at a steady level-flight condition. Inertial aerolastic effects that occur in maneuvering flight must be accounted for theo
9、retically. A comparison of the aerodynamic characteristics of one free-flying model with those of a model of different scale or a full-scale aircraft at the same Mach number may not be Provided by IHSNot for Resale-,-,-I I1 I 11111W11111Il11111 11 I appropriate if Froude number similitude requiremen
10、ts are not met. A difference in Froude number could result in dissimilar angles of attack. Although there are many references of limited scope in the literature on similitude, a comprehensive report is needed to clarify and summarize the many techniques for wind-tunnel and free-flight model testing
11、with regard to similitude requirements , test objectives , and comparison of model and full-scale results. The fulfillment of this need is particularly appropriate in that remotely controlled, subscale , powered and unpowered models of advanced aircraft are currently being used to investigate stabil
12、ity, control , and handling qualities at routine as well as high-risk flight conditions. One of the prime factors necessary to determine the limitations of data obtained from a model is the degree to which the similitude requirements have been met. This report provides a comprehensive review of the
13、similitude requirements for the most general test conditions , from low-speed incompressible flow conditions to high-speed supersonic conditions. The fluid is considered to be a continuum that obeys the perfect gas laws for a fixed value of the adiabatic gas constant. The similitude requirements are
14、 considered in relation to the scaling requirements, test technique , test conditions, and test objectives. Limitations in test techniques are indicated, with emphasis on the free-flying model. Scaling procedures are illustrated for free-flying models in incompressible and compressible flow. For inc
15、ompressible flow, the kinematic properties are preserved by using velocities scaled from Froude number similitude requirements (Froude scaling). For compressible flow, the compressibility effects are pre- served by using velocities scaled from Mach number similitude requirements (Mach scaling). In a
16、ddition , summary tables and nomographs are presented to facilitate a rapid assessment of the scaling requirements for free-flying models and of the extent to which the requirements are satisfied for both Froude and Mach number similitude. Although this report covers parameters encountered in dynami
17、c model tests, it does not include discussions of other similarity effects that may be important in individual cases , such as the scaling of a viscous damper in the control system of a model with free control surfaces or, a more remote example, the scaling of physical parameters for an icing test.
18、To prepare for such situations, the experimenter should refer to books on dimensional analysis, such as references 1 and 2. SYMBOLS Physical quantities in this report are given in the International System of Units (SI) and U .S . Customary Units. Details concerning the use of SI are given in referen
19、ce 3. 2 Provided by IHSNot for Resale-,-,-2 generalized linear acceleration, m/sec2 (ft/sec ) a - a n normal load factor, g b wingspan, m (ft) aerodynamic drag, lift, and side-force coefficients, respectively CD CL crossflow drag coefficient mQX cL maximum lift coefficient = v- acL au U cL - acL - a
20、a a cL acL a6e - - cL6 e C1 C” cn aerodynamic rolling moment, pitching moment, and yawing moment coefficients, respectively P - - aP 3 Provided by IHSNot for Resale-,-,-Cp q m U m a m C mti m 6e C e “6 0 m r n n. r C C C nb 2 m C aCm = v- au - acm aa - a (6,C/ 217) zero-lift pitching moment coeffici
21、ent - acn - aP airfoil section lift and pitching moment coefficients, respectively (fig. 2) 4 Provided by IHSNot for Resale-,-,- C E EI F G GJ 8 I I IY J k 1 M M Mrl m NFr fu Pg N N NRe mean aerodynamic chord, m (ft) tensile and compressive modulus of elasticity, N/cm2 (lb/in 1 bending stiffness, N-
22、cm (lb-in ) force, N (lb) shear modulus of elasticity, N/cm 2 2 2 2 2 (lb/in ) 2 2 torsional stiffness, N-cm (lb-in ) 2 acceleration of gravity, m/sec2 (ft/sec 2 2 mass moment of inertia, kg-m (slug-ft ) 4 second bending moment of area, cm4 (in ) 2 2 mass moment of inertia about pitch axis, kg-m (sl
23、ug-ft ) 44 second torsional moment of area, cm (in ) radius of gyration, m (ft) characteristic dimension, m (ft) Mach number moment, m-N (ft-lb) crossflow Mach number (figs. 3 and 4) mass, kg (slugs) V2 Froude number, - lg number of fundamental units in dimensional analysis number of physical quanti
24、ties considered in dimensional analysis P v1 Reynolds number, f=vl IIV 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHSNstr n P, 9. r Pf - 9 R s RV S t U il V vs W crossflow Reynolds number (fig. 3) COl Strouhal number, scale of model roll, pitch, and y
25、aw rates, respectively, rad/sec 2 2 fluid pressure, N/m (lb/ft ) 2 pitch acceleration, rad/sec dynamic pressure, 1/2pfV2, N/m 2 (lb/ft 2 1 stagnation pressure in compressible flow (eq. (9) , N/m 2 (lb/ft 2 ) spin or turn radius, m velocity of sound ratio, 2 wing area, m2 (ft free-stream and stagnati
26、on temperatures, respectively, K (OR) time, see linear velocity along the x-axis, m/sec (ft/sec) linear acceleration along the x-axis, m/sec velocity, m/sec (ft/sec) velocity of sound, m/sec (ft/sec) 2 2 (ft/sec ) 3 3 specific volume, m /kg (ft /slug) weight, N (lb) normal linear acceleration, m/sec
27、 (ftlsec ) angle of attack, deg or rad 2 2 6 Provided by IHSNot for Resale-,-,-viscosity, F; and elasticity as defined by its velocity of sound, Vs . The pertinent properties of the aircraft include its configuration, represented by a characteristic dimension I; attitude relative to the fluid, a; ma
28、ss, M; mass inertia, I; and elastic bending and torsional rigidity, EI and GJ , respectively (based on beam theory) . The pertinent rate quantities include linear velocity, V; angular velocity, L2; and periodic oscillations typified by frequemy , w . The pertinent accelerations include linear and an
29、gular accelerations, a and a, respectively. Gravitational effects are characterized by the acceleration of gravity, g. Time, t, and angular displacement of a control surface , 6 , are also pertinent parameters. These quantities can be 8 Provided by IHSNot for ResaleNo reproduction or networking perm
30、itted without license from IHSsummarized as M = f(pf, p, VS 2, a, v, a, 6, a, h, o, g, t, m, I, EI, GJ) (1b) where F is force and M is moment. To determine the dependence of the forces and moments on the quantities on the right side of the respective equations, dimensional homogeneity is established
31、 through a dimensional analysis . The resulting dimensionless combination of the physical quantities constitutes the similitude requirements for model testing. Three fundamental units are involved in the mechanics of forces and moments: the unit of length, 2; the unit of time, t; and the unit of mas
32、s, m. All the quantities in equations (la) and (lb) can be expressed in terms of these fundamental units , as indicated in table 1. A dimensional analysis of equations (la) and (lb) using the Lord Rayleigh method (see appendix) and the dimensions of the physical quantities listed in table 1 results
33、in the following equations in which the force F and moment M are the stipulated phenomena that have been expressed as aerodynamic coefficients. The aerodynamic coefficients are functions of the fourteen dimensionless param- eters , which represent the requirements for complete static and dynamic sim
34、ilitude of the model relative to the airplane. Table 2 identifies the individual similitude parameters , defines them in general terms , and gives examples of their normally applied definitions. The equations of motion of an airplane, in their customary dimensionless form, are defined in terms of th
35、ese nondimensional parameters. Thus , for the lift equation, 9 Provided by IHSNot for Resale-,-,-where for dimensional homogeneity cL = v(acL/aU) U cL =acL/aa a = acL/a6, cL6 e For the pitching moment equation where and where the derivatives of the moment coefficients have the same format as those o
36、f the lift coefficient. Although the reduced linear velocity Ty is not included in table 2 it is readily obtained from the product of the reduced linear acceleration and time parameters. Thus 10 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHSThe equation
37、s of motion , as exemplified by equations (3) and (4) , include (eq. (3b); relative mass moment of inertia, rn the relative density factor, PfSC/2 I (eq. (4b); aircraft attitude; control surface position; and reduced PfS(C/2) velocity and acceleration parameters. The force equations also include Fro
38、ude number , gc/2V dependence of the aerodynamic coefficients and their derivatives on Reynolds number , Strouhal number, Mach number, and the aeroelastic bending and torsion parameters. -2 (eq . (3b) . Not evident from the equations of motion is the In the following sections , the implications of s
39、everal key similitude requirements are discussed. Reynolds Number Reynolds number is the ratio of the fluids inertia forces to the viscous forces in the boundary layer of the fluid. It is an important parameter in deter- mining the dynamic similarity of flow around models and full-scale aircraft. Wh
40、en the model data are obtained at much lower Reynolds numbers than those encoun- tered at full-scale conditions , the inertia forces of the fluid on the model are much lower in proportion to the viscous forces than those on the full-scale airplane. As a consequence, the flow conditions are no longer
41、 dynamically similar. The point of transition from laminar to turbulent flow, the thickness of and velocity in the boundary layer at any streamwise station on a surface, and the angle of attack at which the flow field separates from the surface are all functions of Reynolds number. The boundary-laye
42、r (viscous flow) conditions on any configuration affect the drag coefficient throughout the angle of attack range and the maximum lift and stall characteristics of the aircraft. The precise effect depends on the particular airfoil and planform used, and on the interference effects of the fuselage an
43、d nacelies or pods. As Reynolds number increases, the point on the surface along the flow line at which the boundary layer changes from laminar to turbulent moves forward. The precise point or locus of transition is affected by the geometry of the surface or body and by the resulting pressure distri
44、bution, surface roughness or wavi- ness , and the magnitude of the velocity fluctuations in the airstream. As a result , it is difficult to extrapolate model test results of natural transition effects obtained in present test facilities to full-scale Reynolds numbers. Efforts are frequently made to
45、simulate flow conditions typical of higher-than-test Reynolds numbers by artificially fixing the transition using strips of roughness particles (grit) or other flow-tripping devices. The test results at several Mach numbers are then extrapolated to full-scale Reynolds numbers . 11 I. Provided by IHS
46、Not for ResaleNo reproduction or networking permitted without license from IHSThe effect of Reynolds number on stability derivatives and aerodynamic loads at other than near-stall conditions poses problems that have been recognized only in recent years and are only partly understood. Prior to the mi
47、d-1960s Reynolds number was thought to have little effect in the transonic region where the charac- teristics of the flow were thought to be primarily determined by Mach number. However both Reynolds and Mach numbers are important in the transonic region as was effectively shown during the developme
48、nt of the C-141 airplane. Data were obtained in wind-tunnel tests where the Reynolds number based on mean aero- 6 dynamic chord was as high as 8.5 X 10 in the transonic region for both natural and artificial boundary-layer transitions. However when extrapolated independently these data offered littl
49、e guidance in the prediction of full-scale values of approx- imately 50 X 10 . Figure 1 (from ref. 4) shows the variation of Cm as a function of Reynolds number and free and fixed transition for the C-141 airplane at a Mach number of 0.825. The data for natural and artificial transitions appear to converge with increasing Reynolds number; however an extra
