From a mere knowledge of the number of significant physical parameters, dimensional analysis can be made to determine the characteristic dimensionless parameters that govern the dynamic similarity (see § 5.2). Without further information, a flutter model and its prototype must be geometrically similar, have similar mass and stiffness distribution, and have the same geometrical attitude relative to the flow. The scale factors must be such that the density ratio ajp, the reduced frequency k, the Mach number M, and the Reynolds number R be the same for the model as for the prototype. These requirements are, of course, exceedingly severe, and hard to be met.
When more specific information about a physical phenomenon is known, certain conditions of geometric, mass, and elastic similarity may be dispensed with, without loss of exactness. Consider the simple example of the bending deflection of a beam. In this case only the flexural stiffness El is of significance. Hence, in constructing a beam model for the purpose of deflection measurements, only the El distribution needs to be simulated; the cross-sectional shape can be distorted if desired. Such freedom in distorting a model greatly simplifies the model design and testing, and will be discussed in greater detail below.
If the differential or integral equation, or equations, governing a physical phenomenon are known, they provide a deeper insight into the laws of similarity than a mere knowledge of the variables that enter the problem, and offer ways in which a distorted model may be used. Illustrations of this point can be found in many problems.5-62 As a classical example, let us quote the problem of George Stokes, who, in 1850, introduced the term “dynamic similarity” into the literature. Stokes’ problem is the motion of a pendulum in a viscous fluid. The basic equation is the Navier-Stokes equation which, for a two-dimensional flow of an incompressible viscous fluid with fixed boundaries, may be written as
where и and v are velocity components in the x and у directions, v is the kinematic viscosity, and со = (dv/dx) — (du/dy) is the vorticity of the fluid. The same differential equation applies to the model:
where the primes refer to model. Let us introduce the scale factors KL, Kt, etc., between dimensions of the model and the prototype, so that
x’ = KLx, yf = K^, t’ = Ktt, / = K, v
со’ = Кшы, и’ = KYu, v’ — KyV
These scale factors are subject to the kinematic similarity imposed by the relations
The first of Eq. 6 is a kinematic similarity equation 4. The second of Eq. 6 gives the Stokes’ rule for similarity for flows with similar boundary conditions. Using the first of Eqs. 6, the second equation may be written
which means, of course, in present terminology, that both the prototype and the model must have the same Reynolds number.*
An alternate procedure, which differs from the previous one only in form and not in basic reasoning, is also commonly used. The idea is to express the differential equations in dimensionless form. Introduce a characteristic length L, a characteristic time T, a characteristic velocity V, a characteristic vorticity £l, and a characteristic number for the kinematic viscosity N. Let x, у, t, etc., be dimensionless quantities so that
x = xL, у = yL, и = HL/T, v = vL/T (8)
t — IT, Q. = dj/T, v — Nv
Then Eq. 1 may be written
Эй VT t Эй. Эй NT. /Э2й, Э2Й
зі +Т г Ш + VW ~ иудх2 1 w
Now, for dynamically similar systems, the dimensionless values x, у, l, etc., have the same value for the model as for the prototype. Hence the coefficients VTjL and NT/L2 must be the same for the two systems; i. e.,
VT ГГ NT N’T’
U~ L’2 ( )
where primes refer to model.
The results obtained in Eqs. 6 and 10 are of course identical. The first method has the advantage of requiring fewer notations.
* Stokes anticipated Osborne Reynolds’ results by thirty years. See Langhaar, Ref. 5.62.
As a second example, consider the flexural oscillation of a beam. The governing equation is
where p is the density of the beam material, A is the area of beam cross section, and El is the flexural rigidity. Let
where A0, rg are the cross-section area and radius of gyration at a reference section, respectively, and / is the length of the beam,/г,/2 being dimensionless functions involving only the dimensionless parameter ж//. Then
Introducing scale factors
x — KLx, l = К if, w — KLw, r q = Kr q^o
p’ = Kpp, E’ — KeE, t’ = Ktt
where primes refer to model, we have
[ , (Л avi K2KEKr£ p’ /яЛ Elw’
Эж’2 lJl 1/7 Эж’2.] + KpKLl E’r0’2h 1/7 3t’2 Since Eq. 13 applies as well to the model, we must have
‘K? KEKr 2 KPKL*
This result can be expressed in more familiar form if we write m as the frequency of oscillation and notice that, according to the definition со’ = Кшсо, the scale factor K, n must be the reciprocal of Kt. Then Eq. 16 may be written as
к, лг IX = (17)
Kr о * KE
i. e., the scale factor of the dimensionless parameter
— P (IB)
r0 * E
must be equal to unity. This remains true for all systems expressible by means of /, r0, p, E, f^x/l), and /2(ж//). No restriction to any particular
shape of the cross sections is imposed. Thus the design of a model for the purpose of measuring oscillation modes needs only to simulate /iWO. ЛС*//)» and the parameter (18) above, and is left with complete freedom in selecting cross-sectional shape. Without the auxiliary information contained in the differential equation, other dimensionless parameters such as li’rQ would have appeared, and the conclusions would have been more restrictive.
The same reasoning applies to aeroelastic models. As an example, consider the torsion-bending flutter of a cantilever wing in an incompressible fluid. If one is satisfied with Theodorsen’s approximation of characterizing flutter condition by the conditions at a “typical” section, the equations describing flutter are given by Eqs. 11 of § 6.9, which are already written in dimensionless form, and hence must apply equally well to the model as to the prototype. An examination of these equations shows that at the flutter condition P0 = Q0 = 0 the following ten dimensionless ratios are involved:
/л, xa, cojcoz, ah, ra, cojco, a)a/co, k, hjb, a0 (19)
k, (oJa), and hjba^ are determined by the condition of existence of flutter: the vanishing of the flutter determinant. Hence a model must simulate the first five parameters which are to be evaluated at a typical section of the wing. Any wing having these same dimensionless ratios can be considered as a flutter model of the prototype. In particular, the flutter of a cantilever wing may be simulated by a two-dimensional wing model.
Clearly, model testing intended to conform with more comprehensive theories would require more restrictive similarity laws.