A necessary condition for a minimum solution Application of the variational method to the minimum drag problem. As the free stream speed increases, the length and width of the cavity grow indefinitely and the flow can be idealized by. This will simplify matters by considering only that part of the body which is in contact with the liquid.
The lift and drag acting on the body and the shape of the cavity are determined only by this part. The far flow pressure is equal to the cavity pressure, which is assumed to be the safe vapor pressure of the liquid. The condition p ~ p is a clear statement of the fact that c . the vapor pressure of the liquid is the minimum pressure in the stream.
As the current splits at the nose of the plate, there is a jump in the angle of the current. Furthermore, we have because the flow approaches free-stream flow at large distances from the plate. We will use ( )2 for the positive root of the positive quantity inside the parentheses.
The integration from s' = 0 to s' = s can be taken on any path in the upper half plane due to the analyticity of the integrand.
OPTIMIZATION OVER A FINITE PARAMETER SPACE
The plate portion S'OS is convex or concave when viewed from the flow as a positive or. The condition that the pressure is greater than the vapor pressure of the liquid places an upper limit on the value of a. This expression for x is found by integrating the expression for x from Appendix A for the case N = 1.
A necessary condition for the solution of the three equations (3. 14) in two unknowns, >. and >. , it is the determinant. This corresponds to maximum drag profiles if a < 0 and plates of stationary drag {neither maximum nor. The fourth possibility, A= 0, provides a connection between. a and K • If we are only interested in the drag coefficient,. and the ratio of arc length to chord, k = s 0 /y ' 0 this ratio is all that is needed to complete the solution, since the factor.
Since K -o {the ratio of the length of the arc to the chord goes to infinity), it can be shown that a is the root of the transcendental. These results are not contradictory, as the two families of profiles were found to minimize drag under different isoperimetric constraints; however, it is surprising that the tile shapes are as close as they are.
THE METHOD OF CALCULUS OF VARIATIONS
In the next section, it is shown that a necessary condition for the existence of a minimizing solution is that A{x) + C{x) > 0 for In the case where f and g art· required to be continuous of Holder, a natural approach is to expand f and g in orthogonal series. In this particular example we can set the coefficients of the even sine and cosine terms to zero; that is
A special advantage of the result of the variational technique is that it allows It should be noted that the higher-order coefficients ak do not seem to converge but fluctuate wildly as N increases. With this assumption, it is allowed to replace (4.26) in (4.25) and reverse the order of integration of the variables t and x.
The second line follows by substituting Eq. 4. 26), and the third line is the result of the reverse order of integration. Using similar operations and using the Poincare-Bertrand formula, we can write the third term on the right-hand side of (4. 28). By choosing from, the limits of integration in the expression for 6?1 are given by Eq.
We now reduce the interval over which Or is non-zero by letting E go to zero. If F is quadratic in f and g, as in the previous section, condition (4. 33) is independent of the extremum solution; i.e. oz.I> 0 .required.
APPLICATION OF THE VARIATIONAL TECHNIQUE TO THE MINIMUM DRAG PROBLEM
This means that the plate shape that solves the variation problem must have a sharp nose. A third relation is given by (5. 19 ), since k is given a specific value; however, it seems better to leave one parameter free and let it determine k. The results of the parametric problem for N = 1 (Chapter III, § 2) would indicate that this free parameter must be c, the parameter that determines how much of the slab is a free line.
If k - l+ we expect the plate shape to approximate that of a flat plate without a free streamline (c=l). An analytical solution of the system of equations in the previous section seems excluded due to their extreme nonlinearity, our only recourse is a numerical solution, with corrective steps such as increasing the order of the Gaussian quadrature and forcing r to zero.
If this result has anything to do with the solutions of the variational problem, the linearization of the system of equations in section 3 of this chapter seems warranted; i.e., we neglect r 2, r 13, 13 z., etc., and. This situation is partly remedied by approximating r by a function which vanishes at the endpoints (for example, by taking some Fourier sine components of I'); how-. From this we conclude that the linearization of the nonlinear equations of the variational problem is not justified.
DISCUSSION AND CONCLUSION
Since no solution to this integral equation has been found, we can only speculate about the result of the variational method. If it sounds pessimistic, the solution may turn out to have no direct physical application; for example, it may not satisfy the pressure condition r{g) ~ 0, or r (±1) 'f 0, in which case 13. Nevertheless, such a solution (if it actually reduces I ) would provide an absolute lower bound on the resistance that can then be used as a basis for comparison with other physically relevant least resistance profiles (such as those determined by the parameter method}. A final judgment of the utility of the variational technique as a design tool should await a more thorough investigation (most likely numerically) of the equations of Chap.
These equations and the equation s /y = k (k, a given number greater than unity) are all we need. This procedure was performed for the N = 1 case and was found to give acceptable results, although it is difficult to say how good they are. To do this, one must either solve a variational problem (corresponding essentially to N = oo) or use the method of parameters for N = 2,3, etc., as was done in the two cases discussed in Chap.
The parameter problem for N > l will most likely require the use of the computer, although it may be possible to find analytical solutions by series expansion for K near unity. Apart from investigating the equations of variation and extending the parameter method to N > 1, several areas for further study include the following: (1) An extension of the calculus of variations method of Ch. IV to deal with inequality constraints and constraints on the values of f (or g) at the endpoints, the case f(±l) = 0 is particularly important;.
(2) Applying the methods of this paper to the minimum drag problem and the hydrofoil problem with possible use of more complex finite cavity flow models and; (3) Application of variational technique Ch. Due to the linearity of the integral equations for the case of quadratic functionals, this method is particularly suitable for use in problems involving energy constraints. 1962), A flow model for free flow theory. Asterisks on the integral signs mean that the singular parts of the integrands must be combined; i.e. we now propose to solve this differential equation for K close to 1 using a Taylor series. the elements of the determinant of A are given by.
