Integro-Differential Equation Concept

.2 .2 Л -ДДО ♦ – Ддо ♦ -^-ДО дх2 Эу2 Э*

3х+Тх*

An attractive property of integral equations is that the influence of the boundary conditions is transmitted throughout the flow – field instantaneously in the case of a linear flow problem. Even for non-linear flow problems the influence of the boundary conditions is transmitted throughout the flow-field extremely quickly as compared to the partial differential equation models where the finite difference or finite element discretization allows the influence of the boundary condi­tions to be transmitted at most one grid cell per dunng each iteration. Л very fast and versatile 3-D aerodynamic shape inverse design algonthm was developed and is w idely utilized in several countries I I58J-[ 161]. It can accept any available 3-D flow-field analysis code as a large subrou­tine to analyze the flow around the intermediate 3-D configurations. The configurations arc up­dated using a fast intcgro-diffcrential formulation where a velocity potential perturbation OU. y./) around an initial 3-D configuration zw.(x. y) can be obtained from, for example, a Navi – er-Stokes code [ 1611. Here, the subscripts ♦/- refer to the upper and lower surfaces of the flight vehicle. Transonic 3-D small perturbation equation is

Here, differentially small potential perturbation is Atyx. y.z) and x. y.z coordinates have been sealed via Prandtl-Glaucrt transformation, V^, is the free stream magnitude, while

(77)

Here. M is the local Mach number and a^, is the free stream speed of sound. The floss tangency condition is then

iU*Uy.*M» = Vm |^wU. y)

Since —Дф(лг, у, ♦/■О) can be obtained from equation (75). the 3-D geometry is

dZ

readily updated from

bz^ (x, y) ш Jj-(Ae+(x. jr)^ Az (jr, y)l</jf± | J^{Ae4,(jr. y)-Ae,(Jby)I</» (79)

Since equation (75) is linear, it can be reformulated using Green’s theorem as an intc – gro-dilTcrential equation. The Г term on the right hand side of equation (75) would require vol­ume integration which can be avoided if Г is prescribed as smoothly decreasing away from the 3-D flight vehicle surface where it is known. Then, the problem can be very efficiently solved using the 3-D boundary clement method This inverse shape design concept has been success­fully applied to a vancly of planar wings [ 158J-[ 160] and wing-body configurations including the H-II Orbiting Plane with wingleis (161), f 162] where 3-D flow-field analysis codes were of the full potential. Euler and Navier-Stokes type. The method typically requires 10-30 flow anal­ysis runs with an arbitrary flow solver and as many solutions of the linearized inlcgro-diffcrcn – tial equation.

10.2 Conclusions

Several prominent and proven methods that arc applicable to inverse design of 3-D aerodynamic shapes have been briefly surveyed. The design computer codes based on these methods can be readily developed by modifying solid boundary condition subroutines in most of the existing flow-field analysis codes. Thus, all of the design methods surv ey ed are computationally econom­ical since they require typically only a few dozen calls to the 3-D flow – field analysis code. Al­though the inverse shape design methods generate only point-designs, it was pointed out that at least two of the methods arc conceptually capable of inverse shape design for unsteady flow con­ditions.