# 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.

## Leave a reply