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MATHEMATICA BOHEMICA, Vol. 126, No. 2, pp. 293-305 (2001)
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# The far-field modelling of transonic

compressible flows

## C. A. Coclici, I. L. Sofronov, W. L. Wendland

* C. A. Coclici*, Mathematisches Institut A, Universität Stuttgart, 70569 Stuttgart, Germany, e-mail: ` cristi@mathematik.uni-stuttgart.de`, * I. L. Sofronov*, Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya sq. 4, 125047 Moscow, Russia, e-mail: ` sofronov@spp.keldysh.ru`, * W. L. Wendland*, Mathematisches Institut A, Universität Stuttgart, 70569 Stuttgart, Germany, e-mail: ` wendland@mathematik.uni-stuttgart.de`

**Abstract:**
We present a method for the construction of artificial far-field boundary conditions for two- and three-dimensional exterior compressible viscous flows in aerodynamics. Since at some distance to the surrounded body (e.g. aeroplane, wing section, etc.) the convective forces are strongly dominant over the viscous ones, the viscosity effects are neglected there and the flow is assumed to be inviscid. Accordingly, we consider two different model zones leading to a decomposition of the original flow field into a bounded computational domain (near field) and a complementary outer region (far field). The governing equations as e.g. compressible Navier-Stokes equations are used in the near field, whereas the inviscid far field is modelled by Euler equations linearized about the free-stream flow. By treating the linear model analytically and numerically, we get artificial far-field boundary conditions for the (nonlinear) interior problem. In the two-dimensional case, the linearized Euler model can be handled by using complex analysis. Here, we present a heterogeneous coupling of the above-mentioned models and show some results for the flow around the NACA0012 airfoil. Potential theory is used for the three-dimensional case, leading also to non-local artificial far-field boundary conditions.

**Keywords:** artificial boundary and transmission conditions, compressible transonic flow, linearized Euler equations, integral equations with kernels of Cauchy type, potential theory, domain decomposition

**Classification (MSC2000):** 35Q30, 35Q35, 35M20, 45E05, 65M55, 76N17, 76N20, 76H05, 76M40

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