A Moving-Mesh Finite-Volume Scheme for Compressible Flows
This paper presents a new scheme for calculating compressible viscous flows with traveling wall boundary on moving mesh system. For the moving-mesh system, it is necessary for the scheme to satisfy the geometric conservation laws[1]. To satisfy the geomet
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1
Introduction
This paper presents a new scheme for calculating compressible viscous flows with traveling wall boundary on moving mesh system. For the moving-mesh system, it is necessary for the scheme to satisfy the geometric conservation laws[l]. To satisfy the geometric conservation laws, a finite-volume formulation in the complete space-time (x, y, t) domain is adopted in the scheme. This treatment makes it possible that the scheme completely satisfies the physical and geometrical conservation laws for solving unsteady flow equation on moving-mesh system. The resultant fully implicit scheme is solved iteratively at every time step. This "inner" iteration is performed in an explicit manner through the efficient and highly stable Rational Runge-Kutta scheme, so that the algorithm of the scheme can be treated explicitly notwithstanding the implicit formulation. This approach of the inner iteration is similar to so-called pseudo-time approach. This paper also gives some discussion and numerical interpretation between the inner iteration strategy and the pseudo-time approach. The paper gives an application of the present scheme to compressible Navier-Stokes flows with fluid/body-motion coupled interaction.
2 2.1
Moving-Mesh Finite-Volume Scheme Geverning Equation
For an axisymmetric compressible flow, the non-dimensionalized Navier-Stokes equation may be expressed as [2]
~Q +~ Fv) + at + ~ ax (E - ~Ev) Re ay (F - _1 Re
(z - ~Zv) = Re
O.
(1)
where, x and yare axial and radial coordinates respectively. Here, Q is a vecor of conservative variables, and Fare inviscid fluxes, and are viscous and are vectors appearing in axisymmetric cylindrical coordinate fluxes, system, and Re is the Reynolds number. Equ~tion(l) can be represented using extended divergent operator t7 and tensor F as
Z Zv
E
Ev
Fv
(2) N. Satofuka (ed.), Computational Fluid Dynamics 2000 © Springer-Verlag Berlin Heidelberg 2001
K . MatsUDo et aI.
706
Here,
- (aax ' ay' a ata) ' - = (1 ,,= F
and Z 2.2
=Z -
1)
(3)
E - Re Ev, F - Re Fv, Q ,
Zv/Re.
Moving-Mesh Finite-Volume Scheme
Since the flow is non-stationary and body-wall boundaries move and change their relative positions, the body-fitted mesh system must dynamically change its shape according to the movement of the wall boundaries. The present flow solution method is a finite-volume method and we construct a control volume in the moving-mesh system. In this paper, we assume that both the number of grid points and grid topology do not change regardless of any flow and wallboundary-position changes. Now let (e, '1) be general curvilinear coordinate system. The coordinate system is assumed to be time-dependent. Thus = e(x, y, t) and '1 = '1(x, y, t). The present method is a vertex-centered finite-volume method and we define flow variables at the vertex (i, j, n), where i, j, and n are indices of '1, and t directions respectively. Now let the flow and grid system at the time level n + 1 (t = t n +1 ) be unknown, assuming that the flow variables and grid at the time-level up to n
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