S.Battacharya. D.M.Rafi and S.Tripathi

Transoft International,Epinay/Seine, France

A three dimensional non-structured flow solver FLUIDYN-NS, has been developed for investigation of complex internal and external flows. The code employs second order accurate TVD schemes based on the flux vector splitting concept of van Leer ( 1982 ) and flux difference splitting technique due to Roe ( 1981 ). The system of Navier-Stokes equations are solved over any number of multiply connected domains.

This paper presents the following fluid dynamic applications:

(1) Three dimensional flow through transonic compressor rotor stage
(2) Two dimensional test cases

The flow through compressor rotor is investigated by inverse simulation procedure. The rotor walls along with the hub are held fixed whereas the compressor case wall is given a reverse velocity. Suitable source terms are then added to accurately capture the rotation. For two dimensional test cases, local grid rezoning and adaptation is performed to obtain better resolution in areas of interest. Comparison between adapted and non adapted solutions indicate considerable improvement in shock capturing capability for adapted mesh.

With the advent of computing power and numerical tools in the last two decades, many authors (Warming & Beam (1976), van Leer (1977), Moretti (1979), Chakravathy et al. (1980). Roe (1981, etc.) have formulated second order accurate characteristics based schemes for the solution of Euler equations. However, handling of complex geometry still gives rise to problem and considerable research is diverted in this direction. Even with the concept of multi-block technique, it is not possible in many cases to generate a completely structured mesh without singular points (i.e. with only hexahedronal and quadrilateral elements), let alone controlling mesh fineness and smoothness. Scientists have tackled this problem by constructing non-structured mesh and solver (Jameson et al.(1986). Batina (1989), etc.). But for a purely non-structured mesh, with only triangular and tetrahedral elements, computation of viscous flow and turbulence modeling within the boundary layer poses a problem. Also, many schemes are highly sensitive to grid skewness, stretching and non orthogonality. Keeping this requirement in mind, FLUIDYN-NS has been developed to cater to fluid dynamicists handling complex geometrics. The code provides a non-structured solver that can handle cells of mixed type (a combination off triangular and quadrilateral elements in 2D, and tetrahedral, wedge and hexahedronal elements in 3D).

FORMULATION AND IMPLEMENTATION DETAILS

The system of Reynold's averaged Navier-Stokes equations are decomposed into convective and dissipative parts. Flux splitting techniques [ van Leer (1982) and Roe (1981)] are used, for their robustness and crisp shock capturing capability, to spatially discretize the convective terms. Kappa scheme [ Batina (1991)] is employed to obtain numerical schemes of various orders of accuracy. It has been shown for structured meshes that one can obtain second order central, second order upwind. From and third order (can be shown easily for locally one dimensional flows) schemes using values of 1, -1.0. 1/3 respectively for kappa. A first order Riemann solver has also been provided for fast estimation of flow over complex geometries. The code uses continuously differentiable van Albada flux limiter.

For spatial discretization of the dissipative terms, second order accurate Green-Gauss approach is utilized to adequately resolve the gradients of primitive variables. However, since the cells are of arbitrary shape, it is desired to construct a control volume for computation of the gradients. Hence in FLUIDYN-NS, for each and every face, a pseudo-staggered control volume is generated using the two neighboring cell centers along with the centers of the current face sides and suitable interpolation is devised for approximation of the primitive variables at the face side centers.

K-epsilon turbulence model is currently under development.

Temporal discretization is performed by the use of an explicit second order accurate Runge-Kutta type time integration scheme due to Jameson et al (1981).

The code has provision for inclusion of body forces in the system of Navier-Strokes equations by simply representing them as cell based source terms.

The results for this case are presented in Figs.1.2.3 and 4. For this investigation, the hub and the rotor walls are held fixed. The case wall is moved in the opposite direction with an equivalent velocity (negative component). The rotation is inversely modeled by introduction of a centrifugal reaction term as a cell based source in the entire flow domain. Effect of coriolis forces are also incorporated into the source term (it is significant for high rpm). The hub and the rotor blade walls are given a no slip boundary condition, whereas the case wall is moved with a variable velocity (N-S slip wall with wall velocity) proportional to angular velocity and local normal distance from the axis of rotation. The side walls in the force and the aft region of the rotor blades have a cyclic boundary condition (reentrant), in order to maintain mass flow continuity. The flow enter the rotor zone at a small deviation angle of 9 degrees due to the effect of the inlet guide vanes. At the inlet, a subsonic inflow condition and at the outlet a transmit condition is used. For this simulation, the mesh has 25128 cells, 18656 nodes and 55701 faces.

Isobars on the rotor blade left wall is presented in Fig.1. The direction of rotation is from left wall to the right wall. Isomach contours at a mid-section between the two rotor blades is shown in Fig.2.

Flow past NACA 0012 aerofoil at free stream mach number 0.8 and angle of attack 1.25 degrees is presented in Fig.3. A comparison is made between coarse mesh, fine mesh and adapted mesh results. It is noted that the coarse mesh is able to resolve the flow gradients nearly as good as the fine mesh. However, substantial improvement is obtained by adapting the mesh; a weak shock is captured on the lower surface. The lift coefficients for coarse, fine and adapted mesh have been 0.29, 0.365,0.36 respectively. Fig.4 shows the Cp distribution for flow past RAE 2822 aerofoil at angle of incidence 3 degrees and free stream mach number 0.75. Here a comparison is made between the second order and the third order schemes. The lift coefficients for second order and third order schemes have been found to be 1.06 and 1.08 respectively.

CONDLUDING REMARKS

The results presented indicate that FLUIDYN-NS is a powerful tool for simulation of complex flows in and around complex configurations. Also it is noted that the grid dependence of the solver is limited and it is quite versatile over a range of flow speed.

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