Тезисы докладов

Доклады иностранных участников

Recent investigations of flows in cascades of compressors seem to indicate that shock induced vortex breakdown may be responsible for the stall at high compressor loads. To gain insight into this problem the bursting of a slender vortex, caused by normal and oblique shocks is studied for supersonic flow at free-stream Mach-numbers Ma = 1.6 to Ma = 2.0. Since details of the break-down process are difficult to be obtained in experiments, a numerical solution of both, the Euler equations and the Navier-Stokes equations for time-dependent, three-dimensional, supersonic laminar flow was employed in order to demonstrate the influence of pressure alone, and also of the Stokes stresses and the Fourier heat flux in the burst part of the vortex. In the computations, the free-stream Mach number, the Reynolds-number, the shock strength, the circulation of the vortex, and its axial inflow velocity profiles are varied.

The computational domain is a rectangular box containing approximately 2 million grid points to ensure proper resolution of the flow during the passage through the shock and also in the core of the vortex. For the initiation of the computation an isolated, slender, longitudinal vortex, embedded in the main flow, with its axis parallel to the main flow direction was prescribed. For the in-flow boundary, the circumferential velocity component was assumed to be given by a Burgers type vortex, with a vanishing radial velocity component. The inflow profile of the axial velocity component was assumed to be given by a Gaussian distribution, allowing for an axial velocity defect or overshoot. The lateral boundaries were chosen to be stream surfaces, while non-reflecting boundary conditions were prescribed in the outflow cross-section.

The numerical results show that because of the larger pressure jump across the normal shock breakdown sets in earlier than in the case of the interaction with the oblique shock. It begins with the formation a free stagnation point on the axis a certain distance downstream from the shock, and a small region of reversed flow is formed. The flow reversal makes the shock move up-stream, and a bubble-like flow structure emerges, growing in the axial and radial direction until a stable, slightly oscillating position is reached. For the oblique shock-vortex interaction the flow structure of the burst part of the vortex differs from that for the normal shock. The originally straight shock is distorted into an `S'-shaped front near the axis. But again, stable breakdown occurs only when a stagnation point downstream from the shock can be formed, followed by subsonic flow reversal.

The onset of vortex bursting and its dependence on the characteristic flow parameters is clearly evident from the calculations. For example, a reduction of the axial velocity component or an increase of the azimuthal velocity component enhances breakdown. Also, a large shock intensity and a large circulation gives rise to early breakdown, and consequently, for weaker shocks a larger circulation is necessary. Breakdown will also set in earlier, when the radial profile of the axial velocity component has the shape of a wake. This is so, since the neRecent investigations of flows in cascades of compressors seem to indicate that shock induced vortex breakdown may be responsible for the stall at high compressor loads. To gain insight into this problem the bursting of a slender vortex, caused by normal and oblique shocks is studied for supersonic flow at free-stream Mach-numbers Ma = 1.6 to Ma = 2.0. Since details of the break-down process are difficult to be obtained in experiments, a numerical solution of both, the Euler equations and the Navier-Stokes equations for time-dependent, three-dimensional, supersonic laminar flow was employed in order to demonstrate the influence of pressure alone, and also of the Stokes stresses and the Fourier heat flux in the burst part of the vortex. In the computations, the free-stream Mach number, the Reynolds-number, the shock strength, the circulation of the vortex, and its axial inflow velocity profiles are varied.

The computational domain is a rectangular box containing approximately 2 million grid points to ensure proper resolution of the flow during the passage through the shock and also in the core of the vortex. For the initiation of the computation an isolated, slender, longitudinal vortex, embedded in the main flow, with its axis parallel to the main flow direction was prescribed. For the in-flow boundary, the circumferential velocity component was assumed to be given by a Burgers type vortex, with a vanishing radial velocity component. The inflow profile of the axial velocity component was assumed to be given by a Gaussian distribution, allowing for an axial velocity defect or overshoot. The lateral boundaries were chosen to be stream surfaces, while non-reflecting boundary conditions were prescribed in the outflow cross-section.

The numerical results show that because of the larger pressure jump across the normal shock breakdown sets in earlier than in the case of the interaction with the oblique shock. It begins with the formation a free stagnation point on the axis a certain distance downstream from the shock, and a small region of reversed flow is formed. The flow reversal makes the shock move up-stream, and a bubble-like flow structure emerges, growing in the axial and radial direction until a stable, slightly oscillating position is reached. For the oblique shock-vortex interaction the flow structure of the burst part of the vortex differs from that for the normal shock. The originally straight shock is distorted into an `S'-shaped front near the axis. But again, stable breakdown occurs only when a stagnation point downstream from the shock can be formed, followed by subsonic flow reversal.

The onset of vortex bursting and its dependence on the characteristic flow parameters is clearly evident from the calculations. For example, a reduction of the axial velocity component or an increase of the azimuthal velocity component enhances breakdown. Also, a large shock intensity and a large circulation gives rise to early breakdown, and consequently, for weaker shocks a larger circulation is necessary. Breakdown will also set in earlier, when the radial profile of the axial velocity component has the shape of a wake. This is so, since the necessary deceleration of the axial flow to reach a stagnation point is smaller than for a uniform or even a jet-like profile. Finally, it is mentioned that the numerical and also the few existing experimental results can be correlated in good agreement with a simple criterion that is based on the balance of maximum radial and axial pressure differences. This criterion can be used for estimating as to whether or breakdown occurs, when the characteristic flow parameters such as shock strength, circulation of the vortex, axial flow, etc. are known.cessary deceleration of the axial flow to reach a stagnation point is smaller than for a uniform or even a jet-like profile. Finally, it is mentioned that the numerical and also the few existing experimental results can be correlated in good agreement with a simple criterion that is based on the balance of maximum radial and axial pressure differences. This criterion can be used for estimating as to whether or breakdown occurs, when the characteristic flow parameters such as shock strength, circulation of the vortex, axial flow, etc. are known.

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