Methods for Simplification of the Mathematical Model for the Calculation of Flows in the Flow Path of Hydraulic Turbines

journal article in Web of Science

en

The main concern of this paper is fundamental research into an acute problem arising in the operation of hydraulic turbines, namely, the formation of a vortex core downstream of the impeller. It is noted that, at present, the operating range of these machines has to be extended due to the change in power network loads in daily and seasonal cycles, thereby increasing the time of hydraulic turbines' operation under off-design or undesirable conditions, including those involving the risk of formation of a vortex of varying intensity. To describe this complex fluid flow, a mathematical model of the vortex core flow downstream of the hydraulic turbine's impeller was developed. This paper focuses on the fundamental issues encountered in developing the mathematical model. The methods are presented for simplifying the formulation of equations describing the structure of vortex rope formed in the flowpath. In this case, the equations of mathematical physics containing a dependent variable and taking the form of the Navier-Stokes equations when applied fluids flows were employed. They describe the time changes of the selected parameters in a controlled volume induced by the flow through the volume boundaries. The boundary conditions have been demonstrated to considerably affect the unsteady vortex structures. A special case is formulated for a potential force field using the stress tensor that governs the equilibrium. This makes it possible to describe complex motion in liquids without using an intricate form of the Navier-Stokes equations for vortex structures.

The main concern of this paper is fundamental research into an acute problem arising in the operation of hydraulic turbines, namely, the formation of a vortex core downstream of the impeller. It is noted that, at present, the operating range of these machines has to be extended due to the change in power network loads in daily and seasonal cycles, thereby increasing the time of hydraulic turbines' operation under off-design or undesirable conditions, including those involving the risk of formation of a vortex of varying intensity. To describe this complex fluid flow, a mathematical model of the vortex core flow downstream of the hydraulic turbine's impeller was developed. This paper focuses on the fundamental issues encountered in developing the mathematical model. The methods are presented for simplifying the formulation of equations describing the structure of vortex rope formed in the flowpath. In this case, the equations of mathematical physics containing a dependent variable and taking the form of the Navier-Stokes equations when applied fluids flows were employed. They describe the time changes of the selected parameters in a controlled volume induced by the flow through the volume boundaries. The boundary conditions have been demonstrated to considerably affect the unsteady vortex structures. A special case is formulated for a potential force field using the stress tensor that governs the equilibrium. This makes it possible to describe complex motion in liquids without using an intricate form of the Navier-Stokes equations for vortex structures.

unsteady flow; vortex structure; incompressible liquid; Navier-Stokes equations; unsteady term; divergence theorem; finite-volume method

17.12.2021

PLEIADES PUBLISHING INC

NEW YORK

0040-6015

68

12

906–915

10