Operator-Valued Laplace's Integrals and Stability of the Open Flows of Inviscid Incompressible Fluid

Изучаются спектры краевых задач возникающих при линеаризации уравнений Эйлера идеальной несжимаемой жидкости на стационарных решениях, описывающих течения, в которых жидкость поступает в область течения и выводится из нее через определенные части границы. Такие течения естественно называть открытыми. Спектры таких течений относительно мало изучены, посравнению со случаем полностью непроницаемых границ или условий периодичности. В этой статье мы указываем класс открытых течений, спектры которых состоят из нулей некоторой целой операторнозначной функции, представленной операторным интегралом Лапласа. Вопрос орасположении спектра таких течений сводится, следовательно, к своего рода операторнозначной проблеме Рауса Гурвица для этого интеграла. В ряде интересных частных случаев эту операторную функцию удается выразить как мультипликаторное преобразование рядов Фурье, и тогда проблема Рауса Гурвица становится скалярной, и более того, ее удается решить с помощью теоремы Пойа о нулях интегралов Лапласа. На этой основе мы доказываем принадлежность открытой левой полуплоскости спектров ряда конкретных течений, для которых такие доказательства не были известны.

Axial Force Balance of Supercritical CO2 Radial Inflow Turbine Impeller Through Backface Cavity Design

The supercritical CO2-based power cycle is very promising for its potentially higher efficiency and compactness compared to steam-based power cycle. Turbine is the critical component in the supercritical CO2-based cycle which delivers the power. Compared to the gas turbine or steam turbine of similar power output, the size of the supercritical CO2 radial turbine is much smaller and the axial force on the impeller is much larger. The load on the thrust bearing could be too heavy for long-term safe operation. Therefore, it is necessary to balance the axial force on the impeller through aerodynamic design to reduce the load on the thrust bearing. The impeller backface design with radial pump-out vanes proves to be an effective design to reduce the axial force on the impeller of radial turbomachinery, which is widely used in the pump industry. This work investigates the impeller backface cavity flow of a supercritical CO2 radial turbine and the application of the pump-out vanes to the impeller through computational fluid dynamics simulations. Design variations of the pump-out vane are presented and their performance variations are discussed from the view of viscous compressible fluid, instead of the commonly assumed inviscid incompressible fluid in the pump industry.

Flow topology of helical vortices

Equal coaxial symmetrically located helical vortices translate and rotate steadily while preserving their shape and relative position if they move in an unbounded inviscid incompressible fluid. In this paper, the linear and angular velocities of this set of vortices ($U$ and $\unicode[STIX]{x1D6FA}$ respectively) are computed as the sum of the mutually induced velocities found by Okulov (J. Fluid Mech., vol. 521, 2004, pp. 319–342) and the self-induced velocities found by Velasco Fuentes (J. Fluid Mech., vol. 836 2018). Numerical computations of the velocities using the Helmholtz integral and the Biot–Savart law, as well as numerical simulations of the flow evolution under the Euler equations, are used to verify that the theoretical results are accurate for $N=1,\ldots ,4$ vortices over a broad range of values of the pitch and radius of the vortices. An analysis of the flow topology in a reference system that translates with velocity $U$ and rotates with angular velocity $\unicode[STIX]{x1D6FA}$ serves to determine the capacity of the vortices to transport fluid.

Static rivulet instabilities: varicose and sinuous modes

A static rivulet is subject to disturbances in shape, velocity and pressure fields. Disturbances to interfacial shape accommodate a contact line that is either (i) fixed (pinned) or (ii) fully mobile (free) and preserves the static contact angle. The governing hydrodynamic equations for this inviscid, incompressible fluid are derived and then reduced to a functional eigenvalue problem on linear operators, which are parametrized by axial wavenumber and base-state volume. Solutions are decomposed according to their symmetry (varicose) or anti-symmetry (sinuous) about the vertical mid-plane. Dispersion relations are then computed. Static stability is obtained by setting growth rate to zero and recovers existing literature results. Critical growth rates and wavenumbers for the varicose and sinuous modes are reported. For the varicose mode, typical capillary break-up persists and the role of the liquid/solid interaction on the critical disturbance is illustrated. There exists a range of parameters for which the sinuous mode is the dominant instability mode. The sinuous instability mechanism is shown to correlate with horizontal centre-of-mass motion and illustrated using a toy model.