Analytical and numerical determination of natural oscillations of a stratified fluid in an open axisymmetric container


Аuthors

Zhang Y. , Orlov V. V., Temnov A. N.*

Baumann Moscow State Technical University, 105005, Moscow, 2nd Baumanskaya St., b. 5, c. 1

*e-mail: antt45@mail.ru

Abstract

This paper presents the analytical and numerical investigation of the natural oscillations (surface and internal waves) of a continuously stratified fluid with a free surface in an open cylindrical container, a problem fundamental to the dynamic stability of liquid propellant in aerospace applications. Based on the derived analytical properties of the problem, an efficient numerical model is proposed using the weak formulation of the boundary value problem. For spatial discretization, a comparative analysis of third-order (10 degrees of freedom, C^0 continuity) and fifth-order (21 degrees of freedom, C^1 continuity) Pascal elements is performed. It is shown that fifth-order Pascal elements possess a significant advantage: for the same mesh resolution, these higher-order elements successfully resolve the frequencies of higher-mode surface waves (e.g., at n=3), whereas low-order elements are substantially limited. After the necessary filtering of abundant non-physical "numerical artifacts", the range of reliable solutions is critically narrowed. Attempts to force the extraction of higher modes using such low-order elements lead to an overestimation of the natural frequencies due to excessive mesh stiffness, as well as to strong non-physical oscillations in the mode shapes along the z-axis, rendering their application inefficient. To optimize computational costs and improve the accuracy of discrete mode shapes, a two-stage algorithm that consists of a preliminary calculation of the full spectrum on a coarse mesh with the filtering of physical eigenvalues and a local precise search in the vicinity of the identified values on a refined mesh, is proposed. The obtained results demonstrate the high accuracy of the proposed method in determining the characteristics of both surface and internal waves. This work presents a computational tool for determining natural oscillation frequencies, which serves as a basis for predicting complex fluid-structure interactions and optimizing cryogenic fuel management systems in launch vehicles and spacecraft.

Keywords:

Stratified fluid; free axisymmetric oscillations; eigenvalue problem; finite element method; internal waves; surface waves; Galerkin method

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