Fluid damping of cylindrical liquid storage tanks

Habenberger SpringerPlus (2015)4:515 DOI 10.1186/s40064-015-1302-2 Open Access RESEARCH Fluid damping of cylindrical liquid storage tanks Joerg Habenberger* *Correspondence: joerg.habenberger@ baslerhofmann.ch Basler & Hofmann AG, Forchstrasse 395, 8032 Zurich, Switzerland Abstract A method is proposed in order to calculate the damping effects of viscous fluids in liquid storage tanks subjected to earthquakes. The potential equation of an ideal fluid can satisfy only the boundary conditions normal to the surface of the liquid. To satisfy also the tangential interaction conditions between liquid and tank wall and tank bottom, the potential flow is superimposed by a one-dimensional shear flow. The shear flow in this boundary layer yields to a decrease of the mechanical energy of the shell-liquidsystem. A damping factor is derived from the mean value of the energy dissipation in time. Depending on shell geometry and fluid viscosity, modal damping ratios are calculated for the convective component. Keywords: Fluid damping, Earthquake, Cylindrical liquid storage tanks Background The dynamic behavior of liquid storage tanks and of structures in general is highly influenced by the structural damping. In engineering, an ideal fluid is usually assumed in the realm of dynamic analysis of liquid storage tanks. In doing this, the potential equation of the liquid may be divided into two decoupled components: (1) the impulsive component which describes the interaction of the liquid and the shell and (2) the sloshing motion of the free liquid surface which may be accounted for by the convective component. After the modal decomposition of both components, a viscous damping is introduced to consider the dissipation of mechanical energy. The damping influences the resulting pressures as well as the amplitude of the convective fluid motion. If the response spectra method is used to calculate the dynamic response of the tank-liquid-system the spectral acceleration is determined directly by the damping ratios. The damping of the impulsive component is mainly affected by the damping of the shell, and the fluid damping may be neglected. Depending on the material of the shell, damping ratios between 2 % (steel) and 5 % (reinforced concrete) are suggested for the Serviceability Limit State (Eurocode 8, Part 4 2006 or Scharf 1989). The damping ratios are larger for the Ultimate Limit State (4 % for steel and 7 % for concrete structures). These are typical and well established damping ratios (Stevenson 1980). For the convective component damping ratios from 0 up to 5 % are proposed. The second draft of Eurocode 8, Part 4, e.g., suggests a damping ratio of 0.5 % “for water and other liquids”. Scharf (1989) recommends a value of 0 % independent of the content. It is © 2015 Habenberger. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Habenberger SpringerPlus (2015)4:515 Page 2 of 11 also important to mention that there are no experimental or theoretical justifications of the proposed damping values concerning the sloshing oscillation and they are more or less “best guesses”. With the potential equation of the ideal fluid only boundary and interaction conditions normal to the surface of the fluid can be satisfied. It is not possible to describe the adhesion of a real fluid to the tank wall and bottom by the potential equation. To fulfill the boundary conditions though a one-dimensional shear flow is superimposed on the potential flow of the ideal fluid. For this purpose at first the Navier–Stokes-equation is applied and simplified with respect to the conditions at the boundary layer of the fluid. A solution of the simplified form of the Navier–Stokes-equation is derived which describes the velocity in the boundary layer. The shear flow in the boundary layer leads to a dissipation of mechanical energy. This energy dissipation is related to the damping ratio of the fluid oscillation. Damping ratios for the sloshing component are derived for different shell geometries and fluid viscosities. In Fig. 1 a cross-section of the investigated liquid storage tank with the corresponding material and geometry parameters is shown. Damping effects of a viscous fluid Equations of the incompressible, viscous fluid The fluid is assumed to be incompressible and viscous. The friction pressures are proportional to the velocity of the liquid (Newton’s fluid with kinematic viscosity ν). The condition of incompressibility (with the velocity field of the fluid v = (vζ vϕ vξ )T ) reads as follows (Sommerfeld 1988): ∇ · v = 0. (1) The Navier–Stokes-Equation without consideration of volume forces is: 1 ∂v + (v∇)v = νL �v − ∇p. ∂t ̺L (2) Fig. 1 Definition of material and geometry parameters of the investigated vertical cylindrical liquid storage tanks: R radius, H tank height, HL liquid height, d/E/ν thickness, Young’s-modulus and Poisson ratio of the tank shell Habenberger SpringerPlus (2015)4:515 Page 3 of 11 Under the assumption of small oscillation amplitudes, the contribution of the nonlinear expression (v∇)v in (Eq. 2) can be neglected: 1 ∂v = νL �v − ∇p. ∂t ̺L (3) By using the rotation of the velocity field ω = ∇ × v it is possible to transform the Navier–Stokes-Equation into a simpler form (Landau and Lifschitz 1987): ∂ω = νL �ω. ∂t (4) According to Eq. (4), the rotation of the velocity field ω corresponds to the heat equation. Hence, there is a convective transport of the velocity vortices from the surface into the liquid. This process decays exponentially into the interior of the fluid. It is not possible for conservative forces to produce velocity vortices in a viscous fluid. There must be forces which can not be derived from a potential (Schaefer 1950). In the present case these are shear forces (frictional pressures) occurring at the tank wall. Concerning the oscillation of storage tanks, vortices are produced at the tank wall due to frictional pressure. The velocity vortices decrease exponentially into the interior of the liquid. Because of the exponential damping, the rotational flow occurs practically only in a small layer at the tank wall. The main part of the liquid is an irrotational flow and can be described by the following equation: ∇ × v = 0 ∇ · v = 0. (5) As derived from Eq. (5), one can see that v = 0 (the Navier–Stokes-equation becomes the potential equation). Thus, the liquid behavior is everywhere in such a tank like this of an ideal (incompressible and frictionless) fluid. Only in a thin layer on the tank wall the potential flow is disturbed. The boundary conditions of a viscous liquid require the consistency of the liquid veloci (...truncated)