Fluid damping of cylindrical liquid storage tanks
 Joerg Habenberger^{1}Email author
Received: 9 March 2015
Accepted: 2 September 2015
Published: 17 September 2015
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 onedimensional shear flow. The shear flow in this boundary layer yields to a decrease of the mechanical energy of the shellliquidsystem. 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
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 tankliquidsystem 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 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 onedimensional shear flow is superimposed on the potential flow of the ideal fluid. For this purpose at first the Navier–Stokesequation 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–Stokesequation is derived which describes the velocity in the boundary layer.
Damping effects of a viscous fluid
Equations of the incompressible, viscous fluid
As derived from Eq. (5), one can see that \( \Delta {\mathbf{v}} = 0 \) (the Navier–Stokesequation 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 velocity at the surface and the velocity of the boundary (tank wall and bottom). With the equation of the ideal fluid, the boundary conditions normal to the free liquid surface and the tank wall and bottom can be satisfied. Hence, the normal component of the liquid velocity suffers only slightly from the rotational flow in the thin layer at the boundaries.
The Eq. (5) of the ideal fluid can not satisfy the boundary conditions concerning the consistency of the tangential fluid velocity and the velocity of the boundaries (tank wall and bottom, surface). The solution of the potential Eq. (5) gives tangential fluid velocities at the boundary different from those of the boundary itself. Thus, a significant change of the tangential velocity must occur across the thin boundary layer.
In order to investigate the characteristics and properties of the boundary layer, the simple onedimensional shear flow of a viscous fluid over an oscillating plane is analyzed (Landau and Lifschitz 1987).
Solution of the onedimensional shear flow
The outcome of this equation determines that \( v_{x} = const \). On the \( yz \)plane the boundary condition \( v_{x} = 0 \) at \( x = 0 \) must be considered. Hence, \( v_{x} \) becomes \( 0 \) for all \( x \). All fluid velocities depend only on \( x \) and \( t \). Consequentially the expression \( ({\mathbf{v}}\nabla ){\mathbf{v}} \) can be simplified to \( v_{x} \frac{\partial }{\partial x}{\mathbf{v}} \), and it becomes 0 because of the condition \( v_{x} = 0 \).
The general solution of Eq. (10) is:
The obtained expression describes a propagating harmonic wave. The wave amplitude decreases exponentially with the distance in xdirection. Thus, the solution describes a spatially damped, transversal wave (\( v_{y} \) is perpendicular to the direction of propagation). The imaginary part of the complex wave number \( k_{2} \) determines the damping parameter. In contrast, the real part of \( k_{2} \) is inversely proportional to the wavelength (\( \lambda = \frac{2\pi }{{\Re (k_{2} )}} \)). The penetration depth \( \delta_{L} = \sqrt {\frac{{2\nu_{L} }}{\Omega }} \) shows the decay of the damped oscillation into the fluid. The oscillation amplitude diminishes with the factor \( e = 2.718 \ldots \) at about a distance of \( x = \delta_{L} \). At a wavelength of approximately (\( 2\pi \delta_{L} \)), the amplitude decreases with the factor \( e^{2\pi } \approx 535 \). The damping of the oscillation amplitude increases with the increase of the frequency \( {{\Omega }} \) and is inversely proportional to the viscosity of the fluid. In contrast, the penetration depth decreases with increasing frequency and is directly proportional to the viscosity of the fluid.
Tangential fluid velocity on the boundaries to the tank shell
There are no restrictions to the Reynold’ number. The neglecting of the term \( ({\mathbf{v}}\nabla ){\mathbf{v}} \) of the Navier–Stokesequation (Eq. 2) stems from the following considerations. The operator \( ({\mathbf{v}}\nabla ) \) describes the derivation in the direction of the fluid velocities. On the fluid boundaries, the largest velocities are those which are parallel to the surface. Along these lines a considerable change in velocity occurs only over distances with a length comparable to the dimensions of the tank (\( R \) and \( H_{L} \)). According to this, the following relations are true: \( ({\mathbf{v}}\nabla ){\mathbf{v}}\sim \frac{{{\mathbf{v}}^{2} }}{R}\sim \frac{{q^{2} \Omega^{2} }}{R} \). The velocity has a magnitude of \( v\sim q\Omega \). Thus, the magnitude of the partial derivative of the velocity \( \frac{{\partial {\mathbf{v}}}}{\partial t} \) can be given \( v\Omega \sim q\Omega^{2} \). A comparison of both relations shows that \( ({\mathbf{v}}\nabla ){\mathbf{v}} \ll \frac{{\partial {\mathbf{v}}}}{\partial t} \) actually holds for \( q \ll R \).
Now a small section of the fluid surface is considered to determine the distribution of the tangential velocity. The dimensions of the surface section are small compared to the dimensions of the tank and large compared to the penetration depth \( \delta_{L} \). By doing this, the surface section may be assumed to be a plane surface, and the results developed in the previous section can be used.
The \( x \) axis points toward the normal of the surface.
Damping effect of the viscous fluid
The expression \( \frac{\partial \varPhi }{\partial n} \) indicates the velocity normal to the surface of the liquid (positive inside). The impulsive pressure on the surface is denoted by \( {\varrho }_{L} \varPhi \).
Damping of the convective component
The main part of the energy dissipation results from the friction on the tank wall and bottom. The friction due to the rotational flow on the liquid surface is small and can be neglected. The dissipated energy arises from the friction on the tank wall and tank bottom due to the fluid flow in radial and circumferential direction.

Velocity in axial direction on the tank wall:

Velocity in circumferential direction on the tank wall:

Velocity in circumferential direction on the tank bottom:

Velocity in radial direction on the tank bottom:

Integral of the axial flow at the tank wall:

Integral of the radial flow at the tank bottom:

Integral of the flow in circumferential direction on the tank wall:

Integral of the flow in circumferential direction on the tank bottom:
Conclusion

The presented distinction of fluid damping is especially important for fluids with high viscosity and if the sloshing motion has a remarkable contribution to the overall fluid response (e.g. tanks with small dimensions and high sloshing frequency or if the earthquake is dominated by low frequencies). In most practical cases of tank geometry and fluid properties a damping ratio of 0.5 % (like in Eurocode 8, Part 4) is a too optimistic assumption.

The damping factor decreases with increasing fluid volume because of the descent of the surfacetovolume ratio of the fluid.

The damping factor is frequency dependent; it becomes smaller for higher sloshing modes.
It would also possible to apply the method also to the impulsive pressure component, but the impulsive component is dominated by the structural damping of the tank shell. Using normalized damping coefficients, an easytouse procedure for the calculation of the damping ratios is proposed and it seems to be applicable for seismic codes. It would be interesting and maybe a part of a future work to compare the predicted damping ratios with experimental results and those of more sophisticated numerical calculations.
Declarations
Acknowledgements
The research for this paper was a continuation of the project “Calculation of liquid storage tanks and silo structures subjected to seismic loadings” carried out at the Bauhaus University Weimar and the TU Munich (Wunderlich et al. 1999). It was financially supported by the German National Science Foundation (DFG). The author is grateful for this support.
Compliance with ethical guidelines
Competing interests The author declares that he has no competing interests.
Open AccessThis 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.
Authors’ Affiliations
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