Numerical Modelling of Cylindrical Fluid Filled Tank

KAMILA KOTRASOVÁ1,*, PETR FRANTÍK2, EVA KORMANÍKOVÁ1

1Institute of Structural Engineering and Transportation Structures,

Faculty, Faculty of Civil Engineering,

Technical University of Košice,

Vysokoškolská 4, 042 00 Košice,

SLOVAKIA

2Institute of Structural Mechanics,

Faculty of Civil Engineering,

Brno University of Technology,

Veveří 331/95, 602 00 Brno,

CZECH REPUBLIC

*Corresponding Author

Abstract: - The demand for drinking and service water storage is rising with changing climate conditions and

increasing life expectancy. The tanks are commonly used to store large volumes of liquids and materials in

various fields of the economy. This paper presents the model of the numerical simulation for the steel tank

filled with fluid, using the finite element method. The results of the tank filled with water are presented, by the

results: the pressure of the fluid the effective stress, and the maximum deformation of the tank solid domain.

The correctness of the pressure values was verified by the simple calculation of the fluid pressure. Finally, the

paper documents the results for various fluid fillings with a considered range of fluid densities. The influence of

the fluid filling height on the behavior of the solid domain of the fluid filling container loaded by the static

loading as well as the effect of the width of the tank on the behavior of the solid domain of the fluid filling

container.

Key-Words: - Fluid, tank, FEM, static, analysis, density.

Received: June 19, 2023. Revised: May 21, 2024. Accepted: July 13, 2024. Published: September 3, 2024.

1 Introduction

As climate patterns shift and life expectancy

increases, the need for storing drinking and service

water is on the rise. Tanks serve as vital reservoirs

for large quantities of liquids and materials across

various sectors of the economy. However,

categorizing them can pose challenges due to their

diverse shapes, intended uses, and construction

materials, [1].

The cylindrical tanks present advantages in terms

of the pressure and tension stress management on

their exteriors, as well as the material efficiency, [2],

[3]. Yet, their construction demands intricate

formwork, [4], [5].

The Solving of the problem of the reservoir filled

with liquid also includes a wide range of problems,

[6], [7], [8], [9], [10], [11], such as:

- the interaction of the fluid filling with the

solid domain of the reservoir,

- the interaction of the solid domain of the

reservoir with the foundation,

- the interaction of the fluid-filled reservoir

and the foundation situated on the real

subsoil.

From the point of view of the solutions, there are

different levels of the solutions available, ranging

from analytical problem-solving to the numerical

simulations, [12], [13], [14], [15].

The finite element method (FEM) is used to

solve a wide range of problems, [16], FEM is being

developed and constantly improved thanks to the

development of high-performance computing

techniques, [17], [18], [19], [20].

Today, FEM is the most widely used calculation

tool in many branches of engineering and science,

also used for solving, [21], [22], [23].

2 Problem Formulation

The finite element method (FEM) is the powerful

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numerical approach used to solve the complex

problems in the engineering and the applied

sciences, [24].

FEM has gained wide application in practice due

to its versatility in solving a large number of

problems, from simple structures to complex

systems, [25], [26]. Its applications include various

fields including structural mechanics, fluid

dynamics, heat transfer, and electromagnetic fields,

[27].

The essence of FEM is that it divides complex

systems, and structure domain, into smaller

components known as finite elements, [28], [29],

[30]. These elements are interconnected at

designated points referred to as nodes, as shown in

Figure 1. Each finite element is described by a

system of equations. These equations are then

combined into a set of descriptive equations that

describe the behavior of the analyzed system, [31]

as a whole.

Node Finite element

a) b)

Fig. 1: Solution of structure using FEM, a)

schematic representation of the structure domain, b)

(a) the division of the structure domain into smaller

parts - finite elements

When solving the fluid-structure interaction

problems, the FEM offers two principal

methodologies: the Eulerian and the Lagrangian

approaches, [32].

Within the Eulerian framework, the fluid's

behavior is delineated in terms of a pressure

potential, as elucidated in literature, [33]. This

approach allows for the expression of the fluid

behavior through analytical functions tailored to the

specific geometries or via finite element models

where the nodal pressures serve as the primary

unknowns. Throughout the solution process of the

fluid-structure system, the interaction effects are

enforced through iterative techniques, [34].

Conversely, the Lagrangian approach

characterizes the fluid behavior acting on the

structures, expressing it in terms of the

displacements at the finite element nodes.

Consequently, the equilibrium and the compatibility

conditions are inherently satisfied along the fluid-

structure interface, [35]. In this paradigm, the fluid

element is typically conceptualized as the elastic

solid element possessing the nominal shear modulus

and the volumetric elasticity modulus equivalent to

the fluid's bulk modulus, [36].

The advantage of the Lagrangian approach,

compared to the Eulerian approach, is that the

Lagrangian approach can be easily incorporated into

the general-purpose structural analysis programs for

the solution of the fluid-solid element considered

since there obviates the necessity for the specialized

interface equations in the Lagrangian approach.

The potential-based fluid elements, used for the

meshing of the fluid domain, incorporate the

following assumptions:

 inviscid, irrotational medium with no heat

transfer,

 compressible or almost incompressible

medium,

 relatively small displacements,

 actual fluid flow with velocities below the

speed of sound (subsonic formulation) or no

actual fluid flow (linear formulation).

The potential-based fluid elements can be

coupled with the structural elements by the fluid-

structure interface elements. The fluid-structure

interface elements apply the structural motions to

the potential-based fluid elements and apply the

potential-based fluid element pressures to the

structure. The 3-D fluid element can either be the

displacement-based fluid elements or the potential-

based fluid elements. However, in practice, the use

of displacement-based elements is rather restricted

to special applications in static and dynamic

analyses. The potential-based element is much more

general and is usually the recommended element to

use.

3 FEM Model of the Cylindrical Fluid

Filled Tank

The cylindrical water tank is considered and

subjected to the gravity loading (Figure 2). The

dimensions are

- the inner radius R = 15 m,

- the wall height H = 25 m,

- the wall thickness is 50 mm,

- the thickness of the bottom is 500 mm.

The steel tank material has properties

- Young’s modulus E = 2.071011 N/m2,

- Poisson number



= 0.3,

- the density



s = 7800 kg/m3.

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The fluid filling is

- water,

- the height filling Hf is 20 m,

- the water density



w = 1000 kg/m3,

- the water bulk modulus  = 2.1 N/m2.

The presented model encapsulates one common

comprehensive model that includes both domains,

i.e. the solid domain and the fluid domain. At the

beginning of the numerical simulation, the solid

domain of the fluid-filled tank was created, in which

the cylindrical contours of the tank were defined,

the mesh for the tank solid domain was created, and

the physical and material characteristics of the solid

domain were defined. Once the solid model was

created, the focus turned to modelling the liquid fill,

its shape, and its physical and material

characteristics.

Fig. 2: Computational model of the cylindrical

filled fluid-filled tank

Due to the coexistence of these domains within

the model, the differentiated approach was

necessary. Specifically, the fluid modeling

methodology required avoiding overlapping nodes

between domains. As a result, the fluid domain was

created slightly smaller than its solid counterpart,

ensuring an accurate 1 mm distance between their

respective nodes. The dimensions of the fluid

domain were therefore carefully calibrated to reflect

this deliberate reduction.

The 4-node shell element was used for meshing

the solid domain of the tank. Integration through

Shell thickness was used Gauss approach.

The 8-node brick Linear Potential-Based fluid

element was used for meshing of the fluid domain.

Whereas the cylindrical tank boasts a radius of

15 meters, the fluid fill radius was intricately

adjusted to 14.999 m, maintaining the requisite gap.

Similarly, meticulous consideration was given to the

vertical dimension, necessitating the modeling of a 1

mm clearance between the solid tank domain and

the fluid fill domain. Hence, the height of the fluid

domain was precisely given at 19.999 m.

Following the delineation of the fluid fill's shape,

the meshing of its domain was executed,

complemented by the comprehensive specification

of the fluid's physical and material attributes.

Through this rigorous approach, the model achieves

a holistic representation, meticulously capturing the

dynamic interplay between the solid structure and

the fluid contents it houses.

A numerical simulation, considered the static

solution, was executed on the specified tank model,

with its cavity filled with fluid, under the

assumption of the tank resting upon the rigid solid

foundation.

The pressure exerted by the water filling upon

the tank's solid domain has been documented and

graphically depicted in Figure 3. Notably, the

highest pressure manifests at the base of the fluid

domain, peaking at 195,823 Pa.

In an effort to validate these findings, a thorough

verification of the calculations was undertaken.

Utilizing the formula 𝑝 = 𝜌𝑔𝐻

𝑓, where ρ represents

the density of the fluid, g denotes the gravitational

acceleration, and Hf signifies the height of the fluid

column, the pressure was recalculated. Employing a

fluid density of 1000 kg/m3 and the gravitational

acceleration of 9.81 m/s2, the analytical given

pressure yields 196,190 Pa. This recalculated

pressure aligns closely with the originally

documented value, reaffirming the robustness and

accuracy of the numerical simulation results.

Consequently, these findings serve to verification of

the model and its utility in analyzing the structural

behavior of the tank under varying conditions.

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Fig. 3: The pressure of the fluid domain

Fig. 4: The effective stress of the tank solid domain

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a) b)

Fig. 5: The results of behavior tank solid domain, a) the maximum deformation in the direction of axis y, b)

the meshing of the deformed and undeformed solid domain of the tank, c) the strain of the tank solid domain

The effective stress of the tank solid domain by

the numerical simulation in the software Adina, is

documented in Figure 4. The maximum value of the

effective stress of the tank solid domain pressure is

in the bottom of the tank domain and its value is

1.199108 Pa.

The behavior of the solid domain of analyzed

solid domain of the fluid-filled container, loaded by

gravity loading, the maximum deformation, the

shape of the container, and the strain shown in

Figure 5, Figure 5a) the maximum deformation in

the direction of axis y, Figure 5b) the state of

meshing of the original nonloaded solid domain of

container and the deformed solid domain of the

cylindrical water-filled tank subjected to the gravity

loading, and the Figure 5c) the strain of the tank

solid domain.

3.1 The Effect of the Fluid Filling Density

The density of fluid is given by values in the range

of 600 - 1600 kg/m3. The numerical simulation was

performed for the consideration of the fluid filling

with the densities 600 kg/m3, 800 kg/m3, 1000

kg/m3, 1200 kg/m3, and 1400 kg/m3.

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The cylindrical filled fluid-filled tank is

considered, and subjected to gravity loading with

the dimensions:

- the inner radius R = 15 m,

- the wall height H = 25 m,

- the wall thickness is 50 mm,

- the thickness of the bottom is 500 mm.

The steel tank material has properties:

- Young’s modulus E = 2.071011 N/m2,

- Poisson number



= 0.3,

- The density



s = 7800 kg/m3.

Figure 6 documented the results of the behavior

of the numerical simulation of the fluid-filled

container for the considered range of the fluid

densities. Figure 6a) documents the maximum

pressure of the fluid domain for the considered

densities of the fluid filling, and the trend line is

linear. The resulting maximum effective stress of

the tank solid domain for the considered densities of

the fluid filling is presented in Figure 6b). The trend

line of the resulting maximum effective stress of the

tank solid domain is given by the polynomial

function of the second degree. Figure 6c) shows the

maximum deformation of the tank solid domain

depending on the fluid filling density.

Figure 7 and Figure 8 are documented of the

selected results processed in graphs in Figure 6. The

effective stress of the tank solid domain for the

stored fluid filling of the density 600 kg/m3 is

documented in Figure 7a) and for the stored fluid

filling of the density 1400 kg/m3 in Figure 7b). The

maximum deformation of the tank solid domain for

the stored fluid filling of the density 600 kg/m3 in

the direction of the axis y is documented in Figure

8a) and for the stored fluid filling of the density

1200 kg/m3 in the direction of the axis y in Figure

8b). The maximum deformation of the tank solid

domain for the stored fluid filling of the selected

density in the direction of axis x gives the same

value of the maximum deformation as in the

direction of the axis y. Along the circumference at

the same height, the horizontal deformations are the

same at the same height.

Fig. 6: The results of numerical solution depending

on fluid filling density, a) The resulting maximum

pressure of the fluid domain, b) the effective stress

of the tank solid domain, c) the maximum

deformation of the tank solid domain

y = 0,1954x + 0,4583

100

150

200

250

300

500 700 900 1100 1300 1500

pressure

[kPa]

[kgm-3]

y = 3E-05x2+ 0,0574x + 36,597

100

120

140

160

180

500 700 900 1100 1300 1500

eff.stress

[MPa]

[kgm-3]

y = -3E-08x2+ 0,0042x + 0,023

500 700 900 1100 1300 1500

[kgm-3]

max. deformatiom [mm]

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Fig. 7: The effective stress of the tank solid domain for the fluid filling density, a) 600 kg/m3, b) 1400 kg/m3

Fig. 8: The maximum deformation of the tank solid domain, a) for the fluid filling density 600 kg/m3, b) for

the fluid filling density 1200 kg/m3

3.2 The Effect of the Fluid Filling Height

In the next part of the numerical experiments, the

influence of the height of the liquid filling on the

behavior of the solid domain of the fluid filling

container loaded by the static loading was

monitored. The fluid filling with the height Hf of 5

m, 10 m, 15 m, and 20 m was analyzed.

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Fig. 9: The results of the effective stress of the

tank solid domain, a) for the height of the fluid

filling 15 m, b) for the height of the fluid filling 10

m, c) for the height of the fluid filling 5 m

Fig. 10: The results of the maximum

deformation of the tank solid domain, a) for the

height of the fluid filling 15 m, b) for the height of

the fluid filling 10 m, c) for the height of the fluid

filling 5 m

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The cylindrical water tank is considered, and

subjected to gravity loading with the dimensions

- the inner radius R = 15 m,

- the wall height H = 25 m,

- the wall thickness is 50 mm,

- the thickness of the bottom is 500 mm.

The steel tank material has properties:

- Young’s modulus E = 2.071011 N/m2,

- Poisson number



= 0.3,

- The density



s = 7800 kg/m3.

The fluid filling is:

- water,

- the water density is



w = 1000 kg/m3,

- the water bulk modulus  = 2.1 N/m2.

Fig. 11: The behavior of the solid domain of the

fluid filling container loaded by the static loading

depending on the fluid filling height, a) the effective

stress of the tank solid domain, b) the maximum

deformation of the tank solid domain

The results of the effective stress of the tank

solid domain for the height of fluid filling 15 m, 10

m, and for the height of fluid filling 5 m are

presented in Figure 9, Figure 9a) for the height of

the fluid filling 15 m, Figure 9b) for the height of

the fluid filling 10 m, Figure 9c) for the height of

the fluid filling 5 m, Figure 4 for the fluid filling

with the height of 20 m.

Figure 10 shows the results of the maximum

deformation of the tank solid domain for the height

of fluid filling 15 m, 10 m, and 5 m, Figure 10a) for

the height of the fluid filling 15 m, Figure 10b) for

the height of the fluid filling 10 m, Figure 10c) for

the height of the fluid filling 5 m. The maximum

deformation of the tank solid domain for the fluid

filling with a height of 20 m is presented in Figure

5a. The behavior of the solid domain of the fluid

filling container loaded by the static loading

depending on the fluid filling height is summarized

in Figure 11. The results of the effective stress of

the tank solid domain depending on the height of the

fluid filling is presented in Figure 11a), and the

results of the maximum deformation of the tank

solid domain in depending on the height of the fluid

filling is in Figure 11b).

3.3 The Effect of the Fluid Filling Width

In the last part of the numerical experiments, the

influence of the width on the behavior of the solid

domain of the fluid filling container loaded by the

static loading was monitored. The width of the

reservoir with the diameter D is 10 m, 20 m, and 30

m for 20 m fluid filling were analysed.

The cylindrical water tank is considered, and

subjected to gravity loading with the dimensions

- the wall height H = 25 m,

- the wall thickness is 50 mm,

- the thickness of the bottom is 500 mm.

The steel tank material has properties

- Young’s modulus E = 2.071011 N/m2,

- Poisson number



= 0.3,

- The density



s = 7800 kg/m3.

The fluid filling is

- water,

- the height filling Hf is 20 m,

- the water density is



w = 1000 kg/m3,

- the water bulk modulus  = 2.1 N/m2.

The results of the effective stress of the tank

solid domain for the tank diameters 10 m, and 20 m

are presented in Figure 12, and for the tank diameter

30 m in Figure 4. The results of the maximum

deformation of the tank solid domain for the tank

diameter 10 m, 20 m are presented in Figure 13 and

for the tank diameter 30 m in Figure 5a).

100

120

140

160

180

200

510 15 20

eff.stress

[MPa]

Hf[m]

5 8 11 14 17 20

Hf[m]

max. deformatiom [mm]

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Fig. 12: The results of the effective stress of the tank

solid domain, a) for the tank diameter of 20 m, b) for

the tank diameter of 10 m

Fig. 13: The results of the maximum

deformation of the tank solid domain, a) for the

tank diameter of 20 m, b) for the tank diameter

of 10 m

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Fig. 14: The behavior of the solid domain of the

fluid filling container loaded by the static loading

depending on the width of the tank, a) the effective

stress of the tank solid domain, b) the maximum

deformation of the tank solid domain

The behavior of the solid domain of the fluid

filling container loaded by the static loading

depending on the fluid filling width is summarized

in Figure 14. The results of the effective stress of

the tank solid domain depending on the diameter of

the tank is documented in Figure 14a), and the

results of the maximum deformation of the tank

solid domain in depending on the diameter of the

tank is documented in Figure 14b).

4 Conclusion

This paper presents the possibilities of modeling of

the steel tank filled with fluid, using numerical

simulation by the finite element method. The results

of the tank filled with the liquid are presented by the

pressure of the fluid filling domain the effective

stress and the maximum deformation of the tank

solid domain. The correctness of the pressure value

was verified by the simple analytical calculation of

the fluid pressure. Finally, the paper documents the

results for various fluid fillings with different values

of the density, considering range of the fluid

densities. The influence of the fluid filling height on

the behavior of the solid domain of the fluid filling

container loaded by the static loading as well as the

effect of the width of the tank on the behavior of the

solid domain of the fluid filling container.

Acknowledgements:

This research was supported by the Scientific Grant

Agency of the Ministry of Education of Slovak

Republic and the Slovak Academy of Sciences,

Project VEGA 1/0307/23, Project VEGA 1/0642/24,

and MeMoV II CZ.02.2.69/0.0/0.0/18_053

/0016962.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Kamila Kotrasova carried out the supervision,

simulation, organization, and writing – review &

editing.

- Petr Frantik has participated in conceptualization,

writing – original draft, visualization, and

validation.

- Eva Kormanikova has executed for methodology,

investigation, formal analysis, and funding.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

The authors have been supported by the Scientific

Grant Agency of the Ministry of Education of

Slovak Republic and the Slovak Academy of

Sciences under Project VEGA 1/0307/23, and

MeMoV II CZ.02.2.69/0.0/0.0/18_053 /0016962.

Conflict of Interest

The authors have no conflict of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on FLUID MECHANICS

DOI: 10.37394/232013.2024.19.24

Kamila Kotrasová, Petr Frantík, Eva Kormaníková

E-ISSN: 2224-347X

269

Volume 19, 2024