cfd determination of inside fluid reservors movements and wall loads earthquake 1698

7/28/2019 CFD Determination of Inside Fluid Reservors Movements and Wall Loads Earthquake 1698

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Computer fluid dynamics determination of inside fluid reservoirs

movements and walls loads during and earthquakes

Lecturer Dr. Eng. Ioan Sorin Leoveanu1

1University “Transilvania” from Braşov

Rezumat

Rezervoarele sunt constructii destinate depozitarii unei mari varietati de fluide, gaze lichefiate sau diverse gaze.

In timpul proceselor de umplere, golire sau in cazul unui cutremur lichidele stocate in acestea vor dezvolta un

sistem de incarcari dinamice care poate afecta integritatea structurii. Lucrarea propsa are ca scop utilizarea

analiza numerica bazata pe calculul dinamicii fluidului (CFD) pentru a determina dinamica presiunii exercitate defluid asupra rezervoarelor in timpul manevrarii acestora. In lucrare s-a pus un accent deosebit pe efectul pe care

o miscare seismica avind caracteristicile marilor cutremure din Vrancea. Accelerogramele fiind ale ultimului mare

cutremur. Se considera ca sistemul de forte distribuite produse de miscarea fluidului este format din forte

distribuite normale pe suprafata in contact cu fluidul (presiune impulsiva) si intr-un sistem de presiuni distribuite

tangential pe suptafata de contact (presiune convectiva). Lucrarea isi propune sa puna in evidenta complexitatea

unor astfel de probleme.

Cuvinte cheie - Dinamica fluidelor, cutemure de pamint, rezervoare, sisteme de incarcare

Abstract

Ground-supported reservoirs are used for a large variety of fluids and liquefied gases storages. During the

processes of filling, empting and other exterior exceptional conditions the liquid develop a particular system of

dynamic loading that can affect the integrity of the structure. The present papers consider a CFD analysis for

establishing that effects of loading history of fluids that can take place on the walls of the reservoirs in the

conditions of filling, empting and in the earthquake time. The pressure distribution on the walls is transformed in

normal pressure (impulsive pressure) and in the tangent pressure with the walls (convective pressure) and

applied to the structure of the reservoir. The verifications of the mathematical modeling were done using the

acceleration spectra specifically to an earthquake with 7.4 magnitudes on Richter scale acting on both directions.

This paper provides theoretical background for investigation of hydrodynamic pressure that is being developing

during an earthquake in the liquid storage ground-supported rectangular container.

Key words - computer fluid dynamics, earthquake, reservoirs, loading systems

1. Introduction

Satisfactory performance of tanks during strong ground shaking is crucial for modern facilities. Tanksthat were inadequately designed or detailed have suffered extensive damage during past earthquakes [2-7] or external exceptional loads. The knowledge of forces and pressure acting on the tanks walls duringthe earthquake, explosions, tsunami and other natural or military exceptional loads plays essential role

in reliable and durable design of structure resistance tanks, which are made from steel or concrete andworking at the soil level, inside the soil or over the soil level. From the last big earthquake theknowledge of the fluid movement inside the waste reservoirs and tank become more important untilnow. The use of prescriptions and codes get there limits in the cases of extreme conditions functional of



that important buildings. The problems consist in loads system establishing for the geometry other thatcircular or rectangular and in the case of waves breaking walls inside the reservoir. The paper works tryto overcome those problems by using the CFD modelling. The numerical methods frequently used for incompressible fluids are based on VOF or MAC method [8-10] and a large number of versions basedon the two methods exist. In the paper we use the MAC method with some modifications from the

boundary condition. We consider for the beginning the simple 2D analyses for a good and quick estimation of the differences between an analytic and numerical solution. The algorithm developedconsider the free surface too facilities and the small walls deformations. The results obtained put inevidence the complexity of the problem and the importance of knowledge the liquid movement in thecase when for some applications the fluid must be pumped from the reservoir in the moment when aquake take place.

1. Numerical analysis

In this paper we try to establish the pressure distribution dynamics on the walls of reservoirs using theconstitutive Navier-Stokes system of differential equations for liquids and the fluid dynamics inside thetank based on a given acceleration spectra of a quake. In this case the earth movement processesimposed by quake modify the equilibrium of fluid stored in the tank and the fluid speed, shape and

pressure change in accord with the external loads. The geometry and system of loading acting on theliquid volume stored in the tank is getting in the figure 1 for an analyzed more simple case. The shapeof free surface of liquid was modeled using marker and cell method (MAC) in accord with theMarangony flow and the resulting maps pressures on the walls are in accord with the complex movingfluid conditions. The governing equations of fluid, considered in present work and the work hypothesisare:

l) The flow is 2-D, incompressible and laminar.2) Each thermal property of incompressible fluid is constant.3) The walls deformations are small and the structure move between the earthquakes with the instantacceleration of the quake.4) The time computed effects of the quake on the tank is double5) The heat dissipation and the turbulent indices are calculated only in the fluid control volume.Then the general differential equation for conservative lows can be given as:

( ) ( )S

x x x x x

v

x

v

t j jii j

j

i

i +

Γ +

Γ =++

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂ φφφφφ(1)

The definitions of the letters, φ , Γ and S are done in the table 1 where µ is the viscosity of the fluid, g x

and g y the gravitational acceleration, λ /( ρ c) the heat diffusivity and v x , v y , v z , ρ and T the fluid speed onx, y and z direction, the density and the temperature of the fluid walls will have the same acceleration asthe walls cells.

1.1. Boundary conditions.

In accord with the figure 1, the fluid in contact with the walls will be accelerating with the sameacceleration that acts on the wall. In our case, with the consideration that the walls are rigid and there

deformation is neglected, the fluid in the cells near the walls will have the same speed like the wall. Thegeneral equations used are given in table 1.Table 1.



1.1.1. Boundary condition on cells in contact with the walls.

For fluid flow analysis, either the slip wall condition or the no-slip wall condition is adopted [9, 10],according to the size of a cell and the magnitude of velocity. When the walls move, the convectioncoefficient between fluid and solid boundary is calculated, based on Re x and Pr numbers in the fluidcells. The heat flow equation is used only to calculate the fluid temperature distribution variation in thequake action. For water and other liquids the temperature is not important but for oils and liquefiedgases the knowledge of fluid temperature and pressure becomes important. The equations for boundary

domain in contact with solid walls become:

k w jviuv

aw z

pw

z

wv

y

wu

x

w

t

w

av y

pw

z

vv

y

vu

x

v

t

v

au x

pw

x

uv

y

uu

x

u

t

u

Z

Y

X

⋅+⋅+⋅=

⋅+∆+∂

∂−

∂

∂−

∂

∂−

∂

∂−=

∂

∂

⋅+∆+∂

∂−

∂

∂−

∂

∂−

∂

∂−=

∂

∂

⋅+∆+∂

∂−

∂

∂−

∂

∂−

∂

∂−=

∂

∂

ρ µ ρ

ρ µ ρ

ρ µ ρ

(2)

Where a X ,aY and a Z are the spectrum of quake accelerations transmitted to the walls. In accord with the3) hypothesis, that accelerations will be take in calculus equal with the earthquake accelerations.

1.1.2 The free boundary of the fluid.

On free surface, the sum of tangential stresses must be zero and the sum of normal stresses must beequal to the applied stresses or pressures. The tangential and normal stress conditions are applied onfree surface [7, 10]. Where u, v, w are the x, y and z component of velocity; n x , n y and n z are the unitvectors which refer to the local tangential and normal direction of the surface, respectively.

The definition of φ follows the former expression, that is, φa = (pext )/ ρ + γT / Rm; while μ represents

kinematics viscosity; γT is the surface tension function of temperature; pext is the pressure of gas phaseinside the tank and, Rm is the local mean radius of the free surface. The free surface temperature timevariation is considerate nulle.Tangential stress condition for 2D modelling:

ϕ Γ SContinuity ρ 0 0

Momentvi

velocity on

xi direction

μi x

i

g x+

∂

∂φ

v j

velocity on

x j direction

μ j x

j

g x

+

∂

∂φ

wk velocity

on xk

direction

µ k x

k

g x

+

∂

∂φ

Energy Ttemperatur

e

cρ

λ q



t

T

x

v

y

unn

y

v

x

unn T x y y x ji ∂

∂)(2 22 γ µ =

∂∂

+∂∂

−+

∂∂

−∂∂

(3)

Normal stress condition for 2D simulation:

2∂

∂

∂

∂

∂

∂

∂

∂ 22 a

y y x x y

vn x

v

y

unn x

un ji

φ µ =

+

++ (4)

Tangential stress condition for 3D modelling:

t

T

x

w

z

unn

x

v

y

unn

y

v

x

unn Txy x z x y y x ji ∂

∂2

∂

∂

∂

∂)(2 22

γ µ =

∂∂

−∂∂

+

+−+

∂∂

−∂∂

(5)

t

T

x

v

y

unn

x

w

z

unn

z

w

x

unn Txz y x x z y x ji ∂

∂2

∂

∂

∂

∂)(2 22 γ µ =

∂∂

−∂∂

+

+−+

∂∂

−∂∂

(6)

t

T

y

w

z

vnn

z

v

y

wnn

z

w

y

vnn Tyz y z y z z y ji ∂

∂2

∂

∂

∂

∂)(2 22

γ µ =

∂∂

−∂∂

+

+−+

∂∂

−∂∂

(7)

Normal stress condition for 3D simulation:

a z y z x y x z y x y

w

z

vnn

x

w

z

unn

x

v

y

unn

z

wn

y

vn

x

un φ µ =

∂∂

+∂∂

+

∂∂

+∂∂

+

++

∂∂

+∂∂

+ 22∂

∂

∂

∂2

∂

∂ 222 (8)

1.1.3 Initial and load conditions

Because the problem is time dependent, the initial conditions consist in imposing null speed value for v x

and v y and the gas pressure equal with the atmospheric pressure at sea level. The temperature wasconsiderate equal with 200C on the air interface and 150C on the walls interfaces and in these conditionsthe buoyancy movements inside the fluid is considerate. After the first time step, the quake event isconsiderate and the spectra given in the figure 2 was applied onthe liquid/solid boundary accordingly with figure 1 and the inside initial fluid speed distribution isconsiderate in accord with figure 2.



Figure 2. The acceleration specters considerate in the problem modelling

3.Solution and results

As an example case we will assume the ground supported rectangular endlessly tank with L = 24 mlength and H = 12 m height and with uniform concrete thickness of 0.25 m. The tank is filled with water until h=6 m in the first application and h = 11.50 m in the second application. .

a) t = 1s b) t=2s c) t=3s d) t=4s

e) t=1s f) t=2s g) t=3s h) t=6s



Figure 3. a)-d) Pressure distribution for filled tank H=L, e)-h) pressure distribution for half filled tank.

a) t = 1s T = 2 s

c) t = 3 s d) t = 4 s

e) t = 5 s f) t = 6 s

g) t = 7 s h) t = 8 s



i) t = 9 s j) t = 10 s

h) t = 11 s l) t = 12 s

n) t = 16 sm) t = 19 s

o) t = 22 s p) = 23 s

Figure 4. The u speed distribution in the fluid during the analyzed earthquake



a) t = 1 s b) t = 2 s

c) t = 3 s d) t = 4 s

e) t = 5 s f) t = 6 s

g) t = 14 s h) t = 19 s



i) t = 23 s j) t = 29 sFigure 5. The internal heat distribution inside fluid

a) t=1s b) t=2s c) t=3s d) t=4s

e) t=5s f) t=6s g) t=7s h) t=8s

i) t=9s j) t=10s h) t=35s k) t=40sFigure 6. The pressure distribution inside the liquid area a)-d) L=24 m, H=12 m and e)-h) L=48 m,H=12 m. e)-f) speed v evolution inside the fluid area.

a) t=1s b) t=2s c) t=4s d) t=5s

e) t=1s f) t=7s g) t=14s h) t=40sFigure 5. a)-d). The distribution of viscous heat generation. e)-h) The speed v x distribution in the liquid.



physical processes involved in welding phenomenons using Finite Volume and Finite Elements Method for Modelling theExceptional Loads induced in Builings by Earth Quakes, Wind and Explosions.

cfd determination of inside fluid reservors movements and wall loads earthquake 1698

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