RD-E: 0601 Fluid Structure Coupling

Sloshing inside a fuel tank by simulating the fluid structure coupling. The tank deformation is achieved by applying an imposed velocity on the left corners. Water and air inside the tank are modeled with the ALE formulation. The tank container is described using a Lagrangian formulation.

A numerical simulation of fluid-structure coupling is performed on sloshing inside a deformable fuel tank. This example uses the ALE (Arbitrary Lagrangian Eulerian) formulation and the hydrodynamic bi-material law (/MAT/LAW37) to model interaction between water, air and the tank container.

Options and Keywords Used

Fluid structure coupling simulation, and ALE formulation
Shell and brick elements
Hydrodynamic and bi-phase liquid gas material (/MAT/LAW37 (BIPHAS))
ALE boundary conditions (/ALE/BCS)
J. Donea Grid Formulation (/ALE/GRID/DONEA)
Boundary conditions (/BCS)
Gravity (/GRAV)
Imposed velocity (/IMPVEL)
ALE material formulation (/ALE/MAT)

Velocities (/IMPVEL) are imposed on the left corners in the X direction.

Table 1. Imposed Velocity versus Time Curve
Velocity (ms^-1)	0	5	0	0
Time (ms)	0	12	12.01	50

Regarding the ALE boundary conditions, constraints are applied on:

Material velocity
Grid velocity

All nodes, except those on the border have grid (/ALE/BCS) and material (/BCS) velocities fixed in the Z-direction. The nodes on the border only have a material velocity (/BCS) fixed in the Z-direction.

Both the ALE materials air and water must be declared ALE using /ALE/MAT.

Note: Lagrangian material is automatically declared Lagrangian.

The /ALE/GRID/DONEA option activates the J. Donea grid formulation to compute the grid velocity. See the Radioss Theory Manual for further explanations about this option.

Input Files

Refer to Access the Model Files to download the required model file(s).

The model files used in this example include:

TANK_*.rad

Model Description

A rectangular tank made of steel is partially filled with water, the remainder being supplemented by air. The initial distribution pressure is known and supposed homogeneous. The tank container dimensions are 460 mm x 300 mm x 10 mm, with thickness being at 2 mm.

Deformation of the tank container is generated by an impulse made on the left corners of the tank for analyzing the fluid-structure coupling.

The steel container is modeled using the elasto-plastic model of Johnson-Cook law (/MAT/LAW2) with the following parameters:

Material Properties
Density: 0.0078 $[\frac{g}{m m^{3}}]$ $[\frac{g}{m m^{3}}]$
Young's modulus: 210000 $[MPa]$ $[MPa]$
Poisson's ratio: 0.29
Yield stress: 180 $[MPa]$ $[MPa]$
Hardening parameter: 450 $[MPa]$ $[MPa]$
Hardening exponent: 0.5

The material air-water bi-phase is described in the hydrodynamic bi-material liquid-gas law (/MAT/LAW37). Material LAW37 is specifically designed to model bi-material liquid gas.

The equations used to describe the state of viscosity and pressure are:

Viscosity: $S_{i j} = 2 ρ v {\dot{e}}_{i j}$ $S_{i j} = 2 ρ v {\dot{e}}_{i j}$; $σ_{k k} = λ {\dot{ε}}_{k k}$ $σ_{k k} = λ {\dot{ε}}_{k k}$
Liquid EOS: $P_{l} = P_{0} + C_{l} μ$ $P_{l} = P_{0} + C_{l} μ$; Where, $C_{l} = ρ_{0}^{l} {(c_{0}^{l})}^{2}$ $C_{l} = ρ_{0}^{l} {(c_{0}^{l})}^{2}$ and $μ = \frac{ρ}{ρ_{0}} - 1$ $μ = \frac{ρ}{ρ_{0}} - 1$
Gas EOS: $P v^{γ} = P_{0} v_{0}^{γ} = c o n s t a n t$ $P v^{γ} = P_{0} v_{0}^{γ} = c o n s t a n t$; Where, $v = \frac{V}{V_{0}}$ $v = \frac{V}{V_{0}}$ as special volume

The equilibrium is defined by: $Δ P_{l} = Δ P_{g}$ $Δ P_{l} = Δ P_{g}$

Where,

$S_{i j}$ $S_{i j}$: Deviatoric stress tensor
$e_{i j}$ $e_{i j}$: Deviatoric strain tensor

Material Parameters - For Liquid
Liquid reference density, $ρ_{l 0}$ $ρ_{l 0}$: 0.001 $[\frac{g}{m m^{3}}]$ $[\frac{g}{m m^{3}}]$
Liquid bulk modulus, $C_{l}$ $C_{l}$: 2089 N/mm²
Initial mass fraction liquid proportion, $a_{l}$ $a_{l}$: 100%
Shear kinematic viscosity $ν_{l}$ $ν_{l}$ , $= μ / ρ_{0}^{l}$ $= μ / ρ_{0}^{l}$: 0.001 mm²/ms

Material Parameters - For Gas
Gas reference density, $ρ_{0}^{g}$ $ρ_{0}^{g}$: 1.22x10^-6 $[\frac{g}{m m^{3}}]$ $[\frac{g}{m m^{3}}]$
Shear kinematic viscosity $ν_{g}$ $ν_{g}$ , $= μ / ρ_{0}^{g}$ $= μ / ρ_{0}^{g}$: 0.00143 mm²/ms
Constant perfect gas, $γ$ $γ$: 1.4
Initial pressure reference gas, $P_{0}$ $P_{0}$: 0.1 N/mm²

The main solid TYPE14 properties for air/water parts are:

Properties
Quadratic bulk viscosity/linear bulk viscosity: 10^-20
Hourglass bulk coefficient: 10^-5

Model Method

Air and water are modeled using the ALE formulation and the bi-material law (/MAT/LAW37). The tank container uses a Lagrangian formulation and an elasto-plastic material law (/MAT/LAW2).

Using the ALE formulation, the brick mesh is only deformed by tank deformation the water flowing through the mesh. The Lagrangian shell nodes still coincide with the material points and the elements deform with the material: this is known as a Lagrangian mesh. For the ALE mesh, nodes on the boundaries are fixed in order to remain on the border, while the interior nodes are moved.

Results

Curves and Animations

Fluid - Structure Coupling

Kinematic conditions generate oscillations of the structure.

Table 2. Fluid Structure Coupling - Time = 0 ms
Density
Velocity

Table 3. Fluid Structure Coupling - Time = 12 ms
Density
Velocity

Table 4. Fluid Structure Coupling - Time = 42 ms
Density
Velocity