M. Khalili1, M. Larsson2, B. Müller1

1Norwegian University of Science and Technology Department of Energy and Process Engineering

2SINTEF Materials and Chemistry, Trondheim, Norway

COUPLED PROBLEMS, San Servolo Island, Venice, Italy, May 2015

Numerical Simulation of Fluid-Structure Interaction for a Simplified Model of the Soft Palate

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Introduction Motivation Anatomy of upper airway Obstructive Sleep Apnea / Snoring Assumptions Computational Challenges

Fluid Governing Equations Computational Model

Structure Governing Equation Computational Model

FSI ALE Formulation Algorithm

Results Conclusions and Outlook

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Outline

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The current study is motivated by the intention to further understand upper airway dynamics aiming to explore computational modelling as a tool in diagnosis and treatment of Obstructive Sleep Apnea Syndrome (OSAS).

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OSAS affects an estimated 2-4% of the adults and about 10% of snorers being at risk of obstructive sleep apnea.

Motivation

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Anatomy of upper airway

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OSAS / Snoring

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You can find the difference between Snoring and Obstructive Sleep Apnea in below link. https://www.youtube.com/watch?v=inmop4Kv8PI

https://www.youtube.com/watch?v=inmop4Kv8PIhttps://www.youtube.com/watch?v=inmop4Kv8PI

Assumptions

Two dimensional (2D) Air: Perfect gas, Newtonian fluid, Compressible laminar flow

Soft Palate: Cantilevered flexible plate, Thin-plate mechanics

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Computational Model

Flow Compressible flow Very low Mach number Multi-block structured grid approach The exchange of data between neighboring blocks is achieved

by Message Passing Interface (MPI)

Structure Thin plate mechanics Pressure driven

Interaction

ALE formulation Two-way coupled model

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Fluid Compressible Navier-Stokes equations Conservation law form Conservative variables Flux vectors

( )TEρ ρ ρU , u,=

( ) ( ) ( )0

2u

U uu I , U, Uu u

C VF p FH T

ρρ τρ τ κ

= + ∇ = ⋅ + ∇

( ) ( ) ( )1C VF Ft

∂+∇ ⋅ = ∇ ⋅ ∇

∂U U U, U

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Computational Model

Equations solved in a non-dimensional perturbation form.

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4th order explicit Runge-Kutta method in time.

6th order explicit filter applied at the end of each time step to suppress undamped modes.

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Computational Model

6th order finite difference discretization in space with summation by parts (SBP) operators.

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Structure

Motion of the elastic plate is modelled by classical thin-plate mechanics.

( )3xxxxM d C d B d p+ + =− ∆ Displacement d is constrained to be vertical.

( )sM h density thicknessof plateρ= ⋅ = ⋅Plate’s mass

C

( )3

212 1EhB

υ=

−

Damping

Flexural rigidity E, elastic modulus υ, Poisson ratio

External force: fluid pressure ( )upper lowerp p p∆ = −

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Computational Model

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Newmark time integration method is employed for solving implicit transient dynamics in the finite difference discretization.

Central difference discretization is used for dxxxx .

( ) ( )

( )2

1 4

1 52

0 25 0 5. , .

t t t t t t

t t t t t t t

d d d d t

d d d t d d t

γ γ

β β

β γ

+∆ +∆

+∆ +∆

= + − + ∆ = + ∆ + − + ∆

= =

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Computational Model

Boundary conditions: Clamped configuration for the leading edge. Free end configuration for the trailing edge, i.e. bending moment

and shear force is zero at the last node.

11 0

ddx

∂= =∂

2 3

2 3 0i id d

x x∂ ∂

= =∂ ∂

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Fluid-Structure Coupling

ALE formulation for Navier-Stokes equations. The grid point velocity is subtracted from the material velocity u inside the convective derivative.

û

( )ˆu u uchanged intoD D

Dt t Dt t∂ ∂

= + ⋅∇ → = + − ⋅∇∂ ∂

Solving flow problems in a moving mesh described in an ALE framework needs the numerical scheme involved in the solver to obey the Geometric Conservation Law (GCL) for mathematical consistency.

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Fluid-Structure Coupling

In mesh updating, the positions and velocities of the grid points in the fluid domain are linear interpolation of the positions and velocities of the structure.

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Algorithm

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The displacement and velocities are matched.

yf and ds represent the fluid grid and structural displacement at the interface. and are the grid velocity and structural velocity at the interface.

f s

f s

y d

y d

=

=

The aerodynamic traction is simply the pressure acting on the plate surface.

sd

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fy

Fluid-Structure Coupling

( ) ( )upper lowerp p p f external forceonthe plate∆ = − =17

Results

An inlet velocity of 0.32 m/s is applied at Re = 378, which has been shown to promote cantilever plate instability in channel flow, corresponding to previous work in Balint & Lucey [2].

[2] Balint, T.S. & Lucey, A.D. 2005 Instability of a cantilevered flexible plate in viscous channel flow. Journal of Fluids and Structures 20, 893-912.

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Results

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The frequency obtained is between 75-91 Hz.

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Results The fluid velocity

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Results

The fluid vorticity

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uω =∇×

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Conclusions and Outlook

Conclusions

It is found that the numerical model used here is suitable to investigate elastic deformation of the plate.

The numerical model is verified by previous work.

Outlook Validation is needed. More realistic soft palate properties need to be applied for

better correlation with clinically measured data.

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End

Thanks for your attention.

Acknowledgments The current research is part of a larger research project entitled “Modeling of obstructive sleep apnea by fluid-structure interaction in the upper airways” which has been funded by the Research Council of Norway.

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Appendix A

0U F Gt x y′ ′ ′+ + =Conservative form in perturbation variables

Flux vectors

F F F , G G GC V C V′ ′ ′ ′′ ′= − = +

Solution vector

( ) ( )U , u ,T

Eρ ρ ρ ′ ′′ ′=

Perturbation form

( ) ( ) ( )0 , u , E E Eρ ρ ρ ρ ρ ρ ρ′ ′ ′= − = −24

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