A 1D model for unsteady fluid structure interactions in a

A 1D model for unsteady fluid–structure interactions in a soft-walled microchannelIntroduction 1D model derivation Numerical scheme FSI dynamics FSI at steady-state Summary
A 1D model for unsteady fluid–structure
interactions in a soft-walled microchannel
Tanmay C. Inamdar & Ivan C. Christov http://tmnt-lab.org/
School of Mechanical Engineering Purdue University
Focus Session: The Physics of Microscale Fluid Structure Interactions: Fully Coupled Flow and Deformation Mechanics II (D04.00009)
71st Annual Meeting of the APS DFD
Atlanta, Georgia
November 18, 2018
Inamdar & Christov (Purdue) Unsteady micro FSI APS DFD 2018 1 / 14
PURDUE UNIVERSITY COLLEGE OF ENGINEERING BRAND MANUAL page 31
THE SIGNATURE Always use the correct artwork for the schools of Engineering signatures. The left side is always the university name as independent usage or the name of the division level for lockup usage with Frutiger Black font. The right side can be changed to the name of the College or divisions with Frutiger Light font.
VISUAL IDENTITY
Microscale unsteady FSIs: Examples
Cross section of buckled tube

û
LdownLLup
pext
b pext
a
a
a
Figure 1 (a) Sketch of the Starling resistor, a thin-walled elastic tube, mounted on two rigid tubes, and enclosed in a pressure chamber, and (b) its 2D equivalent, a channel in which part of one wall is replaced by a prestressed elastic membrane. (c) Illustration of the sloshing mechanism. The flow is decomposed into its mean and oscillatory components, u and u, respectively. The wall motion creates oscillatory sloshing flows in the upstream and downstream rigid sections. At sufficiently high frequency, the sloshing flows have a blunt inviscid core, with thin Stokes layers near the walls.
Early collapsible-tube experiments, reviewed in Bertram (2003), identified a large number of different types of oscillations, ranging from high-frequency flutter to low-frequency milking. However, the mechanisms responsible for the onset of these oscillations remain poorly understood; in particular, it remains unclear whether they are related to flow-induced instabilities, such as the traveling-wave-flutter or static-divergence mechanisms that successfully explain the onset of instabilities in other fluid-elastic systems (see, e.g., Carpenter & Garrad 1985, 1986).
Most early theoretical analyses of flow in the Starling resistor were based on lumped-parameter or spatially 1D models (reviewed in Heil & Jensen 2003, Pedley & Luo 1998). Such models are still widely used to describe networks of collapsible tubes, e.g., Bull et al.’s (2005) study of flow limitation in liquid-filled lungs; Fullana & Zaleski’s (2009) analyses of flow in the venous network; and Venugopal et al.’s (2009) investigations of the lymphatic system, in which the collapsible vessels’ stiffnesses are adjusted to simulate active pumping. The relative simplicity of these models facilitates rigorous mathematical analysis, which often aids the identification of mechanisms that explain the systems’ behavior. However, a shortcoming of this approach is the need to provide closure assumptions, e.g., to capture the effects of viscous dissipation within the framework of a 1D, cross-sectionally averaged flow model, or to represent the 3D wall mechanics in terms of a so-called tube law: a postulated functional relationship between the vessel’s cross-sectional area and the local transmural pressure. Results obtained from models involving such closure assumptions must be treated with caution. Indeed, many models of flow in the Starling resistor that used plausible, but nonetheless ad hoc, closure relations were able to predict the occurrence of self-excited oscillations, despite the fact that the underlying assumptions were later discredited. For instance, Cancelli & Pedley’s (1985) widely used assumption that the majority of the viscous dissipation arises in the separated-flow region downstream of the tube’s most strongly collapsed cross section (the throat) was found to be unsupported when subsequent Navier-Stokes simulations by Luo & Pedley (1996) showed that most of the dissipation arises in boundary layers on the upstream tube walls.
www.annualreviews.org • Fluid-Structure Interaction in Internal Physiological Flows 143
A nn
u. R
ev . F
lu id
M ec
h. 2
01 1.
43 :1
41 -1
62 . D
ow nl
oa de
d fr
om w
w w
.a nn
ua lre
vi ew
s.o rg
by P
rin ce
to n
U ni
ve rs
ity L
ib ra
ry o
n 11
/2 4/
12 . F
or p
er so
Bioinspired MF (Small, 2018).
A lung-on-chip (Huh et al., Science, 2010).
Introduction 1D model derivation Numerical scheme FSI dynamics FSI at steady-state Summary
Governing equations of the solid mechanics problem
xx
y

Fluid
Solid
Figure: Schematic of an undeformed 2D channel base geometry (width w out of page).
I The dimensionless solid mech. eq. is
∂2UY
= P loading
• E = Young’s modulus [Pa] • I = second moment of inertia [m4] • ρs = mass per unit length [kg/m]
• uy = displacement in y -direction [m]; u′y = wp0` 4/(EI )
Governing equations of the fluid mechanics problem
I Dimensionless Navier–Stokes [lubrication approx. ε 1, εRe ∼ 1]:
∂VX
I Pressure scale and 2 key dimensionless fluids groups:
p0 = ρf νq0`
= ε
√ EI
ν .
• ν = kinematic viscosity [m2/s] • ρf = mass density [kg/m3] • q0 = inlet area flow rate [m2/s]
Reducing the fluid model to 1D and coupling
I Kinematic boundary condition at top wall: St ∂H∂T = VY |Y=H .
I Integrate over Y and introduce Q(X ,T ) = ∫ H
0 VXdY , then von
Karman–Polhausen approx., VX = 6QY [H(X ,T )− Y ]/H(X ,T )3, yields
∂Q
I Fluid→solid coupling through load on the wall.
I Solid→fluid coupling through change in height: H = 1 + βUY , where the FSI parameter β is
β = ρf νq0w`
.
Summary of the reduced-order (1D) model
∂2UY
X=0
X=1
P|X=1 = 0 (outflow).
ICs: U|T=0 = 0 (flat wall); Q|T=0 = 1 (fully-developed flow).
Introduction 1D model derivation Numerical scheme FSI dynamics FSI at steady-state Summary
Computational unsteady FSIs: Segregated approach
Fluid eqs.
Solid eqs.
I Constructed a numerical scheme for our 1D unsteady FSI problem.
I Fully-implicit time stepping: • required for stability, • requires Picard iterations due to nonlinearity
and pressure coupling, • check flow and deformation residuals for
convergence in FSI loop.
I Complete implementation details available in our preprint.
Channel inflation from a flat state
(ε = 0.02, St = 1, εRe = 1.15, β = 918)
0 1 2 3 4 5 T
1
2
3
4
)
Figure: Example time history of the inlet pressure P(0,T ) and outlet flow rate Q(1,T ).
[Backup video link]
z_displacement50t.mp4
Intermediate states at larger Re
(ε = 0.02, St = 1, εRe = 13.8, β = 918)
0 5 10 15 20 25 30 T
−10
0
10
20
30
)
Figure: Example time history of the inlet pressure P(0,T ) and outlet flow rate Q(1,T ) and moderate Re.
[Backup video link]
z_displacement600t.mp4
Scaling of the max. deformation with Reynolds #
I Define an Re? based on maximum channel deformation at steady state reaching 10× h0f [max(H) = 10].
10−2 10−1 100 101 102 103 104
Re
2
4
6
8
10
12
14
Σ = 1.0063× 104
Σ = 1.0063× 105
Σ = 1.0063× 106
Σ = 1.0063× 107
Σ = 1.0063× 108
Σ = 1.0063× 104
Σ = 1.0063× 105
Σ = 1.0063× 106
Σ = 1.0063× 107
Σ = 1.0063× 108
Figure: Maximum height of the microchannel top wall at steady-state, i.e., max0≤X≤1 H(X ,T 1), vs. Re for different values of Σ.
Reynolds # vs. dimensionless Young’s modulus
I Re? as defined previously correlates very well with ∝ Σ3/4, similar to microchannel instability problems (Verma & Kumaran, JFM, 2011, 2013).
103 104 105 106 107 108 109
Σ
Re*, tension (Simulation) 3/4 Scaling
Figure: Without tension (α = 0, triangles) and with tension (α 6= 0, squares).
Scaling of the maximum channel height
I How much deformation occurs in steady state, given Re & Σ?
10−7 10−6 10−5 10−4
Re/Σ0.9
101
Σ = 1.0063× 104
Σ = 1.0063× 105
Σ = 1.0063× 106
Σ = 1.0063× 107
Σ = 1.0063× 108
Σ = 1.0063× 104
Σ = 1.0063× 105
Σ = 1.0063× 106
Σ = 1.0063× 107
Σ = 1.0063× 108
Figure: Maximum steady-state deformation max0≤X≤1 H(X ,T 1) without tension (α = 0, filled symbols) and with tension (α 6= 0, empty symbols).
Scaling of the inlet pressure
I What is force required to maintain steady state shape, given Re & Σ?
10−7 10−6 10−5 10−4
Re/Σ0.9
100
101
Σ = 1.0063× 104
Σ = 1.0063× 105
Σ = 1.0063× 106
Σ = 1.0063× 107
Σ = 1.0063× 108
Σ = 1.0063× 104
Σ = 1.0063× 105
Σ = 1.0063× 106
Σ = 1.0063× 107
Σ = 1.0063× 108
Figure: Inlet pressure at steady-state P(0): without tension (α = 0, filled symbols) and with tension (α 6= 0, empty symbols).
Summary (preprint at arXiv:1808.03954)
I Developed 1D (reduced) model for transient FSI in a microchannel. • Can simulate the highly nonlinear dynamics of inflation. • For a Re/Σ combination, Hmax and Pinlet data collapses into possible
universal scaling.
−10
0
10
20
30
Re/Σ0.9
101
Thank you for your attention!
PURDUE UNIVERSITY COLLEGE OF ENGINEERING BRAND MANUAL page 31
THE SIGNATURE Always use the correct artwork for the schools of Engineering signatures. The left side is always the university name as independent usage or the name of the division level for lockup usage with Frutiger Black font. The right side can be changed to the name of the College or divisions with Frutiger Light font.
VISUAL IDENTITY
Supported, in part, by CBET-1705637

Documents

A 1D model for unsteady fluid structure interactions in a