Computational analysis of flow-driven string dynamics in turbomachinery

Accepted Manuscript

Computational Analysis of Flow-Driven String Dynamics in Turbomachinery Kenji Takizawa, Tayfun E. Tezduyar, Hitoshi Hattori PII: DOI: Reference:

S0045-7930(16)30043-3 10.1016/j.compfluid.2016.02.019 CAF 3104

To appear in:

Computers and Fluids

Received date: Accepted date:

22 December 2015 23 February 2016

Please cite this article as: Kenji Takizawa, Tayfun E. Tezduyar, Hitoshi Hattori, Computational Analysis of Flow-Driven String Dynamics in Turbomachinery, Computers and Fluids (2016), doi: 10.1016/j.compfluid.2016.02.019

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Highlights • A method for computational analysis of flow-driven string dynamics is

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developed, targeting turbomachinery. • The components of the method are ST-VMS and ST-SI methods for the

fluid dynamics, and a one-way-dependence model and the IGA for the string dynamics.

• Two cases of pilot computations, including a ventilating fan, demonstrate

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the potential for the method in flow-driven string dynamics analysis

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Computational Analysis of Flow-Driven String Dynamics in Turbomachinery

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Kenji Takizawaa , Tayfun E. Tezduyarb,∗, Hitoshi Hattoria a Department

of Modern Mechanical Engineering, Waseda University 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan b Mechanical Engineering, Rice University 6100 Main Street, Houston, TX 77005, USA

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Abstract

We focus on computational analysis of flow-driven string dynamics. The objective is to understand how the strings carried by a fluid interact with the solid surfaces present and get stuck on or around those surfaces. Our target application is turbomachinery, such as understanding how strings get stuck on or around the blades of a fan. The components of the method we developed for

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this purpose are the Space–Time Variational Multiscale (ST-VMS) and ST Slip Interface (ST-SI) methods for the fluid dynamics, and a one-way-dependence

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model and the Isogeometric Analysis (IGA) for the string dynamics. The STVMS method is the core computational technology and it also has the features of a turbulence model. The ST-SI method allows in a consistent fashion slip at

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the interface between the mesh covering a spinning solid surface and the mesh covering the rest of the domain, and with this, we maintain high-resolution representation of the boundary layers near spinning solid surfaces such as fan

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blades. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing

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the influence of the string on the flow, which we expect to be small. The IGA for the string dynamics gives us not only a higher-order method and smoothness in the structure shape, but also smoothness in the fluid dynamics forces calculated ∗ Corresponding

author. Tel.: +1-713-348-6051. Email address: [email protected] (Tayfun E. Tezduyar)

Preprint submitted to Computers & Fluids

March 17, 2016

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on the string. To demonstrate how the method can be used in computational analysis of flow-driven string dynamics, we present the pilot computations we carried out, for a duct with cylindrical obstacles and for a ventilating fan.

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Keywords: Turbomachinery, fan, string dynamics, Space–Time Variational

Multiscale method, ST-VMS, ST Slip Interface method, ST-SI, Isogeometric Analysis, IGA, higher-order functions

1. Introduction

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In turbomachinery, objects carried by the fluid can sometimes have a neg-

ative effect on the rotor. For example, a piece of string carried by the fluid can get stuck on or around the blades of a fan, possibly hindering the rotor motion. In this article, we focus on computational analysis of flow-driven string dynamics. Our objective is to enable a better understanding of how the strings carried by a fluid interact with the solid surfaces present and get stuck on or

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around those surfaces. The main components of the method we developed for this purpose are the Space–Time Variational Multiscale (ST-VMS) and ST Slip Interface (ST-SI) methods for the fluid dynamics, and a one-way-dependence

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model and the Isogeometric Analysis (IGA) for the string dynamics. Because a string is a very thin object, its influence on the flow will be very

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small. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow. The one-way-dependence model has been

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used in other contexts of computational engineering analysis. The examples we are familiar with are calculating the aerodynamic forces acting on the suspension

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lines of spacecraft parachutes <1; 2; 3> and calculating the forces acting on the particles in particle-laden flows <4; 5>. In the first example the suspension lines are assumed to have no influence on the flow, and in the second example the particles are assumed to have no influence on the flow. In our case here, we first compute the flow field and store the time-dependent flow data, and then compute several patterns of string dynamics.

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In computing the flow field, we use the ST-VMS method <6; 7>. This is the VMS version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method <8; 9; 10>. The DSD/SST method is a moving-mesh method. The

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VMS components of the ST-VMS method are from the residual-based VMS (RBVMS) method given in <11; 12; 13; 14>. The ALE-VMS method <15; 16>

is the VMS version of the Arbitrary Lagrangian–Eulerian (ALE) finite element method <17>. The ALE-VMS method was first presented in <18>. The RB-

VMS and ALE-VMS methods have been used successfully for different classes of problems (see, for example, <19; 20; 18; 21; 22; 23; 24; 25; 16; 26; 27; 3;

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28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39>). The stabilization components of the original DSD/SST method are the Streamline-Upwind/Petrov-Galerkin

(SUPG) <40> and Pressure-Stabilizing/Petrov-Galerkin (PSPG) <8> stabilizations, and for that the method is now also called “ST-SUPS” (the acronym was coined in <3>).

The ST-VMS method is essentially an “augmented”

version of the ST-SUPS method, with two additional stabilization terms be-

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yond the three that the ST-SUPS method has (see <6; 7; 3>). The classes of problems the ST-VMS method has been successfully applied to include wind-

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turbine aerodynamics <24; 41; 42; 3; 43; 44; 45; 31; 46>, flapping-wing aerodynamics <47; 48; 49; 3; 50; 51; 52; 44; 45; 53>, cardiovascular fluid mechanics <54; 55; 51; 56; 44; 45; 57; 58>, spacecraft aerodynamics <59; 60>, thermo-

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fluid analysis of ground vehicles and their tires <61>, and flow analysis for the turbine part of a turbocharger <62>.

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One of the desirable features of the ST methods is being able to use higherorder basis functions in time, including the non-uniform rational B-spline (NURBS) basis functions. There are now many publications reporting successful use of

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NURBS as spatial basis functions (for examples of use in conjunction with the RBVMS and ALE-VMS methods, see again <19; 20; 18; 21; 22; 23; 24; 25; 16; 26; 27; 3; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39>). The ST-VMS

method (and also the ST-SUPS method), when combined with higher-order NURBS basis functions in time, provides a more accurate representation of the motion of the solid surfaces, and a mesh motion consistent with that. It 4

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also provides more efficiency in temporal representation of the motion and deformation of the volume meshes, and better efficiency in remeshing. These desirable features have enabled computational analysis in flapping-wing aero-

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dynamics <47; 48; 49; 3; 50; 51; 52; 44; 45; 53>, separation aerodynamics of spacecraft <59>, wind-turbine aerodynamics <43; 44; 45; 31; 46>, and thermo-

fluid mechanics of ground vehicles and their tires <61>. The ST framework and

NURBS in time also enable, with the “ST-C” method, extracting a continuous representation from the computed data and, in large-scale computations, efficient data compression <63; 61>.

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With the moving-mesh methods, by moving the fluid mechanics mesh to follow a fluid–solid interface, we can control the mesh resolution near the interface, have high-resolution representation of the boundary layers, and obtain accurate solutions in such critical flow regions. We want to be able to do that also in flow problems with a spinning solid surface, such as the rotor of a fan.

We want the mesh covering the spinning solid surface to spin with it so that

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we maintain the high-resolution representation of the boundary layers. That requires something special at the interface between the spinning mesh and the

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rest of the mesh. That was first accomplished in the ST framework with the Shear–Slip Mesh Update Method (SSMUM) <64; 65; 66>, which was introduced in <64; 65> and named “SSMUM” in <66>. Later it was accomplished

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also with the ST/NURBS Mesh Update Method (STNMUM), which was introduced in <47; 48; 49> and named in <43>. The STNMUM is more general

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than the SSMUM and simpler to use. It was successfully used in <43> in STVMS computation of flow past a wind-turbine rotor, with the tower included in the model. In the STNMUM, NURBS basis functions are used for the tempo-

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ral representation of the spinning motion, mesh motion and also in remeshing. The rotor motion is represented by quadratic temporal NURBS basis functions, with sufficient number of temporal patches for a full rotation. With that, we can represent the circular paths associated with the rotor motion exactly. With an added “secondary mapping” <6; 47; 7; 3>, we can also specify a constant angular velocity corresponding to the invariant speeds along those paths. 5

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The ST-SI method is the most recent ST method where the mesh covering a spinning solid surface spins with it and maintains the high-resolution representation of the boundary layers. The starting point in the development

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of the ST-SI method was the version of the ALE-VMS method designed for computations with “sliding interfaces” <27; 67>. This ALE-VMS version has been used successfully in a number of computations with spinning solid surfaces

<27; 33; 29; 32; 67>. In the ST-SI method, interface terms similar to those in the ALE-VMS version are added to the ST-VMS formulation to account for

the compatibility conditions for the velocity and stress. While having high-

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resolution representation of the boundary layers near a spinning solid surface,

by using NURBS functions in temporal representation of the spinning motion, the ST-SI method has exact representation of the circular paths associated with the spinning.

As we compute the flow field, we store the computed time-dependent data with a special data compression method based on the ST-C method <63>. With

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the ST-C method, we can represent the data with fewer temporal control points, resulting in reduced computer storage cost. In one of the two ST-C versions in-

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troduced in <63>, the continuous representation is extracted by projection from a solution already computed. Because we use a successive-projection technique (SPT), with a small number of temporal NURBS basis functions at each pro-

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jection, the extraction can take place as the original solution is being computed, without the need to first complete the computation and store all that data. This

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version was named “ST-C-SPT” in <63>. In the work reported in this article, the large time-history data from the flow field computation is stored using the

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ST-C-SPT method. The IGA is used in the string dynamics with higher-order basis functions.

This gives us a higher-order method and smoothness in the structure shape. It also gives us smoothness in the fluid dynamics forces calculated on the string. Furthermore, although the bending effect is small for thin filaments considered in this work, it was shown in <68> that the use of higher-order smooth basis functions enables one to incorporate bending action into the string formula6

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tion without introducing rotational degrees of freedom. These bending terms, although not considered in this work, can provide additional numerical stabilization for the string.

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We describe the ST-VMS and ST-SI methods in Section 2. In Section 3, we present the pilot computations we carried out, for a duct with cylindrical obstacles and for a ventilating fan. The concluding remarks are given in Section 4.

2. Method

This section is mostly a condensed version of the related parts of the methods

(Pn )h

Qn

e=1

Qen

X Z

τSUPS ρ

(nel )n e=1

Qen

X Z

(nel )n

−

e=1

Qen

X Z

−


τSUPS wh · rM (uh , ph ) · ∇ uh dQ

2
τSUPS rM (uh , ph ) · ∇ wh · rM (uh , ph )dQ = 0, ρ

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(nel )n

νLSIC∇ · wh ρrC (uh )dQ

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+

Ωn

    ∂wh ρ + uh · ∇ wh + ∇ q h · rM (uh , ph )dQ ∂t

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X Z

(nel )n

+

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presented in <46>. The flow field is computed with the ST-VMS method:  h  Z Z ∂u wh · ρ + uh · ∇ uh − f h dQ + ε (wh ) : σ (uh , ph )dQ ∂t Qn Qn Z Z Z
h h h h h + h − − w · h dP + q ∇ · u dQ + (wh )+ n · ρ (u )n − (u )n dΩ

e=1

Qen

(1)

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where

h

h

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rM (u , p ) = ρ



∂uh + uh · ∇ uh − f h ∂t

rC (uh ) = ∇ · uh



− ∇ · σ (uh , ph ),

(2) (3)

are the residuals of the momentum equation and incompressibility constraint. Here, ρ, u, p, f , σ , ε , and h are the density, velocity, pressure, external force, stress tensor, strain rate tensor, and the traction specified at the boundary. The test functions associated with the velocity and pressure are w and q. A 7

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superscript “h” indicates that the function is coming from a finite-dimensional space. The symbol Qn represents the ST slice between time levels n and n + 1, (Pn )h is the part of the lateral boundary of that slice associated with the traction

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boundary condition h, and Ωn is the spatial domain at time level n. The superscript “e” is the ST element counter, and nel is the number of ST elements. The functions are discontinuous in time at each time level, and the superscripts

“−” and “+” indicate the values of the functions just below and just above the time level. There are various ways of defining the stabilization parameters τSUPS

and νLSIC . See <9; 10; 43; 61; 46> for the stabilization parameter definitions

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used here. For more ways of calculating the stabilization parameters in finite element computation of flow problems, see <69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 4; 5>.

In the presence of spinning solid surfaces, we use the ST-SI method. In describing the ST-SI method, we use the labels “Side A” and “Side B” to represent the two sides of the SI. In the ST-SI version of the formulation given by Eq. (1),

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we have added boundary terms corresponding to the SI. The boundary terms h h for the two sides are first added separately, using test functions wA and qA and

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h h . Then, putting together the terms added to each side, the complete and qB wB

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set of terms added becomes Z
1 h
h h − qB nB − qA nA · uB − uhA dP 2 (Pn )SI Z


1 h − ρwB · FBh − FBh uhB − FBh − FBh uhA dP 2 (Pn )SI Z


h 1 − ρwA · FAh − FAh uhA − FAh − FAh uhB dP 2 (Pn )SI Z

h h 1 + nB · wB + nA · wA phB + phA dP 2 (Pn )SI Z


h h ˆ B · µ ε (uhB ) + ε (uhA ) dP − wB − wA · n (Pn )SI

−

+

Z

(Pn )SI

Z

(Pn )SI





h h ˆ B · µ ε wB n + ε wA · uhB − uhA dP

µC h h wB − wA · uhB − uhA dP, h 8

(4)

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where
h FBh = nB · uhB − vB ,
h FAh = nA · uhA − vA ,

(5)

h = min(hB , hA ),

hB = 2

n ent n ens X X

α=1 a=1

hA = 2

n ens ent n X X

α=1 a=1

(7)

!−1

|nB · ∇ Naα |

|nA · ∇ Naα |

!−1

(for Side B),

(8)

(for Side A),

(9)

nB − nA . knB − nA k

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ˆB = n

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(6)

(10)

Here, (Pn )SI is the SI in the ST domain, n is the unit normal vector, v is the mesh velocity, nens and nent are the number of spatial and temporal element nodes, Naα is the basis function associated with spatial and temporal nodes a enough for stability.

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and α, and C is a nondimensional penalty constant. Usually C = 1 is large

A number of remarks were provided in <46> to explain the added terms

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and to comment on related interpretations. We refer the reader interested in such details to <46>.

As we compute the flow field, the computed data is stored with the ST-C-

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SPT method <63>. This method, which serves as an efficient data compression method in large-scale computations, is one of the versions of the ST-C method introduced in <63>. In the ST-C method, the ST framework and NURBS in

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time are used in extracting a continuous representation from the computed data. With the ST-C method, we can represent the data with fewer temporal control

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points, resulting in reduced computer storage cost. In the ST-C-SPT method, the continuous representation is extracted by projection from a solution already computed. Because we use a successive-projection technique, with a small number of temporal NURBS basis functions at each projection, the extraction can take place as the original solution is being computed, without the need to first complete the computation and store all that data.

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The string is modeled with truss elements, using the IGA <19> with cubic NURBS basis functions. This gives us a higher-order method and smoothness in the structure shape. It also gives us smoothness in the fluid dynamics

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forces calculated on the string. Figure 1 shows a schematic example of a string

Figure 1:

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modeled with truss elements and cubic NURBS basis functions. In the one-way-

A schematic example of a string modeled with truss elements and cubic NURBS

basis functions. The number of control points (red dots) is 9, and the number of elements (colored blue and black alternately) is 6.

dependence model, the fluid mechanics forces acting on the string are calculated

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with the method described in <1; 2; 3> for computing the aerodynamic forces acting on the suspension lines of spacecraft parachutes. Contact between the

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string and solid surfaces is handled with the Surface-Edge-Node Contact Tracking (SENCT-FC) method <89; 2; 3>, which is a newer version of the SENCT

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method introduced in <10; 90>. 3. Computations

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We present two cases of pilot computations. In the first case a string is

falling in an air duct with cylindrical obstacles. In the second case a string is falling into a ventilating fan placed horizontally. In both cases the air density

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and kinematic viscosity are 1.205 kg/m3 and 1.512×10−5 m2 /s, and the string diameter, density, and Young’s modulus are 1 mm, 713 kg/m3 , and 30 MPa. In both cases the problem setup is based on an experimental setup <91>. 3.1. String dynamics in an air duct Figure 2 shows the air duct, which is 3 m long. The cross-section shape 10

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Figure 2: String dynamics in an air duct. Problem setup.

is changing gradually from circular to rectangular, and back to circular. The diameter of the circular segment of the duct is 130 mm, and the cross-sectional dimensions of the rectangular segment are 74 mm × 86 mm. The rectangular segment is 600 mm long and has 5 cylindrical obstacles. The cylinder diameter

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and length are 8 and 74 mm. The fluid mechanics mesh is made of 1,618,536 nodes, 1,516,494 hexahedral elements, 262,222 tetrahedral elements, and 11,426

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pyramid elements. The hexahedral elements are used in the circular and rectangular segments, tetrahedral elements in the transition segments, and the pyramid elements in transition between the hexahedral and tetrahedral elements.

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Figure 3 shows the mesh in the rectangular segment. The flow is from top to bottom, with an inflow speed of 9.4 m/s. The boundary conditions are uniform flow at the inflow, zero stress at the outflow, and no-slip at all solid surfaces.

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The time-step size is 3.925×10−5 s, the number of nonlinear iterations per time step is 4, and the number of GMRES <92> iterations per nonlinear iteration

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are 200, 300, 400, and 500. The string is 50 mm long. As the initial condition for the string, we use 3

angles (−30◦ , 0◦ and 30◦ ) and 3 horizontal positions (−20, 0 and 20 mm from

the center), all 0.5 m down from the inflow. In all 9 cases, the initial speed is 8 m/s downward. The string NURBS mesh is made of 33 control points and 30 elements. The time-step size is 3.925×10−6 s, the number of nonlinear iterations 11

String dynamics in an air duct. Two views of the fluid mechanics mesh in the

rectangular segment of the duct.

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Figure 3:

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per time step is 5, and it is a full GMRES search. Figure 4 shows the 9 cases at 4 instants during the string dynamics computa-

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tion. In the cases with tilted initial orientation, the string moves opposite to the tilt direction, because the force acting on the string is in the normal direction of the relative velocity (see <1; 2>). After hitting a cylinder the string gets stuck

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for a while and then falls. Overall the string behavior is very similar to what was reported in the experiments <91>, however, no quantitative comparison

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has been made yet due to the extensive randomness seen in the experiments. 3.2. String dynamics in a ventilating fan Figure 5 shows the fan and its dimensions. The rotor speed is 700 rpm,

counterclockwise viewed from above. Figure 6 shows the problem setup. The computational domain is 1.8 m high, and the lateral dimensions are 1.2 m × 12

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1.2 m. The fan is at 0.8 m from the top boundary. The external flow is from top to bottom, at 0.06628 m/s. This value is based on the experiment reported in <91>. The boundary conditions are uniform flow at the inflow, zero stress

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at the outflow, slip on the side boundaries, and no-slip on all fan surfaces. The fluid mechanics mesh is made of 1,845,956 nodes and 10,726,390 tetrahedral

elements. Figure 7 shows the mesh used and a part of the SI. The time-step size is 4.28×10−4 s, the number of nonlinear iterations per time step is 3, and

the number of GMRES iterations per nonlinear iteration are 200, 300, and 400.

The string is 80 mm long. The string NURBS mesh is made of 53 control

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points and 50 elements. The time-step size is 4.28×10−5 s, the number of nonlinear iterations per time step is 5, and it is a full GMRES search. Figure 8 shows the flow velocity at the beginning of the string dynamics computation, and the string at 4 instants during the computation. The leading edge of a blade hits the string and drags it for less than 1/5 rotation. Then the lower surface of the next blade hits the string, and the string falls. These results

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4. Concluding Remarks

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are reasonable, however, at this stage our observations remain qualitative.

To understand how the strings carried by a fluid interact with the solid sur-

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faces present and get stuck on or around those surfaces, we have developed a computational analysis method for flow-driven string dynamics. Currently our target application is turbomachinery, such as understanding how strings get

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stuck on or around the blades of a fan. The main components of the method are ST-VMS and ST-SI methods for the fluid dynamics, and a one-way-dependence model and the IGA for the string dynamics. The ST-VMS method is the core

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computational method, and it also serves as a turbulence model. In the ST-SI method, the mesh covering a spinning solid surface spins with it, maintaining high-resolution representation of the boundary layers near spinning solid surfaces such as the blades in turbomachinery. The slip at the interface between the mesh covering the spinning surface and the mesh covering the rest of the

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domain is dealt with in a consistent fashion. In the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the task of computing the influence of the string on the flow, which should be

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rather small. The IGA for the string dynamics gives us a higher-order method, smoothness in the structure shape, and smoothness in the fluid dynamics forces

calculated on the string. We have presented the pilot computations we carried

out, for a duct with cylindrical obstacles and for a ventilating fan, demonstrating that the method has good potential for computational analysis of flow-driven

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string dynamics.

Acknowledgement

This work was supported (first and third authors) in part by Grant-in-Aid for Young Scientists (B) 24760144 from Japan Society for the Promotion of Science (JSPS); Grant-in-Aid for Scientific Research (S) 26220002 from the Ministry

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of Education, Culture, Sports, Science and Technology of Japan (MEXT); and Rice–Waseda research agreement. This work was also supported (second author)

References

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in part by ARO Grant W911NF-12-1-0162.

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action modeling”, Archives of Computational Methods in Engineering, 19 (2012) 171–225, doi: 10.1007/s11831-012-9071-3.

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[16] Y. Bazilevs, M.-C. Hsu, K. Takizawa, and T.E. Tezduyar, “ALE-VMS and ST-VMS methods for computer modeling of wind-turbine rotor aerodynamics and fluid–structure interaction”, Mathematical Models and Methods in

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Applied Sciences, 22 (2012) 1230002, doi: 10.1142/S0218202512300025. [17] T.J.R. Hughes, W.K. Liu, and T.K. Zimmermann, “Lagrangian–Eulerian

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finite element formulation for incompressible viscous flows”, Computer Methods in Applied Mechanics and Engineering, 29 (1981) 329–349.

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[18] Y. Bazilevs, V.M. Calo, T.J.R. Hughes, and Y. Zhang, “Isogeometric fluid– structure interaction: theory, algorithms, and computations”, Computational Mechanics, 43 (2008) 3–37.

[19] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs, “Isogeometric analysis:

CAD, finite elements, NURBS, exact geometry, and mesh re-

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finement”, Computer Methods in Applied Mechanics and Engineering, 194 (2005) 4135–4195. [20] Y. Bazilevs, V.M. Calo, Y. Zhang, and T.J.R. Hughes, “Isogeometric fluid–

Computational Mechanics, 38 (2006) 310–322.

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structure interaction analysis with applications to arterial blood flow”,

[21] Y. Bazilevs and T.J.R. Hughes, “NURBS-based isogeometric analysis for the computation of flows about rotating components”, Computational Mechanics, 43 (2008) 143–150.

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[22] Y. Bazilevs, J.R. Gohean, T.J.R. Hughes, R.D. Moser, and Y. Zhang,

“Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device”, Computer Methods in Applied Mechanics and Engineering, 198 (2009) 3534–3550.

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[23] Y. Bazilevs, M.-C. Hsu, D. Benson, S. Sankaran, and A. Marsden, “Computational fluid–structure interaction: Methods and application to a total

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cavopulmonary connection”, Computational Mechanics, 45 (2009) 77–89. [24] Y. Bazilevs, M.-C. Hsu, I. Akkerman, S. Wright, K. Takizawa, B. Henicke, T. Spielman, and T.E. Tezduyar, “3D simulation of wind turbine rotors

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at full scale. Part I: Geometry modeling and aerodynamics”, International Journal for Numerical Methods in Fluids, 65 (2011) 207–235, doi:

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10.1002/fld.2400.

[25] Y. Bazilevs, M.-C. Hsu, J. Kiendl, R. W¨ uchner, and K.-U. Bletzinger, “3D

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simulation of wind turbine rotors at full scale. Part II: Fluid–structure interaction modeling with composite blades”, International Journal for Nu-

merical Methods in Fluids, 65 (2011) 236–253.

[26] M.-C. Hsu, I. Akkerman, and Y. Bazilevs, “Wind turbine aerodynamics using ALE-VMS: Validation and role of weakly enforced boundary conditions”, Computational Mechanics, 50 (2012) 499–511. 17

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[27] M.-C. Hsu and Y. Bazilevs, “Fluid–structure interaction modeling of wind turbines: simulating the full machine”, Computational Mechanics, 50 (2012) 821–833.

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[28] Y. Bazilevs, K. Takizawa, and T.E. Tezduyar, “Challenges and di-

rections in computational fluid–structure interaction”, Mathematical

Models and Methods in Applied Sciences, 23 (2013) 215–221, doi: 10.1142/S0218202513400010.

[29] A. Korobenko, M.-C. Hsu, I. Akkerman, J. Tippmann, and Y. Bazilevs,

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“Structural mechanics modeling and FSI simulation of wind turbines”, Mathematical Models and Methods in Applied Sciences, 23 (2013) 249–272. [30] A. Korobenko, M.-C. Hsu, I. Akkerman, and Y. Bazilevs, “Aerodynamic simulation of vertical-axis wind turbines”, Journal of Applied Mechanics, 81 (2013) 021011, doi: 10.1115/1.4024415.

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[31] Y. Bazilevs, K. Takizawa, T.E. Tezduyar, M.-C. Hsu, N. Kostov, and S. McIntyre, “Aerodynamic and FSI analysis of wind turbines with the

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ALE-VMS and ST-VMS methods”, Archives of Computational Methods in Engineering, 21 (2014) 359–398, doi: 10.1007/s11831-014-9119-7. [32] Y. Bazilevs, A. Korobenko, X. Deng, J. Yan, M. Kinzel, and J.O. Dabiri,

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“FSI modeling of vertical-axis wind turbines”, Journal of Applied Mechanics, 81 (2014) 081006, doi: 10.1115/1.4027466.

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[33] M.-C. Hsu, I. Akkerman, and Y. Bazilevs, “Finite element simulation of wind turbine aerodynamics: Validation study using NREL Phase VI ex-

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periment”, Wind Energy, 17 (2014) 461–481.

[34] C.C. Long, M. Esmaily-Moghadam, A.L. Marsden, and Y. Bazilevs, “Computation of residence time in the simulation of pulsatile ventricular assist devices”, Computational Mechanics, 54 (2014) 911–919, doi: 10.1007/s00466-013-0931-y.

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[35] C.C. Long, A.L. Marsden, and Y. Bazilevs, “Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk”, Computational Mechanics, 54 (2014) 921–932, doi: 10.1007/s00466-013-0967-z.

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[36] M.-C. Hsu, D. Kamensky, Y. Bazilevs, M.S. Sacks, and T.J.R. Hughes,

“Fluid–structure interaction analysis of bioprosthetic heart valves:

significance of arterial wall deformation”, Computational Mechanics, 54 (2014) 1055–1071, doi: 10.1007/s00466-014-1059-4.

[37] B. Augier, J. Yan, A. Korobenko, J. Czarnowski, G. Ketterman,

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and Y. Bazilevs, “Experimental and numerical FSI study of compli-

ant hydrofoils”, Computational Mechanics, 55 (2015) 1079–1090, doi: 10.1007/s00466-014-1090-5.

[38] Y. Bazilevs, A. Korobenko, J. Yan, A. Pal, S.M.I. Gohari, and S. Sarkar, “ALE–VMS formulation for stratified turbulent incompressible flows with

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applications”, Mathematical Models and Methods in Applied Sciences, 25 (2015) 2349–2375, doi: 10.1142/S0218202515400114.

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[39] Y. Bazilevs, K. Takizawa, and T.E. Tezduyar, “New directions and challenging computations in fluid dynamics modeling with stabilized and multiscale methods”, Mathematical Models and Methods in Applied Sciences,

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25 (2015) 2217–2226, doi: 10.1142/S0218202515020029. [40] A.N. Brooks and T.J.R. Hughes, “Streamline upwind/Petrov-Galerkin for-

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mulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”, Computer Methods in Applied

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Mechanics and Engineering, 32 (1982) 199–259.

[41] K. Takizawa, B. Henicke, T.E. Tezduyar, M.-C. Hsu, and Y. Bazilevs, “Stabilized space–time computation of wind-turbine rotor aerodynamics”, Computational Mechanics, 48 (2011) 333–344, doi: 10.1007/s00466-0110589-2.

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[42] K. Takizawa, B. Henicke, D. Montes, T.E. Tezduyar, M.-C. Hsu, and Y. Bazilevs, “Numerical-performance studies for the stabilized space–time computation of wind-turbine rotor aerodynamics”, Computational Mechan-

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ics, 48 (2011) 647–657, doi: 10.1007/s00466-011-0614-5. [43] K. Takizawa, T.E. Tezduyar, S. McIntyre, N. Kostov, R. Kolesar, and C. Habluetzel, “Space–time VMS computation of wind-turbine rotor and

tower aerodynamics”, Computational Mechanics, 53 (2014) 1–15, doi: 10.1007/s00466-013-0888-x.

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[44] K. Takizawa, Y. Bazilevs, T.E. Tezduyar, M.-C. Hsu, O. Øiseth,

K.M. Mathisen, N. Kostov, and S. McIntyre, “Engineering analysis and design with ALE-VMS and space–time methods”, Archives of Computational Methods in Engineering, 21 (2014) 481–508, doi: 10.1007/s11831-014-91130.

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[45] K. Takizawa, “Computational engineering analysis with the new-generation space–time methods”, Computational Mechanics, 54 (2014) 193–211, doi:

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10.1007/s00466-014-0999-z.

[46] K. Takizawa, T.E. Tezduyar, H. Mochizuki, H. Hattori, S. Mei, L. Pan, and K. Montel, “Space–time VMS method for flow computations with slip

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interfaces (ST-SI)”, Mathematical Models and Methods in Applied Sciences, 25 (2015) 2377–2406, doi: 10.1142/S0218202515400126.

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[47] K. Takizawa, B. Henicke, A. Puntel, T. Spielman, and T.E. Tezduyar, “Space–time computational techniques for the aerodynamics of flap-

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ping wings”, Journal of Applied Mechanics, 79 (2012) 010903, doi: 10.1115/1.4005073.

[48] K. Takizawa, B. Henicke, A. Puntel, N. Kostov, and T.E. Tezduyar, “Space–time techniques for computational aerodynamics modeling of flapping wings of an actual locust”, Computational Mechanics, 50 (2012) 743– 760, doi: 10.1007/s00466-012-0759-x.

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[49] K. Takizawa, N. Kostov, A. Puntel, B. Henicke, and T.E. Tezduyar, “Space–time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle”, Computational Mechanics, 50 (2012) 761–

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778, doi: 10.1007/s00466-012-0758-y. [50] K. Takizawa, B. Henicke, A. Puntel, N. Kostov, and T.E. Tezduyar, “Computer modeling techniques for flapping-wing aerodynam-

ics of a locust”, Computers & Fluids, 85 (2013) 125–134, doi: 10.1016/j.compfluid.2012.11.008.

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[51] K. Takizawa, T.E. Tezduyar, A. Buscher, and S. Asada, “Space–time

interface-tracking with topology change (ST-TC)”, Computational Mechanics, 54 (2014) 955–971, doi: 10.1007/s00466-013-0935-7.

[52] K. Takizawa, T.E. Tezduyar, and N. Kostov, “Sequentially-coupled space– time FSI analysis of bio-inspired flapping-wing aerodynamics of an MAV”,

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Computational Mechanics, 54 (2014) 213–233, doi: 10.1007/s00466-0140980-x.

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[53] K. Takizawa, T.E. Tezduyar, and A. Buscher, “Space–time computational analysis of MAV flapping-wing aerodynamics with wing clapping”, Computational Mechanics, 55 (2015) 1131–1141, doi: 10.1007/s00466-014-1095-0.

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[54] K. Takizawa, K. Schjodt, A. Puntel, N. Kostov, and T.E. Tezduyar, “Patient-specific computer modeling of blood flow in cerebral arteries with

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aneurysm and stent”, Computational Mechanics, 50 (2012) 675–686, doi: 10.1007/s00466-012-0760-4.

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[55] K. Takizawa, K. Schjodt, A. Puntel, N. Kostov, and T.E. Tezduyar, “Patient-specific computational analysis of the influence of a stent on the unsteady flow in cerebral aneurysms”, Computational Mechanics, 51 (2013) 1061–1073, doi: 10.1007/s00466-012-0790-y.

[56] K. Takizawa, Y. Bazilevs, T.E. Tezduyar, C.C. Long, A.L. Marsden, and K. Schjodt, “ST and ALE-VMS methods for patient-specific cardiovascular 21

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fluid mechanics modeling”, Mathematical Models and Methods in Applied Sciences, 24 (2014) 2437–2486, doi: 10.1142/S0218202514500250. [57] H. Suito, K. Takizawa, V.Q.H. Huynh, D. Sze, and T. Ueda, “FSI analysis

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of the blood flow and geometrical characteristics in the thoracic aorta”,

Computational Mechanics, 54 (2014) 1035–1045, doi: 10.1007/s00466-0141017-1.

[58] K. Takizawa, T.E. Tezduyar, A. Buscher, and S. Asada, “Space–time fluid

mechanics computation of heart valve models”, Computational Mechanics,

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54 (2014) 973–986, doi: 10.1007/s00466-014-1046-9.

[59] K. Takizawa, D. Montes, M. Fritze, S. McIntyre, J. Boben, and T.E. Tezduyar, “Methods for FSI modeling of spacecraft parachute dynamics and cover separation”, Mathematical Models and Methods in Applied Sciences, 23 (2013) 307–338, doi: 10.1142/S0218202513400058.

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[60] K. Takizawa, D. Montes, S. McIntyre, and T.E. Tezduyar, “Space– time VMS methods for modeling of incompressible flows at high

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Reynolds numbers”, Mathematical Models and Methods in Applied Sciences, 23 (2013) 223–248, doi: 10.1142/s0218202513400022. [61] K. Takizawa, T.E. Tezduyar, and T. Kuraishi, “Multiscale ST methods

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for thermo-fluid analysis of a ground vehicle and its tires”, Mathematical Models and Methods in Applied Sciences, 25 (2015) 2227–2255, doi:

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10.1142/S0218202515400072. [62] Y. Otoguro, T. Terahara, K. Takizawa, T.E. Tezduyar, T. Kuraishi, and

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H. Hattori, “A higher-order ST-VMS method for turbocharger analysis”, in Proceedings of 13th Asian International Conference on Fluid Machinery,

Paper No. AICFM13-153, Tokyo, Japan, (2015).

[63] K. Takizawa and T.E. Tezduyar, “Space–time computation techniques with continuous representation in time (ST-C)”, Computational Mechanics, 53 (2014) 91–99, doi: 10.1007/s00466-013-0895-y. 22

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[64] V. Kalro and T.E. Tezduyar, “Parallel finite element computation of 3D incompressible flows on MPPs”, in W.G. Habashi, editor, Solution Techniques for Large-Scale CFD Problems, John Wiley & Sons, 1995.

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[65] T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro, and M. Litke,

“Flow simulation and high performance computing”, Computational Mechanics, 18 (1996) 397–412, doi: 10.1007/BF00350249.

[66] M. Behr and T. Tezduyar, “The Shear-Slip Mesh Update Method”, Computer Methods in Applied Mechanics and Engineering, 174 (1999) 261–274,

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doi: 10.1016/S0045-7825(98)00299-0.

[67] Y. Bazilevs, A. Korobenko, X. Deng, and J. Yan, “Novel structural modeling and mesh moving techniques for advanced FSI simulation of wind turbines”, International Journal for Numerical Methods in Engineering, 102 (2015) 766–783, doi: 10.1002/nme.4738.

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[68] S.B. Raknes, X. Deng, Y. Bazilevs, D.J. Benson, K.M. Mathisen, and T. Kvamsdal, “Isogeometric rotation-free bending-stabilized cables: Stat-

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ics, dynamics, bending strips and coupling with shells”, Computer Methods in Applied Mechanics and Engineering, 263 (2013) 127–143. [69] T.E. Tezduyar and D.K. Ganjoo, “Petrov-Galerkin formulations with

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weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusion problems”, Computer

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Methods in Applied Mechanics and Engineering, 59 (1986) 49–71, doi: 10.1016/0045-7825(86)90023-X.

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[70] G.J. Le Beau, S.E. Ray, S.K. Aliabadi, and T.E. Tezduyar, “SUPG finite element computation of compressible flows with the entropy and conservation variables formulations”, Computer Methods in Applied Mechanics and Engineering, 104 (1993) 397–422, doi: 10.1016/0045-7825(93)90033-T.

[71] T.E. Tezduyar, “Finite elements in fluids: Stabilized formulations and mov-

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ing boundaries and interfaces”, Computers & Fluids, 36 (2007) 191–206, doi: 10.1016/j.compfluid.2005.02.011. [72] T.E. Tezduyar and M. Senga, “Stabilization and shock-capturing pa-

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rameters in SUPG formulation of compressible flows”, Computer Meth-

ods in Applied Mechanics and Engineering, 195 (2006) 1621–1632, doi: 10.1016/j.cma.2005.05.032.

[73] T.E. Tezduyar and M. Senga, “SUPG finite element computation of inviscid supersonic flows with YZβ shock-capturing”, Computers & Fluids,

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36 (2007) 147–159, doi: 10.1016/j.compfluid.2005.07.009.

[74] T.E. Tezduyar, M. Senga, and D. Vicker, “Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZβ shock-capturing”, Computational Mechanics, 38 (2006) 469–481, doi: 10.1007/s00466-005-0025-6.

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[75] T.E. Tezduyar and S. Sathe, “Enhanced-discretization selective stabilization procedure (EDSSP)”, Computational Mechanics, 38 (2006) 456–468,

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doi: 10.1007/s00466-006-0056-7.

[76] A. Corsini, F. Rispoli, A. Santoriello, and T.E. Tezduyar, “Improved discontinuity-capturing finite element techniques for reaction effects in tur-

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bulence computation”, Computational Mechanics, 38 (2006) 356–364, doi: 10.1007/s00466-006-0045-x.

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[77] F. Rispoli, A. Corsini, and T.E. Tezduyar, “Finite element computation of turbulent flows with the discontinuity-capturing directional

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dissipation (DCDD)”, Computers & Fluids, 36 (2007) 121–126, doi:

10.1016/j.compfluid.2005.07.004.

[78] T.E. Tezduyar, S. Ramakrishnan, and S. Sathe, “Stabilized formulations for incompressible flows with thermal coupling”, International Journal for Numerical Methods in Fluids, 57 (2008) 1189–1209, doi: 10.1002/fld.1743.

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[79] F. Rispoli, R. Saavedra, A. Corsini, and T.E. Tezduyar, “Computation of inviscid compressible flows with the V-SGS stabilization and YZβ shock-capturing”, International Journal for Numerical Methods in Fluids,

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54 (2007) 695–706, doi: 10.1002/fld.1447. [80] Y. Bazilevs, V.M. Calo, T.E. Tezduyar, and T.J.R. Hughes, “YZβ

discontinuity-capturing for advection-dominated processes with application to arterial drug delivery”, International Journal for Numerical Methods in Fluids, 54 (2007) 593–608, doi: 10.1002/fld.1484.

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[81] A. Corsini, C. Menichini, F. Rispoli, A. Santoriello, and T.E. Tezduyar, “A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reactionlike terms”, Journal of Applied Mechanics, 76 (2009) 021211, doi: 10.1115/1.3062967.

[82] F. Rispoli, R. Saavedra, F. Menichini, and T.E. Tezduyar, “Computation

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of inviscid supersonic flows around cylinders and spheres with the V-SGS stabilization and YZβ shock-capturing”, Journal of Applied Mechanics,

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76 (2009) 021209, doi: 10.1115/1.3057496. [83] A. Corsini, C. Iossa, F. Rispoli, and T.E. Tezduyar, “A DRD finite element formulation for computing turbulent reacting flows in gas turbine combus-

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tors”, Computational Mechanics, 46 (2010) 159–167, doi: 10.1007/s00466009-0441-0.

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[84] M.-C. Hsu, Y. Bazilevs, V.M. Calo, T.E. Tezduyar, and T.J.R. Hughes, “Improving stability of stabilized and multiscale formulations in flow simu-

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lations at small time steps”, Computer Methods in Applied Mechanics and Engineering, 199 (2010) 828–840, doi: 10.1016/j.cma.2009.06.019.

[85] A. Corsini, F. Rispoli, and T.E. Tezduyar, “Stabilized finite element computation of NOx emission in aero-engine combustors”, International Journal for Numerical Methods in Fluids, 65 (2011) 254–270, doi: 10.1002/fld.2451.

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[86] A. Corsini, F. Rispoli, and T.E. Tezduyar, “Computer modeling of waveenergy air turbines with the SUPG/PSPG formulation and discontinuitycapturing technique”, Journal of Applied Mechanics, 79 (2012) 010910, doi:

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10.1115/1.4005060. [87] A. Corsini, F. Rispoli, A.G. Sheard, and T.E. Tezduyar, “Computational analysis of noise reduction devices in axial fans with stabilized finite element formulations”, Computational Mechanics, 50 (2012) 695–705, doi: 10.1007/s00466-012-0789-4.

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[88] P.A. Kler, L.D. Dalcin, R.R. Paz, and T.E. Tezduyar, “SUPG and

discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems”, Computational Mechanics, 51 (2013) 171– 185, doi: 10.1007/s00466-012-0712-z.

[89] K. Takizawa, T. Spielman, and T.E. Tezduyar, “Space–time FSI modeling

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and dynamical analysis of spacecraft parachutes and parachute clusters”, Computational Mechanics, 48 (2011) 345–364, doi: 10.1007/s00466-011-

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0590-9.

[90] S. Sathe and T.E. Tezduyar, “Modeling of fluid–structure interactions with the space–time finite elements: Contact problems”, Computational Me-

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chanics, 43 (2008) 51–60, doi: 10.1007/s00466-008-0299-6. [91] H. Kudo, T. Kawahara, H. Kanai, K. Miyagawa, S. Saito, M. Isono,

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M. Nohmi, H. Uchida, and M. Kawai, “Study on clogging mechanism of fibrous materials in a pump by experimental and computational approaches”,

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in Design and Optimization of Hydraulic Machines, volume 22, Montreal,

Canada, (2014) 012011.

[92] Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems”, SIAM Journal of Scientific and Statistical Computing, 7 (1986) 856–869.

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Figure 4: String dynamics in an air duct. Flow-velocity magnitude and string position for the

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9 cases at 4 instants during the string dynamics computation. The picture blocks correspond to t = 0.0 s, 0.041 s, 0.083 s, and 0.125 s. In each block, the thin picture on the left shows the viewing location for the three pictures on the right, which correspond to the angles −30◦ , 0◦

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and 30◦ . We visualize three cases at a time, as red, blue and yellow strings, corresponding to the three initial horizontal positions of −20, 0 and 20 mm from the center.

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Figure 5: String dynamics in a ventilating fan. Fan geometry.

Figure 6:

String dynamics in a ventilating fan. Problem setup. The yellow, blue and green

surfaces are the inflow, outflow and side boundaries.

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Figure 7: String dynamics in a ventilating fan. Two views of the fluid mechanics mesh and,

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in red, two of the surfaces enclosing the mesh spinning with the rotor in the ST-SI method.

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Figure 8: String dynamics in a ventilating fan. Flow-velocity magnitude and string position at the beginning of the string dynamics computation, and the string at t = 0.300 s, 0.329 s,

0.347 s, and 0.359 s (from left to right and top to bottom).

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