Wind Turbine and Turbomachinery Computational Analysis with the ALE and Space(cid:21)Time Variational Multiscale Methods and Isogeometric Discretization

. The challenges encountered in computational analysis of wind turbines and tur-bomachinery include turbulent rotational (cid:29)ows, complex geometries, moving boundaries and interfaces, such as the rotor motion, and the (cid:29)uid(cid:21)structure interaction (FSI), such as the FSI between the wind turbine blade and the air. The Arbitrary Lagrangian(cid:21)Eulerian (ALE) and Space(cid:21)Time (ST) Variational Multiscale (VMS) methods and isogeometric discretization have been e(cid:27)ective in addressing these challenges. The ALE-VMS and ST-VMS serve as core computational methods. They are supplemented with special methods like the Slip Inter-face (SI) method and ST Isogeometric Analysis with NURBS basis functions in time. We describe the core and special methods and present, as examples of challenging computations performed, computational analysis of horizontal-and vertical-axis wind turbines and (cid:29)ow-driven string dynamics in pumps.


Introduction
Complexity level and reliability of computational analysis of wind turbines and turbomachinery dene the practical value of the computations.The Arbitrary LagrangianEulerian (ALE) and SpaceTime (ST) Variational Multiscale (VMS) methods and isogeometric discretization are now enabling in wind turbine and turbomachinery computational analysis a complexity level that reects the actual conditions, with reliable results (see, for example, [14]).
The computational challenges encountered in this class of problems include turbulent rotational ows, complex geometries, moving boundaries and interfaces, such as the rotor motion, and the uidstructure interaction (FSI), such as the FSI between the wind turbine blade and the air.c 2020 Journal of Advanced Engineering and Computation (JAEC) 1 Our core methods in addressing the computational challenges are the ALE-VMS [5] and ST-VMS [6].We have a number special methods used in combination with them.The special methods used in combination with the ST-VMS include the ST Slip Interface (ST-SI) method [1,7], ST Isogeometric Analysis (ST-IGA) [6,8,9], ST/NURBS Mesh Update Method (STNMUM) [8], a general-purpose NURBS mesh generation method for complex geometries [10,11], and a one-way-dependence model for the string dynamics [12].The special methods used in combination with the ALE-VMS include weak enforcement of no-slip boundary conditions [1315] and sliding interfaces [16,17] (the acronym SI will also indicate that).
We will provide an overview of the core and special methods and present examples of challenging computations performed with these methods, including computational analysis of horizontal-and vertical-axis wind turbines (HAWTs and VAWTs) and ow-driven string dynamics in pumps.Much of the material presented in this review article has been extracted from [18] and the earlier articles written by the authors.
We provide the governing equations in Sec-

Governing equations 2.1. Incompressible ow
Let Ω t ⊂ R n sd be the spatial domain with boundary Γ t at time t ∈ (0, T ), where n sd is the number of space dimensions.The subscript t indicates the time-dependence of the domain.
The NavierStokes equations of incompressible ows are written on Ω t and ∀t ∈ (0, T ) as where ρ, u and f are the density, velocity and body force.The stress tensor σ σ σ(u, p) = −pI + 2µε ε ε(u), where p is the pressure, I is the identity tensor, µ = ρν is the viscosity, ν is the kinematic viscosity, and the strain rate ε ε ε(u) = ∇ ∇ ∇u + (∇ ∇ ∇u) T /2.The essential and natural boundary conditions for Eq. ( 1) are represented as u = g on (Γ t ) g and n•σ σ σ = h on (Γ t ) h , where n is the unit normal vector and g and h are given functions.A divergence-free velocity eld u 0 (x) is specied as the initial condition.

Structural mechanics
In this article we will not provide any of our formulations requiring uid and structure denitions simultaneously; we will instead give reference to earlier journal articles where the formulations were presented.Therefore, for notation simplicity, we will reuse many of the symbols used in the uid mechanics equations to represent their counterparts in the structural mechanics equations.To begin with, Ω t ⊂ R n sd and Γ t will represent the structure domain and its boundary.The structural mechanics equations are then written, on Ω t and ∀t ∈ (0, T ), as where y and σ σ σ are the displacement and Cauchy stress tensor.The essential and natural boundary conditions for Eq. ( 3) are represented as y = g on (Γ t ) g and n • σ σ σ = h on (Γ t ) h .The Cauchy stress tensor can be obtained from where F and J are the deformation gradient tensor and its determinant, and S is the second PiolaKirchho stress tensor.It is obtained from the strain-energy density function ϕ as follows: where E is the GreenLagrange strain tensor: and C is the CauchyGreen deformation tensor: From Eqs. ( 5) and ( 6), S = 2 ∂ϕ ∂C . (8)

Fluidstructure interface
In an FSI problem, at the uidstructure interface, we will have the velocity and stress compatibility conditions between the uid and structure parts.The details on those conditions can be found in Section 5.1 of [19].

ST-VMS and ST-SUPS
The Moving the uid mechanics mesh to follow an interface enables mesh-resolution control near the interface and, consequently, high-resolution boundary-layer representation near uidsolid interfaces.Because of the higher-order accuracy of the ST framework (see [6,28]), the ST-SUPS and ST-VMS are desirable also in computations without MBI.
The ST-SUPS and ST-VMS have been applied to many classes of challenging FSI, MBI and uid mechanics problems (see [29] for a comprehensive summary of the computations prior to July 2018).
For more on the ST-VMS and ST-SUPS, see [19].In the ow analyses presented here, the ST framework provides higher-order accuracy in a general context.The VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady ow.The moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface.The advection equation involved in the residence time computation associated with ow-driven string dynamics in pumps is solved with the ST-SUPG method.
Recent advances in stabilized and multiscale methods may be found for stratied incompressible ows in [124], for divergence-conforming discretizations of incompressible ows in [125], and for compressible ows with emphasis on gas-turbine modeling in [126].
For more on the ALE-VMS, RBVMS and ALE-SUPS, see [19].In the ow analyses presented here, the VMS feature of the ALE-VMS addresses the computational challenges associated with the multiscale nature of the unsteady ow.
The moving-mesh feature of the ALE framework enables high-resolution computation near the rotor surface.

ALE-SI and ST-SI
The ALE-SI was introduced in [16,17]  Interface terms similar to those in the ALE-SI are added to the ST-VMS to accurately connect the two sides of the solution.An ST-SI version where the SI is between uid and solid domains was also presented in [1].The SI in this case is a uidsolid SI rather than a standard uiduid SI and enables weak enforcement of the Dirichlet boundary conditions for the uid.The ST-SI introduced in [7] for the coupled incompressible-ow and thermal-transport equations retains the high-resolution representation of the thermo-uid boundary layers near spinning solid surfaces.These ST-SI methods have been applied to aerodynamic analysis of vertical-axis wind turbines [1,4,44], thermo-uid analysis of disk brakes [7], ow-driven string dynamics in turbomachinery [3,68,69], ow analysis of turbocharger turbines [911, 70,71], ow around tires with road contact and deformation [60,7275], uid lms [75,76], aerodynamic analysis of ram-air parachutes [77], and ow analysis of heart valves [56,58,6164].
For more on the ST-SI, see [1,7].In the computations here, with the ALE-SI and ST-SI the mesh covering the rotor spins with it and we retain the high-resolution representation of the boundary layers.

Stabilization parameters
The ST-SUPS, ALE-SUPS, RBVMS, ALE-VMS and ST-VMS all have some embedded stabilization parameters that play a signicant role 4 c 2020 Journal of Advanced Engineering and Computation (JAEC) (see [19]).These parameters involve a measure of the local length scale (also known as element length) and other parameters such as the element Reynolds and Courant numbers.There are many ways of dening the stabilization parameters.Some of the newer options for the stabilization parameters used with the SUPS and VMS can be found in [1,8,39,40,67,74,127130].
Some of the earlier stabilization parameters used with the SUPS and VMS were also used in computations with other SUPG-like methods, such as the computations reported in [115,131142].
The stabilization-parameter denitions used in the computations reported in this article can be found from the references cited in the sections where those computations are described.

ST-IGA
The ST-IGA is the integration of the ST framework with isogeometric discretization, motivated by the success of NURBS meshes in spatial discretization [5,16,98,143].It was introduced in [6].Computations with the ST-VMS and ST-IGA were rst reported in [6] in a 2D context, with IGA basis functions in space for ow past an airfoil, and in both space and time for the advection equation.Using higher-order basis functions in time enables getting full benet out of using higher-order basis functions in space (see the stability and accuracy analysis given in [6] for the advection equation).
For more on the ST-IGA, see [9,19,45,77].In eas at a reasonable level.In addition, we benet from the mesh generation exibility provided by using SIs.In computational analysis of uid lms [75,76], the ST-SI-IGA enabled solution with a computational cost comparable to that of the Reynolds-equation model for the comparable solution quality [76].With that, narrow gaps associated with the road roughness [75] can be accounted for in the ow analysis around tires.
An SI provides mesh generation exibility in a general context by accurately connecting the two sides of the solution computed over nonmatching meshes.This type of mesh generation exibility is especially valuable in complex-geometry ow computations with isogeometric discretization, removing the matching requirement between the NURBS patches without loss of accuracy.This feature was used in the ow analysis of heart valves [56,58,6164], turbocharger turbines [911, 70,71], and spacecraft parachute compressible-ow analysis [79].
For more on the ST-SI-IGA, see [77].In the computations presented here, the ST-SI-IGA is used for the reasons given and as described in the rst paragraph of this section.

General-purpose NURBS mesh generation method
While the IGA provides superior accuracy and high-delity solutions, to make its use even more practical in computational ow analysis with complex geometries, NURBS volume mesh generation needs to be easier and more automated.
The general-purpose NURBS mesh generation method introduced in [10] serves that purpose.
The method is based on multi-block-structured mesh generation with established techniques, projection of that mesh to a NURBS mesh made Because good techniques and software for generating multi-block-structured meshes are easy to nd, the method makes general-purpose NURBS mesh generation relatively easy.
Mesh-quality performance studies for 2D and 3D meshes, including those for complex models, were presented in [11].A test computation for a turbocharger turbine and exhaust manifold was also presented in [11], with a more detailed computation in [70].The mesh generation method was used also in the pump-ow analysis part of the ow-driven string dynamics presented in [3] and in the aorta ow analysis presented in [56,57].The performance studies, test computations and actual computations demonstrated that the general-purpose NURBS mesh generation method makes the IGA use in uid mechanics computations even more practical.
For more on the general-purpose NURBS mesh generation method, see [10,11].In the computations presented here, the method is used for the VAWT and for the pump-ow part of the ow-driven string dynamics.
10.Other computational methods

String dynamics
The string in the ow-driven string dynamics is modeled with bending-stabilized cable elements [152], using the IGA 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 uid forces acting on the string.Because a string is a very thin structure, its inuence on the ow will be very small.
In the one-way-dependence model, we compute the inuence of the ow on the string dynamics, while avoiding the formidable task of computing the inuence of the string on the ow.The uid mechanics forces acting on the string are calculated with the method described in [12] for computing the aerodynamic forces acting on the sus-pension lines of spacecraft parachutes.Contact between the string and solid surfaces is handled with the Surface-Edge-Node Contact Tracking (SENCT-FC) method [153], which is a later version of the SENCT introduced in [22].

Particle residence time
In ow-driven string dynamics in pumps, the residence time computations help us to have a simplied but quick understanding of the string behavior.The computation is based on solving a time-dependent advection equation with a unit source term.For more on the computation method, see [3].

Rotation representation with constant angular velocity
We use quadratic NURBS functions, as described in [8], to represent a circular-arc trajectory.The secondary mapping concept, introduced in [6], enables us specify a constant velocity along that trajectory.For more on this method, see [6,8].

ST computation: ow-driven string dynamics in a pump
This section is from [3].

Flow analysis of the pump
We use a vortex pump with 6 blades, including

String dynamics in the pump
The string has 1.5 mm diameter and circularshape cross-section.We compute with three dif- (see Figure 4).In all three cases the string rst hits the top of the blade, and then moves to the edge of the pump casing.

Residence time for the pump
The computation is carried out with a time-step size of 4.9×10 −4 s, which is 5 times larger than the time-step size used in the ow computation.
The number of nonlinear iterations per time step is 2, and the number of GMRES iterations per nonlinear iteration is 30.
The ow-rate-averaged residence time over the outlet is shown in Figure 8.After 1.2 s it reaches the maximum value.Figure 9 shows the spatial distribution of the residence time at the end of the computation.The residence time under the rotor is much higher than the residence time at the outlet, which is around 0.4 s.This means that this region is not connected to the main ow.

Discussion
We discuss the relationship between the string dynamics and the residence time.Figure 10 show, for the string with length 70 mm, the time histories of the string centroid positions in radius and height.We see some strings moving in circles along the bottom edges of the casing.
These strings tend to stay there and cannot rise up.Therefore they stay in the pump forever.
This can be correlated with the high residence time at the bottom of the pump (Figure 9).

ST computation: aerodynamics of a VAWT
We present our preliminary test computations with 2D model of the aerodynamics of a VAWT.
The wind turbine has four support columns at the periphery.Figure 11 shows the wind turbine.
The design is modeled after the wind turbine in [154].The rotor diameter is 16 m, and the ma-  With that orientation, the ow speed seen by a blade can be calculated as where λ is the tip-speed ratio (TSR).The symbol T will denote the rotation cycle.
The computational-domain size is 62.5 times the rotor diameter in the wind direction, with a distance of 18.75 times the rotor diameter between the upstream boundary and the center of the rotor.In the cross-wind direction, the domain size is 37.5 times the rotor diameter.
The mesh position is represented by quadratic NURBS in time.There are three patches that are 120 • each, and the secondary mapping in- troduced in [8] is used to achieve the constant angular velocity.The free-stream velocity is We use two dierent meshes.We start with Mesh 1, and obtain the other mesh by knot insertion.We halve the knot spacing to get Mesh 2.
Figure 14 shows Mesh 1.The number of control points and elements are shown in Table 1.We

ALE computation: HAWT FSI with rotortower coupling
Dynamic coupling of a spinning rotor with exible blades to a deformable tower presents a challenge for standalone structural, as well as coupled FSI simulations.In this section we address this challenge by using a penalty-based approach that allows load transfer between the spinning rotor and tower (see Figure 17).This approach presents an alternative technique to that proposed in [92], and naturally accommodates coupling of distinct structural models (e.g., shells and solids) and discretizations (e.g., nite elements and IGA).

Formulation of the rotortower penalty coupling
In a wind turbine, the rotor hub is connected to the nacelle by the main shaft that transfers the rotational motion of the rotor hub to the gearbox.Since we do not wish to model the drivetrain operation directly, a simplied rotortower coupling strategy is required.We develop such a strategy by exploiting a penalty-based technique.For this, we rst dene the regions on both the rotor and nacelle surfaces that interact with one other, and denote them by Γ 1 (rotor side) and Γ 2 (nacelle side).These regions, which are assumed to have a circular shape, are highlighted using distinct colors in Figure 18.We then design the penalty operator, which precludes all relative motion between Γ 1 and Γ 2 except for relative rotation about the rotor axis.This is achieved, conceptually, by using an overconstrained truss-like system to link the two interaction surfaces.More specically, the change of distance between a point on one surface and every point on the opposing surface, as shown in Figure 18 (a), is penalized.With these considerations, the potential form of the penalty term becomes where β is the penalty constant, x 1 and x 2 are the current positions of the two interaction surfaces, and X r 1 and X r 2 are the reference posi- tions of the two interaction surfaces after taking their relative rotation into account.To arrive at the contribution of the penalty term to the weak form of the structural mechanics problem, we take a variation of Π p with respect to x 1 and x 2 to obtain  In the discrete setting, the above integrals are approximated using numerical quadrature.
Because only quadrature-point locations and weights are needed to formulate the method, it is well suited for coupling of distinct models and discretizations for the dierent structural components, which we do in this work.

Rotor and tower models and meshes
A 3D model of the Hexcrete tower is constructed parametrically using the computer-aided design (CAD) software Rhinoceros 3D and the Grasshopper algorithmic modeling plugin for Rhincoeros (see [155] for details of the parametric modeling methodology).The prole of the tower is hexagonal with smaller hexagonal columns at each corner (see Figure 19).
The tower is comprised of two prismatic sections, located at the top and bottom of the structure, and two intermediate sections with unique rates of taper (see Figure 19).

Results
The FSI simulation is performed at the rated wind speed of 11.4 m/s. Figure 21 shows the ow visualization of the full wind turbine con-guration, and the deection of the tower and blades.
The gure clearly demonstrates that the rotor and tower displacements are coupled while the rotor is spinning.To assess the penalty-coupling error E int we dene it as and plot it a function of time in Figure 22.The gure clearly shows that the coupling error, dened as a relative, dimensionless quantity, is very small.

Concluding remarks
We have described how the challenges encountered in computational analysis of wind turbines

tion 2 .
The core and special methods and other methods are described in Sections 3-10.In Sections 11 and 12, as examples of the ST computations, we present ow-driven string dynamics in a pump and aerodynamics of a VAWT.In Section 13, as an example of the ALE computations, we present FSI of a HAWT with rotortower coupling.The concluding remarks are given in Section 14.

2 c 2020
Journal of Advanced Engineering and Computation (JAEC) ST-VMS and ST-SUPS are versions of the Deforming-Spatial-Domain/Stabilized ST (DS-D/SST) method [2022], which was introduced for computation of ows with moving boundaries and interfaces (MBI), including FSI.The ST-SUPS is a new name for the original version of the DSD/SST, with SUPS reecting its stabilization components, the Streamline-Upwind/Petrov-Galerkin (SUPG ) [23] and Pressure-Stabilizing/Petrov-Galerkin (PSPG) [20] stabilizations.The ST-VMS is the VMS version of the DSD/SST.The VMS components of the ST-VMS are from the residual-based VMS (RBVMS) method [24 27].The ve stabilization terms of the ST-VMS include the three that the ST-SUPS has, and therefore the ST-VMS subsumes the ST-SUPS.In MBI computations the ST-VMS and ST-SUPS function as a moving-mesh methods.
to retain the desirable moving-mesh features of the ALE-VMS in computations with spinning solid sur-faces, such as a turbine rotor.The mesh covering the spinning surface spins with it, retaining the high-resolution representation of the boundary layers.The method was in the context of incompressible-ow equations.Interface terms added to the ALE-VMS to account for the compatibility conditions for the velocity and stress at the SI accurately connect the two sides of the solution.The ST-SI was introduced in [1], also in the context of incompressible-ow equations, to retain the desirable moving-mesh features of the ST-VMS and ST-SUPS in computations with spinning solid surfaces.The starting point in its development was the ALE-SI.

[ 6 , 8 ,
28,45,47], a more accurate representation of the motion of the solid surfaces and a mesh motion consistent with that.It also enables more ecient temporal representation of the motion and deformation of the volume meshes, and more ecient remeshing.These motivated the development of the STNMUM[8,40,45,47].The STNMUM has a wide scope that includes spinning solid surfaces.With the spinning motion represented by quadratic NURBS in time, and with sucient number of temporal patches for a full rotation, the circular paths are represented exactly.A secondary mapping[6,8,19,28] enables also specifying a constant angular velocity for invariant speeds along the circular paths.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, ecient data compression[3,7,60, 6769,144].The STN-MUM and the ST-IGA with IGA basis functions in time have been used in many 3D computations.The classes of problems solved are the computational ow analyses presented here, the ST-IGA enables more accurate representation of the turbine and turbomachinery geometries, increased accuracy in the ow solution, and using larger time-step sizes.Integration of the ST-SI with the ST-IGA enables a more accurate representation of the rotor motion and a mesh motion consistent with that, and we will describe the ST-SI-IGA in Section 8. c 2020 Journal of Advanced Engineering and Computation (JAEC) 8. ST-SI-IGA The ST-SI-IGA is the integration of the ST-SI and ST-IGA.The turbocharger turbine ow [911, 70, 71] and ow-driven string dynamics in turbomachinery [3, 69] were computed with the ST-SI-IGA.The IGA basis functions were used in the spatial discretization of the uid mechanics equations and also in the temporal representation of the rotor and spinning-mesh motion.That enabled accurate representation of the turbine geometry and rotor motion and increased accuracy in the ow solution.The IGA basis functions were used also in the spatial discretization of the string structural dynamics equations.That enabled increased accuracy in the structural dynamics solution, as well as smoothness in the string shape and uid dynamics forces computed on the string.The ram-air parachute analysis [77] and spacecraft parachute compressible-ow analysis [79] were conducted with the ST-SI-IGA, based on the ST-SI version that weakly enforces the Dirichlet conditions and the ST-SI version that accounts for the porosity of a thin structure.The ST-IGA with IGA basis functions in space enabled, with relatively few number of unknowns, accurate representation of the parafoil and parachute geometries and increased accuracy in the ow solution.The volume mesh needed to be generated both inside and outside the parafoil.Mesh generation inside was challenging near the trailing edge because of the narrowing space.The spacecraft parachute has a very complex geometry, including gores and gaps.Using IGA basis functions addressed those challenges and still kept the element density near the trailing edge of the parafoil and around the spacecraft parachute at a reasonable level.In the heart valve analysis [56, 58, 6164], the ST-SI-IGA, beyond enabling a more accurate representation of the geometry and increased accuracy in the ow solution, kept the element density in the narrow spaces near the leaet contact areas at a reasonable level.In computational analysis of ow around tires with road contact and deformation [7375], the ST-SI-IGA enables a more accurate representation of the geometry and motion of the tire sur-faces, a mesh motion consistent with that, and increased accuracy in the ow solution.It also keeps the element density in the tire grooves and in the narrow spaces near the contact ar- of patches that correspond to the blocks, and recovery of the original model surfaces.The recovery of the original surfaces is to the extent 6 c 2020 Journal of Advanced Engineering and Computation (JAEC) they are suitable for accurate and robust computations.The method targets retaining the renement distribution and element quality of the multi-block-structured mesh that we start with.
two higher-height blades.The rotor diameter is roughly 150 mm.We are unable to provide more details due to the industrial-partner restrictions.The quadratic NURBS mesh used in the computation is shown in Figure 1.The number of control points and elements are 838,222 and 544,466.The pump is used for water, the density is 998.2 kg/m 3 , and the kinematic viscosity 8.7×10 −7 m 2 /s.The rotation speed is c 2020 Journal of Advanced Engineering and Computation (JAEC) 7 2,544 rpm.The boundary conditions are shown in Figure 2.

Fig. 2 :
Fig. 2: Boundary conditions.Flow velocity at the inlet (red), zero-stress at the outlet (blue), and noslip on the wall and rotor (green).The circular interface (yellow ) is the SI.

Figure 3 Fig. 3 :
Figure 3 shows the second invariant of the velocity gradient tensor.The turbulent nature of the ow is well represented.The solution is compared to the experimental data from Professor Kazuyoshi Miyagawa's group (Waseda University).The conditions here are close to those corresponding to the best-eciency operating point, and the relative error in the eciency compared to the experimental data is less than 1.5 %.The computed ow eld from rotations 17 through 21 is stored with the ST-C [144] as the data compression method and is used repeatedly in the string dynamics and residence time computations.

ferent string lengths, 10 ,Fig. 4 :
Fig. 4: The initial positions of the strings at the inlet plane.

Tab. 1 :
2D VAWT.Number of control points (nc) and elements (ne).dierent time-step sizes.The two time-step sizes selected translate to ∆φ = 2 • and ∆φ = 1 • per time step.The number of nonlinear iterations per time step is 5, and the number of GMRES iterations per nonlinear iteration is 300.The rst three nonlinear iterations are based on the ST-SUPS, and the last two the ST-VMS.The stabilization parameters are those given by Eqs.(4)(8), and (10) in [70].In the ST-SI, we set C = 2.

c 2020
Journal of Advanced Engineering and Computation (JAEC) (a) A set of distances between a point on a surface and points on another surface.(b) A set of distances in the reference conguration.(c) A set of distances in the current conguration.(d) Total rotation angle.

Fig. 19 :
Fig. 19: CAD model of the Hexcrete tower (left) and a section of the tower solid mesh (right).

Fig. 21 :
Fig. 21: Air speed contours at a planar cut (left) and wind-turbine deected shape (right).The undeformed structure is shown in gray and the deformed structure is shown in light green.

Fig. 22 :
Fig. 22: Penalty coupling error as a function of time.

20 c 2020
Journal of Advanced Engineering and Computation (JAEC) and turbomachinery are being addressed by the ALE-VMS and ST-VMS methods and isogeometric discretization.The computational challenges include turbulent rotational ows, complex geometries, MBI, such as the rotor motion, and the FSI, such as the FSI between the wind turbine blade and the air.The ALE-VMS and ST-VMS serve as the core computational methods.They are supplemented with special methods like the ST-ALE and ST-SI, weak enforcement of the no-slip boundary conditions, and ST-IGA with NURBS basis functions in time.We described the core methods and some of the special methods.We presented, as examples of challenging computations performed, computational analysis of a HAWT, a VAWT and owdriven string dynamics in pumps.The examples demonstrate the power and scope of the core and special methods in computational analysis of wind turbines and turbomachinery.Bazilevs, Y. (2017).Compressible ows on moving domains: Stabilized methods, weakly enforced essential boundary conditions, sliding interfaces, and application to gas-turbine modeling.Computers & Fluids, 158, 201220.[127] Hsu, M.-C., Bazilevs, Y., Calo, V. M., Tezduyar, T. E., & Hughes, T. J. R. (2010).Improving Stability of Stabilized and Multiscale Formulations in Flow Simulations at Small Time Steps.Computer Methods in Applied Mechanics and Engineer-Otoguro, Y., Takizawa, K., & Tezduyar, T. E. (2020).Element Length Calculation in B-Spline meshes for Complex Geometries.Computational Mechanics, published online, DOI: 10.1007/s00466-019-01809-w.[131] Corsini, A., Menichini, C., Rispoli, F., Santoriello, A., & Tezduyar, T. E. (2009).A Multiscale Finite Element Formulation with Discontinuity Capturing for Turbulence Models with Dominant Reactionlike Terms.Journal of Applied Mechanics, 76, 021211.