Numerical-performance studies for the stabilized space–time computation of wind-turbine rotor aerodynamics

We present our numerical-performance studies for 3D wind-turbine rotor aerodynamics computation with the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation. The computation is challenging because of the large Reynolds numbers and rotating turbulent flows, and computing the correct torque requires an accurate and meticulous numerical approach. As the test case, we use the NREL 5MW offshore baseline wind-turbine rotor. We compute the problem with both the original version of the DSD/SST formulation and the version with an advanced turbulence model. The DSD/SST formulation with the turbulence model is a recently-introduced space–time version of the residual-based variational multiscale method. We include in our comparison as reference solution the results obtained with the residual-based variational multiscale Arbitrary Lagrangian–Eulerian method using NURBS for spatial discretization. We test different levels of mesh refinement and different definitions for the stabilization parameter embedded in the “least squares on incompressibility constraint” stabilization. We compare the torque values obtained.     

Space–time VMS computation of wind-turbine rotor and tower aerodynamics

We present the space–time variational multiscale (ST-VMS) computation of wind-turbine rotor and tower aerodynamics. The rotor geometry is that of the NREL 5MW offshore baseline wind turbine. We compute with a given wind speed and a specified rotor speed. The computation is challenging because of the large Reynolds numbers and rotating turbulent flows, and computing the correct torque requires an accurate and meticulous numerical approach. The presence of the tower increases the computational challenge because of the fast, rotational relative motion between the rotor and tower. The ST-VMS method is the residual-based VMS version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method, and is also called “DSD/SST-VMST” method (i.e., the version with the VMS turbulence model). In calculating the stabilization parameters embedded in the method, we are using a new element length definition for the diffusion-dominated limit. The DSD/SST method, which was introduced as a general-purpose moving-mesh method for computation of flows with moving interfaces, requires a mesh update method. Mesh update typically consists of moving the mesh for as long as possible and remeshing as needed. In the computations reported here, NURBS basis functions are used for the temporal representation of the rotor motion, enabling us to represent the circular paths associated with that motion exactly and specify a constant angular velocity corresponding to the invariant speeds along those paths. In addition, temporal NURBS basis functions are used in representation of the motion and deformation of the volume meshes computed and also in remeshing. We name this “ST/NURBS Mesh Update Method (STNMUM).” The STNMUM increases computational efficiency in terms of computer time and storage, and computational flexibility in terms of being able to change the time-step size of the computation. We use layers of thin elements near the blade surfaces, which undergo rigid-body motion with the rotor. We compare the results from computations with and without tower, and we also compare using NURBS and linear finite element basis functions in temporal representation of the mesh motion.     

Sequentially-coupled space–time FSI analysis of bio-inspired flapping-wing aerodynamics of an MAV

We present a sequentially-coupled space–time (ST) computational fluid–structure interaction (FSI) analysis of flapping-wing aerodynamics of a micro aerial vehicle (MAV). The wing motion and deformation data, whether prescribed fully or partially, is from an actual locust, extracted from high-speed, multi-camera video recordings of the locust in a wind tunnel. The core computational FSI technology is based on the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) formulation. This is supplemented with using NURBS basis functions in temporal representation of the wing and mesh motion, and in remeshing. Here we use the version of the DSD/SST formulation derived in conjunction with the variational multiscale (VMS) method, and this version is called “DSD/SST-VMST.” The structural mechanics computations are based on the Kirchhoff–Love shell model. The sequential-coupling technique is applicable to some classes of FSI problems, especially those with temporally-periodic behavior. We show that it performs well in FSI computations of the flapping-wing aerodynamics we consider here. In addition to the straight-flight case, we analyze cases where the MAV body has rolling, pitching, or rolling and pitching motion. We study how all these influence the lift and thrust.     

Wall shear stress calculations in space–time finite element computation of arterial fluid–structure interactions

The stabilized space–time fluid–structure interaction (SSTFSI) technique was applied to arterial FSI problems soon after its development by the Team for Advanced Flow Simulation and Modeling. The SSTFSI technique is based on the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation and is supplemented with a number of special techniques developed for arterial FSI. The special techniques developed in the recent past include a recipe for pre-FSI computations that improve the convergence of the FSI computations, using an estimated zero-pressure arterial geometry, Sequentially Coupled Arterial FSI technique, using layers of refined fluid mechanics mesh near the arterial walls, and a special mapping technique for specifying the velocity profile at inflow boundaries with non-circular shape. In this paper we introduce some additional special techniques, related to the projection of fluid–structure interface stresses, calculation of the wall shear stress (WSS), and calculation of the oscillatory shear index. In the test computations reported here, we focus on WSS calculations in FSI modeling of a patient-specific middle cerebral artery segment with aneurysm. Two different structural mechanics meshes and three different fluid mechanics meshes are tested to investigate the influence of mesh refinement on the WSS calculations.     

Space–time fluid mechanics computation of heart valve models

Fluid mechanics computation of heart valves with an interface-tracking (moving-mesh) method was one of the classes of computations targeted in introducing the space–time (ST) interface tracking method with topology change (ST-TC). The ST-TC method is a new version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method. It can deal with an actual contact between solid surfaces in flow problems with moving interfaces, while still possessing the desirable features of interface-tracking methods, such as better resolution of the boundary layers. The DSD/SST method with effective mesh update can already handle moving-interface problems when the solid surfaces are in near contact or create near TC, if the “nearness” is sufficiently “near” for the purpose of solving the problem. That, however, is not the case in fluid mechanics of heart valves, as the solid surfaces need to be brought into an actual contact when the flow has to be completely blocked. Here we extend the ST-TC method to 3D fluid mechanics computation of heart valve models. We present computations for two models: an aortic valve with coronary arteries and a mechanical aortic valve. These computations demonstrate that the ST-TC method can bring interface-tracking accuracy to fluid mechanics of heart valves, and can do that with computational practicality.     

Space–time techniques for computational aerodynamics modeling of flapping wings of an actual locust

We present the special space?Ctime computational techniques we have introduced recently for computational aerodynamics modeling of flapping wings of an actual locust. These techniques have been designed to be used with the deforming-spatial-domain/stabilized space?Ctime (DSD/SST) formulation, which is the core computational technique. The DSD/SST formulation was developed for flow problems with moving interfaces and was elevated to newer versions over the years, including the space?Ctime version of the residual-based variational multiscale (VMS) method, which is called ??DSD/SST-VMST?? and used in the computations reported here. The special space?Ctime techniques are based on using, in the space?Ctime flow computations, NURBS basis functions for the temporal representation of the motion and deformation of the locust wings. The motion and deformation data is extracted from the high-speed, multi-camera video recordings of a locust in a wind tunnel. In addition, temporal NURBS basis functions are used in representation of the motion and deformation of the volume meshes computed and also in remeshing. These ingredients provide an accurate and efficient way of dealing with the wind tunnel data and the mesh. The computations demonstrate the effectiveness of the core and special space?Ctime techniques in modeling the aerodynamics of flapping wings, with the wing motion and deformation coming from an actual locust.     

Space–time computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle

We present a detailed computational analysis of bio-inspired flapping-wing aerodynamics of a micro aerial vehicle (MAV). The computational techniques used include the Deforming-Spatial-Domain/Stabilized Space?CTime (DSD/SST) formulation, which serves as the core computational technique. The DSD/SST formulation is a moving-mesh technique, and in the computations reported here we use the space?Ctime version of the residual-based variational multiscale (VMS) method, which is called ??DSD/ SST-VMST.?? The motion and deformation of the wings are based on data extracted from the high-speed, multi-camera video recordings of a locust in a wind tunnel. A set of special space?Ctime techniques are also used in the computations in conjunction with the DSD/SST method. The special techniques are based on using, in the space?Ctime flow computations, NURBS basis functions for the temporal representation of the motion and deformation of the wings and for the mesh moving and remeshing. The computational analysis starts with the computation of the base case, and includes computations with increased temporal and spatial resolutions compared to the base case. In increasing the temporal resolution, we separately test increasing the temporal order, the number of temporal subdivisions, and the frequency of remeshing. In terms of the spatial resolution, we separately test increasing the wing-mesh refinement in the normal and tangential directions and changing the way node connectivities are handled at the wingtips. The computational analysis also includes using different combinations of wing configurations for the MAV and investigating the beneficial and disruptive interactions between the wings and the role of wing camber and twist.     

Models for space–time random functions

Models are developed for random functions $Q(x,t)$ of space $x\in D$ and time $t\in [0,\tau ]$ from samples of these functions and any other information when available. Most of the models in the paper can be viewed as extensions of Karhunen–Loève (KL) representations for random fields. Their samples are linear forms of basis functions with random coefficients which are extracted from samples of $Q(x,t)$ by singular value decomposition. The coefficients of these forms are stochastic processes rather than random variables. The proposed models can be used to generate large sets of samples whose statistics are similar to those of target random functions. Theoretical arguments and numerical examples are presented to establish properties of the proposed models, assess their accuracy, and illustrate their implementation.     

Multiscale space–time fluid–structure interaction techniques

We present the multiscale space–time techniques we have developed for fluid–structure interaction (FSI) computations. Some of these techniques are multiscale in the way the time integration is performed (i.e. temporally multiscale), some are multiscale in the way the spatial discretization is done (i.e. spatially multiscale), and some are in the context of the sequentially-coupled FSI (SCFSI) techniques developed by the Team for Advanced Flow Simulation and Modeling (T bigstar AFSM){({rm T} bigstar {rm AFSM})}. In the multiscale SCFSI technique, the FSI computational effort is reduced at the stage we do not need it and the accuracy of the fluid mechanics (or structural mechanics) computation is increased at the stage we need accurate, detailed flow (or structure) computation. As ways of increasing the computational accuracy when or where needed, and beyond just increasing the mesh refinement or decreasing the time-step size, we propose switching to more accurate versions of the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation, using more polynomial power for the basis functions of the spatial discretization or time integration, and using an advanced turbulence model. Specifically, for more polynomial power in time integration, we propose to use NURBS, and as an advanced turbulence model to be used with the DSD/SST formulation, we introduce a space–time version of the residual-based variational multiscale method. We present a number of test computations showing the performance of the multiscale space–time techniques we are proposing. We also present a stability and accuracy analysis for the higher-accuracy versions of the DSD/SST formulation.     

Space–time SUPG finite element computation of shallow-water flows with moving shorelines

We show that combination of the Deforming-Spatial-Domain/Stabilized Space–Time and the Streamline-Upwind/Petrov–Galerkin formulations can be used quite effectively for computation of shallow-water flows with moving shorelines. The combined formulation is supplemented with a stabilization parameter that was originally introduced for compressible flows, a compressible-flow shock-capturing parameter adapted for shallow-water flows, and remeshing based on using a background mesh. We present a number of test computations and provide comparisons to theoretical results, experimental data and results computed with nonmoving meshes.     

Space–time computation techniques with continuous representation in time (ST-C)

We introduce space–time computation techniques with continuous representation in time (ST-C), using temporal NURBS basis functions. This gives us a temporally smooth, NURBS-based solution, which is desirable in some cases, and a more efficient way of dealing with the computed data. We propose two versions of ST-C. In the first version, the smooth solution is extracted by projection from a solution computed with a different temporal representation, typically a discontinuous one. We use a successive projection technique with a small number of temporal NURBS basis functions at each projection, and therefore the extraction can take place as the solution with discontinuous temporal representation is being computed, without storing a large amount of time-history data. This version is not limited to solutions computed with ST techniques. In the second version, the solution with continuous temporal representation is computed directly by using a small number of temporal NURBS basis functions in the variational formulation associated with each time step.     

A new visco–elasto-plastic model via time–space fractional derivative

To characterize the visco–elasto-plastic behavior of metals and alloys we propose a new constitutive equation based on a time–space fractional derivative. The rheological representative of the model can be analogous to that of the Bingham–Maxwell model, while the dashpot element and sliding friction element are replaced by the corresponding fractional elements. The model is applied to describe the constant strain rate, stress relaxation and creep tests of different metals and alloys. The results suggest that the proposed simple model can describe the main characteristics of the experimental observations. More importantly, the model can also provide more accurate predictions than the classic Bingham–Maxwell model and the Bingham–Norton model.     

Optical activities as computing resources for space–time symmetries

It is known that optical activities can perform rotations. It is shown that the rotation, if modulated by attenuations, can perform symmetry operations of Wigner's little group which dictates the internal space–time symmetries of elementary particles.     

Computational engineering analysis with the new-generation space–time methods

This is an overview of the new directions we have taken the space–time (ST) methods in bringing solution and analysis to different classes of computationally challenging engineering problems. The classes of problems we have focused on include bio-inspired flapping-wing aerodynamics, wind-turbine aerodynamics, and cardiovascular fluid mechanics. The new directions for the ST methods include the variational multiscale version of the Deforming-Spatial-Domain/Stabilized ST method, using NURBS basis functions in temporal representation of the unknown variables and motion of the solid surfaces and fluid meshes, ST techniques with continuous representation in time, and ST interface-tracking with topology change. We describe the new directions and present examples of the challenging problems solved.     

A scalable time–space multiscale domain decomposition method: adaptive time scale separation

This paper deals with the scalability of a time–space multiscale domain decomposition method in the framework of time-dependent nonlinear problems. The strategy which is being studied is the multiscale LATIN method, whose scalability was shown in previous works when the distinction between macro and micro parts is made on the spatial level alone. The objective of this work is to propose an explanation of the loss-of-scalability phenomenon, along with a remedy which guarantees full scalability provided a suitable macro time part is chosen. This technique, which is quite general, is based on an adaptive separation of scales which is achieved by adding the most relevant functions to the temporal macrobasis automatically. When this method is used, the numerical scalability of the strategy is confirmed by the examples presented.     

Variational modelling of wave–structure interactions with an offshore wind-turbine mast

We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. We develop a linearized model of the fluid–structure or wave–mast coupling, which is a linearization of the variational principle for the fully coupled nonlinear model. Our numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The energy exchange between the subsystems is seen to be in balance, yielding a total energy that shows only small and bounded oscillations amplitude of which tends to zero with the second-order convergence as the timestep approaches zero. Similar second-order convergence is observed for spatial mesh refinement. The linearized model so far developed can be extended to a nonlinear regime.     

A space–time analysis of blood flow in a 3D vessel with multiple aneurysms

In this study, 3D unsteady flow dynamics in an arterial vessel with two asymmetric aneurysms in series have been numerically investigated under pulsatile flow conditions for a full cycle of period T. The non-linear partial differential equations governing the conservation of mass and momentum of viscous incompressible fluid have been numerically analysed by a time accurate cell centered finite volume method in implicit Euler time marching setting. The influence of Reynolds' number (Re), Strouhal's number (St), and degree of dilations (¹ ₁, ² ₁, ¹ ₂, ² ₂) on wall shear stress (WSS), wall pressure (WP) and velocities (u, v, w) have been investigated. While the WSSs increase with either increasing St or increasing Re, WPs are seen to get lowered. During the systolic phase high WPs are seen at the distal end of aneurysms, especially at the distal tip of the larger aneurysm. Decreasing St is seen to delay the flow separation process but the vortex shedding with 3D features is always noticed towards the end of the diastolic phase of the flow cycle.     

Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method in computation of non-Newtonian fluid flow and heat transfer with moving boundaries

This work presents an extension of the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) method to non-Newtonian fluid flow and heat transfer with moving boundaries. In this method, the variational formulation is written over the space–time domain. Three sets of stabilization parameters are used for the continuity, momentum and thermal energy equations. The more efficient solution for highly non-linear problems is achieved by using the Newton–Raphson iterative method for non-linear terms and the generalized minimal residual method for algebraic equations. This work makes the computations feasible with third-order accuracy in time, which is higher then most versions of the FEM. To validate this method, it is used to solve the well-known benchmark problems such as channel-confined flow, lid-driven cavity, flow around a cylinder, and flow in channel with wavy wall, where the non-Newtonian fluid rheological behaviour is incorporated. In particular, the results in terms of the Nusselt number, wall shear stress (WSS), vorticity fields and streamlines are discussed. It shows that the flow and heat transfer characteristics are quite different if the moving boundaries are taken into account. In summary, this work provides an effective extension of the DSD/SST method to hydrodynamics and heat transfer problems involving complex fluids and moving boundaries.