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The superior accuracy isogeometric analysis (IGA) brought to computations in fluid and solid mechanics has been yielding higher fidelity in computational aerodynamics. The increased accuracy we achieve with the IGA is in the flow solution, in representing the problem geometry, and, when we use the IGA basis functions also in time in a space–time (ST) framework, in representing the motion of solid surfaces. It is of course as part of a set of methods that the IGA has been very effective in computational aerodynamics, including complex-geometry aerodynamics. The set of methods we have been using can be categorized into those that serve as a core method, those that increase the accuracy, and those that widen the application range. The core methods are the residual-based variational multiscale (VMS), ST-VMS and arbitrary Lagrangian–Eulerian VMS methods. The IGA and ST-IGA are examples of the methods that increase the accuracy. The complex-geometry IGA mesh generation method is an example of the methods that widen the application range. The ST Topology Change method is another example of that. We provide an overview of these methods for IGA-based computational aerodynamics and present examples of the computations performed. In computational flow analysis with moving solid surfaces and contact between the solid surfaces, it is a challenge to represent the boundary layers with an accuracy attributed to moving-mesh methods and represent the contact without leaving a mesh protection gap.

We present method-evaluation incompressible-flow computations with Space–Time Isogeometric Analysis (ST-IGA) and higher-order basis functions in time. The computational-methods platform is made of the ST Variational Multiscale (ST-VMS) method, the ST-IGA with IGA basis functions in space and time, and local length scales and stabilization parameters targeting isogeometric discretization. The computations are for 2D flow past a circular cylinder at Reynolds number 100, which has an easily discernible vortex shedding frequency and widely published lift and drag coefficients and Strouhal number. We compute with quadratic basis functions in space and polynomial orders of 1, 2, 3, and 4 in time, for four different time-step sizes, and with six different sets of expressions for the stabilization parameters. The computations yield a comprehensive set of method-evaluation data that can serve as reference. They also show computational-cost efficiency in using higher-order functions in time.

We present an extensive study on boundary layer mesh resolution in flow computation with the Space–Time Variational Multiscale (ST-VMS) method and isogeometric discretization. The study is in the context of 2D flow past a circular cylinder, at Reynolds number ranging from [Formula: see text] to [Formula: see text]. It was motivated by the need to have in tire aerodynamics a better understanding of the mesh resolution requirements near the tire surface. The focus in the study is on the normal-direction element length for the first layer of elements near the cylinder, with that length varying by a refinement factor ranging from 2 to 40. The evaluation is based mostly on the velocity profile near the cylinder. As the element length for the first layer is varied, the element lengths for the other layers of the disk-shaped inner mesh are adjusted, with no increase in the number of elements for the refinement factors 2, 3, and 4, and with modest increases only in the radial direction for refinement factors beyond that. The computations are performed with quadratic NURBS basis functions in space and linear basis functions in time. The expressions for the stabilization parameters used in the ST-VMS and for the related local lengths scales are those targeting isogeometric discretization, introduced in recent years. The mesh resolution study is based mostly on the strong enforcement of the Dirichlet boundary conditions on the cylinder, but also includes some computations with the weakly-enforced conditions. We expect that the data generated and observations made will be helpful in setting proper near-surface mesh resolution in VMS-based computations with isogeometric discretization, not only for cylindrical shapes but also for comparable geometries. We furthermore expect that although the data generated and observations made are based on computations with nonmoving meshes, they will also be applicable to computations with moving meshes where the mesh around the solid surface rotates with the surface in the framework of the ST Slip Interface method.

We are introducing the Carrier-Domain Method (CDM) for high-resolution computation of time-periodic long-wake flows, with cost-effectives that makes the computations practical. The CDM is closely related to the Multidomain Method, which was introduced 24 years ago, originally intended also for cost-effective computation of long-wake flows and later extended in scope to cover additional classes of flow problems. In the CDM, the computational domain moves in the free-stream direction, with a velocity that preserves the outflow nature of the downstream computational boundary. As the computational domain is moving, the velocity at the inflow plane is extracted from the velocity computed earlier when the plane’s current position was covered by the moving domain. The inflow data needed at an instant is extracted from one or more instants going back in time as many periods. Computing the long-wake flow with a high-resolution moving mesh that has a reasonable length would certainly be far more cost-effective than computing it with a fixed mesh that covers the entire length of the wake. We are also introducing a CDM version where the computational domain moves in a discrete fashion rather than a continuous fashion. To demonstrate how the CDM works, we compute, with the version where the computational domain moves in a continuous fashion, the 2D flow past a circular cylinder at Reynolds number 100. At this Reynolds number, the flow has an easily discernible vortex shedding frequency and widely published lift and drag coefficients and Strouhal number. The wake flow is computed up to 350 diameters downstream of the cylinder, far enough to see the secondary vortex street. The computations are performed with the Space–Time Variational Multiscale method and isogeometric discretization; the basis functions are quadratic NURBS in space and linear in time. The results show the power of the CDM in high-resolution computation of time-periodic long-wake flows.

In this second part of a two-part article, we provide an overview of the heart valve flow analyses conducted with boundary layer and contact representation, made possible with the space–time (ST) computational methods described in the first part. With these ST methods, we are able to represent the boundary layers near moving solid surfaces, including the valve leaflet surfaces, with the accuracy one gets from moving-mesh methods and without the need for leaving a mesh protection gap between the surfaces coming into contact. The challenge of representing the contact between the leaflets without giving up on high-resolution flow representation near the leaflet surfaces has been overcome. The other challenges that have been overcome include the complexities of a near-actual valve geometry, having in the computational model a left ventricle with an anatomically realistic motion and an aorta from CT scans and maintaining the flow stability at the inflow of the ventricle-valve-aorta sequence, where we have a traction boundary condition during part of the cardiac cycle.

In this second part of a two-part article, we present extensive studies on spatial and temporal resolution in wind turbine wake computation with the computational framework presented in the first part. The framework, which is made of the Space–Time Variational Multiscale (ST-VMS) method, ST isogeometric discretization, and the Multidomain Method (MDM), enables accurate representation of the turbine long-wake vortex patterns in a computationally efficient way. Because of the ST context, the framework has higher-order accuracy to begin with; because of the VMS feature of the ST-VMS, it addresses the computational challenges associated with the multiscale nature of the flow; with the isogeometric discretization, it provides increased accuracy in the flow solution; and with the MDM, a long wake can be computed over a sequence of subdomains, instead of a single, long domain, thus reducing the computational cost. Also with the MDM, the computation over a downstream subdomain can start several turbine rotations after the computation over the upstream subdomain starts, thus reducing the computational cost even more. In the computations presented here, the velocity data on the inflow plane comes from a previous wind turbine computation, extracted by projection from a plane located 10 m downstream of the turbine, which has a diameter of 126 m. The resolution studies involve three different spatial resolutions and two different temporal resolutions. The studies show that the computational framework provides, with a practical level of efficiency, high-fidelity solutions in wind turbine long-wake computations.

In this first part of a two-part article, we present a framework for wind turbine wake computation. The framework is made of the Space–Time Variational Multiscale (ST-VMS) method, ST isogeometric discretization, and the Multidomain Method (MDM). The ST context provides higher-order accuracy in general, and the VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the flow. The ST isogeometric discretization enables increased accuracy in the flow solution. With the MDM, a long wake can be computed over a sequence of subdomains, instead of a single, long domain, thus somewhat reducing the computational cost. Furthermore, with the MDM, the computation over a downstream subdomain can start several turbine rotations after the computation over the upstream subdomain starts, thus reducing the computational cost even more. All these good features of the framework, in combination, enable accurate representation of the turbine long-wake vortex patterns in a computationally efficient way. In the computations we present, the velocity data on the inflow plane comes from a previous wind turbine computation, extracted by projection from a plane located 10 m downstream of the turbine, which has a diameter of 126 m. The results show the effectiveness of the framework in wind turbine long-wake computation.

The Taylor–Couette flow is a classical fluid mechanics problem that exhibits, depending on the Reynolds number, a range of flow patterns, with the interesting ones having small-scale structures, and sometimes even wavy nature. Accurate representation of these flow patterns in computational flow analysis requires methods that can, with a reasonable computational cost, represent the circular geometry accurately and provide a high-fidelity flow solution. We use the Space–Time Variational Multiscale (ST-VMS) method with ST isogeometric discretization to address these computational challenges and to evaluate how the method and discretization perform under different scenarios of computing the Taylor–Couette flow. We conduct the computational analysis with different combinations of the Reynolds numbers based on the inner and outer cylinder rotation speeds, with different choices of the reference frame, one of which leads to rotating the mesh, with the full-domain and rotational-periodicity representations of the flow field, with both the convective and conservative forms of the ST-VMS, with both the strong and weak enforcement of the prescribed velocities on the cylinder surfaces, and with different mesh refinements. The ST framework provides higher-order accuracy in general, and the VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the flow. The ST isogeometric discretization enables exact representation of the circular geometry and increased accuracy in the flow solution. In computations where the mesh is rotating, the ST/NURBS Mesh Update Method, with NURBS basis functions in time, enables exact representation of the mesh rotation, in terms of both the paths of the mesh points and the velocity of the points along their paths. In computations with rotational-periodicity representation of the flow field, the periodicity is enforced with the ST Slip Interface method. With the combinations of the Reynolds numbers used in the computations, we cover the cases leading to the Taylor vortex flow and the wavy vortex flow, where the waves are in motion. Our work shows that all these ST methods, integrated together, offer a high-fidelity computational analysis platform for the Taylor–Couette flow and for other classes of flow problems with similar features.

A recently introduced NURBS mesh generation method for complex-geometry Isogeometric Analysis (IGA) is applied to building a high-quality mesh for a gas turbine. The compressible flow in the turbine is computed using the IGA and a stabilized method with improved discontinuity-capturing, weakly-enforced no-slip boundary-condition, and sliding-interface operators. The IGA results are compared with the results from the stabilized finite element simulation to reveal superior performance of the NURBS-based approach. Free-vibration analysis of the turbine rotor using the structural mechanics NURBS mesh is also carried out and shows that the NURBS mesh generation method can be used also in structural mechanics analysis. With the flow field from the NURBS-based turbine flow simulation, the Courant number is computed based on the NURBS mesh local length scale in the flow direction to show some of the other positive features of the mesh generation framework. The work presented further advances the IGA as a fully-integrated and robust design-to-analysis framework, and the IGA-based complex-geometry flow computation with moving boundaries and interfaces represents the first of its kind for compressible flows.

The space–time (ST) computational method “ST-SI-TC-IGA” has recently enabled computational analysis of tire aerodynamics with actual tire geometry, road contact and tire deformation. The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). These ST methods played their parts in overcoming the computational challenges, including (i) the complexity of an actual tire geometry with longitudinal and transverse grooves, (ii) the spin of the tire, (iii) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flow-domain topology change created by the road contact, and (iv) the turbulent nature of the flow. The combination of the ST-VMS, ST-SI and the ST-IGA has also recently enabled solution of fluid film problems with a computational cost comparable to that of the Reynolds-equation model for the comparable solution quality. This was accomplished with the computational flexibility to go beyond the limitations of the Reynolds-equation model. Here we include and address the computational challenges associated with the road roughness and the fluid film between the tire and the road. The new methods we add to accomplish that include a remedy for the trapped fluid, a method for reducing the number of control points as a space occupied by the fluid shrinks down to a narrow gap, and a method for representing the road roughness. We present computations for a 2D test problem with a straight channel, a simple 2D model of the tire, and a 3D model with actual tire geometry and road roughness. The computations show the effectiveness of our integrated set of ST methods targeting tire aerodynamics.

We present a space–time (ST) computational flow analysis method with built-in Reynolds-equation limit. The method enables solution of lubrication fluid dynamics problems with a computational cost comparable to that of the Reynolds-equation model for the comparable solution quality, but with the computational flexibility to go beyond the limitations of the Reynolds-equation model. The key components of the method are the ST Variational Multiscale (ST-VMS) method, ST Isogeometric Analysis (ST-IGA), and the ST Slip Interface (ST-SI) method. The VMS feature of the ST-VMS serves as a numerical stabilization method with a good track record, the moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces, and the higher-order accuracy of the ST framework strengthens both features. The ST-IGA enables more accurate representation of the solid-surface geometries and increased accuracy in the flow solution in general. With the ST-IGA, even with just one quadratic NURBS element across the gap of the lubrication fluid dynamics problem, we reach a solution quality comparable to that of the Reynolds-equation model. The ST-SI enables moving-mesh computation when the spinning solid surface is noncircular. The mesh covering the solid surface spins with it, retaining the high-resolution representation of the flow near the surface, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. We present detailed 2D test computations to show how the method performs compared to the Reynolds-equation model, compared to finite element discretization, at different circumferential and normal mesh refinement levels, when there is an SI in the mesh, and when the no-slip boundary conditions are weakly-enforced.

Tire aerodynamics with actual tire geometry, road contact and tire deformation pose tough computational challenges. The challenges include (1) the complexity of an actual tire geometry with longitudinal and transverse grooves, (2) the spin of the tire, (3) maintaining accurate representation of the boundary layers near the tire while being able to deal with the flow-domain topology change created by the road contact and tire deformation, and (4) the turbulent nature of the flow. A new space–time (ST) computational method, “ST-SI-TC-IGA,” is enabling us to address these challenges. The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). The VMS feature of the ST-VMS addresses the challenge created by the turbulent nature of the flow, the moving-mesh feature of the ST framework enables high-resolution flow computation near the moving fluid–solid interfaces, and the higher-order accuracy of the ST framework strengthens both features. The ST-SI enables moving-mesh computation with the tire spinning. The mesh covering the tire spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-TC enables moving-mesh computation even with the TC created by the contact between the tire and the road. It deals with the contact while maintaining high-resolution flow representation near the tire. Integration of the ST-SI and ST-TC enables high-resolution representation even though parts of the SI are coinciding with the tire and road surfaces. It also enables dealing with the tire–road contact location change and contact sliding. By integrating the ST-IGA with the ST-SI and ST-TC, in addition to having a more accurate representation of the tire geometry and increased accuracy in the flow solution, the element density in the tire grooves and in the narrow spaces near the contact areas is kept at a reasonable level. We present computations with the ST-SI-TC-IGA and two models of flow around a rotating tire with road contact and prescribed deformation. One is a simple 2D model for verification purposes, and one is a 3D model with an actual tire geometry and a deformation pattern provided by the tire company. The computations show the effectiveness of the ST-SI-TC-IGA in tire aerodynamics.

An earlier version of this article included a number of typesetting mistakes. These were corrected on October 16, 2020. The publisher apologizes for the errors made during production. The symbol “Λn” was incorrectly published as “Γn” in equations 46 and 47. The correct equations are provided in this correction.

We focus on turbocharger computational flow analysis with a method that possesses higher accuracy in spatial and temporal representations. In the method we have developed for this purpose, we use a combination of (i) the Space–Time Variational Multiscale (ST-VMS) method, which is a stabilized formulation that also serves as a turbulence model, (ii) the ST Slip Interface (ST-SI) method, which maintains high-resolution representation of the boundary layers near spinning solid surfaces by allowing in a consistent fashion slip at the interface between the mesh covering a spinning surface and the mesh covering the rest of the domain, and (iii) the Isogeometric Analysis (IGA), where we use NURBS basis functions in space and time. The basis functions are spatially higher-order in all representations, and temporally higher-order in representation of the solid-surface and mesh motions. The ST nature of the method gives us higher-order accuracy in the flow solver, and when combined with temporally higher-order basis functions, a more accurate representation of the surface motion, and a mesh motion consistent with that. The spatially higher-order basis functions give us again higher-order accuracy in the flow solver, a more accurate, in some parts exact, representation of the surface geometry, and better representation in evaluating the second-order spatial derivatives. Using NURBS basis functions with a complex geometry is not trivial, however, once we generate the mesh, the computational efficiency is substantially increased. We focus on the turbine part of a turbocharger, but our method can also be applied to the compressor part and thus can be extended to the full turbocharger.

The Space–Time Variational Multiscale (ST-VMS) method was introduced to function as a moving-mesh method. It is the VMS version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method. It has reasonably good turbulence modeling features and serves as a core computational method. The ST Slip Interface (ST-SI) method was introduced to addresses the challenge involved in high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it and thus maintains the high-resolution representation near it. The ST-TC method was introduced for moving-mesh computation of flow problems with topology changes, such as contact between solid surfaces. It deals with the TC while maintaining high-resolution boundary layer representation near solid surfaces. The “ST-SI-TC” method we introduce here integrates the ST-SI and ST-TC methods in the ST-VMS framework. It enables accurate flow analysis when we have a spinning solid surface that is in contact with a solid surface. We present two test computations with the ST-SI-TC method, and they are both with models of flow around a rotating tire with road contact and prescribed deformation, one with a 2D model, and one with a 3D model.

We present computational thermo-fluid analysis of a disk brake, including thermo-fluid analysis of the flow around the brake and heat conduction analysis of the disk. The computational challenges include proper representation of the small-scale thermo-fluid behavior, high-resolution representation of the thermo-fluid boundary layers near the spinning solid surfaces, and bringing the heat transfer coefficient (HTC) calculated in the thermo-fluid analysis of the flow to the heat conduction analysis of the spinning disk. The disk brake model used in the analysis closely represents the actual configuration, and this adds to the computational challenges. The components of the method we have developed for computational analysis of the class of problems with these types of challenges include the Space---Time Variational Multiscale method for coupled incompressible flow and thermal transport, ST Slip Interface method for high-resolution representation of the thermo-fluid boundary layers near spinning solid surfaces, and a set of projection methods for different parts of the disk to bring the HTC calculated in the thermo-fluid analysis. With the HTC coming from the thermo-fluid analysis of the flow around the brake, we do the heat conduction analysis of the disk, from the start of the breaking until the disk spinning stops, demonstrating how the method developed works in computational analysis of this complex and challenging problem.

We present the core and special multiscale space–time (ST) methods we developed for thermo-fluid analysis of a ground vehicle and its tires. We also present application of these methods to thermo-fluid analysis of a freight truck and its rear set of tires. The core multiscale ST method is the ST variational multiscale (ST-VMS) formulation of the Navier–Stokes equations of incompressible flows with thermal coupling, which is multiscale in the way the small-scale thermo-fluid behavior is represented in the computations. The special multiscale ST method is spatially multiscale, where the thermo-fluid computation over the global domain with a reasonable mesh refinement is followed by a higher-resolution computation over the local domain containing the rear set of tires, with the boundary and initial conditions coming from the data computed over the global domain. The large amount of time-history data from the global computation is stored using the ST computation technique with continuous representation in time (ST-C), which serves as a data compression technique in this context. In our thermo-fluid analysis, we use a road-surface temperature higher than the free-stream temperature, and a tire-surface temperature that is even higher. We also include in the analysis the heat from the engine and exhaust system, with a reasonably realistic representation of the rate by which that heat transfer takes place as well as the surface geometry of the engine and exhaust system over which the heat transfer occurs. We take into account the heave motion of the truck body. We demonstrate how the spatially multiscale ST method, with higher-refinement mesh in the local domain, substantially increases the accuracy of the computed heat transfer rates from the tires.

A new space–time (ST) computational method, “ST-SI-TC-IGA,” is enabling us to address the challenges faced in computational analysis of tire aerodynamics with actual geometry, road contact and tire deformation. The core component of the ST-SI-TC-IGA is the ST Variational Multiscale (ST-VMS) method, and the other key components are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and the ST Isogeometric Analysis (ST-IGA). The VMS feature of the ST-VMS addresses the challenge created by the turbulent nature of the flow, the moving-mesh feature of the ST framework enables high-resolution computation near the moving fluid–solid interfaces, and the higher-order accuracy of the ST framework strengthens both features. The ST-SI enables high-resolution representation of the boundary layers near the tire. The mesh covering the tire spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-TC enables moving-mesh computation even with the TC created by the contact between the tire and the road. It deals with the contact while maintaining high-resolution representation near the tire. Integration of the ST-SI and ST-TC enables high-resolution representation even though parts of the SI are coinciding with the tire and road surfaces. It also enables dealing with the tire-road contact location change and contact sliding. By integrating the ST-IGA with the ST-SI and ST-TC, in addition to having a more accurate representation of the tire surfaces and increased accuracy in the flow solution, the element density in the tire grooves and in the narrow spaces near the contact areas is kept at a reasonable level. We present computations with the ST-SI-TC-IGA and two models of flow around a rotating tire with road contact and prescribed deformation. One is a simple 2D model, and one is a 3D model with an actual tire geometry that includes the longitudinal and transverse grooves. The computations show the effectiveness of the ST-SI-TC-IGA in tire aerodynamics.