The first reported computation of the Taylor–Couette flow in the ST framework goes back to 1993 [
7], where the computations were carried out with the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method [
8] and finite element discretization. The stabilization components of the DSD/SST constituted a VMS precursor. The Reynolds numbers used in the computations led to the Taylor vortex flow and the wavy vortex flow, where the waves are in motion. The first reported computation with the isogeometric discretization, in the semi-discrete framework, was in 2010 [
9]. We see advantages in computing the Taylor–Couette flow, and other classes of flow problems that have similar features, with the isogeometric discretization in the ST framework, for reasons summarized in the previous paragraph and explained more in the later parts of the article.
1.1 Stabilized and VMS ST computational methods
The stabilized and VMS ST computational methods started with the DSD/SST, which was introduced for computation of flows with moving boundaries and interfaces (MBI), including fluid–structure interaction (FSI). In flow computations with MBI, the DSD/SST functions as a moving-mesh method. Moving the fluid mechanics mesh to follow an interface enables mesh-resolution control near the interface and, consequently, high-resolution boundary-layer representation near fluid–solid interfaces.
Stabilized and VMS methods have for decades been playing a core-method role in flow analysis with semi-discrete and ST computational methods. The incompressible-flow Streamline-Upwind/Petrov-Galerkin (SUPG) [
10,
11] and compressible-flow SUPG [
12‐
14] methods are two of the earliest and most widely used stabilized methods. The Pressure-Stabilizing/Petrov-Galerkin (PSPG) method [
8], with its Stokes-flow version introduced in [
15], is also among the earliest and most widely used. These methods bring numerical stability in computation of flow problems at high Reynolds or Mach numbers and when using equal-order basis functions for velocity and pressure in incompressible flows. Because the methods are residual-based, the stabilization is achieved without loss of accuracy. The residual-based VMS (RBVMS) [
9,
16‐
18], which is also widely used now, subsumes its precursor SUPG/PSPG.
Because the stabilization components of the original DSD/SST are the
SUPG and
PSPG stabilizations, it is now also called “ST-SUPS.” The ST-VMS is the VMS version of the DSD/SST. The VMS components of the ST-VMS are from the RBVMS. The ST-VMS, which subsumes its precursor ST-SUPS, has two more stabilization terms beyond those in the ST-SUPS, and the additional terms give the method better turbulence modeling features. The ST-SUPS and ST-VMS, because of the higher-order accuracy of the ST framework (see [
1,
19]), are desirable also in computations without MBI.
As a moving-mesh method, the DSD/SST is an alternative to the Arbitrary Lagrangian–Eulerian (ALE) method, which is older (see, for example, [
20]) and more commonly used. The ALE-VMS method [
21,
22] is the VMS version of the ALE. It succeeded the ST-SUPS and ALE-SUPS [
23] and preceded the ST-VMS. The ALE-SUPS, RBVMS and ALE-VMS have been applied to many classes of FSI, MBI and fluid mechanics problems. The classes of problems include ram-air parachute FSI [
23], wind turbine aerodynamics and FSI [
24‐
33], more specifically, vertical-axis wind turbines (VAWTs) [
31,
34,
35], floating wind turbines [
36], wind turbines in atmospheric boundary layers [
30,
31,
37‐
39], and fatigue damage in wind turbine blades [
40], patient-specific cardiovascular fluid mechanics and FSI [
21,
41‐
46], biomedical-device FSI [
47‐
54], ship hydrodynamics with free-surface flow and fluid–object interaction [
55,
56], hydrodynamics and FSI of a hydraulic arresting gear [
57,
58], hydrodynamics of tidal-stream turbines with free-surface flow [
59], bioinspired FSI for marine propulsion [
60,
61], bridge aerodynamics and fluid–object interaction [
62‐
64], and mixed ALE-VMS/Immersogeometric computations [
50‐
52,
65,
66] in the framework of the Fluid–Solid Interface-Tracking/Interface-Capturing Technique. Recent advances in stabilized and multiscale methods may be found for stratified incompressible flows in [
67], for divergence-conforming discretizations of incompressible flows in [
68], and for compressible flows with emphasis on gas-turbine modeling in [
69].
In flow computations with FSI or MBI, the ST-SUPS and ST-VMS require a mesh moving method. Mesh update has two components: moving the mesh for as long as it is possible, which is the core component, and full or partial remeshing when the element distortion becomes too high. The key objectives of a mesh moving method should be to maintain the element quality near solid surfaces and to minimize remeshing frequency. A number of well-performing mesh moving methods were developed in conjunction with the ST-SUPS and ST-VMS. The first one, introduced in [
7,
70], was the Jacobian-based stiffening, which is now called, for reasons explained in [
71], “mesh-Jacobian-based stiffening.” The most recent ones are the element-based mesh relaxation [
72], where the mesh motion is determined by using the large-deformation mechanics equations and an element-based zero-stress-state (ZSS), a mesh moving method [
73] based on fiber-reinforced hyperelasticity and optimized ZSS, and a linear-elasticity-based mesh moving method with no cycle-to-cycle accumulated distortion [
71,
74].
The ST-SUPS and ST-VMS have also been applied to many classes of FSI, MBI and fluid mechanics problems (see [
75] for a comprehensive summary of the work prior to July 2018). The classes of problems include spacecraft parachute analysis for the landing-stage parachutes [
22,
72], cover-separation parachutes [
76] and the drogue parachutes [
77], wind turbine aerodynamics for horizontal-axis wind turbine (HAWT) rotors [
22], full HAWTs [
78] and VAWTs [
6,
31‐
33,
79], flapping-wing aerodynamics for an actual locust [
4,
5,
22], bioinspired MAVs [
80] and wing-clapping [
81,
82], blood flow analysis of cerebral aneurysms [
83], stent-blocked aneurysms [
84,
85], aortas [
53,
54,
86‐
88], heart valves [
53,
54,
81,
87,
89‐
92], ventricle-valve-aorta sequences [
71], and spacecraft aerodynamics [
93], thermo-fluid analysis of ground vehicles and their tires [
2,
38,
39,
90], thermo-fluid analysis of disk brakes [
94], flow-driven string dynamics in turbomachinery [
32,
33,
95,
96], flow analysis of turbocharger turbines [
3,
97,
98], flow around tires with road contact and deformation [
90,
99‐
101], fluid films [
102], ram-air parachutes [
38,
39,
103], and compressible-flow spacecraft parachute aerodynamics [
104,
105].
1.2 ST-SI
The ST-SI was introduced in [
6] in the context of incompressible-flow equations, to retain the desirable moving-mesh features of the ST-VMS and ST-SUPS in computations involving spinning solid surfaces, such as a turbine rotor. The mesh covering the spinning surface spins with it, retaining the high-resolution representation of the boundary layers, while the mesh on the other side of the SI remains unaffected. This is accomplished by adding to the ST-VMS formulation interface terms similar to those in the version of the ALE-VMS for computations with sliding interfaces [
106,
107]. The interface terms account for the compatibility conditions for the velocity and stress at the SI, accurately connecting the two sides of the solution. An ST-SI version where the SI is between fluid and solid domains was also presented in [
6]. The SI in that case is a “fluid–solid SI” rather than a standard “fluid–fluid SI” and enables weak enforcement of the Dirichlet boundary conditions for the fluid. The ST-SI introduced in [
94] for the coupled incompressible-flow and thermal-transport equations retains the high-resolution representation of the thermo-fluid boundary layers near spinning solid surfaces. These ST-SI methods have been applied to aerodynamic analysis of VAWTs [
6,
31‐
33,
79], thermo-fluid analysis of disk brakes [
94], flow-driven string dynamics in turbomachinery [
32,
33,
95,
96], flow analysis of turbocharger turbines [
3,
97,
98], flow around tires with road contact and deformation [
90,
99‐
101], fluid films [
102], aerodynamic analysis of ram-air parachutes [
38,
39,
103], and flow analysis of heart valves [
53,
54,
81,
87,
89‐
92] and ventricle-valve-aorta sequences [
71]. In the ST-SI version presented in [
6] the SI is between a thin porous structure and the fluid on its two sides. This enables dealing with the porosity in a fashion consistent with how the standard fluid–fluid SIs are dealt with and how the Dirichlet conditions are enforced weakly with fluid–solid SIs. This version also enables handling thin structures that have T-junctions. This method has been applied to incompressible-flow aerodynamic analysis of ram-air parachutes with fabric porosity [
38,
39,
103].
1.3 ST Isogeometric Analysis
The success with Isogeometric Analysis (IGA) basis functions in space [
21,
41,
106,
108] motivated the integration of the ST methods with isogeometric discretization, which is broadly called “ST-IGA.” The ST-IGA was introduced in [
1]. Computations with the ST-VMS and ST-IGA were first reported in [
1] in a 2D context, with IGA basis functions in space for flow past an airfoil, and in both space and time for the advection equation. Using higher-order basis functions in time enables deriving full benefit from using higher-order basis functions in space. This was demonstrated with the stability and accuracy analysis given in [
1] for the advection equation.
The ST-IGA with IGA basis functions in time enables a more accurate representation of the motion of the solid surfaces and a mesh motion consistent with that. This was pointed out in [
1,
19] and demonstrated in [
4,
5]. It also enables more efficient temporal representation of the motion and deformation of the volume meshes, and more efficient remeshing. These motivated the development of the STNMUM [
4,
5], with the name coined in [
78]. The STNMUM has a wide scope that includes spinning solid surfaces. With the spinning motion represented by quadratic NURBS in time, and with sufficient number of temporal patches for a full rotation, the circular paths are represented exactly. A “secondary mapping” [
1,
4,
19,
22] 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, efficient data compression [
2,
90,
94‐
96,
109]. The STNMUM and the ST-IGA with IGA basis functions in time have been used in many 3D computations. The classes of problems solved are flapping-wing aerodynamics for an actual locust [
4,
5,
22], bioinspired MAVs [
80] and wing-clapping [
81,
82], separation aerodynamics of spacecraft [
76], aerodynamics of horizontal-axis [
22,
78] and vertical-axis [
6,
31‐
33,
79] wind turbines, thermo-fluid analysis of ground vehicles and their tires [
2,
38,
39,
90], thermo-fluid analysis of disk brakes [
94], flow-driven string dynamics in turbomachinery [
32,
33,
95,
96], and flow analysis of turbocharger turbines [
3,
97,
98].
The ST-IGA with IGA basis functions in space enables more accurate representation of the geometry and increased accuracy in the flow solution. It accomplishes that with fewer control points, and consequently with larger effective element sizes. That in turn enables using larger time-step sizes while keeping the Courant number at a desirable level for good accuracy. It has been used in ST computational flow analysis of turbocharger turbines [
3,
97,
98], flow-driven string dynamics in turbomachinery [
32,
33,
95,
96], ram-air parachutes [
38,
39,
103], spacecraft parachutes [
105], aortas [
53,
54,
87,
88], heart valves [
53,
54,
87,
91,
92], ventricle-valve-aorta sequences [
71], tires with road contact and deformation [
99‐
101], fluid films [
102], and VAWTs [
6,
31‐
33,
79]. The image-based arterial geometries used in patient-specific arterial FSI computations do not come from the ZSS of the artery. Using IGA basis functions in space is now a key part of some of the newest ZSS estimation methods [
53,
110‐
112] and related shell analysis [
113]. The IGA has also been successfully applied to the structural analysis of wind turbine blades [
114‐
118].
1.4 Stabilization parameters and element lengths
In all the semi-discrete and ST stabilized and VMS methods discussed in Sect.
1.1, an embedded stabilization parameter, known as “
\(\tau \),” plays a significant role (see [
22]). This parameter involves a measure of the local length scale (also known as “element length”) and other parameters such as the element Reynolds and Courant numbers. The interface terms in the ST-SI also involve element length, in the direction normal to the interface. Various element lengths and
\(\tau \)s were proposed, starting with those in [
10,
11] and [
12‐
14], followed by the ones introduced in [
119,
120]. In many cases, the element length was seen as an advection length scale, in the flow-velocity direction. The
\(\tau \) definition introduced in [
120], which is for the advective limit and is now called “
\(\tau _{\mathrm {SUGN1}}\)” (and the corresponding element length is now called “
\(h_{\mathrm {UGN}}\)”), automatically yields lower values for higher-order finite element basis functions.
Calculating the
\(\tau \)s based on the element-level matrices and vectors was introduced in [
121] in the context of the advection–diffusion equation and the Navier–Stokes equations of incompressible flows. These definitions are expressed in terms of the ratios of the norms of the matrices or vectors. They automatically take into account the local length scales, advection field and the element Reynolds number. The definitions based on the element-level vectors were shown [
121] to address the difficulties reported at small time-step sizes. A second element length scale, in the solution-gradient direction and called “
\(h_{\mathrm {RGN}}\),” was introduced in [
122,
123]. Recognizing this as a diffusion length scale, a new stabilization parameter for the diffusive limit, “
\(\tau _{\mathrm {SUGN3}}\),” was introduced in [
123,
124], to be used together with
\(\tau _{\mathrm {SUGN1}}\) and “
\(\tau _{\mathrm {SUGN2}}\),” the parameters for the advective and transient limits. For the stabilized ST methods, “
\(\tau _{\mathrm {SUGN12}}\),” representing both the advective and transient limits, was also introduced in [
122,
123].
Some new options for the stabilization parameters used with the SUPS and VMS were proposed in [
2,
4,
78,
125]. These include a fourth
\(\tau \) component, “
\(\tau _{\mathrm {SUGN4}}\)” [
2], which was introduced for the VMS, considering one of the two extra stabilization terms the VMS has compared to the SUPS. They also include stabilization parameters [
2] for the thermal-transport part of the VMS for the coupled incompressible-flow and thermal-transport equations.
Some of the stabilization parameters described in this subsection were also used in computations with other SUPG-like methods, such as the computations reported in [
126,
127].
The stabilization parameters and element lengths discussed in this subsection so far were all originally intended for finite element discretization but quite often used also for isogeometric discretization. The element lengths and stabilization parameters introduced in [
128] target isogeometric discretization but are also applicable to finite element discretization. They were introduced in the context of the advection–diffusion equation and the Navier–Stokes equations of incompressible flows. The direction-dependent element length expression was outcome of a conceptually simple derivation. The key components of the derivation are mapping the direction vector from the physical ST element to the parent ST element, accounting for the discretization spacing along each of the parametric coordinates, and mapping what has been obtained in the parent element back to the physical element. The test computations presented in [
128] for pure-advection cases, including those with discontinuous solution, showed that the element lengths and stabilization parameters proposed result in good solution profiles. The test computations also showed that the “UGN” parameters give reasonably good solutions even with NURBS basis functions. The stabilization parameters given in [
100], which were mostly from [
128], were the latest ones designed in conjunction with the ST-VMS.
In general, we decide what parametric space to use based on reasons like numerical integration efficiency or implementation convenience. Obviously, choices based on such reasons should not influence the method in substance. We require the element lengths, including the direction-dependent element lengths, to have node-numbering invariance for all element types, including simplex elements. The direction-dependent element length expression introduced in [
129] meets that requirement. This is accomplished by using in the element length calculations for simplex elements a preferred parametric space instead of the standard integration parametric space. The element length expressions based on the two parametric spaces were evaluated in [
129] in the context of simplex elements. It was shown that when the element length expression is based on the integration parametric space, the variation with the node numbering could be by a factor as high as 1.9 for 3D elements and 2.2 for ST elements. It was also shown that the element length expression based on the integration parametric space could overestimate the element length by a factor as high as 2.8 for 3D elements and 3.2 for ST elements.
Targeting B-spline meshes for complex geometries, new direction-dependent element length expressions were introduced in [
130]. These latest element length expressions are outcome of a clear and convincing derivation and more suitable for element-level evaluation. The new expressions are based on a preferred parametric space, instead of the standard integration parametric space, and a transformation tensor that represents the relationship between the integration and preferred parametric spaces. We do not want the element splitting to influence the actual discretization, which is represented by the control or nodal points. Therefore, the local length scale should be invariant with respect to element splitting. That invariance is a crucial requirement in element definition, because unlike the element definition choices based on implementation convenience or computational efficiency, it influences the solution. It was proven in [
131] that the local-length-scale expressions introduced in [
130] meet that requirement.
The direction-dependent local-length-scale expressions introduced in [
128,
130] have been used in computational flow analysis of turbocharger turbines [
98], compressible-flow spacecraft parachutes [
105], tires with road contact and deformation [
100,
101], fluid films [
102], ventricle-valve-aorta sequences [
71], and tsunami-shelter VAWTs [
79].
1.5 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.
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. The computations show 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.