Stabilized and VMS methods have been an important component of the ST computational methods, starting with the original Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method [
6,
18,
19] in 1990. The DSD/SST [
6,
20,
21] 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. 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 [
22‐
24] 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 [
22,
23]), 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, [
25]) and more commonly used. The ALE-VMS method [
26‐
32] is the VMS version of the ALE. It succeeded the ST-SUPS and ALE-SUPS [
33] and preceded the ST-VMS. To increase their scope and accuracy, the ALE-VMS and RBVMS are often supplemented with special methods, such as those for weakly-enforced Dirichlet boundary conditions [
34‐
36] and “sliding interfaces” [
37,
38]. 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 [
33], wind-turbine aerodynamics and FSI [
39‐
49], more specifically, vertical-axis wind turbines [
48‐
51], floating wind turbines [
52], wind turbines in atmospheric boundary layers [
47‐
49,
53], and fatigue damage in wind-turbine blades [
54], patient-specific cardiovascular fluid mechanics and FSI [
26,
55‐
60], biomedical-device FSI [
61‐
66], ship hydrodynamics with free-surface flow and fluid–object interaction [
67,
68], hydrodynamics and FSI of a hydraulic arresting gear [
69,
70], hydrodynamics of tidal-stream turbines with free-surface flow [
71], passive-morphing FSI in turbomachinery [
72], bioinspired FSI for marine propulsion [
73,
74], bridge aerodynamics and fluid–object interaction [
75‐
77], and mixed ALE-VMS/Immersogeometric computations [
64‐
66,
78,
79] in the framework of the Fluid–Solid Interface-Tracking/Interface-Capturing Technique [
80]. Recent advances in stabilized and multiscale methods may be found for stratified incompressible flows in [
81], for divergence-conforming discretizations of incompressible flows in [
82], and for compressible flows with emphasis on gas-turbine modeling in [
83].
The ST-SUPS and ST-VMS have also been applied to many classes of FSI, MBI and fluid mechanics problems (see [
84] for a comprehensive summary). The classes of problems include spacecraft parachute analysis for the landing-stage parachutes [
29,
85‐
88], cover-separation parachutes [
89] and the drogue parachutes [
90‐
92], wind-turbine aerodynamics for horizontal-axis wind-turbine rotors [
29,
39,
93,
94], full horizontal-axis wind-turbines [
45,
95‐
97] and vertical-axis wind-turbines [
48,
49,
98], flapping-wing aerodynamics for an actual locust [
29,
99‐
101], bioinspired MAVs [
96,
97,
102,
103] and wing-clapping [
104,
105], blood flow analysis of cerebral aneurysms [
96,
106], stent-blocked aneurysms [
106‐
108], aortas [
109‐
113], heart valves [
97,
104,
111,
113‐
117] and coronary arteries in motion [
118], spacecraft aerodynamics [
89,
119], thermo-fluid analysis of ground vehicles and their tires [
24,
115], thermo-fluid analysis of disk brakes [
120], flow-driven string dynamics in turbomachinery [
121‐
123], flow analysis of turbocharger turbines [
124‐
128], flow around tires with road contact and deformation [
115,
129‐
132], fluid films [
132,
133], ram-air parachutes [
134], and compressible-flow spacecraft parachute aerodynamics [
135,
136].