Mathematical Models and Methods in Applied Sciences

In this lead paper of the special issue, we provide a brief summary of the stabilized and multiscale methods in fluid dynamics. We highlight the key features of the stabilized and multiscale scale methods, and variational methods in general, that make these approaches well suited for computational analysis of complex, unsteady flows encountered in modern science and engineering applications. We mainly focus on the recent developments. We discuss application of the variational multiscale (VMS) methods to fluid dynamics problems involving computational challenges associated with high-Reynolds-number flows, wall-bounded turbulent flows, flows on moving domains including subdomains in relative motion, fluid–structure interaction (FSI), and complex-fluid flows with FSI.

We present methods for computation of flow-driven string dynamics in a pump and related residence time. The string dynamics computations help us understand how the strings carried by a fluid interact with the pump surfaces, including the blades, and get stuck on or around those surfaces. The residence time computations help us to have a simplified but quick understanding of the string behavior. The core computational method is the Space–Time Variational Multiscale (ST-VMS) method, and the other key methods are the ST Isogeometric Analysis (ST-IGA), ST Slip Interface (ST-SI) method, ST/NURBS Mesh Update Method (STNMUM), a general-purpose NURBS mesh generation method for complex geometries, and a one-way-dependence model for the string dynamics. The ST-IGA with NURBS basis functions in space is used in both fluid mechanics and string structural dynamics. The ST framework provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the turbulent nature of the unsteady flow, and the moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor 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-IGA enables more accurate representation of the pump geometry and increased accuracy in the flow solution. The IGA discretization also enables increased accuracy in the structural dynamics solution, as well as smoothness in the string shape and fluid dynamics forces computed on the string. The STNMUM enables exact representation of the mesh rotation. The general-purpose NURBS mesh generation method makes it easier to deal with the complex geometry we have here. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small.

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.

We present an immersogeometric analysis (IMGA) approach for the simulation of compressible flows around complex geometries. In this method, compressible flow simulations are performed directly on various boundary representations (B-reps) of mechanical designs, circumventing the labor-intensive and time-consuming cleanup of complex geometric models. A new formulation for the weak imposition of essential boundary conditions in the context of non-body-fitted meshes is proposed. The formulation employs the non-symmetric Nitsche method, which yields good performance especially when the penalty parameters are difficult to estimate. We test the proposed immersogeometric formulation on benchmark problems for a wide range of Reynolds and Mach numbers, showing its robustness and accuracy. Finally, the methodology is applied to the simulation of a UH-60 Black Hawk helicopter in forward flight, illustrating the ability of the proposed approach to support the design of real-world engineering systems through high-fidelity aerodynamic analysis.

Buffeting analysis plays an important role in the wind-resistant design of long-span bridges. While computational methods have been widely used in the study of self-excited forces on bridge sections, there is very little work on applying advanced simulation to buffeting analysis. In an effort to address this shortcoming, we developed a framework for the buffeting simulation of bridge sections subjected to turbulent flows. We carry out simulations of a rectangular bridge section with aspect ratio 10 and compute its aerodynamic admittance functions. The simulations show good agreement with airfoil theory and experimental observations. It was found that inflow turbulence plays an important role in obtaining accurate wind loads on the bridge sections. The proposed methodology is envisioned to have practical impact in wind engineering of structures in the future.

Computational fluid–structure interaction (FSI) and flow analysis now have a significant role in design and performance evaluation of turbomachinery systems, such as wind turbines, fans, and turbochargers. With increasing scope and fidelity, computational analysis can help improve the design and performance. For example, it can help add a passive morphing attachment (MA) to the blades of an axial fan for the purpose of controlling the blade load and section stall. We present a stabilized Arbitrary Lagrangian–Eulerian (ALE) method for computational FSI analysis of passive morphing in turbomachinery. The main components of the method are the Streamline-Upwind/Petrov–Galerkin (SUPG) and Pressure-Stabilizing/Petrov–Galerkin (PSPG) stabilizations in the ALE framework, mesh moving with Jacobian-based stiffening, and block-iterative FSI coupling. The turbulent-flow nature of the analysis is handled with a Reynolds-Averaged Navier–Stokes (RANS) model and SUPG/PSPG stabilization, supplemented with the “DRDJ” stabilization. As the structure moves, the fluid mechanics mesh moves with the Jacobian-based stiffening method, which reduces the deformation of the smaller elements placed near the solid surfaces. The FSI coupling between the blocks of the fully-discretized equation system representing the fluid mechanics, structural mechanics, and mesh moving equations is handled with the block-iterative coupling method. We present two-dimensional (2D) and three-dimensional (3D) computational FSI studies for an MA added to an axial-fan blade. The results from the 2D study are used in determining the spanwise length of the MA in the 3D study.

We consider a computational model for binary-fluid–solid interaction based on an arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes–Korteweg equations, and we assess the predictive capabilities of this model. Due to the presence of two distinct fluid components, the stress tensor in the binary-fluid exhibits a capillary component in addition to the pressure and viscous-stress components. The distinct fluid–solid surface energies of the fluid components moreover lead to preferential wetting at the solid substrate. Compared to conventional FSI problems, the dynamic condition coupling the binary-fluid and solid subsystems incorporates an additional term associated with the binary-fluid–solid surface tension. We consider a formulation of the Navier–Stokes–Korteweg equations in which the free energy associated with the standard van-der Waals equation of state is replaced by a polynomial double-well function to provide better control over the diffuse-interface thickness and the surface tension. For the solid subsystem, we regard a standard hyperelastic model. We explore the main properties of the binary-fluid–solid interaction problem and establish a dissipation relation for the aggregated system. In addition, we present numerical results based on a fully monolithic approach to the complete nonlinear system. To validate the computational model, we consider the elasto-capillary interaction of a sessile droplet on a soft solid substrate and compare the numerical results with a corresponding solid model with fabricated fluid loads and with experimental data.