Accuracy of high‐order density‐based compressible methods in low Mach vortical flows

At low Mach numbers, Godunov‐type approaches, based on the method of lines, suffer from an accuracy problem. This paper shows the importance of using the low Mach number correction in Godunov‐type methods for simulations involving low Mach numbers by utilising a new, well‐posed, two‐dimensional, two‐mode Kelvin–Helmholtz test case. Four independent codes have been used, enabling the examination of several numerical schemes. The second‐order and fifth‐order accurate Godunov‐type methods show that the vortex‐pairing process can be captured on a low resolution with the low Mach number correction applied down to 0.002. The results are compared without the low Mach number correction and also three other methods, a Lagrange‐remap method, a fifth‐order accurate in space and time finite difference type method based on the wave propagation algorithm, and fifth‐order spatial and third‐order temporal accurate finite volume Monotone Upwind Scheme for Conservation Laws (MUSCL) approach based on the Godunov method and Simple Low Dissipation Advection Upstream Splitting Method (SLAU) numerical flux with low Mach capture property. The ability of the compressible flow solver of the commercial software, ANSYS FLUENT , in solving low Mach flows is also demonstrated for the two time‐stepping methods provided in the compressible flow solver, implicit and explicit. Results demonstrate clearly that a low Mach correction is required for all algorithms except the Lagrange‐remap approach, where dissipation is independent of Mach number. © 2013 Crown copyright. International Journal for Numerical Methods in Fluids. © 2013 John Wiley & Sons, Ltd.     

Joint L1/Lp-regularized minimization in video recovery of remote sensing based on compressed sensing

L₁ regularization and L_p regularization are proposed for processing recovered images based on compressed sensing (CS). L₁ regularization can be solved as a convex optimization problem but is less sparse than L_p (0 < p < 1). L_p regularization is sparser than L₁ regularization but is more difficult to solve. This paper proposes joint L₁/L_p (0 < p < 1) regularization, which combines L_p regularization and L₁ regularization. This joint regularization is applied to recover video of remote sensing based on CS. Joint regularization is sparser than L₁ regularization but is as easy to solve as L₁ regularization. A linearized Bregman reweighted iteration algorithm is proposed to solve the joint L₁/L_p regularization problem. The performance and capabilities of the linearized Bregman algorithm and linearized Bregman reweighted algorithm for solving the joint L₁/L_p regularization model are analyzed and compared through numerical simulations.     

Stability of algebraic multigrid for Stokes problems

An investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle‐point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed‐order and equal‐order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh‐dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed‐order interpolation to permit stable smoothing and therefore to deliver mesh‐independent convergence is explained. Two expedient options are suggested for obtaining mesh‐independent convergence for those AMG codes that are based on an equal‐order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.     

Evaluation of RBFs for volume data interpolation in CFD

A greedy method for choosing an optimum reduced set of control points is integrated with RBF interpolation and evaluated for the purpose of interpolating large‐volume data sets in CFD. Given a function defined at a set of points, the greedy method selects a small subset of these points that is sufficient to keep the interpolation error at all the remaining points below a chosen bound. This is equivalent to a type of data compression and would have useful storage, post‐processing, and computational applications in CFD. To test the method in terms of both the point selection scheme and the suitability of reduced control point volume interpolation, a trial application of the interpolation to velocity fields in CFD volume meshes is considered. To optimise the point selection process, and attempt to be able to capture multiple length scales, a variable support radius formulation has also been included. Structured and unstructured mesh cases are considered for aerofoils, a wing case and a wing‐body case. For smooth volume functions, the method is shown to work well, producing accurate velocity interpolations using a very small number of the cells in the mesh. For general complex fields including large gradients, the method is still shown to be effective, although large gradients require more interpolation points to be used.Copyright © 2013 John Wiley & Sons, Ltd.     

Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L^∞ norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.     

GPU‐accelerated direct numerical simulations of decaying compressible turbulence employing a GKM‐based solver

Gas Kinetic Method‐based flow solvers have become popular in recent years owing to their robustness in simulating high Mach number compressible flows. We evaluate the performance of the newly developed analytical gas kinetic method (AGKM) by Xuan et al. in performing direct numerical simulation of canonical compressible turbulent flow on graphical processing unit (GPU)s. We find that for a range of turbulent Mach numbers, AGKM results shows excellent agreement with high order accurate results obtained with traditional Navier–Stokes solvers in terms of key turbulence statistics. Further, AGKM is found to be more efficient as compared with the traditional gas kinetic method for GPU implementation. We present a brief overview of the optimizations performed on NVIDIA K20 GPU and show that GPU optimizations boost the speedup up‐to 40x as compared with single core CPU computations. Hence, AGKM can be used as an efficient method for performing fast and accurate direct numerical simulations of compressible turbulent flows on simple GPU‐based workstations. Copyright © 2016 John Wiley & Sons, Ltd.     

An operator splitting strategy for fluid–structure interaction problems with thin elastic structures in an incompressible Newtonian flow

We present a computational framework based on the use of the Newton and level set methods to model fluid–structure interaction problems involving elastic membranes freely suspended in an incompressible Newtonian flow. The Mooney–Rivlin constitutive model is used to model the structure. We consider an extension to a more general case of the method described in Laadhari (2017) to model the elasticity of the membrane. We develop a predictor–corrector finite element method where an operator splitting scheme separates different physical phenomena. The method features an affordable computational burden with respect to the fully implicit methods. An exact Newton method is described to solve the problem, and the quadratic convergence is numerically achieved. Sample numerical examples are reported and illustrate the accuracy and robustness of the method.     

Application of Reynolds stress transport turbulence closure models to flows affected by Lorentz and buoyancy forces

The effect of magnetic field strength and orientation on two types of electromagnetically influenced turbulent flows was studied numerically under the Reynolds averaged Navier–Stokes (RANS) framework. Previous work (Wilson et al., 2014) used an electromagnetically extended linear eddy-viscosity model, whilst the current paper focuses on the performance of a more advanced Reynolds stress transport type model both with and without electromagnetic modifications proposed by Kenjereš et al. (2004). First, a fully-developed 2D channel flow is considered with a magnetic field imposed in either the wall-normal or streamwise direction. Both forms of the RSM gave good agreement with the DNS data for the wall-normal magnetic field across the range of Hartmann numbers with the additional electromagnetic terms providing a small, but noticeable, difference. For the streamwise magnetic field, where electromagnetic influence is only through the turbulence, the electromagnetically extended RSM performed well at moderate Hartmann numbers but returned laminar flow at the highest Hartmann number considered, contrary to the DNS. The RSM results were, however, significantly better than the previous eddy-viscosity model predictions. The second case is that of unsteady 3D Rayleigh–Bénard convection with a magnetic field imposed in either a horizontal or vertical direction. Results revealed that a significant reorganization of the flow structures is predicted to occur. For a vertically oriented magnetic field, the plume structures increase in number and become thinner and elongated along the magnetic field lines, leading to an increase in thermal mixing within the core in agreement with Hanjalić and Kenjereš (2000). With a horizontal magnetic field, the structures become two-dimensional and a striking realignment of the roll cells’ axes with the magnetic field lines occurs. The results demonstrate the capability of the Reynolds stress transport approach in modelling MHD flows that are relevant to industry and offer potential for those wishing to control levels of turbulence, heat transfer or concentration without recourse to mechanical means.     

A sharp‐interface immersed boundary framework for simulations of high‐speed inviscid compressible flows

A new finite‐volume flow solver based on the hybrid Cartesian immersed boundary (IB) framework is developed for the solution of high‐speed inviscid compressible flows. The IB method adopts a sharp‐interface approach, wherein the boundary conditions are enforced on the body geometry itself. A key component of the present solver is a novel reconstruction approach, in conjunction with inverse distance weighting, to compute the solutions in the vicinity of the solid‐fluid interface. We show that proposed reconstruction leads to second‐order spatial accuracy while also ensuring that the discrete conservation errors diminish linearly with grid refinement. Investigations of supersonic and hypersonic inviscid flows over different geometries are carried out for an extensive validation of the proposed flow solver. Studies on cylinder lift‐off and shape optimisation in supersonic flows further demonstrate the efficacy of the flow solver for computations with moving and shape‐changing geometries. These studies conclusively highlight the capability of the proposed IB methodology as a promising alternative for robust and accurate computations of compressible fluid flows on nonconformal Cartesian meshes.