Viscous Fluxes Research Articles

An efficient cell-centered finite-volume method with face-averaged nodal-gradients for triangular grids

In this paper, a face-averaged nodal-gradient approach is proposed as an efficient gradient method for a second-order cell-centered finite-volume discretization on triangular grids. The gradients needed in the linear reconstruction are computed in two steps: (1) compute gradients at nodes from solutions stored at cells, and (2) compute the gradient at each face by averaging the nodal gradients over the nodes forming the face. A unique feature of the proposed approach is that the same face-averaged gradient is used in linear reconstructions from both the left and right cells to the face centroid. This approach offers advantages over a conventional cell-gradient-based finite-volume method: reduced memory requirement for the gradients, smaller residual stencils, direct evaluations of viscous stresses and heat fluxes at solid boundaries, and less interpartition communication. The method is demonstrated for various inviscid and viscous problems from low to high Mach numbers on triangular grids.

A modal discontinuous Galerkin method for simulating dusty and granular gas flows in thermal non-equilibrium in the Eulerian framework

A modal discontinuous Galerkin method was developed for computing compressible rarefied gaseous flows interacting with rigid particles and granular medium. In contrast to previous particle-based models that were developed to handle rarefied flows or solid phase particles, the present computational method employs full continuum-based models. This work is one of the first attempts to apply the modal discontinuous Galerkin method to a two-fluid model framework, which covers a wide range of gas and solid phase regimes, from a continuum to non-equilibrium gas, and from dusty to collisional regimes. The rarefaction effects were taken into account by applying the second-order Boltzmann-Curtiss-based constitutive relationship in a two-fluid system of equations. For the dust phase, computational models were developed based on the kinetic theory of the granular flows. Due to the orthogonal property of the basis functions in the method, no specific treatment of the source terms, commonly necessary in the conventional finite volume method, was required. Moreover, a high-fidelity approach was selected to treat the non-strictly hyperbolic equations of a dusty gas. This allows the same inviscid numerical flux functions to be applied to both the gaseous Euler and solid pressureless-Euler system of equations. Further, we observed that, for the discretization of the viscous fluxes in multiphase cases, the local discontinuous Galerkin is superior to the first method by Bassi and Rebay. After a verification and validation study, the new computational model was used to simulate the impingement of an underexpanded jet on a dusty surface in a rarefied condition. A surface erosion model based on viscous erosion associated with aerodynamic entrainment was implemented at a solid surface. Simulation cases in the near-field of the nozzle flow were tested to evaluate the capabilities of the present computational model in handling the challenging problems of multi-scale multiphase flows.

An entropy–stable discontinuous Galerkin approximation for the incompressible Navier–Stokes equations with variable density and artificial compressibility

We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier–Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressibility, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split–form nodal Discontinuous Galerkin (DG) approximation that satisfies the summation–by–parts simultaneous–approximation–term (SBP–SAT) property. The scheme and the stability proof are presented for general curvilinear three–dimensional hexahedral meshes. We use the exact Riemann solver and the Bassi–Rebay 1 (BR1) scheme at the inter–element boundaries for inviscid and viscous fluxes respectively, and an explicit low storage Runge–Kutta RK3 scheme to integrate in time. We assess the accuracy and robustness of the method by solving a manufactured solution, the Kovasznay flow, a lid driven cavity, the inviscid Taylor–Green vortex, and the Rayleigh–Taylor instability.

Weak formulation and scaling properties of energy fluxes in three-dimensional numerical turbulent Rayleigh–Bénard convection

We apply the weak formalism on the Boussinesq equations, to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy flux in very high resolution numerical simulations. The local Bolgiano-Oboukhov length $L_{BO}$ is investigated and it is found that its value may change of an order of magnitude through the domain, in agreement with previous results. We investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman-Howarth-Monin and Yaglom equations. We have not found the classical Bolgiano-Oboukhov picture, but evidence of a mixture of Bolgiano-Oboukhov and Kolmogorov scalings. In particular, all the terms are compatible with a Bolgiano-Oboukhov local H\"older exponent for the temperature and a Kolmogorov 41 for the velocity. This behavior may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of Bolgiano-Oboukhov local scales. The scale-by-scale analysis allows us also to compare the theoretical Bolgiano-Oboukhov length $L_{BO}$ computed from its definition with that empirically extracted through scalings obtained from weak analysis. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties.

A Two-Zone Shear-Induced Red Blood Cell Migration Model for Blood Flow in Microvessels

The Fahraeus and Fahraeus-Lindqvist effects are both associated with the concentration of red blood cells (RBCs) in the core region of microvessels. The annular region is a cell-free layer. Blood flow dynamics and both effects are related to the hematocrit level profile. The aim is to propose a model for blood flow in microvessels that is not compute intensive like many other models such as those using finite element methods. Modeling blood flow requires solving for both the hematocrit level and velocity profiles as blood viscosity depends on the hematocrit level. The two-zone shear-induced model for blood flow is adopted while including an annular cell-free layer, as in the marginal zone theory and in consistency with experimental observations. In the core region, the hematocrit level is not considered to be uniform, and the concentration and viscous fluxes are equal in magnitude and opposite in directions in the fully developed velocity and concentration profiles case. The momentum and hematocrit balance equations are solved. Both analytical and numerical solutions for the velocity and hematocrit level profiles are determined. The numerical results are found to exactly match the analytical solutions, and to be in very good agreement with published experimental data for the cell-free layer thickness, the velocity profile, and the hematocrit ratio.

Intrinsic rotation in axisymmetric devices

Toroidal rotation is critical for fusion in tokamaks, since it stabilizes instabilities that can otherwise cause disruptions or degrade confinement. Unlike present-day devices, ITER might not have enough neutral-beam torque to easily avoid these instabilities. We must therefore understand how the plasma rotates ‘intrinsically,’ that is, without applied torque. Experimentally, torque-free plasmas indeed rotate, with profiles that are often non-flat and even non-monotonic. The rotation depends on many plasma parameters including collisionality and plasma current, and exhibits sudden bifurcations (‘rotation reversals’) at critical parameter values.Since toroidal angular momentum is conserved in axisymmetric systems, and since experimentally inferred momentum transport is much too large to be neoclassical, theoretical work has focused on rotation drive by nondiffusive turbulent momentum fluxes. In the edge, intrinsic rotation relaxes to a steady state in which the total momentum outflux from the plasma vanishes. Ion drift orbits, scrape-off-layer flows, separatrix geometry, and turbulence intensity gradient all play a role. In the core, nondiffusive and viscous momentum fluxes balance to set the rotation gradient at each flux surface. Although many mechanisms have been proposed for the nondiffusive fluxes, most are treated in one of two distinct but related gyrokinetic formulations. In a radially local fluxtube, appropriate for , the lowest-order gyrokinetic formulations exhibit a symmetry that prohibits nondiffusive momentum flux for nonrotating plasmas in an up-down symmetric magnetic geometry with no shear. Many symmetry-breaking mechanisms have been identified, but none have yet been conclusively demonstrated to drive a strong enough flux to explain commonly observed experimental rotation profiles. Radially global gyrokinetic simulations naturally include many symmetry-breaking mechanisms, and have shown cases with experimentally relevant levels of nondiffusive flux. These promising early results motivate further work to analyze, verify, and validate.This article provides a pedagogical introduction to intrinsic rotation in axisymmetric devices. Intended for both newcomers to the topic and experienced practitioners, the article reviews a broad range of topics including experimental and theoretical results for both edge and core rotation, while maintaining a focus on the underlying concepts.

A diffuse‐interface immersed boundary method for simulation of compressible viscous flows with stationary and moving boundaries

SummaryIn this paper, a diffuse‐interface immersed boundary method (IBM) is proposed for simulation of compressible viscous flows with stationary and moving boundaries. In the method, the solution of flow field and the implementation of boundary conditions are decoupled into two steps by applying the fractional step technique, ie, the predictor step and the corrector step. Firstly, in the predictor step, the intermediate flow field is resolved by a recently developed gas kinetic flux solver (GKFS) without consideration of the solid boundary. The GKFS is a finite volume approach that solves the Navier‐Stokes equations for the flow variables at cell centers. In GKFS, the inviscid and viscous fluxes are evaluated as a single entity by reconstructing the local solution of continuous Boltzmann equation. Secondly, in the corrector step, the intermediate flow field is corrected by the present diffuse‐interface IBM. During this process, the velocity field is firstly corrected by the implicit boundary condition–enforced IBM so that the no‐slip boundary condition can be accurately satisfied. After that, the density correction is made by an iterative approach with the help of the continuity equation. Finally, the correction of the temperature field is made in the same way as that of the velocity field. Good agreements between the present simulations and the reference data in literature demonstrate the reliability of the proposed method.

A high order least square-based finite difference-finite volume method with lattice Boltzmann flux solver for simulation of incompressible flows on unstructured grids

This paper presents a high order least square-based finite difference-finite volume (LSFD-FV) method together with lattice Boltzmann flux solver for accurate simulation of incompressible flows on unstructured grids. Within each control cell, a high order polynomial, which is based on Taylor series expansion, is applied to approximate the solution function. The derivatives in the Taylor series expansion are approximated by the functional values at the centers of neighboring cells using the mesh-free least square-based finite difference (LSFD) method developed by Ding et al. (2004) [36]. In the high order finite volume method, the recently developed lattice Boltzmann flux solver (LBFS) is applied to evaluate the inviscid and viscous fluxes physically and simultaneously at the cell interface by local reconstruction of lattice Boltzmann solution. Compared with traditional k-exact high-order finite volume method, the present method is more accurate and efficient. Various benchmark examples are tested to validate the high-order accuracy, high computational efficiency and flexibility of the proposed method on unstructured grids.

A shock-capturing meshless scheme using RBF blended interpolation and moving least squares

Solving the full Navier-Stokes equations requires the use of numerical methods, especially for engineering application involving complex geometry. Numerical methods commonly used today all require a mesh before a solution attempt. Developing the mesh can require expensive preprocessing time, and the quality of the mesh can have major effects on the solution. Meshless methods have become a research interest due to the simplicity of using scattered data points. Radial basis functions (RBF) interpolation is a class of meshless method, and many of these methods have been developed to solve partial differential equations. Schemes using RBF interpolation are capable of multivariate interpolation over scattered data even for data with discontinuities. A meshless method scheme is presented capable of solving the compressible Navier-Stokes equations. A blended RBF interpolation method switching between low and high shape parameter interpolation is used to approximate the inviscid fluxes, while the viscous fluxes are approximated using moving least squares (MLS).

Five Theorems of Turbulence and Their Practical Application

Five theorems of turbulence are stated on the basis of experimental and theoretical research. These theorems permit simpler mathematical description of the gas dynamics of viscous, nonsteady, nonequilibrium, and compressible fluxes. Some conclusions are reached regarding the gas dynamics of combustion chambers and nozzles in rocket engines.

An embedded boundary approach for efficient simulations of viscoplastic fluids in three dimensions

We present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modeled by generalized Newtonian rheological equations. It is implemented in a highly efficient framework for massively parallelizable computations on block-structured grids. In this context, geometric features are handled by the embedded boundary approach, which requires specialized treatment only in cells intersecting or adjacent to the boundary. This constitutes the first published implementation of an embedded boundary algorithm for simulating flow of viscoplastic fluids on structured grids. The underlying algorithm employs a two-stage Runge-Kutta method for temporal discretization, in which viscous terms are treated semi-implicitly and projection methods are utilized to enforce the incompressibility constraint. We augment the embedded boundary algorithm to deal with the variable apparent viscosity of the fluids. Since the viscosity depends strongly on the strain rate tensor, special care has been taken to approximate the components of the velocity gradients robustly near boundary cells, both for viscous wall fluxes in cut cells and for updates of apparent viscosity in cells adjacent to them. After performing convergence analysis and validating the code against standard test cases, we present the first ever fully three-dimensional simulations of creeping flow of Bingham plastics around translating objects. Our results shed new light on the flow fields around these objects.

Magnetodrag in the hydrodynamic regime: Effects of magnetoplasmon resonance and Hall viscosity

In this work we study magnetotransport properties in electronic double layers of strongly correlated electron liquids. For sufficiently clean high-mobility samples, the high-temperature regime of transport in these systems can be described in pure hydrodynamic terms. We concentrate on the magnetic field dependence of longitudinal drag effect mediated by the interlayer Coulomb scattering and identify several mechanisms of transresistance which is caused by viscous flows, magnetoplasmon resonance, and dissipative thermal fluxes. In particular, we elucidate how Hall viscosity enters magnetodrag and modifies its temperature dependence in the magnetic field.

Static and Dynamic Throughput of a Hose Valve in Pipeline Regulation

A method is proposed for determining the throughput of a hose valve in systems for regulating the pressure and stabilizing the speed of viscous liquid fluxes in a pipe. The static and dynamic characteristics are determined for valves with different parameters.

High-order two-fluid plasma solver for direct numerical simulations of plasma flows with full transport phenomena

The two-fluid plasma equations for a single ion species, with full transport terms, including temperature and magnetic field dependent ion and electron viscous stresses and heat fluxes, frictional drag force, and ohmic heating terms, have been implemented in the CFDNS code and solved by using sixth-order non-dissipative compact finite differences for plasma flows in several different regimes. In order to be able to fully resolve all the dynamically relevant time and length scales, while maintaining computational feasibility, the assumptions of infinite speed of light and negligible electron inertia have been made. Non-dimensional analysis of the two-fluid plasma equations shows that, by varying the characteristic/background number density, length scale, temperature, and magnetic strength, the corresponding Hall, resistive, and ideal magnetohydrodynamic equations can be recovered as limiting cases. The accuracy and robustness of this two-fluid plasma solver in handling plasma flows in different regimes have been validated against four canonical problems: Alfven and whistler dispersion relations, electromagnetic plasma shock, and magnetic reconnection. For all test cases, by using physical dissipation and diffusion, with negligible numerical dissipation/diffusion, fully converged Direct Numerical Simulation (DNS)-like solutions are obtained when the ion Reynolds number based on the grid size is smaller than a threshold value which is about 2.3 in this study. For the magnetic reconnection problem, the results show that the magnetic flux saturation time and value converge when the ion and magnetic Reynolds numbers are large enough. Thus, the DNS-like results become relevant to practical problems with much larger Reynolds numbers.

Development of a structured overset Navier–Stokes solver with a moving grid and full multigrid method

An unsteady Reynolds averaged Navier–Stokes solver with a structured overset grid method has been developed. Velocity–pressure coupling is achieved using an artificial compressibility approach, spatial discretization is based on a finite-volume method, and convective fluxes are evaluated by a third-order upwind scheme based on flux-difference splitting. Body motions are considered using the grid deformation technique and grid velocities in the convective term. Viscous fluxes are evaluated via second-order centered differencing. The full multigrid (FMG) method is applied to obtain fast convergence. The cell flag on a coarse grid level is determined using the cell flag on a fine grid level. In the coarse grid-level calculations at the FMG stage, the data are interpolated until the finest grid level is achieved at an overset update interval. Then, the data are updated based on the overset relations at the finest grid level and then transferred to a coarser grid level. For free surface treatments, a single-phase localized level-set method is employed. The computations for flows around a hull form, including an unsteady simulation with regular waves, are demonstrated.

The barely implicit correction algorithm for low-Mach-Number flows

A new Barely Implicit Correction (BIC) algorithm is presented for the simulation of low-Mach-number flows. This new algorithm is based on the original, introduced by G. Patnaik et al. [G. Patnaik, R. H. Guirguis, J. P. Boris and E. S. Oran, A barely implicit correction for flux-corrected transport. In: Journal of Computational Physics 71.1 (1987), pp. 120], which was a solution procedure including an explicit predictor step to solve the convective portion of the Navier–Stokes equations and an implicit corrector step to remove the acoustic limit on the integration time-step. The explicit predictor uses a high-order monotone algorithm while the implicit corrector solves an elliptic equation for a pressure correction to equilibrate acoustic waves. In this paper, we develop and extend BIC for multidimensional viscous flows. We introduce a new filter to further stabilize the algorithm and clarify the solution procedure for the inclusion of the viscous fluxes. The new algorithm is examined in three test problems with successively increased difficulty. First, a two-dimensional lid-driven cavity flow is simulated to demonstrate the ability of BIC on solving steady-state swirling flows. Using time steps at least 100 times larger than the explicit limit, good agreements are obtained for solutions when compared with an incompressible calculation by a prior work. A two-dimensional (2D) doubly periodic shear layer flow is simulated to examine the algorithm on solving a transient flow with strong vorticity gradients. Finally, vortex breakdown in three-dimensional (3D) swirling flows are used to further test the stability and performance of the new BIC algorithm. Comparisons of explicit and implicit BIC calculations of both the 2D doubly periodic shear layer and 3D vortex breakdown are presented side by side. They demonstrate that the new BIC algorithm is able to predict accurate and robust solutions using time steps varying from near the explicit stability limit to tens and hundreds of times larger. Excellent agreement is also obtained when compared with results from other algorithms. We discuss our observations of these computations and features which were found to be critical for robustly simulating low-speed, highly dynamic flows.

Hybrid discretization of multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces

Multi-phase flow in porous media in the presence of viscous, gravitational, and capillary forces is described by advection diffusion equations with nonlinear parameters of relative permeability and capillary pressures. The conventional numerical method employs a fully implicit finite volume formulation. The phase-potential-based upwind direction is commonly used in computing the transport terms between two adjacent cells. The numerical method, however, often experiences non-convergence in a nonlinear iterative solution due to the discontinuity of transmissibilities, especially in transition between co-current and counter-current flows. Recently, Lee et al. (Adv. Wat. Res. 82, 27–38, 2015) proposed a hybrid upwinding method for the two-phase transport equation that comprises viscous and gravitational fluxes. The viscous part is a co-current flow with a one-point upwinding based on the total velocity and the buoyancy part is modeled by a counter-current flow with zero total velocity. The hybrid scheme yields C1-continuous discretization for the transport equation and improves numerical convergence in the Newton nonlinear solver. Lee and Efendiev (Adv. Wat. Res. 96, 209–224, 2016) extended the hybrid upwind method to three-phase flow in the presence of gravity. In this paper, we present the hybrid-upwind formula in a generalized form that describes two- and three-phase flows with viscous, gravity, and capillary forces. In the derivation of the hybrid scheme for capillarity, we note that there is a strong similarity in mathematical formulation between gravity and capillarity. We thus greatly utilize the previous derivation of the hybrid upwind scheme for gravitational force in deriving that for capillary force. Furthermore, we also discuss some mathematical issues related to heterogeneous capillary domains and propose a simple discretization model by adapting multi-valued capillary pressures at the end points of capillary pressure curves. We demonstrate this new model always admits a consistent solution that is within the discretization error. This new generalized hybrid scheme yields a discretization method that improves numerical stability in reservoir simulation.

Numerical simulation of flow past a bluff body of two different shapes using gradient smoothing method with unstructured grids

In this paper, flow across a bluff cylinder with circular and equilateral triangular sections shapes at low Reynolds number (10 ≤ Re ≤ 250) is numerically investigated using the gradient smoothed method (GSM). The method was originally developed based on unstructured grids, which could be generated easily for complicated domains. For solving the incompressible flow, artificial compressibility terms are introduced in Navier–Stokes equations. The spatial derivatives of convective and viscous fluxes are obtained using the gradient smoothing operation. And the time marching is implemented based on the dual time stepping technique. The situations of steady flows with Re = 10–40 and unsteady flows with Re = 50∼250 are simulated. Analysis of the drag coefficients, root mean square (rms) value of lift coefficients and Strouhal number of circular and triangular cylinders has been carried out. Compared with both experimental and numerical reference solutions in the literatures, the accuracy of GSM results has been demonstrated.

Development of a coupled matrix-free LU-SGS solver for turbulent compressible flows

The paper deals with the development of the lower-upper symmetric Gauss–Seidel (LU-SGS) matrix-free finite volume solver for the simulation of compressible flows within the framework of the OpenFOAM package. The solver evaluates the convective fluxes using approximate Riemann solvers with limited piece-wise linear reconstructions whereas the viscous fluxes are approximated using a central scheme. The time evolution is realised through the backward differentiation formula of first or second order. The system of non-linear equations is then solved with the help of the matrix-free LU-SGS method. The developed solver is used to solve several flow problems and compared to a pressure-based segregated solver. Our numerical experiments indicate that the LU-SGS solver is more efficient for flows with higher Mach numbers and provides better resolution of shock waves. Moreover the LU-SGS solver benefits from the low memory footprint and does not use any problem specific setup.

A reconstructed direct discontinuous Galerkin method for simulating the compressible laminar and turbulent flows on hybrid grids

A reconstructed direct discontinuous Galerkin method is developed and evaluated for simulating the compressible laminar and turbulent flows on hybrid grids. The spatial discretization is implemented through the combination of a highly efficient high-order hybrid reconstructed discontinuous Galerkin method together with a newly developed direct discontinuous Galerkin method for computing viscous fluxes. A variety of typical test cases are selected to evaluate the performance of the proposed hybrid reconstructed direct discontinuous Galerkin method for simulating both laminar flows and turbulent flows. Compared with the traditional discontinuous Galerkin method, the hybrid reconstructed direct discontinuous Galerkin method demonstrates its superior potential of being more accurate and more efficient under the same number of degrees of freedom, thus, shows its promise for further practical applications.