Viscous Terms Research Articles

Stability of Implicit Difference Schemes for a Linearized Hyperbolic Quasi-Gasdynamic System of Equations

We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized on a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study the implicit three-level weighted and two-level vector difference schemes. The important domination property of the operator of viscous terms (with no allowance for the relaxation parameter) over the operator of convective terms is derived. We apply this property to prove by the energy method that, regardless of the Mach number, our implicit schemes on a nonuniform rectangular mesh (without any conditions on the mesh steps) are stable with respect to the initial data and the right-hand side uniformly in time and the relaxation parameter.

Effects of the viscous dissipation on the Darcy-Brinkman flow: Rigorous derivation of the higher-order asymptotic model

Abstract We consider the non-isothermal fluid flow through a thin domain occupied by a saturated porous medium. The flow is governed by the Darcy-Brinkman system and the heat equation including the viscous dissipation term proposed by Al-Hadhrami et al. (2003). Using asymptotic analysis with respect to domain’s thickness, a second–order asymptotic approximation of the problem is built. The formally derived model is rigorously justified via error estimate.

Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy

We discuss how to combine exponential time differencing technique with multi-step method to develop higher order in time linear numerical scheme that are energy stable for certain gradient flows with the aid of a generalized viscous damping term. As an example, a stabilized third order in time accurate linear exponential time differencing (ETD) scheme for the epitaxial thin film growth model without slope selection is proposed and analyzed. An artificial stabilizing term $$A\tau ^3\frac{\partial \Delta ^3 u}{\partial t}$$ is added to ensure energy stability, with ETD-based multi-step approximations and Fourier pseudo-spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long-time energy stability and an $$\ell ^{\infty }(0,T; \ell ^2)$$ error analysis are provided, based on the energy method. In addition, a few numerical experiments are presented to demonstrate the energy decay and convergence rate.

Parameter Identification of Fiber Orientation Models Based on Direct Fiber Simulation with Smoothed Particle Hydrodynamics

The behavior of fiber suspensions during flow is of fundamental importance to the process simulation of discontinuous fiber reinforced plastics. However, the direct simulation of flexible fibers and fluid poses a challenging two-way coupled fluid-structure interaction problem. Smoothed Particle Hydrodynamics (SPH) offers a natural way to treat such interactions. Hence, this work utilizes SPH and a bead chain model to compute a shear flow of fiber suspensions. The introduction of a novel viscous surface traction term is key to achieve full agreement with Jeffery’s equation. Careful modelling of contact interactions between fibers is introduced to model suspensions in the non-dilute regime. Finally, parameters of the Reduced-Strain Closure (RSC) orientation model are identified using ensemble averages of multiple SPH simulations implemented in PySPH and show good agreement with literature data.

An implicit, sharp numerical treatment of viscous terms at arbitrarily shaped liquid-gas interfaces in evaporative flows

We present a novel methodology for accurate treatment of viscous terms in evaporation problems. The proposed scheme is an extension of the sharp viscous treatment of Kang et al. (2000) [7] to 3D phase change problems. To ensure accuracy and grid converging solutions, a new implicit approach to computing viscous fluxes across the phase interfaces is proposed, which was previously unavailable in fixed grid numerical schemes. Analytical relations were derived for the jump in velocity gradients across the 2D and 3D phase interfaces, an important constituent of the proposed scheme. The relations show a non-vanishing jump in the tangential gradients across the phase interface that are associated with evaporative flux and interfacial curvature. The proposed methodology demonstrated first order accuracy in canonical test cases. It is general and applicable to arbitrarily oriented interfaces, and can be readily implemented in existing evaporation flow solvers.

A reconstructed discontinuous Galerkin method for incompressible flows on arbitrary grids

The discontinuous Galerkin (DG) methods have attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. However, the DG methods have their own weakness due to the high computational costs and storage requirements. In this work, we develop a high-order hybrid reconstructed DG (rDG) method for solving the INS equations on arbitrary grids. To be specific, the inviscid term of the INS equations is discretized by applying the third-order hybrid rDG(P1P2) method with a simplified artificial compressibility flux, while the viscous term of the INS equations is discretized by using the simple direct DG (DDG) method. A number of incompressible flow problems, in both steady and unsteady forms, for a variety flow conditions are computed to numerically assess the performance of the hybrid rDG(P1P2) method, which confirm its ability to achieve the optimal third order of accuracy at a significantly reduced computational costs. Furthermore, a detailed comparison of a variety of different reconstructed strategies is performed and presented. Numerical results demonstrate that the hybrid rDG(P1P2) method outperforms the rDG(P1P2) method based on either the original least-squares reconstruction or the Green-Gauss reconstruction for solving the INS equations.

Numerical investigation of porous stack for a solar-powered thermoacoustic refrigerator

This article proposes a numerical analysis for performance improvement of the stack, which represents a crucial element on the solar-powered thermoacoustic refrigerator. The stack is considered as a saturated parallelepiped homogeneous porous media. Numerical simulation of the flow and the heat transfer in the thermoacoustic refrigerator is carried out. The physical flow is governed by the modified Darcy–Brinkmann–Forchheimer model. The governing equations are solved numerically using the lattice Boltzmann method. Furthermore, the local thermal equilibrium assumption is applied to examine heat transfer. Particular attention is paid a new form of the lattice Boltzmann equation system describing the flow and the heat transfer in porous media and fluid regions. The effects of several parameters characterizing the thermal behaviour in the porous medium are studied. The parametric results lead to the optimization of the porous media form. The presence of the viscous dissipation term in the heat transfer formulation within the thermoacoustic system is particularly highlighted, due to its significant effects introduced by the Eckert number.

An analysis of a linearly extrapolated BDF2 subgrid artificial viscosity method for incompressible flows

This report extends the mathematical support of a subgrid artificial viscosity (SAV) method to simulate the incompressible Navier-Stokes equations to better performing a linearly extrapolated BDF2 (BDF2LE) time discretization. The method considers the viscous term as a combination of the vorticity and the grad-div stabilization term. SAV method introduces global stabilization by adding a term, then anti-diffuses through the extra mixed variables. We present a detailed analysis of conservation laws, including both energy and helicity balance of the method. We also show that the approximate solutions of the method are unconditionally stable and optimally convergent. Several numerical tests are presented for validating the support of the derived theoretical results.

Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow

This article is concerned with the nanofluid flow in a rotating frame under the simultaneous effects of thermal slip and convective boundary conditions. Arrhenius activation energy is another important aspect of the present study. Flow phenomena solely rely on the Darcy–Forchheimer-type porous medium in three-dimensional space to tackle the symmetric behavior of viscous terms. The stretching sheet is assumed to drive the fluid. Buongiorno’s model is adopted to see the features of Brownian diffusion and thermophoresis on the basis of symmetry fundamentals. Governing equations are modeled and transformed into ordinary differential equations by suitable transformations. Solutions are obtained through the numerical RK45-scheme, reporting the important findings graphically. The outputs indicate that larger values of stretching reduce the fluid velocity. Both the axial and transverse velocity fields undergo much decline due to strong retardation produced by the Forchheimer number. The thermal radiation parameter greatly raises the thermal state of the field. The temperature field rises for a stronger reaction within the fluid flow, however reducing for an intensive quantity of activation energy. A declination in the concentration profile is noticed for stronger thermophoresis. The Forchheimer number and porosity factors result in the enhancement of the skin friction, while both slip parameters result in a decline of skin friction. The thermal slip factor results in decreasing both the heat and mass flux rates. The study is important in various industrial applications of nanofluids including the electro-chemical industry, the polymer industry, geophysical setups, geothermal setups, catalytic reactors, and many others.

Solar activity phase dependence of the magnetospheric processes and relativistic electron flux at geostationary orbit

This study aims to investigate the effects of the solar activity phases on the magnetospheric processes in relation to daily relativistic electron ($E>2$ MeV) dynamics at geosynchronous orbit (GEO) during Solar Cycles 23–24. GOES observations indicate that in the descending phase the electron fluxes are seasonally dependent with largest flux in equinoctial periods, relatively high, and 27-day recurrent, while in the maximum phases they are relatively low and nonrecurrent. The electron fluxes are relatively low and partially recurrent in the ascending phases. The cross correlation coefficients (c.c.s) of daily $K_{\textrm{p}}$–$V_{\textrm{sw}}$, $K_{\textrm{p}}$–$AE$, $K_{\textrm{p}}$–$AL$ evolve in the similar trend for both solar cycles: highest during the descending phases, lower in the ascending phases, and lowest in the maximum and minimum phases. The correlation of $K_{\textrm{p}}$–viscous term is strongest during descending phases and weakest around the maximum phases. The correlation of $K_{\textrm{p}}$–merging term slightly varies in a chaotic way from one solar phase to another, but drops to its lowest value in the solar minimum. The correlation analysis results signify the role of the solar activity in controlling the solar wind–magnetosphere couplings. Two case studies during maximum (2000) and descending (2017) phases indicate that the solar activity dependence of the magnetospheric processes and relativistic electrons is characterized by substorm activity and solar wind drivers. Most of the substorms induced by coronal mass ejections in the maximum phase exhibit strong but short and non-repetitive features that associate with low electron fluxes at GEO. In contrast, high intensity, prolonged, and repetitive substorm activities induced by high-speed solar winds are the main cause of the relativistic electron enhancements during the descending phases. The enhancements strongly depend on $V_{\textrm{sw}}$ and $\Sigma K_{\textrm{p}}$ in which only appropriate $V_{\textrm{sw}}$ and $\Sigma K_{\textrm{p}}$ are required. In the descending phase, the requirements for the flux enhancements to $\geq 10^{3}$ cm−2 sr−1 s−1 are the simultaneity of $V_{\textrm{sw}}>500$ km/s and the prolongation of $\Sigma K_{\textrm{p}}>220$ and to $\geq 10^{4}$ cm−2 sr−1 s−1 when $V_{\textrm{sw}}\geq 630$ km/s and $\Sigma K_{\textrm{p}}>227$. Furthermore, the correlations of $\Sigma K_{\textrm{p}}$ and log-electron fluxes are stronger in the descending/ascending phases than in the maximum phase, while the time lag between them is longer in the maximum phase. The remarkable correlations of $V_{\textrm{sw}}$–$K_{\textrm{p}}$ and in the descending phase indicate appropriate magnetospheric convection in association with the repetitive substorms that can effectively trigger the stochastic mechanisms of electron acceleration. The results are extensively discussed in the light of observations and current theories of radiation belt dynamics.

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, different from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier–Stokes–Fourier equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gauge-invariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

Meso-scale drag model designed for coarse-grid Eulerian-Lagrangian simulation of gas-solid flows

Coarse-grid Eulerian-Lagrangian approaches are promising for simulating fluidized gas-solid flows due to their ability to track the trajectories and collisions of individual particles. Like coarse-grid Eulerian-Eulerian methods, these approaches also require drag corrections to account for subgrid inhomogeneities. This work develops novel meso-scale models that make use of the detailed information of the resolved positions and velocities of the solid phase. Each coarse grid is divided into three-dimensional subgrids. The gas velocities at subgrids are computed using simplified mass and momentum conservation equations for the fluid phase. The viscous term representing the shear force between different subgrids are included in the momentum equation. Priori analysis shows that the proposed meso-scale models give significantly better predictions than the best available model designed for coarse-grid Eulerian-Eulerian methods. The effectiveness of the proposed models is also demonstrated by posteriori tests.

A compressible viscous three-phase model for porous media flow based on the theory of mixtures

In this paper we focus on a general model to describe compressible and immiscible three-phase flow in porous media. The underlying idea is to replace Darcy’s law by more general momentum balance equations. In particular, we want to account for viscous coupling effects by introducing fluid-fluid interaction terms. In [Qiao, et al. (2018) Adv Water Resour 112: 170–188] a water-oil model based on the theory of mixtures was explored. It was demonstrated how the inclusion of viscous coupling effects could allow the model to better capture flow regimes which involve a combination of co-current and counter-current flow. In this work we extend the model in different aspects: (i) account for three phases (water,oil,gas) instead of two; (ii) deal with both the compressible and incompressible case; (iii) include viscous terms that represent frictional forces within the fluid (Brinkman type). A main objective of this work is to explore this three-phase model, which appears to be more realistic than standard formulation, in the context of petroleum related applications. We first provide development of stable numerical schemes in a one-dimensional setting which can be used to explore the generalized water-oil-gas model, both for the compressible and incompressible case. Then, several numerical examples with waterflooding in a gas reservoir and water alternating gas (WAG) experiments in an oil reservoir are investigated. Differences and similarities between the compressible and incompressible model are highlighted, and the fluid-fluid interaction effect is illustrated by comparison of results from the generalized model and a conventional model formulation.

Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium

A linear stability analysis is performed on a tilted parallel wake in a strongly stratified fluid at low Reynolds numbers. A particular emphasis of the present study is given to the understanding of the low-Froude-number mode observed by the recent experiment (Meunier in J Fluid Mech 699:174–197, 2012). In the limit of low Froude number, the linearised equations of motion can be reduced to the Orr–Sommerfeld equation on the horizontal plane, except the viscous term that contains vertical dissipation. Based on this equation, it is proposed that the low-Froude-number mode would be a horizontal inflectional instability and should remain two-dimensional at small tilting angles as long as the Reynolds number is sufficiently low. To support this claim, the asymptotic regime where this analysis is strictly valid is subsequently discussed in relation to previous work on the proper vertical length scale. The absolute and convective instability analysis of parallel wake is further performed, showing qualitatively good agreement with the experimental result. The low-Froude-number mode is found to be stabilised on increasing Froude number, as in the experiment. It is shown that the emergence of small vertical velocity at finite Froude number, the size of which is proportional to the square of Froude number, plays the key role in the stabilisation by modifying the inflectional instability and paradoxically creating stabilising buoyancy effect with the increase of Froude number.

An immersed boundary method for fluid–structure–acoustics interactions involving large deformations and complex geometries

This paper presents an immersed boundary (IB) method for fluid–structure–acoustics interactions involving large deformations and complex geometries. In this method, the fluid dynamics is solved by a finite difference method where the temporal, viscous and convective terms are respectively discretized by the third-order Runge–Kutta scheme, the fourth-order central difference scheme and a fifth-order W/TENO (Weighted/Targeted Essentially Non-oscillation) scheme. Without loss of generality, a nonlinear flexible plate is considered here, and is solved by a finite element method based on the absolute nodal coordinate formulation. The no-slip boundary condition at the fluid–structure interface is achieved by using a diffusion-interface penalty IB method. With the above proposed method, the aeroacoustics field generated by the moving boundaries and the associated flows are inherently solved. In order to validate and verify the current method, several benchmark cases are conducted: acoustic waves scattered from a stationary cylinder in an inviscid quiescent flow, sound generation by a stationary and a rotating cylinder in a uniform flow, sound generation by an insect in hovering flight, deformation of a red blood cell induced by acoustic waves and acoustic waves scattered by a stationary sphere. The comparison of the sound scattered by a cylinder shows that the present IB–WENO scheme, a simple approach, has an excellent performance which is even better than the implicit IB–lattice Boltzmann method. For the sound scattered by a sphere, the IB–TENO scheme has a lower dissipation compared with the IB–WENO scheme. Applications of this technique to model fluid–structure-acoustics interactions of flapping foils mimicking an insect wing section during forward flight and flapping foil energy harvester are also conducted, considering the effects of foil shape and flexibility. The difference of the force and sound generations of the foils are compared. For wing during forward flight, the results show that flexible wing generates larger thrust with higher acoustic pressure. In terms of the energy harvester, the current results show that the geometrical shape has no significant effects on the force and sound generation, and the flexibility of the plate tends to deteriorate the power extraction efficiency. The flexible plate also induces larger fluctuating pressure at the frequency of 2f (f is the flapping frequency) and weaker sound at the frequencies of f and 3f. The successful validations and applications show that the IB method handled by delta function, whose accuracy is generally lower than that of the internal flow solver, is accurate for predicting the dilatation and acoustics, and thus is an attractive alternative for modeling fluid–structure–acoustics interactions involving large deformations and complex geometries.

Investigation of flow past a cylinder embedded on curved and flat surfaces

A discrete exterior calculus investigation of flow past a stationary cylinder of circular, triangular, or square cross section embedded on a spherical or cylindrical surface shows insignificant effect of the embedding surface curvature. The contributing parameters, such as the magnitude of the Gaussian curvature term relative to the magnitude of the viscous term around the cylinder and the effect of the geometrical constraint, are insignificant. Is the dynamics of the flow past bluff bodies universal?

An approximate similarity solution for spatial fractional boundary-layer flow over an infinite vertical plate

In this paper, an approximate similarity solution of a fractional laminar boundary layer of viscous fluid flow over an infinite vertical plate has been presented. We introduce new fractional similarity transformations to convert the partial differential equation into a similarity ordinary differential equation with a fractional context. The viscous term of the flow momentum equation is written based on the Caputo fractional derivative. The proposed analytical solution has been developed using fractional power-series technique, and the convergence of the solution has been examined. The velocity distributions with the similarity variable are plotted against different values of the fractional order to illustrate the effects of fractional argument on the velocity behavior.

Effects of Stefan Blowing and Slip Conditions on Unsteady MHD Casson Nanofluid Flow Over an Unsteady Shrinking Sheet: Dual Solutions

In this article, the magnetohydrodynamic (MHD) flow of Casson nanofluid with thermal radiation over an unsteady shrinking surface is investigated. The equation of momentum is derived from the Navier–Stokes model for non-Newtonian fluid where components of the viscous terms are symmetric. The effect of Stefan blowing with partial slip conditions of velocity, concentration, and temperature on the velocity, concentration, and temperature distributions is also taken into account. The modeled equations of partial differential equations (PDEs) are transformed into the equivalent boundary value problems (BVPs) of ordinary differential equations (ODEs) by employing similarity transformations. These similarity transformations can be obtained by using symmetry analysis. The resultant BVPs are reduced into initial value problems (IVPs) by using the shooting method and then solved by using the fourth-order Runge–Kutta (RK) technique. The numerical results reveal that dual solutions exist in some ranges of different physical parameters such as unsteadiness and suction/injection parameters. The thickness of the velocity boundary layer is enhanced in the second solution by increasing the magnetic and velocity slip factor effect in the boundary layer. Increment in the Prandtl number and Brownian motion parameter is caused by a reduction of the thickness of the thermal boundary layer and temperature. Moreover, stability analysis performed by employing the three-stage Lobatto IIIA formula in the BVP4C solver with the help of MATLAB software reveals that only the first solution is stable and physically realizable.

Wave damping of a sloshing wave by an interacting turbulent vortex flow.

We report on the enhancement of the hydrodynamic damping of gravity waves at the surface of a fluid layer as they interact with a turbulent vortex flow in a sloshing experiment. Gravity surface waves are excited by oscillating horizontally a square container holding our working fluid (water). At the bottom of the container, four impellers in a quadrupole configuration generate a vortex array at moderate to high Reynolds number, which interact with the wave. We measure the surface fluctuations using different optical nonintrusive methods and the local velocity of the flow. In our experimental range, we show that as we increase the angular velocity of the impellers, the gravity wave amplitude decreases without changing the oscillation frequency or generating transverse modes. This wave dissipation enhancement is contrasted with the increase of the turbulent velocity fluctuations from particle image velocimetry measurements via a turbulent viscosity. To rationalize the damping enhancement a periodically forced shallow water model including viscous terms is presented, which is used to calculate the sloshing wave resonance curve. The enhanced viscous dissipation coefficient is found to scale linearly with the measured turbulent viscosity. Hence, the proposed scheme is a good candidate as an active surface gravity wave dampener via vortex flow reconfiguration.

Viscous heating effects on heat transfer characteristics of laminar compressible channel flow

The present paper investigates viscous heating effects on heat transfer characteristics of laminar Newtonian compressible channel flow. At first, incompressible Poiseuille flow case is addressed analytically by including viscous dissipation term in energy conservation equation. Temperature distributions in the channel cross-section and local Nusselt numbers are derived as functions of the Brinkman number, highlighting the role of the viscous term. The analysis is numerically supported by a set of CFD simulations. Additional CFD results are employed to extend the analysis to compressible flow case resulting in dedicated local Nusselt number correlations, function of Brinkman and Mach numbers, and expansion-altered temperature profiles. The study assumes no-slip flow and no temperature-jump boundary conditions at the channel wall (Knudsen number Kn<10−3), and addresses both fluid heating and cooling, both uniform heat flux (UHF) and uniform wall temperature (UWT) boundary condition (BC), and both circular (CR) and parallel-plate (PP) channel cross-section.