Viscous Terms Research Articles

Hybrid nanofluid flow past a stretching/shrinking sheet with thermal radiation and mass transpiration

In this article, an exact analytical solution of a hybrid nanofluid (HNF) flow over a stretching/shrinking sheet in the presence of mass transpiration (suction/injection) and radiation is investigated. The governing PDEs are first transformed into ODEs with the help of similarity variables. Further, an exact analytical solution of this problem, including a general formulation of the stretching/shrinking sheet problems, is established. HNFs have a significant feature in increasing the heat transfer process and promoting the production in industrial applications. The analysis demonstrated that the upper branch of the analytical solution procedure is stable, although unstable for the lower branch. Momentum and impact of thermal radiation and heat generation on the flow of an incompressible HNF over a stretching/shrinking sheet with mass transfer in fluid flow have been considered. Al2O3 and Cu nanoparticles are immersed in H2O to form an Al2O3 – Cu/H2O HNF. The thermal radiation is also considered in this analysis. Additionally, the rate of heat transfer of the HNF is lower than that of the base fluid (water). We also observe the interesting phenomena of exponentially and algebraically decaying solutions. After neglecting the viscous dissipation term, the heat equation is solved analytically and the effect of various parameters are discussed with the help of graphs. The present problem contains numerous applications in industrial processes namely paper production, metal spinning, extrusion of polymer etc.

Estimation of Turbulent Triplet Covariances for Bora Flows

Bora is a strong or severe, relatively cold, gusty wind that usually blows from the northastern quadrant at the east coast of the Adriatic Sea. In this study bora’s turbulence triplet covariances were analysed, for the first time, for bora flows. The measurements used were obtained from the measuring tower on Pometeno brdo (“Swept-Away Hill”), in the hinterland of the city of Split, Croatia. From April 2010 until June 2011 three components of wind speed and sonic temperature were measured. The measurements were performed on three heights, 10, 22 and 40 m above the ground with the sampling frequency of 5 Hz. During the observed period, total of 60 bora episodes were isolated. We analyse the terms in prognostic equations for turbulence variances. In that respect, the viscous dissipation term was calculated using two approaches: (i) inertial dissipation method (εIDM) and (ii) direct approach from the prognostic equations for variances of turbulence (εEQ). We determine that the direct approach can successfully reproduce the shape of the curve, but the values are for several orders of magnitudes smaller compared to the real data. Further, linear relationship between εIDM and εEQ was obtained. Using the results for εEQ, viscous dissipation rate in longitudinal, transversal and vertical direction was determined. It is shown that viscous dissipation has the greatest impact on bora’s longitudinal direction. The focus is on the turbulence transport term, i.e., the triplet covariance term. For the first time, it is found that turbulence transport is very significant for the intensity of near−surface bora flows. Furthermore, turbulence transport can be both positive and negative, yet intensive. It is mostly negative at the upper levels and positive at the lower levels. Therefore, turbulence transport, in most cases, takes away turbulence variance from the upper levels and brings it down to the lower ones. This is one of the main findings of this study; it adds to the understanding of peculiarities of bora wind, and perhaps some other severe winds.

Scaling of global properties of fluctuating streamwise velocities in pipe flow: Impact of the viscous term

We extend the procedure outlined in Basse [“Scaling of global properties of fluctuating and mean streamwise velocities in pipe flow: Characterization of a high Reynolds number transition region,” Phys. Fluids 33, 065127 (2021)] to study global, i.e., radially averaged, scaling of streamwise velocity fluctuations. A viscous term is added to the log-law scaling, which leads to the existence of a mathematical abstraction, which we call the “global peak.” The position and amplitude of this global peak are characterized and compared to the inner and outer peaks. A transition at a friction Reynolds number of order 10 000 is identified. Consequences for the global peak scaling, length scales, non-zero asymptotic viscosity, turbulent energy production/dissipation, and turbulence intensity scaling are appraised along with the impact of including an additional wake term.

Two-stage fourth-order gas kinetic solver-based compact subcell finite volume method for compressible flows on triangular meshes

To meet the demand for complex geometries and high resolutions of small-scale flow structures, a two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques for compressible flows on (unstructured) triangular meshes was developed to improve the compactness and efficiency. Compared to the fourth-order GKS-based traditional finite volume (FV) method, the proposed method realizes compactness effectively by subdividing each cell into a set of subcells or control volumes (CVs) and selecting only face-neighboring cells for high-order compact reconstruction. Because a set of CVs in a main cell share the same reconstruction, it is more efficient than traditional FV-GKS, where the solution polynomial on each CV needs to be separately reconstructed. Unlike in the single-stage third-order SCFV-GKS, both accuracy and efficiency are improved significantly by two-stage fourth-order temporal discretization, for which only a second-order gas distribution function is needed to simplify the construction of the flux function and reduce computational costs. For viscous flows, it is not necessary to compute the viscous term with GKS. Compared to the fourth-stage Runge–Kutta method, one half of the stage is saved for achieving fourth-order time accuracy, which also helps to improve the efficiency. Therefore, a new high-order method with compactness, efficiency, and robustness is proposed by combining the SCFV method with the two-stage gas-kinetic flux. Several benchmark cases were tested to demonstrate the performance of the method in compressible flow simulations.

IMEX‐DG Schemes for Advection‐Diffusion Problems using Term‐based or Domain‐based IMEX Splitting

AbstractIMEX partitioning may be based on a decomposition of the computational domain into two subgrids by implicitly discretizing the unknowns on the smallest cells while explicit time stepping is applied to moderately sized elements. Alternatively, advection‐diffusion IMEX splitting applies implicit time discretization only to the viscous terms while inviscid terms are discretized explicitly. Analytical investigations have shown that a careful choice of both the specific IMEX scheme and the particular DG approach yield an additional stability property of such schemes resulting in time step restrictions independent of grid refinement. In this contribution, we will compare the domain‐based IMEX partitioning to advection‐diffusion splitting with respect to time step restrictions and efficiency.

Theoretical and experimental insight into the homogeneous expansion of wet particles in a fluidized bed

The presence of liquid in a fluidized bed induces a homogeneous bed expansion at incipient fluidization, which improves the fluidization quality by enhancing the gas-holding capacity of emulsion phase. This paper made an insight into the expansion of wet bed through theoretical and experimental approaches. The bed expansion ratio increases with particle diameter and liquid surface tension, and hardly changes with liquid viscosity due to the absence of viscous term in liquid bridge force. The expansion ends up as the tensile strength of wet bed equals to the shear stress from gas drag force. The superficial gas velocity at this point is the complete fluidization velocity, Ucf. Theoretical calculation shows that Ucf sharply increases then gradually saturates with increasing liquid amount. The tensile strength of wet bed is most sensitive to the superficial gas velocity, followed by the particle diameter. The effect of liquid amount is the smallest. From an optimization perspective, increasing particle diameter and liquid surface tension can enhance the gas-holding capacity of a wet bed during fluidization.

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.

Heterogeneous CPU+GPU parallelization for high-accuracy scale-resolving simulations of compressible turbulent flows on hybrid supercomputers

A heterogeneous parallel algorithm for simulation of compressible turbulent flows and its portable software implementation are presented. The underlying numerical method is based on a family of higher accuracy edge-based reconstruction schemes on unstructured mixed-element meshes. The proposed parallel solution can engage a large number of computing devices of most of the existing computing architectures used in modern supercomputers, including manycore CPUs and GPUs. It is capable of co-execution on both CPUs and accelerators simultaneously. The multilevel parallel algorithm combines: MPI for distributing workload among hybrid cluster nodes and between devices inside nodes; OpenMP for manycore CPUs and other supporting devices, such as Intel Xeon Phi; OpenCL for massively-parallel accelerators, such as GPUs of various vendors, including NVIDIA, AMD, Intel. The main focus is on the adaptation of the numerical method and its computational algorithm to the stream processing parallel paradigm. The very limited device memory inherent in GPU computing is also taken into account. A detailed description of the parallel algorithm is presented, as well as the techniques used for its efficient parallel implementation. Special attention is paid to implicit time integration with its linear solver and calculation of convective fluxes and viscous terms. The use of mixed floating-point precision and overlapping communications and computations is also discussed. Parallel performance is demonstrated in practical applications on different kinds of supercomputers using up to 10 thousand cores and multiple GPUs of comparable overall performance.

Numerical study of a point absorber wave energy converter with different power take-off systems

Wave energy resources are huge and widely distributed all over the world. Wave energy converters (WECs) are employed to harness this energy. In this paper, a heaving point absorber WEC with three kinds of Power Take-Off (PTO) systems is investigated. A fully nonlinear numerical model of the WEC has been developed, taking into consideration viscosity. A Computational Fluid Dynamics (CFD) analysis is performed in order to determine the quadratic viscous term coefficient. A physical model test has also been carried out and used to validate the numerical model. The response and energy capture characteristics of the WEC with various PTO systems are analyzed in both regular and irregular wave. The differences among the WECs with different PTOs are assessed. Finally, the annual wave power assessment and suggestions for optimizing the WEC in the real sea state are given.

Simulation of Counter-Current Spontaneous Imbibition Based on Momentum Equations with Viscous Coupling, Brinkman Terms and Compressible Fluids

A numerical model is investigated representing counter-current spontaneous imbibition of water to displace oil or gas from a core plug. The model is based on mass and momentum conservation equations in the framework of the theory of mixtures. We extend a previous imbibition model that included fluid–rock friction and fluid–fluid drag interaction (viscous coupling) by including fluid compressibility and Brinkman viscous terms. Gas compressibility accelerated recovery due to gas expansion from high initial non-wetting pressure to ambient pressure at typical lab conditions. Gas compressibility gave a recovery profile with two characteristic linear sections against square root of time which could match tight rock literature experiments. Brinkman terms decelerated recovery and delayed onset of imbibition. Experiments where this was prominent were successfully matched. Both compressibility and Brinkman terms caused recovery deviation from classical linearity with the square root of time. Scaling yielded dimensionless numbers when Brinkman term effects were significant.Article HighlightsSpontaneous imbibition with viscous coupling, compressibility and Brinkman terms.Viscous coupling reduces spontaneous imbibition rate by fluid–fluid friction.Brinkman terms delay early recovery and explain seen delayed onset of imbibition.Gas compressibility accelerates recovery and can be significant at lab conditions.Gas compressibility gives recovery with two root of time lines as seen for shale.

A compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations

Following the development of a third-order compact gas-kinetic scheme (GKS) for the Euler and Navier-Stokes equations (Ji et al. (2020) [40]), in this paper a higher-order compact GKS up to sixth order of accuracy will be constructed for the shock and acoustic wave computation on unstructured mesh. The compactness is defined by the physical domain of dependence for an unstructured triangular cell, which may involve the closest neighbors of neighboring cells. The compactness and high-order accuracy of the scheme are coming from the consistency between the high-order initial reconstruction and the high-order gas evolution model under GKS framework. The high-order evolution solution at a cell interface provides not only a time-accurate flux function, but also the time-evolving flow variables. Therefore, the cell-averaged flow variables and their gradients can be explicitly updated at the next time level from the moments of the same time-dependent gas distribution function. Based on the cell averaged flow variables and their cell-averaged derivatives, the nonlinear high-order compact reconstruction can be obtained for these variables in the evaluation of local equilibrium and non-equilibrium states. The current nonlinear reconstruction is a combination of WENO and ENO methodology, which is specifically suitable for compact GKS on unstructured mesh with a high-order (≥4) accuracy. The initial piecewise discontinuous reconstruction is used for the determination of non-equilibrium state and an evolved smooth reconstruction for the equilibrium state. The evolution model in GKS is based on a relaxation process from non-equilibrium to equilibrium state. The time-accurate gas distribution function provides the Navier-Stokes (NS) flux function directly without separating the inviscid and viscous terms, which simplifies the numerical discretization of the NS equations directly on unstructured mesh. Based on the time-accurate flux solver, the two-stage fourth-order time discretization can be applied to get a fourth-order time-accurate solution with only two stages, which reduces two reconstructions in comparison with the same order Runge-Kutta time stepping method. The current high-order GKS can uniformly capture acoustic and shock waves without identifying troubled cells and implementing additional limiting procedure. In addition, the fourth- up to sixth-order compact GKS can use almost the same time step as a second-order shock capturing scheme in most flow problems. The fourth-order GKS on unstructured mesh will be used in the computations from low speed incompressible viscous flow to the high Mach number shock interaction. The accuracy, efficiency, and robustness of the scheme have been validated. The main conclusion of the paper is that beyond the first-order Riemann solver, the use of high-order gas evolution model seems have advantages in the development of high-order schemes.

New governing equations for fluid dynamics

The difference in the governing equation between inviscid and viscous flows is the introduction of viscous terms. Traditional Navier–Stokes (NS) equations define stress based on Stokes’s assumptions. In NS equations, stress is supposedly proportional to strain, and both strain and stress tensors are symmetric. There are several questions with NS equations, which include the following: 1. Both symmetric shear terms and stretching terms in strain and stress are coordinate-dependent and thus not Galilean invariant. 2. The physical meaning of both diagonal and off-diagonal elements is not clear, which is coordinate-dependent. 3. It is hard to measure strain and stress quantitatively, and viscosity is really measured by vorticity, not by symmetric strain. 4. There is no vorticity tensor in NS equations, which plays an important role in fluid flow, especially for turbulent flow. The newly proposed governing equations for fluid dynamics use the vorticity tensor only, which is anti-symmetric. The advantages include the following: 1. Both shear and stress are anti-symmetric, which are Galilean invariants and independent of coordinate rotation. 2. The physical meaning of off-diagonal elements is clear, which is anti-symmetric shear stress. 3. Viscosity coefficients are obtained by experiments, which use vorticity. 4. The vorticity term can be further decomposed into rigid rotation and anti-symmetric shear, which are important to turbulence research. 5. The computation cost for the viscous term is reduced to half as the diagonal terms are all zero and six elements are reduced to three. Several computational examples are tested, which clearly demonstrate both NS and new governing equations have exactly the same results. As shown below, the new governing equation is identical to NS equations in mathematics, but the new one has lower cost and the several advantages mentioned above, including the possibility to study turbulent flow better. It is recommended to use the new governing equation instead of NS equations. The unique definition and operation of vectors and tensors by matrix and matrix operation are also discussed in this paper.

Numerical simulation of non-Newtonian Carreau fluid in a lid-driven cavity

In this work, the 2D lid-driven cavity flow of non-Newtonian Carreau fluids has been studied by finite difference method on a staggered grid. A finite-difference algorithm on staggered grid based on projection method is adopted to solve the lid-driven cavity flow, which includes a second-order central difference scheme for the non-Newtonian viscous stress term. This study has been conducted for the certain pertinent parameters of Reynolds number (Re=100-1000), power-law index (n=0.6-1.4). The results show that as the Reynolds number increases, the influence of the power-law index on the flow increases. As the power-law index decreases, the flow field becomes more complicated.

Stochastic Forcing in Hydrodynamic Models with Non-local Interactions

The hydrodynamical model of the collective behavior of animals consists of the Euler equation with additional non-local forcing terms representing the repulsive and attractive forces among individuals. This paper deals with the system endowed with an additional white-noise forcing and an artificial viscous term. We provide a proof of the existence of a dissipative martingale solution—a cornerstone for a subsequent analysis of the system with stochastic forcing.

Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

In a Hilbert space mathcal{H}, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A= nabla f +B , where ∇f is the gradient of a continuously differentiable convex function f and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as tto +infty of the generated trajectories towards the zeros of nabla f +B. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

High-Resolution Viscous Terms Discretization and ILW Solid Wall Boundary Treatment for the Navier–Stokes Equations

High-Resolution Viscous Terms Discretization and ILW Solid Wall Boundary Treatment for the Navier–Stokes Equations

Application of projection methods to simulating mass transport in reverse osmosis systems

Reverse Osmosis has important applications to seawater desalination and advanced water treatment. Its efficiency depends, however, on unsteady fluid flow and solute transport that are challenging to simulate. The challenges arise due to interactions between solute boundary layers and unsteady vortical flow structures generated by complicated geometries. These flow structures also interact with semi-permeable membranes through which the permeate flow depends on the local pressure. We show that this additional pressure coupling causes the temporal accuracy of traditional projection methods to drop to first-order. We track the source of this accuracy drop to the treatment of viscous terms in the derivation of the Poisson equation used to update the velocity and pressure fields. This allows us to propose a modified projection method that recovers second-order temporal accuracy. Finally, we show that the modified projection method can be coupled to convective outlet conditions and immersed boundary conditions to simulate reverse osmosis in steady and unsteady flow regimes.

Influence of Structural Stiffness and Loss Factor on Railroad Vehicle Comfort

This paper presents a new formulation for analyzing a beam on elastic supports traveling on irregular profiles. The model is a first approximation of a passenger railway vehicle car body. The main difference with previous works is the use of a complex modulus to represent structural damping rather than relying on equivalent viscous terms. The formulation groups rigid body modes with flexible modes and proposes a matrix form that is easy to interpret and solve in the frequency domain. Comfort indexes are readily obtained from weighted response spectral densities. The model is used to assess the influence of structural damping and stiffness on comfort. It will be shown that the evolution of comfort with stiffness is non-monotonic and, therefore, comfort does not always improve as stiffness increases.

A pencil distributed direct numerical simulation solver with versatile treatments for viscous term

We present an efficient second- or fourth-order finite difference direct numerical simulation (DNS) solver using pencil-like domain decomposition parallel strategy, with the ability to handle different boundary conditions. The viscous term is treated implicitly, partial implicitly, or explicitly. The runtimes for different viscous treatments and different boundary conditions are evaluated quantitatively, which can help us to get a fast computational speed for specific flow cases. FFT-based method is used for solving Pressure Poisson equation, and alternating direction implicit approach is adopted for Helmholtz equations during the implicit viscous treatment. The numerical results achieve good match with those obtained from spectral methods. In addition, the resulting solver exhibits quite good parallel efficiency, and demonstrates good computational speedup and versatility than other pencil-like DNS solvers reported in the literature. The source code is freely available at https://github.com/GongZheng-Justin/Channel3d.

Nonlinear Longitudinal Vibrations in Accreting One-Dimensional Nanostructures

Abstract The investigation of an accreting one-dimensional chain of atoms was carried out in this paper. A governing partial differential equation that describes the accreting nanocontinuum containing an attachment of infinite atoms was derived and analyzed. We transformed the space variable u(t,r)→v(τ,x) (for a governing partial differential equation formulated in a previous publication) and introduced a function of linear growth. The boundary conditions were also transformed in terms of the new variables. We assumed that the left end of the accreting nanostructure was free u(t,r=0)=0 while the right end was fixed ∂u∂r|r=l[1+εf(t)]=0. The method of lines was employed to obtain numerical approximate solutions for the nonlinear partial differential equations because their exact solutions proved difficult to obtain. The approximate solutions were expressed using graphical plots in Mathematica®. The viscous damping term, δ, was introduced into the nanocontinuum and analysis was carried out by considering the damped continuum as an accreting nanostructure. Numerical simulations for both cases illustrated linear growth, which was observed via numerical simulation plots.