Simulation Of Viscous Flow Research Articles

Numerical investigation of shear-flow free-surface turbulence and air entrainment at large Froude and Weber numbers

We investigate two-phase free-surface turbulence (FST) associated with an underlying shear flow under the condition of strong turbulence (SFST) characterized by large Froude ($Fr$) and Weber ($We$) numbers. We perform direct numerical simulations of three-dimensional viscous flows with air and water phases. In contrast to weak FST (WFST) with small free-surface distortions and anisotropic underlying turbulence with distinct inner/outer surface layers, we find SFST to be characterized by large surface deformation and breaking accompanied by substantial air entrainment. The interface inner/outer surface layers disappear under SFST, resulting in nearly isotropic turbulence with ${\sim}k^{-5/3}$ scaling of turbulence kinetic energy near the interface (where $k$ is wavenumber). The SFST air entrainment is observed to occur over a range of scales following a power law of slope $-10/3$. We derive this using a simple energy argument. The bubble size spectrum in the volume follows this power law (and slope) initially, but deviates from this in time due to a combination of ongoing broad-scale entrainment and bubble fragmentation by turbulence. For varying $Fr$ and $We$, we find that air entrainment is suppressed below critical values $Fr_{cr}$ and $We_{cr}$. When $Fr^{2}>Fr_{cr}^{2}$ and $We>We_{cr}$, the entrainment rate scales as $Fr^{2}$ when gravity dominates surface tension in the bubble formation process, while the entrainment rate scales linearly with $We$ when surface tension dominates.

An implicit dual-time stepping spectral difference lattice Boltzmann method for simulation of viscous compressible flows on structured meshes

In this work, the spectral difference lattice Boltzmann method (SDLBM) is extended and applied for accurately computing two-dimensional viscous compressible flows on structured meshes. Here, the compressible form of the discrete Boltzmann-BGK equation with the Watari model is considered and the numerical solution of the resulting LB equation is obtained by using the spectral difference method. The main benefit of the use of the LB method in simulating compressible flows is that a same formulation can be applied to compute the inviscid and viscous portions of the flowfield. Note that the LB formulation for simulating the viscous flows is the same as that used for the inviscid ones, however, the wall boundary conditions are changed in a manner to consider the no-slip conditions. Here, the SDLBM is also extended to use curved-edge cells for properly representing curved wall boundaries. In addition, to enhance the solution of the SDLBM the time integration is efficiently performed by implementing an implicit dual-time stepping method which does not require a matrix inversion. Both steady and unsteady flows are simulated. Different test cases including the Couette flow, the viscous shock–vortex interaction, the flow over a circular cylinder and the flow over a NACA-0012 airfoil are simulated to assess the accuracy and robustness of the solution procedure proposed based on the SDLBM in computing steady and unsteady viscous compressible flows. The present results obtained by applying the SDLBM exhibit good agreement compared to the analytical and available high-order accurate solutions of the LB and Navier–Stokes equations.

High-order accurate large-eddy simulations of compressible viscous flow in cylindrical coordinates

Dynamic interactions of shock waves and turbulent structures occur in a wide range of applications. The accurate simulation of turbulence in these flows requires a numerical scheme with minimal dissipation whereas the shock-capturing requires a dissipative scheme to stabilise the solution in the shock-wave discontinuous regions. These contradictory requirements make simulations of these flows extremely challenging. A compatible implementation of the solid boundary conditions at the point of incepting the acoustic waves and their internalisation into hydrodynamic instabilities is also challenging.Here we present, a high-fidelity, massively parallel and accurate solver for both direct numerical simulations and large-eddy simulations of supersonic turbulent flows in cylindrical coordinates. A hybrid WENO/ high order central difference scheme is used for the spatial discretisation with a fourth-order five-stage 2N-storage Runge–Kutta for the time integration. The least square contraction of Lilly [1] is utilised for large-eddy subgrid-scale modelling. The solid surface boundary condition is implemented using the offset wall and ghost cell technique.The numerical implementation is validated through a wide range of test cases from simple to more complex cases. The numerical results show good agreement with analytical solutions and available experimental results. In the large-eddy simulation of a supersonic under-expanded impinging jet, it is observed that the conventional Ducros sensor leads to a more dissipative hybrid scheme. In this paper, a new shock sensor based on a WENO smoothness indicator is proposed and it is demonstrated to perform better especially in the simulation of the supersonic under-expanded impinging jet by limiting the expensive WENO scheme to the discontinuous regions and reducing the computational cost by approximately 10%.

Algebraic-volume meshfree method for application in finite volume solver

Nowadays the finite volume (FV) method remains the most popular choice for Computational Fluid Dynamics (CFD). But few meshfree methods have been developed based on the framework of existing FV codes which can boost the development of the meshfree method. An algebraic volume (AV) concept for the meshfree method is proposed to mimic the geometric control volume (CV) of the FV method via a meshfree point cloud. Similar to the CV, each AV is closed by algebraic faces which are constructed via the weighted least-squares method. Only the bias fluxes are solved at the midpoint of the current point and its neighbor. The other flow variables of Navier–Stokes (NS) equations are solved in the same way as the traditional FV method. So it automatically solves the compressible flow with shock wave without additional treatment. The AV meshfree method only requires to modify the calculation method of volume and face area vector from a geometric way to an algebraic solution in pre-processing, therefore it is straightforward to implement in a traditional FV solver. The numerical methods of upwind schemes for convective flux, corrected central difference for viscous flux and implicit temporal discretization are identical for both the mesh-based FV method and AV meshfree method. The AV meshfree method has been integrated into an FV code and applied to high Reynolds number, transonic, viscous flow simulations. The accuracy and convergence of both the FV method and AV meshfree method are verified via 2D low-speed flow and transonic 3D flow. The results of AV meshfree method agree quite well with those of the FV method and experimental data. And the comparison shows good point size convergence. Furthermore, a demo of hybrid FV method and AV meshfree method is presented to investigate the potential of this new meshfree method. The trajectory results obtained by CFD and experimental data are in good agreement.

ESTIMATION OF THE STRESS-STRAIN STATE OF THE BLADE BASED ON INTERPROGRAM INTERACTION OF PROGRAM COMPLEXES

Basis of numerical simulation of turbulent viscous gas flow, using the Navier-Stokes equations averaged the Reynolds (RANS-model), a static analysis of the gas turbine compressor blade was performed. The simulation of flow parameters in three-dimensional formulation was carried out. For the solution of system of the equations the iterative differential scheme was built. The initial equations are integrated numerically by use of the iterative explicit and implicit differential scheme with a second order approximation. The differential two-parametrical SST Menter's model is used as a model of turbulence. As a result, distributed pressure and velocities in the blades channel were obtained and unfavorable flow zones were identified. At the second stage, was developed algorithm by which the interrelation of the gas-dynamic and mechanical calculation program was established. As a result, distributed pressure was applied to the finite-element model of the blade and a static analysis was performed. Static calculation makes it possible to calculate the stress-strain state of the construction.

Large-Amplitude Intermittent Swimming in Viscous and Inviscid Flows

Recently, it was demonstrated in an inviscid flow that burst-and-coast or intermittent swimming can save energy when compared with continuous swimming and that the energy savings are maximized for large-amplitude pitching motions where flow separation is likely to occur. This paper examines the effects of flow separation on altering inviscid flow predictions. As such, viscous and inviscid flow simulations are presented of a hydrofoil pitching intermittently with large-amplitude motions. It is observed that leading-edge vortex formation and shedding in a viscous flow significantly alter the wake dynamics from the inviscid flow solutions where only trailing-edge shedding is modeled. Moreover, the inviscid flow solutions predict higher peak force production, lower cost of transport, and lower optimal duty cycles than the viscous flow solutions. Despite these differences, the trends in the force production and energetics seen in a viscous flow are well captured by the inviscid flow simulations. Importantly, both predict energy savings on the order of 10–30% for intermittent swimming, and the energy savings increase when the amplitude of motion is increased even when significant leading-edge separation occurs.

Switching mechanism investigation for the supersonic jet element: Deflection, attachment and adjustment stages

Based on the unsteady viscous flow simulation, the flow characteristics inside the supersonic jet Element have been investigated numerically. The results have revealed that once the specific structures of the supersonic jet element are finalized, even if the boundary conditions remain unchanged, the corresponding internal flow will also shows strong unsteady in a certain range of primary gas source pressure. The instability of the main vortex center is a main reason to make the output thrust of the supersonic jet components fluctuate all the time in the attached wall condition. At the deflection of the jet stage, once the static pressure for the right side of the wedge exceeds the left side, the transverse expansion for the stripping zone near the right output channel entrance can play a significant role in making the primary jet deflect successfully. When the jet starts to attach the wall layer, due to the “Coanda” effect, the jet also can attach the layer successfully even though the corresponding control flow is totally closed. When the jet enters the adjustment stage, the thrust changes process for the left and right outputs will experience two typical stages: adjusting and adjusted stages. The corresponding vortex structures at different switching time together with the force variations etc. inside the jet Element have been obtained computationally and analyzed in details.

Estimating lift from wake velocity data in flapping flight

The application of the Kutta–Joukowski (KJ) theorem to estimating the lift of a flying animal based on wake velocity fields often leads to significant underprediction of the lift, which is known as the wake momentum paradox. This work attempts to answer the puzzling question on whether the KJ theorem is legitimate in its use for complex viscous unsteady wakes generated by flapping wings. The limitations in applying the KJ theorem to flapping wings are quantitatively examined through numerical simulations of viscous incompressible flows over three flapping wing models. The three flapping wing models studied in this work are a flapping wing with a fixed wingspan, a flapping wing with a dynamically changing wingspan and a dihedral flapping wing. The KJ theorem fails to give a satisfactory prediction of the time-averaged lift unless an effective span length is correctly computed. We propose a wake-sectional Kutta–Joukowski (WS-KJ) model to predict the time-averaged lift, where the effective span length is computed based on the time-averaged distance between the streamwise vorticity centroids in the right and left half sides of the Trefftz plane. The WS-KJ model incorporates the spatial evolutionary effects of the complex vortex structures in the wake and significantly improves the prediction of the time-averaged lift. The physical foundation for such improvement is explored. In addition, the time-dependent amplitude and phase changes of the unsteady lift are discussed as the fluid acceleration effect.

A boundary thickening-based direct forcing immersed boundary method for fully resolved simulation of particle-laden flows

A boundary thickening-based direct forcing (BTDF) immersed boundary (IB) method is proposed for fully resolved simulation of incompressible viscous flows laden with finite size particles. By slightly thickening the boundary thickness, the local communications between the Lagrangian points on the solid boundary and their neighboring fluid Eulerian grids are improved, based on an implicit direct forcing (IDF) approach and a partition-of-unity condition for the regularized delta function. This strategy yields a simple, yet much better imposition of the non-slip and non-penetration boundary conditions than the conventional direct forcing (DF) technique. In particular, the present BTDF method can achieve a numerical accuracy comparable with other representative improved methods, such as multi-direct forcing (MDF), implicit velocity correction (IVC) and the reproducing kernel particle method (RKPM), while its computing cost remains much lower than them and nearly equivalent to the conventional DF method. The dependence of the optimum thickness value of boundary thickening on the form of the different regularized delta functions is also revealed. By coupling the lattice Boltzmann method (LBM) with BTDF-IB, the robustness of the present BTDF IB method is demonstrated using numerical simulations of creeping flow (Re=0.1), steady vortex separating flow (Re=40) and unsteady vortex shedding flow (Re=200) around a circular cylinder. The accuracy and robustness of the present method for moving particle-laden flows are performed on three benchmark cases, such as free sedimentation of single circular cylinder, DKT sedimentation of two particles and Rayleigh-Taylor instability of 504 particles sedimentation in an enclosed cavity. The computational efficiency between the present BTDF method and the MDF method is also compared at last.

Construction of knotted vortex tubes with the writhe-dependent helicity

We propose a feasible method for constructing knotted vortex tubes with the finite thickness and arbitrary complexity and develop an accurate algorithm to implement this method in numerical simulations. The central axis of the knotted vortex tube is determined by the parametric equation of a given smooth and non-degenerate closed curve. The helicity of the vortex tube is only proportional to the writhe of the vortex axis, a geometric measure for coiling of vortex tubes. This vortex construction can facilitate the investigation of the conversion of writhe to twist in the helicity evolution of knotted vortex tubes. As examples, we construct velocity–vorticity fields of trefoil, cinquefoil, and septafoil vortex knots. These vortex knots are used as initial conditions in the direct numerical simulation of viscous incompressible flows in a periodic box. In the evolution of vortex knots from simple flows to turbulent-like flows, all the knots are first untied. Then the vortex topology is invariant and the helicity is almost conserved for the trefoil knot, whereas the helicity decays rapidly during the breakdown and coaxial interactions of pinch-off vortex rings for cinquefoil and septafoil knots.

Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics

Abstract We propose a numerical algorithm for simulations of two-component viscous compressible isothermal flows with surface effects in 3D domains of complex shape with voxel representation of geometry. The basic mathematical model is the regularized system of Navier–Stokes–Cahn–Hilliard equations. Simulations of droplet spreading over a flat base and displacement of one liquid by another one in a pore space of real rock sample are carried out. The simulation results demonstrate the applicability and good efficiency of the used system of equations, the corresponding difference scheme, and its implementation algorithms for numerical solution of the considered class of problems.

On the permeability of colloidal gels

We reexamine and refine analytical theories for permeability in colloidal networks, with particular focus on constants and identification of approximations. The new theories are compared against numerical simulations of Stokes flow through the networks and reveal nearly quantitative power-law predictions for both pore size and permeability at low volume fractions, with systematic deviations observed only at high volume fractions. Comparison with two previously published experimental data sets yields mixed results: in one case, very good agreement is found, while in the other, only the scaling is correctly predicted. In fractal gel networks, the permeability is commonly modeled as a power-law function of volume fraction, with the fractal dimension of the network determining the power-law exponent. To quantitatively probe the influence of gel structure on permeability, we investigate this relation in structures generated by diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA) and, for contrast, non-overlapping uniform random dispersions of particles. Geometric analyses are used to determine network pore size distributions, fractal dimensions, and percolation characteristics. High-fidelity simulations of the slow viscous flow of Newtonian fluids are used to obtain first-principles-based velocity and fields and hence network permeabilities. Interestingly, the effective pore size that determines permeability is found to be somewhat larger than that measured by a method based on the insertion of spherical probes. Empirical inclusion of a fractal dimension dependence on volume fraction is found to yield quantitative results for permeabilities over the entire volume fraction range studied, in both DLCA and RLCA materials.

Grid‐Free Modelling Based on the Finite Particle Method for Incompressible Viscous Flow Problems

In this paper, we present a grid‐free modelling based on the finite particle method for the numerical simulation of incompressible viscous flows. Numerical methods of interest are meshless Lagrangian finite point scheme by the application of the projection method for the incompressibility of the Navier–Stokes flow equations. The moving least squares method is introduced for approximating spatial derivatives in a meshless context. The pressure Poisson equation with Neumann boundary condition is solved by the finite particle method in which the fluid domain is discretized by a finite number of particles. Also, a continuous particle management has to be done to prevent particles from moving into configurations problematic for a numerical approximation. With the proposed finite particle technique, problems associated with the viscous free surface flow which contains the study on the liquid sloshing in tanks with low volumetric fluid type, solitary waves movement, and interaction with a vertical wall in numerical flume as well as the vortex patterns of the ship rolling damping are circumvented. These numerical models are investigated to validate the presented grid‐free methodology. The results have revealed the efficiency and stability of the finite particle method which could be well handled with the incompressible viscous flow problems.

Investigation on tip leakage flow characteristics of radial inflow micro turbine used in helium liquefying system

Tip clearance leakage flow with different axial and radial clearance in a radial inflow helium micro turbine impeller used in helium liquefying system under the stage environment is investigated using a three-dimensional transient viscous flow simulation. The comparison of the numerical results with the experimental results is proved to verify the accuracy of the numerical method. The results indicate the formation and development mechanism of the tip leakage flow. The leakage flow effects in the helium turbine passage have slightly transient fluctuations. Then leakage vortex develops in the impeller passage and causes large leakage flow loss. Rotational speed and expansion ratio have great impact on leakage flow characteristics. Furthermore, there is a notable difference of the performance affected by axial clearance and radial clearance. The leakage mass flow and leakage flow loss are more affected by the radial clearance than the axial clearance. In addition, empirical formulas are obtained to describe the leakage mass flow and flow loss caused by the axial clearance and radial clearance. Finally, a modified semi-empirical formula is obtained based on the numerical results to calculate leakage flow loss correctly. The results in this paper can give a further understanding of leakage flow in radial inflow micro turbine and provide a method to estimate the tip clearance effect on performance and flow loss quantitatively.

Numerical and experimental assessment of nominal wake for a bulk carrier

In this paper, numerical and experimental assessment of nominal wake for a handymax bulk carrier at model and full-scale is performed. A circumferential averaged non-dimensional axial velocity distribution and integral value of nominal wake are obtained based on the measurements performed with the Pitot tube in the towing tank of Brodarski institute. Reynolds averaged Navier–Stokes equations along with three different turbulence models are utilized within double body simulations. The influence of the application of wall functions is investigated at model scale. Since the obtained differences between numerical solutions with and without wall functions are not large, full-scale simulations are performed with wall functions to avoid large cell number. The obtained numerical results are validated at model scale and satisfactory agreement is achieved. The significant scale effects are noticed in both integral value of nominal wake and circumferential averaged non-dimensional axial velocity distribution. Therefore, full-scale numerical simulations of viscous flow offer important benefits, especially in the optimization of the stern flow.

A conservative scheme for simulation of free-surface turbulent and wave flows

A numerical scheme with good conservation properties is developed for the simulation of free-surface turbulent and viscous wave flows using a surface-fitted curvilinear grid. The Navier–Stokes equations are written in a strong conservative formulation with respect to the curvilinear coordinates, and are discretized by a pseudo-spectral method in the horizontal directions and a finite-difference method in the vertical direction. Large-eddy simulation (LES) is implemented with the conservative scheme to extend the simulation capability to turbulent flows with higher Reynolds numbers. Fully nonlinear kinematic and dynamic boundary conditions are implemented at the free surface. The numerical scheme is validated using a variety of wave and vortical flow test cases. The results show good agreement with previous theoretical and numerical predictions, whereas the present scheme achieves significant improvement in the conservation of mass and momentum over the non-conservative scheme developed by Yang & Shen [1]. Meanwhile, the present conservative scheme is found to be more stable than the non-conservative scheme for the simulation of sideband waves and broadband waves. The effect of viscous dissipation on the long-term nonlinear wave evolution is also captured by the present scheme. The ability of the present scheme for simulating long-term wave-current-turbulence interaction is demonstrated by the computation of Langmuir circulation, for which the non-conservative scheme produces significant errors in mass and momentum conservation and the simulation fails. Flow features of the Langmuir circulation, such as the counter-rotating vortices and converging-diverging zones, have been successfully captured with our numerical scheme. The turbulence statistics also agree with the characteristics of Langmuir circulation.

Automatic and efficient hybrid viscous mesh generation based on clipped Voronoi diagrams

SummaryThe commonly used advancing layers method to generate hybrid meshes suffers from many drawbacks. The generation of isotropic meshes for far‐field domains with irregular and complex boundary subdivisions after boundary layers advancing is time consuming and, in some cases, is not robust in 3D. To address these difficulties, this paper presents a novel method to generate hybrid polygonal meshes in 2D and polyhedral meshes in 3D for viscous flow simulations. In the proposed method, first, we generate a full Voronoi diagram for the appropriate distribution of generators that avoids the extra mesh generation required for the remaining holes in the advancing layers method. To recover the inner solid boundaries, we implement a robust boundary cell cutting process. Because the generators are located layer by layer near the boundaries, there is no requirement to consider all of the Voronoi cells. Only the first layer Voronoi cells must be cut, making the calculation very efficient. We have generated hybrid meshes using the present method for many viscous flow cases. The results show close agreement between the computations and the experimental results, thus indicating the reliability and effectiveness of the hybrid mesh generated by our method.

An Overview of Numerical Methods for Incompressible Viscous Flow with Moving Particles

The importance of the fluid–particle interaction problem is considerable. Tremendous attention has been paid by researchers in the past two decades. This paper presents an overview of the representative methods for the simulation of incompressible viscous flow with moving boundaries based on conforming or nonconforming meshes. These methods include the distributed Lagrange multiplier/fictitious domain method, the immersed boundary method, the lattice Boltzmann method, and the arbitrary Lagrange–Eulerian method. The classification of the selected methodology is made to create a firm foundation for the future. The latest developments in the field during past few years are also reviewed. The opportunities and limits of all the methods are discussed to improve the research of particulate flow. Finally, the paper points out several directions of future researches.

Numerical simulations of unsteady viscous incompressible flows using general pressure equation

In fluid dynamics, an important problem is linked to the knowledge of the fluid pressure. Recently, another approach to study incompressible fluid flow was suggested. It consists in using a general pressure equation (GPE) derived from compressible Navier–Stokes equation. In this paper, GPE is considered and compared with the Chorin's artificial compressibility method (ACM) and the Entropically damped artificial compressibility (EDAC) method. The three methods are discretized in a staggered grid system with second order centered schemes in space and a third order Runge–Kutta scheme in time. Three test cases are realized: two-dimensional Taylor–Green vortex flow, the traveling wave and the doubly periodic shear layers. It is demonstrated that GPE is accurate and efficient to capture the correct behavior for unsteady incompressible flows. The numerical results obtained by GPE are in excellent agreement with those obtained by ACM, EDAC and a classical finite volume method with a Poisson equation. Furthermore, GPE convergence is better than ACM convergence. The proposed general pressure equation (GPE) allows to solve general, time-accurate, incompressible Navier–Stokes flows. Finally, the parametric study realized in terms of Mach and Prandtl numbers shows that the velocity divergence can be limited by an arbitrary maximum and that acoustic waves can be damped.

Homogenization of a Large Eddy Simulation Model for Turbulent Fluid Motion Near a Rough Wall

In this paper, we are interested in the numerical simulation of a turbulent viscous flow in the vicinity of a periodic rough wall. To this aim, we consider a large eddy simulation (LES) model, and perform its asymptotic analysis, letting the period $$\varepsilon $$ of the roughness tend to zero. Relying on an homogenization method closely related to the two-scale convergence method (Allaire in SIAM J Math Anal 23(6):1482–1518, 1992; Nguetseng in SIAM J Math Anal 20(3):608–623, 1989), we exhibit a critical scaling associated with parameter $$\lambda :=\lim _{\varepsilon \rightarrow 0}{a_\varepsilon }/{\varepsilon ^{5/3}}$$ , where $$a_\varepsilon $$ stands for the amplitude of the roughness. In the critical regime $$0<\lambda <\infty $$ , the roughness effect is described by an additional, nonlinear friction term, that depends locally on the solution to a nonlinear auxiliary problem of boundary layer type. In the particular case of riblets, which is of major practical interest, we show that the directional invariance of the geometry results in a simplified expression of this extra term, and we propose a numerical method to simulate the homogenized LES model, based on that observation. Finally, we study the influence of parameter $$\lambda $$ on the flow approximated by the homogenized model, using different commonly used riblet shapes. Our simulations demonstrate that the extra friction term can be handled numerically, and may have an important influence on the results of LES in presence of roughness.