Two-dimensional Flow Problems Research Articles

On immiscibility preservation conditions of material interfaces in the generic five-equation model

Interfaces separating pure materials and mixtures tend to be severely smeared with interface-capturing methods for compressible multi-material flows, necessitating the requirement of various interface-sharpening techniques. However, these techniques have various problems related to consistency, conservation, and thermodynamic compatibility. In this work, we derive a general theoretical formulation of interface-sharpening techniques for the generic five-equation model. This theoretical formulation is not only conservative in mass, momentum, and total energy but also asymptotically compatible with the thermodynamic mixture laws of the mixture model upon which it is constructed, and is independent of various specific numerical algorithms. We further propose a general numerical method to solve this theoretical formulation. The proposed method is consistent and conservative, and it prevents spurious errors at the interfaces. Examples of one- and two-dimensional multimaterial compressible flow problems, including shocks and interfaces, are considered to verify the analysis and demonstrate the efficiency of the method.

Numerical discretisation of hyperbolic systems of moment equations describing sedimentation in suspensions of rod-like particles

We present a numerical discretisation of the coupled moment systems, previously introduced in Dahm and Helzel [1], which approximate the kinetic-fluid model by Helzel and Tzavaras [2] for sedimentation in suspensions of rod-like particles for a two-dimensional flow problem and a shear flow problem. We use a splitting ansatz which, during each time step, separately computes the update of the macroscopic flow equation and of the moment system. The proof of the hyperbolicity of the moment systems in [1] suggests solving the moment systems with standard numerical methods for hyperbolic problems, like LeVeque's Wave Propagation Algorithm [3]. The number of moment equations used in the hyperbolic moment system can be adapted to locally varying flow features. An error analysis is proposed, which compares the approximation with 2N+1 moment equations to an approximation with 2N+3 moment equations. This analysis suggests an error indicator which can be computed from the numerical approximation of the moment system with 2N+1 moment equations. In order to use moment approximations with a different number of moment equations in different parts of the computational domain, we consider an interface coupling of moment systems with different resolution. Finally, we derive a conservative high-resolution Wave Propagation Algorithm for solving moment systems with different numbers of moment equations.

Exact formulas for estimating the area of flow regions in free boundary problems

An effective technique is proposed for obtaining exact formulas for estimating the area of flow regions in two-dimensional fluid flow problems with free boundaries, that allow an exact solution in terms of elliptic functions. The effectiveness of the technique is demonstrated using a specific example of the problem of capillary waves on the surface of a liquid of finite depth. This example is characterized by mirror symmetry of the flow region, but the technique can be generalized to the case of other symmetry of the flow region.

A variational data assimilation approach for sparse velocity reference data in coarse RANS simulations through a corrective forcing term

The Reynolds-averaged Navier–Stokes (RANS) equations provide a computationally efficient method for solving fluid flow problems in engineering applications. However, the use of closure models to represent turbulence effects can reduce their accuracy. To address this issue, recent research has explored data-driven techniques such as data assimilation and machine learning.An efficient variational data assimilation (DA) approach is presented to enhance steady-state eddy viscosity based RANS simulations. To account for model deficiencies, a corrective force term is introduced in the momentum equation. In the case of only velocity reference data, this term can be represented by a potential field and is divergence-free. The DA implementation relies on the discrete adjoint method and approximations for efficient gradient evaluation.The implementation is based on a two-dimensional coupled RANS solver in OpenFOAM, which is extended to allow the computation of the adjoint velocity and pressure as well as the adjoint gradient. A gradient-based optimizer is used to minimize the difference between the simulation results and the reference data. To evaluate this approach, it is compared with alternative data assimilation methods for canonical stationary two-dimensional turbulent flow problems. For the data assimilation, sparsely distributed reference data from averaged high-fidelity simulation results are used.The results suggest that the proposed method achieves the optimization goal more efficiently compared to applying data assimilation for obtaining the eddy viscosity, or a field modifying the eddy viscosity, directly. The method works well for different reference data configurations and runs efficiently by leveraging coarse meshes.

Conformal and approximate analysis of the effect of cutoff wall thickness on finite depth seepage characteristics

ABSTRACT An analytical solution to the problem of two-dimensional steady-state seepage flow beneath a water-retaining structure is presented using conformal mapping and complex variables. This paper addresses the question of to what extent the cut-off thickness affects the values of seepage characteristics. The closed-form analytical equations are calculated numerically. The analytical results have been validated by comparing them with the available exact solutions for finite-depth seepage. The resulting implicit equations, involving elliptic integrals, are used to create design diagrams. Increasing the thickness can cause a reduction in hydraulic gradient, seepage discharge, and uplift pressure at the left tip of the cut-off. A customized fragment type considering the cut-off’s thickness is developed using the analytical results to estimate seepage quantities and exit gradients. The form factor for this fragment is obtained in the form of dimensionless charts, which predict the flow rate and exit gradient more precisely than the existing solutions.

A method to calculate inverse solutions for steady open channel free-surface flow

The inverse problem of steady two-dimensional open channel free-surface flow is considered, with the focus on determining two types of disturbances: a surface pressure distribution and solid channel bottom topography. A closed-form expression for the inverse surface pressure is derived, and a linear Fredholm equation of the first kind is shown to describe the inverse topography problem, which then needs to be descretised and solved numerically. However, the equation for the channel bottom is prone to instability, so the truncated singular value decomposition (TSVD) method is proposed as a way to stabilise the associated discrete solution. The effectiveness of the TSVD method is demonstrated through several numerical examples, and its performance in the presence of error-contaminated input data is also examined. The results show that the TSVD method can recover the topography accurately from the forward free-surface problem, and provide good approximations even with noisy input data.

A parameter-free staggered-grid Lagrangian scheme for two-dimensional compressible flow problems

The present study aims to develop a parameter-free staggered-grid Lagrangian scheme that avoids the empirical parameters needed for artificial viscosity and anti-hourglass force. The artificial viscosity is equal to the pressure jump, which is constructed by the relationship between pressure and velocity. The anti-hourglass force is established using pressure compensation, which is the difference in pressure between sub-cells belonging to the same primary cell. The scheme maintains the conservation of total mass, momentum, and energy. Numerical experiments show that the scheme is highly robust for extreme flow problems with different fine meshes such as Sedov, Noh, Saltzmann triple point and Rayleigh–Taylor instability. The robust parameter-free scheme is well suited for multi-physics problems and engineering applications.

Study of Nonclassical Transport by Applying Numerical Methods for Solving the Boltzmann Equation

This paper overviews the state of the art in the study of nonequilibrium gas flows with nonclassical transport, in which the Stokes and Fourier laws are violated (and, accordingly, the Chapman–Enskog method is inapplicable). For a reliable validation of anomalous transport effects, we use computational methods of different nature: the direct solution of the Boltzmann equation and direct simulation Monte Carlo. Nonclassical anomalous transport is manifested on scales of 5–10 mean free paths, which confirms the fact that a highly nonequilibrium flow is a prerequisite for the detection of the effects. Two-dimensional flow problems are considered, namely, the supersonic flow over a flat plate in the transient regime and the supersonic flow through membranes (lattices), where the flow behind the lattice corresponds to the spatially nonuniform relaxation problem. In this region, nonequilibrium distributions demonstrating anomalous transport are formed. The relationship of the effect with the second law of thermodynamics is discussed, the possibilities of experimental verification are considered, and the prospects of creating new microdevices on this basis are outlined.

A novel approach to micropolar fluid flow between a non-porous disk and a porous disk with slip

This paper considers the problem of steady, laminar, incompressible, and two-dimensional micropolar fluid flow between two disks. The top disk is considered porous, while the lower one is not. The body forces and body couples were neglected, and the flow was assumed to be fully developed. The governing equations of the problem were reduced to a set of ordinary differential equations (ODEs) by Von-Karman's similarity transformations. Since the obtained governing ODEs have not been solved analytically according to the previous studies, in this article, the Modified Akbari-Ganji method (Modified AGM) and the hybrid analytical and numerical method (HAN-method) have been chosen to solve these equations analytically, which is one of the novelties of this study. However, most of this article's novelty is related to the physical results obtained from the analytical solution of these equations. The effects of various slip coefficients, Reynolds number, and micropolar parameters of vortex viscosity, spin gradient viscosity, and microinertia density on profiles of normal velocity, streamwise velocity, and microrotation. The validity of these two analytical solutions was proved by comparison with previously published results. The results of the two methods are almost the same in all cases, which can be seen as an indirect sign of the validity of the results of this study.

MHD natural convection flow in a porous medium-filled corrugated enclosure: Effect of heat sources with different heights

A two-dimensional natural convection flow problem in a porous medium-filled enclosure with electrically conducting fluids and solid heat sources of different heights is numerically investigated in this study. The top boundary is cold, and the vertical side boundaries are insulated, while rectangular-shaped corrugations at the lower boundary are generated by placing heat sources. Three distinctive cases are considered concerning the height of the solid rectangular-shaped heat sources. The nonlinear coupled transport equations are discretized by using a higher-order compact (HOC) finite difference scheme. First, existing numerical and experimental data are used to validate the developed code. Then, the effect of key parameter values, like Hartmann number (25–150), Rayleigh number (103–106), Darcy number (10−5–10−1), and Prandtl number (0.015–10), on the convective flow in the corrugated enclosure is analysed. The influence of the permeability of the porous medium is abundantly seen for a range of significant parameter values with a fixed high Rayleigh-Darcy number. The formation of primary, secondary, tertiary, and quaternary vortices affects the thickness of the thermal boundary layer on both hot and cold surfaces in both symmetric and asymmetric patterns. In addition, possible multiple steady states are observed in the Case 2 at Pr=10.0. The local and average Nusselt numbers remains constant on the bottom cold surfaces of the corrugated enclosure, while significant variations are observed on the upper surfaces of the heat sources. These findings are analysed by conducting stream-function (ψ), isotherms (T), local (Nuh,Nuv), and average (Nū) Nusselt number plots. The intensities of heat transfer rate are also examined by analysing Nusselt numbers as a function of Prandtl, Darcy, Rayleigh and Hartmann numbers. Analysing natural convective heat transfer fluid flow in an enclosure with heat sources is prominent for ample engineering applications such as solar collectors, designing nuclear reactors, chemical energy production systems, building constructions and microelectronic equipment, to name a few.

Nonlinear gravity waves in the channel covered by broken ice

The two-dimensional nonlinear problem of a steady flow in a channel covered by broken ice with an arbitrary bottom topography including a semi-circular obstruction is considered. The mathematical model is based on the velocity potential theory accounting for nonlinear boundary conditions on the bottom of the channel and at the interface between the liquid and the layer of the broken ice, which are coupled through a numerical procedure. A mass loading model together with a viscous layer model is used to model the ice cover. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. The coupled problem is reduced to a system of integral equations in the unknown velocity magnitude at the interface, which is solved numerically using a collocation method. Case studies are conducted both for the subcritical and for the supercritical flow regimes in the channel. For subcritical flows, it is found that the ice cover allows for generating waves with amplitudes larger than those that may exist in the free surface case; the ice cover prevents the formation of a cusp and extends the solution to larger obstruction heights on the bottom. For supercritical flow regimes, the broken ice significantly affects the waveform of the soliton wave making it gentler. The viscosity factor of the model apparently governs the wave attenuation.

A new model of time-dependent fractional second grade fluid for two-dimensional channel flow with heat transfer

The goal of the current paper is to create a novel model for the two-dimensional flow (TDF) of a second-grade fluid and the transfer of heat due to free convection. The second-grade fluid travels through the porous media and is electrically conducted. The governing equation for fluid velocity and temperature is formulated as a set of partial differential equations jointly. The model is developed in fractional form by applying the Caputo-Fabrizio (CF) fractional derivative approach. The two-dimensional flow problem is modeled in terms of one-sided coupled fractional partial differential equations with corresponding physical conditions imposed on the fluid flow and temperature. The problem is initially converted to a dimensionless form and then solved for the exact solutions by applying joint Fourier and Laplace transforms. Exact solutions are examined for fluid velocity and temperature and then in limiting cases are reduced to some well-known fluid motions. Some special cases are also obtained from the general solutions. For the numerical results, Mathcad-15 software is used and the results are plotted in various graphs. The temperature, and velocity profiles are discussed and highlighted for embedded flow parameters. The model obtained in this work is new and can be extended for other fluid models hence this article provides a base for future research.

On Exact Solutions to Problems of Steady Flow of Incompressible Viscous Fluid in a Channel and in a Circular Pipe

Turning to the well-known Couette flow, we again, but now in a new formulation, consider the steady motion of an incompressible viscous fluid in a channel with two parallel flat walls, one of which is stationary while the other moves at a constant speed in its plane. In the classical formulation of this problem on a layered (laminar) fluid flow in a channel, the wall velocity and the constant pressure drop along the channel are independent quantities and are considered predetermined. However, when a solid body moves along the fluid surface, the fluid particles are set in motion. This occurs due to the arising pressure generated in the fluid caused by the movement of the solid body. Therefore, here the pressure or pressure drop is not predetermined but is determined as a result of solving the corresponding two-dimensional boundary value problem of a steady flow of a viscous fluid in a channel. For this, the corresponding boundary value problem of a fluid flow in the channel is formulated based on the linearized Navier-Stokes equations obtained from the general nonlinear Navier-Stokes equations for a steady flow of a viscous fluid neglecting the convective terms, which is true for small Reynolds numbers. Using the Fourier integral transform method, an exact (closed) solution to this boundary value problem is constructed; the velocity and pressure components are determined. Then it is shown that the resulting solution, different from the known Couette flow solution, also satisfies the original nonlinear Navier-Stokes equations, the continuity equation, and boundary conditions. A comparative analysis of the new solution with the Couette flow solution is carried out. The well-known Hagen-Poiseuille problem is also discussed from the point of view of establishing the conditions for the laminar axisymmetric and steady flow of an incompressible viscous fluid in a straight circular pipe According to the study's results, to implement laminar flow under the indicated conditions, the pressure drop along the pipe must be constant. The study demonstrates that this condition is simultaneously sufficient, and therefore, the necessary and sufficient condition is established for a laminar flow in a straight circular pipe.

Analytical approach for micropolar fluid flow in a channel with porous walls

This article aims to analyze and reach new physical achievements in a well-known nonlinear problem of two-dimensional, steady, and steady micropolar fluid flow between two parallel permeable porous walls. Micropolar fluids are considered a class of non-Newtonian fluids. The nonlinear governing equations of this problem were analyzed using two analytical methods. The achievements and novelty of this article, apart from showing the application of two mathematical methods in solving non-linear equations, are more about achieving new results that are obtained from solving the governing equations of this article. From the results, it is obvious that with the raise of the coupling parameter, other dimensionless parameters increase, except for the dimensionless microrotation. Also, when the spin gradient viscosity parameter increases, all the average values of the mentioned dimensionless parameters increase. From the problem's solution, it is clear that when the micro-inertia density ratio increases across the channel, all the average values of the dimensionless parameters decrease except for the dimensionless flow function characteristics. Also, when the Peclet number for the heat diffusion parameter is changed, it only affects the dimensionless temperature profile. When the Peclet number for the mass diffusion parameter is changed, it only affects the dimensionless concentration profile.

A novel artificial neural network-based interface coupling approach for partitioned fluid–structure interaction problems

This paper presents a novel interface technique employing Artificial Neural Networks (ANN) for efficient data transfer between incompressible fluid and deformable solid domains in a partitioned fluid–structure interaction (FSI) framework. Governing differential equations (GDE) with potential flow assumptions are solved to obtain pressure values at computational nodes in the fluid domain. Pressure values are then used to train an ANN-based model to interpolate pressure loads for application at non-collocated nodes on the solid boundary. Both finite element method (FEM) and meshfree element free Galerkin (EFG) formulations are used to calculate the solid deformation. Proposed methodology is applied to solve a two-dimensional (2D) unidirectional flow problem, consisting of a flexible cantilever beam immersed in a laminar, steady, and inviscid fluid. Overwhelming mathematical intricacies of computational solvers are avoided for the sake of simplistic interface treatment, flux transfer and associated phenomena. Dedicated partitioned solvers are developed for both solid and fluid domains, which are individually benchmarked for established problems from the literature. The presented ANN-based interpolation scheme provides an alternative to higher-order polynomial algorithms at the interface boundary. The proposed scheme is simple to employ, computationally efficient, and offers competitive accuracy as well as stability.

Nonlinear Free Surface Flow past a Wedge in Channel

In this paper, the two-dimensional problem of irrotational flow past a wedge located in the center of the channel is considered. Assuming that the fluid is incompressible and non-viscous, the influence of gravity is ignored but the surface tension is considered. The problem which is characterized by the nonlinear boundary conditions on the free surface of the unknown equation is solved numerically by the series truncation technique. The results show that for all given wedge configurations, there is a critical value for the Weber number, for which there is no solution for every Weber number value smaller than this. In addition, the obtained results extend the work done by Gasmi and Mekias [2].

Applications of triadic hybridized-cross nanomaterials suspended in engine oil using quadratic and linear convection with magnetic dipole

This work comprises the contribution of buoyancy forces, linear and quadratic convection on cross Nano fluid under the effects of dipole effects with chemical reaction. The flow of a cross model (CM) with mass and thermal transport has been modeled in Cartesian coordinates for the two-dimensional fluid flow problem. Temperature gradient effects have been incorporated in mass transport and concentration gradient impacts in thermal transport phenomenon with chemical reaction. The governing laws are derived under boundary layer principle (BLP) in the conformation of coupled nonlinear partial differential equations. Afterwards, coupled partial differential equations are converted into a scheme of ordinary differential equations (ODEs) by incorporating the similarity transformation. The nonlinear modeled equations are solved numerically with the aid of finite element scheme as well as analysis of grid independent is reported based on the reliability of the utilized approach. Several graphs and tables have been prepared to observe the bearing of multiple emerging parameters regarding velocity, temperature as well as concentration fields. Comparison has been made to confirm the efficiency of a finite element scheme. It is perceived that higher values of a Schmidt number control the fluid concentration. Moreover, velocity is diminished against higher values of Weissenberg parameter.

A New Multi-Level Grid Multiple-Relaxation-Time Lattice Boltzmann Method with Spatial Interpolation

The traditional multi-level grid multiple-relaxation-time lattice Boltzmann method (MRT-LBM) requires interpolation calculations in time and space. It is a complex and computationally intensive process. By using the buffer technique, this paper proposes a new multi-level grid MRT-LBM which requires only spatial interpolation calculations. The proposed method uses a center point format to store multi-level grid information. The grid type determination in the flow field calculation domain is done using the axis aligned bounding box (AABB) triangle overlap test. According to the calculation characteristics of MRT-LBM, the buffer grid is proposed for the first time at the interface of different levels of grids, which is used to remove the temporal interpolation calculation and simplify the spatial interpolation calculation. The corresponding multi-level grid MRT-LBM algorithm is also presented for two-dimensional and three-dimensional flow field calculation problems. For the two-dimensional problem of flow around a circular cylinder, the simulation results show that a four-level grid MRT-LBM proposed in this paper can accurately obtain the aerodynamic coefficients and Strouhal number at different Reynolds numbers, and it has about 1/9 of the total number of grids as a single-level grid MRT-LBM and is 6.76 times faster. For the three-dimensional flow calculation problem, the numerical experiments of flow past a sphere are simulated to verify the numerical precision of the presented method at Reynolds numbers = 100, 200, 250, 300, and 1000. With the streamlines and velocity contours, it is demonstrated that the multi-level grid MRT-LBM can be calculated accurately even at the interface of different size grids.

Numerical analysis of two-dimensional unsaturated soil water flow problems with two-grid finite element methods

<p style='text-indent:20px;'>In this paper, we investigate a two-grid finite element method (FEM) for two-dimensional nonlinear unsaturated soil water flow problems. The application of two-grid methods can transform the solution of nonlinear problems on the fine grid to the solution of nonlinear problems on the coarse grid and the solution of linear problems on the fine grid. Based on the error results of standard FEM, we derive the continuous and discrete time error estimates in the energy norm for the two-grid method. Numerical experiments are carried out with the linear element as an example to illustrate that the two-grid method can achieve the similar optimal accuracy as standard FEM with less time as long as the two mesh sizes satisfy <inline-formula><tex-math id="M1">\begin{document}$ H = O(\sqrt{h}) $\end{document}</tex-math></inline-formula>.</p>

Multiscale model reduction technique for fluid flows with heterogeneous porous inclusions

Numerical treatment of the problem of two-dimensional viscous fluid flow in and around circular porous inclusions is considered. The mathematical model is described by Navier–Stokes equation in the free flow domain Ωf and nonlinear convective Darcy–Brinkman–Forchheimer equations in porous subdomains Ωp. It is well-known that numerical solutions of the problems in such heterogeneous domains require a very fine computational mesh that resolve inclusions on the grid level. The size alteration of the relevant system requires model reduction techniques. Here, we present a multiscale model reduction technique based on the Generalized Multiscale Finite Element Method (GMsFEM). We discuss construction of the multiscale basis functions for the velocity fields based on the solution of the local problems with and without oversampling strategy. Three test cases are considered for a given choice of the three key model parameters, namely, the Reynolds number (Re), the Forchheimer coefficient (C) and the Darcy number (Da). For the test runs, the Reynolds number values are taken to be Re=1,10,100 while the Forchheimer coefficient and Darcy number are chosen as C=1,10 and Da=10−5,10−4,10−3, respectively. We numerically study the convergence of the method as we increase the number of multiscale basis functions in each domain, and observe good performance of the multiscale method.