Time-dependent Incompressible Flow Research Articles

Effect of the Non-Linear Radiative Unsteady Mixed Convective Flow over a Curved Stretching Surface with Soret and Dufour Effects: A Numerical Study

The subject of unsteady convective flow with non-linear thermal radiation has become an important issue of research, due to its implications in advanced energy conversion systems operating at high temperature, solar energy technology and chemical process at high operation temperature. Due to the importance of this issue, a time dependent incompressible viscous fluid flow, heat and mass transfer over a curved stretching surface has been numerically analysed by taking into account the heat flux due to concentration gradient and mass flux due to temperature gradient. Together with this the Rosseland approximation is being employed for the nonlinear thermal radiation impact in presence of thermal slip. With the aid of non-dimensional variables and the corresponding physical boundary conditions, the leading nonlinear momentum, energy, and species equations are converted into a set of coupled ordinary differential equations. These equations are then resolved using the MATLAB bvp4c solver. The stability of the numerical technique has been verified and compared with available literatures. The resultant parameters of engineering interest and the boundary layer flow field parameters and have been presented using tables and graphically plots. The study concludes that for lesser curvature parameter (0.5≤K≤0.7) the surface drag force, heat and mass transfer rates can improve by about 9.59%, 2.87% and 1.67% each respectively. The presence of the temperature ratio parameter and the non-linear thermal radiation are found to greatly influence the temperature profile and the heat transfer rate of the system. Results show that the heat transfer rate improves by about 24.39% and 16.66% for varying non-linear thermal radiation (1≤Rd≤1.5) and temperature ratio parameter (1.2≤θw≤1.4) respectively. Results obtained also show that improving the thermal slip parameter (0.4≤L≤0.6) can reduce heat transfer rate by about 13.62% and reduce the surface temperature profile.

An efficient tree-topological local mesh refinement on Cartesian grids for multiple moving objects in incompressible flow

This paper develops a tree-topological local mesh refinement (TLMR) method on Cartesian grids for the simulation of bio-inspired flow with multiple moving objects. The TLMR nests refinement mesh blocks of structured grids to the target regions and arrange the blocks in a tree topology. The method solves the time-dependent incompressible flow using a fractional-step method and discretizes the Navier-Stokes equation using a finite-difference formulation with an immersed boundary method to resolve the complex boundaries. When iteratively solving the discretized equations across the coarse and fine TLMR blocks, for better accuracy and faster convergence, the momentum equation is solved on all blocks simultaneously, while the Poisson equation is solved recursively from the coarsest block to the finest ones. When the refined blocks of the same block are connected, the parallel Schwarz method is used to iteratively solve both the momentum and Poisson equations. Convergence studies show that the algorithm is second-order accurate in space for both velocity and pressure, and the developed mesh refinement technique is benchmarked and demonstrated by several canonical flow problems. The TLMR enables a fast solution to an incompressible flow problem with complex boundaries or multiple moving objects. Various bio-inspired flows of multiple moving objects show that the solver can save over 80% computational time, proportional to the grid reduction when refinement is applied.

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.

Effects of viscous dissipation and thermal radiation on time dependent incompressible squeezing flow of [formula omitted]water hybrid nanofluid between two parallel plates with variable viscosity

In view of the dominant properties of hybrid nanofluid such as high thermal and electrical conductivity in addition to enhanced heat transfer rate, efforts had been strengthened by many researchers to upgrade the thermal behavior of the base fluid through different approaches. In this study, viscous dissipation and thermal radiation effects on unsteady incompressible squeezing flow conveying CuO−Al2O3/water hybrid nanoparticles between two aligned surfaces with variable viscosity is examined. The fluid model is transformed to ordinary differential equations by incorporating appropriate similarity transformation. The numerical simulation is carried out in MATLAB software package via shooting procedure coupled with 4th order Runge–Kutta integration scheme. The limiting case is found to be in accord relative to the preceding reports. The outcomes of the scrutiny are unveiled in tables and graphs. It was revealed that the velocity and temperature augment with increasing viscosity variation and squeezing fluid parameters. Meanwhile, increasing viscous dissipation and thermal radiation parameters decrease the temperature distribution with no significant change in the fluid velocity.

Low Reynolds Number Flow Past Square Cylinder in the Vicinity of a Plane Wall

The characteristics of flow over a square cylinder in the vicinity of a plane stationary wall have been numerically investigated. 2D time-dependent incompressible flow at low-Reynolds numbers of 100 and 200 has been calculated using the finite volume method. CFRUNS scheme is used for pressure-velocity coupling in numerical calculation. The authors have tried to make an attempt to analyze the features of the complex flow field through flow streamlines and the vorticity contours. The impact of the gap between the cylinder and the plane wall upon flow pattern behavior is also studied. An intense interaction between vorticity in the boundary layer generated over the plane wall and the vorticity associated with the shear layer emanating from the separation points on the cylinder surface has been observed producing a very complex flow field in the cylinder wake and the gap between the cylinder and the wall.

A vector penalty-projection approach for the time-dependent incompressible magnetohydrodynamics flows

In this paper, we study a fully discrete vector penalty-projection method for the time-dependent incompressible magnetohydrodynamics flows. This fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization and first-order backward Euler for temporal discretization. Moreover, unconditionally energy stable is established, and error estimates for the fully discrete scheme is also derived. Finally, some numerical experiments are provided to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.

Time-dependent incompressible viscous flows around a rigid body: Estimates of spatial decay independent of boundary conditions

We consider the incompressible time-dependent Navier-Stokes system with Oseen term and terms arising in stability problems, in a 3D exterior domain. No boundary conditions are imposed. We consider L2-strong solutions, that is, the velocity u is an L∞-function in time and Lκ-integrable in space for some κ∈[1,3) and some κ∈(3,∞), the spatial gradient ∇xu is L2-integrable in space and in time, and the nonlinearity (u⋅∇x)u is L2-integrable in time and L3/2-integrable in space. It is shown that if the right-hand side of the equation and the initial data decay pointwise in space sufficiently fast, then u and ∇xu also decay pointwise in space, with rates which are higher than those provided by previous theories.

Evaluating the accuracy of hybrid finite element/particle-in-cell methods for modelling incompressible Stokes flow

SUMMARY Combining finite element methods for the incompressible Stokes equations with particle-in-cell methods is an important technique in computational geodynamics that has been widely applied in mantle convection, lithosphere dynamics and crustal-scale modelling. In these applications, particles are used to transport along properties of the medium such as the temperature, chemical compositions or other material properties; the particle methods are therefore used to reduce the advection equation to an ordinary differential equation for each particle, resulting in a problem that is simpler to solve than the original equation for which stabilization techniques are necessary to avoid oscillations. On the other hand, replacing field-based descriptions by quantities only defined at the locations of particles introduces numerical errors. These errors have previously been investigated, but a complete understanding from both the theoretical and practical sides was so far lacking. In addition, we are not aware of systematic guidance regarding the question of how many particles one needs to choose per mesh cell to achieve a certain accuracy. In this paper we modify two existing instantaneous benchmarks and present two new analytic benchmarks for time-dependent incompressible Stokes flow in order to compare the convergence rate and accuracy of various combinations of finite elements, particle advection and particle interpolation methods. Using these benchmarks, we find that in order to retain the optimal accuracy of the finite element formulation, one needs to use a sufficiently accurate particle interpolation algorithm. Additionally, we observe and explain that for our higher-order finite-element methods it is necessary to increase the number of particles per cell as the mesh resolution increases (i.e. as the grid cell size decreases) to avoid a reduction in convergence order. Our methods and results allow designing new particle-in-cell methods with specific convergence rates, and also provide guidance for the choice of common building blocks and parameters such as the number of particles per cell. In addition, our new time-dependent benchmark provides a simple test that can be used to compare different implementations, algorithms and for the assessment of new numerical methods for particle interpolation and advection. We provide a reference implementation of this benchmark in aspect (the ‘Advanced Solver for Problems in Earth’s ConvecTion’), an open source code for geodynamic modelling.

Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number

In this paper, we consider up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared using a high Reynolds number vortex dynamics simulation. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov–Galerkin (SUPG) method together with grad–div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulation of a high Reynolds number flow with vortical structures on relatively coarse grids are showcased, by focusing on a two-dimensional plane mixing-layer flow. Both Inf–Sup Stable (ISS) and Equal Order (EO) H1-conforming FE pairs are explored using a second-order semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies conducted, it is concluded that the SUPG method using EO–FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend the SUPG method by the higher order terms of the RB-VMS method.

Unsteady time-dependent incompressible Newtonian fluid flow between two parallel plates by homotopy analysis method (HAM), homotopy perturbation method (HPM) and collocation method (CM)

Analytical and numerical analyses have performed to study the problem of the flow of incompressible Newtonian fluid between two parallel plates approaching or receding from each other symmetrically. The Navier–Stokes equations have been transformed into an ordinary differential equation using a similarity transformation. The powerful analytical methods called collocation method (CM), the homotopy perturbation method (HPM), and the homotopy analysis method (HAM) have been used to solve nonlinear differential equations. It has been attempted to show the capabilities and wide-range applications of the proposed methods in comparison with a type of numerical analysis as fourth-order Runge–Kutta numerical method in solving this problem. Also, velocity fields have been computed and shown graphically for various values of physical parameters. The objective of the present work is to investigate the effect of Reynolds number and suction or injection characteristic parameter on the velocity field.

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, $Re$-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $\nabla u \in L^1(0,T;L^\infty(\Omega))$ which is discussed in detail. In the sense of best practice, we review and establish pressure- and $Re$-semi-robust estimates for pointwise divergence-free $H^1$-conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free $H$(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow

Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow

Divergence-Free H(div)-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics

In this article, we consider exactly divergence-free $H$(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free $H^1$-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented here whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, $H$(div)-FEM are applied to the simulation of full nonlinear Navier-Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin-Helmholtz instabilities and freely decaying two-dimensional turbulence.

A hybrid incremental projection method for thermal-hydraulics applications

A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya–Babuška–Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie–Chow interpolation or by using a Petrov–Galerkin finite element formulation. The use of a co-velocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advection method is shown to deliver second-order spatial convergence on mixed element topology meshes, and the implicit advective treatment significantly increases the stable time-step size. The algorithm is robust and extensible, permitting the incorporation of features such as porous media flow, RANS and LES turbulence models, and semi-/fully-implicit time stepping. A series of verification and validation problems are used to illustrate the convergence properties of the algorithm. The temporal stability properties are demonstrated on a range of problems with 2 ≤ C F L ≤ 100 . The new flow solver is built using the Hydra multiphysics toolkit. The Hydra toolkit is written in C++ and provides a rich suite of extensible and fully-parallel components that permit rapid application development, supports multiple discretization techniques, provides I/O interfaces, dynamic run-time load balancing and data migration, and interfaces to scalable popular linear solvers, e.g., in open-source packages such as HYPRE, PETSc, and Trilinos. • A new second-order hybrid finite-element/finite-volume projection algorithm for transient viscous flow has been introduced. • The hybrid discretization prevents pressure modes without using Rhie–Chow interpolation or a Petrov–Galerkin formulation. • A monotonicity-preserving advection method shown to deliver second-order accuracy on mixed element topology meshes. • Verification studies demonstrate hybrid projection solver accuracy and temporal stability for super-CFL conditions.

NUMERICAL IMPLEMENTATION OF THE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATION

In this paper, we briefly review and describe a projection algorithm for numerically computing the two-dimensional time-dependent incompressible Navier?Stokes equation. The projection method, which was originally introduced by Alexandre Chorin [A.J. Chorin, Numerical solution of the Navier?Stokes equations, Math. Comput., 22 (1968), pp. 745?762], is an effective numerical method for solving time-dependent incompressible fluid flow problems. The key advantage of the projection method is that we do not compute the momentum and the continuity equations at the same time, which is computationally difficult and costly. In the projection method, we compute an intermediate velocity vector field that is then projected onto divergence-free fields to recover the divergence-free velocity. Numerical solutions for flows inside a driven cavity are presented. We also provide the source code for the programs so that interested readers can modify the programs and adapt them for their own purposes.

유연한 경동맥 분지관에서 분지각이 혈액의 유동에 미치는 영향에 관한 연구

To investigate the effect of the flexible artery wall on the blood flow, three-dimensional numerical simulations were carried out for analyzing the time-dependent incompressible flows of Newtonian fluids constrained by a flexible wall. The Navier-Stokes equations for fluid flow were solved using the P2P1 Galerkin finite element method, and mesh movement was achieved using an arbitrary Lagrangian-Eulerian formulation. The Newmark method was employed for solving the dynamic equilibrium equations for the deformation of a linear elastic solid. To avoid complexity due to the necessity of additional mechanical constraints, we used a combined formulation that includes both the fluid and structure equations of motion to produce a single coupled variational equation. The results showed that the flexibility of the carotid wall significantly affects flow phenomena during the pulse cycle. The flow field was also found to be strongly influenced by the bifurcation angle.

An artificial compressibility based fractional step method for solving time dependent incompressible flow equations. Temporal accuracy and similarity with a monolithic method

In this note, an artificial compressibility based fractional step method is analysed against a monolithic scheme for solving incompressible flow equations. The artificial compressibility (AC) procedure presented in this paper is stabilized via a characteristic based split (CBS), and thus it is referred to as the AC-CBS method. The monolithic method used for comparison in the present study is the pressure stabilized Petrov–Galerkin (PSPG) method. It is shown that the AC-CBS and PSPG procedures are identical in structure, except for the stabilization parameters. For unsteady problems, a dual time stepping algorithm is employed in the AC-CBS scheme. Unlike classical fractional step methods, this dual time stepping mechanism circumvents the temporal pressure splitting error, and thus provides the anticipated temporal accuracy. The temporal accuracy of the AC-CBS method is demonstrated via a standard benchmark problem. Up to fourth order time accurate schemes are introduced for a thorough analysis of the AC-CBS scheme.

ORIENTATIONAL STRESS TENSOR OF POLYMER SOLUTION WITH APPLICATIONS TO BLOOD FLOW

Blood circulating efficiently inside the veins and arteries, provides essential nutrients and oxygen to tissues and organs in the entire body. To highlight the fundamental properties of blood and gain insight into the regularizing effect of various formulations, we need to develop mathematical models. In order to do this, first we present the polymer dynamics in terms of an ensemble of Hookean dumbbells with Brownian configuration fields to derive the orientation stress tensor. Then, we describe the continuity and the momentum equations for time-dependent incompressible flow and the Oldroyd-B model. Finally, we present our numerical analysis of the model and give the effect of the orientation stress tensor in a blood vessel. The development of mathematical models and numerical procedures for the approximation of the blood specific flow equations have enabled us to understand the complex behaviors of blood flows inside the vessel.

Assessment of pressure field calculations from particle image velocimetry measurements

This paper explores the challenges associated with the determination of in-field pressure from DPIV (digital particle image velocimetry)-measured planar velocity fields for time-dependent incompressible flows. Several methods that have been previously explored in the literature are compared, including direct integration of the pressure gradients and solution of different forms of the pressure Poisson equations. Their dependence on grid resolution, sampling rate, velocity measurement error levels and off-axis recording was quantified using artificial data of two ideal sample flow fields—a decaying vortex flow and pulsatile flow between two parallel plates, and real DPIV and pressure data from oscillating flow through a diffuser. The need for special attention to mitigate the velocity error propagation in the pressure estimation is also addressed using a physics-preserving approach based on proper orthogonal decomposition (POD). The results demonstrate that there is no unique or optimum method for estimating the pressure field and the resulting error will depend highly on the type of the flow. However, the virtual boundary, omni-directional pressure integration scheme first proposed by Liu and Katz (2006 Exp. Fluids 41 227–40) performed consistently well in both synthetic and experimental flows. Estimated errors can vary from less than 1% to over 100% with respect to the expected value, though in contrast to more traditional smoothing algorithms, the newly proposed POD-based filtering approach can reduce errors for a given set of conditions by an order of magnitude or more. This analysis offers valuable insight that allows optimizing the choice of methods and parameters based on the flow under consideration.

New stabilized finite element method for time‐dependent incompressible flow problems

AbstractA new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P1−P0 and Q1−P0 elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.