A locally conservative,discontinuous least‐squares finite element method for the Stokes equations

Conventional least‐squares finite element methods (LSFEMs) for incompressible flows conserve mass only approximately. For some problems, mass loss levels are large and result in unphysical solutions. In this paper we formulate a new, locally conservative LSFEM for the Stokes equations wherein a discrete velocity field is computed that is point‐wise divergence free on each element. The central idea is to allow discontinuous velocity approximations and then to define the velocity field on each element using a local stream‐function. The effect of the new LSFEM approach on improved local and global mass conservation is compared with a conventional LSFEM for the Stokes equations employing standard C⁰ Lagrangian elements. Copyright © 2011 John Wiley & Sons, Ltd.     

A locally DIV‐free projection scheme for incompressible flows based on non‐conforming finite elements

We present a projection scheme whose end‐of‐step velocity is locally pointwise divergence free, using a continuous ?₁ approximation for the velocity in the momentum equation, a first‐order Crouzeix–Raviart approximation at the projection step, and a ?₀ approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming ?₁ interpolant for the velocity. Optimal estimates for the velocity error in L²‐ and H¹‐norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous ?₁ approximation for the velocity locks. Since the end‐of‐step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright © 2005 John Wiley & Sons, Ltd.     

Unsteady flow and heat transfer past an axisymmetric permeable shrinking sheet with radiation effect

In this paper, the problem of unsteady axisymmetric boundary layer flow and heat transfer induced by a permeable shrinking sheet in the presence of radiation effect is studied. The transformed boundary layer equations are solved numerically by an implicit finite‐difference scheme known as the Keller‐box method. The influence of radiation, unsteadiness and mass suction parameters on the reduced skin friction coefficient f′′(0) and the heat transfer coefficient ?θ′(0), as well as the velocity and temperature profiles are presented and discussed in detail. It is found that dual solutions exist and suction parameter delays the separation of boundary layer. Copyright © 2010 John Wiley & Sons, Ltd.     

Impact of mass lumping on gravity and Rossby waves in 2D finite‐element shallow‐water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite‐element velocity/surface‐elevation pairs that are used to approximate the linear shallow‐water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P₀?P₁, RT₀ and P?P₁ pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results. Copyright © 2008 John Wiley & Sons, Ltd.     

A non‐conforming least‐squares finite element method for incompressible fluid flow problems

In this paper, we develop least‐squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity–vorticity–pressure Stokes system, which uses a piecewise divergence‐free basis for the velocity and standard C⁰ elements for the vorticity and the pressure. The new method, which we term dV‐VP improves upon our previous discontinuous stream‐function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second‐order terms from the least‐squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one‐half of the dimension of a stream‐function element of equal accuracy. In two dimensions, the discontinuous stream‐function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV‐VP formulation, which significantly reduces the condition number of the LSFEM problem. Published 2012. This article is a US Government work and is in the public domain in the USA.     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element study of free surface turbulent flow in rotating channels

Under certain conditions of liquid flow through rotating channels, the Coriolis force can induce a free surface to be formed. This problem is of practical importance in a Coriolis wear tester, which is used for determining the sliding wear coefficient of wear materials in slurry handling equipment. A deforming Galerkin finite element method is presented for predicting two‐dimensional turbulent free surface mean flow in rotating channels. Reynolds‐averaged Navier–Stokes (RANS) equations are cast into weak(algebraic) form using primitive variables (velocity and pressure). Eddy viscosity is determined via a mixing length model. Velocity is interpolated biquadratically, while pressure is interpolated bilinearly. The kinematic condition is used to form the Galerkin residual for the free surface. The free surface is represented by Hermite polynomials of zeroeth order for continuity of position and slope. Combined Newton's iteration is used to simultaneously solve for the free surface and the field variables. Results of velocity and pressure fields, as well as the free surface are shown to converge with mesh‐size refinement. There is excellent respect for mass conservation. Results are presented for various values of Rossby number (Ro) and height‐based Reynolds number (Re_H). Parameter continuation in Ro and Re_H space is used to compute solutions at higher values of flow rate and angular velocity. Copyright © 2002 John Wiley & Sons, Ltd.     

An evaluation of a 3D free‐energy‐based lattice Boltzmann model for multiphase flows with large density ratio

In this paper, the 3D Navier–Stokes (N–S) equation and Cahn–Hilliard (C–H) equations were solved using a free‐energy‐based lattice Boltzmann (LB) model. In this model, a LB equation with a D3Q19 velocity model is used to recover continuity and N–S equations while another LB equation with D3Q7 velocity model for solving C–H equation (Int. J. Numer. Meth. Fluids, 2008; 56 :1653–1671) is applied to solve the 3D C–H equation. To avoid the excessive use of computational resources, a moving reference frame is adopted to allow long‐time simulation of a bubble rising. How to handle the inlet/outlet and moving‐wall boundary conditions are suggested. These boundary conditions are simple and easy for implementation. This model's performance on two‐phase flows was investigated and the mass conservation of this model was evaluated. The model is validated by its application to simulate the 3D air bubble rising in viscous liquid (density ratio is 1000). Good agreement was obtained between the present numerical results and experimental results when Re is small. However, for high‐Re cases, the mass conservation seems not so good as the low‐Re case. Copyright © 2009 John Wiley & Sons, Ltd.     

Axisymmetric stagnation flow obliquely impinging on a moving circular cylinder with uniform transpiration

Laminar stagnation flow, axisymmetrically yet obliquely impinging on a moving circular cylinder, is formulated as an exact solution of the Navier–Stokes equations. Axial velocity is time‐dependent, whereas the surface transpiration is uniform and steady. The impinging free stream is steady with a strain rate k?. The governing parameters are the stagnation‐flow Reynolds number Re=k?a²/2ν, and the dimensionless transpiration S=U₀/k?a. An exact solution is obtained by reducing the Navier–Stokes equations to a system of differential equations governed by Reynolds number and the dimensionless wall transpiration rate, S. The system of Boundary Value Problems is then solved by the shooting method and by deploying a finite difference scheme as a semi‐similar solution. The results are presented for velocity similarity functions, axial shear stress and stream functions for a variety of cases. Shear stresses in all cases increase with the increase in Reynolds number and suction rate. The effect of different parameters on the deflection of viscous stagnation circle is also determined. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

This paper presents results on the combined effect of thermo‐solutal buoyancy forces on the recirculatory flow behavior in a horizontal channel with backward‐facing step and the ensuing impact on heat and mass transfer phenomena. The governing equations for double diffusive mixed convection are represented in velocity–vorticity form of momentum equations, velocity Poisson equations, energy and concentration equations. Galerkin's finite‐element method has been employed to solve the governing equations. Recirculatory flow fields with heat and mass transfer are simulated for opposing and aiding thermo‐solutal buoyancy forces by assuming suitable boundary conditions for energy and concentration equations. The effect of Richardson number (0.1?Ri?10) and buoyancy ratio (?10?N?10) on the recirculation bubble and Nusselt and Sherwood numbers are studied in detail. For Richardson number greater than unity, distinct variations in the gradients of Nusselt number and Sherwood number with buoyancy ratio are observed for flow regimes with opposing and aiding buoyancy forces. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of the squeezing flow of dusty fluids

The current investigation deals with the study of the effect of introducing a small fraction of dust, by volume, to the fluid in a squeeze film on the viscous resistance to a steady moving disc. Expressions are obtained for the fluid-phase and the dust-phase velocity distributions and the dust particle number density. Analysis based on an iterative procedure indicates that the resistance to motion experienced by the moving disc increases due to the presence of dust.Nomenclature A arbitrary function of integration - B bulk concentration - F resistance to motion experienced by the disc (dusty fluid case) - F _c resistance to motion experienced by the disc (clean fluid case) - F* difference in resistance between the clean fluid and dusty fluid films - f mass concentration - h thickness of the squeeze film - K Stokes coefficient of resistance - m mass of a single dust particle - fluid viscosity coefficient - N dust particles number density - N ₀ dust particles number density at r=R - n iteration level - p fluid pressure in the squeeze film - P pressure in the surrounding - R radius of the disc - fluid density - (r, , y) cylindrical coordinates - t time - U fluid-phase velocity vector - V dust-phase velocity vector - ₁ fluid-phase radial velocity component - U ₂ dust-phase radial velocity component     

Simulation of shockwave propagation with a thermal lattice Boltzmann model

A two‐dimensional 19‐velocity (D2Q19) lattice Boltzmann model which satisfies the conservation laws governing the macroscopic and microscopic mass, momentum and energy with local equilibrium distribution order O(u⁴) rather than the usual O(u³) has been developed. This model is applied to simulate the reflection of shockwaves on the surface of a triangular obstacle. Good qualitative agreement between the numerical predictions and experimental measurements is obtained. As the model contains the higher‐order terms in the local equilibrium distribution, it performs much better in terms of numerical accuracy and stability than the earlier 13‐velocity models with the local equilibrium distribution accurate only up to the second order in the velocity u. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent impinging jet on a rotating disk

The calculations of quasi‐three‐dimensional momentum equations were carried out to study the influence of wall rotation on the characteristics of an impinging jet. The pressure coefficient, the mean velocity distributions and the components of Reynolds stress are calculated. The flow is assumed to be steady, incompressible and turbulent. The finite volume scheme is used to solve the continuity equation, momentum equations and k–ε model equations. The flow characteristics were studied by varying rotation speed ω for 0?ω?167.6 rad/s, the distance from nozzle to disk (H/d) was (3, 5, 8 and 10) and the Reynolds number Re base on V_J and d was 1.45 × 10⁴. The results showed that, the radial velocity and turbulence intensity increase by increasing the rotation speed and decrease in the impingement zone as nozzle to disk spacing increases. When the centrifugal force increases, the radial normal stresses and shear stresses increase. The location of maximum radial velocity decreases as the local velocity ratio (α) increases. The pressure coefficient depends on the centrifugal force and it decreases as the distance from nozzle to plate increases. In impingement zone and radial wall jet, the spread of flow increases as the angular velocity decreases The numerical results give good agreement with the experiment data of Minagawa and Obi (Int. J. of Heat and Fluid Flow 2004; 25 :759–766). Copyright © 2006 John Wiley & Sons, Ltd.     

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

A 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.     

Shoreline tracking and implicit source terms for a well balanced inundation model

The HyFlux2 model has been developed to simulate severe inundation scenario due to dam break, flash flood and tsunami‐wave run‐up. The model solves the conservative form of the two‐dimensional shallow water equations using the finite volume method. The interface flux is computed by a Flux Vector Splitting method for shallow water equations based on a Godunov‐type approach. A second‐order scheme is applied to the water surface level and velocity, providing results with high accuracy and assuring the balance between fluxes and sources also for complex bathymetry and topography. Physical models are included to deal with bottom steps and shorelines. The second‐order scheme together with the shoreline‐tracking method and the implicit source term treatment makes the model well balanced in respect to mass and momentum conservation laws, providing reliable and robust results. The developed model is validated in this paper with a 2D numerical test case and with the Okushiri tsunami run up problem. It is shown that the HyFlux2 model is able to model inundation problems, with a satisfactory prediction of the major flow characteristics such as water depth, water velocity, flood extent, and flood‐wave arrival time. The results provided by the model are of great importance for the risk assessment and management. Copyright © 2009 John Wiley & Sons, Ltd.     

Benchmark computations of diffuse interface models for two‐dimensional bubble dynamics

Diffuse interface models for incompressible two‐phase flow with large density ratios are tested on benchmark configurations for a two‐dimensional bubble rising in liquid columns. The benchmark quantities circularity, center of mass, and mean rise velocity are compared with reference solutions from Hysing et al. Copyright © 2011 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical studies of slow viscous rotating flow past a sphere. III

The flow of steady incompressible viscous fluid rotating about the z-axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Re_ω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non-linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’. The results obtained by solving the non-linear partial differential equations are compared with the results obtained by linearizing the equations for very small values of Re_ω. Streamlines are plotted for Ψ = 0·05, 0·2, 0·5 for both linear and non-linear cases. The magnitude of the vorticity vector near the body, i.e. at z = 0·2, is plotted for Re_ω = 0·05, 0·24, 0·5. The correction to the Stokes drag as a result of rotation of the fluid is calculated.     

Role of the inclination of an inverted‐V upper plate on the heat and flow behavior of trapped gases in a modified Rayleigh–Bénard cavity

Thermal buoyant air inside a modified Rayleigh–Bénard (RB) cavity bounded by a lower flat plate and an inverted‐V upper plate has been investigated numerically using the finite‐volume method. The second‐order‐accurate QUICK and SIMPLE schemes were used for the discretization of the convective terms and the pressure–velocity coupling in the set of conservation equations, respectively. The problem under study is controlled by two parameters: (1) the Rayleigh number ranging from 10³ to 10⁶ and (2) the relative height of the vertical sidewalls d. In reference to the latter, it varies from one limiting case corresponding to the standard RB cavity (a rectangle with d = 1) to another limiting case represented by an isosceles triangular cavity where d = 0. The numerical results for the velocity and temperature fields are presented in terms of streamlines, isotherms, local and mean heat fluxes. An additional effort was devoted to determine the critical Ra values characterizing the transition from symmetrical to asymmetrical buoyant airflow responsive to incremental changes in Ra. For purposes of engineering design, a general correlation equation for the Nusselt number in terms of the pertinent Ra and d was constructed using nonlinear multiple regression theory. Copyright © 2007 John Wiley & Sons, Ltd.     

Measurements of turbulent inclined plane dual jets

Measurements of mean velocities, flow direction, velocity fluctuations and Reynolds shear stress were made with a split film probe of hot wire anemometer to investigate the interactions created by two air jets issuing from two identical plane inclined nozzles. The reverse flow was detected by using the split film probe and observed by flow visualization. Experimental results with an inclined angle of 9° are presented in the paper. Some experimental results with an inclined angle of 27° are presented to investigate the effect of inclination on the flow field.Mean velocities approach self-preservation in both the converging region and the combining region. Velocity fluctuations and Reynolds shear stress approach self-preservation in the combining region only. The spreads of jet and the square of the decay of maximum mean velocity increase linearly as the distance from the nozzle exit increases.List of symbols D nozzle width - h nozzle height - J momentum of jet - J ₀ momentum of jet at nozzle exit - M mass flow rate - M ₀ mass flow rate at nozzle exit - S nozzle spacing - U, V mean velocities in the X and Y axis respectively - U _m maximum axial velocity - U ₀ axial velocity at nozzle exit - u, v velocity fluctuations in the X and Y axis respectively - u, v r.m.s. of u and v - Reynolds shear stress - X, Y Cartesian coordinates - X _m , Y _m coordinates at the location of maximum axial velocity - y _0.5 distance from the location of maximum axial velocity to the location where the velocity is half of maximum axial velocity - inclined angle - yY/S - Reynolds stress correlation coefficient - C.P combining point - max maximum value - M.P merging point - o nozzle exit plane - V.C vortex center