Vorticity Variables Research Articles

Boundary element method for natural convection in non-Newtonian fluid saturated square porous cavity

The main purpose of this work is to present the use of the Boundary Element Method (BEM) in the analysis of the natural convection in the square porous cavity saturated by the non-Newtonian fluid. The results of hydrodynamic and heat transfer evaluations are reported for the configuration in which the enclosure is heated from a side wall while the horizontal walls are insulated. The flow in the porous medium is modelled using the modified Brinkman extended Darcy model taking into account the non-Darcy viscous effects. The governing equations are transformed by the velocity–vorticity variables formulation enabling the computation scheme to be partitioned into kinematic and kinetic parts. To analyse the effects of the available non-Newtonian viscosity and to evaluate the presented approach, the power law model for shear thinning fluids ( n<1), for shear thickening fluids ( n>1) and in the limit for the Newtonian fluids ( n=1) is considered. Numerical model is tested also for the Carreau model adequate for many non-Newtonian fluids. Solutions for the flow and temperature fields and Nusselt numbers are obtained in terms of a modified Rayleigh number Ra ∗, Darcy number Da, and the non-Newtonian model parameters. The agreement between the results obtained with finite difference method is very good indicating that BEM can be efficiently used for solving transport phenomena in saturated porous medium.

Modeling of Rapidly Rotating Thermal Convection Using Vorticity and Vector Potential

We have used a numerical scheme based on higher-order finite differences to investigate effects of adiabatic heating and viscous dissipation on 3-D rapidly rotating thermal convection in a Cartesian box with an aspect-ratio of 2⊗2⊗1. Although we omitted coupling with the magnetic field, which can play a key role in the dynamics of the Earth's core, the understanding of non-linear rotating convection including realistic thermodynamic effects is a necessary prerequisite for understanding the full complexity of the Earth's core dynamics. The system of coupled partial differential equations has been solved in terms of the principal variables vorticity ω, vector potential A and temperature T. The use of the vector potential A allows the velocity field to be calculated with one spatial differentiation in contrast to the spheroidal and toroidal function approach. The temporal evolution is governed by a coupled time-dependent system consisting of ω and T. The equations are discretized in all directions by using an eighth-order, variable spaced scheme. Rayleigh number Ra of 106, Taylor number Ta of 108 and a Prandtl number Pr of 1 have been employed. The dissipation number of the outer core was taken to be 0.2. A stretched grid has been employed in the vertical direction for resolving the thin shear boundary layers at the top and bottom. This vertical resolution corresponds to around 240 regularly spaced points with an eighth-order accuracy. For the regime appropriate to the Earth's outer core, the dimensionless surface temperature T 0 takes a large value, around 4. This large value in the adiabatic heating/cooling term is found to cause stabilization of both the temperature and velocity fields.

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.

Vorticity dynamics of dilute two-way-coupled particle-laden mixing layers

The two-way coupling mechanisms in particle-laden mixing layers are investigated, with and without particle settling, and with an emphasis on the resulting modifications to the fluid vorticity field. The governing equations are interpreted with respect to the production and cancellation of vorticity. These mechanisms are shown to be related to the misalignment of the concentration gradient and the slip velocity, as well as to the difference in fluid and particle vorticities. Preliminary insight into the physics is obtained from an analysis of the unidirectional base flow. For this model problem, the conditions are established under which the particle velocity remains a single-valued function of space for all times. The resulting simplified set of two-way-coupled equations governing the vorticity of the fluid and particulate phases, respectively, is solved numerically. The formation of a decaying travelling wave solution is demonstrated over a wide range of parameters. Interestingly, the downward propagation of the fluid vorticity field is not accomplished through convection, but rather by the production and loss of vorticity on opposite sides of the mixing layer. For moderate settling velocities, the simulation results reveal an optimal coupling mechanism between the fluid and particle vorticities at intermediate values of the mass loading parameter. For large settling velocities and intermediate mass loadings, more than one local maximum is seen to evolve in the vorticity field. A scaling law for the downward propagation rates of the vorticity fronts is derived.Two-dimensional particle-laden mixing layers are investigated by means of a mixed Lagrangian–Eulerian approach which is based on the vorticity variable. For uniformly seeded mixing layers, the simulations confirm some of the features observed by Druzhinin (1995b) for the model problem of a two-way-coupled particle-laden Stuart vortex, as well as by Dimas &amp; Kiger (1998) in a linear stability analysis. For small values of the Stokes number, a mild destabilization of the mixing layer is observed. At moderate and large Stokes numbers, on the other hand, the transport of vorticity from the braids into the core of the evolving Kelvin–Helmholtz vortices is seen to be slowed by the two-way coupling effects. As a result, the particle ejection from the vortex cores is weakened. For constant mass loadings, the two-way coupling effects are strongest at intermediate Stokes number values. For moderately large Stokes numbers, the formation of two bands of high particle concentration is observed in the braids, which reflects the multi-valued nature of the particle velocity field. For mixing layers in which only one stream is seeded, the particle concentration gradient across the mixing layer leads to strong vorticity production and loss, which results in an effective net motion of the vortex in the flow direction of the seeded stream. Under particle settling, the vortex propagates downward as well. For the parameter range explored here, its settling velocity agrees well with the scaling law derived from the unidirectional flow analysis.

Solution of the Navier-Stokes equations in velocity-vorticity form using a Eulerian-Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection-dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two-dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.

Steady incompressible fluid flow over a bundle of cylinders at moderate reynolds numbers

AbstractThe complete Navier‐Stokes equations describing the steady flow of incompressible Newtonian fluids normal to an array of long cylinders have been solved numerically using the finite difference method in terms of the stream function and vorticity variables. The inter‐cylinder interactions have been mimicked using the well known free surface cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, stagnation pressure, stream line and iso‐vorticity line plots, as well as on the values of the integral quantities, such as pressure, friction and total drag coefficients, have been obtained under wide ranges of conditions as follows: 0.3 ≤ ϵ 0.99 and 0.01 ≤ Re ≤ 100. The numerical results presented herein have been validated using the appropriate theoretical and experimental results available in the literature; the match between the present predictions and the scant experimental results is good.

Vorticity-velocity formulations of the Stokes problem in 3D

This work studies the three-dimensional Stokes problem expressed in terms of vorticity and velocity variables. We make general assumptions on the regularity and the topological structure of the flow domain: the boundary is Lipschitz and possibly non-connected and the flow domain may be multiply connected. Upon introducing a new variational space for the vorticity, five weak formulations of the Stokes problem are obtained. All the formulations are shown to lead to well-posed problems and to be equivalent to the primitive variable formulation. The various formulations are discussed by interpreting the test functions for the vorticity (resp. velocity) equation as vector potentials for the velocity (resp. vorticity). Of the five sets of boundary conditions derived in the paper, three are already known, but only for domains with a trivial topological structure, while the remaining two are new. Copyright © 1999 John Wiley & Sons, Ltd.

多波長マイクロ波放射計データの直接的データ同化とTOGA COARE領域でのメソ数値予報モデルへのインパクトについて

Direct assimilation using 1-dimensional variational method in the vertical (1D-VAR) was developed to incorporate vertically polarized brightness temperatures (TB's), and rain flag data (index of existence of precipitation) from the special sensor microwave imager (SSM/I), into a mesoscale numerical weather prediction (NWP) model. We used a radiative transfer model (RTM) developed by Liu (1998) to calculate TB's from NWP model variables. Observational residuals of TB's are assumed to be nonlinear functions of precipitation rates in rainy areas, and of other thermodynamic variables in rain-free areas. Quasiequilibrium assumptions on humidity and convective instability in model precipitation areas are used to assimilate TB's in rainy areas and rain flag data, which are functions of precipitation. TB's in rain-free areas are assimilated into total water content (cloud water content + humidity). Smith's method (1990) is used to calculate cloud water content and humidity from total water content. To simplify 1D-VAR, the following approximations are introduced: 1) In the tropics, variations in temperature are much smaller than variations in humidity. 2) Variations in divergence dominates the rainy areas, compared to vorticity and mass variables. Based on the above approximation, we assumed that the background error covariance of relative vorticity and the unbalanced component of mass variables is negligible, so we divided 1D-VAR into that for total water content and that for divergence. Newtonian iteration is used to solve these 1D-VAR problems. To study the assimilated variables, assimilation was applied to cases during 19-20 Dec. 1992 over the Tropical Ocean Global Atmosphere Coupled Ocean-Atmosphere Response Experiment (TOGA COARE) domain. Results indicate: 1) Precipitable water content (PWC) obtained by assimilation agreed well with PWC calculated from radiosonde data and PWC retrieved from TB's using Shibata's algorithm (1994). 2) Consistency between humidity and cloud water content profiles was attained by Smith's method (1990), although no significant improvement was seen in humidity profile accuracy by assimilation. 3) Assimilation of rain flag and TB's in rainy areas successfully produced model precipitation areas agreeing with radar observation data by Short et al. (1997). To test the impact of assimilation on NWP forecasts, experiments were conducted assimilating TB data for 19 Dec. 1992. Results indicate: 1) Assimilation modified large-scale model humidity distribution, improving large-scale humidity and precipitation forecasts for 48 hours. 2) Assimilation of rain flag and TB's in rainy areas reduced spin-up error of precipitation and positional error of mesoscale precipitation patterns. 3) The boundary between subtropical northeasterly, and equatorial weak wind areas, shifted with the change in model precipitation areas by assimilation.

A vorticity-velocity variational formulation for the exterior stokes problem in weighted sobolev spaces

In this article, we present a mixed variational formulation for the exterior Stokes problem in terms of the velocity and vorticity variables in the two and three-dimensional cases. The existence and the uniqueness of solutions of the variational problem are proved, as well as the equivalence with the primitive variables formulation.

Least-squares methods for Stokes equations based on a discrete minus one inner product

The purpose of this paper is to develop and analyze least-squares approximations for Stokes and elasticity problems. The major advantage of the least-squares formulation is that it does not require that the classical Ladyzhenskaya-Babǔska-Brezzi (LBB) condition be satisfied. We provide two methods. The first is posed in terms of the velocity-pressure pair without the introduction of additional variables. The second adds a vorticity variable. In both cases, we employ least-squares functionals which involve a discrete inner product which is related to the inner product in H −1 (Ω) (the Sobolev space of order minus one on Ω). The use of such inner products (applied to second order problems) was proposed in an earlier paper by Bramble, Lazarov and Pasciak (1994).

On the potential of assimilating meteorological analyses in a global climate model for the purpose of model validation

A simple four‐dimensional assimilation technique, called Newtonian relaxation, has been applied to the Hamburg climate model (ECHAM), to enable comparison of model output with observations for short periods of time. The prognostic model variables vorticity, divergence, temperature, and surface pressure have been relaxed toward European Center for Medium‐Range Weather Forecasts (ECMWF) global meteorological analyses. Several experiments have been carried out, in which the values of the relaxation coefficients have been varied to find out which values are most usable for our purpose. To be able to use the method for validation of model physics or chemistry, good agreement of the model simulated mass and wind field is required. In addition, the model physics should not be disturbed too strongly by the relaxation forcing itself. Both aspects have been investigated. Good agreement with basic observed quantities, like wind, temperature, and pressure is obtained for most simulations in the extratropics. Derived variables, like precipitation and evaporation, have been compared with ECMWF forecasts and observations. Agreement for these variables is smaller than for the basic observed quantities. Nevertheless, considerable improvement is obtained relative to a control run without assimilation. Differences between tropics and extratropics are smaller than for the basic observed quantities. Results also show that precipitation and evaporation are affected by a sort of continuous spin‐up which is introduced by the relaxation: the bias (ECMWF‐ECHAM) is increasing with increasing relaxation forcing. In agreement with this result we found that with increasing relaxation forcing the vertical exchange of tracers by turbulent boundary layer mixing and, in a lesser extent, by convection, is reduced.

Vortex simulation of axisymmetrical flows in cylindrical geometries. Part I: Numerical algorithm

The Random Vortex Method extended to an axisymmetrical flow is used in the study of the flow field inside circular geometries characterized by an axis of symmetry. The method uses a formulation in vorticity variables expressed in cylindrical coordinates with the assumption of an axisymmetrical flow. The algorithm uses a semi-Lagrangian approach for the simulation of the vortex dynamics. In this paper, the algorithm of the computational method, along with a description of the method of vorticity generation on solid walls, are presented in detail.

Steady flow with separation past a thin airfoil at high Reynolds numbers

The results of calculating the steady separated flow of a viscous incompressible fluid past a 12% sinusoidal airfoil with and without a splitter plate are presented. The Navier—Stokes equations written in stream-function—vorticity variables are approximated in accordance with a central difference scheme and solved simultaneously by means of the direct method. The results obtained are analyzed from the standpoint of their consistency with the asymptotic theory based on the well-known three-deck scheme.

Numerical simulation and physical analysis of high Reynolds number recirculating flows behind sudden expansions

This work presents the results of numerical simulations of unsteady recirculating flows at high Reynolds number. The two geometries investigated are a two-dimensional channel that incorporates a sudden expansion in the form of a single backward-facing step and a two-dimensional channel that incorporates a sudden expansion in the form of a double symmetrical backward-facing step. The random vortex method (RVM) is used in this study. This grid-free Lagrangian method solves the unsteady, incompressible Navier–Stokes equations and the continuity equation, with the appropriate physical boundary conditions, using a formulation in vorticity variables. In order to show the ability of the RVM an extensive set of numerical results is presented and compared with experimental results from the literature. In particular, the dissymmetrical behavior of the flow in the double expansion channel, as observed experimentally, is simulated accurately. Frequency analyses and autocorrelation analyses show that the flows are characterized by dominant frequencies and turbulent length scales that are function of the position inside the channels. Those frequencies and turbulent length scales are related to the dynamics of the flow fields.

Finite Element Solution of the 3D Compressible Navier-Stokes Equations by a Velocity-Vorticity Method

This paper presents an alternative formulation to the primitive variable form of the Navier-Stokes equations. The approach is based on the use of the velocity and vorticity variables as a logical extension of the stream function-vorticity method which is quite effective for two-dimensional flows. Second-order equations are obtained for the variables and discretization is through a weak-Galerkin finite element method, followed by Newton linearization. The velocity and vorticity components are solved simultaneously by a direct solver. The scheme is demonstrated for two-dimensional and three-dimensional incompressible and subsonic internal enclosed flow problems.

A multigrid method for solution of vorticity—velocity form of 3‐D navier—stokes equations

AbstractA multigrid method has been developed and applied to the simulation of unsteady three‐dimensional incompressible flow, using the Navier—Stokes equations in terms of vorticity and velocity variables. A system of equations for the velocity problem, consisting of the continuity equation and three curl equations, is first formulated as a uniquely determined, non‐singular system. An efficient and robust technique, namely the multigrid distributive Gauss—Seidel scheme, is developed for solution of this system of algebraic equations. The method is applied to study 3‐D flow in a shear‐driven cavity.

A multigrid solver for the vorticity-velocity Navier-Stokes equations

This paper provides a multigrid incremental line-Gauss-Seidel method for solving the steady Navier-Stokes equations in two and three dimensions expressed in terms of the vorticity and velocity variables. The system of parabolic and Poisson equations governing the scalar components of the vector unknowns is solved using centred finite differences on a non-staggered grid. Numerical results for the two-dimensional driven cavity problem indicate that the spatial discretization of the equation defining the value of the vorticity on the boundary is extremely critical to obtaining accurate solutions. In fact, a standard one-sided three-point second-order-accurate approximation produces very inaccurate results for moderate-to-high values of the Reynolds number unless an exceedingly fine mesh is employed. On the other hand, a compact two-point second-order-accurate discretization is found to be always satisfactory and provides accurate solutions for Reynolds number up to 3200, a target impossible heretofore using this formulation and a non-staggered grid.

A numerical case study of a non‐Newtonian flow problem

AbstractThis paper addresses some of the theoretical aspects involved in the numerical study of non‐Newtonian flow problems. We consider the second‐order Rivlin–Erickson constitutive model due to the simple differential form that emerges for the system of equations that govern the flow when expressed in stream function–vorticity variables. This model describes slightly elastic fluids that exhibit a constant viscosity behaviour. A steady two‐dimensional flow is studied through a planar contraction geometry.An auxiliary variable is introduced into the problem formulation producing a non‐linear system of differential equations comprising two elliptic equations and one hyperbolic equation. This system is discretized by finite difference methods and the resulting system of non‐linear algebraic equations is solved iteratively by successive substitutions. The simple structure of this iteration permits a convergence analysis which is presented in Section 2. This analysis is performed prior to the spatial discretization and establishes the dependence of the iteration upon the material parameters. At the discrete linearized equation level a combination of inner iterations for elliptic equations and direct marching for the hyperbolic equation is used. The stability of the marching scheme is considered in Section 4.3 and a discussion on the results is given in Section 5.

Neutral-surface potential vorticity

The use of potential density in ocean models does not overcome all the annoying problems caused by the compressible nature of seawater. For example, the reduced gravity for layered ocean models is not proportional to the difference of potential density across an interface. The appropriate reduced gravity is derived and it is shown to be relatively easy to incorporate in future layered models. Potential-density surfaces are commonly inclined with respect to neutral surfaces and so lateral motion along neutral surfaces gives rise to unwanted diapycnal velocities that are unrelated to vertical mixing processes. This unphysical part of the diapycnal motion is quantified, emphasizing the need for performing ocean-circulation studies in the neutral-surface framework. If there were no variations of θ and S along neutral surfaces in the ocean, then lateral mixing would occur along potential-density surfaces. A perfectly respectable potential vorticity variable would then be isopycnal potential vorticity, IPV, defined by -gf[∂ρ θ/∂z]/ρ θ. However, there is mixing in the ocean, and neutral surfaces do not coincide with potential density surfaces. Since the efficient lateral mixing processes occur along neutral trajectories, and vertical velocities through neutral surfaces occur only in response to mixing processes of a vertical nature (including the two dianeutral advection processes, thermobaricity and cabbeling), it is logical to seek a potential vorticity variable that is proportional to f/h, where h is the vertical distance between adjacent neutral surfaces. This form of potential vorticity is here referred to as “neutral-surface potential vorticity”, NSPV. Its epineutral variations can be significantly different to those of the two other commonly used potential vorticity variables, IPV, and fN 2. The difference between NSPV and fN 2 is shown to result from the path-dependence inherent in integrating N 2 in the vertical plane between two neutral trajectories. The β-spiral equations are derived for a fully non-linear equation of state, and neutral surfaces emerge as the dynamically relevant surfaces for these studies. It is also shown that IPV is not conserved on potential-density surfaces because of the non-linear nature of the equation of state; this problem does not arise in the neutral tangent plane. Mixing processes cause scalar properties to change at different rates on a potential-density surface than in a neutral tangent plane. In particular, potential density itself is mixed and advected by epineutral processes, thereby complicating the interpretation of inversemodels that use potential density surfaces. The dynamical equations governing the ocean circulation take simple and exact forms with respect to the neutral tangent plane at eaach point, but when an interface or surface must be formed over an extended geographical region, the ambiguity associated with defining a neutral surface raises its ugly head. Here it is shown that the approximately neutral surfaces of McDougall and Jackett (1988) are the most appropriate surfaces for this purpose.

Neutral surfaces in the ocean: Implications for modelling

Surfaces of neutral static stability (neutral surfaces) along which lateral ocean mixing occurs, can be quite different to potential density surfaces. For example, the lateral gradients of potential temperature may differ in the two surfaces by up to a factor of four in some extreme but important regions. Vertical mixing processes cause vertical upwelling velocities that can be modelled as vortex‐stretching terms in layered models of ocean circulation. However, since potential density surfaces are commonly inclined with respect to neutral surfaces, lateral motion along neutral surfaces also gives rise to unwanted vertical motion through isopycnals that is not related to vertical mixing processes.When considering the conservation of potential vorticity in the real ocean, the differences between the depths of neutral surfaces and potential density surfaces becomes quite important. Neutral surface potential vorticity (NSPV), is introduced, defined to be proportional to f/h, where f is the Coriolis parameter and h is the vertical distance between adjacent neutral surfaces. NSPV is often significantly different (by factors of 2 or 3) to the two other commonly used potential vorticity variables, isopycnal potential vorticity, IPV, and fN².