Primitive Variable Formulation Research Articles

On pressure boundary conditions for thermoconvective problems

AbstractSolving numerically hydrodynamical problems of incompressible fluids raises the question of handling first order derivatives (those of pressure) in a closed container and determining its boundary conditions. A way to avoid the first point is to derive a Poisson equation for pressure, although the problem of taking the right boundary conditions still remains. To remove this problem another formulation of the problem has been used consisting of projecting the master equations into the space of divergence‐free velocity fields, so pressure is eliminated from the equations. This technique raises the order of the differential equations and additional boundary conditions may be required. High‐order derivatives are sometimes troublesome, specially in cylindrical coordinates due to the singularity at the origin, so for these problems a low order formulation is very convenient. We research several pressure boundary conditions for the primitive variables formulation of thermoconvective problems. In particular we study the Marangoni instability of an infinite fluid layer and we show that the numerical results with a Chebyshev collocation method are highly correspondent to the exact ones. These ideas have been applied to linear stability analysis of the Bénard–Marangoni (BM) problem in cylindrical geometry and the results obtained have been very accurate. Copyright © 2002 John Wiley & Sons, Ltd.

Fluid Dynamics: Theory, Computation, and Numerical Simulation

Fluid Dynamics: Theory, Computation, and Numerical Simulation

Thermal convection in a cylindrical annulus heated laterally

In this paper we study thermoconvective instabilities appearing in a fluid within a cylindrical annulus heated laterally. As soon as a horizontal temperature gradient is applied a convective state appears. As the temperature gradient reaches a critical value a stationary or oscillatory bifurcation may take place. The problem is modelled with a novel method which extends the one described in (numerico). The Navier Stokes equations are solved in the primitive variable formulation, with appropriate boundary conditions for pressure. This is a low order formulation which in cylindrical coordinates introduces lower order singularities. The problem is discretized with a Chebyshev collocation method easily implemented and its convergence has been checked. The results obtained are not only in very good agreement with those obtained in experiments, but also provide a deeper insight into important physical parameters developing the instability, which has not been reported before.

Solutions for the internal boundary layer equations in simultaneously developing flow of power-law fluids within parallel plates channels

The generalized integral transform technique (GITT) is employed in the solution of the boundary layer equations in simultaneously developing laminar flow of power-law non-Newtonian fluids within a parallel plates channel. In the modeling of the related momentum and energy equations within the range of validity of the boundary layer equations, a streamfunction formulation is employed which offers a better computational performance than the primitive-variables formulation. Numerical results for the bulk temperature and Nusselt numbers are established at different axial positions along the channel and for various power-law indices, and critical comparisons with previously reported works in the literature are also performed.

Numerical simulation of fluid flow and heat transfer in a passage with moving boundary

In this paper, a method is presented in detail that can be used to solve the fluid flow and heat transfer in domains with moving boundaries. The primitive variables formulation is adopted and a non-staggered grid, with Cartesian velocity components used as the primary unkowns in the momentum equations, is utilised. Discretisation is carried out using a control-volume method, the simplified QUICK scheme combined with a deferred correction approach is adopted for the convective fluxes and implicit time stepping is used for temporal differencing. The well-known SIMPLE algorithm is employed for handling the velocity?pressure coupling. The computational method is applied for the prediction of fluid flow and heat transfer in a channel with a boundary moving in a prescribed manner. Results show that both the amplitude and Strouhal number have great influences on the characteristics of fluid flow and heat transfer, and in the range studied, the heat transfer rate increases monotonously with the amplitude, whereas the Strouhal number only has a small effect on heat transfer.

A Numerical Investigation of Peristaltic Waves in Circular Tubes

Peristaltic pumping is a process of fluid transport arising from the progressive waves, which travel along the walls of a flexible channel. It is a primary physiological transport mechanism that is inherent in many tubular organs of the human body such as the ureter, the gastro-intestinal tract, the urethra, and so on. Many studies exist in literature with the aim of understanding the characteristics of peristaltic flow under the assumption of low Reynolds number and infinitely long wavelength in a two-dimensional channel. However, peristaltic pumping is also the mechanism used in other industrial applications such as the blood pump for which the Reynolds number has a moderately high value. As studies concerning moderate to high Reynolds number flow in the circular tube are rare in literature, in the present study, the peristaltic flow of an incompressible fluid is numerically simulated using the finite volume method for solving the incompressible Navier-Stokes equations in primitive variable formulation by means of an infinite train of sinusoidal waves traveling along the wall of an axi-symmetric tube. The computational model presented in this work covers a wider range of Reynolds number (0.01-100), wave amplitude (0-0.8), and wavelength (0.01-0.4) than the those attempted in previous studies reported in literature and some new results pertaining to the distribution of velocity, pressure, wall shear stress for different peristaltic flow conditions characterizing flow at moderately higher Reynolds number have been obtained. The effect of the wave amplitude, wavelength, and Reynolds number on the "flow trapping" mechanism induced by peristalsis has also been investigated here for higher ranges of values of the parameters characterizing peristalsis.

Numerical simulation of natural convection in a square cavity by SIMPLE-generalized differential quadrature method

This is the first endeavor whereby the two-dimensional incompressible Navier–Stokes equations in primitive variable form are solved by using the generalized differential quadrature method with SIMPLE strategy. A new approach is proposed to enforce the continuity condition on the boundary and to implement the boundary condition for pressure correction equation. It is demonstrated that accurate numerical results can be obtained by the present method using a few grid points. It is also shown that the `multi-stage' process can enhance the convergence rate of numerical simulation.

A spectral projection method for the simulation of complex three-dimensional rotating flows

In this paper, we present an efficient projection method to solve the three-dimensional time-dependent incompressible Navier–Stokes equations in primitive variables formulation using spectral approximations. This method is based on a modification of the algorithm proposed by Goda [J. Comp. Phys. 30 (1979) 76]. It brings an improvement by introducing a preliminary step for the pressure in order to allow a temporal evolution of the normal pressure gradient at the boundaries. Its efficiency is brought to the fore by comparison with the Goda's algorithm. The modified projection method is then applied to the simulation of complex three-dimensional flows in rotating cavities, involving either a throughflow or a differential rotation.

On Spurious Vortical Structures

The paper investigates the formation of spurious vortical structures in incompressible flow simulations employing Godunov-type methods. The present work is motivated by the earlier studies of Brown and Minion (1995, J. Comput. Phys.122, 165 and 1997, J. Comput. Phys.138, 734) who demonstrated for a variety of numerical schemes (and for the upwind-biased methods in particular) that spurious vortices can occur in underresolved flow simulations. The aim of our work is threefold: (i) to identify deficiencies in various Godunov-type methods leading to spurious flow structures, (ii) to examine the numerical mechanisms responsible for these artifacts, and (iii) to propose modifications of Godunov-type methods in order to recover the correct solutions even under insufficient grid resolution. Our results reveal that the occurrence of spurious solutions depends strongly on the Godunov-type method employed. We show that in addition to the dissipation properties of a scheme—emphasized by Brown and Minion—there are other factors that can also contribute to numerical artifacts. These include a vortical instability arising from the numerical discretization of the advective terms in the primitive variable formulation of the Navier–Stokes equations, the balance of dissipation among the different discretized terms in a Godunov-type method, as well as order of accuracy of the interpolation used to discretize the wave-speed dependent term of the Godunov flux.

A Galerkin Implementation of the Generalized Helmholtz Decomposition for Vorticity Formulations

Vorticity formulations for the incompressible Navier–Stokes equations have certain advantages over primitive-variable formulations including the fact that the number of equations to be solved is reduced. However, the accurate implementation of the boundary conditions seems to continue to be an impediment to the acceptance and use of numerical methods based on vorticity formulations. Velocity boundary conditions can be implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields as described by the generalized Helmholtz decomposition (GHD). This can be accomplished in one of two ways by either solving for boundary vorticity (leading to a Dirichlet boundary condition for the vorticity equation) or solving for boundary vortex sheet strengths (leading to a Neumann condition). In the past, vortex sheet strengths have often been determined by solving an over-specified set of linear equations. The over-specification arose because integral constraints were imposed on the vortex sheet strengths. These integral constraints are not necessary and typically are included to mitigate errors in determining the vortex sheet strengths themselves. Further, the constraints overspecify the linear system requiring least-squares solution techniques. To more accurately satisfy both components of the velocity boundary conditions, a Galerkin formulation is applied to the generalized Helmholtz decomposition. This formulation implicitly satisfies an integral constraint that is more general than many of the integral constraints that have been explicitly imposed. Two implementations of the Galerkin GHD are considered in the current work, one based on determining the boundary vorticity and one based on determining the boundary vortex sheet strengths. A finite element method (FEM) is implemented to solve the vorticity equation along with the boundary data generated from the GHD.

A Fourth-Order-Accurate Finite Volume Compact Method for the Incompressible Navier–Stokes Solutions

This paper presents a finite volume fourth-order-accurate compact scheme for discretization of the incompressible Navier–Stokes equations in primitive variable formulation. The numerical method of integrating the Navier–Stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes. The pressure–velocity coupling is achieved via the coupled solution of the resulting system of equations. The solution of the coupled set of equations is performed with an implicit Newton–Krylov matrix-free method for stationary problems. For simulation of unsteady flows, a standard fourth-order Runge–Kutta method was used for temporal discretization and the velocity–pressure coupling was ensured at each stage also using the matrix-free method. Several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method.

CONTROL OF FLOW PAST BLUFF BODIES USING ROTATING CONTROL CYLINDERS

Computational results for control of flow past a circular cylinder using small rotating cylinders are presented. A well-proven stabilized finite-element method, that has been applied to various flow problems earlier, is utilized to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The formulation is first applied to study flow past an isolated rotating cylinder. Excellent match with experimental results, reported earlier, is observed. It is found that in purely two-dimensional flows, very high lift coefficients can be realized. However, it is observed, via three-dimensional Navier–Stokes simulations, that the end-effects and centrifugal instabilities along the cylinder span lead to a loss of lift and increase in drag. The aspect ratio of the cylinder plays an important role. The flow past a bluff body with two rotating control cylinders is studied using 2-D numerical simulations. The effect of the Reynolds number is studied by carrying out simulations for Re=102and 104. Finite element meshes with an adequate number of grid points are employed to resolve the flow in the gap between the main and control cylinders. Two values of the gap are considered: 0·01D and 0·075 D, where D is the diameter of the main cylinder. It is observed that when the control cylinders rotate at high speed, such that the tip speed is 5 times the free-stream speed, the flow at Re=100 achieves a steady state. For Re=104, even though the flow remains unsteady, the wake is highly organized and narrower compared to the one without control. The results are in good agreement with the flow-visualization studies conducted by other researchers for bluff bodies using similar control concepts. In all the cases, a significant reduction in the overall drag coefficient and the unsteady aerodynamic forces acting on the main cylinder is observed. Results are also presented for the power requirements of the system for translation and rotation. It is found that the coefficient of power required for the rotation of control cylinders is significant for Re=100 but negligible for Re=104flow. The size of the gap is found to be more critical for the Re=104flows. This study brings out the relevance of the gap as a design parameter for such flow control devices.

Control of vortex shedding behind circular cylinder for flows at low Reynolds numbers

SUMMARY It has been observed by researchers in the past that vortex shedding behind circular cylinders can be altered, and in some cases suppressed, over a limited range of Reynolds numbers by proper placement of a second, much smaller, ‘control’ cylinder in the near wake of the main cylinder. Results are presented for numerical computations of some such situations. A stabilized finite element method is employed to solve the incompressible Navier‐Stokes equations in the primitive variables formulation. At low Reynolds numbers, for certain relative positions of the main and control cylinder, the vortex shedding from the main cylinder is completely suppressed. Excellent agreement is observed between the present computations and experimental findings of other researchers. In an effort to explain the mechanism of control of vortex shedding, the streamwise variation of the pressure coefficient close to the shear layer of the main cylinder is compared for various cases, with and without the control cylinder. In the cases where the vortex shedding is suppressed, it is observed that the control cylinder provides a local favorable pressure gradient in the wake region, thereby stabilizing the shear layer locally. Copyright © 2001 John Wiley & Sons, Ltd.

Computation of two-dimensional flows past ram-air parachutes

Computational results for flow past a two-dimensional model of a ram-air parachute with leading edge cut are presented. Both laminar (Re=104) and turbulent (Re=106) flows are computed. A well-proven stabilized finite element method (FEM), which has been applied to various flow problems earlier, is utilized to solve the incompressible Navier-Stokes equations in the primitive variables formulation. The Baldwin-Lomax model is employed for turbulence closure. Turbulent flow computations past a Clarck-Y airfoil without a leading edge cut, for α=7.5°, result in an attached flow. The leading edge cut causes the flow to become unsteady and leads to a significant loss in lift and an increase in drag. The flow inside the parafoil cell remains almost stagnant, resulting in a high value of pressure, which is responsible for giving the parafoil its shape. The value of the lift-to-drag ratio obtained with the present computations is in good agreement with those reported in the literature. The effect of the size and location of the leading edge cut is studied. It is found that the flow on the upper surface of the parafoil is fairly insensitive to the configuration of the cut. However, the flow quality on the lower surface improves as the leading edge cut becomes smaller. The lift-to-drag ratio for various configurations of the leading edge cut varies between 3.4 and 5.8. It is observed that even though the time histories of the aerodynamic coefficients from the laminar and turbulent flow computations are quite different, their time-averaged values are quite similar.

Flow Past Rotating Cylinders: Effect of Eccentricity

Computational results are presented for flows past a translating and rotating circular cylinder. A stabilized finite element method is utilized to solve the incompressible Navier-Stokes equations in the primitive variables formulation. To validate the formulation and its implementation certain cases, for which the flow visualization and computational results have been reported by other researchers, are computed. Results are presented for Re=5, 200 and 3800 and rotation rate, (ratio of surface speed of cylinder to the freestream speed of flow), of 5. For all these cases the flow reaches a steady state. The values of lift coefficient observed for these flows exceed the limit on the maximum value of lift coefficient suggested by Goldstein based on intuitive arguments by Prandtl. These observations are in line with measurements reported, earlier, by other researchers via laboratory experiments. To investigate the stability of the computed steady-state solution, receptivity studies involving an eccentrically rotating cylinder are carried out. Computations are presented for flow past a rotating cylinder with wobble; the center of rotation of the cylinder does not match its geometric center. These computations are also important from the point of view that in a real situation it is almost certain that the rotating cylinder will be associated with a certain degree of wobble. In such cases the flow is unsteady and reaches a temporally periodic state. However, the mean values of the aerodynamic coefficients and the basic flow structure are still quite comparable to the case without any wobble. In this sense, it is found that the two-dimensional solution is stable to purely two-dimensional disturbances.

A fully implicit model for simulating dynamo action in a Cartesian domain

We present a fully implicit numerical method to solve the incompressible MHD equations in a strongly rotating Cartesian domain. The equations are solved in a primitive variable formulation using a finite volume discretization. In order to use massively parallel computers, we applied a domain decomposition approach in space. The performance of this model is compared with an earlier model, which treated the convective terms of the equations in an explicit manner. Our results indicate that although the fully implicit method needs about three times the memory of the implicit–explicit method, it is superior in terms of computational efficiency. As an application of this model, we investigated the influence of the Prandtl number in the range of 0.01–1000 on the dynamics of the dynamo.

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J-G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher-order) finite elements. This method can achieve high-order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.

A cell boundary–element method for viscous laminar flow solutions

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Tan M., Farrant T. and Price W. G. 1999A cell boundary–element method for viscous laminar flow solutionsProc. R. Soc. Lond. A.4554277–4304http://doi.org/10.1098/rspa.1999.0501SectionRestricted accessA cell boundary–element method for viscous laminar flow solutions M. Tan M. Tan School of Engineering Sciences, Ship Science, Highfield, Southampton SO17 1BJ, UK Google Scholar Find this author on PubMed Search for more papers by this author , T. Farrant T. Farrant School of Engineering Sciences, Ship Science, Highfield, Southampton SO17 1BJ, UK Computational Engineering Design Centre, University of Southampton, Highfield, Southampton SO17 1BJ, UK Google Scholar Find this author on PubMed Search for more papers by this author and W. G. Price W. G. Price School of Engineering Sciences, Ship Science, Highfield, Southampton SO17 1BJ, UK Google Scholar Find this author on PubMed Search for more papers by this author M. Tan M. Tan School of Engineering Sciences, Ship Science, Highfield, Southampton SO17 1BJ, UK Google Scholar Find this author on PubMed Search for more papers by this author , T. Farrant T. Farrant School of Engineering Sciences, Ship Science, Highfield, Southampton SO17 1BJ, UK Computational Engineering Design Centre, University of Southampton, Highfield, Southampton SO17 1BJ, UK Google Scholar Find this author on PubMed Search for more papers by this author and W. G. Price W. G. Price School of Engineering Sciences, Ship Science, Highfield, Southampton SO17 1BJ, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:08 December 1999https://doi.org/10.1098/rspa.1999.0501AbstractA cell boundary–element method is developed to solve viscous fluid–structure interaction problems modelled by Navier–Stokes equations. This is achieved through a hybrid approach incorporating boundary–element and finite–element methods. In the proposed scheme, cell equations are generated using the principles of the boundary–element method with global equations derived following the procedures of the finite–element method. A primitive–variable formulation with an unstructured mesh requirement forms the basis of the hybrid approach which can be applied to both two– and three–dimensional problems.The validation of this numerical scheme of study involving analytical and numerical mathematical procedures is carried out using a number of well–documented flow solutions and the accuracy and robustness of the method are demonstrated in these examples. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Ravi A, Tan M and Price W (2008) An analysis of a two cylinder-fluid interaction at critical gap spacing by a cell boundary element method, The European Physical Journal Special Topics, 10.1140/epjst/e2008-00858-8, 165:1, (151-160), Online publication date: 1-Dec-2008. Ravi A, Nair P, Tan M and Price W (2002) Optimal Feedforward and Feedback Control of Vortex Shedding Using Trust-Region Algorithms 1st Flow Control Conference, 10.2514/6.2002-3076, 978-1-62410-105-2, Online publication date: 24-Jun-2002. Tan M, Uzunogĝlu B, Price W and Rogers E (2005) Reduced models for statistically stationary and non-stationary flows with control applications, Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment, 10.1243/147509002320382167, 216:1, (95-102), Online publication date: 1-Jun-2002. Uzuno?lu B, Tan M and Price W (2001) Low-Reynolds-number flow around an oscillating circular cylinder using a cell viscousboundary element method, International Journal for Numerical Methods in Engineering, 10.1002/nme.122, 50:10, (2317-2338), Online publication date: 10-Apr-2001. Farrant T, Tan M and Price W (2001) A cell boundary element method applied to laminar vortex shedding from circular cylinders, Computers & Fluids, 10.1016/S0045-7930(00)00009-8, 30:2, (211-236), Online publication date: 1-Feb-2001. FARRANT T, TAN M and PRICE W (2000) A CELL BOUNDARY ELEMENT METHOD APPLIED TO LAMINAR VORTEX-SHEDDING FROM ARRAYS OF CYLINDERS IN VARIOUS ARRANGEMENTS, Journal of Fluids and Structures, 10.1006/jfls.1999.0275, 14:3, (375-402), Online publication date: 1-Apr-2000. This Issue08 December 1999Volume 455Issue 1992 Article InformationDOI:https://doi.org/10.1098/rspa.1999.0501Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/12/1999Published in print08/12/1999 License: Citations and impact Keywordsnumerical simulationhybrid approachboundary–element methodunstructured meshcomputational fluid dynamicsviscous flow solution

An unsteady incompressible Navier-Stokes solver for large eddy simulation of turbulent flows

An unsteady incompressible Navier–Stokes solver that uses a dual time stepping method combined with spatially high-order-accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence–acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical Study of Natural Convection in Fully Open Tilted Cavities

A numerical simulation of two-dimensional laminar natural convection in a fully open tilted square cavity with an isothermally heated back wall is conducted. The remaining two walls of the cavity are adiabatic. Steady-state solutions are presented for Grashof numbers between 102 and 105 and for tilt angles ranging from 60 to 90 (where 90 represents a cavity with the opening facing down). The fluid properties are assumed to be constant except for the density variation with temperature that gives rise to the buoyancy forces, which is treated by the Boussinesq approximation. The fluid concerned is air with Prandtl number fixed at 0.71. The governing equations are expressed in a normalized primitive variables formulation. Numerical predictions of the velocity and temperature fields are obtained using the finite-volume-based power law (SIMPLER: Semi-Implicit Method for Pressure-Linked Equations Revised) algorithm. For a vertical open cavity (alpha = 0 deg), the algorithm generated results that were in good agreement with those previously published. Flow patterns and isotherms are shown in order to give a better understanding of the heat transfer and flow mechanisms inside the cavity. Effects of the controlling parameters-Grashof number and tilt angle-on the heat transfer (average Nusselt number) are presented and analyzed. The results also revealed that the open-cavity Nusselt number approaches the flat-plate solution when either Grashof number or tilt angle increases. In addition, a correlation of the Nusselt number in terms of the Grashof number and tilt angle is developed and presented; a comparison is made with available data from other literature.