Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

The numerical solution of the unsteady natural convection flow in a square cavity at high Rayleigh number using SADI method

The unsteady natural convection flow in a square cavity at high Rayleigh number Ra=10 ⁷ and 2×10 ⁷ has been computed using cubic spline integration. The required solutions to the two dimensional Navier-Stokes and energy equations have been obtained using two alternate numerical formulations on non-uniform grids. The main features of the transient flow have been briefly discussed. The results obtained by using the present method are in good agreement with the theoretical predictions ^[1,2].The steady state results have been compared with accurate solutions presented recently for Ra=10 ⁷.     

Transient natural convection of low-Prandtl-number fluids in a differentially heated cavity

The transient convective motion in a two-dimensional square cavity driven by a temperature gradient is analysed. The cavity is filled with a low-Prandtl-number fluid and the vertical walls are maintained at constant but different temperatures, while the horizontal boundaries are adiabatic. A control volume approach with a staggered grid is employed to formulate the finite difference equations. Numerically accurate solutions are obtained for Prandtl numbers of 0·001, 0·005 and 0·01 and for Grashof numbers up to 1 × 10⁷. It was found that the flow field exhibits periodic oscillation at the critical Grashof numbers, which are dependent on the Prandtl number. As the Prandtl number is decreased, the critical Grashof number and the frequency of oscillation decrease. Prior to the oscillatory flow, steady state solutions with an oscillatory transient period were predicted. In addition to the main circulation, four weak circulations were predicted at the corners of the cavity.     

Finite element analysis of vortex shedding using equal order interpolations

An operator splitting and element‐by‐element conjugated gradient solver, and equal order interpolations are applied for solving time dependent Navier–Stokes (NS) equations to simulate flow induced vortex shedding in the present study. In addition, the convection term is corrected by balanced tensor diffusivity, which can stabilize the numerical simulation and overcome the numerical oscillations. The evolution of the interested flowing properties with time is analyzed by using spectral analysis. The developed code has been validated by the application of two examples: a driven cavity flow and a flow induced vortex vibration. Results from the first example for Reynolds number Re=10³ and Re=10⁴ are compared with other numerical simulations. Results from the second example, uniform flow past a square rod over a wide range of high Reynolds numbers from Re=10³～10⁵, are compared with experimental data and other numerical studies. Copyright © 2002 John Wiley & Sons, Ltd.     

A least-squares finite element method for time-dependent incompressible flows with thermal convection

The time-dependent Navier–Stokes equations and the energy balance equation for an incompressible, constant property fluid in the Boussinesq approximation are solved by a least-squares finite element method based on a velocity–pressure–vorticity–temperature–heat-flux ( u –P–ω–T– q ) formulation discretized by backward finite differencing in time. The discretization scheme leads to the minimization of the residual in the l²-norm for each time step. Isoparametric bilinear quadrilateral elements and reduced integration are employed. Three examples, thermally driven cavity flow at Rayleigh numbers up to 10⁶, lid-driven cavity flow at Reynolds numbers up to 10⁴ and flow over a square obstacle at Reynolds number 200, are presented to validate the method.     

Developing turbulent flow in smooth pipes

A numerical and experimental investigation of steady incompressible developing turbulent flow in smooth pipes is presented. Finite difference techniques are used to solve simultaneously the vorticity transport and stream function equations utilising a modified form of the Van Driest effective viscosity model. The numerical solutions are verified experimentally using air as a working fluid at pipe Reynolds 1 × 10⁵, 2 × 10⁵ and 3 × 10⁵.     

Flow Bifurcations in a Thin Gap Between Two Rotating Spheres

This paper presents the use of a parameter continuation method and a test function to solve the steady, axisymmetric incompressible Navier–Stokes equations for spherical Couette flow in a thin gap between two concentric, differentially rotating spheres. The study focuses principally on the prediction of multiple steady flow patterns and the construction of bifurcation diagrams. Linear stability analysis is conducted to determine whether or not the computed steady flow solutions are stable. In the case of a rotating inner sphere and a stationary outer sphere, a new unstable solution branch with two asymmetric vortex pairs is identified near the point of a symmetry-breaking pitchfork bifurcation which occurs at a Reynolds number equal to 789. This solution transforms smoothly into an unstable asymmetric 1-vortex solution as the Reynolds number increases. Another new pair of unstable 2-vortex flow modes whose solution branches are unconnected to previously known branches is calculated by the present two-parameter continuation method. In the case of two rotating spheres, the range of existence in the (Re ₁ , Re ₂ ) plane of the one and two vortex states, the vortex sizes as a function of both Reynolds numbers are identified. Bifurcation theory is used to discuss the origin of the calculated flow modes. Parameter continuation indicates that the stable states are accompanied by certain unstable states. Received 26 November 2001 and accepted 10 May 2002 Published online 30 October 2002 Communicated by M.Y. Hussaini     

Time-dependent natural convection in a square cavity: Application of a new finite volume method

A new finite volume (FV) approach with adaptive upwind convection is used to predict the two-dimensional unsteady flow in a square cavity. The fluid is air and natural convection is induced by differentially heated vertical walls. The formulation is made in terms of the vorticity and the integral velocity (induction) law. Biquadratic interpolation formulae are used to approximate the temperature and vorticity fields over the finite volumes, to which the conservation laws are applied in integral form. Image vorticity is used to enforce the zero-penetration condition at the cavity walls. Unsteady predictions are carried sufficiently forward in time to reach a steady state. Results are presented for a Prandtl number (Pr) of 0-71 and Rayleigh numbers equal to 10³, 10⁴ and 10⁵. Both 11 × 11 and 21 × 21 meshes are used. The steady state predictions are compared with published results obtained using a finite difference (FD) scheme for the same values of Pr and Ra and the same meshes, as well as a numerical bench-mark solution. For the most part the FV predictions are closer to the bench-mark solution than are the FD predictions.     

Shock wave-induced vortex loops emanating from nozzles with singular corners

The focus of the current study is to examine experimentally the diffracted shock wave pattern and the consequent vortex loop formation, propagation, and decay from nozzles having singular corners. Non-intrusive qualitative and quantitative techniques: schlieren, shadowgraphy, and particle image velocimetry (PIV) are employed to analyze the induced flow-fields. Eye-shaped nozzles were used with the corner joints representing singularities. The length of the minor axes are a = 6 and 15 mm, with the major axis b = 30 mm for both cases. The experiments are performed for flow Reynolds numbers in the range 0.8 × 10⁵ and 4.6 × 10⁵. Air is used in both driver and driven sections of the shock tube.     

A boundary element method for steady viscous fluid flow using penalty function formulation

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier–Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved. © 1997 John Wiley & Sons, Ltd.     

Numerical study of mixed convection and conduction in a 2-D square ventilated cavity with an inlet at the vertical glazing wall and outlet at the top surface

A steady state numerical study of combined laminar mixed convection and conduction heat transfer in a ventilated square cavity is presented. The air inlet gap is located at the bottom of a vertical glazing wall and air exits the cavity via a gap located at the top surface. Three locations for the opening at the top surface: left (case a), center (case b) and right side (case c) are considered. All the remaining surfaces are considered adiabatic. The mass, momentum and energy conservation equations were solved using the finite volume method for different Rayleigh numbers in the interval of 10⁴ < Ra < 10⁶ and Reynolds number in the interval of 100 < Re < 700. Temperature, flow field, and heat transfer rates are analyzed. The effect of the interaction between ambient conditions outside the glazing and the air inlet gap at the bottom for different air outlet gap positions at the top surface modifies the flow structure and temperature distribution of the air inside the cavity. The Nusselt number as a function of the Reynolds number was determined for the three cases. It was found that configuration for case (a) removes a higher amount of heat entering the cavity compared to cases (b) and (c). This is due to the short distance between the main stream and the glass wall surface. Thus, the forced airflow entering the cavity is assisted by the buoyancy forces, and most of the cavity remains at the inlet flow temperature, which should be appropriate for warm climates. These results may provide useful information about the heat transfer and fluid flow for future studies.     

Transient mixed convection in an enclosure driven by a sliding lid

Transient convection of an incompressible viscous fluid in a square cavity is investigated. The temperature at the top lid is higher than that at the bottom wall, producing a stably stratified overall configuration. The vertical sidewalls are insulated. Flow is initiated by an impulsive start of the sliding motion of the top lid. The transient features of the mixed convection are delineated by procuring numerical solutions in a wide range of parameters, i.e., 400≤Re≤4,000, 1.6×10⁵≤Gr≤1.6×10⁷. Flows and heat transfer characteristics are described both in the interior core and boundary-layer regions. In the large-time limit, the steady state features are depicted. Parallel experimental efforts are made by employing the particle image velocimetry (PIV) to visualize the steady state flow and thermal fields, together with thermocouple measurements.     

A fluid solver based on vorticity–helical density equations with application to a natural convection in a cubic cavity

We study numerically a recently introduced formulation of incompressible Newtonian fluid equations in vorticity–helical density and velocity–Bernoulli pressure variables. Unlike most numerical methods based on vorticity equations, the current approach provides discrete solutions with mass conservation, divergence‐free vorticity, and accurate kinetic energy balance in a simple and natural way. The method is applied to compute buoyancy‐driven flows in a differentially heated cubic enclosure in the Boussinesq approximation for Ra ∈ {10⁴,10⁵,10⁶}. The numerical solutions on a finer grid are of benchmark quality. The computed helical density allows quantification of the three‐dimensional nature of the flow. Copyright © 2011 John Wiley & Sons, Ltd.     

Combined radiation and natural convection in a rectangular cavity with a transparent wall and containing a non-participating fluid

This paper describes a numerical method for the study of combined natural convection and radiation in a rectangular, two-dimensional cavity containing a non-participating (i.e. transparent) fluid. One wall of the cavity is isothermal, being heated either by solar radiation or independently. The opposite wall is partially transparent, permitting radiation exchanges between the cavity and its surroundings and/or the Sun; that wall also exchanges heat by convection from its external surface to the surroundings. The other two walls are adiabatic: convection and radiation there are balanced, so that there is no heat transfer through those walls. The equations of motion and energy are solved by finite difference methods. Coupled to these equations are the radiative flux boundary conditions which are used to determine the temperature distribution along the non-isothermal walls. A two-band radiation model has been employed. Results are presented for a square cavity with a vertical hot wall at 150 °C, the ambient at 20 °C and 10⁴ ? Ra ? 3 × 10⁵, in the absence of direct insolation. The effects on the flow and heat transfer in the cavity of radiation and external convection have been examined. More extensive results will be presented in subsequent papers.     

SUPG finite element method based on penalty function for lid-driven cavity flow up to $$Re = 27500$$

A streamline upwind/Petrov–Galerkin(SUPG)finite element method based on a penalty function is proposed for steady incompressible Navier–Stokes equations.The SUPG stabilization technique is employed for the formulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pressure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.     

Higher-order difference approximations of the Navier-Stokes equations

A discretization scheme is presented which, unlike the standard higher-order finite difference and spline methods, does not give rise to unphysical solution modes and boundary conditions. Practical application of this scheme is achieved via the DCMG algorithm recently developed by the same author, which turns out to be able to find a converged solution of the ψ-ζ Navier-Stokes equations in about the same time for highorder as for low-order discretization schemes. Examples are presented for the driven cavity problem to explore the accuracy of the new method. Finally, a local analysis is performed of the corner singularities which exist in driven cavity flow, and their effect on the overall accuracy of the solutions obtained by polynomial interpolation methods is investigated.     

NUMERICAL SIMULATION OF TWO–DIMENSIONAL BUOYANCY–DRIVEN TURBULENCE IN A TALL RECTANGULAR CAVITY

A computational study of natural convection of air in a tall rectangular cavity with 4:1 aspect ratio is conducted. In an effort to investigate the applicability of the Boussinesq approximation to turbulent flow simulation, the cavity is differentially heated from the sides and is insulated at the ends at a Rayleigh number of 10⁹. Starting from quiescent and isothermal flow conditions, the flow is driven to turbulence without any artificial perturbations. The computer programme developed integrates the two-dimensional, time-dependent Navier–Stokes equations with the Boussinesq approximation and the energy equation by a time-accurate method on a stretched, staggered grid. The simulation proceeds to a statistically steady solution in which large-scale structures are found in the mean. Both mean and fluctuating quantities provide good agreement with experimental results.     

Thermal effects of the micropolar fluid parameters on the laminar free convection in spherical annuli with mixed boundary conditions

The effects of micro-rotation and vortex viscosity in micropolar fluids have been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with specified mixed boundary conditions. Calculations were carried out systematically for several different eccentricities and a range of modified Rayleigh numbers to determine the average Nusslet numbers which are affected by the micropolar parameters (F) of the flow and temperature fields. The skin friction stress on the walls has also been studied and discussed. The governing equations, in terms of vorticity, stream function, temperature and angular momentum are expressed in a spherical polar coordinate system. Results were obtained for steady heat-transfer in spherical annuli at a Prandtl number of 0.7, with the modified Rayleigh number ranging from 10³ to 5 × 10⁵, for a radius ratio of 2.0 and eccentricities varying from −0.625 to +0.625. Comparisons are attempted between the Newtonian fluid and micropolar fluid.     

Effects of surfactant and salt concentrations on capillary flow and its entry flow for wormlike micelle solutions

Dynamic viscoelasticities and flow properties were measured for aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) to examine the effects of surfactant (C _D) and salt (C _S). The relaxation time λ of a single mode Maxwell model was obtained, and the relationship between λ and free NaSal concentration was discussed. The relation between λ and was applied to the classification of flow curves, which were obtained using a capillary rheometer. In the flow curves, a shear rate jump occurred at low shear rates for the solutions with low , while bending was seen at high shear rates for all the flow curves. On the other hand, vortex growth at the salient corner in the entrance region of the capillary was also investigated. Four different flow patterns were identified: Newtonian-like flow (A), steady vortex flow (B), periodically oscillated flow (C), and perfectly unstable flow (D). In the steady vortex of the flow pattern B, the vortex length increased with increasing shear rate. In the flow patterns C and D, white turbidity was observed. Furthermore, the relation between λ and was also applied to the discussion on the development of the vortex.     

Natural convection in inclined two dimensional rectangular cavities

Steady two-dimensional natural convection in fluid filled cavities is numerically investigated. The channel is heated from below and cooled from the top with insulated side walls and the inclination angle is varied. The field equations for a Newtonian Boussinesq fluid are solved numerically for three cavity height based Rayleigh numbers, Ra = 10⁴, 10⁵ and 10⁶, and several aspect ratios. The calculations are in excellent agreement with previously published benchmark results. The effect of the inclination of the cavity to the horizontal with the angle varying from 0° to 180° and the effect of the startup conditions on the flow pattern, temperature distribution and the heat transfer rates have been investigated. Flow admits different configurations at different angles as the angle of inclination is increased depending on the initial conditions. Regardless of the initial conditions Nusselt number Nu exhibits discontinuities triggered by gradual transition from multiple cell to a single cell configuration. The critical angle of inclination at which the discontinuity occurs is strongly influenced by the assumed startup field. The hysteresis effect previously reported is not always present when the calculations are reversed from 90° to 0°. A comprehensive study of the flow structure, the Nu variation with varying angle of inclination, the effect of the initial conditions and the hysteresis effect are presented.