Flows In Cylindrical Coordinates Research Articles

Analytical Analysis of the Bladeless Rotor Turbine Design for Enhanced Performance

The growing desire for clean and efficient energy solutions has fueled considerable advances in turbine technology. This study proposed a conceptual approach to enhancing the performance of bladeless turbines using an analytical technique, and the analytical solutions bear a resemblance to the experimental setup for a fluid dynamics investigation of a rotor for real-flow effects. The Navier-Stokes equations for fluid flow in cylindrical coordinates were simplified to two-dimensional flow by ignoring axial direction and velocity, and direct integration was used in conjunction with series expansion solutions. The experiment data were used to verify the rotor's fluid dynamics analytical solutions. The model's result was compared to previously calculated solutions of the fluid dynamics of a disc rotor in a bladeless turbine. The disc turbine models were used to predict radial velocity, tangential velocity, pressure gradient, volumetric flow rate, rotor torque, shear stress on the inner and outer disc walls, and rotor efficiency. The model was validated using experimental data with an efficiency of 23.9%, the theoretical solution model was 34%, and the analytical efficiency was 24.3%. The efficiency comparison of the analytical solutions model to the theoretical solutions model revealed a substantial difference, however, the correlation between computed theoretical and analytical results is significant. Previous studies used computed solutions for models, but current analytical solutions outperformed them. The model's output will be valuable to engineers building the disc turbine. It demonstrates a strong link between the analytical and experimental research of the bladeless turbine.

Numerical Study of Compressible and Weak Compressible Flows in Channel Based on Artificial Compressibility Method and Fully Artificial Compressibility Method

In this article, a numerical study of compressible and weak compressible Newtonian flows is achieved for a time marching, Galerkin algorithm. A comparison between two numerical techniques for such flows, namely the artificial compressibility method (AC–method) and the fully artificial compressibility method (FAC–method) is performed. In the first artificial compressibility parameter ( is added to the continuity equation, while this parameter is added to both continuity and momentum equations in the second technique. This strategy is implemented to treat the governing equations of Newtonian flow in cylindrical coordinates (axisymmetric). Particularly, this study concerns with the effect of the artificial compressibility parameters on the convergence level of solutions components. To confirm the analysis of these approaches, Poiseuille flow along a circular channel under an isothermal state is used as a simple test problem. The results show that when the AC-method is used there is a significant reduction in the level of time convergence of pressure and axial velocity compared to that with FAC-method. Here, for compressible flow the Tail model of state is employed to relate the pressure to density. In this context, the effect of Tail parameters and Reynolds number on the time convergence of solution components is also investigated in the present study. The results indicate a significant reduction in the time-stepping convergence as increasing in the {B,m}-value. In contrast, more difficulties are faced in the convergence when the level of the Reynolds number is increasing.

Scrutiny the Heat Transfer Effect in an Annulus by Mounting Axial Fins with Different Shapes: Without and with Taylor Number

This study aimed to investigate numerically the heat transfer improvement and pressure drop inside annular channel of a rotor-stator provided with fins mounted on the stator without and with Taylor number. The impact of mounting various types of fins (triangular, rectangular, trapezoidal shapes with small and large base) is studied by varying the fin width b from 0 to 14 mm. In the presence of axial air flow, numerical simulations are carried out by solving the governing continuity, momentum and energy equations of turbulent flow in cylindrical coordinates using the Finite Volume Method. The results obtained by Reynolds Stress Model RSM model have indicated that the heat transfer enhances as the surface area of the fins and the effective Reynolds number increase, while there is an increase in pressure drop. Furthermore, we have shown that the presence of Taylor number has a slight increase in Nusselt number and pressure drop compared to the case without Taylor number. Among the four geometries, it is found that the rectangular cavity is the best geometry which gives maximum heat transfer and minimum pressure loss.

Stratified equatorial flows in cylindrical coordinates

We construct an exact solution modelling the geophysical dynamics of an inviscid and incompressible fluid which possesses a variable density stratification, where the fluid density may vary with both the depth and latitude. Our solution pertains to the large-scale equatorial dynamics of a fluid body with a free surface propagating steadily in a purely azimuthal direction, and is expressed in terms of cylindrical coordinates. Allowing for general fluid stratification greatly complicates the Bernoulli relation—which relates the imposed pressure to the reciprocal fluid distortion at the free-surface—thereby acting as a constraint on the existence of a solution. Employing the implicit function theorem, we establish the existence of solutions and determine that the requisite monotonicity properties hold for the flow solutions we found. Furthermore, since the fluid velocity and pressure are prescribed by explicit formulae in the framework of cylindrical coordinates, our solution is amenable to analysis by the short-wavelength stability approach, which we investigate.

Development of a fast EoS based compositional model for three-phase core flooding

Compositional simulators are of great importance to model core flow experiments and enhanced oil recovery processes. In the present study, a semi-implicit, three dimensional EoS-based formulation is developed for modeling and simulation of core-flooding process. The model treats three-phase flow in cylindrical coordinates with no need to use an iterative numerical method. Overall composition, pressure and water saturation are calculated implicitly for each grid block. Using this approach, a linear set of equations with constant coefficients would be solved at each time step. It is assumed that there is no mass transfer between hydrocarbon phases and water phase. In addition, a new technique has been suggested to calculate transmissibility coefficients in each direction, which is a major problem in compositional modeling. Each transmissibility coefficient is divided into two general parts which are treated differently. The first part is constant and the second part is time-dependent. This model yields a robust and fast calculation procedure, which is validated using Eclipse simulation results. Simulation of several miscible flooding processes were performed using Eclipse core-flood models in order to compare the results with the developed in-house model. Results are matched reasonably while the computing time is considerably decreased using the in-house model. It is shown that using second Stone’s relative permeability model and Advanced Peng–Robinson Equation of State, best results are achieved with an average absolute relative deviation of 7.7% from Eclipse results for ultimate recovery factor. In addition, sensitivity analysis has been performed on several input data including time-step size and block number. Results show that in a reasonable range of these parameters, convergence problems are not encountered. We have developed an in-house simulator based on our model, which can be a basis for other compositional simulation purposes. In addition to decreasing computing time, convergence problems encountered in most implicit techniques would be avoided as well.

High-order accurate large-eddy simulations of compressible viscous flow in cylindrical coordinates

Dynamic interactions of shock waves and turbulent structures occur in a wide range of applications. The accurate simulation of turbulence in these flows requires a numerical scheme with minimal dissipation whereas the shock-capturing requires a dissipative scheme to stabilise the solution in the shock-wave discontinuous regions. These contradictory requirements make simulations of these flows extremely challenging. A compatible implementation of the solid boundary conditions at the point of incepting the acoustic waves and their internalisation into hydrodynamic instabilities is also challenging.Here we present, a high-fidelity, massively parallel and accurate solver for both direct numerical simulations and large-eddy simulations of supersonic turbulent flows in cylindrical coordinates. A hybrid WENO/ high order central difference scheme is used for the spatial discretisation with a fourth-order five-stage 2N-storage Runge–Kutta for the time integration. The least square contraction of Lilly [1] is utilised for large-eddy subgrid-scale modelling. The solid surface boundary condition is implemented using the offset wall and ghost cell technique.The numerical implementation is validated through a wide range of test cases from simple to more complex cases. The numerical results show good agreement with analytical solutions and available experimental results. In the large-eddy simulation of a supersonic under-expanded impinging jet, it is observed that the conventional Ducros sensor leads to a more dissipative hybrid scheme. In this paper, a new shock sensor based on a WENO smoothness indicator is proposed and it is demonstrated to perform better especially in the simulation of the supersonic under-expanded impinging jet by limiting the expensive WENO scheme to the discontinuous regions and reducing the computational cost by approximately 10%.

Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates

Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates

Entry and exit flows in curved pipes

Solutions are presented for both laminar developing flow in a curved pipe with a parabolic inlet velocity and laminar transitional flow downstream of a curved pipe into a straight outlet. Scalings and linearized analyses about appropriate base states are used to show that both cases obey the same governing equations and boundary conditions. In particular, the governing equations in the two cases are linearized about fully developed Poiseuille flow in cylindrical coordinates and about Dean’s velocity profile for curved pipe flow in toroidal coordinates respectively. Subsequently, we identify appropriate scalings of the axial coordinate and disturbance velocities that eliminate dependence on the Reynolds number $Re$ and dimensionless pipe curvature $\unicode[STIX]{x1D6FC}$ from the governing equations and boundary conditions in the limit of small $\unicode[STIX]{x1D6FC}$ and large $Re$. Direct numerical simulations confirm the scaling arguments and theoretical solutions for a range of $Re$ and $\unicode[STIX]{x1D6FC}$. Maximum values of the axial velocity, secondary velocity and pressure perturbations are determined along the curved pipe section. Results collapse when the scalings are applied, and the theoretical solutions are shown to be valid up to Dean numbers of $D=Re^{2}\unicode[STIX]{x1D6FC}=O(100)$. The developing flows are shown numerically and analytically to contain spatial oscillations. The numerically determined decay of the velocity perturbations is also used to determine entrance/development lengths for both flows, which are shown to scale linearly with the Reynolds number, but with a prefactor ${\sim}60\,\%$ larger than the textbook case of developing flow in a straight pipe.

M2Di: Concise and efficient MATLAB 2‐D Stokes solvers using the Finite Difference Method

AbstractRecent development of many multiphysics modeling tools reflects the currently growing interest for studying coupled processes in Earth Sciences. The core of such tools should rely on fast and robust mechanical solvers. Here we provide M2Di, a set of routines for 2‐D linear and power law incompressible viscous flow based on Finite Difference discretizations. The 2‐D codes are written in a concise vectorized MATLAB fashion and can achieve a time to solution of 22 s for linear viscous flow on 10002 grid points using a standard personal computer. We provide application examples spanning from finely resolved crystal‐melt dynamics, deformation of heterogeneous power law viscous fluids to instantaneous models of mantle flow in cylindrical coordinates. The routines are validated against analytical solution for linear viscous flow with highly variable viscosity and compared against analytical and numerical solutions of power law viscous folding and necking. In the power law case, both Picard and Newton iterations schemes are implemented. For linear Stokes flow and Picard linearization, the discretization results in symmetric positive‐definite matrix operators on Cartesian grids with either regular or variable grid spacing allowing for an optimized solving procedure. For Newton linearization, the matrix operator is no longer symmetric and an adequate solving procedure is provided. The reported performance of linear and power law Stokes flow is finally analyzed in terms of wall time. All MATLAB codes are provided and can readily be used for educational as well as research purposes. The M2Di routines are available from Bitbucket and the University of Lausanne Scientific Computing Group website, and are also supplementary material to this article.

Flow, thermal criticality and transition of a reactive third-grade fluid in a pipe with Reynolds' model viscosity

Neglecting the consumption of the material, a steady incompressible flow of an exothermic reacting third-grade fluid with viscous heating in a circular cylindrical pipe is numerically studied for both cases of constant viscosity and Reynolds’ viscosity model. The coupled ordinary differential equations governing the flow in cylindrical coordinates, are transformed into dimensionless forms using appropriate transformations, and then solved numerically. Solutions using Maple are presented in tabular form and given in terms of dimensionless central fluid velocity and temperature, skin friction and heat transfer rate for three parametric values in the Reynolds' case. The numerical results for the velocity and temperature fields are also presented through graphs. Bifurcations are discussed using shooting method. Comparisons are also made between the present results and those of previous work, and thus verify the validity of the provided numerical solutions. Important properties of thermal criticality are provided for variable viscosity parameter and reaction order. Further numerical results are presented in the form of tables and graphs for transition of physical parame- ters, while varying certain flow and fluid material parameters. Also, the flow behaviour of the reactive fluid of third-grade is compared with those of the Newtonian reactive fluid.

Derivation of transport equations for three-dimensional non-isothermal turbulent flow in cylindrical coordinates

Derivation of transport equations for three-dimensional non-isothermal turbulent flow in cylindrical coordinates

High-order pole-treatment in cylindrical coordinates for incompressible flow simulations with finite-difference collocated schemes

A finite-difference collocated scheme is proposed for the simulation of unsteady incompressible flows in cylindrical coordinates. The Rhie–Chow flux-interpolation technique is used to avoid the odd–even decoupling of the pressure field. It is shown that the calculation of the Coriolis and centrifugal acceleration terms based on the cell-center values lead to numerical instabilities near the pole in the inviscid regime. To avoid this issue, a new formulation based on the combination of cell-center and cell-face velocity components is proposed for the calculation of these terms. Furthermore, a new pole treatment methodology is introduced to calculate the viscous terms, resulting single-valued properties at the pole. As regards the extensions of the present scheme to higher orders of accuracy, a fourth-order spatial discretization is also introduced and verified. Moreover, it is shown that for higher-order extensions, the present pole treatment is able to preserve the order of accuracy and the L2-norm and L∞-norm errors are greatly improved in comparison with the previous schemes. The robustness of the present method is assessed via five different numerical tests including laminar flows in a polar cavity, vortex-wall interaction, evolution of a monopole vortex-system to a tripole vortex-system, advection of the lamb dipole, and large eddy simulation (LES) of turbulent pipe flows. Compared to the previous LES results for turbulent pipe flows, significant improvement is obtained for the mean and fluctuating velocities especially at the core region of the channel. Similarly, at the wall region, the fluctuating axial and azimuthal velocities and the skewnesses of the axial and radial velocities are very close to the previous direct numerical simulation and experimental data.

Numerical Simulation Characteristics of Logging Response in Water Injection Well by Reproducing Kernel Method

Reproducing kernel Hilbert space method (RKHSM) is an effective method. This paper, for the first time, uses the traditional RKHSM for solving the temperature field in two phase flows of multilayer water injection well. According to 2D oil-water temperature field mathematical model of two phase flows in cylindrical coordinates, selecting the properly initial and boundary conditions, by the process of Gram-Schmidt orthogonalization, the analytical solution was given by reproducing kernel functions in a series expansion form, and the approximate solution was expressed byn-term summation. The satisfied numerical results were carried out by Mathematica 7.0, showing that the larger the difference between injected water temperature and initial borehole temperature or water injection conditions, the more obvious the indication of water accepting zones. The numerical examples evidence the feasibility and effectiveness of the proposed method of the two phase flows in engineering.

Extending a Hybrid Finite-Volume-Element Method to Solve Laminar Diffusive Flame

We extend a hybrid finite-volume-element (FVE) method to treat the laminar reacting flow in cylindrical coordinates considering the collocation of all chosen primitive variables. To approximate the advection fluxes at the cell faces, we use the upwind-biased physical influence scheme PIS and derive a few new extended expressions applicable in the cylindrical frame. These expressions are derived for both the Navier-Stokes and reactive flow governing equations, of which the latter expressions are considered novel in the finite-volume formulation. To validate our derived expressions, the current results are compared with the experimental data and other available numerical solutions. The results show that the accuracy of the current expressions is better than those derived by the past similar numerical investigators.

Investigation of the Linear Stability Problem of Electrified Jets, Inviscid Analysis

The instability characteristics of a liquid jet discharging from a nozzle into a stagnant gas are investigated using the linear stability theory. Starting with the equations of motion for incompressible, inviscid, axisymmetric flows in cylindrical coordinates, a dispersion relation is obtained, where the amplification factor of the disturbance is related to its wave number. The parameters of the problem are the laminar velocity profile shape parameter, surface tension, fluid densities, and electrical charge of the liquid jet. The dispersion relation is numerically solved as a function of the wave number. The growth of instabilities occurs in two modes, the Rayleigh and atomization modes. For rWe<1 (where We represents the Weber number and r represents the gas-to-liquid density ratio) corresponds to a Rayleigh or long wave instability, where atomization does not occur. On the contrary, for rWe>>1 the waves at the liquid-gas interface are shorter and when they reach a threshold amplitude the jet breaks down or atomizes. The surface tension stabilizes the flow in the atomization regime, while the density stratification and electric charges destabilize it. Additionally, a fully developed flow is more stable compared to an underdeveloped one. For the Rayleigh regime, both the surface tension and electric charges destabilize the flow.

Explicit analytical solutions for fluid flow and temperature field synergy in cylindrical coordinates

The field synergy principle–an approach to find out the best convection condition–has been successfully proposed and applied. However, the case for worst convection contribution will be worthy researched also. In this paper, four groups of algebraically explicit analytical solutions with heat source or mass flow source are derived for 2D incompressible laminar flow in cylindrical coordinate. The concrete physical meanings of the solutions are analyzed. The extraordinary methods used to obtain solutions are named the method of separating variables with addition and other hybrid method. Besides theoretical meaning, the solutions can be benchmark solutions to develop the computational heat transfer.

Some explicit analytical solutions of field synergy for steady 2D incompressible laminar flow in cylindrical coordinates

In this paper, several analytical solutions for the steady 2D incompressible laminar flow including the heat source, the mass flow source or no any sources in cylindrical coordinates are derived and analyzed based on the field synergy theory. The physical meanings of the solutions are discussed and explained, which are valuable to further develop the field synergy principle and understand how to improve or to weaken the field synergy in practice. The method of separating variables with addition and another extraordinary method are used to obtain analytical solutions in the process of derivation. Besides theoretical meaning, the solutions can serve as the benchmark problems for developing the numerical heat transfer.

An integral perturbation model of flow and momentum transport in rotating microchannels with smooth or microstructured wall surfaces

This paper summarizes the development of an integral perturbation solution of the equations governing flow momentum transport and energy conversion in microchannels between disks of multiple-disk drag turbines such as Tesla turbines. Analysis of this type of flow problem is a key element in optimal design of Tesla drag-type turbines for geothermal or solar alternative energy technologies. In multiple-disk turbines, high speed flow enters tangentially at the outer radius of cylindrical microchannels formed by closely spaced parallel disks, spiraling through the channel to an exhaust at a small radius, or at the center of the disk. Previous investigations have generally developed models based on simplifying idealizations of the flow in these circumstances. Here, beginning with the momentum and continuity equations for incompressible and steady flow in cylindrical coordinates, an integral solution scheme is developed that leads to a dimensionless perturbation series solution that retains the full complement of momentum and viscous effects to consistent levels of approximation in the series solution. This more rigorous approach indicates all dimensionless parameters that affect flow and transport and allows a direct assessment of the relative importance of viscous, pressure, and momentum effects in different directions in the flow. The resulting lowest-order equations are solved explicitly and higher order terms in the series solutions are determined numerically. Enhancement of rotor drag in this type of turbine enhances energy conversion efficiency. We also developed a modified version of the integral perturbation analysis that incorporates the effects of enhanced drag due to surface microstructuring. Results of the model analysis for smooth disk walls are shown to agree well with experimental performance data for a prototype Tesla turbine and predictions of performance models developed in earlier investigations. Model predictions indicate that enhancement of disk drag by strategic microstructuring of the disk surfaces can significantly increase turbine efficiency. Exploratory calculations with the model indicate that turbine efficiencies exceeding 75% can be achieved by designing for optimal ranges of the governing dimensionless parameters.

A finite difference method for 3D incompressible flows in cylindrical coordinates

In this work, a finite difference method to solve the incompressible Navier–Stokes equations in cylindrical geometries is presented. It is based upon the use of mimetic discrete first-order operators (divergence, gradient, curl), i.e. operators which satisfy in a discrete sense most of the usual properties of vector analysis in the continuum case. In particular the discrete divergence and gradient operators are negative adjoint with respect to suitable inner products. The axis r = 0 is dealt with within this framework and is therefore no longer considered as a singularity. Results concerning the stability with respect to 3D perturbations of steady axisymmetric flows in cylindrical cavities with one rotating lid, are presented.

Formulation of a Galerkin spectral element–Fourier method for three-dimensional incompressible flows in cylindrical geometries

A primitive-variable formulation for simulation of time-dependent incompressible flows in cylindrical coordinates is developed. Spectral elements are used to discretise the meridional semi-plane, coupled with Fourier expansions in azimuth. Unlike previous formulations where special distributions of nodal points have been used in the radial direction, the current work adopts standard Gauss–Lobatto–Legendre nodal-based expansions in both the radial and axial directions. Using a Galerkin projection of the symmetrised cylindrical Navier–Stokes equations, all geometric singularities are removed as a consequence of either the Fourier-mode dependence of axial boundary conditions or the shape of the weight function applied in the Galerkin projection. This observation implies that in a numerical implementation, geometrically singular terms can be naively treated by explicitly zeroing their contributions on the axis in integral expressions without recourse to special treatments such as l'Hopital's rule. Exponential convergence of the method both in the meridional semi-plane and in azimuth is demonstrated through application to a three-dimensional analytical solution of the Navier–Stokes equations in which flow crosses the axis.