A simple physical approach for deriving the characteristic equations of fluid dynamics

A simple physical approach for deriving the characteristic equations of fluid dynamics is presented. The approach is based on the physical concept that information propagates through a flowfield along pathlines due to particle motion and along wavelines due to acoustic wave motion. The characteristic equations and compatibility equations are derived in vector forms which are valid in any co-ordinate system.     

On the no-slip boundary conditions for squeeze flow of viscous fluids

A typical class of boundary conditions for squeeze flow problems in lubrication approximation is the one in which the squeezing rate and the width between the squeezing plates are constant. This hypothesis is justified by claiming that the plates moves so slowly that they can be considered static. In this short note we prove that this assumption leads to a contradiction and hence cannot be used.     

A concise method for determining a valve flow coefficient of a valve under compressible gas flow

This work presents a novel method of determining a valve flow coefficient for a valve and transient mass flow rate of compressible gas discharged from a reservoir. The proposed method consists of a set of equations to express the physical phenomena and measurement equipment to measure the indispensable data used in the above equations. Regardless of the kind of valve, the valve flow coefficient can be obtained efficiently and feasibly. The results of this study indicate not only that the valve flow characteristics of the diaphragm valve significantly differ from those of the ball valve, but also that the C_v flow equation conventionally used is no longer valid for the diaphragm valve. The valve flow coefficient of the ball valve determined by the proposed method is about 48 and the representative one proposed by the ANSI/ISA is about 56. In addition, the mass flow rate of gas flow through a valve under transient process can be estimated without using a flow meter. Moreover, the cumulated masses discharged predicted by the method proposed herein are consistent well with those of the experimental results. The deviations are smaller than 6%.     

A parallel unstructured dynamic mesh adaptation algorithm for 3‐D unsteady flows

An unstructured dynamic mesh adaptation and load balancing algorithm has been developed for the efficient simulation of three‐dimensional unsteady inviscid flows on parallel machines. The numerical scheme was based on a cell‐centred finite‐volume method and the Roe's flux‐difference splitting. Second‐order accuracy was achieved in time by using an implicit Jacobi/Gauss–Seidel iteration. The resolution of time‐dependent solutions was enhanced by adopting an h‐refinement/coarsening algorithm. Parallelization and load balancing were concurrently achieved on the adaptive dynamic meshes for computational speed‐up and efficient memory redistribution. A new tree data structure for boundary faces was developed for the continuous transfer of the communication data across the parallel subdomain boundary. The parallel efficiency was validated by applying the present method to an unsteady shock‐tube problem. The flows around oscillating NACA0012 wing and F‐5 wing were also calculated for the numerical verification of the present dynamic mesh adaptation and load balancing algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

A spectral collocation method for compressible,non-similar boundary layers

An efficient and highly accurate algorithm based on a spectral collocation method is developed for numerical solution of the compressible, two-dimensional and axisymmetric boundary layer equations. The numerical method incorporates a fifth-order, fully implicit marching scheme in the streamwise (timelike) dimension and a spectral collocation method based on Chebyshev polynomial expansions in the wall-normal (spacelike) dimension. The discrete governing equations are cast in residual form and the residuals are minimized at each marching step by a preconditioned Richardson iteration scheme which fully couples energy, momentum and continuity equations. Preconditioning on the basis of the finite difference analogues of the governing equations results in a computationally efficient iteration with acceptable convergence properties. A practical application of the algorithm arises in the area of compressible linear stability theory, in the investigation of the effects of transverse curvature on the stability of flows over axisymmetric bodies. The spectral collocation algorithm is used to derive the non-similar mean velocity and temperature profiles in the boundary layer of a ‘fuselage’ (cylinder) in a high-speed (Mach 5) flow parallel to its axis. The stability of the flow is shown to be sensitive to the gradual streamwise evolution of the mean flow and it is concluded that the effects of transverse curvature on stability should not be ignored routinely.     

A study on numerical methods for transient rotating flow induced by starting blades

Based on the finite volume method, three methods for rotational region treatment were presented and validated by simulating two-dimensional accelerating rotational flows. Separate transient incompressible flows induced by cross-shaped blades during starting process were simulated using the dynamic mesh, sliding mesh and dynamic reference frame methods. The computing performance and stability of the three methods were evaluated by comparing numerical results, and the transient characteristics of the accelerating rotational flow were analysed numerically. Results showed that the starting process affected the flow structure and transient characteristics of the accelerating rotational flows. The sliding mesh method showed higher computational efficiency and accuracy compared with other methods, and could easily be extended to solve three-dimensional transient flows in hydraulic machineries under transient operations, such as start-up and shutdown.     

A new approach to potential flow around axisymmetric bodies

A new method is introduced to solve potential flow problems around axisymmetric bodies. The approach relies on expressing the infinite series expansion of the Laplace equation solution in terms of a finite sum which preserves the Laplace solution for the potential function under a Neumann-type boundary condition. Then the coefficients of the finite sum are calculated in a least squares approximation sense using the Gram-Schmidt orthonormalization method. Sample benchmark problems are presented and discussed in some detail. The solutions are accurate and converged faster when a rather small number of terms were used. The method is simple and can be easily programmed.     

Periodic array of traction-free interface cracks subjected to far-field uniform heat flow

A periodic array of interface cracks is subjected to a uniform heat flow in the far field. The crack opening displacements and complex stress intensity factors are determined, using analytic function theory, for the case that the upper half-space is less distortive than the lower half-space.     

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k–ε model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k–ω model.     

The interplay between boundary conditions and flow geometries in shear banding: Hysteresis, band configurations, and surface transitions

We study shear banding flows in models of wormlike micelles or polymer solutions, and explore the effects of different boundary conditions for the viscoelastic stress. These are needed because the equations of motion are inherently non-local and include “diffusive” or square-gradient terms. Using the diffusive Johnson–Segalman model and a variant of the Rolie-Poly model for entangled micelles or polymer solutions, we study the interplay between different boundary conditions and the intrinsic stress gradient imposed by the flow geometry. We consider prescribed gradient (Neumann) or value (Dirichlet) of the viscoelastic stress tensor at the boundary, as well as mixed boundary conditions in which an anchoring strength competes with the gradient contribution to the stress dynamics. We find that hysteresis during shear rate sweeps is suppressed if the boundary conditions favor the state that is induced by the sweep. For example, if the boundaries favor the high shear rate phase then hysteresis is suppressed at the low shear rate edges of the stress plateau. If the boundaries favor the low shear rate state, then the high shear rate band can lie in the center of the flow cell, leading to a three-band configuration. Sufficiently strong stress gradients due to curved flow geometries, such as that of cylindrical Couette flow, can convert this to a two-band state by forcing the high shear rate phase against the wall of higher stress, and can suppress the hysteresis loop observed during a shear rate sweep.     

A formal finite element approach for open boundaries in transport and diffusion ground-water problems

In the application of the finite element method to diffusion and convection-dispersion equations over a ground-water domain, the Galerkin technique was used to incorporate Neumann (or second-type) and Cauchy (or third-type) boundary conditions. While mass movement through open boundaries is a priori unknown, these boundaries are usually treated as a zero Neumann condition at some far distance from the domain of interest. Nevertheless, cheaper and better solutions can be obtained if these unknown conditions are adequately incorporated in the weak formulation and in the transient solution schemes (open boundary condition). Theoretical and numerical proofs are given of the equivalences between this approach and a ‘well-posed’ problem in a semi-infinite domain with a zero Neumann condition at a boundary placed at infinity. Transport and diffusion equations were applied in one dimension to show the numerical performances and limitations of this procedure for some linear and non-linear problems. No a priori limitations are foreseen in order to find similar solutions in two or three dimensions. Thus the spatial discretization in the proximity of open boundaries could be drastically reduced to the domain of interest.     

A finite volume approach to large eddy simulation of compressible,homogeneous, isotropic,decaying turbulence

Large eddy simulation of compressible, homogeneous, isotropic, decaying turbulence in a rectangular box is performed using finite volume techniques. An analysis of the energy spectra obtained from the simulations shows that an agreement with the Kolmogorov law for the inertial range is found only when an appropriate spatial discretization method is used. This agreement is obtained both for a low (0·05) and a moderate (0·6) Mach number when Smagorinsky's subgrid model is employed.     

A lattice model of foam flow in porous media: A percolation approach

Because fluid flow in porous media is opaque to most observational techniques simulations of the processes occurring in porous media have become important. Typical reservoir simulations treat the flow as taking place in some averaged (Darcy-scale) medium but simulations can also be carried out at the level of the network of pores and throats of the porous medium. We report the results of a pore-scale investigation of mechanisms for the alteration of mobility by foam lamella blockage in a network of these spaces and channels of porous media. Saturation and relative permeability curves are obtained using well-known power-law expressions of percolation theory and a rescaling of the percolation parameter readily permits a number of lamella-blocking mechanisms to be treated. An explanation of the shift in breakthrough gas saturation and the deformation of the shape of permeabilityvs saturation curves upon introduction of foam is provided for a variety of blocking mechanisms. The qualitatively different features seen in experimental studies of modification of gas mobility by foam can be rationalized using only two parameters which characterize the throat-size at which blockage commences and the degree of blockage.     

A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow

A boundary integral method for the simulation of the time-dependent deformation of Newtonian or non-Newtonian drops suspended in a Newtonian fluid is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. The implementation of the boundary conditions is facilitated by rewriting the domain integral by means of the Gauss divergence theorem. To apply the divergence theorem smoothness assumptions are made concerning the non-Newtonian stress tensor. The correctness of these assumptions in actual simulations is checked with a numerical validation procedure. The method appears mathematically correct and the numerical algorithm is second order accurate. Besides this validation we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for Newtonian and non-Newtonian drops in two dimensions indicate that the steady state deformation is quite independent of the drop-fluid. The deformation process, however, appears to be strongly dependent on the drop-fluid. For the non-Newtonian drop a mechanical model is developed to describe the time-dependent deformation of the cylinder for small capillary numbers.     

An analytical approach to estimate local liquid heights in horizontal diabatic slug flow

As the intermittent (slug) flow pattern was recently shown to be, along with the stratified flow regime, responsible for circumferential anisothermality of horizontal steam generating tubes operating at moderate steam qualities, the ability to estimate the local liquid levels in such tubes and to compare them with the position of the circumferentially maximum value of the externally applied heat loading appears to be of great practical importance for boiler designers. While the procedure of the estimation of minimum liquid heights (h_L) in horizontal stratified flows was suggested in previous papers, this study presents an analytical approach for engineering evaluations of h_L in the horizontal, diabatic slug flow pattern. It is, importantly, shown that the use of the stratified flow-based approach to evaluate h_L in slug flows results in the overestimation of actual liquid heights which may be detrimental for boiler tubes, especially under circumferentially nonuniform heat loading.     

A novel finite point method for flow simulation

A novel finite point method is developed to simulate flow problems. The mashes in the traditional numerical methods are supplanted by the distribution of points in the calculation domain. A local interpolation based on the properties of Taylor series expansion is used to construct an approximation for unknown functions and their derivatives. An upwind‐dominated scheme is proposed to efficiently handle the non‐linear convection. Comparison with the finite difference solutions for the two‐dimensional driven cavity flow and the experimental results for flow around a cylinder shows that the present method is capable of satisfactorily predicting the flow separation characteristic. The present algorithm is simple and flexible for complex geometric boundary. The influence of the point distribution on computation time and accuracy of results is included. Copyright © 2002 John Wiley & Sons, Ltd.     

A new approach to solving Poisson system for free surface nonhydrostatic flow simulations

A nonhydrostatic finite volume model is presented to simulate free surface flow in a two‐dimensional vertical plane. The algorithm is based on a projection method including the solution of the pressure Poisson equation (PPE). The model is developed in a Cartesian grid in which the size of all the cells in the computational domain, excluding those of the top layer, is constant in time. To simulate the variable water surface, the heights of the top layer cells are variable and proportional to the local water elevation. Taking the layout of the grid system into consideration, a new method is proposed to solve the PPE derived in Cartesian coordinates. In this method, the system of pressure equations is divided into two subsystems, namely a subsystem for the upper layer cells and another for the remaining cells. The coefficient matrix of the former is variable with respect to time, whereas that of the latter remains constant. Therefore, the coefficient matrix of the latter subsystem can be inversed once and saved throughout the simulation. The application of this procedure reduces the computational cost compared with other PPE solvers in certain conditions. The model is applied to simulate a series of numerical tests including strong vertical accelerations and is verified against analytical and experimental results, demonstrating satisfactory performance. Copyright © 2011 John Wiley & Sons, Ltd.     

Application of the Dorodnitsyn finite element method to swirling boundary layer flow

The Dorodnitsyn finite element method for turbulent boundary layer flow with surface mass transfer is extended to include axisymmetric swirling internal boundary layer flow. Turbulence effects are represented by the two-layer eddy viscosity model of Cebeci and Smith¹ with extensions to allow for the effect of swirl. The method is applied to duct entry flow and a 10 degree included-angle conical diffuser, and produces results in close agreement with experimental measurements with only 11 grid points across the boundary layer. The introduction of swirl (w_e/u_e = 0.4) is found to have little effect on the axial skin friction in either a slightly favourable or adverse pressure gradient, but does cause an increase in the displacement area for an adverse pressure gradient. Surface mass transfer (blowing or suction) causes a substantial reduction (blowing) in axial skin friction and an increase in the displacement area. Both suction and the adverse pressure gradient have little influence on the circumferential velocity and shear stress components. Consequently in an adverse pressure gradient the flow direction adjacent to the wall is expected to approach the circumferential direction at some downstream location.