Unified compact ADI methods for solving nonlinear viscous and nonviscous wave equations

In this paper, two unified alternating direction implicit (ADI) methods, based on the combination of fourth-order compact difference for the approximations of the second spatial derivatives with approximation factorization of difference operators, are presented for solving a two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations, respectively. By the discrete energy method, it is shown that their solutions converge to exact solutions with an order of two in time and four in space in L²- and H¹-norms. Finally, numerical findings testify the computational efficiency of the algorithms.     

Non-Formation of Vacuum States for Compressible Navier–Stokes Equations

We prove that weak solutions of the Navier–Stokes equations for compressible fluid flow in one space dimension do not exhibit vacuum states, provided that no vacuum states are present initially. The solutions and external forces that we consider are quite general: the essential requirements are that the mass and energy densities of the fluid be locally integrable at each time, and that the L ² _loc-norm of the velocity gradient be locally integrable in time. Our analysis shows that, if a vacuum state were to occur, the viscous force would impose an impulse of infinite magnitude on the adjacent fluid, thus violating the hypothesis that the momentum remains locally finite. Received: 20 March 2000 / Accepted: 16 July 2000     

Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method

The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769–795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air–water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier–Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model’s capability in predicting the interaction dynamics of rising bubbles.     

Long-Time Behavior for the Two-Dimensional Motion of a Disk in a Viscous Fluid

In this article, we study the long-time behavior of solutions of the two-dimensional fluid-rigid disk problem. The motion of the fluid is modeled by the two-dimensional Navier–Stokes equations, and the disk moves under the influence of the forces exerted by the viscous fluid. We first derive L ^p ?L ^q decay estimates for the linearized equations and compute the first term in the asymptotic expansion of the solutions of the linearized equations. We then apply these computations to derive time-decay estimates for the solutions to the full Navier–Stokes fluid-rigid disk system.     

Integro-differential equation method in the hydrodynamics of an incompressible viscous fluid

An integro-differential equation method is proposed to describe the motion of an incompressible viscous fluid. The method uses an analogy between the hydrodynamic equations for an incompressible viscous fluid and the magnetostatic equations. An analysis is made of the flow of an incompressible viscous fluid round an object as a specific application. The solution automatically satisfies the boundary conditions at the surface of the object and at infinity. Zh. éksp. Teor. Fiz. 112, 1332–1339 (October 1997)     

Viscous Bianchi type II universes with intergalactic magnetic field

Homogeneous Bianchi type II models are studied for universes containing a viscous fluid and a large scale magnetic field. The present values of the magnetic field, the anisotropy in the microwave radiation background and the Hubble parameters are obtained by numerical integration of the Einstein and Maxwell equations from the beginning of the lepton era (10¹² K) up to the present (2.7 K). It is found that an inflationary scenario combined with a Bianchi type II geometry leads to data that conflict with observation.     

Topological evolution near the turbulent/non-turbulent interface in turbulent mixing layer

The topological evolution near the turbulent/non-turbulent interface (TNTI) in turbulent mixing layer is studied by means of statistical analysis of the invariants of velocity gradient tensor (VGT) based on direct numerical simulation data. The dynamics of topological evolution is investigated in terms of the source terms of the evolution equations for the invariants, including the pressure effect term, viscous effect term and interaction term among the invariants. It is found that the local topology of fluid particles at the TNTI evolves from non-focal region to focal region in the plane of the second (Q) and the third (R) invariants of the VGT. The topological evolution is mainly associated with the pressure effect term in the TNTI region. According to the analysis of the evolution of enstrophy and dissipation, the enstrophy increase and the dissipation decrease are revealed in the TNTI region, which are caused by viscous vorticity diffusion near the TNTI. A weak correlation between the strain rate and the rotation rate is found in the TNTI region which is related to the reduction of enstrophy production. The alignments between vorticity and strain near the TNTI are investigated and a strong alignment of the vorticity with the extensive eigenvector direction is identified in the TNTI region.     

Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet

This Letter endeavours to complete an earlier numerical analysis for flow and heat transfer in a viscous fluid over a sheet nonlinearly stretched by extending the investigation in two directions. On one side, the effects of thermal radiation are included in the energy equation, and, on the other hand, the prescribed wall heat flux case (PHF case) is also analyzed. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The variations of dimensionless surface temperature as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include a nonlinearly stretching sheet, thermal radiation, viscous dissipation and power-law index of the wall temperature parameters, are graphed and tabulated.     

Piecewise parabolic method on a local stencil for magnetized supersonic turbulence simulation

Stable, accurate, divergence-free simulation of magnetized supersonic turbulence is a severe test of numerical MHD schemes and has been surprisingly difficult to achieve due to the range of flow conditions present. Here we present a new, higher order-accurate, low dissipation numerical method which requires no additional dissipation or local “fixes” for stable execution. We describe PPML, a local stencil variant of the popular PPM algorithm for solving the equations of compressible ideal magnetohydrodynamics. The principal difference between PPML and PPM is that cell interface states are evolved rather that reconstructed at every timestep, resulting in a compact stencil. Interface states are evolved using Riemann invariants containing all transverse derivative information. The conservation laws are updated in an unsplit fashion, making the scheme fully multidimensional. Divergence-free evolution of the magnetic field is maintained using the higher order-accurate constrained transport technique of Gardiner and Stone. The accuracy and stability of the scheme is documented against a bank of standard test problems drawn from the literature. The method is applied to numerical simulation of supersonic MHD turbulence, which is important for many problems in astrophysics, including star formation in dark molecular clouds. PPML accurately reproduces in three-dimensions a transition to turbulence in highly compressible isothermal gas in a molecular cloud model. The low dissipation and wide spectral bandwidth of this method make it an ideal candidate for direct turbulence simulations.     

Self-similar solutions for implosion and reflection of coalesced shocks in a plasma: Spherical and cylindrical geometries

Approximate analytic solutions to the self similar equations of gas dynamics for a plasma, treated as an ideal gas with specific heat ratioγ = 5/3, are obtained for the implosion and subsequent reflection of various types of shock sequences in spherical and cylindrical geometries. This is based on the lowest-order polynomial approximation, in the reduced fluid velocity, for a suitable nonlinear function of the sound velocity and the fluid velocity. However, the method developed here is powerful enough to be extended analytically to higher order polynomial approximations, to obtain successive approximations to the exact self-similar solutions. Also obtained, for the first time, are exact asymptotic solutions, in analytic form, for the reflected shocks. Criteria are given that may enable one to make a choice between the two geometries for maximising compression or temperature of the gas. These solutions should be useful in the study of inertial confinement of a plasma. An erratum to this article is available at .     

Effects of thermal radiation on the MHD flow past a vertical plate

The effects of thermal radiation on the MHD flow over a vertical and porous plate of an optically thin gray, electrically conducting, viscous and incompressible fluid are studied. The differential equations and their boundary conditions, describing the problemunder consideration, are dimensionalized and the numerical solution is obtained. The numerical results for the velocity and temperature profiles are shown for different dimensionless parameters entering the problem under consideration, such as the radiation parameter S, Grashof number G, Prandtl number P, and the magnetic parameter M.     

Relativistic Mechanics of Continuous Media

In this work we study the relativistic mechanics of continuous media on a fundamental level using a manifestly covariant proper time procedure. We formulate equations of motion and continuity (and constitutive equations) that are the starting point for any calculations regarding continuous media. In the force free limit, the standard relativistic equations are regained, so that these equations can be regarded as a generalization of the standard procedure. In the case of an inviscid fluid we derive an analogue of the Bernoulli equation. For irrotational flow we prove that the velocity field can be derived from a potential. If in addition, the fluid is incompressible, the potential must obey the d'Alembert equation, and thus the problem is reduced to solving the d'Alembert equation with specific boundary conditions (in both space and time). The solutions indicate the existence of light velocity sound waves in an incompressible fluid (a result known in previous literature ⁽¹⁹⁾ ). Relaxing the constraints and allowing the fluid to become linearly compressible one can derive a wave equation, from which the sound velocity can again be computed. For a stationary background flow, it has been demonstrated that the sound velocity attains its correct values for the incompressible and nonrelativistic limits. Finally viscosity is introduced, bulk and shear viscosity constants are defined, and we formulate equations for the motion of a viscous fluid.     

A new viscous sublayer influx (VSI) concept for near-wall turbulent momentum,heat and mass transfer

This paper presents a new viscous sublayer influx (VSI) concept to describe near-wall turbulent momentum, heat and mass transfer. Based on visual studies, this concept takes account of a viscous sublayer adjacent to the wall, which is not directly affected by the bursts occurring in the wall region. Fluid penetrates only due to a wallward flow into this viscous sublayer. Thus, in contrast to the known surface renewal concept, the new VSI concept is consistent with visual flow studies and, in addition, makes it possible to meet the experimentally found limiting condition Sh ∼ ³√Sc_Sc→∞ for mass transfer. In this work, two models have been developed from the new VSI concept. The simplified viscous sublayer influx model follows the known models in literature and provides analytical equations for the profiles in the wall region. This model gives an explanation for the varying experimental results on the time intervals between successive bursts and predicts them in quantity by using measured Sherwood numbers at very high Schmidt numbers. The second, more detailed viscous sublayer influx model approximates the wallward flow in the viscous sublayer with a spherical stagnation point flow. The profiles are calculated from two ordinary differential equations. Using measured Sherwood numbers at very high Schmidt numbers, this model provides normal velocity fluctuations at the wall that agree well with experimental data. Furthermore, both models provide axial velocity fluctuations near the wall and Nusselt/Sherwood numbers in the range 0.5 ≤ Pr, Sc≤ 10⁵ that both correspond with experimental data.     

Physiological Flow of Jeffrey Six Constant Fluid Model due to Ciliary Motion

The main purpose of this article is to present a mathematical model of ciliary motion in an annulus. In this analysis, the peristaltic motion of non-Newtonian Jeffrey six constant fluid is observed in an annulus with ciliated tips in the presence of heat and mass transfer. The effects of viscous dissipation are also considered. The flow equations of non-Newtonian fluid for the two-dimensional tube in cylindrical coordinates are simplified using the low Reynolds number and long wave-length approximations. The main equations for Jeffrey six constant fluid are considered in cylindrical coordinates system. The resulting nonlinear problem is solved using the regular perturbation technique in terms of a variant of small dimensionless parameter α. The results of the solutions for velocity, temperature and concentration field are presented graphically. B_k is Brinkman number, ST is soret number, and SH is the Schmidth number. Outcome for the longitudinal velocity, pressure rise, pressure gradient and stream lines are represented through graphs. In the history, the viscous-dissipation effect is usually represented by the Brinkman number.     

Variational principle in canonical variables,Weber transformation,and complete set of the local integrals of motion for dissipation-free magnetohydrodynamics

The intriguing problem of the “missing” MHD integrals of motion is solved in this paper; i.e., analogues of the Ertel, helicity, and vorticity invariants are obtained. The two latter have been discussed earlier in the literature only for specific cases, and the Ertel invariant is presented for the first time. The set of ideal MHD invariants obtained appears to be complete: to each hydrodynamic invariant corresponds its MHD generalization. These additional invariants are found by means of the fluid velocity decomposition based on its representation in terms of generalized potentials. This representation follows from the discussed variational principle in Hamiltonian (canonical) variables, and it naturally decomposes the velocity field into the sum of “hydrodynamic” and “ magnetic” parts. The “missing” local invariants are expressed in terms of the “ hydrodynamic” part of the velocity and therefore depend on the (nonunique) velocity decomposition; i.e., they are gauge-dependent. Nevertheless, the corresponding conserved integral quantities can be made decomposition-independent by the appropriate choice of the initial conditions for the generalized potentials. It is also shown that the Weber transformation of MHD equations (partial integration of the MHD equations) leads to the velocity representation coinciding with that following from the variational principle with constraints. The necessity of exploiting the complete form of the velocity representation in order to deal with general-type MHD flows (nonbarotropic, rotational, and with all possible types of breaks as well) in terms of single-valued potentials is also under discussion. The new basic invariants found allow one to widen the set of the local invariants on the basis of the well-known recursion procedure.     

Modeling of steady Herschel–Bulkley fluid flow over a sphere

Characteristics of the incompressible flow of Herschel–Bulkley fluid over a sphere were studied via systematic numerical modeling. A steady isothermal laminar flow mode was considered within a wide range of flow parameters: the Reynolds number 0 < Re ≤ 200, the Bingham number 0 ≤ Bn ≤ 100, and the power index 0.3 ≤ n ≤ 1. The numerical solution to the hydrodynamic equations was obtained using the finite volume method in the axisymmetric case. The changes in flow structures, pressure and viscous friction distribution, and integral drag as a function of the flow rate and fluid rheology are shown. Depending on whether plastic or inertial effects dominate in the flow, the limiting cases were identified. The power law and Bingham fluid flows were studied in detail as particular cases of the Herschel–Bulkley rheological model. Based on the modeling results, a new correlation was developed that approximates the calculated data with an accuracy of about 5% across the entire range of the input parameters. This correlation is also applicable in the particular cases of the power law and Bingham fluids.     

On linear and non-linear hydromagnetic fluid motion generated by oscillating solid boundaries. Part I

In the present paper we start an investigation of the generation of second order vorticity (streaming) in an electric conducting, viscous and compressible fluid embedded in an external magnetic field. The equations used are those pertaining to the one-fluid model. First, fairly general equations for the generation of fluid vorticity as well as magnetic vorticity are derived. By using the method of successive approximations, the systems of equations of first and second order are developed. By the use of dimensional analysis the importance in the vorticity generation of the different generating terms is briefly investigated. Next the linear equations for two-dimensional problems are solved in the case of fluid motion above a flexible plate, vibrating in an external magnetic field.     

Comparison of Several Spatial Discretizations for the Navier–Stokes Equations

Grid convergence studies for subsonic and transonic flows over airfoils are presented in order to compare the accuracy of several spatial discretizations for the compressible Navier–Stokes equations. The discretizations include the following schemes for the inviscid fluxes: (1) second-order-accurate centered differences with third-order matrix numerical dissipation, (2) the second-order convective upstream split pressure scheme (CUSP), (3) third-order upwind-biased differencing with Roe's flux-difference splitting, and (4) fourth-order centered differences with third-order matrix numerical dissipation. The first three are combined with second-order differencing for the grid metrics and viscous terms. The fourth discretization uses fourth-order differencing for the grid metrics and viscous terms, as well as higher-order approximations near boundaries and for the numerical integration used to calculate forces and moments. The results indicate that the discretization using higher-order approximations for all terms is substantially more accurate than the others, producing less than two percent numerical error in lift and drag components on grids with less than 13,000 nodes for subsonic cases and less than 18,000 nodes for transonic cases. Since the cost per grid node of all of the discretizations studied is comparable, the higher-order discretization produces solutions of a given accuracy much more efficiently than the others.     

Bound states of topological defects in parametrically excited capillary waves

Bound states of topological defects arising in a tetragonal lattice formed by two orthogonal standing parametrically excited capillary surface waves are investigated. Such bound states are shown to consist either of two topological charges of one sign (type 1) or of topological charges having opposite signs (type 2). It was found that bound states of type 1 move primarily along wave fronts, and type 2 bound states move at an angle of 45^○ to the wave fronts forming a tetragonal lattice. A system of four coupled Ginzburg–Landau equations is proposed to model bound states. Numerical modeling of this system gave solutions corresponding to type 1 bound states observed in experiment.     

Stability of axisymmetric swirl flows of viscous incompressible fluid

A new method of solution to the problem of stability of the swirl flow of viscous incompressible fluid is developed. The method based on expansion of the required function into power series of radial coordinate allows an avoidance of difficulties related to numerical integration of the system of differential equations with a singular point. Stability of the Poiseuille flow in a rotating pipe is considered as an example.