Narrow Results

In his ground breaking paper [4], Leray proves that if the velocity $w(t)$ is a smooth solution to the 3-dimensional, incompressible, Navier–Stokes equations, that blows up in finite time $T$, then $\parallel w(t){\parallel }_{\infty }\ge C/\sqrt{T-t}$. This provides a lower bound for possible blowouts of solutions to the Navier–Stokes equations. The possibility of a blowout occurring at this minimal level was explored in [3, 5]. This paper supplements and extends these results, looking in more detail at the concept of ancient solutions.

The aerodynamic performance of a moving flapping wing is investigated numerically using a fluid–solid interaction model. The flow field is assumed to be two-dimensional, viscous, unsteady, and incompressible. Galerkin-least/squares finite element method is used to account for the convective nature of the flow field. The flow solver is coupled with the rigid body equations of motion describing the movement of the flapping wing. An interface-capturing technique is developed and tailored to the present model to capture the flapping wing during its movement. The developed algorithm is validated against published data with a great degree of success. Then, simulations for a flapping wing with a stationary center of gravity and free center of gravity motion, are performed. This is followed by a parametric study through which the effects of the wing’s initial position, maximum flapping angle, Reynolds number, the mass of the wing, the gravitational acceleration, and the critical frequency for each Reynolds number are studied. Finally, the voluntary vertical takeoff of a fruit fly, Drosophila Melanogaster is simulated using the proposed model.

This paper presents the results of an experiment to study the performance of multitasking techniques on the CRAY Y-MP using a single algorithm. The algorithm is a compact difference scheme for the solution of the incompressible, two-dimensional, time-dependent Navier-Stokes equations. Three implementations of multitasking on the CRAY Y-MP are considered. These are: macrotasking (parallelism at the subroutine level), microtasking (parallelism at the do-loop level), and autotasking (automatic multitasking). The three techniques are briefly described. The implementation of the algorithm is discussed in relation to these techniques and the results for four problem sizes are presented. The timing results for the three techniques using few processors are in general comparable. For the eight processor case, this algorithm achieved a speedup of seven and a processing rate of over one GFLOPS using macrotasking. The results are also compared to earlier results on the CRAY-2.

Since the validity of the Navier-Stokes equations is well established, any fluid dynamic phenomenon could be calculated if methods for solving them correctly are obtained. A fair portion of this dream seems to have come true through the remarkable development of supercomputers and solution algorithms that have made the simulation of high-Reynolds-number flows possible. For understanding the underlying flow mechanism, means of properly visualizing the computed flow field are needed and have been developed. On the whole, computer simulation is becoming the most effective tool for the study of fluid dynamics.

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus T^d of any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍⁿ(T^d), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍⁿ distance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

A multigrid (MG) numerical solution procedure for a mathematical model for epitaxial growth in metalorganic chemical vapor deposition reactors is developed. The model is governed by the full compressible Navier–Stokes equations for two-dimensional (plane and axisymmetric) laminar flows extended by transport equations for the chemical species, the energy equations and the equation of state, together with boundary conditions providing information on the reactor geometry and experimental conditions. The MG numerical solution is implemented in a finite element procedure that uses the same linear finite elements to discretize both convection and diffusion fluxes by adding to the model a stabilizing term, resulting in easy implementation of the procedure. A deposition process for a model of GaAs growth is calculated with our procedure showing consistency with experimental data.

A multiple-relaxation-time (MRT) collision operator is introduced into the author's rectangular lattice Boltzmann method for simulating fluid flows. The model retains both the advantages and the standard procedure of using a constant transformation matrix in the conventional MRT scheme on a square lattice, leading to easy implementation in the algorithm. This allows flow problems characterized by dominant feature in one direction to be solved more efficiently. Two numerical tests have been carried out and shown that the proposed model is able to capture complex flow characteristics and generate an accurate solution if an appropriate lattice ratio is used. The model is found to be more stable compared to the original rectangular lattice Boltzmann method using the single relaxation time.

In the present work, the meshless local Petrov–Galerkin vorticity-stream function (MLPG-VF) method is extended to solve two-dimensional laminar fluid flow and heat transfer equations for high Reynolds and Rayleigh numbers. The characteristic-based split (CBS) scheme which uses unity test function is employed for discretization, and the moving least square (MLS) method is used for interpolation of the field variables. Four test cases are considered to evaluate the present algorithm, namely lid-driven cavity flow with Reynolds numbers up to and including $10^4$, flow over a backward-facing step at Reynolds number of 800, natural convection in a square cavity for Rayleigh numbers up to and including $10^8$, and natural convection in a concentric square outer cylinder and circular inner cylinder annulus for Rayleigh numbers up to and including $10^7$. In each case, the result obtained using the proposed algorithm is either compared with the results from the literatures or with those obtained using conventional numerical techniques. The present algorithm shows stable results at lower or equal computational cost compared to the other upwinding schemes usually employed in the MLPG method. Close agreements between the compared results as well as higher accuracy of the proposed method show the ability of this stabilized algorithm.

The conventional meshless local Petrov–Galerkin method is modified to enable the method to solve turbulent convection heat transfer problems. The modifications include developing a new computer code which empowers the method to adopt nonlinear equations. A source term expressed in terms of turbulent viscosity gradients is appended to the code to optimize the accuracy for turbulent flow domains. The standard $\kappa -\epsilon$ transport equations, one of the most applicable two equation turbulent viscosity models, is incorporated, appropriately, into the developed code to bring about both versibility and stability for turbulent natural heat transfer applications. The amenability of the new developed technique is tested by applying the modified method to two conventional turbulent fluid flow test cases. Upon the obtained acceptable results, the modified technique is, next, applied to two conventional natural heat transfer test cases for their turbulent domain. Based on comparing the results of the new technique with those of the available experimental or conventional numerical methods, the proposed method shows good adaptability and accuracy for both the fluid flow and convection heat transfer applications in turbulent domains. The new technique, now, furthers the applicability of the mesh-free local Petrov-Galerkin (MLPG) method to turbulent flow and heat transfer problems and provides much closer results to those of the available experimental or conventional numerical methods.

The asymmetric vortex flows about slender bodies at high angle of attack are investigated numerically. 3D O-O type multi-block structured grids are adopted for the computation, and a series of small spatial disturbances are imposed on the leeward body surface near the tip to trigger the asymmetry. The unsteady Navier-Stokes equations are solved using a dual-time method based on the cell-centered finite-volume scheme. Comparisons between numerical results and experimental data demonstrate that the asymmetric vortex pattern observed in the experiment is successfully simulated. In a certain small perturbation level, the magnitude of the side-force is independent of the size of the spatial perturbation, but its direction is dependent on the direction of the spatial perturbation, i.e. a counterclockwise or clockwise circumferential angle from the leeward meridian. When perturbation is removed, the flow field returns to its original asymmetric shape after a period of oscillation.

A three-dimension numerical wave model (3DWAVE) has been developed to simulate free surface flows. The model solves Navier-Stokes equations for two-phase flows of air and water. The level set method is employed to track water surfaces. The model is tested for water sloshing in a 3-D confined tank. The relative error in the mass and total energy computation is less than 1%. Excellent agreements between numerical results and analytical solution are obtained for free surface calculation. The nonlinearity in the 3-D fluid sloshing is analyzed. These have laid a foundation on research of breaking waves.

In this paper, the BR2 high-order Discontinuous Galerkin (DG) method is used to discretize the 2D Navier-Stokes (N-S) equations. The nonlinear discrete system is solved using a Newton method. Both preconditioned GMRES methods and block Gauss-Seidel method can be used to solve the resulting sparse linear system at each nonlinear step in low-order cases. In order to save memory and accelerate the convergence in high-order cases, a linear p-multigrid is developed based on the Taylor basis instead of the GMRES method and the block Gauss-Seidel method. Numerical results indicate that highly accurate solutions can be obtained on very coarse grids when using high order schemes and the linear p-multigrid works well when the implicit backward Euler method is employed to improve the robustness.

The purpose of this study is to implement a new analytical method (the DTM-Padé technique, which is a combination of the differential transform method (DTM) and the Padé approximation) for solving Navier–Stokes equations. In this letter, we will consider the DTM, the homotopy perturbation method (HPM) and the Padé approximant for finding analytical solutions of the three-dimensional viscous flow near an infinite rotating disk. The solutions are compared with the numerical (fourth-order Runge–Kutta) solution. The results illustrate that the application of the Padé approximants in the DTM and HPM is an appropriate method in solving the Navier–Stokes equations with the boundary conditions at infinity. On the other hand, the convergence of the obtained series from DTM-Padé is greater than HPM-Padé.

We have performed a comparative computational study of magnetic particle alignment in a strong magnetic field and ambient viscous fluid using the first-principles Navier–Stokes equation as well as numerical modeling using the finite element method (FEM). FEM solution has been compared with the solution of the unsteady rotations of a magnetic particle in a viscous fluid during the alignment process described with nonlocal integro-differential equations (IDEs) for torque and angular velocity. The assumption of nonlocality comes from the history term of acceleration torque with a non-Basset kernel function, which has its origin in the presence of vortices in the flow of a particle fluid environment due to its unsteady rotation and ambient fluid inertia and friction. The flow vortices of the ambient fluid are explicitly shown in the solution of the FEM model. Moreover, the solution of the time evolution of the alignment angle and angular velocity for the FEM model are in good agreement with an IDE model which we have recently developed. The obtained results are justification for interchangeability of the FEM and IDE models, which may have important consequences for large-scale simulations of magnetic microparticles.

In this paper, the authors consider the two-dimensional Navier–Stokes equations defined on Ω = ℝ × (-L, L) and , where is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ω_m → Ω as m → ∞. Let and be the global attractors of the equations corresponding to Ω and Ω_m, respectively, we establish that for any neighborhood of , the global attractor enters if m is large enough.

We analyze a nonoverlapping domain decomposition method for the treatment of two-dimensional compressible viscous flows around airfoils. Since at some distance to the given profile the inertial forces are strongly dominant, there the viscosity effects are neglected and the flow is assumed to be inviscid. Accordingly, we consider a decomposition of the original flow field into a bounded computational domain (near field) and a complementary outer region (far field). The compressible Navier–Stokes equations are used close to the profile and are coupled with the linearized Euler equations in the far field by appropriate transmission conditions, according to the physical properties and the mathematical type of the corresponding partial differential equations. We present some results of flow around the NACA0012 airfoil and develop an a posteriori analysis of the approximate solution, showing that conservation of mass, momentum and energy are asymptotically attained with the linear model in the far field.

This paper deals with problems arising in the sensitivity analysis for fluid-structure interaction systems. Our model consists of a fluid described by the incompressible Navier–Stokes equations interacting with a solid under large deformations. We obtain a linearized problem which allow us to compute the derivative of the state variable with respect to a given boundary parameter. We use a particular definition of the first-order correction for the perturbed state and consider a weak arbitrary Euler–Lagrange formulation for the coupled system.

We consider a splitting approximation of the magnetohydrodynamic equations which consists of decoupling the Navier–Stokes and Maxwell equations which form the magnetohydrodynamic system, on small time intervals. To compute the approximate solutions we start with a pair of initial velocity and magnetic fields and alternately solve Navier–Stokes and dynamo equations. The main result of the paper states that for sufficiently smooth initial data, the proposed scheme converges in L^p norm with p > 3. Besides the applications in numerical analysis, this scheme can also be a valuable tool in understanding the physical and mathematical structure of the magnetohydrodynamic equations.

We establish some existence and uniqueness results for a nonlinear elliptic equation. The problem has a diffusion matrix A(x, u) such that A(x, s)ξξ ≥ β(s)|ξ|², with β : (s₀, + ∞) ↦ ℝ a continuous, strictly positive function which goes to infinity when s is near s₀. On the other hand, . Also, the right-hand side f belongs to L¹(Ω). We make use of the concept of renormalized solutions adapted to our problem.

We derive a new relaxation method for the compressible Navier–Stokes equations endowed with general pressure and temperature laws compatible with the existence of an entropy functional and Gibbs relations. Our method is an extension of the energy relaxation method introduced by Coquel and Perthame for the Euler equations. We first introduce a consistent splitting of the diffusion fluxes as well as a global temperature for the relaxation system. We then prove that under the same subcharacteristic conditions as for the relaxed Euler equations and for a specific form of the global temperature and the heat flux splitting, the stability of the relaxation system may be obtained from the non-negativity of a suitable entropy production. A first-order asymptotic analysis around equilibrium states confirms the stability result. Finally, we present a numerical implementation of the method allowing for a straightforward use of Navier–Stokes solvers designed for ideal gases as well as numerical results illustrating the accuracy of the proposed algorithm.