Nonlinear Navier-Stokes Equations Research Articles

Numerical spectral method for flows through anevrisms

A numerical Tau-Chebyshev spectral method is used for solving the set of nonlinear Navier-Stokes equations which describes Newtonian unsteady flow. The special case of a tube with expansion of the cross-section is studied. The results show the influence of the hydrodynamic, geometrical and rheological parameters upon the evolution of a ‘separation area’. Good agreement of experimental results (these being based on LDA and dataprocessing techniques) and numerical results was obtained.

Development of the Tollmin — Schlichting wave in a boundary layer, on a plate

The development of three-dimensional perturbations of constant frequency in a boundary layer on a semi-infinite plate is studied within the framework of the Navier-Stokes (NS) equations for an incompressible fluid. A case in which the Tollmin-Schlichting (TS) /1, 2/ wave has reached a point on the plate corresponding to the lower branch of the neutral stability curve (NSC), obtained by solving the eigenvalue problem for the Orr-Sommerfeld equation, is discussed. An asymptotic solution of the non-linear NS equations at large Reynolds numbers in given. According to the result obtained, first we have a non-linear process taking place within the NSC near its lower branch, for the separated TS wave with an amptitude that is not too small, leading to gradual reduction in the wave amplitude. Since the Blasius boundary layer is not parallel, the process changes when the amplitude increases. Thus the point at which the amplitude of the TS wave is at a minimum, lies within the loop of the NSC. Therefore, when the experiment is compared with the linear theory based on the Orr-Sommerfeld equation, the theory must be corrected.

The Effect of the Earth’s Rotation on Channel Flow

The influence that the rotation of the earth has on laminar channel flow is investigated theoretically. The full nonlinear Navier-Stokes equations relative to a reference frame rotating with the earth are solved numerically for laminar flow in a rectangular channel whose axis is aligned east-west: the orientation which yields the most drastic effect. It is demonstrated that for channels of moderate width (less than 1 ft for the flow of most liquids), the rotation of the earth can give rise to a roll instability which has a severe distortional effect on the classical parabolic velocity profile. Consequently, the usual assumption made of neglecting the effect of the earth’s rotation in the calculation of channel flow can lead to serious errors unless the channel is substantially smaller than this size. It is briefly shown that similar effects would be expected for turbulent channel flow when the channel width is approximately an order of magnitude larger.

Bifurcation of developed flow in rectangular channels rotating about the transverse axis

The linear problem of the stability of Poiseuille flow between two infinite plates rotating about an axis parallel to the plates and normal to the flow direction was studied in [1, 2]. It was established that the flow is least stable with respect to disturbances in the form of standing waves known as Taylor eddies. The experimental data of [1, 3] and the results [4] of a numerical integration of the Navier-Stokes equations for channels with cross sections highly elongated in the direction of the axis of rotation are in good agreement with the conclusions of the linear theory. In the case of channels with a side ratio of the order of unity, which is of greater practical interest, the primary flow becomes essentially three-dimensional and evolves with variation of the governing criteria: the Reynolds and Rossby numbers. Obviously, this seriously complicates the use of the methods of the linear theory. The effect of the ratio of the sides of the cross section on the stability of the primary flow regime was studied experimentally in [3]. The present article describes the results of an investigation of the problem based on a numerical method of integrating the nonlinear Navier-Stokes equations. Moreover, an asymptotic estimate of the stability limit of the primary regime, based on a local condition of inviscid instability of rotating flows, is presented.

A generalization of Uzawa's algorithm for the solution of the Navier‐Stokes equations

AbstractUsawa's algorithm provides an efficient method for solving the divergence‐free Stokes problem. On the other hand, the Newton–Raphson scheme is very popular for the solution of the nonlinear Navier–Stokes equations. We propose here a new method that combines these two algorithms and converges to a divergence‐free solution of the nonlinear Navier–Stokes equations.

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.

On finite element approximations of problems having inhomogeneous essential boundary conditions

The analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered. The results are given for linear second order elliptic partial differential equations and for the nonlinear stationary Navier-Stokes equations. For certain easily implemented boundary treatments, optimal error estimates and numerical examples are provided for problems posed on polyhedral domains.

Numerical Simulation of the Solar Granulation

AbstractThe solar granulation has been simulated by numerical solution of the multidimensional, time-dependent, nonlinear Navier-Stokes equations applied to the solar atmosphere. Granules may be explained as buoyantly rising bubbles created at the level where T = 8000 K, and which have collapsed into vortex rings. The calculation is in quantitative agreement with observations and has a number of implications for solar physics and convection theory.

Asymptotic method for peristaltic transport.

The purpose of this paper is to justify an asymptotic method developed for the study of peristaltic transport in a tube of arbitrary cross section. Within the framework of long wave approximation, the three-dimensional nonlinear Navier-Stokes equations are reduced to a sequence of two-dimensional linear boundary value problems of Laplace and biharmonic operators. It is shown that, if a Reynolds number is less than some constant, the solution of the approximate equations is indeed an asymptotic approximation to the exact solution of the problem as the ratio of the maximum radius of the tube to the wave length of the peristaltic motion of the wall tends to zero, and the error estimates are expressed in L 2 norms. Furthermore, under the same condition the exact solution is shown to be unique and stable under arbitrary perturbation of spatially periodic disturbance. Application of the stability condition to peristaltic transport in a tube of circular cross section is given.

Numerical simulation of the solar granulation

The solar granulation has been simulated by numerical solution of the multidimensional, time-dependent, nonlinear Navier-Stokes equations applied to the solar atmosphere. Granules may be explained as buoyantly rising bubbles created at the level where T = 8000 K, and which have collapsed into vortex rings. The calculation is in quantitative agreement with observations and has a number of implications for solar physics and convection theory.

Computational simulation of transient blast loading on three-dimensional structures

Abstract A computing technique is described for the numerical solution of three-dimensional, time-dependent fluid dynamic problems with irregularly shaped and movable boundaries (or internal obstacles) confining the flow. This paper presents in detail the method used in treating the three-dimensional boundary conditions. The numerical solution technique for the full nonlinear Navier-Stokes equations has been presented elsewhere. The general method is illustrated here through comparison of computed surface pressure time histories with the corresponding experimental data for a plane-shock wave (5 psi overpressure) passing over a rectangular parallelepiped.

Isolated Singularities in Steady State Fluid Flow

New results concerning isolated singularities for classical and distribution solutions of the nonlinear stationary Navier-Stokes equations are established. Also new results are established for the equations of Stokes and Oseen. Some counterexamples are given. In particular, it is shown that the nonlinear result for distribution solutions in dimension 2 is, in a certain sense, a best possible result.

Generalized and Classical Solutions of the Nonlinear Stationary Navier-Stokes Equations

Generalized and Classical Solutions of the Nonlinear Stationary Navier-Stokes Equations

Generalized and classical solutions of the nonlinear stationary Navier-Stokes equations

New regularity results in domains of Euclidean 3-space are established for the generalized solutions of the nonlinear stationary Navier-Stokes equations in terms of Dini criteria on the external force.

The bid method for the steady-state Navier-Stokes equations

The Biharmonic Driver (BID) method uses direct (non-iterative) linear biharmonic solvers to obtain solutions to the nonlinear two-dimensional steady-state incompressible Navier-Stokes equations. Low Reynolds number recirculating flows are solved in 6–8 non-time-like iterations. Modifications are suggested for flow-through problems, and comparisons are made with other semidirect methods.

Calculating three-dimensional fluid flows at all speeds with an Eulerian-Lagrangian computing mesh

A computing technique is presented for the calculation of three-dimensional, time-dependent fluid dynamics problems. The full nonlinear Navier-Stokes equations are solved with a finite-difference scheme based upon an Arbitrary Lagrangian-Eulerian (ALE) computing mesh with vertices that may move with the fluid (Lagrangian), remain fixed (Eulerian), or move in any prescribed manner. The method is applicable to three-dimensional flows at all speeds, employing an implicit formulation similar to the Implicit Continuous-Fluid Eulerian (ICE) technique. Marker particles are used that may follow exactly the motion of the fluid to aid in flow visualization, or they can represent particulate matter whose behavior is affected by inertia, drag, gravity, and molecular and turbulent diffusion. Calculational examples are shown in the form of a variety of perspective view plots.

Anomalous shear viscosity near a critical point in multicomponent systems

A derivation of the fluctuating force appearing in the nonlinear Navier-Stokes equation for a multicomponent fluid system neat a critical point is presented. We point out that this nonlocal fluctuating force leads to an anomalous shear viscosity at any kind of critical point, in particular also in pure fluids.

Isolated singularities for solutions of the nonlinear stationary Navier-Stokes equations

The notion for (u, p) to be a distribution solution of the nonlinear stationary Navier-Stokes equations in an open set is defined, and a theorem concerning the removability of isolated singularities for distribution solutions in the punctured open ball B ( 0 , r 0 ) − { 0 } B(0,{r_0}) - \{ 0\} is established. This result is then applied to the classical situation to obtain a new theorem for the removability of isolated singularities. In particular, in two dimensions this gives a better than expected result when compared with the theory of removable isolated singularities for harmonic functions.

Isolated Singularities for Solutions of the Nonlinear Stationary Navier-Stokes Equations

Isolated Singularities for Solutions of the Nonlinear Stationary Navier-Stokes Equations

Capacity and the Nonlinear Navier–Stokes Equations

A classical result concerning capacity theory and removable sets for harmonic functions of finite energy is extended to solutions of the nonlinear Navier–Stokes equations. The theory of multiple trigonometric series is used in proving the basic lemma, and a new theorem concerning capacity theory and removable sets for first order systems is also established.