Showing papers on "Navier–Stokes equations published in 1976"

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

Abstract: The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Solutions of the Navier-Stokes equations for vortex breakdown

Abstract: Steady solutions of the Navier-Stokes equations, in terms of velocity and pressure, for breakdown in an unconfined viscous vortex are obtained numerically using the artificial compressibility technique of Chorin combined with an ADI finite-difference scheme. Axisymmetry is assumed and boundary conditions are carefully applied at the boundaries of a large finite region in an axial plane while resolution near the axis is maintained by a coordinate transformation. The solutions, which are obtained for Reynolds numbers up to 200 based on the free-stream axial velocity and a characteristic core radius, show that breakdown results from the diffusion and convection of vorticity away from the vortex core which, because of the strong coupling between the circumferential and axial velocity fields in strongly swirling flows, can lead to stagnation and reversal of the axial flow near the axis.

Numerical Boundary Conditions for Viscous Flow Problems

Abstract: A method for treating certain troublesome boundary conditions in the numerical solution of time-dependent incompressible viscous flow problems is presented. This method is developed on the basis of an integral representation for the velocity vector which contains the entire kinematics of the problem, including the boundary conditions of concern. It is shown that for the exterior flow problem the freestream condition is satisfied at infinity exactly, and the need to treat a farfield condition is removed by the use of the integral representation. The distribution of a nonvelocity variable on the solid boundary, i.e., the "extraneous" boundary condition needed for both the exterior and the interior flows, are shown to be governed by the kinematics of the problem. The method is shown to accurately follow the local generation of vorticity on the solid boundary computationally.

Computation of Separated Transonic Turbulent Flows

Abstract: The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-

Fluid velocities in induction melting furnaces: Part I. Theory and laboratory experiments

Abstract: Fluid flow in an induction furnace due to electromagnetic stirring forces is predicted theoretically from furnace design parameters by the simultaneous solution of the Maxwell and Navier Stokes equations. Streamline plots and velocity profiles are obtained and compared with surface velocities measured experimentally. The measurements were made on a mercury pool stirred inductively by a Tocco 30 kW 3 kHz induction melting unit. The agreement between the experimental measurements and theoretical predictions was good considering that no curve fitting by manipulation of adjustable parameters was involved. It is believed that such a model would be of value in the design and development of induction furnaces.

Numerical Methods for Separated Flow Solutions around a Circular Cylinder

Abstract: Numerical solutions of the Navier-Stokes equations were obtained for separated flows around a circular cylinder at Reynolds numbers 40, 80, and 200. The flowfields were obtained by using three finite-difference techniques. The implicit scheme solved by matrix factorizations gave the best accuracy and used the least computer time. The flow pattern in the recirculating region of a circular cylinder begins to oscillate as the Reynolds number exceeds 40. The calculated drag coefficients, separation angles, and Strouhal numbers were compared with available experimental data. Computational inaccuracy resulting from numerical approximations needs to be identified before a complicated flow phenomenon can be realistically analyzed.

Computation of Two-Dimensional, Viscous Nozzle Flow

Abstract: The calculation of viscous nozzle flows can be accomplished by either solving the inviscid-core and viscous-boundary-layer equations separately or by solving the viscous equations for the entire flowfield. In the inviscid-core, boundary-layer approach, the assumption is made that the boundary layer is thin when compared to the nozzle diameter. However, for Reynolds numbers on the order of 10/sup 3/ based on the throat diameter, this assumption is questionable. On the other hand, while the viscous equation approach is physically desirable, the computations tend to be rather lengthy. Therefore, the object of this research was to modify an efficient inviscid code, to solve the viscous equations. This new code, called VNAP (Viscous Nozzle Analysis Program), is then used to calculate the flow in a chemical laser nozzle. This numerical solution is compared with both the inviscid-core, boundary-layer solution and experimental data.

Approximation of the Eigenvalues of a Nonselfadjoint Operator Arising in the Study of the Stability of Stationary Solutions of the Navier–Stokes Equations

Abstract: In this paper, rate of convergence estimates are established for the approximate calculation by the finite element method of the eigenvalues of the linearized operator which arises in the analysis of the stability of stationary solutions of the Navier–Stokes equations in a convex polygon.

Optimal Control of a System Govebned by the Navier-Stokes Equations Coupled with the Heat Equation

Abstract: Publisher Summary This chapter focuses on optimal control of a system governed by the navier-stokes equations coupled with the heat equation; and discusses the problem of free convection, thermal diffusivity, kinematic viscosity, the problem of optimal control, iterative method for the construction of an element that satisfies the necessary condition for optimality, the method of perturbation (often called the method of artificial compressibility) a convergence theorem which proves the solution of the perturbed system corresponding to a optimal control converges, optimal control of the system of equations, formulation of an adjoint system, the original system of equations in which a system of Cauchy-Kowaleska type is obtained, some preliminary results for a fixed control, various theorems, and lemmas.

The Fluid Motion Due to a Rotating Disk

Abstract: The solution of the Navier--Stokes equation for fluid motion due to a rotating disk includes characteristic parameters. Accurate values are presented for three of these parameters and are compared with values obtained by Newman's numerical integration technique. (DLC)

A numerical study of viscous flow around an airfoil

Abstract: An integrodifferential method, previously formulated in terms of velocity and vorticity vectors, is reformulated in terms of stream function and vorticity for two-dimensional incompressible viscous flows The reformulated integrodifferential method is shown to retain the distinguishing feature of the previous formulation in permitting the confinement of the solution field to the viscous region of the flow and consequently offers great computational advantages The application of this procedure in a study of an incompressible flow around an impulsively started 9% thick symmetric Joukowski airfoil at an angle of attack of 15 deg and a Reynolds number of 1000 is discussed Numerical results are presented and compared with available finite-difference results

Studies of the merging of vortices

Abstract: Finite difference equations of the two‐dimensional unsteady incompressible Navier Stokes equations are constructed to study the merging of vortices. Comparison with asymptotic solutions show that the latter remain as good approximations even when the viscous cores of the vortices overlap each other.

Navier-Stokes Solutions of the Flowfield in an Internal Combustion Engine

Abstract: The flowfield inside the cylinder of a reciprocating internal combustion engine is calculated by solving the complete Navier--Stokes equations by use of a time-dependent finite-difference technique. The main problem is to solve these equations, including multicomponent diffusion and finite-rate chemical reactions (for H-C-O-N chemistry), and obtain a complete solution for the velocity, pressure, temperature, and chemical composition of the flowfield throughout the four-stroke cycle of a piston-cylinder arrangement. At present, this total problem is a horrendous task, and its exact numerical solution will most likely be an evolutionary process over a long period. The flowfield between the top of the cylinder and the face of the piston is computed as a function of space and time as the piston moves through the conventional 4-stroke cycle, with the inlet and exhaust valves opening and closing appropriately. Combustion is not considered in detail. Solutions are obtained for 2-D geometry (infinite aspect ratio engine) and low Reynolds number.

Inviscid Solutions of the Flowfield in an Internal Combustion Engine

Abstract: Theme T paper is a direct companion to that of Griffin et al. In Ref. 1, the complete Navier-Stokes equations were applied to the solution of the viscous flowfield inside an internal combustion engine for a complete 4-stroke cycle-intake, compression, power, and exhaust. The major problem with such solutions is their restriction to low Reynolds numbers in order to remain within practical computer times. In contrast, the present paper investigates the inviscid flowfield in an 1C engine. Some advantages of an inviscid solution are: 1) full-scale engines running under atmospheric conditions can be readily treated in reasonably practical computer times; 2) the true flowfield inside real 1C engines might be reasonably modeled via a large inviscid core plus a viscous boundary layer at the walls, as conventionally done in aerodynamics; and 3) an inviscid solution might be the most practical vehicle for initial studies of the interaction between the flowfield and the combustion processes.

On the Navier-Stokes equations with constant total temperature

Abstract: For various applications in fluid dynamics, it is assumed that the total temperature is constant. Therefore, the energy equation can be replaced by an algebraic relation. The resulting set of equations in the inviscid case is analyzed. It is shown that the system is strictly hyperbolic and well posed for the initial value problems. Boundary conditions are described such that the linearized system is well posed. The Hopscotch method is investigated and numerical results are presented.

Turbulence and Navier Stokes equations: Proceedings of the conference held at the University of Paris-Sud, Orsay, 12-13 June 1975

Abstract: Finite-time regularity for bounded and unbounded ideal incompressible fluids using holder estimates.- Modified dissipativity for a non linear evolution equation arising in turbulence.- A generic property of the set of stationary solutions of Navier stokes equations.- Two strange attractors with a simple structure.- Direct bifurcation of a steady solution of the Navier-stokes equations into an invariant torus.- Factorization theorems for the stability of bifurcating solutions.- Mesures et dimensions.- Singular perturbation and semigroup theory.- Les equations spectrales en turbulence homogene et isotrope. Quelques resultats theoriques et numeriques.- Intermittent turbulence and fractal dimension: Kurtosis and the spectral exponent 5/3+B.- The Lorenz attractor and the problem of turbulence.- Pattern formation in convective phenomena.- Turbulence and Hausdorff dimension.- Local existence of ?? solutions of the euler equations of incompressible perfect fluids.

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

Abstract: 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.