Showing papers on "Navier–Stokes equations published in 1989"

Lattice Gas Automata for the Navier-Stokes Equations. A New Approach to Hydrodynamics and Turbulence

Abstract: A very brief presentation of how lattice gas hydrodynamics is made. It includes key references.

Numerical Solution of the Navier-Stokes Equations*

Abstract: A finite-difference method for solving the time-dependent Navier-Stokes equations for an incompressible fluid is introduced. This method uses the primitive variables, i.e. the velocities and the pressure, and is equally applicable to problems in two and three space dimensions. Test problems are solved, and an application to a three-dimensional convection problem is presented.

Spectral element methods for the incompressible Navier-Stokes equations

Abstract: Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergence rate of spectral schemes. The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems, as well as the efficient solution of the resulting discrete equations by rapidly convergent tensor-product-based iterative procedures. Several examples of spectral element simulation of moderate Reynolds number unsteady flow in complex geometry are presented.

The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity

Abstract: A particle method for convection-diffusion equations based on the approximation of diffusion operators by integral operators and the use of a particle method to solve integro-differential equations previously described is presented and studied. The isotropic diffusion operators are dealt with first. Two approximation possibilities are obtained, depending on whether or not the integral operator is positive. An extension of the method to anisotropic diffusion operators follows. The consistency and the accuracy of the method require much more complex conditions on the cutoff functions than in the isotropic case. After detailing these conditions, several examples of cutoff functions which can be used for practical computations are given. A detailed error analysis is then performed. 24 refs.

Solution of the 2D Navier-Stokes equations on unstructured adaptive grids

Abstract: This paper presents a solution adaptive scheme for solving the Navier-Stokes equations on a n unstructured mixed grid of triangles and quadrilaterals. The solution procedure uses an explicit Itunge-Kutta finite volun~e time marching scheme. The solution is begun on a coarse grid and points are added adaptively during the solution procedure using criteria such as pressure and velocity gradients. In viscous regions the gradients are essentially one dimensional, and we use quadrilateral elements in these regions to facilitate the one dimensional refinement required for the efficient resolution of boundary layers and wakes. The effect of turbulence is modeled by the inclusion of a K E tubulence rnodel. When used for analyzing flows in turbomachinery blade rows, terms rrpresenting the effects of changes in strearnsheet lhicknrss and radius, and the effects of rotation are included. Axisymnletric flows with swirl can also be analyzed. Solutions are presented for several examples that illustrate the capability of the algorithm.

Finite element analysis of the compressible Euler and Navier-Stokes equations

Abstract: A space-time finite element method is presented for solving the compressible Euler and Navier-Stokes equations. The proposed formulation includes the variational equation, predictor multi-corrector algorithms, boundary conditions, and solution strategies. The variational equation is based on the time-discontinuous Galerkin method, in which the physical entropy variables are employed. A least-squares operator and a discontinuity-capturing operator are added, resulting in a high-order accurate and unconditionally stable method. Implicit/explicit predictor multi-corrector algorithms, applicable to steady as well as unsteady problems, are presented; techniques are developed to enhance their efficiency. Implementation of boundary conditions is addressed; in particular, a technique is introduced to satisfy nonlinear essential boundary conditions, and a consistent method is presented to calculate boundary fluxes. A multi-element group, domain decomposition algorithm is presented for solving the nonsymmetric linear systems. This algorithm employs an iterative strategy based on the generalized minimal residual (GMRES) procedure. A two-level preconditioning technique is presented, which significantly accelerates the convergence of the GMRES procedure. Numerical results are presented to demonstrate the performance of the method.

Incompressible Navier-Stokes and Euler Limits of the Boltzmann Equation

Abstract: We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2, 3, For macroscopic times τ = t/ϵN, ϵ « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ϵN−1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t0, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t0 which are close, to O(ϵ2) in a suitable norm, to the local Maxwellian [p/(2πT)d/2]exp{−[v − ϵu(x,t)]2/2T} with constant density p and temperature T. This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.

Extending hydrodynamics via the regularization of the Chapman-Enskog expansion.

Abstract: A regularization procedure of the Chapman-Enskog expansion is introduced. It extends the hydrodynamical description into higher gradients domains. This was intended but never accomplished satisfactorily by the Burnett equations. The resulting macroscopic system formally has the same form (and complexity) as the Navier-Stokes equations, with the new transport coefficients being wavelength dependent. It reduces in the long-wavelength limit to the Burnett equations. In agreement with experimental evidence, strong shock layers are narrower in comparison with the predictions of the Navier-Stokes theory.

Exact Solutions of the Unsteady Navier-Stokes Equations

Abstract: The unsteady Navier-Stokes equations are a set of nonlinear partial differential equations with very few exact solutions. This paper attempts to classify and review the existing unsteady exact solutions. There are three main categories: parallel, concentric and related solutions, Beltrami and related solutions, and similarity solutions. Physically significant examples are emphasized.

Efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations

Abstract: A cell-vertex scheme for the three-dimensional Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing with locally varying coefficients, a multigrid method and carefully controlled dissipative terms, very good convergence rates are obtained for two- and three-dimensional flows. Details of the acceleration techniques, which are important for convergence on meshes with high aspect-ratio cells, are discussed. Emphasis is put on the analysis of the stability properties of the implicit smoothing of the explicit residuals with coefficients, which depend on cell aspect ratios.

Multi-grid methods for stokes and navier-stokes equations

Abstract: In the present paper we introduce transforming iterations, an approach to construct smoothers for indefinite systems. This turns out to be a convenient tool to classify several well-known smoothing iterations for Stokes and Navier-Stokes equations and to predict their convergence behaviour, epecially in the case of high Reynolds-numbers. Using this approach, we are able to construct a new smoother for the Navier-Stokes equations, based on incomplete LU-decompositions, yielding a highly effective and robust multi-grid method. Besides some qualitative theoretical convergence results, we give large numerical comparisons and tests for the Stokes as well as for the Navier-Stokes equations. For a general convergence theory we refer to [29].

Flow-thermal-structural study of aerodynamically heated leading edges

Abstract: A finite element approach for integrated fluid-thermal-structural analysis of aerodynamically heated leading edges is presented. The Navier-Stokes equations for high speed compressible flow, the energy equation, and the quasi-static equilibrium equations for the leading edge are solved using a single finite element approach in one integrated, vectorized computer program called LIFTS. The fluid-thermal-structural coupling is studied for Mach 6.47 flow over a 3-in diam cylinder for which the flow behavior and the aerothermal loads are calibrated by experimental data. Issues of the thermal-structural response are studied for hydrogen-cooled, super thermal conducting leading edges subjected to intense aerodynamic heating.

Are fem solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!)

Abstract: SUMMARY It is generally accepted that mixed and penalty finite element methods can routinely solve the incompressible Navier-Stokes equations. This paper shows by means of simple examples that problems can arise even for the simpler Stokes equations. The causes of the problem fall in either of two categories: round-off and ill conditioning, or a poor choice of pressure discretization. Nonsensical solutions can be obtained. Computation of the discrete divergence of the flow field is a simple and powerful tool to diagnose such conditions. In the first part of the paper several simple techniques for minimizing the effect of round-off are reviewed. In the second part it is shown that, for coupled flow problems, care must be exercised in the choice of the pressure approximation. A unified treatment of various observations by different workers is presented. This should prove useful for general users of the finite element method.

Pressure-based Navier-Stokes solver using the multigrid method

Abstract: APRESSURE-BASED implicit procedure to solve steadystate Navier-Stokes equations is developed. A multistep pressure correction procedure with an implicit density treatment is used to establish the pressure and velocity fields. A multigrid relaxation scheme is used to solve the scalar matrices resulting from the finite-volume formulation. The algorithm is valid for all Mach number flows ranging from incompressible to supersonic flow regimes.

Navier-Stokes analysis of solid propellant rocket motor internal flows

Abstract: A multidimensional implicit Navier-Stokes analysis that uses numerical solution of the ensemble-averaged Navier-Stokes equations in a nonorthogonal, body-fitted, cylindrical coordinate system has been applied to the simulation of the steady mean flow in solid propellant rocket motor chambers. The calculation procedure incorporates a two-equation (k-epsilon) turbulence model and utilizes a consistently split, linearized block-implicit algorithm for numerical solution of the governing equations. The code was validated by comparing computed results with the experimental data obtained in cylindrical-port cold-flow tests. The agreement between the computed and experimentally measured mean axial velocities is excellent. The axial location of transition to turbulent flow predicted by the two-equation (k-epsilon) turbulence model used in the computations also agrees well with the experimental data. Computations performed to simulate the axisymmetric flowfield in the vicinity of the aft field joint in the Space Shuttle solid rocket motor using 14,725 grid points show the presence of a region of reversed axial flow near the downstream edge of the slot.

The linearization method in hydrodynamical stability theory

Abstract: Estimates of solutions of the linearized Navier-Stokes equations Estimates of integral operators in $L_p$ Some estimates of solutions of evolution equations Estimates of the ""leading derivatives"" of solutions of evolution equations Applications to parabolic equations and imbedding theorems The linearized Navier-Stokes equations An estimate of the resolvent of the linearized Navier-Stokes operator Estimates of the leading derivatives of a solution of the linearized steady-state Navier-Stokes equations Stability of fluid motion Stability of the motion of infinite-dimensional systems Conditions for stability Conditions for instability. Conditional stability Stability of periodic motions Formulation of the problem The problem with initial data A condition for asymptotic stability A condition for instability Conditional stability Stability of auto-oscillatory regimes Instability of cycles Damping of the leading derivatives.

Numerical solution of the incompressible Navier-Stokes equations for steady-state and time-dependent problems

Abstract: An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a higher-order flux-difference splitting technique for the convective terms and a second-order central difference for the viscous terms. The steady-state solution of flow through a square duct with a 90 deg bend is computed and the results are compared with experimental data. Good agreement is observed. A comparison with an analytically known exact solution is then performed to verify the time accuracy of the algorithm. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Newton solution of inviscid and viscous problems

Abstract: The application of Newton iteration to inviscid and viscous airfoil calculations is examined. Spatial discretization is performed using upwind differences with split fluxes. The system of linear equations which arises as a result of linearization in time is solved directly using either a banded matrix solver or a sparse matrix solver. In the latter case, the solver is used in conjunction with the nested dissection strategy, whose implementation for airfoil calculations is discussed. The boundary conditions are also implemented in a fully implicit manner, thus yielding quadratic convergence. Complexities such as the ordering of cell nodes and the use of a far field vortex to correct freestream for a lifting airfoil are addressed. Various methods to accelerate convergence and improve computational efficiency while using Newton iteration are discussed. Results are presented for inviscid, transonic nonlifting and lifting airfoils and also for laminar viscous cases.

Finite Analytic Solution of Convective Heat Transfer for Tube Arrays in Crossflow: Part I—Flow Field Analysis

Abstract: The convective motion in two types of tube array is solved numerically by the Finite Analytic Method. The Finite Analytic Method utilizes the local analytic solution of governing differential equations in obtaining its discretized algebraic representation. Both in-line tube arrays and staggered tube arrays with longitudinal and transverse pitches of 2 are studied. The geometries are expressed in boundary-fitted coordinates on which the Navier-Stokes equations and energy equation are solved. Solutions for Reynolds numbers of 40, 120, 400, and 800 are obtained. Differences in stream function, vorticity function, and location of separation and reattachment for flow past in-line tube arrays and staggered tube array are predicted and compared. The zone of separation for both arrays tends to increase with increasing Reynolds number. The predicted results on flow field and heat transfer are shown to agree with available experimental measurements.

Non‐Gaussian statistics in isotropic turbulence

Abstract: Several measures of non‐Gaussian behavior in simulations of decaying isotropic turbulence are compared with predictions of the direct‐interaction approximation (DIA) at an initial Rλ≈35. The quantities studied include the variances and wavenumber power spectra of (a) the total nonlinear term in the Navier–Stokes equation, (b) the time derivative of the velocity at a point, (c) pressure fluctuations, and (d) vorticity and dissipation fluctuations. The direct‐interaction approximation gives a good quantitative prediction of the variance of the time derivative and the variance of total nonlinear term, and a fair qualitative prediction of the power spectrum associated with the latter. But DIA totally fails to capture the non‐Gaussian statistics associated with pressure fluctuation and vorticity spottiness. Some discussion is given of demands that vorticity and dissipation statistics place upon theories of tubulence at moderate and high Reynolds numbers.

A point implicit unstructured grid solver for the euler and Navier–Stokes equations

Abstract: An upwind finite element technique that uses cell centered quantities and implicit and/or explicit time marching has been developed for computing hypersonic laminar viscous flows using adaptive unstructured triangular grids. A structured grid of quadrilaterals is laid out near the body surface. For inviscid flows the method is stable at Courant numbers of over 100,000. A first order basic scheme and a higher order flux corrected transport (FCT) scheme have been implemented. This technique has been applied to the problem of predicting type III and IV shock wave interactions on a cylinder, with a view of simulating the pressure and heating rate augmentation caused by an impinging shock on the leading edge of a cowl lip of an engine inlet. The predictions of wall pressure and heating rates compare very well with experimental data. The flow features are very distinctly captured with a sequence of adaptively generated grids. The adaptive mesh generator and the upwind Navier-Stokes solver are combined in a set of programs called LARCNESS, an acronym for Langley Adaptive Remeshing Code and Navier-Stokes Solver.

Two‐particle description of turbulence, Markov property, and intermittency

Abstract: It is shown that the hypothesis of independent increments for velocity, which is widely used by many authors [e.g., A. M. Obukhov, Adv. Geophys. 6, 113 (1959)] in the Lagrangian description of turbulence, is inconsistent with the Navier–Stokes equations in a fundamental way. A more general Lagrangian description of turbulent velocity such as the Markov process with dependent increments, which recognizes the condition of incompressibility and the important phenomenon of intermittency, is proposed. A model of intermittent relative motion of fluid particles in turbulent flow is presented. The high‐order Lagrangian moments and the probability distribution are obtained. The distribution for the intermittent vorticity is also proposed.

On the validity of Taylor’s hypothesis for wall‐bounded flows

Abstract: The results of large eddy simulation (LES) of the Navier–Stokes equations are used to evaluate the validity of Taylor’s hypothesis of frozen turbulence, which states that the time derivative of some instantaneous quantity is proportional to its derivative in the streamwise direction, for incompressible plane channel flow. Time and space derivatives in the streamwise direction of the velocity components are, in fact, found to be well correlated. Root‐mean‐square fluctuations of the terms in Taylor’s hypothesis also support the validity of this hypothesis above the buffer layer. The good agreement between LES and experimental results indicates that errors in the evaluation of derivatives in the streamwise direction are due mostly to insufficient resolution.