Showing papers on "Navier–Stokes equations published in 1971"

A numerical study of laminar separation bubbles using the Navier-Stokes equations

Abstract: The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ₃

Abstract: It is shown here that a unique solution to the Navier-Stokes equations exists in R3 for a small time interval independent of the viscosity and that the solutions for varying viscosities converge uniformly to a function that is a solution to the equations for ideal flow in R3. The existence of the solutions is shown by transforming the Navier-Stokes equations to an equivalent system solvable by applying fixed point methods with estimates derived from using semigroup theory. Introduction. We wish to find a solution, local in time, to the Cauchy problem for the Navier-Stokes equations for viscous incompressible flow in R3 and show that the solutions of the Navier-Stokes equations for various viscosities converge, as the viscosity goes to zero, to a function that is a solution to the Euler equations for an ideal (inviscid) fluid. The Navier-Stokes equations are (E') av/lt + (v * grad) v-vAv = -grad P + B, VV = v O, with constraints lim v(x, t) = 0 and v(x, 0) = Cx), Ixl +o where x = (x1, X2, X3) is a point in R3; t is in some time interval [0, T]; the velociy v(x, t)=(v1(x, t), v2(x, t), v3(x, t)); the pressure is P(x, t); the force is B(x, t) = (B1(x, t), B2(x, t), B3(x, t)); and the constant v>0 is the viscosity (the coefficient of kinematic viscosity). The Euler equations for ideal flow differ from the Navier-Stokes equations (E') only in that the viscosity term vAv does not occur in the Euler equations. Uniqueness and existence of a solution to the Navier-Stokes equations in R3 has been shown for both bounded and unbounded domains: in both cases existence has been shown only for a sufficiently small time interval. The first results are those of C. W. Oseen [11] and Jean Leray [8]. The time interval where the solution is shown to exist must be small enough to satisfy a condition of form T? Kv, where K is an appropriate constant and v is the viscosity. Thus the length of the time interval Received by the editors January 28, 1970 and, in revised form, June 25, 1970. AMS 1968 subject classifications. Primary 7635, 7646; Secondary 3536, 4750.

Vortex computation by the method of weighted residuals using exponentials

Abstract: A method of weighted residuals for the computation of rotationally symmetric quasicylindrical viscous incompressible vortex flow is presented. The method approximates the axial velocity and circulation profiles by series of exponentials having (N + 1) and TV free parameters, respectively. Exponentials are also used as weighting functions. Formal integration results in a set of (2N + 1) ordinary differential equations for the free parameters. The governing equations are shown to have an infinite number of discrete singularities. Sample solutions for different swirl parameters and three typical vortex flows (initially uniform axial flow, leading edge vortex, and trailing vortex) are presented, and the effects of external axial velocity and circulation gradients are investigated. The computations point to the controlling influence of the inner core flow on vortex behavior. They also confirm the existence of two particular critical swirl parameter values: So, which separates vortex flow which decays smoothly from vortex flows which eventually "breaks down," and Si, the first singularity of the quasi-cylindrical system, at which point physical vortex breakdown is thought to occur. The results are close to the inviscid values for S0 and Si[(2)lf2 and 3.8317/2 for initially uniform axial flow].

Vortex flow over a flat surface with suction

Abstract: Introduction B the use of momentum integral methods, Taylor and Cooke calculated the laminar boundary-layer development on the interior surface of a frustum of a cone, with a potential vortex as the outer flow. Taylor's swirl atomizer problem has also been examined by means of similarity transformations of the boundary-layer equations by Moore, Mager, and Rott and Lewellen, and in Ref. 3 it is concluded that no valid solution can be found. However, in Ref. 4 it is noted that the valid solution may be found if the velocity profile of secondary flow changes sign somewhere in the boundary layer. Contrary to this, in Ref. 5 the existence of similarity solution is denied by a simple and clear consideration. Here, for simplicity we restrict our attention to the vortex flow over an infinite flat surface, i.e., to the case when the vertex angle of a cone is equal to 180°. For this problem, even the complete Navier-Stokes equations have a similarity solution only for a limited range of the Reynolds number." In this Note, we attempt to solve the boundary-layer equations under the boundary condition of distributed suction, and show without any speculation to the singularity on the vortex axis that a formal similarity solution can be obtained for suction parameter greater than a certain value.

Evolution of the Laminar Wake behind a Flat Plate and Its Upstream Influence

Abstract: Analysis of laminar, two-dimensional, viscous flow of an incompressible fluid over the trailing edge of a vanishingly thin flat plate is presented. A coordinate transformation is introduced which admits sufficient scaling of the problem to enable detailed study of the separating flow and its interactions with the inviscid stream, as well as with a presumed boundary-layer flow in an intermediate region. Second-order boundary-layer theory is applied in an iterative manner to extract a first approximation to the displacement thickness and associated induced pressure distribution. It is found that the near-wake region is nonisobaric, the classical isobaric result of Blasius-Goldstein for displacement thickness being slightly high just prior to the trailing edge and about 10% low downstream. Local corrections to the velocity and pressure distributions are obtained by numerically solving the full Navier-Stokes equations, using the second-order boundary-layer results to establish Dirichlet boundary conditions over a rectangular region enclosing the trailing edge point. It is found that the physical extent of this region is of order R~3/4. Within the region, shear at the plate increases with x and becomes very large at the trailing edge. No significant correction to the displacement thickness was found on resolving the second-order boundary-layer problem, using the Navier-Stokes results as inner boundary conditions.

Stability of Normal Shock Waves with Viscosity and Heat Conduction

Abstract: The stability problem, for small arbitrary one‐dimensional disturbances, of a normal shock wave with viscosity and heat conduction in a thermodynamically perfect gas with a Prandtl number of 0.75 is treated, and is formulated explicitly as an eigenvalue problem involving ordinary linear differential equations with polynomial coefficients in a fixed finite domain whose end points are singular points of the differential equations. It is shown by a simple general type of mathematical argument that one possible mode shape for the perturbations is a translation of the shock structure, and that such a disturbance is neutrally stable. For the limiting case of a weak‐shock structure, the equations developed here are shown to reduce systematically to a perturbed form of Burgers' equation. The weak shock structure is shown to be stable for any Prandtl number and general equation of state, and a complete solution for the disturbance eigenvalues and eigenfunctions in this case is derived and discussed.

Calculation of supersonic flow past blunt bodies using the complete Navier-Stokes equations

Abstract: A numerical method is presented for solving the steady problem of supersonic flow of a low density gas past blunt bodies. In this flow regime the effects of compressibility, viscosity, and heat conduction of the gas are significant. The study of the flow field is made using the two-dimensional Navier-Stokes equations for a compressible gas, which are integrated in a finite region near a blunt body. The solution of the boundary value problem is sought by the asymptotic method. An explicit difference scheme [1] is used to approximate the unsteady NavierStokes system. The primary objective of the study is to check on the computation method. The nature of the asymptotic approach to the steady-state solution and the peculiarities of the calculation using the selected difference scheme are clarified; the effect on some surfaces within the flow is analyzed; the computation accuracy is evaluated.

Laminar Incompressible Flow in the Entrance Region of Ducts of Arbitrary Cross Section

Abstract: A combination of the numerical technique of Chorin for the solution of the Navier-Stokes equations and a transformation of the initial value problem to a boundary value problem is shown to allow calculation of the laminar hydrodynamic entrance region of ducts of arbitrary cross section. Numerical examples consisting of the solution for ducts of square and triangular cross sections are presented along with the associated friction factors.

Numerical METHODS FOR Two- and Three-Dimensional Viscous Flow Problems: Applications to Hypersonic Leading Edge Equations,

Abstract: : Several explicit and implicit finite-difference methods, useful for treating two- and three-dimensional viscous flow problems, are compared. These techniques are applied to the single-layer equations previously developed by the authors for continuum leading-edge studies. Stability and accuracy of different schemes, effects of linearization, boundary conditions, coordinate systems and grid size, and the need for iteration are discussed. Solutions are presented for equilibrium and rotational non-equilibrium flow fields and comparisons with experimental data are provided. Different models for the pressure gradient (px) term in the streamwise momentum equation are discussed and it is shown that the effects of upstream influence appear in certain px representations that may be useful when these effects are important. The relationship to so-called sub- or super-critical flows is demonstrated. For three-dimensional geometries, a new predictor-corrector method is devised and tested for stability properties. For a right-angle corner geometry, solutions are compared with explicit results obtained with step sizes three orders of magnitude smaller. The need for iteration in obtaining accurate and consistent results is emphasized. The use of these techniques for two-dimensional unsteady Navier-Stokes solutions is discussed. (Author)

Stability of Uniform Solutions of the Navier-Stokes Equations in n-Dimensions.

Abstract: : The stability of the uniform solutions of the Navier-Stokes equations on all of R sup n, n =or> 3, is proved for a class of perturbations that are small but not necessarily square integrable. Uniform estimates on the rate of decay are given. (Author)

On the Motion of a Viscous Incompressible Fluid in a Tube with Permeable and Deformable Wall

Abstract: : The problem the author considers consists in the study of the motion of a viscous incompressible fluid in a tube with deformable and permeable wall; this problem is, for instance, encountered in the investigation of the flow of blood in artificial arteries. The two-dimensional case is studied, i.e. the plane flow of the fluid, since all the results given refer to this case and cannot be extended directly to flows in three or more dimensions. In order, moreover, to avoid formal complications, the problem is as much implied as possible, although the results obtained hold also for more general cases.

A numerical solution of the Navier-Stokes equations for supercritical fluid thermodynamic analysis

Abstract: An explicit numerical solution of the compressible Navier-Stokes equations is applied to the thermodynamic analysis of supercritical oxygen in the Apollo cryogenic storage system. The wave character is retained in the conservation equations which are written in the basic fluid variables for a two-dimensional Cartesian coordinate system. Control-volume cells are employed to simplify imposition of boundary conditions and to ensure strict observance of local and global conservation principles. Non-linear real-gas thermodynamic properties responsible for the pressure collapse phenomonon in supercritical fluids are represented by tabular and empirical functions relating pressure and temperature to density and internal energy. Wall boundary conditions are adjusted at one cell face to emit a prescribed mass flowrate. Scaling principles are invoked to achieve acceptable computer execution times for very low Mach number convection problems. Detailed simulations of thermal stratification and fluid mixing occurring under low acceleration in the Apollo 12 supercritical oxygen tank are presented which model the pressure decay associated with de-stratification induced by an ordinary vehicle maneuver and heater cycle operation.

Computer simulation of a viscous channel flow

Abstract: A numerical solution of the two — dimensional Navier — Stokes equations for the flow in a duct between parallel plates in the presence of a diaphragm with an orifice at Reynolds number Re=1800 is presented. The plots of the resulting streamlines are illustrated and the computed discharge coefficient is compared with the experimental results.

Laminar Flows Past an Infinitely Thin Disk.

Abstract: : Numerical computations of axisymmetric laminar flows past an infinitely thin disk strongly indicate that the solutions of the Navier-Stokes equations are weakly singular at the tip points. This means that vorticity and pressure at the surface of the disk become infinite with the inverse square root of the distance from the tip. Complete numerical instability which was encountered in previous studies is avoided by excluding the tip from the grid system. Computer output is displayed and discussed for Reynolds numbers 70 and 100.

An Outline of Methods Applicable to Viscous Fluid Flow Problems

Abstract: : Given the Navier-Stokes equations in general orthogonal curvilinear coordinates, specific equations for particular cases are indicated. In the notation of singular perturbation theory, the first approximation to the inner expansion (boundary-layer) is solved in simple cases. Some remarks are also made concerning shock-wave boundary-layer interactions; in particular, no approximation to the equations is found in the immediate vicinity of the interaction. Base flows are also mentioned. (Author)

Time-dependent viscous flow in a straight channel

Abstract: An infinite straight channel, filled with an incompressible viscous fluid, is closed at one end by a piston. This is set in motion with finite acceleration and then maintained at constant velocity until the flow pattern in the fluid reaches a steady state. The development of velocity profiles, stream lines, and streak lines is investigated by direct numerical solution of the complete Navier Stokes equations. It is found that the nonconvex velocity profiles found in previous work on steady-state problems appear from the beginning, and their development is studied. In the downstream region alternative methods can be used which serve as a check on the accuracy of the numerical procedures. The asymptotic behaviour downstream is studied in some detail.

Rotor Blade Boundary Layer Calculation Programs

Abstract: : The development of the turbulent compressible boundary layer on two typical helicopter rotors, for a range of hover conditions, has been calculated using two different analytical methods: the differential method, which uses the differential form of the boundary layer momentum equations and solves for the local velocity gradients, and the integral method, which uses the integrated form of the momentum equations and solves for the development of the characteristic boundary layer thickness parameters and skew angle. Both methods decouple the chordwise and spanwise boundary layer equations without making any small crossflow assumptions. The effects of rotational speed, vortex-induced crossflows, surface curvature, and applied chordwise pressure gradients were evaluated separately and in combination to simulate rotor airfoil boundary layer growth. (Author)

Numerical solution of the unsteady Navier- Stokes equations and application to flow in a rectangular cavity with a moving wall

Abstract: Computer program for numerical solution of unsteady Navier-Stokes equations and application to flow in rectangular cavity with moving wall

The Stability of Cellular Branching Solutions of the Navier-Stokes Equations

Abstract: In 1958 J. T. Stuart [1] introduced an energy balance method for the study of the growth of Taylor vortices for supercritical Reynolds numbers where linear stability theory does not apply anymore. Besides the classical energy method leading to sufficient stability limits (cf. [2]), this was the first systematic attempt toward an understanding of nonlinear phenomena in hydrodynamic stability theory.