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A sequence of approximate solutions for the nonlinear, nonstationary Navier-Stokes equations for a two-dimensional domain, from which explicit error estimates and rates of convergence are obtained, is described. This sequence of approximate solutions is based primarily on the Newton-Kantorovich method.
A research effort was conducted with the goal of reducing computer time of a Navier Stokes Computer Code for prediction of viscous flow fields about lifting bodies. A two-dimensional, time-dependent, laminar, transonic computer code (STOKES) was modified to incorporate a non-uniform timestep procedure. The non-uniform time-step requires updating of a zone only as often as required by its own stability criteria or that of its immediate neighbors. In the uniform timestep scheme each zone is updated as often as required by the least stable zone of the finite difference mesh. Because of less frequent update of program variables it was expected that the nonuniform timestep would result in a reduction of execution time by a factor of five to ten. Available funding was exhausted prior to successful demonstration of the benefits to be derived from the non-uniform time-step method.
An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.
A new numerical method used to drastically reduce the computation time required to solve the Navier-Stokes equations at flight Reynolds numbers is described. The new method makes it possible and practical to calculate many important three-dimensional, high Reynolds number flow fields on computers.
An uncoupled time asymptotic alternating direction implicit method for solving the Navier-Stokes equations was tested on two laminar parallel mixing flows. A constant total temperature was assumed in order to eliminate the need to solve the full energy equation; consequently, static temperature was evaluated by using algebraic relationship. For the mixing of two supersonic streams at a Reynolds number of 1,000, convergent solutions were obtained for a time step 5 times the maximum allowable size for an explicit method. The solution diverged for a time step 10 times the explicit limit. Improved convergence was obtained when upwind differencing was used for convective terms. Larger time steps were not possible with either upwind differencing or the diagonally dominant scheme. Artificial viscosity was added to the continuity equation in order to eliminate divergence for the mixing of a subsonic stream with a supersonic stream at a Reynolds number of 1,000.
A method of automatic body-fitted curvilinear coordinate generation is described and used to construct a finite-difference solution of the full incompressible time-dependent Navier-Stokes equations for the unsteady laminar viscous flow arbitrary two-dimensional airfoils or any other two-dimensional body. A method of controlling the spacing of the coordinate lines encircling the body is developed in order to treat high Reynolds number flows, since the coordinate lines must concentrate near the surface to a greater degree as the Reynolds number increases. Multiple airfoils and submerged hydrofoils are treated as illustrative examples. The solution shows good agreement with the Blasius boundary layer solution for the flow past a semi-infinite flat plate.
The aerodynamic sound described by the Lighthill-Curle solution is reexamined using a method of matched asymptotic expansions. The governing Navier-Stokes equations written in nondimensional form are expanded for a small Mach number. First- and second-order solutions for the pressure field are obtained, and the singular nature of the expansion at large distances is indicated. The nearfield pressure is governed by the Poisson equation, whereas the farfield equations describe a linear wave system in a dissipative medium. The pseudosound is related to the incompressible Reynolds stresses associated with a solenoidal velocity field, the velocity, the pressure perturbation, and their derivatives on the boundaries. A uniformly valid first-order solution for the pressure is obtained. It is shown that viscosity, thermal conductivity, and entropy in the flow do not contribute to the first-order noise generation, while the viscous stress contributes to noise only from some boundaries. The application of the proposed perturbation method to a subsonically moving surface and a hot jet is discussed.
Comparisons are made over a range of Mach numbers from 1.5 to 7.5 at a fixed Reynolds number for various amounts of mass removal (boundary-layer bleed). Results indicate that boundary-layer theory can be used for design purposes in the range of bleed rates required to eliminate separation in an isolated, strong, adverse pressure gradient. Only about 5% to 10% of the boundary-layer mass flow must be removed to prevent such separations. Direct comparisons of bleed rates required to eliminate separation predicted by boundary-layer and Navier-Stokes equations for an M = 3.5-engine inlet centerbody show comparable results.
A numerical investigation of vortex breakdown has been undertaken in an attempt to understand its properties, and the mechanisms responsible for it. Solutions of the full steady axisymmetric Navier-Stokes equations for breakdown in an unconfined viscous vortex have been obtained for core Reynolds numbers up to 200, for a two-parameter family of assumed upstream velocity distributions. Diffusion and convection of vorticity away from the vortex core, and the strong coupling between the circumferential and axial velocity fields in highly-swirling flows, are shown to lead to stagnation and reversal of the axial flow near the axis. The various theories of vortex breakdown are considered in light of the present numerical solutions.
A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.
The structure and decay of a trailing vortex were analyzed during the numerical solutions of the full Navier-Stokes equations. Unsteady forms of the governing equations were recast in terms of circulation, vorticity, and stream function as dependent variables, and a second upwind finite difference scheme was used to integrate them with prescribed initial and boundary conditions. The boundary conditions at the outer edge and at the outflow section of the trailing vortex were considered. Different models of the flow were postulated, and solutions were obtained describing the development of the flow as integration proceeds in time. A parametric study was undertaken with a view to understanding the various phenomena that may possibly occur in the trailing vortex. Using the Hoffman and Joubert law of circulation at the inflow section, the results of this investigation were compared with experimental data for a Convair 990 wind model and a rectangular wing. With an exponentially decaying law of circulation at the inflow section and an adverse pressure gradient at the outer edge of the trailing vortex, solutions depict vortex bursting through the sudden expansion of the core and/or through the stagnation and consequent reversal of the flow on the axis. It was found that this bursting takes place at lower values of the swirl ratio as the Reynolds number increases.
The results of a comprehensive numerical investigation of the basic capabilities of the finite element method (FEM) for numerical solution of compressible flow problems governed by the two-dimensional and axis-symmetric Navier-Stokes equations in primitive variables are presented. The strong and weak points of the method as a tool for computational fluid dynamics are considered. The relation of the linear element finite element method to finite difference methods (FDM) is explored. The calculation of free shear layer and separated flows over aircraft boattail afterbodies with plume simulators indicate the strongest assets of the method are its capabilities for reliable and accurate calculation employing variable grids which readily approximate complex geometry and capably adapt to the presence of diverse regions of large solution gradients without the necessity of domain transformation.
Problems encoutered in investigations of high Reynolds number, incompressible flow are reviewed. A numerical solution computer program was modified to solve the stream function-vorticity form of the Navier-Stokes equations. Using a body fitted coordinate system with a U-shaped outer boundary, a simulation of incompressible flow at a Reynolds number of one million and a body angle of attack of zero was achieved.
The proposed approach to the derivation of the Navier-Stokes equation is thought to be more plausible and easier to understand than other derivations that can be found in works on fluid mechanics. The tensor character of the stress is central to the derivation. In particular, a linear relation between stress and strain rate is assumed only for the shear, rather than for the full stress tensor as is done in most other derivations. An assumption for the shear is naturally simpler and easier to verify experimentally. The use of tensor analysis is shown to greatly simplify the derivation.
This paper describes an efficient numerical method for solving the steady incompressible Navier-Stokes equations. The method is a fully implicit method based on the generalized Galerkin method, and the resulting system of equations is solved in a sweeping mode by iterative line relaxation. Results of the present method are compared with published results for separating and reattaching flows, and parametric studies showing the effects of step size, boundary locations, and Reynolds number are presented. The present method is substantially faster than previously published methods; typical run times range from 10 sec to 1 min of 7600 CPU time; and, based on results obtained to date, it is stable at any Reynolds number.
An implicit finite difference scheme is developed for the numerical solution of the compressible Navier-Stokes equations in conservation-law form. The algorithm is second-order-time accurate, noniterative, and spatially factored. In order to obtain an efficient factored algorithm, the spatial cross-derivatives are evaluated explicitly. However, the algorithm is unconditionally stable and, although a three-time-level scheme, requires only two-time-levels of data storage. The algorithm is constructed in a 'delta' form (i.e., increments of the conserved variables and fluxes) that provides a direct derivation of the scheme and leads to an efficient computational algorithm. In addition, the delta form has the advantageous property of a steady-state (if one exists) independent of the size of the time step. Numerical results are presented for a two-dimensional shock boundary-layer interaction problem.
A fine-mesh method incorporating two new operators, which drastically reduces the computation time, has been developed for solving the time-dependent Navier-Stokes equations at flight Reynolds numbers. The approach time-splits the equations into a hyperbolic part and a parabolic part, solves the hyperbolic part by a new explicit numerical method based on characteristics theory, and solves the parabolic part by a new efficient implicit parabolic method. The method has reduced the computation time by one and two orders of magnitude from that required previously to solve for the interaction of a shock wave with a boundary layer on a flat plate.
An implicit finite difference scheme based on the Newton-Kantorovich technique was developed for the numerical solution of the nonsteady, incompressible, two-dimensional Navier-Stokes equations in conservation-law form. The algorithm was second-order-time accurate, noniterative with regard to the nonlinear terms in the vorticity transport equation except at the earliest few time steps, and spatially factored. Numerical results were obtained with the technique for a circular cylinder at Reynolds number 15. Results indicate that the technique is in excellent agreement with other numerical techniques for all geometries and Reynolds numbers investigated, and indicates a potential for significant reduction in computation time over current iterative techniques.