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The past decade has seen considerable activity in algorithm development for the Navier-Stokes equations. This has resulted in a wide variety of useful new techniques. Some examples for the numerical solution of the Navier-Stokes equations are presented, divided into two parts. One is devoted to the incompressible Navier-Stokes equations, and the other to the compressible form.
A numerical simulation of bubble-type vortex breakdown using a unique discrete form of the full 3-D, unsteady incompressible Navier-Stokes equations was performed. The Navier-Stokes equations were written in a vorticity-velocity form and the physical problem was not restricted to axisymmetric flow. The problem was parametized on a Rossby- Reynolds-number basis. Utilization of this parameter duo was shown to dictate the form of the free-field boundary condition specification and allowed control of axial breakdown location within the computational domain. The structure of the breakdown bubble was studied through time evolution plots of planar projected velocity vectors as well as through plots of particle traces and vortex lines. These results compared favorably with previous experimental studies. In addition, profiles of all three velocity components are presented at various axial stations and a Fourier analysis was performed to identify the dominant circumferential modes. The dynamics of the breakdown process were studied through plots of axial variation of rate of change of integrated total energy and rate of change of integrated enstrophy, as well as through contour plots of velocity, vorticity and pressure.
A numerical analysis procedure useful in the propulsion-airframe integration problem has been established. Flow around a generic hypersonic vehicle forebody is solved using Parabolized Navier-Stokes equations and Thin Layer Navier-Stokes equations. Forebody cross sectional geometry corresponds to a two-ellipse configuration. Effect of forebody geometry on the flow structure, especially at the engine inlet location, is analyzed.
Recent success in applying the Burnett equations to the one-dimensional shock-structure problem has raised the issue of whether the full Burnett equations can be used replace the Navier-Stokes equations for solving boundary-value problems in rarefied gasdynamics. As is familiar from the classical rarefied gasdynamics literature, the Burnett equations, if not solved as a successive approximation to the Navier-Stokes equations for a small Knudsen number, would require more boundary conditions than those in the Navier-Stokes system, owing to the presence of the higher-order derivatives. In this paper, this issue is examined with concrete solution examples for the steady Couette flows, addressing specifically whether solutions to the full Burnett equations can be uniquely determined without adding more boundary conditions than those in the Navier-Stokes system. The analysis, supported by detailed numerical solutions, confirms that additional boundary conditions are needed as long as the Knudsen number is not identically zero, lest the solution to the Burnett equations is not unique.
In this article, we present a reduced order modeling approach suitable for active control of fluid dynamical systems based on proper orthogonal decomposition (POD). The rationale behind the reduced order modeling is that numerical simulation of Navier-Stokes equations is still too costly for the purpose of optimization and control of unsteady flows. We examine the possibility of obtaining reduced order models that reduce computational complexity associated with the Navier-Stokes equations while capturing the essential dynamics by using the POD. The POD allows extraction of certain optimal set of basis functions, perhaps few, from a computational or experimental data-base through an eigenvalue analysis. The solution is then obtained as a linear combination of these optimal set of basis functions by means of Galerkin projection. This makes it attractive for optimal control and estimation of systems governed by partial differential equations. We here use it in active control of fluid flows governed by the Navier-Stokes equations. We show that the resulting reduced order model can be very efficient for the computations of optimization and control problems in unsteady flows. Finally, implementational issues and numerical experiments are presented for simulations and optimal control of fluid flow through channels.
During the past five years, numerous pioneering archival publications have appeared that have presented computer solutions of the mass-weighted, time-averaged Navier-Stokes equations for transonic problems pertinent to the aircraft industry. These solutions have been pathfinders of developments that could evolve into a major new technological capability, namely the computational Navier-Stokes technology, for the aircraft industry. So far these simulations have demonstrated that computational techniques, and computer capabilities have advanced to the point where it is possible to solve forms of the Navier-Stokes equations for transonic research problems. At present there are two major shortcomings of the technology: limited computer speed and memory, and difficulties in turbulence modelling and in computation of complex three-dimensional geometries. These limitations and difficulties are the pacing items of the continuing developments, although the one item that will most likely turn out to be the most crucial to the progress of this technology is turbulence modelling. The objective of this presentation is to discuss the state of the art of this technology and suggest possible future areas of research. We now discuss some of the flow conditions for which the Navier-Stokes equations appear to be required. On an airfoil there are four different types of interaction of a shock wave with a boundary layer: (1) shock-boundary-layer interaction with no separation, (2) shock-induced turbulent separation with immediate reattachment (we refer to this as a shock-induced separation bubble), (3) shock-induced turbulent separation without reattachment, and (4) shock-induced separation bubble with trailing edge separation.
Detailed aeroelastic analyses of the SemiSpan SuperSonic Transport (S4T) wind tunnel model at Mach 0.95 with a 1.75deg fixed angle of attack are presented. First, a numerical procedure using the Computational Fluids Laboratory 3-Dimensional (CFL3D) Version 6.4 flow solver is investigated. The mesh update method for structured multi-block grids was successfully applied to the Navier-Stokes simulations. Second, the steady aerodynamic analyses with a rigid structure of the S4T wind tunnel model are reviewed in transonic flow. Third, the static analyses were performed for both the Euler and Navier-Stokes equations. Both the Euler and Navier-Stokes equations predicted a significant increase of lift forces, compared to the results from the rigid structure of the S4T wind-tunnel model, over various dynamic pressures. Finally, dynamic aeroelastic analyses were performed to investigate the flutter condition of the S4T wind tunnel model at the transonic Mach number. The condition of flutter was observed at a dynamic pressure of approximately 75.0-psf for the Navier-Stokes simulations. However, it was observed that the flutter condition occurred a dynamic pressure of approximately 47.27-psf for the Euler simulations. Also, the computational efficiency of the aeroelastic analyses for the S4T wind tunnel model has been assessed.
A three-dimensional spectral method is developed for the solution to the incompressible Navier-Stokes equations in an unbounded domain. The spectral method relies on divergence-free basis functions as proposed by Leonard (1981). The basis functions are formed using vector spherical harmonics and Jacobi polynomials together with a mapping in the radial direction. An axisymmetric code was written and is verified using an exact solution of the Stokes equations. Preliminary results for the evolution of a vortex ring according to the Navier-Stokes equations are presented.
Computational fluid dynamics has an increasingly important role in the design and analysis of aircraft as computer hardware becomes faster and algorithms become more efficient. Progress is being made in two directions: more complex and realistic configurations are being treated and algorithms based on higher approximations to the complete Navier-Stokes equations are being developed. The literature indicates that linear panel methods can model detailed, realistic aircraft geometries in flow regimes where this approximation is valid. As algorithms including higher approximations to the Navier-Stokes equations are developed, computer resource requirements increase rapidly. Generation of suitable grids become more difficult and the number of grid points required to resolve flow features of interest increases. Recently, the development of large vector computers has enabled researchers to attempt more complex geometries with Euler and Navier-Stokes algorithms. The results of calculations for transonic flow about a typical transport and fighter wing-body configuration using thin layer Navier-Stokes equations are described along with flow about helicopter rotor blades using both Euler/Navier-Stokes equations.
Vortex shedding frequency caused by the protrusion of inhibitors into the flow field of a solid rocket motor is investigated by experimental and mathematical models. The time dependent Navier-Stokes equations are solved using a finite difference technique assuming incompressible, two-dimensional flow under both laminar and turbulent flow conditions. For laminar flow, explicit solutions are obtained using a vorticity-transport equation in place of the Navier-Stokes equations. For turbulent flow, a two-equation (k-epsilon) model is used for turbulent modeling and the SIMPLE algorithm is employed as the computational scheme. Cold flow tests were conducted to confirm the basic flow structure and to determine the vortex shedding frequency under both laminar and turbulent flow conditions. The vortex shedding frequencies were determined using a stroboscope to measure the oscillating frequency of yarn tufts which were fastened to one inhibitor in the models. A hot-film anemometer established the velocity history behind the inhibitor. Good agreement between the theoretical results and measurements of the vortex shedding frequencies is demonstrated.
This two-part paper presents laser anemometer measurements of the unsteady velocity field within the stator row of a transonic axial-flow fan. The objective is to provide additional insight into unsteady blade-row interactions within high speed compressors which affect stage efficiency, energy transfer, and other design considerations. Part 1 describes the measurement and analysis techniques used for resolving the unsteady flow field features. The ensemble-average and variance of the measured velocities are used to identify the rotor wake generated and unresolved unsteadiness, respectively. (Rotor wake generated unsteadiness refers to the unsteadiness generated by the rotor wake velocity deficit and the term unresolved unsteadiness refers to all remaining contributions to unsteadiness such as vortex shedding, turbulence, mass flow fluctuations, etc.). A procedure for calculating auto and cross correlations of the rotor wake generated and unresolved unsteady velocity fluctuations is described. These unsteady-velocity correlations have significance since they also result from a decomposition of the Navier-Stokes equations. This decomposition of the Navier-Stokes equations resulting in the velocity correlations used to describe the unsteady velocity field will also be outlined in this paper.
This two-part paper presents laser anemometer measurements of the unsteady velocity field within the stator row of a transonic axial-flow fan. The objective is to provide additional insight into unsteady blade-row interactions within highspeed compressors which affect stage efficiency, energy transfer, and other design considerations. Part 1 describes the measurement and analysis techniques used for resolving the unsteady flow field features. The ensemble-average and variance of the measured velocities are used to identify the rotor wake generated and unresolved unsteadiness, respectively. (Rotor wake generated unsteadiness refers to the unsteadiness generated by the rotor wake velocity deficit and the term unresolved unsteadiness refers to all remaining contributions to unsteadiness such as vortex shedding, turbulence, mass flow fluctutions, etc.). A procedure for calculating auto and cross correlations of the rotor wake generated and unresolved unsteady velocity fluctuations is described. These unsteady-velocity correlations have significance since they also result from a decomposition of the Navier-Stokes equations. This decomposition of the Navier-Stokes equations resulting in the velocity correlations used to describe the unsteady velocity field will also be outlined in this paper.
The problem of steady incompressible viscous flow past prolate spheroids at incidence is formulated using the unsteady incompressible and compressible thin-layer Navier-Stokes equations. The two sets of Navier-Stokes equations are solved using a pseudotime stepping of the implicit flux-difference splitting scheme on a curvilinear grid, which is generated by a transfinite grid generator. The Baldwin and Lomax (1978) algebraic eddy-viscosity model is used to model the turbulent flow. The computational applications cover a 6:1 prolate spheroid at different angles of attack and Reynolds numbers. The results are compared with experimental data.
The state of the art in computer simulations for transonic flowfields requiring solutions for the Navier-Stokes equations is assessed. It is noted that current simulations of transonic flowfields require comparisons with experimental results because the simulations are not free from discretization errors. Cases of turbulence are treated with weighted variables in a time-averaged scheme to yield Reynolds averaged Navier-Stokes equations. The turbulence is modeled in a first-order approach with a Reynolds stress tensor or a second-order approach where the tensor is obtained from the Navier-Stokes equations. The choice of coordinate systems and dependent variables for simulating the flow around airfoils is discussed, with particular attention to curvilinear coordinates. The determinations of boundary conditions is examined, along with numerical methods related to physical phenomena
A finite volume method to solve the Navier-Stokes equations at all flow velocities (e.g., incompressible, subsonic, transonic, supersonic and hypersonic flows) is presented. The numerical method is based on a finite volume method that incorporates a pressure-staggered mesh and an incremental pressure equation for the conservation of mass. Comparison of three generally accepted time-advancing schemes, i.e., Simplified Marker-and-Cell (SMAC), Pressure-Implicit-Splitting of Operators (PISO), and Iterative-Time-Advancing (ITA) scheme, are made by solving a lid-driven polar cavity flow and self-sustained oscillatory flows over circular and square cylinders. Calculated results show that the ITA is the most stable numerically and yields the most accurate results. The SMAC is the most efficient computationally and is as stable as the ITA. It is shown that the PISO is the most weakly convergent and it exhibits an undesirable strong dependence on the time-step size. The degenerated numerical results obtained using the PISO are attributed to its second corrector step that cause the numerical results to deviate further from a divergence free velocity field. The accurate numerical results obtained using the ITA is attributed to its capability to resolve the nonlinearity of the Navier-Stokes equations. The present numerical method that incorporates the ITA is used to solve an unsteady transitional flow over an oscillating airfoil and a chemically reacting flow of hydrogen in a vitiated supersonic airstream. The turbulence fields in these flow cases are described using multiple-time-scale turbulence equations. For the unsteady transitional over an oscillating airfoil, the fluid flow is described using ensemble-averaged Navier-Stokes equations defined on the Lagrangian-Eulerian coordinates. It is shown that the numerical method successfully predicts the large dynamic stall vortex (DSV) and the trailing edge vortex (TEV) that are periodically generated by the oscillating airfoil. The calculated streaklines are in very good comparison with the experimentally obtained smoke picture. The calculated turbulent viscosity contours show that the transition from laminar to turbulent state and the relaminarization occur widely in space as well as in time. The ensemble-averaged velocity profiles are also in good agreement with the measured data and the good comparison indicates that the numerical method as well as the multipletime-scale turbulence equations successfully predict the unsteady transitional turbulence field. The chemical reactions for the hydrogen in the vitiated supersonic airstream are described using 9 chemical species and 48 reaction-steps. Consider that a fast chemistry can not be used to describe the fine details (such as the instability) of chemically reacting flows while a reduced chemical kinetics can not be used confidently due to the uncertainty contained in the reaction mechanisms. However, the use of a detailed finite rate chemistry may make it difficult to obtain a fully converged solution due to the coupling between the large number of flow, turbulence, and chemical equations. The numerical results obtained in the present study are in good agreement with the measured data. The good comparison is attributed to the numerical method that can yield strongly converged results for the reacting flow and to the use of the multiple-time-scale turbulence equations that can accurately describe the mixing of the fuel and the oxidant.
The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri=-2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.
The scaling properties of plane homogeneous turbulent shear flows in a rotating frame are examined mathematically by a direct analysis of the Navier-Stokes equations. It is proved that two such shear flows are dynamically similar if and only if their initial dimensionless energy spectrum E star (k star, 0), initial dimensionless shear rate SK sub 0/epsilon sub 0, initial Reynolds number K squared sub 0/nu epsilon sub 0, and the ration of the rotation rate to the shear rate omega/S are identical. Consequently, if universal equilibrium states exist, at high Reynolds numbers, they will only depend on the single parameter omega/S. The commonly assumed dependence of such equilibrium states on omega/S through the Richardson number Ri = -2(omega/S)(1-2 omega/S) is proven to be inconsistent with the full Navier-Stokes equations and to constitute no more than a weak approximation. To be more specific, Richardson number similarity is shown to only rigorously apply to certain low-order truncations of the Navier-Stokes equations (i.e., to certain second-order closure models) wherein closure is achieved at the second-moment level by assuming that the higher-order moments are a small perturbation of their isotropic states. The physical dependence of rotating turbulent shear flows on omega/S is discussed in detail along with the implications for turbulence modeling.
A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.