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Numerical experiments show that use of a potential enstrophy conserving scheme drastically improves numerical simulation of flow near steep mountains.
The VANS successive approximation numerical method was extended to the computation of three dimensional, viscous, transonic flows in turbomachines. A cross-sectional computer code, which conserves mass flux at each point of the cross-sectional surface of computation was developed. In the VANS numerical method, the cross-sectional computation follows a blade-to-blade calculation. Numerical calculations were made for an axial annular turbine cascade and a transonic, centrifugal impeller with splitter vanes. The subsonic turbine cascade computation was generated in blade-to-blade surface to evaluate the accuracy of the blade-to-blade mode of marching. Calculated blade pressures at the hub, mid, and tip radii of the cascade agreed with corresponding measurements. The transonic impeller computation was conducted to test the newly developed locally mass flux conservative cross-sectional computer code. Both blade-to-blade and cross sectional modes of calculation were implemented for this problem. A triplet point shock structure was computed in the inducer region of the impeller. In addition, time-averaged shroud static pressures generally agreed with measured shroud pressures. It is concluded that the blade-to-blade computation produces a useful engineering flow field in regions of subsonic relative flow; and cross-sectional computation, with a locally mass flux conservative continuity equation, is required to compute the shock waves in regions of supersonic relative flow.
This paper deals with the development of a space-marching method for incompressible flows. The method solves the continuity and momentum equations as a coupled system at each streamwise station. The character of the system of equations has been changed from elliptic to hyperbolic/parabolic in order to enable the equations to be marched in space. The present method has many advantages compared to the existing parabolic or space-marching methods for incompressible flow: (1) it avoids the solution of Poisson equations, (2) it conserves the mass flow with no additional computation, (3) it does not require the specification of an assumed pressure field when used in the prediction of duct flows. The present method can capture strong secondary velocities and strong transverse pressure gradients. Predictions of the flow through straight and curved ducts are in good agreement with analytical and experimental results.
For the one-dimensional nozzle flow equations, the number of analytical boundary conditions that may be applied corresponds to the number of exterior characteristics reaching the boundary points. Since the slopes of the characteristics are the characteristic speeds u, u+c, and u-c, one concludes that two analytical boundary conditions are required at a subsonic inflow boundary and one at a subsonic outflow boundary. However, using this guideline alone can lead to nonunique solutions in subsonic cases. This paper presents a rule for selecting analytical boundary conditions for the dependent variables which will yield unique solutions if they exist.
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The conditions of electron-impact excitation in flowing oxygen downstream from a discharge are reinterpreted. It is shown that total oxygen mass flow is not conserved as the discharge is turned on and off, even though the inlet flow rate remains constant. The total amount of gas is diminished when the discharge is turned on, apparently due to collection of atomic oxygen by the system walls. A revised total cross section is reported for the excitation of 1304-A radiation.
A nonlinear method based on the full potential equation in conservation form, cast in an arbitrary coordinate system, has been developed to treat predominantly supersonic flows with embedded subsonic regions. This type of flow field occurs frequently near the fuselage-canopy junction area and wing leading edge regions for a moderately swept fighter configuration. The method uses the theory of characteristics to accurately monitor the type-dependent flow field. A conservative switching scheme is developed to transition from the supersonic marching algorithm to a subsonic relaxation procedure, and vice versa. An implicit approximate factorization scheme is employed to solve the finite-differenced equation. Results are shown for a few configurations, including a wing-body-wake realistic fighter model having embedded subsonic regions.
An alternating-direction implicit algorithm is presented for solving the conservative, full-potential equation for unsteady, transonic flow. A new development is the time-linearization of the density function. This linearization reduces the solution process from one of solving a system of two equations at each mesh point to one of solving a single equation. Two sample cases are computed. First, a one-dimensional traveling shock wave is computed and compared with the analytic solution. Second, a two-dimensional case is computed of a flow field that results from a thickening and subsequently thinning airfoil. The resulting flow field, which includes a traveling shock wave, is compared to the flow field obtained from the low-frequency, small-disturbance, transonic equation.
A numerical method based on the conservation form of the full potential equation has been applied to the problem of three-dimensional supersonic flows with embedded subsonic regions. The governing equation is cast in a nonorthogonal coordinate system, and the theory of characteristics is used to accurately monitor the type-dependent flow field. A conservative switching scheme is employed to transition from the supersonic marching procedure to a subsonic relaxation algorithm and vice versa. The newly developed computer program can handle arbitrary geometries with fuselage, canard, wing, flow through nacelle, vertical tail and wake components at combined angles of attack and sideslip. Results are obtained for a variety of configurations that include a Langley advanced fighter concept with fuselage centerline nacelle, Rockwell's Advanced Tactical Fighter (ATF) with wing mounted nacelles, and the Shuttle Orbiter configuration. Comparisons with available experiments were good.
Borovsky et al. have shown that Earth's magnetotail plasma sheet is strongly turbulent. More recently, Borovsky and Funsten have shown that eddy turbulence dominates and have suggested that the eddy turbulence is driven by fast flows that act as jets in the plasma. Through basic considerations of energy and magnetic flux conservation, these fast flows are thought to be localized to small portions of the total plasma sheet and to be generated by magnetic flux reconnection that is similarly localized. Angelopoulos et al., using single spacecraft Geotail data, have shown that the plasma sheet turbulence exhibits signs of intermittence and Weygand et al., using four spacecraft Cluster data, have confirmed and expanded on this conclusion. Uritsky et al., using Polar UVI image data, have shown that the evolution of bright, nightside, UV auroral emission regions is consistent with many of the properties of systems in self-organized criticality (SOC). Klimas et al. have suggested that the auroral dynamics is a reflection of the dynamics of the fast flows in the plasma. sheet. Their hypothesis is that the transport of magnetic fludenergy through the magnetotail is enabled by scale-free avalanches of localized reconnection whose SOC dynamics are reflected in the auroral UV emission dynamics. A corollary of this hypothesis is that the strong, intermittent, eddy turbulence of the plasma sheet is closely related to its critical dynamics. The question then arises: Can in situ evidence for the SOC dynamics be found in the properties of the plasma sheet turbulence? A 2-dimensional numerical driven current-sheet model of the central plasma sheet has been developed that incorporates an idealized current-driven instability with a resistive MHD system. It has been shown that the model can evolve into SOC in a physically relevant parameter regime. Initial results from a study of intermittent turbulence in this model and the relationship of this turbulence to the model's known SOC dynamics will be discussed.
The purpose of this research is to construct accurate finite difference schemes for incompressible unsteady flow simulations such as LES (large-eddy simulation) or DNS (direct numerical simulation). In this report, conservation properties of the continuity, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discretized equations. Existing finite difference schemes in staggered grid systems are checked for satisfaction of the requirements. Proper higher order accurate finite difference schemes in a staggered grid system are then proposed. Plane channel flow is simulated using the proposed fourth order accurate finite difference scheme and the results compared with those of the second order accurate Harlow and Welch algorithm.
The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.
A self-consistent derivation of the conservation laws is given for flows of a fluid-solid mixture. A unified analytical framework for obtaining constitutive relations is provided. This analysis uses a control volume/control surface approach that is widely used in fluid mechanics. All terms in the governing equations and the constitutive relations are written in terms of the mass-weighted averages except solid concentration. It is believed that the mass-weighted average is the natural bridge between micromechanics and constitutive relations. The derived momentum equations contain terms that differ from all existing models except that of Prosperetti and Jones (1984). However, their assumptions are not needed here. Special attention is given to the solid phase pressure. The physical basis of the previously assumed form for this pressure (Givler 1987) becomes clear. A number of related phenomena are also discussed. These include the anti-diffusion and anisotropic normal stresses. The energy equations are also different from existing models.
The high-resolution wave-propagation method for computing the nonhydrostatic atmospheric flows on meso- and micro-scales is described. The design and implementation of the Riemann solver used for computing the Godunov fluxes is discussed in detail. The method uses a flux-based wave decomposition in which the flux differences are written directly as the linear combination of the right eigenvectors of the hyperbolic system. The two advantages of the technique are: 1) the need for an explicit definition of the Roe matrix is eliminated and, 2) the inclusion of source term due to gravity does not result in discretization errors. The resulting flow solver is conservative and able to resolve regions of large gradients without introducing dispersion errors. The methodology is validated against exact analytical solutions and benchmark cases for non-hydrostatic atmospheric flows.
Implicit approximate factorization techniques (AF) are investigated for the solution of matrix equations resulting from finite-difference approximations to the full potential equation in conservation form. For transonic flows, an artificial viscosity, required to maintain stability in supersonic regions, is introduced by an upwind bias of the density. Two implicit AF procedures are presented, and their convergence performance is compared with that of the standard transonic solution procedure: successive line overrelaxation (SLOR). Subcritical and supercritical test cases are considered. Results indicate a substantial improvement in convergence rate for AF schemes relative to SLOR.
Modern numerical techniques employing properties of flux Jacobian matrices are extended to general, nonequilibrium flows. Generalizations of the Beam-Warming scheme, Steger-Warming and van Leer Flux-vector splittings, and Roe's approximate Riemann solver are presented for 3-D, time-varying grids. The analysis is based on a thermodynamic model that includes the most general thermal and chemical nonequilibrium flow of an arbitrary gas. Various special cases are also discussed.
Modern numerical techniques employing properties of flux Jacobian matrices are extended to general, nonequilibrium flows. Generalizations of the Beam-Warming scheme, Steger-Warming and van Leer Flux-vector splittings, and Roe's approximate Riemann solver are presented for 3-D, time-varying grids. The analysis is based on a thermodynamic model that includes the most general thermal and chemical nonequilibrium flow of an arbitrary gas. Various special cases are also discussed.
Design of fluidized-bed reactor containing multiple, horizontally arrayed stages offers advantages over present multistage fluidized bed reactors with horizontal, vertical, or slanted stacking. In new reactor, solids proceed from one stage to next by flow over weir. Design promotes conservation of energy and optimization of flows and residence times.