A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables

doi:10.1016/0021-9991(82)90032-8

Journal of Computational Physics

Volume 48, Issue 1, October 1982, Pages 1-22

https://doi.org/10.1016/0021-9991(82)90032-8 Get rights and content

Abstract

Solution methods for compact finite-difference schemes are applied to the vorticity-velocity form of the two-dimensional unsteady Navier-Stokes equations. Numerical experiments for stagnation point and driven cavity flows are described.

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the Navier–Stokes equations, which are recast into their vorticity–velocity form. Through this formulation, a numerical solution can be found without resorting to staggered grids [21,22]. Dirichlet conditions are imposed at the lower boundary of the domain to match temperature, velocity and mixture composition to the experimental values (see Table 1).
The structure of a JP-8 coflow flame is investigated by applying laser diagnostic techniques to three different fuel surrogates. The results are compared against theoretical predictions from numerical simulations; very good agreement is obtained for temperature and major species.
Rayleigh and Raman scattering are used to measure temperature and major species mole fractions in the flame (oxygen, carbon dioxide, water, and fuel molecules). Quantitative laser diagnostic techniques are particularly challenging when applied to jet-fuel flames; the presence of aromatic molecules in the fuel mixtures and the formation of polyaromatic compounds inside the flame generate spectrally broad fluorescence signals that interfere with the measurement. A polarized/depolarized subtraction technique combined with a post-processing filter based on least-squares fitting is used to mitigate this undesired effect. The proposed technique tries to match the experimental signal against previously calculated spectra and has proved to be a very efficient filter at rejecting polyaromatic fluorescence.
Numerical simulations play a fundamental role in this study. Computer predictions are used not only to compare experimental data, but as an active component of the data post-processing. For example, numerically calculated cross-section maps are used to refine the measured temperature for both the Rayleigh and Raman experiments.
The use of cubic interpolation method for transient hydrodynamics of solid particles
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As the main vortex gradually increases in size with the particle trapped in it, the wall vortex propagates to the lower part of the cavity and eventually joins with the lower corner vortex. Gatski, Grosch, and Rose (1982) and Goodrich and Soh (1988) predicted the same phenomena using the conventional finite different solution to the Navier–Stokes equation but failed to reproduce the right wall vortex in their investigations. Recently, Quartapelle (1981) predicted the transient flow in a square cavity at Re = 1000 and showed that the solution is unstable and the vortex near the moving lid exhibits a large oscillatory behavior.
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A transport-flux-based directed relation graph method for the spatially inhomogeneous instantaneous reduction of chemical kinetic mechanisms
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A novel, flux-based, directed relation graph (DRG) method is proposed for adaptive and automatic generation of skeletal chemical mechanisms. Unlike many previous approaches for static mechanism reduction, the proposed method does not rely on the complete solution of a set of test problems using the detailed mechanism, but rather it simplifies the model “on the fly,” analyzing the local reaction states with an extended directed relation graph. Compared to the original DRG method of Lu and Law [T. Lu, C.K. Law, Proc. Combust. Inst. 30 (2005)], the flux-based DRG explicitly considers the effect of transport fluxes, as they provide a way to quantify the coupling of the governing equations among adjacent grid cells. The resulting numerical scheme operates on a cell-by-cell basis, so that different chemical submodels are applied in different regions of the flame. This approach leads to very large reduction ratios, in which most of the flame structure is resolved using about 30% of the chemical species employed in the full mechanism, while a more detailed model that considers roughly 70–80% of the total species is used only at ignition. The flux-based DRG method has been employed in the two-dimensional simulation of axisymmetric coflow flames for a variety of fuels and chemical kinetic mechanisms (three steady flames and one oscillating flame), with a speedup factor of about 20 obtained for the largest problem — a JP-8 flame.
A high accuracy hybrid method for two-dimensional Navier-Stokes equations
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A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.
A mass-conserving vorticity-velocity formulation with application to nonreacting and reacting flows
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In a commonly implemented version of the vorticity–velocity formulation, the governing equations for the fluid dynamics are expressed as two Poisson-like velocity equations together with the vorticity transport equation. However, for some flows with large vorticity gradients, spurious mass loss or gain can be observed. In order to conserve mass, a modification to the vorticity–velocity formulation is proposed, involving the substitution of the kinematic definition of vorticity in certain terms of the fluid-dynamic equations. This modified formulation results in a broader computational stencil when the equations are in a second-order-accurate discretized form, and a stronger coupling between the predicted vorticity and the curl of the predicted velocity field. The resulting system of elliptic equations – which includes the energy and species transport equations for the reacting flow case – is discretized with finite differences on a nonstaggered grid and is then solved using Newton’s method. Both the unmodified and modified vorticity–velocity formulations are applied to two problems with high vorticity gradients: (1) incompressible, axisymmetric fluid flow through a suddenly expanding pipe and (2) a confined, axisymmetric laminar flame with detailed chemistry and multicomponent transport, generated on a burner whose inner tube extends above the burner surface. The modified formulation effectively eliminates the spurious mass loss in the two test cases to within an acceptable tolerance. The two cases demonstrate the broader range of applicability of the modified formulation, as compared with the unmodified formulation.

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^∗: Research was supported under NASA Contract No. NASI-15810 at ICASE, NASA Langley Research Center, Hampton, Va. 23665. The third author was also supported under NASA Contract No. NASI-16394.

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A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables

Abstract

J. Comput. Phys.

An Introduction to Fluid Dynamics

AIAA J.

Numerical Solution of the Complete Navier-Stokes Equations for the Simulation of Unsteady Flows

Bull. Acad. Pol. Sci. Lett., A