Numerical solution of steady Navier-Stokes problems using integral representations

Wang, C. M.; Wu, J. C.

doi:10.2514/3.9436

Cited by 23 publications

(9 citation statements)

References 10 publications

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“…This becomes unacceptable for the stability analysis undertaken in this study. However if we limit the size of the computational domain, the``spectral'' integral constraints introduced by Wang and Wu (1986) need be modi®ed, as indicated below. We discretize the integral constraint conditions for the vorticity harmonics considering the values of a and b only at the discretization nodes r k .…”

Section: Discussionmentioning

confidence: 99%

“…Finally, as shown by Wu (1976), in the case of external two-dimensional¯ows, the conservation of total vorticity 2 It should be noted that for two-dimensional¯ows, one may use either the normal or the tangential boundary condition (see Wu 1976Wu , 1982, whereas for three-dimensional¯ows only one condition is required, and therefore the normal boundary condition appears the only possible choice (for a detailed discussion the reader is referred to Morino 1986). All the results presented here are obtained using a projection technique (related to the work of Wang and Wu 1986) which yields the same algorithm for both the tangential and the normal approach.…”

Section: Mathematical Formulationmentioning

confidence: 96%

“…For two-dimensional¯ows, the formulation used here is a modi®cation of the vorticity-stream function method, in which the w-f relationship r 2 w Àf 2 is inverted to yield an integral representation of the velocity in terms of the vorticity. In contrast to Wu (1976Wu ( , 1982, in order to invert Eq. 2 and obtain the desired integral form of the w-f relationship, we use an in®nite-space formulation and assume that Eq.…”

Section: Mathematical Formulationmentioning

confidence: 96%

“…For the sake of simplicity, we limit ourselves to two-dimensional ows (in this case, the formulation is similar to that of Wu 1976Wu , 1982; the differences are pointed out throughout the text). Also, we assume the¯uid to be initially at rest, so that the region where the vorticity is not negligible (vortical region) is bounded.…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…The postcritical behavior is analyzed using the approach introduced by 1 Indeed, for the application considered here (incompressible¯ow around an in®nite cylinder), the ®rst-instability is two-dimensional (see, e.g., Noack and Eckelmann 1994b). It may be worth noting that the methodology of Morino (1986) is valid for threedimensional¯ows and is closely related to that developed for twodimensional¯ows by Wu (1976Wu ( , 1982) (see Morino 1986) for analysis of the differences). The three-dimensional extension of the present work may be obtained by combining this formulation with that of Morino (1986).…”

Section: Introductionmentioning

confidence: 99%

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A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation

Cossu¹,

Morino²

1997

Computational Mechanics

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A vorticity-only formulation is used in order to study the behavior of the solutions of the Navier-Stokes equations for two-dimensional incompressible¯ows as the Reynolds number is increased. This approach allows one to limit the numerical-solution domain to the vortical region of the¯ow, thereby reducing the number of the state variables of the system. The vorticity-only formulation is obtained from a vorticity-stream function formulation by inverting the Poisson equation relating the vorticity to the stream function and substituting the expression for the velocity in the vorticity-transport equation. The vorticity at the solid boundary is determined from the boundary conditions. The resulting dynamical system consists of a set of ®rst-order ordinary differential equations having only quadratic nonlinearities. This system is then used to address the behavior of the solution beyond the stability boundary, within the context of the theory of dynamical systems. This part of the paper is general and is based on the use of a singular-perturbation technique, known as the method of multiple time scales; formulae for the nonlinear analysis near a Hopf bifurcation are given explicitly in terms of the coef®cients of the dynamical system. Preliminary numerical results for the validation of the formulation are presented for the limited case of twodimensional¯ows around a circular cylinder. These results include the steady-state solution with varying Reynolds number, the eigenvalues and the eigenvectors of the related stability matrix, and the characteristics of the corresponding limit cycle.

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Section: Discussionmentioning

confidence: 99%

Section: Mathematical Formulationmentioning

confidence: 96%

Section: Mathematical Formulationmentioning

confidence: 96%

Section: Mathematical Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation

Cossu¹,

Morino²

1997

Computational Mechanics

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Some uses of Green's theorem in solving the Navier–Stokes equations

Dennis

Quartapelle

1989

Numerical Methods in Fluids

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SUMMARYThis paper gives a review of methods where Green's theorem may be employed in solving numerically the Navier-Stokes equations for incompressible fluid motion. They are based on the concept of using the theorem to transform local boundary conditions given on the boundary of a closed region in the solution domain into global, or integral, conditions taken over it. Two formulations of the Navier-Stokes equations are considered: that in terms of the streamfunction and vorticity for two-dimensional motion and that in terms of the primitive variables of the velocity components and the pressure. In the first formulation overspecification of conditions for the streamfunction is utilized to obtain conditions of integral type for the vorticity and in the second formulation integral conditions for the pressure are found. Some illustrations of the principle of the method are given in one space dimension, including some derived from two-dimensional flows using the series truncation method. In particular, an illustration is given of the calculation of surface vorticity for two-dimensional flow normal to a flat plate. An account is also given of the implementation of these methods for general two-dimensional flows in both of the mentioned formulations and a numerical illustration is given.

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Dynamic vorticity condition: Theoretical analysis and numerical implementation

Jiang

et al. 1994

Numerical Methods in Fluids

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SUMMARYThe dynamic boundary conditions for vorticity, derived from the incompressible Navier-Stokes equations, are examined from both theoretical and computational points of view. It is found that these conditions can be either local (Neumann type) or global (Dirichlet type), both containing coupling with the boundary pressure, which is the main difficulty in applying vorticity-based methods. An integral formulation is presented to analyse the structure of vorticity and pressure solutions, especially the strength of the coupling. We find that for high-Reynolds-number flows the coupling is weak and, if necessary, can be effectively bypassed by simple iteration. In fact, even a fully decoupled approximation is well applicable for most Reynolds numbers of practical interest. The fractional step method turns out to be especially appropriate for implementing the decoupled approximation. Both integral and finite difference methods are tested for some simple cases with known exact solutions. In the integral approach smoothed heat kernels are used to increase the accuracy of numerical quadrature. For the more complicated problem of impulsively started flow over a circular cylinder at Re = 9500 the finite difference method is used. The results are compared against numerical solutions and fine experiments with good agreement. These numerical experiments confirm our thoeretical analysis and show the advantages of the dynamic condition in computing high-Reynolds-number flows.

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Numerical solution of steady Navier-Stokes problems using integral representations

Cited by 23 publications

References 10 publications

A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation

A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation

Some uses of Green's theorem in solving the Navier–Stokes equations

Dynamic vorticity condition: Theoretical analysis and numerical implementation

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