Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

Nikitin, N.

doi:10.1016/j.jcp.2006.01.036

Cited by 97 publications

(49 citation statements)

References 41 publications

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“…The corresponding Taylor microscale Reynolds numbers are Re λ = 50 and 110, respectively. The Navier-Stokes equations are solved with a finite differences scheme and with time advancement computed by a semi-implicit Runge-Kutta method (Nikitin 2006). The resolution is 256×256×256 grid points in x 1 , x 2 and x 3 direction.…”

Section: Methodsmentioning

confidence: 99%

Acceleration, pressure and related quantities in the proximity of the turbulent/non-turbulent interface

Holzner¹,

Lüthi²,

Tsinober³

et al. 2009

J. Fluid Mech.

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This paper presents an analysis of flow properties in the proximity of the turbulent/ non-turbulent interface (TNTI), with particular focus on the acceleration of fluid particles, pressure and related small scale quantities such as enstrophy, ω 2 = ω i ω i , and strain, s 2 = s ij s ij . The emphasis is on the qualitative differences between turbulent, intermediate and non-turbulent flow regions, emanating from the solenoidal nature of the turbulent region, the irrotational character of the non-turbulent region and the mixed nature of the intermediate region in between. The results are obtained from a particle tracking experiment and direct numerical simulations (DNS) of a temporally developing flow without mean shear. The analysis reveals that turbulence influences its neighbouring ambient flow in three different ways depending on the distance to the TNTI: (i) pressure has the longest range of influence into the ambient region and in the far region non-local effects dominate. This is felt on the level of velocity as irrotational fluctuations, on the level of acceleration as local change of velocity due to pressure gradients, Du/Dt ∂ u/∂t −∇p/ρ, and, finally, on the level of strain due to pressureHessian/strain interaction, (D/Dt)(s 2 /2) (∂/∂t)(s 2 /2) −s ij p, ij > 0; (ii) at intermediate distances convective terms (both for acceleration and strain) as well as strain production -s ij s jk s ki > 0 start to set in. Comparison of the results at Taylor-based Reynolds numbers Re λ = 50 and Re λ = 110 suggests that the distances to the far or intermediate regions scale with the Taylor microscale λ or the Kolmogorov length scale η of the flow, rather than with an integral length scale; (iii) in the close proximity of the TNTI the velocity field loses its purely irrotational character as viscous effects lead to a sharp increase of enstrophy and enstrophy-related terms. Convective terms show a positive peak reflecting previous findings that in the laboratory frame of reference the interface moves locally with a velocity comparable to the fluid velocity fluctuations.

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Section: Methodsmentioning

confidence: 99%

Acceleration, pressure and related quantities in the proximity of the turbulent/non-turbulent interface

Holzner¹,

Lüthi²,

Tsinober³

et al. 2009

J. Fluid Mech.

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“…The infinitesimal partial motion of M with respect to axis, i.e., keeping constant the position of M from and axes, yields motion at M , where MM corresponds to the Mathematical Problems in Engineering 3 infinitesimal partial change of position r 0 given as the vector variation r 0 , where r 0 = i + j + k = cℎ 3 . The vector r 0 + r 0 defines the new location M of point M. Paying attention to the sketch in Figure 1, vectors r 0 and r 0 + r 0 can be described by the following relationships:…”

Section: Development Of a Curvilinear Orthogonal Coordinatementioning

confidence: 99%

“…Compared to integral methods with a curvilinear coordinate system, their advantage is the generalized possibility of applications and their disadvantage is the necessary more detailed validation, which is accompanied by longer computational times (from some hours to some days) and much more computer memory and power. Alternatively, depending on the selection of the solution procedure and in case that the specific particles are in curvilinear motion, the above equations may be preferable to be expressed at a curvilinear coordinate system [3]. Although, several buoyant jet phenomena can be analyzed by the governing equations written in orthogonal Cartesian or cylindrical coordinate systems [4][5][6][7][8], there exist some specific phenomena that the use of a curvilinear orthogonal coordinate system is mandatory [9].…”

Section: Introductionmentioning

confidence: 99%

Curvilinear Coordinate System for Mathematical Analysis of Inclined Buoyant Jets Using the Integral Method

Bloutsos

Yannopoulos

2018

Mathematical Problems in Engineering

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The development of a local system of orthogonal curvilinear coordinates, which is appropriate to monitor the flow of an inclined buoyant jet with reference to the basic Cartesian coordinate system is presented. Such a system is necessary for the correct application of the integral method, since the well-known Gaussian profiles should be integrated on the cross-sectional area of inclined buoyant jet, where they are valid. This is the major advantage of the present work compared to all other integral methods using Cartesian coordinate systems. Consequently, the flow and mixing governing partial differential equations (PDE), i.e., continuity, momentum, buoyancy, and/or tracer conservation, are written in the local orthogonal curvilinear coordinate system and, then, the Reynolds substitution regarding mean and fluctuating components of all dependent variables is applied. After averaging with respect to time, the mean flow PDEs are taken, omitting second-order terms, as the dynamic pressure and molecular viscosity, compared to the mean flow and mixing contributions of turbulent terms. The latter are introduced through empirical coefficients. The Boussinesq's approximation regarding small density differences is taken into consideration. The system of PDEs is closed by assuming known spreading coefficients along with Gaussian similarity profiles. The methodology is applied in the inclined two-dimensional buoyant jet; thus, PDEs are integrated on the jet cross-sectional area resulting in ordinary differential equations (ODE), which are appropriate to be solved by applying the 4th order Runge-Kutta algorithm coded in either FORTRAN or EXCEL. The numerical solution of ODEs, concerning trajectory of the inclined two-dimensional buoyant jet, as well as longitudinal variations of the mean axial velocity, mean concentration, minimum dilution, and entrainment velocity or entrainment coefficient, occurs quickly, saving computer memory and effort. The satisfactory agreement of results with experimental data available in the literature empowers the usefulness of the proposed methodology in inclined buoyant jets.

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“…These equations were numerically solved in a spherical coordinate system where the impermeability and noslip conditions for the azimuthal u φ , radial u r , and polar u θ components of the velocity have the form u φ (r = r 1,2 ) = Ω 1,2 (t)r 1,2 sinθ, u r (r = r 1,2 ) = 0, u θ (r = r 1,2 ) = 0, where subscripts 1 and 2 correspond to the inner and outer spheres, respectively. We used an algorithm of numerical solution [18] based on a conservative finite difference scheme of the discretization of the Navier-Stokes equations in space and semi-implicit Runge-Kutta scheme of the third order integration accuracy in time. Discretization in space was performed on grids nonuniform in r and θ directions with concentration near the boundaries and equatorial plane and the total number of nodes 5.76 × 10 5 .…”

Section: Methods Of Calculationmentioning

confidence: 99%

The Different Types of Turbulence in Rotating Spherical Layers

Zhilenko

Krivonosova

Gritsevich

2018

J. Phys.: Conf. Ser.

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Abstr act. Turbulent flows of a viscous incompressible fluid in a layer between rotating concentric spheres under the action of the modulation of the velocity of one of the spheres have been studied experimentally and numerically. The form of spectra of turbulent pulsations of the azimuthal velocity depends on the sphere whose rotational velocity is modulated, as well as on the amplitude and frequency of modulation. The possibility of the formation of turbulence with spectra qualitatively similar to spectra obtained in measurements in the upper atmosphere is established: with the slope close to -3 at low frequencies and close to -5/3 at high frequencies and with the negative longitudinal velocity structure function of the third order. It has been shown that such spectra are formed in the regions of a flow that are strongly synchronized under the action of the modulation of the rotational velocity.

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Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

Cited by 97 publications

References 41 publications

Acceleration, pressure and related quantities in the proximity of the turbulent/non-turbulent interface

Acceleration, pressure and related quantities in the proximity of the turbulent/non-turbulent interface

Curvilinear Coordinate System for Mathematical Analysis of Inclined Buoyant Jets Using the Integral Method

The Different Types of Turbulence in Rotating Spherical Layers

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