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Dynamic properties of two-dimensional and quasi-geostrophic turbulence
KTH, School of Engineering Sciences (SCI), Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

Two codes have been developed and implemented for use on massively parallelsuper computers to simulate two-dimensional and quasi-geostrophic turbulence.The codes have been found to scale well with increasing resolution and width ofthe simulations. This has allowed for the highest resolution simulations of twodimensionaland quasi-geostrophic turbulence so far reported in the literature.The direct numerical simulations have focused on the statistical characteristicsof turbulent cascades of energy and enstrophy, the role of coherent vorticesand departures from universal scaling laws, theoretized more than 40 yearsago. In particular, the investigations have concerned the enstrophy and energycascades in forced and decaying two-dimensional turbulence. Furthermore, theapplicability of Charney’s hypotheses on quasi-geostrophic turbulence has beentested. The results have shed light on the flow evolution at very large Reynoldsnumbers. The most important results are the robustness of the enstrophycascade in forced and decaying two-dimensional turbulence, the sensitivity toan infrared Reynolds number in the spectral scaling of the energy spectrumin the inverse energy cascade range, and the validation of Charney’s predictionson the dynamics of quasi-geostrophic turbulence. It has also been shownthat the scaling of the energy spectrum in the enstrophy cascade is insensitiveto intermittency in higher order statistics, but that corrections apply to the”universal” Batchelor-Kraichnan constant, as a consequence of large-scale dissipationanomalies following a classical remark by Landau (Landau & Lifshitz1987). Another finding is that the inverse energy cascade is maintained bynonlocal triad interactions, which is in contradiction with the classical localityassumption.

##### Place, publisher, year, edition, pages
Stockholm: KTH , 2010. , ix, 54 p.
##### Series
Trita-MEK, ISSN 0348-467X ; 2010:06
##### Keyword [en]
two-dimensional turbulence, decaying turbulence, quasi-geostrophic turbulence, direct numerical simulation (DNS), coherent vortices, energy cascade, enstrophy cascade, intermittency, massively parallel simulations, locality iii
##### National Category
Other Physics Topics
##### Identifiers
ISBN: 978-91-7415-763-5 (print)OAI: oai:DiVA.org:kth-25712DiVA: diva2:359545
##### Public defence
2010-11-19, D1, Lindstedtsvägen 17, Stockholm, 10:15 (English)
##### Note
QC 20101029Available from: 2010-10-29 Created: 2010-10-28 Last updated: 2011-03-24Bibliographically approved
##### List of papers
1. The enstrophy cascade in forced two-dimensional turbulence
Open this publication in new window or tab >>The enstrophy cascade in forced two-dimensional turbulence
2011 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 671, 168-183 p.Article in journal (Refereed) Published
##### Abstract [en]

We carry out direct numerical simulations of two-dimensional turbulence with forcing at different wave numbers and resolutions up to 32768^2 gridpoints. In the absence of large scale drag, a state is reached where enstrophy is quasistationary while energy is growing. In the enstrophy cascade range the energy spectrum has the form $E(k) = K \epsilon_{\omega} ^{2/3} k^{-3}$, without any logarithmic correction, where$\epsilon_{\omega}$ is the enstrophy dissipation and K is of the order of unity. However, K is varying between different simulations and is thus not a perfect constant. This variation can be understood as a consequence of large-scale dissipation intermittency, following the argument by Landau (Landau \& Lifshitz 1959).  In the presence of a large scale drag, we obtain a slightly steeper spectrum. When forcing is applied at a scale which is somewhat smaller than the computational domain no vortices are formed and the statistics remain close to Gaussian in the enstrophy cascade range. When forcing is applied at a smaller scale, long lived coherent vortices form at larger scales  than the forcing scale and intermittency measures become very large at all scales, including the scales of the enstrophy cascade. We conclude that the enstrophy cascade with a $k^{-3}$-spectrum, is a robust feature of the two-dimensional Navier-Stokes equations. However, there is a complete lack of universality of higher order statistics of vorticity increments in the enstrophy cascade range.

##### Keyword
Two-dimensional turbulence, enstrophy cascade, intermittency, dissipation, coherent vortices
##### National Category
Other Physics Topics
##### Identifiers
urn:nbn:se:kth:diva-25707 (URN)10.1017/S0022112010005562 (DOI)000288100100007 ()2-s2.0-79952816018 (Scopus ID)
##### Note
QC 20101029. Uppdaterat från submitted till published 20110404Available from: 2010-10-28 Created: 2010-10-28 Last updated: 2017-12-12Bibliographically approved
2. Testing Batchelor's similarity hypotheses for decaying two-dimensional turbulence
Open this publication in new window or tab >>Testing Batchelor's similarity hypotheses for decaying two-dimensional turbulence
2010 (English)In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 22, no 9, 091704- p.Article in journal (Refereed) Published
##### Abstract [en]

We carry out three high resolution direct numerical simulations of the two-dimensional Navier-Stokes equation to test Batchelor's similarity hypotheses of an equilibrium spectral range and an inertial subrange where the enstrophy wave number spectrum has the form Phi(k)=C chi(2/3)k(-1), where chi is the mean enstrophy dissipation rate and C is a constant. We use very different initial conditions in the three simulations and find that Batchelor's hypotheses are well satisfied in each simulation. However, there is a small but significant difference between the equilibrium range spectrum of one of the simulations as compared to the spectra of the other two. We suggest that the difference is linked to the stronger degree of large scale variation of the enstrophy dissipation which is observed in this simulation as compared to the other two.

##### Keyword
flow simulation, Navier-Stokes equations, turbulence
##### National Category
Fluid Mechanics and Acoustics
##### Identifiers
urn:nbn:se:kth:diva-25760 (URN)10.1063/1.3488997 (DOI)000282437100017 ()2-s2.0-79251574687 (Scopus ID)
##### Note
QC 20101029Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2017-12-12Bibliographically approved
3. Infrared Reynolds number dependency of the two-dimensional inverse energy cascade
Open this publication in new window or tab >>Infrared Reynolds number dependency of the two-dimensional inverse energy cascade
2011 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 667, 463-473 p.Article in journal (Refereed) Published
##### Abstract [en]

High resolution simulations of forced two-dimensional turbulence reveal that the inverse cascade range is sensitive to an infrared Reynolds number $Re_{\alpha}=k_f/k_{\alpha}$, where $k_f$ is the forcing wave number and $k_{\alpha}$ is a frictional wave number based on linear friction. In the limit of high $Re_{\alpha}$, the classic $k^{-5/3}$-scaling is lost and we obtain steeper energy spectra. The sensitivity is traced to the formation of vortices in the inverse energy cascade range. Thus, it is hypothesized that the dual limit $Re_{\alpha} \rightarrow \infty$ and $Re_{\nu}=k_d/k_f \rightarrow \infty$, where $k_d$ is the small-scale dissipation wave number, will lead to a steeper energy spectrum than $k^{-5/3}$ in the inverse energy cascade range. It is also found that the inverse energy cascade is maintained by nonlocal triad interactions.

##### Keyword
Two-dimensional turbulence, inverse energy cascade, coherent vortices, non-locality
##### National Category
Other Physics Topics
##### Identifiers
urn:nbn:se:kth:diva-25709 (URN)10.1017/S0022112010005628 (DOI)000287052900018 ()2-s2.0-79951619915 (Scopus ID)
##### Funder
Knut and Alice Wallenberg Foundation
##### Note
QC 20101029Available from: 2010-10-28 Created: 2010-10-28 Last updated: 2017-12-12Bibliographically approved
4. Charney isotropy and equipartition in quasi-geostrophic turbulence
Open this publication in new window or tab >>Charney isotropy and equipartition in quasi-geostrophic turbulence
2010 (English)In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 656, 448-457 p.Article in journal (Refereed) Published
##### Abstract [en]

High-resolution simulations of forced quasi-geostrophic (QG) turbulence reveal that Charney isotropy develops under a wide range of conditions, and constitutes a preferred state also in beta-plane and freely decaying turbulence. There is a clear analogy between two-dimensional and QG turbulence, with a direct enstrophy cascade that is governed by the prediction of Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525) and an inverse energy cascade following the classic k(-5/3) scaling. Furthermore, we find that Charney's prediction of equipartition between the potential and kinetic energy in each of the two horizontal velocity components is approximately fulfilled in the inertial ranges.

##### Keyword
rotating turbulence, turbulence simulation, turbulence theory
##### National Category
Fluid Mechanics and Acoustics
##### Identifiers
urn:nbn:se:kth:diva-25763 (URN)10.1017/S0022112010002703 (DOI)000280868600021 ()2-s2.0-77957156534 (Scopus ID)
##### Note
QC 20101029Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2017-12-12Bibliographically approved
5. Simulations of two-dimensional and quasi-geostrophic turbulence: Internal Report
Open this publication in new window or tab >>Simulations of two-dimensional and quasi-geostrophic turbulence: Internal Report
2010 (English)Report (Other academic)
##### Abstract [en]

This report is devoted to the details of the two codes that have been developed aimed at studies of large-scale turbulent flows. For this purpose, a first approach has been to derive and implement a code (PNSE2D) that simulates two-dimensional turbulence by numerically solving the two-dimensional incompressible Navier-Stokes equation in a doubly periodic square domain. The second code (QGE3D) adds complexity by taking into account background rotation and a stable stratification. This latter code solves the QG equation for the potential vorticity derived by Charney (1971), and can be considered an extension from PNSE2D into three dimensions. The motivation for developing new codes instead of using existing codes is that it gives us full control of the codes while they solve the equations in a very simple geometry (periodic boundary conditions). Furthermore, the codes are portable and have been developed to give a high degree of flexibility. The two codes have in common, among other things, the need for high resolution, which required the codes to be parallelized for utilization on parallel machines. Experiments have shown that good speed-up is obtained when increasing the number of processes (essentially increasing the number of cpu:s). The codes rely on a message passing interface (MPI) and a fast FFT-library, FFTW. The codes have been validated by the conservation of the inviscid invariants, i.e. energy and enstrophy in two-dimensional turbulent flows, and energy and potential enstrophy in QG turbulence.

##### Place, publisher, year, edition, pages
Stockholm: KTH, 2010. 35 p.
##### Keyword
Parallelisation, performance, MPI, Runge-Kutta, quasi-geostrophic, two-dimensional, turbulence
Computer Science
##### Identifiers
urn:nbn:se:kth:diva-25711 (URN)
##### Note
QC 20101029Available from: 2010-11-01 Created: 2010-10-28 Last updated: 2010-11-01Bibliographically approved

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