The Decay of Energy and Scalar Variance in Axisymmetric Turbulence
We review the recent progress in our understanding of the large scales in homogeneous (but anisotropic) turbulence. We focus on turbulence which emerges from Saffman-like initial conditions, in which the vortices possess a finite linear impulse. Such turbulence supports long-range velocity correlati...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Atmosphere |
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| Online Access: | https://www.mdpi.com/2073-4433/14/6/1019 |
| _version_ | 1797596162026373120 |
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| author | Peter A. Davidson |
| author_facet | Peter A. Davidson |
| author_sort | Peter A. Davidson |
| collection | DOAJ |
| description | We review the recent progress in our understanding of the large scales in homogeneous (but anisotropic) turbulence. We focus on turbulence which emerges from Saffman-like initial conditions, in which the vortices possess a finite linear impulse. Such turbulence supports long-range velocity correlations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced close="⟩" open="⟨"><mrow><msub><mi>u</mi><mi>i</mi></msub><msub><msup><mi>u</mi><mo>′</mo></msup><mi>j</mi></msub></mrow></mfenced><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, where <b>u</b> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle mathvariant="bold" mathsize="normal"><msup><mi>u</mi><mo>′</mo></msup></mstyle></semantics></math></inline-formula> are separated by a distance <i>r</i>, and these long-range interactions dominate the dynamics of large eddies. We show that, for axisymmetric turbulence, the energy and integral scales evolve as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>u</mi><mo>⊥</mo><mn>2</mn></msubsup><mo>~</mo><msubsup><mi>u</mi><mrow><mo>/</mo><mo>/</mo></mrow><mn>2</mn></msubsup><mo>~</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>6</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">l</mi><mo>⊥</mo><mrow></mrow></msubsup><mo>~</mo><msubsup><mi mathvariant="script">l</mi><mrow><mo>/</mo><mo>/</mo></mrow><mrow></mrow></msubsup><mo>~</mo><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>⊥</mo></semantics></math></inline-formula> and // indicate directions that are perpendicular and parallel to the symmetry axis, respectively. These predictions are consistent with the evidence of direct numerical simulations. Similar results are obtained for the passive scalar variance, where we find that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced close="⟩" open="⟨"><mrow><msup><mi>θ</mi><mn>2</mn></msup></mrow></mfenced><mo>~</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>6</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></semantics></math></inline-formula>. The primary point of novelty in our discussion of passive scalar decay is that it is based in real (rather than spectral) space, making use of an integral invariant which is a generalization of the isotropic Corrsin integral. |
| first_indexed | 2024-03-11T02:46:41Z |
| format | Article |
| id | doaj.art-505c14e516a9402c91336395d61350d6 |
| institution | Directory Open Access Journal |
| issn | 2073-4433 |
| language | English |
| last_indexed | 2024-03-11T02:46:41Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Atmosphere |
| spelling | doaj.art-505c14e516a9402c91336395d61350d62023-11-18T09:15:05ZengMDPI AGAtmosphere2073-44332023-06-01146101910.3390/atmos14061019The Decay of Energy and Scalar Variance in Axisymmetric TurbulencePeter A. Davidson0Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UKWe review the recent progress in our understanding of the large scales in homogeneous (but anisotropic) turbulence. We focus on turbulence which emerges from Saffman-like initial conditions, in which the vortices possess a finite linear impulse. Such turbulence supports long-range velocity correlations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced close="⟩" open="⟨"><mrow><msub><mi>u</mi><mi>i</mi></msub><msub><msup><mi>u</mi><mo>′</mo></msup><mi>j</mi></msub></mrow></mfenced><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><msup><mi>r</mi><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, where <b>u</b> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle mathvariant="bold" mathsize="normal"><msup><mi>u</mi><mo>′</mo></msup></mstyle></semantics></math></inline-formula> are separated by a distance <i>r</i>, and these long-range interactions dominate the dynamics of large eddies. We show that, for axisymmetric turbulence, the energy and integral scales evolve as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>u</mi><mo>⊥</mo><mn>2</mn></msubsup><mo>~</mo><msubsup><mi>u</mi><mrow><mo>/</mo><mo>/</mo></mrow><mn>2</mn></msubsup><mo>~</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>6</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">l</mi><mo>⊥</mo><mrow></mrow></msubsup><mo>~</mo><msubsup><mi mathvariant="script">l</mi><mrow><mo>/</mo><mo>/</mo></mrow><mrow></mrow></msubsup><mo>~</mo><msup><mi>t</mi><mrow><mn>2</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>⊥</mo></semantics></math></inline-formula> and // indicate directions that are perpendicular and parallel to the symmetry axis, respectively. These predictions are consistent with the evidence of direct numerical simulations. Similar results are obtained for the passive scalar variance, where we find that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced close="⟩" open="⟨"><mrow><msup><mi>θ</mi><mn>2</mn></msup></mrow></mfenced><mo>~</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>6</mn><mo>/</mo><mn>5</mn></mrow></msup></mrow></semantics></math></inline-formula>. The primary point of novelty in our discussion of passive scalar decay is that it is based in real (rather than spectral) space, making use of an integral invariant which is a generalization of the isotropic Corrsin integral.https://www.mdpi.com/2073-4433/14/6/1019scalar varianceSaffman turbulenceaxisymmetric turbulence |
| spellingShingle | Peter A. Davidson The Decay of Energy and Scalar Variance in Axisymmetric Turbulence Atmosphere scalar variance Saffman turbulence axisymmetric turbulence |
| title | The Decay of Energy and Scalar Variance in Axisymmetric Turbulence |
| title_full | The Decay of Energy and Scalar Variance in Axisymmetric Turbulence |
| title_fullStr | The Decay of Energy and Scalar Variance in Axisymmetric Turbulence |
| title_full_unstemmed | The Decay of Energy and Scalar Variance in Axisymmetric Turbulence |
| title_short | The Decay of Energy and Scalar Variance in Axisymmetric Turbulence |
| title_sort | decay of energy and scalar variance in axisymmetric turbulence |
| topic | scalar variance Saffman turbulence axisymmetric turbulence |
| url | https://www.mdpi.com/2073-4433/14/6/1019 |
| work_keys_str_mv | AT peteradavidson thedecayofenergyandscalarvarianceinaxisymmetricturbulence AT peteradavidson decayofenergyandscalarvarianceinaxisymmetricturbulence |