The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number
The mean dissipation rate of turbulent energy reaches a constant value at high Taylor–Reynolds numbers (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>λ</mi></msub...
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2024-12-01
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| author | Paolo Orlandi |
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| description | The mean dissipation rate of turbulent energy reaches a constant value at high Taylor–Reynolds numbers (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>λ</mi></msub></semantics></math></inline-formula>). This value is associated with the well-scaling dissipation spectrum in Kolmogorov units, where the maximum corresponds to the bottleneck peak. Even the scalar dissipation rate at the high <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>λ</mi></msub></semantics></math></inline-formula> considered in the present direct numerical simulations attains a constant value as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> increases. In this scenario, the maximum of the scalar dissipation spectra reaches its peak within the bottleneck, starting at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi><mo>></mo><mn>0.5</mn></mrow></semantics></math></inline-formula>. A qualitative explanation for the formation of the two bottlenecks is related to the blockage of energy transfer from large to small scales in the inertial ranges. Within the bottleneck, the self-similar, ribbon-like structures transition into the rod-like structures characteristic of the exponential decay range. Investigating the viscous dependence of the bottleneck’s amplitude may be aided by examining the evolution of a passive scalar. As <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> decreases, the scalar spectra undergo changes across the wave number <i>k</i> range. The bottleneck is dismantled, and at very low <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> values, the spectrum tends towards Batchelor’s theoretical prediction, diminishing proportionally to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>k</mi><mrow><mo>−</mo><mn>17</mn><mo>/</mo><mn>3</mn></mrow></msup></semantics></math></inline-formula>. To comprehend the flow structures responsible for the bottleneck, visualizations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><msup><mo>∇</mo><mn>2</mn></msup><mi>θ</mi></mrow></semantics></math></inline-formula> and probability density functions at various <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> values are presented and compared with those of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>i</mi></msub><msup><mo>∇</mo><mn>2</mn></msup><msub><mi>u</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>. The numerical method employed for generating three-dimensional spectra and quantities such as energy and scalar variance dissipation in physical space must be accurate, particularly in resolving small scales. This paper additionally demonstrates that the second-order finite difference scheme conserving kinetic energy and scalar variance in the inviscid limit in viscous simulations accurately predicts the exponential decay range in one-dimensional and three-dimensional turbulent kinetic energy and scalar variance spectra. |
| format | Article |
| id | doaj-art-88dbd5de2a0043deaa572e2dad8bd7ba |
| institution | DOAJ |
| issn | 2311-5521 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Fluids |
| spelling | doaj-art-88dbd5de2a0043deaa572e2dad8bd7ba2025-08-20T02:50:53ZengMDPI AGFluids2311-55212024-12-0191228510.3390/fluids9120285The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt NumberPaolo Orlandi0Dipartimento di Ingegneria Meccanica e Aerospaziale, Università La Sapienza, Via Eudossiana 16, I-00184 Roma, ItalyThe mean dissipation rate of turbulent energy reaches a constant value at high Taylor–Reynolds numbers (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>λ</mi></msub></semantics></math></inline-formula>). This value is associated with the well-scaling dissipation spectrum in Kolmogorov units, where the maximum corresponds to the bottleneck peak. Even the scalar dissipation rate at the high <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>λ</mi></msub></semantics></math></inline-formula> considered in the present direct numerical simulations attains a constant value as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> increases. In this scenario, the maximum of the scalar dissipation spectra reaches its peak within the bottleneck, starting at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi><mo>></mo><mn>0.5</mn></mrow></semantics></math></inline-formula>. A qualitative explanation for the formation of the two bottlenecks is related to the blockage of energy transfer from large to small scales in the inertial ranges. Within the bottleneck, the self-similar, ribbon-like structures transition into the rod-like structures characteristic of the exponential decay range. Investigating the viscous dependence of the bottleneck’s amplitude may be aided by examining the evolution of a passive scalar. As <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> decreases, the scalar spectra undergo changes across the wave number <i>k</i> range. The bottleneck is dismantled, and at very low <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> values, the spectrum tends towards Batchelor’s theoretical prediction, diminishing proportionally to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>k</mi><mrow><mo>−</mo><mn>17</mn><mo>/</mo><mn>3</mn></mrow></msup></semantics></math></inline-formula>. To comprehend the flow structures responsible for the bottleneck, visualizations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>θ</mi><msup><mo>∇</mo><mn>2</mn></msup><mi>θ</mi></mrow></semantics></math></inline-formula> and probability density functions at various <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>c</mi></mrow></semantics></math></inline-formula> values are presented and compared with those of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>i</mi></msub><msup><mo>∇</mo><mn>2</mn></msup><msub><mi>u</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>. The numerical method employed for generating three-dimensional spectra and quantities such as energy and scalar variance dissipation in physical space must be accurate, particularly in resolving small scales. This paper additionally demonstrates that the second-order finite difference scheme conserving kinetic energy and scalar variance in the inviscid limit in viscous simulations accurately predicts the exponential decay range in one-dimensional and three-dimensional turbulent kinetic energy and scalar variance spectra.https://www.mdpi.com/2311-5521/9/12/285passive scalarturbulencedirect numerical simulation |
| spellingShingle | Paolo Orlandi The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number Fluids passive scalar turbulence direct numerical simulation |
| title | The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number |
| title_full | The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number |
| title_fullStr | The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number |
| title_full_unstemmed | The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number |
| title_short | The Bottleneck in the Scalar Dissipation Rate Spectra: Dependence on the Schmidt Number |
| title_sort | bottleneck in the scalar dissipation rate spectra dependence on the schmidt number |
| topic | passive scalar turbulence direct numerical simulation |
| url | https://www.mdpi.com/2311-5521/9/12/285 |
| work_keys_str_mv | AT paoloorlandi thebottleneckinthescalardissipationratespectradependenceontheschmidtnumber AT paoloorlandi bottleneckinthescalardissipationratespectradependenceontheschmidtnumber |