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Kevin Zumbrun - One of the best experts on this subject based on the ideXlab platform.
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Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude Number or “weakly unstable” limit $$F\rightarrow 2^+$$ F → 2 + and for general values of the Froude Number F , including the limit $$F\rightarrow +\infty $$ F → + ∞ . In the former, $$F\rightarrow 2^+$$ F → 2 + , limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto–Sivashinsky (KdV–KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $$F\rightarrow 2^+$$ F → 2 + is equivalent to stability of the corresponding KdV–KS waves in the KdV limit. Together with recent results obtained for KdV–KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around $$F=2.3$$ F = 2.3 from weakly unstable to different, large- F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $$2.5\le F\le 6.0$$ 2.5 ≤ F ≤ 6.0 .
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Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude Number or “weakly unstable” limit F → 2+ and for general values of the Froude Number F , including the limit F → +∞. In the former, F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto- Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues– Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable to different, large-F behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0.
Blake Barker - One of the best experts on this subject based on the ideXlab platform.
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Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude Number or “weakly unstable” limit $$F\rightarrow 2^+$$ F → 2 + and for general values of the Froude Number F , including the limit $$F\rightarrow +\infty $$ F → + ∞ . In the former, $$F\rightarrow 2^+$$ F → 2 + , limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto–Sivashinsky (KdV–KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $$F\rightarrow 2^+$$ F → 2 + is equivalent to stability of the corresponding KdV–KS waves in the KdV limit. Together with recent results obtained for KdV–KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around $$F=2.3$$ F = 2.3 from weakly unstable to different, large- F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $$2.5\le F\le 6.0$$ 2.5 ≤ F ≤ 6.0 .
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Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude Number or “weakly unstable” limit F → 2+ and for general values of the Froude Number F , including the limit F → +∞. In the former, F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto- Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues– Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable to different, large-F behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0.
Pascal Noble - One of the best experts on this subject based on the ideXlab platform.
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Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude Number or “weakly unstable” limit $$F\rightarrow 2^+$$ F → 2 + and for general values of the Froude Number F , including the limit $$F\rightarrow +\infty $$ F → + ∞ . In the former, $$F\rightarrow 2^+$$ F → 2 + , limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto–Sivashinsky (KdV–KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $$F\rightarrow 2^+$$ F → 2 + is equivalent to stability of the corresponding KdV–KS waves in the KdV limit. Together with recent results obtained for KdV–KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around $$F=2.3$$ F = 2.3 from weakly unstable to different, large- F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $$2.5\le F\le 6.0$$ 2.5 ≤ F ≤ 6.0 .
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Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude Number or “weakly unstable” limit F → 2+ and for general values of the Froude Number F , including the limit F → +∞. In the former, F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto- Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues– Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable to different, large-F behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0.
Mathew A. Johnson - One of the best experts on this subject based on the ideXlab platform.
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Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, L. Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll wave solutions of the viscous St. Venant equations modeling inclined shallow water flow, both at onset in the small Froude Number or “weakly unstable” limit $$F\rightarrow 2^+$$ F → 2 + and for general values of the Froude Number F , including the limit $$F\rightarrow +\infty $$ F → + ∞ . In the former, $$F\rightarrow 2^+$$ F → 2 + , limit, the shallow water equations are formally approximated by a Korteweg-de Vries/Kuramoto–Sivashinsky (KdV–KS) equation that is a singular perturbation of the standard Korteweg-de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $$F\rightarrow 2^+$$ F → 2 + is equivalent to stability of the corresponding KdV–KS waves in the KdV limit. Together with recent results obtained for KdV–KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around $$F=2.3$$ F = 2.3 from weakly unstable to different, large- F behavior, with stability determined by simple power-law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $$2.5\le F\le 6.0$$ 2.5 ≤ F ≤ 6.0 .
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Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude Number or “weakly unstable” limit F → 2+ and for general values of the Froude Number F , including the limit F → +∞. In the former, F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto- Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues– Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable to different, large-F behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0.
Luis Miguel Rodrigues - One of the best experts on this subject based on the ideXlab platform.
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Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
Journal of Nonlinear Science, 2017Co-Authors: Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Rodrigues, Kevin ZumbrunAbstract:We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude Number or “weakly unstable” limit F → 2+ and for general values of the Froude Number F , including the limit F → +∞. In the former, F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto- Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues– Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude Number increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable to different, large-F behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0.
