dorsal/arxiv
View SchemaLong-wave instability and growth rate of the inviscid shear flows
| Authors | Liang Sun |
|---|---|
| Categories | |
| ArXiv ID | physics/0601112 |
| URL | https://arxiv.org/abs/physics/0601112 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
In this paper, we studied the long-wave instability of the shear flows. When the wavenumber of perturbation is larger than the critical value, the flow is always neutrally stable. First, we obtain a new upper bound for the neutral wavenumber $k_1\leq (p^2-1)\mu_1$, where $p>1$ and $\mu_1$ is the smallest eigenvalue of Poincar\'{e}'s problem. Second, we find a new upper bound for the imaginary part of the complex phase velocity $c_i \leq k_1 \Delta U/\sqrt{\mu_1}$, where $\Delta U$ is the variance of the velocity. The new bound is finite for all $k>0$ similar to the Howard's semicircle theorem, while the previous ones by Craik and Banerjee et al would be infinity as $k\rightarrow 0$. Third, we find a new upper bound of growth rate $\omega_i \leq (p-1) \sqrt{\mu_1} \Delta U$. All the new bounds are much more strict than the previous ones by H{\o}iland, Howard, Craik and Banerjee et al. Our results also extend the inverse energy cascade theory by Kraichnan. As shear instability is due to long-wave instability, it implies that the truncation of long-waves may change the instability of shear flows.
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"abstract": "In this paper, we studied the long-wave instability of the shear flows. When\nthe wavenumber of perturbation is larger than the critical value, the flow is\nalways neutrally stable. First, we obtain a new upper bound for the neutral\nwavenumber $k_1\\leq (p^2-1)\\mu_1$, where $p\u003e1$ and $\\mu_1$ is the smallest\neigenvalue of Poincar\\\u0027{e}\u0027s problem. Second, we find a new upper bound for the\nimaginary part of the complex phase velocity $c_i \\leq k_1 \\Delta\nU/\\sqrt{\\mu_1}$, where $\\Delta U$ is the variance of the velocity. The new\nbound is finite for all $k\u003e0$ similar to the Howard\u0027s semicircle theorem, while\nthe previous ones by Craik and Banerjee et al would be infinity as\n$k\\rightarrow 0$. Third, we find a new upper bound of growth rate $\\omega_i\n\\leq (p-1) \\sqrt{\\mu_1} \\Delta U$. All the new bounds are much more strict than\nthe previous ones by H{\\o}iland, Howard, Craik and Banerjee et al. Our results\nalso extend the inverse energy cascade theory by Kraichnan. As shear\ninstability is due to long-wave instability, it implies that the truncation of\nlong-waves may change the instability of shear flows.",
"arxiv_id": "physics/0601112",
"authors": [
"Liang Sun"
],
"categories": [
"physics.flu-dyn",
"physics.ao-ph"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Long-wave instability and growth rate of the inviscid shear flows",
"url": "https://arxiv.org/abs/physics/0601112"
},
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