Rotational Steady Waves in a Low-pressure Region

Nonlinear steady rotational waves in a low-pressure region are investigated. The problem is formulated in a simplified canonical domain through the use of a conformal mapping, which flattens the free surface. Steady waves are computed numerically using a Newton’s method and classified into three types. Besides, our results indicate that there is a region in which steady waves do not exist. The thickness of this region is compared with the one predicted by the weakly nonlinear, weakly dispersive regime.

Keywords:
steady waves; rotational waves; shear flow; Euler equations

Authorship

M. V. FLAMARION

Universidade Federal Rural de Pernambuco (UFRPE), UACSA/Unidade Acadêmica do Cabo de Santo Agostinho, BR 101 Sul, 5225, 54503-900, Ponte dos Carvalhos, Cabo de Santo Agostinho, PE, Brazil - E-mail: marcelo.flamarion@ufrpe.brUniversidade Federal Rural de PernambucoBrazilCabo de Santo Agostinho, PE, BrazilUniversidade Federal Rural de Pernambuco (UFRPE), UACSA/Unidade Acadêmica do Cabo de Santo Agostinho, BR 101 Sul, 5225, 54503-900, Ponte dos Carvalhos, Cabo de Santo Agostinho, PE, Brazil - E-mail: marcelo.flamarion@ufrpe.br

https://orcid.org/0000-0001-5637-7454

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (5)
Tables (1)
Formulas (21)

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Figure 1:
The inverse conformal mapping. The free surface is flattened out in the canonical domain.

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Figure 2:
Top (left): typical depression steady solitary wave for F = 1.3 and Ω = −0.4. Top (right): typical nanopteron steady solution for F = 0.9 and Ω = −0.5. Bottom: non-physical solution for F = 1.2 and Ω = −0.4.

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Figure 3:
Steady solution for F = 1.2 and Ω = −0.3.

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Figure 4:
Amplitude of the steady waves as a function of the Froude number for different values of Ω.

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Figure 5:
Critical curve on the plane Ω × F.

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Table 1:
The ratios of the nonlinear and weakly nonlinear models.

(2.1)

(2.2)

(2.3)

(3.1)

(3.2)

(4,1)

Figure 1: The inverse conformal mapping. The free surface is flattened out in the canonical domain.

Figure 2: Top (left): typical depression steady solitary wave for F = 1.3 and Ω = −0.4. Top (right): typical nanopteron steady solution for F = 0.9 and Ω = −0.5. Bottom: non-physical solution for F = 1.2 and Ω = −0.4.

Figure 3: Steady solution for F = 1.2 and Ω = −0.3.

Figure 4: Amplitude of the steady waves as a function of the Froude number for different values of Ω.

Figure 5: Critical curve on the plane Ω × F.

Table 1: The ratios of the nonlinear and weakly nonlinear models.

ε	Nonlinear ratio	Weakly Nonlinear ratio
0.01	0.003	0.000
0.25	0.005	0.001
0.05	0.016	0.005
0.10	0.051	0.022
0.20	0.1228	0.087
0.25	0.152	0.135

F = \frac{U_{0}}{\sqrt{g h_{0}}},

1 - {(\frac{9 δ}{4 \sqrt{2}})}^{2 / 3} < F < 1 + {(\frac{9 δ}{8 \sqrt{2}})}^{2 / 3},

1 - {(\frac{9 |δ|}{4 \sqrt{2}})}^{2 / 3} < F < 1 .

\nabla \tilde{ϕ} (x, y, t) + (- Ω_{0} y, 0),

(2.1)

Δ \tilde{ϕ} = 0 for - 1 < y < \tilde{ζ} (x, t), {\tilde{ϕ}}_{y} = 0 at y = - 1, {\tilde{ζ}}_{t} - Ω \tilde{ζ} {\tilde{ζ}}_{x} + {\tilde{ϕ}}_{x} {\tilde{ζ}}_{x} - {\tilde{ϕ}}_{y} = 0 at y = \tilde{ζ} (x, t), {\tilde{ϕ}}_{t} + \frac{1}{2} ({\tilde{ϕ}}_{x}^{2} + {\tilde{ϕ}}_{y}^{2}) - Ω \tilde{ζ} {\tilde{ϕ}}_{x} + \tilde{ζ} + Ω \tilde{ψ} = - P (x + F t) at y = \tilde{ζ} (x, t),

\tilde{ζ} (x - F t, t) = \bar{ζ} (x, t), \tilde{ϕ} (x - F t, y, t) = \bar{ϕ} (x, y, t) .

(2.2)

Δ \bar{ϕ} = 0 for - 1 < y < \bar{ζ} (x, t), {\bar{ϕ}}_{t} = 0 at y = - 1, {\bar{ζ}}_{t} + (F - Ω \bar{ζ} + {\bar{ϕ}}_{x}) {\bar{ζ}}_{x} - {\bar{ϕ}}_{y} = 0 at y = \bar{ζ} (x, t), {\bar{ϕ}}_{t} + \frac{1}{2} ({\bar{ϕ}}_{x}^{2} + {\bar{ϕ}}_{y}^{2}) + (F - Ω \bar{ζ}) {\bar{ϕ}}_{x} + \bar{ζ} + Ω \bar{ψ} = - P (x) at y = \bar{ζ} (x, t) .

\bar{ζ} = \bar{ζ} (x) and \bar{ϕ} = \bar{ϕ} (x, y) .

(2.3)

Δ ϕ = 0 for - 1 < y < \bar{ζ} (x), {\bar{ϕ}}_{y} = 0 at y = - 1, (F - Ω \bar{ζ} + {\bar{ϕ}}_{x}) {\bar{ζ}}_{x} - {\bar{ϕ}}_{y} = 0 at y = \bar{ζ} (x), \frac{1}{2} ({\bar{ϕ}}_{x}^{2} + {\bar{ϕ}}_{y}^{2}) + (F - Ω \bar{ζ}) {\bar{ϕ}}_{x} + \bar{ζ} + Ω \bar{ψ} = - P (x) at y = \bar{ζ} (x) .

f (ξ - i η) = x (ξ, η) + i y (ξ, η),

y (ξ, 0) = 0 and y (ξ, - D) = - 1 .

ϕ (ξ, η) = \bar{ϕ} (x (ξ, η), y (ξ, η))

(3.1)

Φ_{ξ} + F Y_{ξ} - Ω Y Y_{ξ} = 0 Y + \frac{1}{2 J} (Φ_{ξ}^{2} - Ψ_{ξ}^{0}) + \frac{1}{J} (F - Ω Y) X_{ξ} Φ_{ξ} + Ω Ψ + P (X) = 0,

(3.2)

X_{ξ} = 1 - C [Y_{ξ}],

F_{k_{j}} [g (ξ)] = \hat{g} (k_{j}) = \frac{1}{2 L} \int_{- L}^{L} g (ξ) e^{- i k_{j} ξ} d ξ, F_{k_{j}}^{- 1} [\hat{g} (k_{j})] (ξ) = g (ξ) = \sum_{j = - \infty}^{\infty} \hat{g} (k_{j}) e^{i k_{j} ξ},

G_{n} (Y_{1}, Y_{2}, \dots, Y_{N}) : = Y_{n} + \frac{1}{2 J} (Φ_{ξ, n}^{2} - Ψ_{ξ, n}^{2}) + \frac{1}{J} (F + Ω Y_{n}) X_{ξ, n} Φ_{ξ, n} + Ω Ψ_{n} + P_{n} .

\frac{\partial G_{n}}{\partial Y_{l}} = \frac{G_{n} (Y_{1}, Y_{2}, \dots, Y_{l} + Δ Y, \dots, Y_{N}) - G_{n} (Y_{1}, Y_{2}, \dots, Y_{l}, \dots, Y_{N})}{Δ Y}

\frac{\sum_{j = 1}^{N} |G_{n} (Y_{1}, Y_{2}, \dots, Y_{N})|}{J} < δ,

P (x) = - ε e x p (- x^{2} / w),

R = {(\frac{9 ε^{3}}{4 \sqrt{2}})}^{2 / 3},

(4,1)

F_{Ω} = - \frac{Ω}{2} + \frac{\sqrt{Ω^{2} + 4}}{2},

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Rotational Steady Waves in a Low-pressure Region

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