Nonlinear steady two-layer interfacial flow about a submerged point vortex

Wang, Zhen; Zou, Li; Liang, Hui; Zong, Zhi

doi:10.1007/s10665-016-9859-5

Cited by 3 publications

(20 citation statements)

References 12 publications

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“…These waveless solutions are related to the design of drag-free objects. In this work, we extend the works in [ 18 ] and infer that trapped waves due to two vortices may occur in certain cases.…”

Section: Introductionsupporting

confidence: 70%

“…According to setting the pressure equally on the interface, the fluid interface satisfies Bernoulli's equation

1.

This is obtained as follows [ 18 ]: for the upper-layer fluids Bernoulli's equation is

and for the lower-layer fluids that is

under the assumption that the pressure along the separating streamlines at infinity is zero. Thus the condition of pressure continuity, namely p upper = p lower along the interface, gives ( 2.7 ).…”

Section: Model Of the Problemmentioning

confidence: 99%

See 1 more Smart Citation

Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction

Wang

Liu

2022

R. Soc. open sci.

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In this work, we consider steady two-dimensional interfacial waves in a two-layer stratified fluid, which is induced by a vortex pair located in the lower layer of the fluids. A mathematical model based on the boundary integral equation method and the potential-flow theory is established. The linear analytical solution for the linearized model is given in the form of Cauchy integral and then asymptotic behaviour for large x is presented. The fully nonlinear model is solved by the Jacobian-free Newton–Krylov (JFNK) method numerically. Nonlinear characteristics of wave profiles are identified compared with the linear results under different vortex strengths and the distance between the vortex pair. The amplitude of steady downstream waves is found to vary periodically with respect to the distance of the vortex pair, which can be regarded as the interference between waves produced by each vortex. For equal-strength counter- and co-rotating pairs, the downstream wave heights of linear solutions can be eliminated for some special values of the distance between point vortices, namely, the destructive interference occurs. Meanwhile, the wave only exists between the vortex pair like trapped waves. So does the nonlinear counterpart for counter-rotating pairs, but it could not be diminished with any distance.

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Section: Introductionsupporting

confidence: 70%

“…According to setting the pressure equally on the interface, the fluid interface satisfies Bernoulli's equation

1.

This is obtained as follows [ 18 ]: for the upper-layer fluids Bernoulli's equation is

and for the lower-layer fluids that is

Section: Model Of the Problemmentioning

confidence: 99%

Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction

Wang

Liu

2022

R. Soc. open sci.

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“…For example, the number of mesh points typically used is roughly 500 < N < 2000. [30][31][32][33][34][35][36][37] The increased number of points can allow for a solution with much higher resolution or over a much larger domain. In an attempt to quantify the benefits of using a large number of grid points, we perform a brief convergence study.…”

Section: Flow Past a Pressure Distributionmentioning

confidence: 99%

“…This approach with N = 120, 001 represents a substantial increase in the number of collocation points used for two-dimensional free-surface flow problems in recent times. For example, the number of mesh points typically used is roughly 500 < N < 2000 [21,22,30,46,50,54,55,56]. The increased number of points can allow for a solution with much higher resolution or over a much larger domain.…”

Section: Flow Past a Pressure Distributionmentioning

confidence: 99%

Efficient computation of two‐dimensional steady free‐surface flows

Pethiyagoda

Moroney

McCue

2017

Numerical Methods in Fluids

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Summary We consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.

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“…A point source and vortex induced internal waves are studied using the boundary integral methods. [27][28][29][30] Forbes 31 modelled the free surface wave due to a moving hydrofoil using the potential flow theory coupled with complex analyses. It results in an integral-differential equation, which was simulated by using Newton's method.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear internal wave at the interface of two-layer liquid due to a moving hydrofoil

Wang

Zou

et al. 2017

Physics of Fluids

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This paper is concerned with the internal wave at the interface of two layers of liquids due to a hydrofoil in the lower layer liquid. The two-layer fluid is assumed moving parallel to the interface at different velocities. The stratified flow is modeled based on the incompressible potential flow theory, with the nonlinear boundary conditions at the interface. Boundary integral equations are formulated for the fully nonlinear interfacial wave generated by the hydrofoil. The numerical model results in a set of nonlinear algebra equations, which are solved using the quasi-Newton method. We show that the quasi-Newton method is more efficient than Newton's method, which is often used for solving these types of equations in the literature. The wave profiles were analyzed in terms of the location and thickness of the hydrofoil, the Froude number, and the ratio of the densities of the two fluids. The computations show that the interfacial wave amplitude showed a trend first of increase and then of decrease with the distance between the hydrofoil and the still interface.

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Nonlinear steady two-layer interfacial flow about a submerged point vortex

Cited by 3 publications

References 12 publications

Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction

Interference of internal waves due to two point vortices: linear analytical solution and nonlinear interaction

Efficient computation of two‐dimensional steady free‐surface flows

Nonlinear internal wave at the interface of two-layer liquid due to a moving hydrofoil

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