分层流体中点涡对非线性界面波的影响分析
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摘要
在分层流体模型中,两层流体分别以不同的速度平行于界面流动.文中探讨当点涡分别位于上层和下层流体中时,对界面波的影响.当点涡位于上层流体中时,应用势流理论和边界积分方程建立完全非线性的数学模型,构造关于非线性界面波的积分微分方程组,通过基于拟牛顿法的数值方法得到界面波的数值结果.与点涡在下层流体时引起的界面波比较,数值结果显示在等强度的点涡和相同的来流条件下,点涡位于上层流体中所引起的界面波振幅远小于点涡位于下层流体中所引起的界面波振幅,并且它们的振幅比值随着上层和下层流体密度比的增大而增大,但总是小于1.
This paper is concerned with the interfacial wave of stratified fluiddue to a point vortex in the upper layer liquid or in the upper layer liquid,respectively,when the two-layer of fluids are moving parallel to theinterface at different velocities. When the point vortex is locatedin the upper layer,the stratified flow is modeled based on the incompressiblepotential flow theory,with the nonlinear boundary conditions at the interface. An integral-differential equation is formulated for the nonlinear interfacial wave. The numerical results for nonlinear interfacial waves are given by solving nonlinearboundary integral equations based on the quasi-Newton method,which are compared with the fully nonlinear numerical results for the interfacial wave when the pointvortex is located in the lower layer. It is found that when the point vortex strength and thestratified flow condition are kept same,the interfacial wave amplitude for the point vortex locatedin the upper layer is far less than that for the point vortex locatedin the lower layer. And its amplitude ratio increases as the density of the upper layer and lower layer,but it's always less than 1.
This paper is concerned with the interfacial wave of stratified fluiddue to a point vortex in the upper layer liquid or in the upper layer liquid,respectively,when the two-layer of fluids are moving parallel to theinterface at different velocities. When the point vortex is locatedin the upper layer,the stratified flow is modeled based on the incompressiblepotential flow theory,with the nonlinear boundary conditions at the interface. An integral-differential equation is formulated for the nonlinear interfacial wave. The numerical results for nonlinear interfacial waves are given by solving nonlinearboundary integral equations based on the quasi-Newton method,which are compared with the fully nonlinear numerical results for the interfacial wave when the pointvortex is located in the lower layer. It is found that when the point vortex strength and thestratified flow condition are kept same,the interfacial wave amplitude for the point vortex locatedin the upper layer is far less than that for the point vortex locatedin the lower layer. And its amplitude ratio increases as the density of the upper layer and lower layer,but it's always less than 1.
引文
[1]LIGHTHILL M J S.Waves in fluids[M].Cambridge University Press,2001.
[2]ALFORD MH,PEACOCK T,MACKINNON JA,et al.Corrigendum:The formation and fate of internal waves in the South China Sea[J].Nature,2015,528(7580):152.DOI:10.1038/nature16157.
[3]CHIA-SHUN Yih.Stratified Flows[M].Academic Press,1980.
[4]李家春.水面下的波浪——海洋内波[J].力学与实践,2005,27(2):1-6.DOI:10.3969/j.issn.1000-0879.2005.02.001.LI Jiachun.Billow under the sea surface-internal waves in the ocean[J].Mechanics in Engineering,2005,27(2):1-6.DOI:10.3969/j.issn.1000-0879.2005.02.001.(in Chinese)
[5]杜涛,吴巍,方欣华.海洋内波的产生与分布[J].海洋科学,2001,25(4):24-28.DOI:10.3969/j.issn.1000-3096.2001.04.008.DU Tao,WU Wei,FANG Xinhua.The generation and distribution of ocean internal waves[J].Marine Scineces,2001,25(4):24-28.DOI:10.3969/j.issn.1000-3096.2001.04.008.(in Chinese)
[6]Vanden-Broeck J M.Gravity-capillary free-surface flows[M].Gravity-capillary free-surface flows/.Cambridge University Press,2010:128.
[7]耿新,李海涛.海洋内波的生成研究[J].科技创新导报,2015(35):186-187.DOI:10.16660/j.cnki.1674-098X.2015.35.186.GENG Xin,LI Haitao.Study on the generation of internal waves[J].Science and Technology Innovation Herald,2015(35):186-187.DOI:10.16660/j.cnki.1674-098X.2015.35.186.(in Chinese)
[8]魏岗,戴世强.分层流体中运动源生成的内波研究进展[J].力学进展,2006,36(1):111-124.DOI:10.3321/j.issn:1000-0992.2006.01.016.WEI Gang,DAI Shiqiang.Advances in internal waves due to moving body in stratified fluid systems[J].Advances in Mechanics,2006,36(1):111-124.DOI:10.3321/j.issn:1000-0992.2006.01.016.(in Chinese)
[9]HELFRICH K R,MELVILLE W K.Long nonlinear internal waves[J].Annual Review of Fluid Mechanics,2006,38(1):395-425.DOI:10.1146/annurev.fluid.38.050304.092129.
[10]ALAM M R,LIU Y,YUE D K P.Waves due to an oscillating and translating disturbance in a two-layer density-stratified fluid[J].Journal of Engineering Mathematics,2009,65(2):179-200.DOI:10.1007/s10665-009-9303-1.
[11]FORBES L K.On the effects of non-linearity in freesurface flow about a submerged point vortex[J].Journal of Engineering Mathematics,1985,19(2):139-155.DOI:10.1007/bf00042737.
[12]FORBES L K.A numerical method for non-linear flow about a submerged hydrofoil[J].Journal of Engineering Mathematics,1985,19(4):329-339.DOI:10.1007/bf00042877.
[13]HOCKING G C,FORBES L K.Super-critical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth[J].Journal of Fluid Mechanics,2001,428:333-348.DOI:10.1017/s0022112000002780.
[14]FORBES L K,COSGROVE J M.A line vortex in a two-fluid system[J].Journal of Engineering Mathematics,2013,84(1):181-199.DOI:10.1007/s10665-012-9606-5.
[15]STOKES T E,HOCKING G C,FORBES L K.Steady free surface flows induced by a submerged ring source or sink[J].Journal of Fluid Mechanics,2012,694:352-370.DOI:10.1017/jfm.2011.551.
[16]WANG Z,ZOU L,LIANG H,et al.Nonlinear steady two-layer interfacial flow about a submerged point vortex[J].Journal of Engineering Mathematics,2016,103(1):39-53.DOI:10.1007/s10665-016-9859-5.
[17]BROYDEN C G.A class of methods for solving nonlinear simultaneous equations[J].Mathematics of Computation,1965,19(92):577.DOI:10.2307/2003941.
[2]ALFORD MH,PEACOCK T,MACKINNON JA,et al.Corrigendum:The formation and fate of internal waves in the South China Sea[J].Nature,2015,528(7580):152.DOI:10.1038/nature16157.
[3]CHIA-SHUN Yih.Stratified Flows[M].Academic Press,1980.
[4]李家春.水面下的波浪——海洋内波[J].力学与实践,2005,27(2):1-6.DOI:10.3969/j.issn.1000-0879.2005.02.001.LI Jiachun.Billow under the sea surface-internal waves in the ocean[J].Mechanics in Engineering,2005,27(2):1-6.DOI:10.3969/j.issn.1000-0879.2005.02.001.(in Chinese)
[5]杜涛,吴巍,方欣华.海洋内波的产生与分布[J].海洋科学,2001,25(4):24-28.DOI:10.3969/j.issn.1000-3096.2001.04.008.DU Tao,WU Wei,FANG Xinhua.The generation and distribution of ocean internal waves[J].Marine Scineces,2001,25(4):24-28.DOI:10.3969/j.issn.1000-3096.2001.04.008.(in Chinese)
[6]Vanden-Broeck J M.Gravity-capillary free-surface flows[M].Gravity-capillary free-surface flows/.Cambridge University Press,2010:128.
[7]耿新,李海涛.海洋内波的生成研究[J].科技创新导报,2015(35):186-187.DOI:10.16660/j.cnki.1674-098X.2015.35.186.GENG Xin,LI Haitao.Study on the generation of internal waves[J].Science and Technology Innovation Herald,2015(35):186-187.DOI:10.16660/j.cnki.1674-098X.2015.35.186.(in Chinese)
[8]魏岗,戴世强.分层流体中运动源生成的内波研究进展[J].力学进展,2006,36(1):111-124.DOI:10.3321/j.issn:1000-0992.2006.01.016.WEI Gang,DAI Shiqiang.Advances in internal waves due to moving body in stratified fluid systems[J].Advances in Mechanics,2006,36(1):111-124.DOI:10.3321/j.issn:1000-0992.2006.01.016.(in Chinese)
[9]HELFRICH K R,MELVILLE W K.Long nonlinear internal waves[J].Annual Review of Fluid Mechanics,2006,38(1):395-425.DOI:10.1146/annurev.fluid.38.050304.092129.
[10]ALAM M R,LIU Y,YUE D K P.Waves due to an oscillating and translating disturbance in a two-layer density-stratified fluid[J].Journal of Engineering Mathematics,2009,65(2):179-200.DOI:10.1007/s10665-009-9303-1.
[11]FORBES L K.On the effects of non-linearity in freesurface flow about a submerged point vortex[J].Journal of Engineering Mathematics,1985,19(2):139-155.DOI:10.1007/bf00042737.
[12]FORBES L K.A numerical method for non-linear flow about a submerged hydrofoil[J].Journal of Engineering Mathematics,1985,19(4):329-339.DOI:10.1007/bf00042877.
[13]HOCKING G C,FORBES L K.Super-critical withdrawal from a two-layer fluid through a line sink if the lower layer is of finite depth[J].Journal of Fluid Mechanics,2001,428:333-348.DOI:10.1017/s0022112000002780.
[14]FORBES L K,COSGROVE J M.A line vortex in a two-fluid system[J].Journal of Engineering Mathematics,2013,84(1):181-199.DOI:10.1007/s10665-012-9606-5.
[15]STOKES T E,HOCKING G C,FORBES L K.Steady free surface flows induced by a submerged ring source or sink[J].Journal of Fluid Mechanics,2012,694:352-370.DOI:10.1017/jfm.2011.551.
[16]WANG Z,ZOU L,LIANG H,et al.Nonlinear steady two-layer interfacial flow about a submerged point vortex[J].Journal of Engineering Mathematics,2016,103(1):39-53.DOI:10.1007/s10665-016-9859-5.
[17]BROYDEN C G.A class of methods for solving nonlinear simultaneous equations[J].Mathematics of Computation,1965,19(92):577.DOI:10.2307/2003941.