Turbulent decay of a passive scalar in the Batchelor limit: Exact results from a quantum-mechanical approach
Turbulent decay of a passive scalar in the Batchelor limit: Exact results from a quantum-mechanical approach
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巴彻勒极限内被动标量的湍流衰变：量子力学方法的精确结果
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发表时间：
1998
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通讯作者：
Dam Thanh Son
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作者：
Dam Thanh Son
我们证明，由巴彻勒极限中相关时间为零的随机不可压缩流平流传播的被动标量 {theta} 的衰减可以精确地映射到具有有限自由度的特定量子力学系统。推导了薛定谔方程，并针对标量一开始具有高斯统计量且相关函数形式为 e{sup {minus}{vert_bar}x{minus}y{vert_bar}{sup 2}} 的情况分析了其解。标量的任何等时相关函数都可以通过封闭代数形式的薛定谔方程的解来表达。我们发现标量在衰减过程中是间歇性的，并且 {vert_bar}{theta}{vert_bar}{sup {alpha}} 的平均值（假设 {theta} 的平均值为零）在大 t 处下降为 e{sup {minus}{gamma}{sub {alpha}}Dt}，其中 D 是流的参数，{gamma}{sub {alpha}}= (1) /(4) {alpha}(6{minus}{alpha}) 0{lt}{alpha}{lt}3，且对于 {alpha}{ge}3，{gamma}{sub {alpha}}= (9) /(4)，与 {alpha} 无关。 {版权} {ital 1999} {ital 美国物理学会}
We show that the decay of a passive scalar {theta} advected by a random incompressible flow with zero correlation time in the Batchelor limit can be mapped exactly to a certain quantum-mechanical system with a finite number of degrees of freedom. The Schr{umlt o}dinger equation is derived and its solution is analyzed for the case where, at the beginning, the scalar has Gaussian statistics with correlation function of the form e{sup {minus}{vert_bar}x{minus}y{vert_bar}{sup 2}}. Any equal-time correlation function of the scalar can be expressed via the solution to the Schr{umlt o}dinger equation in a closed algebraic form. We find that the scalar is intermittent during its decay and the average of {vert_bar}{theta}{vert_bar}{sup {alpha}} (assuming zero mean value of {theta}) falls as e{sup {minus}{gamma}{sub {alpha}}Dt} at large t, where D is a parameter of the flow, {gamma}{sub {alpha}}= (1) /(4) {alpha}(6{minus}{alpha}) for 0{lt}{alpha}{lt}3, and {gamma}{sub {alpha}}= (9) /(4) for {alpha}{ge}3, independent of {alpha}. {copyright} {ital 1999} {ital The American Physical Society}