<p>The linear instability of inviscid, incompressible annular flows, a problem commonly known as the Circular Rayleigh Problem in hydrodynamic stability, is analyzed. Tollmien’s series solutions are presented for the problem under consideration. A neutrally stable eigensolution is demonstrated to exist for wave numbers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k_s > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mi>s</mi> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and it is established that instability arises exclusively for wave numbers within the interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0< k < k_s$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <msub> <mi>k</mi> <mi>s</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, the behavior of the imaginary part of the complex wave speed <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ c $</EquationSource> <EquationSource Format="MATHML"><math> <mi>c</mi> </math></EquationSource> </InlineEquation>, namely <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ c_i $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>, as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ k $</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> approaches <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ k_s^{-} $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>k</mi> <mi>s</mi> <mo>-</mo> </msubsup> </math></EquationSource> </InlineEquation> in the context of the Circular Rayleigh problem, is studied using Tollmien–Lin’s perturbation formula. For singular neutral modes, the Reynolds stress (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>) decreases proportionally to the inverse square of the radial distance, unlike the parallel flow case where it is constant.</p>

On the Reynolds stress and inviscid solution to the hydrodynamic stability problem

期刊论文
发表日期：

2026年5月26日

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Nagarohini Challa,
S. Prakash

摘要

The linear instability of inviscid, incompressible annular flows, a problem commonly known as the Circular Rayleigh Problem in hydrodynamic stability, is analyzed. Tollmien’s series solutions are presented for the problem under consideration. A neutrally stable eigensolution is demonstrated to exist for wave numbers $k_s > 0$ $k_{s} > 0$ , and it is established that instability arises exclusively for wave numbers within the interval $0< k < k_s$ $0 < k < k_{s}$ . Furthermore, the behavior of the imaginary part of the complex wave speed $ c $ $c$ , namely $ c_i $ $c_{i}$ , as $ k $ $k$ approaches $ k_s^{-} $ $k_{s}^{-}$ in the context of the Circular Rayleigh problem, is studied using Tollmien–Lin’s perturbation formula. For singular neutral modes, the Reynolds stress ( $\tau $ $τ$ ) decreases proportionally to the inverse square of the radial distance, unlike the parallel flow case where it is constant.