Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear Instability of the Prandtl Equations via Hypergeometric Functions and the Harmonic Oscillator

Francesco De Anna, Joshua Kortum

Abstract

We establish a deep connection between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schrödinger operator. Our first result concerns an ODE and a spectral condition derived in [10], associated with unstable quasi-eigenmodes of the Prandtl equations. We entirely determine the space of solutions in terms of Kummer's functions. By classifying their asymptotic behaviour, we verify that the spectral condition has a unique, explicitly determined pair of eigenvalue and eigenfunction, the latter being expressible as a combination of elementary functions. Secondly, we prove that any quasi-eigenmode solution of the linearised Prandtl equations around a quadratic shear flow can be explicitly determined from algebraic eigenfunctions of the Schrödinger operator with quadratic potential. We show finally that the obtained analytical formulation of the velocity align with previous numerical simulations in the literature.

Linear Instability of the Prandtl Equations via Hypergeometric Functions and the Harmonic Oscillator

Abstract

We establish a deep connection between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schrödinger operator. Our first result concerns an ODE and a spectral condition derived in [10], associated with unstable quasi-eigenmodes of the Prandtl equations. We entirely determine the space of solutions in terms of Kummer's functions. By classifying their asymptotic behaviour, we verify that the spectral condition has a unique, explicitly determined pair of eigenvalue and eigenfunction, the latter being expressible as a combination of elementary functions. Secondly, we prove that any quasi-eigenmode solution of the linearised Prandtl equations around a quadratic shear flow can be explicitly determined from algebraic eigenfunctions of the Schrödinger operator with quadratic potential. We show finally that the obtained analytical formulation of the velocity align with previous numerical simulations in the literature.

Paper Structure

This paper contains 16 sections, 21 theorems, 165 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The criterion of unstable modes by Kummer's functions
  3. Explicit solutions of the Prandtl equations via harmonic oscillator
  4. Equivalent forms of the linearised Prandtl's System
  5. Rescaling around the critical point
  6. A further reduction through a rotation
  7. Proof of \ref{['thm:explicit-form-of-X']}
  8. Proof of \ref{['thm:TheoremSeparatedSolutions']}
  9. Proof of Corollary \ref{['cor:GrowthOnR']}: the behaviour at infinity
  10. Asymptotic expansions for $\mu \notin 2\mathbb{Z}-1$
  11. Asymptotic expansions for $\mu \in -2\mathbb{N}-1$
  12. Exact formulation for $\mu =-1$
  13. Proof of Corollary \ref{['cor:GrowthOnR']}
  14. Some examples of explicit solutions
  15. An explicit formulation of the shear-layer velocity
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\tau \in \mathbb{C}$ be an arbitrary complex number and define the constants ${\bf a}_\tau, {\bf b}_\tau, {\bf c}_\tau, {\bf d}_\tau \in \mathbb{C}$ as Every solution $X: \mathbb{C} \setminus A_\tau \to \mathbb{C}$ of eq:X-intro can be written explicitely as $X(z) = c_1 X_{\tau, 1}( z) + c_2 X_{\tau, 2}(z)$, where $c_1, c_2 \in \mathbb{C}$ are arbitrary constants, and the functions $X_{\tau,

Figures (3)

  • Figure 1: Plot of the stream function $\phi_k= \phi_k(y)$ defined by \ref{['eq:an-interesting-ups']} for $k = 1$ (dotted line), $k =10^2$ (dashed line) and $k = 10^6$ (full line). With $\sigma = (1-{\rm i})/\sqrt{2}$, thus this profile inflates in time as $\exp(t\sqrt{k}/{2\sqrt{2}})$. The no-slip boundary conditions $\phi_k(0) = \phi_k'(0) = 0$ are not satisfied. Left and right correspond to the real and imaginary parts, respectively.
  • Figure 2: Plot of the stream function $\phi_k= \phi_k(y)$ defined by \ref{['eq:an-interesting-ups2']} for $k = 1$ (full line), $k =10$ (dashed line) and $k = 10^2$ (dotted line). With $\sigma = (1-{\rm i})/\sqrt{2}$, this profile inflates in time as $\exp(t\sqrt{k}/{\sqrt{2}})$. The no-slip boundary conditions $\phi_k(0) = \phi_k'(0) = 0$ are satisfied but generates exponential growth and oscillations in $y$. Left and right correspond to the real and imaginary parts, respectively.
  • Figure 3: Plot of the "shear-layer" function $V$\ref{['eq:V-Gerard-Dormy-explicit']}, explicitly defined in terms of the $\, \mathrm{erf}$ and $\exp$ functions in $z$. It coincides with the shear-layer correction in MR2601044 for the shear layer $U_{\rm sh}(y) = 2y \exp(-y^2)$. Left and right correspond to the real and imaginary parts, respectively.

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • ...and 33 more