[Paper Review] Shifted polynomials in a convection problem

Ioana Dragomirescu|ArXiv.org|Sep 14, 2007

Differential Equations and Numerical Methods18 references3 citations

TL;DR

This paper investigates linear stability in a horizontal fluid layer heated from below under a vertically varying gravity field using Galerkin methods based on shifted Legendre and Chebyshev polynomials. It demonstrates that decreasing gravity fields expand the stability domain, with Rayleigh number thresholds increasing significantly with non-uniform gravity, and finds the shifted Chebyshev polynomial method to be more effective than Legendre polynomials for this eigenvalue problem.

ABSTRACT

The onset of convection in a horizontal layer of fluid heated from below in the presence of a gravity field varying across the layer is investigated. The eigenvalue problem governing the linear stability of the mechanical equilibria of the fluid layer in the case of free boundaries is solved using a Galerkin method based on shifted polynomials (Legendre and Chebyshev polynomials).

Motivation & Objective

• To analyze the influence of a vertically varying gravity field on the onset of thermal convection in a horizontal fluid layer.
• To solve the linear stability eigenvalue problem for free-slip boundary conditions using spectral methods based on orthogonal polynomials.
• To compare the accuracy and efficiency of shifted Legendre and Chebyshev polynomials in approximating the critical Rayleigh number.
• To evaluate how non-uniform gravity fields—specifically h(z) = -z, -z², and z²-2z—affect the stability domain.
• To validate the method against classical results and prior trigonometric Galerkin approaches.

Proposed method

• The study formulates a two-point boundary value problem for the velocity and temperature perturbations governed by ODEs with variable coefficients due to gravity variation H(z) = 1 + εh(z).
• The Galerkin method is applied using shifted orthogonal polynomials: Legendre and Chebyshev, transformed into basis functions satisfying homogeneous boundary conditions.
• A new variable Ψ = (D² - a²)W is introduced to recast the fourth-order system into a first-order system of three coupled ODEs for U = (W, Ψ, Θ).
• The trial functions are constructed as integrals of orthogonal polynomials Q_i(z), forming a complete set in H₀¹(0,1) that satisfy zero boundary conditions.
• The weak form of the eigenvalue problem is derived, leading to a secular equation in matrix form involving inner products of the basis functions.
• Numerical solutions are obtained for n = 4 basis functions, with convergence assessed by comparing results across different ε and a values.

Experimental results

Research questions

• RQ1 How does a non-uniform gravity field, specifically H(z) = 1 + εh(z), affect the critical Rayleigh number for the onset of convection in a fluid layer with free boundaries?
• RQ2 Which orthogonal polynomial basis—shifted Legendre or shifted Chebyshev—yields more accurate and efficient numerical approximations for the eigenvalue problem in this convection setting?
• RQ3 To what extent does a decreasing gravity field (e.g., h(z) = -z, -z²) stabilize the fluid layer compared to the classical uniform gravity case?
• RQ4 How do the results from the polynomial-based Galerkin method compare with classical solutions and prior trigonometric Galerkin results for the same problem?
• RQ5 Does the principle of exchange of stabilities hold under the considered variable gravity configurations with free boundaries?

Key findings

• For ε = 0.75 and h(z) = z² - 2z, the critical Rayleigh number reaches 1251.178 (SCP) and 1276.05 (SLP), indicating a substantial increase in stability compared to the classical case (R² ≈ 657.51).
• With h(z) = -z, the Rayleigh number increases from 657.51 at ε = 0 to 787.36 at ε = 0.33, showing that decreasing gravity enhances stability.
• For h(z) = -z², the Rayleigh number rises from 657.51 at ε = 0 to 993.51 at ε = 0.75, confirming a similar stabilizing trend.
• The shifted Chebyshev polynomial method consistently produced results closer to classical values and showed better convergence behavior than the Legendre-based approach.
• The method based on shifted Chebyshev polynomials was found to be more effective than Legendre polynomials, despite the latter's common use in Galerkin methods.
• Numerical results for ε = 0.0 matched classical values (R² ≈ 657.51), validating the method’s accuracy.