Benchmark AI / Public workspace
CritPt / Challenge_32_main / Consider natural convection in a porous medium modeled by Darcy's law (i.e.,…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
def answer():
r"""
Return the value of the critical Rayleigh number and associated critical horizontal wavenumber
Inputs
----------
None
Outputs
----------
Ra_c: float, the critical Rayleigh number.
k_c: float, the associated critical horizontal wavenumber.
ratio_w2T: float, the value of w(z)/T(z) at z=0.67365.
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
Ra_c = ...
k_c = ...
ratio_w2T = ...
# ---------------------------------------------------------------
return Ra_c, k_c, ratio_w2Tproblem description
# Problem setup:
Consider natural convection in a porous medium modeled by Darcy's law (i.e., Rayleigh-Darcy convection) with horizontal periodic boundary conditions. The boundary condition on the top wall is free (constant pressure) and at constant temperature. The boundary condition on the bottom wall is impermeable and has constant heat flux.
# Main problem:
What is the critical Rayleigh number (allowing error) and associated critical horizontal wavenumber (allowing error), where, above this critical Rayleigh number, the conduction base state is linearly unstable? Moreover, consider eigenfunction of vertical velocity w(z) and temperature T(z) associated with this critical Rayleigh number and crical horizontal wavenumber, what is the value of w(z)/T(z) at z=0.67365, where the bottom wall is at z=0 and the top wall is at z=1?
Plain-text mathematical notation (without MathML)
# Problem setup: Consider natural convection in a porous medium modeled by Darcy's law (i.e., Rayleigh-Darcy convection) with horizontal periodic boundary conditions. The boundary condition on the top wall is free (constant pressure) and at constant temperature. The boundary condition on the bottom wall is impermeable and has constant heat flux. # Main problem: What is the critical Rayleigh number (allowing ±0.05 error) and associated critical horizontal wavenumber (allowing ±0.02 error), where, above this critical Rayleigh number, the conduction base state is linearly unstable? Moreover, consider eigenfunction of vertical velocity w(z) and temperature T(z) associated with this critical Rayleigh number and crical horizontal wavenumber, what is the value of w(z)/T(z) at z=0.67365, where the bottom wall is at z=0 and the top wall is at z=1?
Original LaTeX notation
# Problem setup: Consider natural convection in a porous medium modeled by Darcy's law (i.e., Rayleigh-Darcy convection) with horizontal periodic boundary conditions. The boundary condition on the top wall is free (constant pressure) and at constant temperature. The boundary condition on the bottom wall is impermeable and has constant heat flux. # Main problem: What is the critical Rayleigh number (allowing $\pm 0.05$ error) and associated critical horizontal wavenumber (allowing $\pm 0.02$ error), where, above this critical Rayleigh number, the conduction base state is linearly unstable? Moreover, consider eigenfunction of vertical velocity w(z) and temperature T(z) associated with this critical Rayleigh number and crical horizontal wavenumber, what is the value of w(z)/T(z) at z=0.67365, where the bottom wall is at z=0 and the top wall is at z=1?
Discussion
No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.
Artifacts
Code, notes and reproducible work shared by participants. Files are served from a separate origin.
No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.
Source and history
initial import