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For plane channel flow, thermal stratification resulting from a wall-normal temperature gradient together with an opposing gravitational field can lead to buoyancy-driven instability of three-dimensional waves. Moreover, viscosity-driven instability can lead to the amplification of two-dimensional Tollmien-Schlichting waves. Temporal stability simulations considering different combinations of Reynolds number and Rayleigh number were performed to investigate both the buoyancy and viscosity-driven instability of Rayleigh-Benard-Poiseuille flow. The investigated cases are either (1) stable, (2) unstable with respect to three-dimensional waves (buoyancy-driven instability), or (3) unstable with respect to two-dimensional waves (viscosity-driven instability). Two new and highly accurate computational fluid dynamics codes have been developed for solving the full and linearized unsteady compressible Navier-Stokes equations in Cartesian coordinates. The codes employ fifth-order-accurate upwind-biased compact finite differences for the convective terms and fourth-order-accurate compact finite differences for the viscous terms. For the case with buoyancy-driven instability, strong linear growth is observed for a broad range of spanwise wavenumbers and the wavelength of the spanwise mode with the strongest non-linear growth is gradually decreasing in time. For the case with viscosity-driven instability, the linear growth rates are lower and the first mode to experience non-linear growth is a higher harmonic with half the wavelength of the primary wave. The present results are consistent with the neutral curves from the linear stability theory analysis by Gage and Reid.
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Higher‐order‐accurate numerical method for temporal stability simulations of Rayleigh‐Bénard‐Poiseuille flows
For Rayleigh-Bénard-Poiseuille flows, thermal stratification resulting from a wall-normal temperature gradient together with an opposing gravitational field can lead to buoyancy-driven instability. Moreover, for sufficiently large Reynolds numbers, viscosity-driven instability can occur. Two higher-order-accurate methods based on the full and linearized Navier-Stokes equations were developed for investigating the temporal stability of such flows. The new methods employ a spectral discretization in the homogeneous directions. In the wall-normal direction, the convective and viscous terms are discretized with fifth-order-accurate biased and fourth-order-accurate central compact finite differences. A fourth-order-accurate explicit Runge-Kutta method is employed for time integration. To validate the methods, the primary instability was investigated for different combinations of the Reynolds and Rayleigh number. The results from these primary stability investigations are consistent with linear stability theory results from the literature with respect to both the onset of the instability and the dependence of the temporal growth rate on the wave angle. For the cases with buoyancy-driven instability, strong linear growth is observed for a broad range of spanwise wavenumbers. The largest growth rates are obtained for a wave angle of 90deg. For the cases with viscosity-driven instability, the linear growth rates are lower and the first mode to experience nonlinear growth is a higher harmonic with half the wavelength of the fundamental.
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- Award ID(s):
- 1510179
- PAR ID:
- 10180771
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Fluids
- ISSN:
- 0271-2091
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/fld.4877
