A Non-normal-Mode Marginal State of Convection in a Porous Rectangle
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- Chapel Hill, NC (16ha) Field measurement CTAG (CFD) Steffens et al.(Steffens et al., 2012) Chapel Hill, NC and generic near-road environments (16 ha) Field measurement CTAG (CFD).
- Age of leaf Pollutant concentration; Rate of pollutant interaction, pollutant composition, Rdc Solar radiation; Solar elevation angle Slope of the local terrain - - Rcl - - Bark area index; Porosity; Areal density; Stem area.
- The monetary benefits of these air pollutant removal on human health were estimated equal to 6.3 million USD.
- The analysis estimated that Woodland’s tree canopy annually removing 40 tons of air pollutants (includes CO, NO2, O3, SO2 and PM10) and save 15.3 million gallons of stormwater.
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Cites background from "A Non-normal-Mode Marginal State of..."
...Tyvand et al. (2019) designed and solved numerically a two-dimensional (2D) problem in a rectangle of the HRL type which is fully non-degenerate, which means that the onset is governed by a non-normal-mode solution where the horizontal and vertical dependencies cannot be separated from one another....
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Cites background from "A Non-normal-Mode Marginal State of..."
...Little is still known on non-separable eigenfunctions of non-normal mode type (Tyvand and Nøland 2019; Tyvand et al. 2019)....
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References
1,461 citations
"A Non-normal-Mode Marginal State of..." refers background in this paper
...A wellknown example of this type is the Moffatt vortices (Moffatt 1964), representing a corner singularity for a driven viscous cavity with no-slip condition along the walls....
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1,234 citations
"A Non-normal-Mode Marginal State of..." refers background in this paper
...The onset of convection in a horizontal porous layer was first studied by Horton and Rogers (1945) and Lapwood (1948)....
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796 citations
"A Non-normal-Mode Marginal State of..." refers background or methods in this paper
...These numerical eigenfunctions contain several novel features that we hope will stimulate more research on this topic, which is a well-established one since the pioneering work by Horton and Rogers (1945)....
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...To solve the problem analytically requires normal modes in the vertical direction, following Horton and Rogers (1945)....
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...The onset of convection in a horizontal porous layer was first studied by Horton and Rogers (1945) and Lapwood (1948)....
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588 citations
"A Non-normal-Mode Marginal State of..." refers background or result in this paper
...This is in good agreement with the asymptotic results derived from Nield (1968), and as expected, the thermal cell wall will be the one that is closest to vertical, simply because it is the cell wall that is the closest neighbor to the left-hand boundary of the porous rectangle....
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...Wang (1998, 1999) exemplified how the cases solved by Nield (1968) can be adjusted to be valid for cylinders with normal-mode-compatible wall conditions....
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...The critical Rayleigh number is 27.749, which is two percent above the asymptotic limit value 27.10 from Nield (1968)....
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250 citations
Additional excerpts
...The details of Wooding’s theory were carried out by Beck (1972) and Zebib (1978), for rectangular and circular cylinders, respectively....
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Frequently Asked Questions (9)
Q2. What is the preferred cell with minimum Rayleigh number?
Yet the preferred cell with minimum Rayleigh number is always expected to be one single flow cell, and with one dominating thermal cell, with an additional smaller and weaker thermal cell on the right hand side.
Q3. What type of conducting walls are open to inflow/outflow?
The Neumann type of impermeable/insulating conditions (Wooding 1959) and Dirichlet type of conducting walls that are open to inflow/outflow.
Q4. What is the general eigenfunction for a rectangular geometry?
Since the biharmonic operator does not separate in orthogonal coordinate systems, there is no general closed-form analytical eigenfunction available, even for a rectangular geometry.
Q5. What is the eigenfunction of the cylinder?
With normal modes valid over the horizontal cylinder cross-section, there is no need for normal modes in the third (vertical) direction in order to derive analytical eigenfunctions.
Q6. What is the assumption of 2D flow in the x, z plane?
The assumption of 2D flow in the x, z plane isvalid also for a 3D porous medium with thickness b in the y direction, if two constraints are met: (i) The thermomechanical wall conditions at y = 0 and y = b are those of impermeable and thermally insulating walls.
Q7. What is the eigenvalue equation for the free vibrations?
Inserting the time dependency of a single oscillatory frequency changes the wave equation to a Helmholtz eigenvalue equation for the free vibrations, with the wave number as the eigenvalue.
Q8. What is the only modification to be made to the local boundary?
The only modification to be made is the transformation of the local coordinates z → (z − 1), resulting in the formulaψ ∼ x(z − 1) +O(x3(z − 1)) +O(x(z − 1)3), (46)valid for the local streamfunction around the corner C.
Q9. Why is the asymmetry between the two isotherms so important?
This behavior occurs because of the mutual asymmetry between the thermomechanical conditions at x = 0 and those at x = L.A general tendency in all the Figures 2-4 is a higher concentration of streamlines than of isotherms near the right-hand side AB of the rectangle.