The Effect of Heater Size, Location, Aspect Ratio, and Boundary Conditions on Two-Dimensional, Laminar, Natural Convection in Rectangular Channels

The effect of localized heating in rectangular channels was studied by solving the partial differential equations for the conservation of mass, momentum, and energy numerically using an unsteady state formulation and the alternating-direction-implicit method. The heating element was a long, horizontal, isothermal strip located in one, otherwise-insulated vertical wall. The opposing wall was maintained at a lower uniform temperature and the upper and lower surfaces were insulated or maintained at the lower temperature. Computations were carried out for Pr = 0.7, 0 ≤ Ra ≤ 105, a complete range of heater widths and locations and a wide range of aspect ratios. Flow visualization studies and comparison with prior computed results for a limiting case confirm the validity of the computed values. The computed rates of heat transfer and circulation provide guidance for locating heaters or coolers.

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The Effect of the Top Wall Temperature on the Laminar Natural Convection in Rectangular Cavities With Different Aspect Ratios

Journal of Heat Transfer ◽

10.1115/1.2993138 ◽

2009 ◽

Vol 131 (5) ◽

Cited By ~ 9

Author(s):

Wenjiang Wu ◽

Chan Y. Ching

Keyword(s):

Natural Convection ◽

Aspect Ratio ◽

Wall Temperature ◽

Vertical Wall ◽

Flow Region ◽

Temperature Profiles ◽

Aspect Ratios ◽

Vertical Walls ◽

C And N ◽

Laminar Natural Convection

The effect of the top wall temperature on the laminar natural convection in air-filled rectangular cavities driven by a temperature difference across the vertical walls was investigated for three different aspect ratios of 0.5, 1.0, and 2.0. The temperature distributions along the heated vertical wall were measured, and the flow patterns in the cavities were visualized. The experiments were performed for a global Grashof number of approximately 1.8×108 and nondimensional top wall temperatures from 0.52 (insulated) to 1.42. As the top wall was heated, the flow separated from the top wall with an undulating flow region in the corner of the cavity, which resulted in a nonuniformity in the temperature profiles in this region. The location and extent of the undulation in the flow are primarily determined by the top wall temperature and nearly independent of the aspect ratio of the cavity. The local Nusselt number was correlated with the local Rayleigh number for all three cavities in the form of Nu=C⋅Ran, but the values of the constants C and n changed with the aspect ratio.

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Alternating direction implicit type preconditioners for the steady state inhomogeneous Vlasov equation

Journal of Plasma Physics ◽

10.1017/s0022377817000101 ◽

2017 ◽

Vol 83 (1) ◽

Cited By ~ 1

Author(s):

Markus Gasteiger ◽

Lukas Einkemmer ◽

Alexander Ostermann ◽

David Tskhakaya

Keyword(s):

Steady State ◽

Numerical Simulations ◽

Vlasov Equation ◽

Numerical Solutions ◽

Splitting Method ◽

Time Integration ◽

Computational Effort ◽

Alternating Direction Implicit ◽

Alternating Direction ◽

Wide Range

The purpose of the current work is to find numerical solutions of the steady state inhomogeneous Vlasov equation. This problem has a wide range of applications in the kinetic simulation of non-thermal plasmas. However, the direct application of either time stepping schemes or iterative methods (such as Krylov-based methods such as the generalized minimal residual method (GMRES) or relaxation schemes) is computationally expensive. In the former case the slowest time scale in the system forces us to perform a long time integration while in the latter case a large number of iterations is required. In this paper we propose a preconditioner based on an alternating direction implicit type splitting method. This preconditioner is then combined with both GMRES and Richardson iteration. The resulting numerical schemes scale almost ideally (i.e. the computational effort is proportional to the number of grid points). Numerical simulations conducted show that this can result in a speed-up of close to two orders of magnitude (even for intermediate grid sizes) with respect to the unpreconditioned case. In addition, we discuss the characteristics of these numerical methods and show the results for a number of numerical simulations.

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Numerical Study of Transient and Steady Induced Symmetric Flows in Rectangular Cavities

Journal of Fluids Engineering ◽

10.1115/1.3448834 ◽

1977 ◽

Vol 99 (3) ◽

pp. 526-530 ◽

Cited By ~ 4

Author(s):

B. S. Jagadish

Keyword(s):

Steady State ◽

Stream Function ◽

Numerical Study ◽

Reynolds Numbers ◽

Alternating Direction Implicit Method ◽

Alternating Direction Implicit ◽

Aspect Ratios ◽

Alternating Direction ◽

Vorticity Transport ◽

Steady State Solutions

Symmetric flows induced in rectangular cavities by a pair of moving walls are studied numerically. Solutions are obtained by solving the coupled transient vorticity transport and stream function relations using the alternating direction implicit method. Steady state solutions are obtained as limiting cases of the transients. The study covers Reynolds numbers of 1 100 and 1000 for cavities having aspect ratios of 0.5 and 1.0.

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Laminar Natural Convection in a Discretely Heated Cavity: I—Assessment of Three-Dimensional Effects

Journal of Heat Transfer ◽

10.1115/1.2836309 ◽

1995 ◽

Vol 117 (4) ◽

pp. 902-909 ◽

Cited By ~ 32

Author(s):

T. J. Heindel ◽

S. Ramadhyani ◽

F. P. Incropera

Keyword(s):

Natural Convection ◽

Vertical Wall ◽

Three Dimensional ◽

Companion Paper ◽

Two Dimensional ◽

Dimensional Model ◽

Three Dimensional Model ◽

Nusselt Numbers ◽

Aspect Ratios ◽

Laminar Natural Convection

Two and three-dimensional calculations have been performed for laminar natural convection induced by a 3 × 3 array of discrete heat sources flush-mounted to one vertical wall of a rectangular cavity whose opposite wall was isothermally cooled. Edge effects predicted by the three-dimensional model yielded local and average Nusselt numbers that exceeded those obtained from the two-dimensional model, as well as average surface temperatures that were smaller than the two-dimensional predictions. For heater aspect ratios Ahtr ≲ 3, average Nusselt numbers increased with decreasing Ahtr. However, for Ahtr ≳ 3, the two and three-dimensional predictions were within 5 percent of each other and results were approximately independent of Ahtr. In a companion paper (Heindel et al., 1995a), predictions are compared with experimental results and heat transfer correlations are developed.

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Viscous lock-exchange in rectangular channels

Journal of Fluid Mechanics ◽

10.1017/s0022112010006208 ◽

2011 ◽

Vol 673 ◽

pp. 132-146 ◽

Cited By ~ 16

Author(s):

J. MARTIN ◽

N. RAKOTOMALALA ◽

L. TALON ◽

D. SALIN

Keyword(s):

Diffusion Coefficient ◽

Rectangular Channel ◽

Gravity Current ◽

Cylindrical Tube ◽

Viscous Damping ◽

Two Fluids ◽

Aspect Ratios ◽

Rectangular Channels ◽

Wide Range ◽

Vertical Height

In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy-driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a two-dimensional (2D) Stokes flow between two parallel horizontal boundaries separated by a vertical height H and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness b and vertical height H) of any aspect ratio (H/b) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios (1/10 to 10). We also discuss the 2D Stokes–Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used.

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Polycarbonate Masters for Soft Lithography

Micromachines ◽

10.3390/mi12111392 ◽

2021 ◽

Vol 12 (11) ◽

pp. 1392

Author(s):

Filippo Amadeo ◽

Prithviraj Mukherjee ◽

Hua Gao ◽

Jian Zhou ◽

Ian Papautsky

Keyword(s):

Soft Lithography ◽

Low Cost ◽

Microfluidic Devices ◽

Silicon Substrates ◽

Large Area ◽

Aspect Ratios ◽

Rectangular Channels ◽

Wide Range ◽

Long Term Preservation

Fabrication of microfluidic devices by soft lithography is by far the most popular approach due to its simplicity and low cost. The approach relies on casting of elastomers, such as polydimethylsiloxane (PDMS), on masters fabricated from photoresists on silicon substrates. These masters, however, can be expensive, complicated to fabricate, and fragile. Here we describe an optimized replica molding approach to preserve the original masters by heat molding of polycarbonate (PC) sheets on PDMS molds. The process is faster and simpler than previously reported methods and does not result in a loss of resolution or aspect ratio for the features. The generated PC masters were used to successfully replicate a wide range of microfluidic devices, including rectangular channels with aspect ratios from 0.025 to 7.3, large area spiral channels, and micropost arrays with 5 µm spacing. Moreover, fabrication of rounded features, such as semi-spherical microwells, was possible and easy. Quantitative analysis of the replicated features showed variability of <2%. The approach is low cost, does not require cleanroom setting or hazardous chemicals, and is rapid and simple. The fabricated masters are rigid and survive numerous replication cycles. Moreover, damaged or missing masters can be easily replaced by reproduction from previously cast PDMS replicas. All of these advantages make the PC masters highly desirable for long-term preservation of soft lithography masters for microfluidic devices.

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The Effect of the Top Wall Temperature on the Laminar Natural Convection in Rectangular Cavities With Different Aspect Ratios

ASME/JSME 2007 Thermal Engineering Heat Transfer Summer Conference, Volume 1 ◽

10.1115/ht2007-32090 ◽

2007 ◽

Author(s):

Wenjiang Wu ◽

Chan Y. Ching

Keyword(s):

Natural Convection ◽

Aspect Ratio ◽

Wall Temperature ◽

Vertical Wall ◽

Flow Region ◽

Temperature Profiles ◽

Aspect Ratios ◽

Vertical Walls ◽

C And N ◽

Laminar Natural Convection

The effect of the top wall temperature on the laminar natural convection in air-filled rectangular cavities driven by a temperature difference across the vertical walls was investigated for three different aspect ratios of 0.5, 1.0 and 2.0. The temperature distributions along the heated vertical wall were measured and the flow patterns in the cavities were visualized. The experiments were performed for a global Grashof number of approximately 1.8×108 and non-dimensional top wall temperatures from 0.52 (insulated) to 1.42. As the top wall was heated, the flow separated from the top wall with an undulated flow region in the corner of the cavity, which resulted in a non-uniformity in the temperature profiles in this region. The location and extent of the undulation in the flow is primarily determined by the top wall temperature, and nearly independent of the aspect ratio of the cavity. The local Nusselt number was correlated to the local Rayleigh number for all three cavities in the form of Nu = C · Ran, but the values of the constants C and n changed with the aspect ratio.

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