Analysis of Coolant Entrance Boundary Shape of Porous Region to Control Cooling Along Exit Boundary

1983 ◽  
Vol 105 (3) ◽  
pp. 513-518 ◽  
Author(s):  
R. Siegel ◽  
A. Snyder
Keyword(s):  
Heat Flux ◽  
Path Length ◽  
Flow Resistance ◽  
Plane Surface ◽  
Three Dimensional ◽  
Flow Distribution ◽  
Surface Shape ◽  
Porous Region ◽  
Exit Boundary ◽  
Spatially Varying

A cooled porous region has a plane surface exposed to a specified spatially varying heat flux. The coolant leaves the region through this surface, and it is desired to control the flow distribution to maintain a specified uniform surface temperature. This is accomplished by having the coolant entrance surface shaped to provide in the region the necessary variation of path length and, hence, flow resistance. The surface shape at the coolant entrance is found by solving a Cauchy boundary value problem. An exact solution is obtained that will deal with a wide variety of heating distributions for both two- and three-dimensional shapes.

scholarly journals Geothermal heat flux anomaly due to a 3D prismoid situated in the second layer of a three-layered Earth

2013 ◽  
Vol 43 (1) ◽  
pp. 39-58
Author(s):  
Milan Hvoždara ◽  
Dušan Majcin
Keyword(s):  
Heat Flux ◽  
Three Dimensional ◽  
Flow Distribution ◽  
Geothermal Field ◽  
Boundary Integral ◽  
Geothermal Heat ◽  
Layer Potential ◽  
Lower Face ◽  
Layered Earth ◽  
Vertical Heat Flux

Abstract We present mathematical modelling of the stationary geothermal field for the three-layered earth which includes a three-dimensional perturbing body below the first layer (over the halfspace substratum). The unperturbed temperature field corresponds to the uniform vertical heat flux. The perturbing body is in the form of 3D prismoid with sloping side faces, while its upper and lower face are rectangles at the planes z = z1, z2. The theoretical formulae are based on the generalized theory of the double-layer potential and boundary integral equation (BIE). Special attention is paid to the quadrilateral prismoids bounded by planar skew faces. The numerical calculations were performed for the 3D prismoids (blocks), the thermal conductivity of which was greater than that in the ambient second layer, while the upper face of the prismoid may be in contact with the upper layer and the lower face may touch the bottom halfspace. Numerous graphs are shown for the disturbance of the temperature and heat flow distribution on the surface of the Earth or inside all three layers.

scholarly journals The Numerical Analysis of the Flow on the Smooth and Nonsmooth Boundaries by IBEM/DBIEM

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Gang Xu ◽  
Guangwei Zhao ◽  
Jing Chen ◽  
Shuqi Wang ◽  
Weichao Shi
Keyword(s):  
Boundary Value ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Tangential Velocity ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Surface Shape ◽  
Normal Vector ◽  
Computational Domain ◽  
Nonsmooth Boundary

The value of the tangential velocity on the Boundary Value Problem (BVP) is inaccurate when comparing the results with analytical solutions by Indirect Boundary Element Method (IBEM), especially at the intersection region where the normal vector is changing rapidly (named nonsmooth boundary). In this study, the singularity of the BVP, which is directly arranged in the center of the surface of the fluid computing domain, is moved outside the computational domain by using the Desingularized Boundary Integral Equation Method (DBIEM). In order to analyze the accuracy of the IBEM/DBIEM and validate the above-mentioned problem, three-dimensional uniform flow over a sphere has been presented. The convergent study of the presented model has been investigated, including desingularized distance in the DBIEM. Then, the numerical results were compared with the analytical solution. It was found that the accuracy of velocity distribution in the flow field has been greatly improved at the intersection region, which has suddenly changed the boundary surface shape of the fluid domain. The conclusions can guide the study on the flow over nonsmooth boundaries by using boundary value method.

scholarly journals Inverse Problem for Looped River Networks – Lower oder River Case Study

2013 ◽  
Vol 39 (1) ◽  
pp. 105-118
Author(s):  
Jacek Kurnatowski
Keyword(s):  
Mathematical Modeling ◽  
Flow Resistance ◽  
Flow Distribution ◽  
River Networks ◽  
Sensitivity Equation ◽  
Flow Measurements ◽  
Sensitivity Equation Method ◽  
Coefficients Identification ◽  
Oder River

Abstract Identification of coefficients determining flow resistance, in particular Manning’s roughness coefficients, is one of the possible inverse problems of mathematical modeling of flow distribution in looped river networks. The paper presents the solution of this problem for the lower Oder River network consisting of 78 branches connected by 62 nodes. Using results of six sets of flow measurements at particular network branches it was demonstrated that the application of iterative algorithm for roughness coefficients identification on the basis of the sensitivity-equation method leads to the explicit solution for all network branches, independent from initial values of identified coefficients.

Laminar Natural Convection About Downward Facing Heated Blunt Bodies to Liquid Metals

1976 ◽  
Vol 98 (2) ◽  
pp. 208-212 ◽  
Author(s):  
G. M. Harpole ◽  
I. Catton
Keyword(s):  
Heat Flux ◽  
Surface Temperature ◽  
Series Expansion ◽  
Surface Heat Flux ◽  
Liquid Metals ◽  
Surface Heat ◽  
Surface Shape ◽  
Boundary Layer Equations ◽  
Stagnation Points ◽  
Shape Dependence

The laminar boundary layer equations for free convection over bodies of arbitrary shape (i.e., a three-term series expansion) and with arbitrary surface heat flux or surface temperature are solved in local Cartesian coordinates. Both two-dimensional bodies (e.g., horizontal cylinders) and axisymmetric bodies (e.g., spheres) with finite radii of curvature at their stagnation points are considered. A Blasius series expansion is applied to convert from partial to ordinary differential equations. An additional transformation removes the surface shape dependence and the surface heat flux or surface temperature dependence of the equations. A second-order-correct, finite-difference method is used to solve the resulting equations. Tables of results for low Prandtl numbers are presented, from which local Nusselt numbers can be computed.

The Lid-Driven Cavity Flow: A Synthesis of Qualitative and Quantitative Observations

1984 ◽  
Vol 106 (4) ◽  
pp. 390-398 ◽  
Author(s):  
J. R. Koseff ◽  
R. L. Street
Keyword(s):  
Heat Flux ◽  
Flow Visualization ◽  
Three Dimensional ◽  
Symmetry Plane ◽  
Lower Boundary ◽  
Thymol Blue ◽  
Width Ratio ◽  
Cavity Depth ◽  
Driven Cavity ◽  
Turbulent Characteristics

A synthesis of observations of flow in a three-dimensional lid-driven cavity is presented through the use of flow visualization pictures and velocity and heat flux measurements. The ratio of the cavity depth to width used was 1:1 and the span to width ratio was 3:1. Flow visualization was accomplished using the thymol blue technique and by rheoscopic liquid illuminated by laser-light sheets. Velocity measurements were made using a two-component laser-Doppler-anemometer and the heat flux on the lower boundary of the cavity was measured using flush mounted sensors. The flow is three-dimensional and is weaker at the symmetry plane than that predicted by accurate two-dimensional numerical simulations. Local three-dimensional features, such as corner vortices in the end-wall regions and longitudinal Taylor-Go¨rtler-like vortices, are significant influences on the flow. The flow is unsteady in the region of the downstream secondary eddy at higher Reynolds numbers (Re) and exhibits turbulent characteristics in this region at Re = 10,000.

Mixed Convection Adjacent to 3-D Backward-Facing Step

2000 ◽  
Author(s):  
A. Li ◽  
B. F. Armaly
Keyword(s):  
Heat Flux ◽  
Nusselt Number ◽  
Mixed Convection ◽  
Buoyancy Force ◽  
Three Dimensional ◽  
The Other ◽  
Convection Flow ◽  
Backward Facing Step ◽  
Grashof Number ◽  
Recirculating Flow

Abstract Results from three-dimensional numerical simulation of laminar, buoyancy assisting, mixed convection airflow adjacent to a backward-facing step in a vertical rectangular duct are presented. The Reynolds number, and duct geometry were kept constant at Re = 200, AR = 8, ER = 2, and S = 1 cm. Heat flux at the wall downstream from the step was kept uniform, but its magnitude was varied to cover a Grashof number (Gr) range between 0.0 to 4000. All the other walls in the duct were kept at adiabatic condition. The flow, upstream of the step, is treated as fully developed and isothermal. The relatively small aspect ratio of the channel is selected specifically to focus on the developments of the three-dimensional mixed convection flow in the separated and reattached flow regions downstream from the step. The presented results focus on the effects of increasing the buoyancy force, by increasing the uniform wall heat flux, on the three-dimensional flow and heat transfer characteristics. The flow and thermal fields are symmetric about the duct’s centerline. Vortex generated near the sidewall, is the major contributor to the three dimensional behavior in the flow domain, and that feature increases as the Grashof number increases. Increasing the Grashof number results in an increase in the Nusselt number, the size of the secondary recirculating flow region, the size of the sidewall vortex, and the spanwise flow from the sidewall toward the center of the channel. On the other hand, the size of the primary reattachment region decreases with increasing the Grashof number. That region lifts away and partially detaches from the downstream wall at high Grashof number flow. The maximum Nusselt number occurs near the sidewalls and not at the center of the channel. The effects of the buoyancy force on the distributions of the three-velocity components, temperature, reattachment region, friction coefficient, and Nusselt number are presented, and compared with 2-D results.

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