An International Conference on Mathematical Modeling And Computer Simulation with Applications IIT Kanpur, December 31, 2013-January 2, 2014
Numerical Simulations of Mixed Convection in a Porous Double Lid Driven Cavity Anirban Chattopadhyay*, Sreejata Sensarma* and Swapan K Pandit*
* Integrated Science Education and Research Centre (ISERC), Visva-bharati, santiniketan-731
235, India, Presenting authors email:
[email protected] Abstract
In this paper, we have applied our recently proposed higher order compact scheme [3] based on 9-point stencil to spatial differencing of the streamfunction-vorticity formulation of the two dimensional incompressible viscous flows governed by Navier-Stokes equations with Darcy-Forchheimer model including the energy transport equations in a two sided lid-driven differentially heated square cavity filled with a fluid saturated porous medium. Three cases are considered depending on the direction of moving walls to analyze the mixed convection with nonuniform heating. The results are analyzed over a range of Richardson number, Darcy numbers and amplitude ratios.
Introduction In recent years, the problem of mixed convection in enclosures with various thermal boundary conditions has been analyzed in a number of studies by several researchers. In addition, the analysis of convective flow and heat transfer in fluid saturated porous media has also attracted the attention of many researchers during the past few decades. This type of flow can be found in grain storage, chemical catalytic reactors, solar collectors, heat exchangers, solidification of casting, separation processes in chemical industries, etc. It is needed to study on convection heat transfer in enclosures with non-uniform heated wall(s) is important in such situations. Saeid [1] studied numerically the natural convection in a porous cavity with sinusoidal temperature variation in the bottom wall. They found that the average Nusselt number sinusoidally changes on increasing the weave number. Very recently, Sivasankaran et.al [2] studied mixed convection flow and heat transfer in a square cavity with top lid moving filled with fluid saturated porous medium with sinusoidal temperature distributions on both side walls. They conclude that the non-uniform heating of both walls is beneficial for improving heat transfer, as compared to the case of uniform heating. In most of the studies, convection heat transfer in a cavity with sinusoidal varying temperature on side wall(s) or natural convection in a porous cavity with nonuniform heating wall(s) is reported. However, mixed convection in a porous cavity with nonuniform heating of one of the moving two vertical wall(s) is not reported so far. The present work deals with the numerical simulation of two dimensional (2D) transient mixed-convection flows in a vertical two sided lid-driven differentially heated square cavity filled with a fluid saturated porous medium. In the geometry, the left and right moving walls are maintained at different temperatures with the consideration of nonuniform heating right wall while upper and bottom walls are adiabatic. Three cases are considered depending on the direction of moving walls. We have used streamfunction-vorticity formulation of DarcyForchheimer model to simulate the momentum transfer in the porous medium using fourth order compact scheme on nonuniform grids presented in [3], which was used earlier only for the streamfunction vorticity form of the 2D Navier-Stokes equations. . Governing Equations The governing equation describing the incompressible viscous flows in a two-sided lid-driven cavity is in terms of nondimensional streamfunction-vorticity (ψ , ς ) formulation as follows:
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An International Conference on Mathematical Modeling And Computer Simulation with Applications IIT Kanpur, December 31, 2013-January 2, 2014 2 2 ∂ψ ∂ψ − + =ς ∂x 2 ∂y 2
∂ς 1 ∂ 2ς 1 ∂ 2ς ∂ς ∂ς 1 Gr ∂T − − +u +v + ς= 2 2 2 ∂t Re ∂x Re ∂y ∂x ∂y Re Da Re ∂x ∂T 1 ∂ 2T 1 ∂ 2T ∂T ∂T − − +u +v =0 2 2 ∂t Re Pr ∂x Re Pr ∂y ∂x ∂y where, Gr , Re and Pr are respectively the Grasshof number, Reynolds number and Prandlt number. The dimensionless boundary conditions are as follows: u = 0, v = 1, or v = −1 and T = 0 for x = 0 and 0 ≤ y ≤ 1 u = 0, v = 1, or v = −1 and T = sin(πy ) for x = 1 and 0 ≤ y ≤ 1 ∂T ∂T u = 0, v = 0, and = 0 for y = 0 and 0 ≤ x ≤ 1 ; u = 0, v = 0, and = 0 for y = 1 and ∂y ∂y 0 ≤ x ≤1. The Local Nusselt number defined as Nu y = −(Tx )w (Th − Tc )
Numerical Results 1
Fig. 1. Centerline velocity for Richardson number Ri =1.0
0.8
0.6
0.2
0 −0.3
Re
Da=0.00001 Da=0.0001 Da=0.001 Da=0.01 Da=0.1
0.4
y
( Ri = Gr2 ) in Case I (Left wall is
−0.2
−0.1
0
0.1
0.2
moving upwards and right wall is moving downwards). 0.3
0.4
u
Summary The present work involves the computation of incompressible flows in a two-sided lid-driven cavity using time dependent compact scheme based on 9-point stencil to spatial differencing of the streamfunction-vorticity formulation of the Darcy-Forchheimer model including the energy transport equations. We have investigated the transient flow as well as steady-state solutions for both parallel and antiparallel motion of the two vertical walls. The solutions reveal that there is a significant change in the uniform and nonuniform heating wall(s). Also, the solutions in case II (Left wall is moving downwards and right wall is moving upwards) are not the mirror image of the solutions of Case I. References [1] N.H. Saeid, Natural convection in porous cavity with sinusoidal bottom wall temperature variation, Int. comm. heat mass transfer 32, 454-463,2005. [2] S. Sivasankaran, and K.L. Pan, Numerical simulation on mixed convection in a porous lid-driven cavity with non-uniform heating on both side wall, Numerical Heat Transfer, Part A , 61, 101-121, 2012. [3] S. K. Pandit, J. C. Kalita, D. C. Dalal, A transient higher order compact schemes for incompressible viscous flows on geometries beyond rectangular, J. Comput. Phys. 225 (2007) 1100-1124.
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