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Modeling Thermal Convection Using Citcom 4 - Boundary Conditions and Initial Conditions |
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For any differential equations, we need to specify boundary conditions to have a unique solution. For time-dependent problems, we also need initial solutions. For our 2-D Cartisian problem, we use free slip boundary conditions (i.e., zero normal velocity and zero shear stress) for the four sides of the box and isothermal boundary condition for the top and bottom boundaries (i.e., T=0 and 1 at the top and bottom respectively) and reflecting (or zero heat flux) boundary condition for the two sidewalls. However, other type of boundary conditions are often used as well, such as prescribed surface velocity or bottom heat flux. Initial temperature is needed as an initial condition for the time-dependent energy equation (unlike wave equations which require both initial displacement and velocity to specify the solution because of their 2nd order derivatives with time, our energy equation only has first order derivative with time and requires only one initial condition). In thermal convection studies, often we are interested in steady state solutions by running models for many time steps. Quite often, these steady state solutions (or statistically steady state) are rather insensitive to initial conditions. In any case, here initial temperature is given as T=z+d*cos(pi/L*x)*sin(pi/D*z). |
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