Natural convection between concentric cylinders - Revision history

Gkosec: /* Governing equations */

2017-12-15T06:46:39Z

‎Governing equations

2017-12-15T06:44:46Z

‎Governing equations

2017-12-15T06:44:17Z

‎Governing equations

2017-12-15T06:38:14Z

‎Governing equations

2017-12-15T06:38:03Z

‎Governing equations

2017-12-15T06:34:59Z

‎Governing equations

2017-12-15T06:33:59Z

‎Governing equations

2017-12-11T18:42:04Z

‎Code

2017-12-11T18:40:18Z

‎Results

2017-12-11T18:00:05Z

‎Sources

@@ Line 27: / Line 27: @@
 \end{equation}
 with knowing that
 \begin{equation}
 \b{v}=\nabla \times \psi
 \end{equation}
 \begin{equation}
 \b{\omega}=\nabla \times \b{v}

@@ Line 26: / Line 26: @@
 \rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T
 \end{equation}
 <!--
 with knowing that
@@ Line 31: / Line 32: @@
 \b{v}=\nabla \times \psi
 \end{equation}
 \begin{equation}
 \b{\omega}=\nabla \times \b{v}
 \end{equation}
 The fluid properties that appear in this problem are the fluid density $\rho$, specific heat $c$, kinematic viscosity $\nu$ and thermal conductivity $k$. The equations can be put into dimensionless form by introducing the following new variables:

@@ Line 26: / Line 26: @@
 \rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T
 \end{equation}
 with knowing that
 \begin{equation}
 \b{v}=\nabla \times \psi
 \end{equation}
 \begin{equation}

@@ Line 32: / Line 32: @@
 \begin{equation}
-\b{omega}=\nabla \times \b{v}
+\b{\omega}=\nabla \times \b{v}
 \end{equation}

@@ Line 17: / Line 17: @@
 To solve this problem we will use the same formulation as Kuehn and Goldstein (1975) only that we will use MLSM to supply the discrete approximations to the PDE operators. Note that in the actual computations it is convenient to use the velocities $\b{v} = \left(u, v\right)^T$ rather than the derivatives of the stream function in the coefficients. The equations to be solved are:
 \begin{equation}
 \Delta \psi = \omega
@@ Line 28: / Line 25: @@
 \begin{equation}
 \rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T
 \end{equation}

@@ Line 134: / Line 134: @@
          ++T_stepper;
          T2 = reshape(T_stepper.value());
-−
          ++W_stepper;
          W2 = reshape(W_stepper.value());
-        // Apply vorticity BC
          W1.swap(W2);

@@ Line 151: / Line 151: @@
 [[File:ra100.png|800px]]
 =Sources=
 Kuehn, T. H., & Goldstein, R. J. (1976). An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders. Journal of Fluid mechanics, 74(4), 695-719.