Natural convection between concentric cylinders - Revision history

Gkosec: /* Governing equations */

2017-12-15T06:46:39Z

Governing equations

← Older revision		Revision as of 06:46, 15 December 2017
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	\end{equation}		\end{equation}

	~~<!--~~
	with knowing that		with knowing that
	\begin{equation}		\begin{equation}
	\b{v}=\nabla \times \psi		\b{v}=\nabla \times \psi
	\end{equation}		\end{equation}
			and having in mind that we are dealing with 2D problem.

			<!--
	\begin{equation}		\begin{equation}
	\b{\omega}=\nabla \times \b{v}		\b{\omega}=\nabla \times \b{v}

2017-12-15T06:44:46Z

Governing equations

← Older revision		Revision as of 06:44, 15 December 2017
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	\rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T		\rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T
	\end{equation}		\end{equation}

	<!--		<!--
	with knowing that		with knowing that
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	\b{v}=\nabla \times \psi		\b{v}=\nabla \times \psi
	\end{equation}		\end{equation}
	~~-->~~

	\begin{equation}		\begin{equation}
	\b{\omega}=\nabla \times \b{v}		\b{\omega}=\nabla \times \b{v}
	\end{equation}		\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:		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:

2017-12-15T06:44:17Z

Governing equations

← Older revision		Revision as of 06:44, 15 December 2017
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	\rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T		\rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T
	\end{equation}		\end{equation}
			<!--
	with knowing that		with knowing that
	\begin{equation}		\begin{equation}
	\b{v}=\nabla \times \psi		\b{v}=\nabla \times \psi
	\end{equation}		\end{equation}
			-->

	\begin{equation}		\begin{equation}

2017-12-15T06:38:14Z

Governing equations

2017-12-15T06:38:03Z

Governing equations

← Older revision		Revision as of 06:38, 15 December 2017
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	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:		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}~~
	~~\b{v}=\nabla \times \psi~~
	~~\end{equation}~~
	\begin{equation}		\begin{equation}
	\Delta \psi = \omega		\Delta \psi = \omega
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	\begin{equation}		\begin{equation}
	\rho c\frac{\partial T}{\partial t} + \b{v}\cdot\nabla T = k \Delta T		\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}
			\b{omega}=\nabla \times \b{v}
	\end{equation}		\end{equation}

2017-12-15T06:34:59Z

Governing equations

← Older revision		Revision as of 06:34, 15 December 2017
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	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:		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}		\begin{equation}
	\b{v}=\nabla \times \~~Psi~~		\b{v}=\nabla \times \psi
	\end{equation}		\end{equation}

	\begin{equation}		\begin{equation}
	\Delta \psi = \omega		\Delta \psi = \omega

2017-12-15T06:33:59Z

Governing equations

← Older revision		Revision as of 06:33, 15 December 2017
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:		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}
			\b{v}=\nabla \times \Psi
			\end{equation}

	\begin{equation}		\begin{equation}
	\Delta \psi = \omega		\Delta \psi = \omega

2017-12-11T18:42:04Z

Code

2017-12-11T18:40:18Z

Results

← Older revision		Revision as of 18:40, 11 December 2017
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	[[File:ra100.png\|800px]]		[[File:ra100.png\|800px]]

			[[File:ra1000.png\|800px]]

	=Sources=		=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.		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.

2017-12-11T18:00:05Z

Sources

← Older revision		Revision as of 18:00, 11 December 2017
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	}		}
	</syntaxhighlight>		</syntaxhighlight>

			=Results=

			[[File:ra100.png\|800px]]

	=Sources=		=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.		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.