# ACU-T: 3000 Enclosed Hot Cylinder: Natural Convection

## Prerequisites

This tutorial provides instructions for setting up, solving and viewing results of a CFD simulation of air flow inside the annulus of a concentric cylinder with a heat source due to natural convection.Prior to starting this tutorial, you should have already run through the introductory tutorial, ACU-T: 1000 HyperWorks UI Introduction

Since the HyperMesh database (.hm file) contains meshed geometry, this tutorial does not include steps related to geometry import and mesh generation.

## Problem Description

The schematic of the problem to be solved is shown in the figure above. The inner cylinder is a solid with internal heat generation and the annular space between the inner and outer cylinders is a fluid volume with air as the fluid. The air in contact with the surface of the inner cylinder is heated up and rises to the upper parts of the annulus due to buoyancy effects and displaces the cold fluid at the top. At the same time, the film of the fluid which was in contact with the hot surface is replaced by the surrounding cold fluid. This process is repeated continuously until a steady state is achieved.

Both cylinders are assumed to be infinitely long and are modeled using half symmetry and periodicity. The cylinders are infinite in z-direction and hence periodicity will be applied along this direction.

## Introduction to Theory

### Natural Convection

Convection is a heat transfer mechanism where the transfer of heat energy happens through the motion of matter. Since the definition of convection involves motion of matter a fluid state is usually present in convection. Usually this type of heat transfer takes place between a hot or a cold surface and a fluid. The film of fluid in contact with the surface absorbs heat from or transfers heat to the surface and is then replaced by a new film. This movement of fluid may either be governed by an external source, such as a fan or pump, or due to internal changes in the fluid properties. When no external sources are responsible for the fluid motion the heat transfer mechanism at work is called the Natural Convection. The driving force for motion of the fluid in a natural convection is density changes in the fluid due to temperature gradients induced in the fluid by heat transfer.

The natural convection mechanism works similarly as described above, whilst discussion of the problem. The fluid which is in contact with the surface absorbs or transfers heat from the surface and becomes hotter or colder than the surrounding fluid. Driven by buoyancy forces due to difference in densities caused by the temperature gradient, the fluid is displaced upwards or downwards. Surrounding fluid fills in the void created by the displaced fluid, which then undergoes the same process again. This gives rise to a convection current which drives the hot fluid to the top and cold fluid to the bottom of the convection cell. Buoyancy effects are driven by gravity, therefore natural convection requires presence of a gravitational force to work. It must be noted, however, that gravity is not the driving force behind the fluid movement. Presence of gravity only enables displacement of the fluid due to the density changes caused by temperature gradients.

Mathematical determination of the onset of natural convection is done through a dimensionless number called the Rayleigh number (Ra). The Rayleigh number is defined as:

$R{a}_{x}=\frac{g\beta}{\alpha \nu}\left({T}_{s}-{T}_{\infty}\right){x}^{3}$

`x`is the characteristic length (m)- $R{a}_{x}$ is the Rayleigh number for characteristic length
`x` - $g$ is acceleration due to gravity (m/s
^{2}) - ${T}_{s}$ is the surface temperature (K)
- ${T}_{\infty}$ is the quiescent temperature (fluid temperature far from the surface of the object) (K)
- $\nu $ is the kinematic viscosity (m
^{2}/s) - α is the thermal diffusivity (m
^{2}/s) - β is the thermal expansion coefficient (equals to $1/T$ for ideal gases where is absolute temperature).

The fluid properties $\nu $, α and β are evaluated at the film temperature, ${T}_{f}$, which is defined as:

${T}_{f}=\frac{{T}_{s}+{T}_{\infty}}{2}$

When the Rayleigh number is below a critical value for the fluid heat transfer is primarily in the form of conduction. When it exceeds this critical value the dominant heat transfer mechanism is convection.

### Boussinesq Density Model

The Boussinesq density model is an approximation method applied to buoyancy driven flows, such as natural convection flows. In the Boussinesq approximation, the density variation terms are neglected everywhere except when multiplied by acceleration due to gravity, $g$. The basis of this approximation is that since temperature changes are small, the resultant changes in density are small as well and thus can be neglected. However, when multiplied by $g$, the resultant term gives rise to forces which no longer are negligible. The Boussinesq approximation is:

$\rho ={\rho}_{0}\left(1-\beta \text{\Delta}T\right)$

- $\rho $ is the instantaneous density at temperature $T$ (kg/m
^{3}) - ${\rho}_{0}$ is the density at reference temperature ${T}_{o}$ (kg/m
^{3}) - $\text{\Delta}T$ is change in temperature $T-{T}_{o}$ (K)

As stated in the approximation, the Boussinesq density model is only applicable when density variations are small. A general guideline is to check for the condition $\left(\beta \text{\Delta}T\ll 1\right)$ to be true. This indirectly puts a limitation on this model to be used to only for cases where expected temperature differences within the fluid are not large.

## Open the HyperMesh Model Database

## Set the General Simulation Parameters

### Set the General Simulation Parameters

### Specify the Solver Settings

### Create Time History Output Points

The Time History Output command enables you to extract the nodal solution at any point within the domain.

## Set the Body Force and Material Model Parameters

### Modify the Material Model Parameters of Air

### Create a Material Model for the Solid

### Define the Gravitational Body Force

### Define the Volumetric Heat Source

## Specify the Boundary Conditions and Initial Conditions

### Specify the Material Properties and Surface Boundary Conditions

### Specify the Periodic Boundary Conditions

Since you are modeling only a portion of the infinitely long domain, you need to define the periodicity of the solution by specifying periodic boundary conditions. The solution is then constrained by periodicity. The periodic boundary condition links the corresponding pairs of nodes on the two surfaces on which the periodicity is defined.

### Specify the Reference Pressure

Since this model set up doesn’t have an inlet and outlet, we need to specify the reference pressure manually. This can be done by specifying a pressure nodal boundary condition at a node in the fluid region.

### Specify the Nodal Initial Conditions

## Compute the Solution

## Post-Process the Results with HyperGraph

## Post-Process the Results with HyperView

### Open HyperView and Load the Model and Results

### Create a Temperature Distribution Contour

### Create a Velocity Vector Plot

## Summary

In this tutorial, you successfully learned how to set up and solve a natural convection problem using AcuSolve in HyperMesh. You started by importing the HyperMesh model database and then set up the simulation parameters and boundary conditions. Once the solution was computed, you post-processed the results using HyperView and created a contour plot of temperature distribution across the domain.