# Buoyant Flow: Natural convection between heated plates¶

## Overview¶

The aim of this test case is to validate the following parameters of steady-state natural convection between two plates maintained at different temperatures. The incompressible, turbulent case is validated with the experimental results of Betts and Bokhari [1] as archived in the ERCOFTAC database [2]. The following parameters have been analysed:

• Velocity Profiles
• Temperature Profiles

The geometry is uploaded on to the SimScale platform and meshed using the snappyHexMesh tool.

This project could be imported from the library upon request.

## Geometry¶

The geometry is constructed based on the reference case [1], as shown in Fig.1. Its dimensions are $$2.18\ m \times 0.076\ m \times 0.52\ m$$, and the face details have been given in Table 1.

Fig.1. Geometry used in the study

Table 1: Domain Details
Face Type
ABCD Bottom
EFGG Top
ABFE Hot Wall
DCGH Cold Wall
BCGF Front

## Analysis type and Domain¶

The snappyHexMesh tool was used to generate a uniform mesh (see Fig.2. and Table 2.).

A typical property of the generated mesh is the $$y^+$$ (“y-plus”) value, which is defined as the non-dimensionalized distance to the wall; it is given by $$y^+ = u^*y/\nu$$. A $$y^+$$ value of 1 would correspond to the upper limit of the laminar sub-layer.

• Explicit resolution of the near-wall region: The first cell lies at most at the boundary of the laminar sub-layer and no further. Here, $$y^+$$ value is 1 or below.
• Use of wall-functions to resolve the near-wall region: There is no need to place cells very close to the laminar sub-layer, and typically $$30 \leqslant y^+ \leqslant 300$$.

A $$y^+$$ value of 30 was used for the inflation layer. The $$k-\omega$$ SST turbulence model was chosen, with wall functions for near-wall treatment of the flow.

Tool Type : OPENFOAM®

Analysis Type : buoyantSimpleFoam

Mesh and Element types :

Table 2: Mesh Metrics
Mesh type Number of volumes Type
snappyHexMesh $$5.95 \times 10^6$$ 3D hex

Fig.2. Mesh used for the SimScale case

## Simulation Setup¶

Fluid:

Table 3 encapsulates the properties of fluids used in the subsonic and supersonic case simulations.

Table 3: Fluid Properties
$$m$$ $$g/mol$$ $$c_p$$ $$J/kgK$$ $$mu$$ $$N/ms$$ $$Pr$$
$$28.9$$ $$1005$$ $$1.831\times 10^{-5}$$ $$0.705$$

The boundary conditions for the simulation are shown in Table 4. Note: FFP stands for Fixed Flux Pressure.

Boundary Conditions:

Table 4: Boundary Conditions for Ahmed Body simulation
Parameter Top and Bottom Front and Back Hot Wall Cold Wall
Velocity $$0.0\ ms^{-1}$$ $$0.0\ ms^{-1}$$ $$0.0\ ms^{-1}$$ $$0.0\ ms^{-1}$$
Modified Pressure FFP ( $$10^5$$ Pa) FFP ( $$10^5$$ Pa) FFP ( $$10^5$$ Pa) FFP ($$10^5$$ Pa)
Temperature Zero Gradient Zero Gradient $$307.85$$ K $$288.25$$ K
$$k$$ Wall Function Wall Function Wall Function Wall Function
$$\omega$$ Wall Function Wall Function Wall Function Wall Function
$$\alpha _t$$ Wall Function Wall Function Wall Function Wall Function
$$\mu _t$$ Wall Function Wall Function Wall Function Wall Function

## Results¶

Velocity Profiles

Shown below are comparisons of velocity profile between the two plates from SimScale simulation results with the reference [1] at different heights. The reference lines are located at the mid-plane normal to the z-direction.

Fig.3.a. Velocity profile at $$h = 872\ mm$$

Fig.3.b. Velocity profile at $$h = 218\ mm$$

Fig.3.c. Velocity profile at $$h = 109\ mm$$

Temperature Profiles

Shown below is the comparison of the temperature profile between the two plates from SimScale simulation results with the reference [1] at a height of $$109\ mm$$. The reference line is located at the mid-plane normal to the z-direction.

Fig.4. Temperature profile at $$h = 872\ mm$$

## Disclaimer¶

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.