Validation Case: Natural Convection Around Cylinder

The aim of this validation is to compare the natural convection around cylinder simulation results performed in SimScale using the compressible flow feature in its proprietary solver, Multi-purpose, with the results obtained analytically$^1$.

This simulation aims at validating the heat loss characteristics from a constant heat flux horizontal cylinder. First, the convective heat transfer coefficient, as well as the corresponding cylindrical surface temperature, is calculated analytically through an iterative approach. After that, the surface temperature values per various surface heat flux values are calculated in the SimScale platform. Finally, the results from the analytical method and the platform are compared.

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Geometry

Analytically, the geometry is nothing but a cylinder in a flow domain. For simulation purposes, reducing the computational requirement, we model only 50% of the cylinder-flow domain.

To be able to capture the flow physics, we need to ensure that the domain is large enough. In other words, flow domain boundaries need to be far enough from the cylinder. How far? This needs to be found by testing different flow domain sizes. Slightly increase the domain size until the results no longer change significantly (domain independence test).

The schematic of the cylinder and the flow domain around it is depicted in Figure 1:

half cylinder domain schematic heat transfer case — Figure 1: Schematic of the half-cylinder and the flow domain with dimensions

The half-cylinder is 0.5 $cm$ in diameter, placed 2.15 $cm$ from the inlet and 6.55 $cm$ from the outlet. The fluid domain is 1.5 $cm$ thick and 3 $cm$ wide. The exact model in SimScale can be observed as follows:

half cylinder symmetry convective heat transfer around cylinder — Figure 2: Symmetric CAD model of a cylinder in a flow domain for the convective heat transfer case used in SimScale

Analysis Type and Mesh

Analysis Type: Steady-state, Multi-purpose with k-epsilon and Compressible model

Mesh and Element Types:

Two meshes were created, Mesh 1 and Mesh 2 — Mesh 2 more refined than the other — to perform the mesh independence study. Both meshes were created with SimScale’s Multi-purpose mesh type, which is a body-fitted structured mesh. An automatic sizing definition was defined with an additional surface refinement at the inlet and the cylinder surfaces.

Mesh	Mesh Type	Global Fineness	Target Cell Size (refinement) $[m]$	Number of Cells	Element Type
Mesh 1	Automatic with surface refinements	8	1e-4	888702	3D Hexahedral
Mesh 2	Automatic with surface refinements	10	1e-4	1600369	3D Hexahedral

Table 1: Mesh data for normal shock inside a diffuser validation case

The resulting Mesh 1 is as observed below. Due to extreme fineness only the part near the cylinder is shown.

mesh cylinder surface refinement — Figure 3: Mesh 1 in detail near the cylinder. A surface refinement is used to accurately capture the temperature profile around it.

Simulation Setup

Material

Fluid:

Water
- Dynamic viscosity $(\mu)$: 0.0010230498 $kg/m.s$
- Molar mass $(M_m)$: 18 $kg/kmol$
- Density $(\rho)$: 998 $kg/m^3$
- Prandlt number $(Pr)$: 6.991
- Specific heat $(C_p)$: 4182 $J/kg.K$

Boundary Conditions

Figure 4 shows the schematic of the boundary conditions applied:

The inlet is set at atmospheric pressure while the outlet at equivalent hydrostatic pressure due to the influence of gravity. The cylinder is assigned a fixed heat flux value while the remaining faces act as a symmetry.

This convective heat transfer validation case is tested for five different values of heat flux. All the values for the boundary conditions are listed in Table 2:

Boundary Condition	Value
Pressure inlet $[Pa]$	101325 (Absolute total pressure with 295 $K$ total temperature)
Pressure outlet $[Pa]$	100424.28 (Hydrostatic absolute static pressure)
External wall heat flux $[W/m^2]$	147634.916 107456.01 70228.18 36856.23 9314.75
Symmetry	All other faces

Table 2: Boundary conditions for the natural convection around cylinder validation case

Reference Solution

The following section explains the calculation for the analytical results using equations as stated in [1].

Natural convection heat transfer is calculated as follows:
$$ q = \alpha (T_s – T_{amb}) \tag{1}$$
Where

$q$ is the heat flux $[W/m^2]$,
$\alpha$ is the convection coefficient $[W/(m^2.K)]$,
$T_s$ is the surface temperature $[K]$,
$T_{amb}$ is the ambient temperature $[K]$.

Convection coefficient $\alpha$ is a function of the material properties and the geometry.
$$ \alpha = Nu \frac{k}{L} \tag{2}$$
Where

$Nu$ is the Nusselt number,
$k$ is the thermal conductivity of the medium $[W/(m.K)]$,
$L$ is the characteristic length of the geometry $[m]$ .

Calculation of the Nusselt number requires an empirical correlation. Natural convection from a horizontal heated cylinder (constant heat flux) is calculated by using the following empirical equation:

empirical correlation nusselt number natural convection — Figure 5: Empirical correlation for the average Nusselt number for natural convection over a horizontal cylinder surface$^1$

To be able to calculate the surface temperature $(T_s)$, we need to know the convection coefficient $(\alpha)$. $\alpha$ is a temperature-dependent characteristic. This means the calculation of surface temperature is an iterative process.

The values for heat flux, convection coefficient, and surface temperature are listed in Table 3 below.

Result Comparison

An average surface temperature over the cylinder surface was calculated per value of the heat flux (see Table 2) for both meshes, Mesh 1 and Mesh 2. The difference between the values evaluated with each mesh is negligible.

The result output from the SimScale simulation is compared against the analytically obtained surface temperature values.

$q$ $[W/m^2]$	$\alpha$ $[W/m^2.K]$	$T_{analytical}$ $[K]$	$T_{simulation}$ $[K]$ Mesh 1	$T_{simulation}$ $[K]$ Mesh 2	Error %, $T_{simulation}$ Mesh 1 and $T_{analytical}$
147634.916	1.64E+03	383.15	386.02	386.45	0.749
107456.01	1.53E+03	363.15	361.19	361.61	-0.539
70228.18	1.40E+03	343.15	338.26	338.53	-1.425
36856.23	1.22E+03	323.15	317.7	317.85	-1.686
9314.75	9.18E+02	303.15	300.74	300.77	-0.795

Table 3: Comparison of the cylinder surface temperature values between analytical and SimScale

SimScale results show a very good agreement with analytical ones. The calculated deviation is less than 2%.

Here onwards, visual results for only Mesh 1 study are shown:

Temperature

The cylinder surface temperature increases as we travel away from the inlet. This happens because the oncoming cold water temperature increases as it travels upwards after coming in contact with the cylinder.

Velocity Vectors

On close observation using the vectors we observe a flow circulation above the cylinder. The reverse water flow helps to reduce the temperature of the cylinder surface closer to the symmetry face.

This also explains why the maximum point of temperature on the cylinder surface is not the farthest from the inlet (see Figure 7).

velocity vectors and contours natural convection flow simscale multi-purpose — Figure 8: Velocity contours and vectors show a natural convection flow pattern as the inlet water current hits the hot cylinder.

Temperature with velocity streamlines

References

Cengel, Y. A., & Ghajar, A. J. (2014). Heat and mass transfer: Fundamentals and applications (5th ed.). McGraw-Hill Professional.

Note

If you still encounter problems validating your simulation, then please post the issue on our forum or contact us.

Last updated: June 23rd, 2025

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Boundary Condition	Value
Pressure inlet \([Pa]\)	101325 (Absolute total pressure with 295 \(K\) total temperature)
Pressure outlet \([Pa]\)	100424.28 (Hydrostatic absolute static pressure)
External wall heat flux \([W/m^2]\)	147634.916 107456.01 70228.18 36856.23 9314.75
Symmetry	All other faces

\(q\) \([W/m^2]\)	\(\alpha\) \([W/m^2.K]\)	\(T_{analytical}\) \([K]\)	\(T_{simulation}\) \([K]\) Mesh 1	\(T_{simulation}\) \([K]\) Mesh 2	Error %, \(T_{simulation}\) Mesh 1 and \(T_{analytical}\)
147634.916	1.64E+03	383.15	386.02	386.45	0.749
107456.01	1.53E+03	363.15	361.19	361.61	-0.539
70228.18	1.40E+03	343.15	338.26	338.53	-1.425
36856.23	1.22E+03	323.15	317.7	317.85	-1.686
9314.75	9.18E+02	303.15	300.74	300.77	-0.795