Flow over a Heated Plate

The incoming uniform air stream has a velocity of 1.5 m/s, a temperature of 293.2 K, and a static pressure of 1 atm, see Figure 1. Thus, the flow Reynolds number defined on the incoming flow characteristics and on the plate length of 0.31 m is equal to 3.1·104, therefore the boundary layer beginning from the plate’s leading edge is laminar (see Ref. 12).

Then, let us consider the following three cases:

• Case #1

  The plate over its whole length (within the computational domain) is 10°C warmer than the incoming air (303.2 K), both the hydrodynamic and the thermal boundary layer begin at the plate's leading edge coinciding with the computational domain boundary;

• Case #2

  The upstream half of the plate (i.e. at x ~ 0.15 m) has a fluid temperature of 293.2 K, and the downstream half of the plate is 10°C warmer than the incoming air (303.2 K), the hydrodynamic boundary layer begins at the plate's leading edge coinciding with the computational boundary;

• Case #3

  Plate temperature is the same as in case #1, the thermal boundary layer begins at the inlet computational domain boundary, whereas the hydrodynamic boundary layer at the inlet computational domain boundary has a non-zero thickness which is equal to that in case #2 at the thermal boundary layer starting.

The calculation goal is to predict the local coefficient of heat transfer from the wall to the fluid, as well as the local skin-friction coefficient.

The geometry model used for calculating the 2D flow over the heated flat plate with Flow Simulation is shown in Figure 2. The problem is solved as internal in order to avoid the conflict situation when the external flow boundary with ambient temperature conditions intersects the wall with a thermal boundary layer.

To avoid any influence of the upper wall on the flow near the heated lower wall, the ideal wall boundary condition has been specified on the upper wall. To solve the internal problem, the incoming fluid velocity is specified at the channel inlet, whereas the fluid static pressure is specified at the channel exit. To specify the external flow features, the incoming stream's turbulent intensity is set to 1% and the turbulent length is set to 0.01 m, i.e., these turbulent values are similar to the default values for external flow problems.

The heat transfer coefficient h and the skin-friction coefficient Cf are Flow Simulation output flow parameters. The theoretical values for laminar flow boundary layer over a flat plate, in accordance with Ref. 12 can be determined from the following equations:

where

k is the thermal conductivity of the fluid, x is the distance along the wall from the start of the hydrodynamic boundary layer,

Nux is the Nusselt number defined on a heated wall as follows:

for a laminar boundary layer if it’s starting point coincides with the thermal boundary layer starting point, and

for a laminar boundary layer if the thermal boundary layer begins at point x0 lying downstream of the hydrodynamic boundary layer starting point, in this case Nux is defined at x > x0 only;

where is the Prandtl number, μ is the fluid dynamic viscosity, Cp is the fluid specific heat at constant pressure, is the Reynolds number defined on x, ρ is the fluid density, and V is the fluid velocity;

at Rex ≤ 5∙105, i.e., with a laminar boundary layer.

As for the hydrodynamic boundary layer thickness d needed for specification at the computational domain boundary in case #3, in accordance with Ref. 12, it has been determined from the following equation: , so d = 0.00575 m in this case. For these calculations all fluid parameters are determined at the outer boundary of the boundary layer.

The Flow Simulation predictions of h and Cf performed at result resolution level 7, and the theoretical curves calculated with the formulas presented above are shown in Figure 3 and Figure 4. It is seen that the Flow Simulation predictions of the heat transfer coefficient and the skin-friction coefficient are in excellent agreement with the theoretical curves.