3.2.4.18. Advective Bar Equation

The advective bar block is used to solve for 1-D temperature, velocity and (optionally) pressure to be used for a convective heat transfer boundary condition on a surface. These variables only vary along the length of the bar, and assume a constant value in the cross sectional area for each segment.

3.2.4.18.1. Advective Bar Velocity

The advective bar assumes a constant mass flow rate over the length of the bar. Typically, the mass flow rate is specified in the advective bar command block. The bar velocity $\v_{bar}$ is then computed as:

(3.79) $\v_{bar} = \frac{\dot{m}}{\density A}\hat{\vector{s}}$

Where $\hat{\vector{s}}$ is the bar coordinate direction, $\dot{m}$ is the mass flow rate, $\density$ is the density and $A$ is the cross section area. The bar velocity is then used to compute a Nusselt number $\mathrm{Nu}$ (details of declaring this can be found in Correlation Heat Transfer Coefficient). The heat transfer correlation coefficient $h$ is then computed as

(3.80) $h = \frac{K}{D_{h}} \mathrm{Nu}$

Where $D_{h}$ and $K$ are the scalar thermal conductivity and characteristic length for the simulation, which MUST be specified in the heat transfer correlation coefficient command block. A convective heat flux ( $q_{conv}$ ) can then be applied to the bar

(3.81) $q_{conv} = h (T_{bar} - T)$

Where $T_{bar}$ and $T$ are the temperature of the bar and the coupled surface, respectively. Note that the flux applied to the bar is equal and opposite to the flux that will be received by the coupled, opposing domain.

3.2.4.18.2. Advective Bar Energy

The energy equation for the 1-D bar block is given as follows:

(3.82) $\density\specificheat\pt{T_{bar}} + \density\specificheat \v_{bar}\bcdot\grad T_{bar} = -\div\vector{q_{bar}} + H_{V,bar}.$

Taking a standard finite element formulation, the G/FEM residual form is:

(3.83) $\symRes_T^i & = \underbrace{\int\limits_\Vol \density\specificheat\pt{T_{bar}}\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \density\specificheat \v_{bar}\bcdot\grad T_{bar}\phi^i\dV}_\mathrm{ADV} - \underbrace{\int\limits_\Vol H_{V,bar} \phi^i\dV}_{\mathrm{SRC}} \\ & - \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{q_{bar}}\dV}_\mathrm{DIFF} + \int\limits_\Surf q_{conv}\phi^i\dS = 0$

where $q_{conv}$ includes any surface heat transfer occurring between the bar and the coupled surfaces (Eq. (3.81)). The diffusion term in Eq. (3.83) ends up being an axial diffusion term; for advection dominant flows, this diffusion term may be dropped and SUPG stabilization is recommended.

Beta Capability

BAR MOMENTUM EQUATION is a beta feature and requires use of the –beta flag. This capability is not thoroughly tested and should be used with caution. Refer to Current Discrepancies with Advective Bars for proper usage.

3.2.4.18.3. Advective Bar Pressure

The velocity is assumed to be known from Eq. (3.79). Assuming steady state flow, a Newtonian fluid and continuity, the G/FEM residual form of the bar pressure $P_{bar}$ is:

(3.84) $R_P^i = \int\limits_\Vol \phi^i \vector{\hat{s}}\bcdot\left(\density \v_{bar}\bcdot\grad \v_{bar} - \density\g + \nabla P_{bar} \right) \dV + \int\limits_\Surf \tau_{w} \phi^i\dS= \vector{0}$

Where $\tau_{w}$ is the wall shear stress. This shear wall shear stress resolves any head loss due to friction along the length of the bar and the connected surface. It should be noted that within Aria, the surface integral of $\tau_{w}$ is re-written into a volumetric form over the advective bar as opposed to a surface integral over the bar-coupled surface (for ease of implementation). Eq. (3.84) becomes:

(3.85) $R_P^i & = \underbrace{\int\limits_\Vol \phi^i \vector{\hat{s}}\bcdot\left(\density \v_{bar}\bcdot\grad \v_{bar} + \nabla P_{bar} \right)\dV }_\mathrm{ADV} \\ & + \underbrace{\int\limits_\Vol \phi^i \left( \frac{W_{p}}{A}\tau_{w} -\vector{\hat{s}}\bcdot\density\g \right) \dV }_\mathrm{SRC} = \vector{0}$

where $W_{p}$ is the wetted perimeter. Thus, any major losses are account for in the source term. Currently, the wall shear stress is modeled using the Darcy-Weisbach equation:

(3.86) $\tau_{w} = \frac{1}{8} f \density \v_{bar} \cdot \v_{bar}$

where $f$ is a non-dimensional friction factor. This friction factor is defined in the Aria material block; for the bar momentum equation, the Haaland friction factor is typically used:

(3.87) $f = \begin{cases} \left[ -1.8\mathrm{log_{10}}\left( \left(\frac{r_{\varepsilon}}{3.7 D}\right)^{1.11} + \frac{6.9}{Re}\right)\right]^{-2} & Re > 2000 \\ \frac{64}{Re} & 0 < Re \le 2000 \end{cases}$

where $D$ is the hydraulic diameter and $r_{\varepsilon}$ is the relative surface roughness. Eq. (3.87) is a fitting to the Moody chart, which is a graph of friction factors for varying Reynolds number $Re$ and surface roughness $r_{\varepsilon}$ . Some notes on defining this friction factor are reviewed in Current Discrepancies with Advective Bars.