Axisymmetric Flow

For fluid flows that are symmetric about a central axis in terms of geometry and flow conditions, Simcenter STAR-CCM+ can model the flow on a 2D solution domain.

The conservation equations for mass and energy are given by Eqn. (664) and Eqn. (666), respectively. The conservation equation for mometum is defined depending on whether or not swirl is involved:

Momentum Equation without Swirl

For the momentum equation without swirl, Simcenter STAR-CCM+ assumes that the circumferential velocity and the circumferential gradients are zero.

The following image displays an example of axisymmetric flow without swirl:

The conservation equation for momentum in cylindrical coordinates is defined as:

$$\begin{array}{c}\frac{\partial }{\partial t}\underset{A}{\int }\rho vrdA+\underset{\partial A}{\overset{}{\oint }}\rho v\otimes v\cdot rds\text{ =}\\ -{\oint }_{\partial A}pI\cdot rds+{\oint }_{\partial A}T\cdot rds+\underset{A}{\int }\frac{1}{r}\left[\begin{array}{c}0\\ \begin{array}{c}p-{\tau }_{\theta \theta }\\ 0\end{array}\end{array}\right]rdA+\underset{A}{\int }{f}_{b}dA+\underset{A}{\int }{s}_{u}dA\end{array}$$

(667)

with:

• $A$ is area.
• $\partial A$ is the contour of $A$ .
• $I$ is the identity matrix.
• $v={\left(v_z{v}_r{v}_{\theta }\right)}^T$
• $v_{\theta }=0$ and $\frac{\partial }{\partial \theta }\left(\right)=0$

$T$ is the stress tensor defined as:

$T=\mu \left[\begin{array}{ccc}2\frac{\partial v_z}{\partial z}-\frac{2}{3}\nabla \cdot v& \frac{\partial v_z}{\partial r}+\frac{\partial v_r}{\partial z}& 0\\ \frac{\partial v_z}{\partial r}+\frac{\partial v_r}{\partial z}& 2\frac{\partial v_r}{\partial r}-\frac{2}{3}\nabla \cdot v& 0\\ 0& 0& 2\frac{v_r}{r}-\frac{2}{3}\nabla \cdot v\end{array}\right]$

Momentum Equation with Swirl

The Axisymmetric Swirl model adds the prediction of swirling or rotating flow to a 2D axisymmetric flow simulation. You can use this model for axisymmetric flows that include a swirling flow from an inlet, such as a fan interface, or rotating walls.

The following image displays an example of axisymmetric flow with swirl:

Under the assumption that the flow shows circumferential velocities but no circumferential gradients, this model solves the momentum conservation equation in cylindrical coordinates as:

$$\begin{array}{c}\frac{\partial }{\partial t}\underset{A}{\int }\rho vrdA+\underset{\partial A}{\overset{}{\oint }}\rho v\otimes v\cdot rds\text{ =}\\ -{\oint }_{\partial A}pI\cdot rds+{\oint }_{\partial A}T\cdot rds+\underset{A}{\int }\frac{1}{r}\left[\begin{array}{c}0\\ \begin{array}{c}p+\rho v_{\theta }^2-{\tau }_{\theta \theta }\\ -\rho v_r{v}_{\theta }+{\tau }_{\theta r}\end{array}\end{array}\right]rdA+\underset{A}{\int }{f}_{b}dA+\underset{A}{\int }{s}_{u}dA\end{array}$$

(668)

with:

• $v_{\theta }\ne 0$ and $\frac{\partial }{\partial \theta }\left(\right)=0$

The stress tensor $T$ is defined as:

$T=\mu \left[\begin{array}{ccc}2\frac{\partial v_z}{\partial z}-\frac{2}{3}\nabla \cdot v& \frac{\partial v_z}{\partial r}+\frac{\partial v_r}{\partial z}& \frac{\partial v_{\theta }}{\partial z}\\ \frac{\partial v_z}{\partial r}+\frac{\partial v_r}{\partial z}& 2\frac{\partial v_r}{\partial r}-\frac{2}{3}\nabla \cdot v& \frac{\partial v_{\theta }}{\partial r}-\frac{v_{\theta }}{r}\\ \frac{\partial v_{\theta }}{\partial z}& \frac{\partial v_{\theta }}{\partial r}-\frac{v_{\theta }}{r}& 2\frac{v_r}{r}-\frac{2}{3}\nabla \cdot v\end{array}\right]$