## 2 Theoretical Assumptions in the Ansys Fluent Program [17; 18; 19; 20; 21; 22]
### Vortex and Rotational Flows  
Many important engineering flows involve spinning or rotation, and Ansys Fluent is well-suited to modeling such flows. Swirl flows are common in the combustion process, and swirl is introduced into the burners and combustion chambers to increase residence time and stabilize the flow pattern. When beginning to analyze swirling or vortex flow, it is important to classify the problem into one of the following five flow categories:  
* Axisymmetric flows with swirling or rotation;
* Fully three-dimensional vortex or rotational flows;
* Flows requiring a moving reference frame;
* Flows requiring multiple moving reference frames or mixing planes;
* Flows requiring sliding grids. * The principle of conservation of momentum for the vortex speed;  
The tangential momentum equation for 2D vortex flows can be written as:  
\[\frac{\partial}{\partial t}(\mathbf{\rho w})+\frac{1}{r}\frac{\partial}{\partial x }(\mathbf{r\rho w})+\frac{1}{r}\frac{\partial}{\partial r}(\mathbf{r\rho vw})=\frac{1 }{r}\frac{\partial}{\partial x}\left[\mathbf{r\mu}\frac{\partial\mathbf{w}}{\partial x }\right]+\frac{1}{r^{2}}\frac{\partial}{\partial r}\left[\mathbf{r}^{3}\mu\frac{ \partial}{\partial r}\left(\frac{\mathbf{w}}{r}\right)\right]-\mathbf{\rho}\frac{\mathbf{ vw}}{r} \tag{7}\]  
where \(x\) is the axial coordinate, \(r\) is the radial coordinate, \(\mathbf{u}\) is the axial velocity, \(\mathbf{v}\) is the radial velocity and \(\mathbf{w}\) is the rotation speed. * Compressible flows ([PERSON]'s law);
* Compressible flows can be characterized by the value of the Mach number: \[\mathbf{M}\equiv\mathbf{u}/\mathbf{c}\] (8)  
Here \(\mathbf{c}\) is the speed of sound in the gas \(\mathbf{c}=\sqrt{\mathbf{\gamma}\mathbf{R}\mathbf{T}}\), while \(\mathbf{\gamma}\) is the specific heat ratio \(\mathbf{\gamma}=\mathbf{c_{p}}/\mathbf{c_{p}}\). Equation (8) applies to an ideal gas. In the general case of real fluids, the speed of sound is defined in terms of isentropic compressibility as:  
\[\mathbf{c}=\mathbf{1}/\sqrt{\left(\frac{\partial\mathbf{\rho}}{\partial\mathbf{\rho}}\right)_ {s}} \tag{9}\]  
where \(\mathbf{\rho}\) is the density of the fluid, \(\mathbf{p}\) is the pressure, and the subscript \(s\) means that the partial derivative of the density \(\mathbf{\rho}\) with respect to the pressure \(\mathbf{p}\) is taken at constant entropy. When the Mach number is less than 1.0, the flow is called subsonic. At Mach numbers much less than 1.0 (\(\mathbf{M}<\mathbf{0.1}\) or thereabouts), compressibility effects are negligible, and changes in gas density with pressure can be safely ignored in flow modeling. As the Mach number approaches 1.0 (which is referred to as the transonic flow regime), compressibility effects become important. When the Mach number exceeds 1.0, the flow is said to be supersonic and may contain shocks and expansion fans that can significantly affect the flow pattern. Ansys Fluent provides a wide range of compressible flow modeling capabilities for subsonic, transonic, and supersonic flows. * Physics of compressible flows;  
Compressible flows are typically characterized by a total pressure \(\mathbf{p}_{0}\) and a total flow temperature \(\mathbf{T_{0}}\). In the case of an ideal gas, these quantities can be related to static pressure and temperature in the following way (assuming a constant \(\mathbf{C_{P}}\) parameter):  
\[\frac{\mathbf{p_{0}}}{\mathbf{p}}=\left(\mathbf{1}+\frac{\mathbf{y-1}}{\mathbf{2}}\mathbf{M}^{2} \right)^{\mathbf{y}/(\mathbf{y-1})} \tag{10}\]  
\[\frac{\mathbf{T_{0}}}{\mathbf{T}}=\mathbf{1}+\frac{\mathbf{y-1}}{\mathbf{2}}\mathbf{M}^{2} \tag{11}\]  
These relationships describe the change in static pressure and temperature in the flow with the change in speed (Mach number) in isentropic conditions. For example, given the ratio of pressures from inlet to outlet (total to static), Equation (10) can be used to estimate the output Mach number that would exist in one-dimensional isentropic flow.