Deviation models

Accurately predicting flow angles is essential for performance analysis because they have a significant influence on the velocity triangles and work output or consumption of the turbomachinery. In general, the relative flow angle at the exit of a cascade is not exactly equal to the metal angle because the blades cannot guide the flow perfectly. This discrepancy is known as the deviation angle, which is formally defined as the difference between a reference geometric angle (\(\beta_g\)) and the relative flow angle (\(\beta\)) at the exit of the cascade:

\[\delta = \beta_g - \beta\]

Axial turbines

For axial turbines, three deviation models are implemtented, varying in complexity.

Zero deviation

The simplest way to model deviation is to assume zero deviation:

\[\beta = \beta_g\]

Ainley and Mathieson

The deviation model presented by Ainley and Mathieson ([Ainley and Mathieson, 1951]) assumes constant deviation at low speed (mach \(<\) 0.5), and a linear deviation between mach = 0.5 and mach = 1.0 (critical point). This result in sharp edges at the points where mach = 0.5 and mach = 1.0. In addition, the model assumes that the critical state occur when mach = 1.0 at the throat, which can lead to unphysical behaviour of the simulated performance ([Anderson et al., 2022]). To correct for this behaviour, the model is adjusted to be interpolated between mach = 0.5 and the critical mach number.

Note that his model defines the gauing angle with respect to axial direction:

\[\beta_g = \cos^{-1}(A_\mathrm{throat} / A_\mathrm{out})\]

For \(\mathrm{Ma_exit} < 0.50\) (low-speed), the deviation is a function of the gauging angle:

\[\delta_0 = \beta_g - (35.0 + \frac{80.0-35.0}{79.0-40.0}\cdot (\beta_g-40.0))\]

For \(0.50 \leq \mathrm{Ma_exit} < \mathrm{Ma_crit}\) (medium-speed), the deviation is calculated by a linear
interpolation between low and critical Mach numbers:

\[\delta = \delta_0\cdot \left(1+\frac{0.5-\mathrm{Ma_exit}}{\mathrm{Ma_crit}-0.5}\right)\]

For \(\mathrm{Ma_exit} \geq \mathrm{Ma_crit}\) (supersonic), zero deviation is assumed:

\[\delta = 0.00\]

The flow angle (\(\beta\)) is then computed based on the deviation and the gauging angle:

\[\beta = \beta_g - \delta\]

Aungier

The deviation model presented by Aungier ([Aungier, 2006]) assumes constant deviation at low speed (Mach \(<\) 0.5), and a fifth order polynomial between mach = 0.5 and mach = 1.0 (critical point). This ensures a smooth evolution of the flow angle with the mach number. However, also this model assumes that the critical state occur when mach = 1.0 at the throat. To correct this behaviour, the model is adjusted to be interpolated between mach = 0.5 and the critical mach number.

Note that his model defines the gauging angle with respect to tangential axis:

\[\beta_g = 90 - \cos^{-1}\left(\frac{A_\mathrm{throat}}{A_\mathrm{out}}\right)\]

For \(Ma_\mathrm{exit} < 0.50\), the deviation is a function of the gauging angle:

\[\delta_0 = \sin^{-1}\left(\frac{A_\mathrm{throat}}{A_\mathrm{out}} \left(1+\left(1-\frac{A_\mathrm{throat}}{A_\mathrm{out}}\right)\cdot\left(\frac{\beta_g}{90}\right)^2\right)\right)\]

For \(0.50 \leq Ma_\mathrm{exit} < Ma_\mathrm{crit}\), the deviation is calculated by a fifth order interpolation between low and critical Mach numbers:

\[\begin{split}\begin{align*} X &= \frac{2\cdot Ma_\mathrm{exit}-1}{2\cdot Ma_\mathrm{crit}-1} \\ \delta &= \delta_0 \cdot (1-10X^3+15X^4-6X^5) \end{align*}\end{split}\]

For \(Ma_\mathrm{exit} \geq Ma_\mathrm{crit}\), zero deviation is assumed:

\[\delta = 0.00\]

The flow angle (\(\beta\)) is then computed based on the deviation and the gauging angle:

\[\beta = 90 - \beta_g - \delta\]