**5.4 Functional relationship between distortion parameters and the aerodynamic stability of fans**

First of all, the calculated examples are supplied using a uniform test design to increase the generality, universality, and reliability of the calculation results. Secondly, the mathematical relations between the stability margin loss and distortion parameters are derived and used as the basis for the next calculation and analysis to strengthen the logic of the calculation and analysis (**Figure 19**).

The changes in the velocity triangle before and after the swirl distortion and mathematical approximation show that stability margin loss Δ*SM* and swirl distortion

#### **Figure 17.**

*Stability margin loss under the total pressure and total temperature distortion. (a) Total pressure distortion parameters as the abscissa. (b) Total temperature distortion parameters as the abscissa.*

**Figure 18.**

*Maximum stability margin loss under the total pressure and total temperature distortion at different spans (from left to right: 99, 80, 50, and 10% of the span). (a) Total pressure distribution. (b) Velocity distribution.*

**Figure 19.**

*Velocity triangles before and after the distortion. (a) Swirl distortion. (b) Total temperature distortion.*

parameter *τ*<sup>87</sup> have an approximate quadratic relationship when loss coefficient *ω* has an approximate quadratic relationship with attack angle *i* and an approximately linear relationship with stability margin *SM* (see Eq. (6)).

Similarly, the changes in the velocity triangle before and after the total temperature distortion and mathematical approximation show that stability margin loss Δ*SM* and total temperature distortion parameter *δT2*FAV have an approximately quartic relationship (see Eq. (7)). Besides, stability margin loss Δ*SM* and total pressure distortion parameter Δ*σ*<sup>0</sup> have an approximately linear relationship (see Eq. (8)).

The approximate error derived from the above mathematical relationship is within the acceptable range through mathematical verification.

$$
\Delta \text{SM}\_{\text{r}} \sim \overline{w} \sim i^2 \sim \tau\_{87}^2 \tag{6}
$$

$$
\Delta \text{SM}\_{\text{tr}} \sim \overline{\alpha} \sim i^2 \sim \left(\frac{\Delta T}{T}\right)^4 \tag{7}
$$

$$
\Delta \text{SM}\_p \sim \overline{w} \sim \frac{\Delta p}{p} \tag{8}
$$

According to the numerical simulation results including the ternary combined distortion examples, linear regression fitting is carried out based on the approximate mathematical relationship between the stability margin loss and each distortion parameter derived above. The obtained fitting equation is used as the prediction model of the stability margin loss of fans based on combined distortion parameters (see Eq. (9)).

$$
\Delta \mathbf{SM} = \text{tr}(\mathbf{AX}) + \mathbf{B} \tag{9}
$$

where

$$A = \begin{bmatrix} a\_1 & 0 & 0 & 0 \\ a\_2 & a\_4 & a\_6 & a\_7 \\ a\_3 & a\_5 & 0 & 0 \end{bmatrix} \tag{10}$$

$$X = \begin{bmatrix} \Delta \overline{\sigma}\_0 & \delta T\_2 & \tau\_{87} \\ \Delta \overline{\sigma}\_0^{\;2} & \delta T\_2^{\;2} & \tau\_{87}^{\;2} \\ \Delta \overline{\sigma}\_0^{\;3} & \delta T\_2^{\;3} & \tau\_{87}^{\;3} \\ \Delta \overline{\sigma}\_0^{\;4} & \delta T\_2^{\;4} & \tau\_{87}^{\;4} \end{bmatrix} \tag{11}$$

$$B = \mathfrak{a}\_0 \tag{12}$$

**Table 3** shows the parameter values in the prediction model.

Compared with the CFD calculation results, the error of the prediction model is generally within �3.1%; the average of the absolute value of the error is 0.801%; the median of the absolute error value is 0.610%. To sum up, the prediction results are generally good.


**Table 3.** *Parameter values of the prediction model.*

**Figure 20** shows the prediction results. Scattered points are CFD calculation data, and the 3D counter is the prediction result. The prediction model has a good prediction effect and can be used as a prediction tool for the stability margin loss.

$$\text{(a)}$$

#### **Figure 20.**

*Prediction of the stability margin loss under the total pressure, total temperature, and swirl distortion based on the prediction model. (a) View of the low-total-pressure and low-total-temperature distortion. (b) View of the high-total-pressure and high-total-temperature distortion.*
