*4.4. Quasi-Circular (Spiral) Whirl*

In this section, the rotor is assumed to perform the circular whirl around seal center with growing whirl radius, i.e., the spiral whirl, in order to validate the suitability of nonlinear dynamic model for quasi-circular whirls no matter eccentricity magnitudes. The whirling speed is r/min and the rotor eccentricity ratio rises linearly to 60% within six whirl periods. The whirl equation is presented in Equation (17) and the whirl orbit is shown in Figure 25.

$$\begin{cases} \mathbf{x} = f \cdot t \cos(\Omega t) \\ \mathbf{y} = f \cdot t \sin(\Omega t) \end{cases} \tag{19}$$

where *T* is the whirl period and *f* = 60% *Cr*/(6 *T*), indicating the eccentricity speed of the rotor.

In actual applications, the spiral whirl in Figure 25 represents the destabilizing process of the rotor. Fluid forces induced by the destabilizing whirl are obtained, respectively, by nonlinear dynamic model and transient CFD simulations. They are shown in Figure 26. The rise of rotor eccentricity with time leads to the increasing fluid force (i.e., the resultant force of *Fx* and *Fy*). The *Fx* and *Fy* from nonlinear model are in good agreement with those from transient CFD simulations. Maximum differences of two approaches are 42.7 N (i.e., relative error 2.3%) for *Fx* and 35.7 N (relative error 2.3%) for *Fy*. Namely, nonlinear dynamic model is reliable in evaluating fluid forces induced by the quasi-circular whirl around seal center without special limitations on eccentricity magnitudes.

**Figure 25.** Spiral whirl orbit of the rotor.

**Figure 26.** Fluid forces induced by the spiral whirl around seal center.

#### **5. Conclusions**

In the paper, dynamic characteristics of the annular plain liquid seal under various large rotor disturbance motions are studied using the transient CFD method based on dynamic mesh technique and nonlinear Muszynska' model.

Force coefficients and leakage rates of annular seal under different static eccentricities are evaluated. The reliability of transient CFD simulation is validated by comparing the force coefficients and leakage rates with those from the Marquette's experiment and bulk flow method. With increasing static eccentricity, these force coefficients show clearly asymmetric behavior and obvious changes. The force coefficients from transient CFD simulations show a high consistency with experimental values despite the different values of stiffness. The error sources are mainly form the influence of upstream and inlet boundary condition due to the drawback of the experimental apparatus for absent inlet control. Leakage rates computed by the CFD method fit better to measured values than those from the bulk flow method, which indicates that leakage rates are insensitive to static eccentricity.

As to the concentric annular seal, its dynamic characteristics are usually supposed to be linear (namely, constant force coefficients) when the rotor disturbance is within 10% *Cr*. However, this conclusion is not suitable for the eccentric annular seal, especially the seal under large static eccentricity. As rotor static eccentricity increases, the force characteristics of annular seal become more sensitive to whirl amplitude, in other words, the linear range of dynamic characteristics narrows gradually. With respect to the concentric seal, the annular seal with large eccentricity is easier to show nonlinear characteristics.

According to the Muszynska's model, a nonlinear dynamic model is presented in the paper for describing nonlinear seal forces induced by rotor large disturbances. The suitability of the nonlinear model for all kinds of rotor disturbances is studied through four forms of rotor motions. The nonlinear dynamic model is suitable for various rotor disturbances when the rotor is under small static eccentricity (e.g., eccentricity ratio under 20%). However, when rotor static eccentricity is large (e.g., eccentricity ratio 50%), the nonlinear dynamic model based on circular whirls around eccentric center becomes incompetent and unsatisfactory. It shows high reliability only for circular or quasi-circular whirls around concentric center. This means that dynamic characteristics of annular seal under large disturbance are related to rotor motion ways. For the annular seals under large dynamic eccentricity (whirl amplitude) and rather small static eccentricity (e.g., static eccentricity ratio under 20% in this case), the nonlinear Muszynska's model performs well when dealing with large rotor disturbances. The range of capability of this nonlinear model depends on the typical parameters of annular seals. It can also explain why Muszynska's model is out of action when rotor–seal system has a large eccentricity ratio in He's research.

On the whole, dynamic characteristics of annular seals under large disturbance are very complex. They are very sensitive to various rotor motion ways including whirl amplitude and static eccentricity. For the seal with large disturbances motion of a small static eccentricity, the nonlinear Muszynska's model performs reliably, which provides a solid basis for the seal–rotor system analysis using nonlinear seal force model. The capability and limitation of nonlinear dynamic model under large disturbances needs further investigation.

**Author Contributions:** Conceptualization, K.Z. and D.W.; methodology, K.Z., X.J. and P.W.; validation, D.W. and P.W.; formal analysis, K.Z. and X.J.; investigation, K.Z., S.L. and P.W.; resources, D.W.; data curation, K.Z., X.J. and S.Y.; writing—original draft preparation, K.Z. and X.J.; writing—review and editing, S.L. and S.Y.; visualization, K.Z.; supervision, P.W. and D.W.; project administration, B.H. and D.W.; and funding acquisition, B.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant numbers 51706198 and 51839010.

**Conflicts of Interest:** The authors declare no conflict of interest.
