**1. Introduction**

Hydraulic machinery such as pumps and turbines is widely applied in various energy fields, playing a significant role in energy development, utilization and transformation. The vibration caused by the fluid forces generated in gap seals of hydraulic machinery tend to have important effects on the efficiency and vibration of rotor system [1]. Due to the rise of safety and efficiency concerns, dynamic characteristics of various annular seals have been studied by researchers [2–5]. Almost all of these studies are based on the assumption of small perturbation, hence linear dynamic characteristics of annular seals can be investigated. Generally, the annular seal is not the supporting element in design. Under the condition of static equilibrium, the rotor is normally concentric with the annular seal. Due to the axial-symmetry of seal geometry, as shown in Figure 1, the force coefficients of concentric seals show symmetric or skew symmetric features, as shown in Equation (1), where *Fx*, *Fy* are the X and Y components of fluid forces respectively; *K* and *k* denote direct and cross stiffness coefficients,

respectively; similarly, direct and cross damping coefficients are expressed as *C* and *c*, respectively; and *M* is direct mass coefficient. These five coefficients can be numerically computed by using the bulk flow model [6], CFD simulations by introducing moving reference frame [7,8] or transient method [9] and measured by perturbing the rotor or the stator [10].

$$
\begin{aligned}
\begin{Bmatrix} -\begin{Bmatrix} F\_X \\ F\_Y \end{Bmatrix} \end{Bmatrix} = \begin{bmatrix} \begin{array}{cccc} K & k \\ -k & K \end{array} \end{bmatrix} \begin{Bmatrix} \begin{Bmatrix} \mathbf{x} \\ \mathbf{y} \end{Bmatrix} \end{Bmatrix} + \begin{bmatrix} \mathbf{C} & c \\ -c & \mathbf{C} \end{bmatrix} \begin{Bmatrix} \dot{\mathbf{x}} \\ \dot{\mathbf{y}} \end{Bmatrix} + \begin{bmatrix} M & 0 \\ 0 & M \end{bmatrix} \begin{Bmatrix} \ddot{\mathbf{x}} \\ \ddot{\mathbf{y}} \end{Bmatrix} \\\\ \mathbf{A} &= \mathbf{A}
\end{aligned} \tag{1}
$$

**Figure 1.** The circular whirl around seal center.

However, under actual condition, the static eccentricity of the rotor may exist in annular seal due to the misalignment during assembly process or the effects of various side loads (e.g., impeller weight). The dynamic characteristics of eccentric annular seals, as shown in Figure 2, were also investigated based on the bulk flow model by Nelson and Nguyen [11]. The fluid force increments (Δ*Fx* and Δ*Fy*) induced by the small perturbation around static eccentricity position are similarly expressed in linearized rotordynamic form, as shown in Equation (2) [12].

$$-\left\{\begin{array}{c} \Delta F\_{x} \\ \Delta F\_{y} \end{array}\right\} = \left[\begin{array}{cc} k\_{\text{xx}} & k\_{\text{xy}} \\ k\_{\text{yx}} & k\_{\text{yy}} \end{array}\right] \left\{\begin{array}{c} \Delta \mathbf{x} \\ \Delta \mathbf{y} \end{array}\right\} + \left[\begin{array}{cc} c\_{\text{xx}} & c\_{\text{xy}} \\ c\_{\text{yx}} & c\_{\text{yy}} \end{array}\right] \left\{\begin{array}{c} \Delta \dot{\mathbf{x}} \\ \Delta \dot{\mathbf{y}} \end{array}\right\} + \left[\begin{array}{cc} m\_{\text{xx}} & 0 \\ 0 & m\_{\text{yy}} \end{array}\right] \left\{\begin{array}{c} \Delta \ddot{\mathbf{x}} \\ \Delta \ddot{\mathbf{y}} \end{array}\right\},\tag{2}$$

where Δ*x* and Δ*y* define the rotor motion relative to the equilibrium position. Unlike concentric seals, the force coefficients of eccentric seals are no longer symmetric or skew symmetric due to rotor misalignment. This brings difficulties to the numerical solutions of force coefficients. Arghir and Frene [13] compared the bulk flow model of concentric seals and eccentric seals, the results showing that the terms of circumferential partial derivatives emerge in all bulk flow equations due to the static eccentricity of flow field. This can result in the coupling effect between circumferential momentum equation and continuity equation and make the solutions of both bulk flow equations and their perturbation equations very complex. As to the CFD method, the seal flow field disturbed by rotor circular whirl is not axisymmetric, as shown in Figure 2, and the steady-state simplified treatment by introducing moving reference frame is no longer applicable [8]. This means that transient simulations are necessary for evaluating force coefficients of eccentric seals.

**Figure 2.** The circular whirl around equilibrium position.

To overcome numerical difficulties in eccentric seal research, Venkataraman and Palazzolo [14] determined the circumferential derivatives through a cubic spline interpolation method and simplified the bulk flow equations of eccentric seals. Athavale and Hendricks [15] presented a small perturbation CFD method for calculation of rotordynamic coefficients of concentric and eccentric seals, and the SCISEAL code along with a modified SIMPLEC algorithm was adopted. Wu et al. [16] developed a new transient CFD method, which is based on rotor's variable-speed whirl; the results show that this new method can keep good accuracy of traditional transient method and save much computational time and cost in the meantime.

The research for fluid force presented above has mainly focused on linear fluid force analysis, and it was performed under the strict restriction and assumption that the whirl amplitude is relatively very small compared to the seal clearance (within 0.1 *Cr*; *Cr* denotes the seal clearance). While large amplitude vibration often occurs during the passage of the critical speed of actual turbomachinery, the linear bulk flow analysis may not be applicable for the accurate fluid force characteristics in such situations with large amplitude. To describe the fluid forces of annular seal induced by large disturbances, the nonlinear dynamic model should be established. San Andres and Jeung [17] presented an orbit analysis method based on extended Reynolds equation to investigate force coefficients valid over a wide frequency range of a squeeze film damper bearing with large amplitude and static eccentricity. Ikemoto et al. [6] investigated the nonlinear fluid forces for the concentric seal with large whirl amplitude up to about a half of the clearance by using extended perturbation analysis of the bulk flow theory. Currently, the Muszynska's model proposed by Bently and Muszynska [18] is commonly used by researchers as a nonlinear dynamic model. Li and Chen [19] adopted the Muszynska's seal force model with the empirical parameters to investigate the 1:2 subharmonic resonance of labyrinth seal–rotor system. These empirical parameters obtained by employing the CFD analysis are used in the subsequent nonlinear analysis, regardless of whether the whirl amplitude is around the concentric position or not. He and Jing [20] indicated that Muszynska's model will not describe the dynamic characteristics of the rotor–seal system well when the rotor–seal system has larger eccentricity ratio. However, the present paper is devoted to develop nonlinear dynamic models of concentric seal with large whirl amplitude or eccentric seal with large static eccentricity and rather small whirl amplitude. The applicability of linear assumption and reliability of nonlinear model for seals under large static eccentricities and disturbance amplitude is rarely discussed in the literature. Thus, an investigation on the applicability of nonlinear Muszynska's model under large eccentricities and disturbances is wished for, particularly in nonlinear rotor–seal system research considering radial loads.

In experimental studies of eccentric seals, Marquette, Childs and Andres [21] measured the force coefficients of a plain liquid annular seal under different static eccentricities, and the results show that the force coefficients were more sensitive to the changes of static eccentricity than theoretically predicted. Childs, Arthur and Mehta [22] measured the net reaction forces of gas annular seals as the eccentricity ratios increased; negative stiffness created by unanticipated eccentricities may lead to over prediction of critical speeds, which illustrates the importance of concentric assembly of annular seals.

In this paper, three-dimensional (3D) transient CFD simulations based on dynamic mesh method are performed to evaluate the static and dynamic characteristics of eccentric annular seals. The obtained force coefficients and leakage rates are compared with Marquette's experiment [21] for validating the reliability of the transient CFD method. The effects of rotor disturbance amplitude on the dynamic characteristics of eccentric annular seals are analyzed to investigate the linear ranges of seal dynamic characteristics. In addition, transient CFD simulations and a nonlinear dynamic model are adopted to study the fluid excitations of annular seals induced by different rotor large motions. The nonlinear dynamic model is based on the famous Muszynska's model [18,23,24] and is obtained by fitting the "nominal" force coefficients of concentric annular seal under different whirl amplitude, as shown in Figure 2. With nonlinear model and transient CFD simulations, fluid excitations under various large disturbances are computed. Based on these fluid excitations, seal dynamic characteristics under large eccentricities and disturbances are investigated in detail, which provides a solid basis for the research of seal–rotor system analysis by using Muszynska' model as nonlinear seal force.

### **2. Numerical Methods**
