Generalized Reynolds equation for fluid film problems with arbitrary boundary conditions: application to double-sided spiral groove thrust bearings

Abstract

Reliable high-performance bearings are essential in rotordynamics. As speeds and loads are continuously increased, the demands on bearing technology grow simultaneously. Detailed calculations of bearing characteristics become more important under these circumstances. In this work, a generalized Reynolds equation for analysing thin film lubrication problems is presented. The equation describes the flow of compressible fluids in arbitrary narrow gaps with general boundary conditions. As an application for this equation, we consider the double-sided spiral groove thrust bearing. A CFD calculation is performed to validate the generalized Reynolds equation derived here. On the basis of a particle swarm optimization method, optimal geometrical bearing parameters are identified. Performance charts for optimized bearing configurations with respect to load carrying capacity and friction coefficient are delivered.

Appendix A : details of the derivation of the two-sided generalized Reynolds equation

In Sect. 2.1, the derivation of the two-sided generalized Reynolds equation has been outlined. Here, the relevant steps to derive Eq. (11) starting from Eq. (9) are presented in detail. Applying to the second and third term of Eq. (9) in the general relation

$$\begin{aligned} \int \limits _{h_{1}}^{h_2} \frac{\partial {}}{\partial {x}} f(x,y,z)\mathrm{{d}}z\,=\, \frac{\partial {}}{\partial {x}} \int \limits _{h_{1}}^{h_2} f(x,y,z) \mathrm{{d}}z- f(x,y,h_2) \frac{\partial {h_2}}{\partial {x}}\,+\, f(x,y,h_1) \frac{\partial {h_1}}{\partial {x}} , \end{aligned}$$

(30)

we get

$$\begin{aligned} \begin{aligned} \int \limits _{h_{1}}^{h_2} \frac{\partial {\rho }}{\partial {t}}\mathrm{{d}}z\,+\,&\underbrace{ \left( \frac{\partial {}}{\partial {x}} \int \limits _{h_{1}}^{h_2} \rho u \mathrm{{d}}z - (\rho u)_2 \frac{\partial {h_2}}{\partial {x}} \,+\, (\rho u)_1 \frac{\partial {h_1}}{\partial {x}} \right) }_{I}+ \\&\underbrace{ \left( \frac{\partial {}}{\partial {y}} \int \limits _{h_{1}}^{h_2} \rho v \mathrm{{d}}z - (\rho v)_2 \frac{\partial {h_2}}{\partial {y}} \,+\, (\rho v)_1 \frac{\partial {h_1}}{\partial {y}} \right) }_{II}\,+\, \int \limits _{h_{1}}^{h_2} \frac{\partial {(\rho w)}}{\partial {z}}\mathrm{{d}}z=0 . \end{aligned} \end{aligned}$$

(31)

Splitting into two terms, expression I can be rewritten as

$$\begin{aligned} I= \underbrace{ \left( \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_2} \rho u \mathrm{{d}}z - (\rho u)_2 \frac{\partial {h_2}}{\partial {x}} \right) }_{I_A} - \underbrace{ \left( \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_1} \rho u \mathrm{{d}}z - (\rho u)_1 \frac{\partial {h_1}}{\partial {x}} \right) }_{I_B} . \end{aligned}$$

(32)

Applying the product rule to the second term of $I_A$ results in

$$\begin{aligned} I_A= \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_2} \rho u \mathrm{{d}}z - \frac{\partial {}}{\partial {x}} \bigg [(\rho u)_2 h_2 \bigg ] + \left[ \frac{\partial {}}{\partial {x}}(\rho u)_2 \right] h_2 . \end{aligned}$$

(33)

Next, we are going to rewrite the first and second term of Eq. (33) by considering the following identity

$$\begin{aligned}{}[\rho z u]_0^{h_2} = \int \limits _{0}^{h_2} \frac{\partial {\rho }}{\partial {z}} zu \mathrm{{d}}z \,+\, \int \limits _{0}^{h_2} \rho \frac{\partial {z}}{\partial {z}} u \mathrm{{d}}z \,+\, \int \limits _{0}^{h_2} \rho z \frac{\partial {u}}{\partial {z}} \mathrm{{d}}z . \end{aligned}$$

(34)

Using $[\rho z u]_0^{h_2} = (\rho u)_2 \cdot h_2$, partial differentiation of Eq. (34) with respect to x and reordering of terms gives

$$\begin{aligned} \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_2} \rho u \mathrm{{d}}z - \left[ \frac{\partial {}}{\partial {x}}(\rho u)_2 h_2 \right] = - \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_2} \frac{\partial {\rho }}{\partial {z}} zu \mathrm{{d}}z - \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_2} \rho z \frac{\partial {u}}{\partial {z}} \mathrm{{d}}z . \end{aligned}$$

(35)

If we insert this result in Eq. (33) and if we perform the manipulations (33)–(35) analogously for $I_B$, expression I becomes

$$\begin{aligned} I= h_2 \left[ \frac{\partial {}}{\partial {x}}(\rho u)_2 \right] - \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_2} \left( \rho z \frac{\partial {u}}{\partial {z}} \,+\, zu \frac{\partial {\rho }}{\partial {z}}\right) \mathrm{{d}}z - h_1 \left[ \frac{\partial {}}{\partial {x}}(\rho u)_1 \right] \,+\, \frac{\partial {}}{\partial {x}} \int \limits _{0}^{h_1} \left( \rho z \frac{\partial {u}}{\partial {z}} \,+\, zu \frac{\partial {\rho }}{\partial {z}}\right) \mathrm{{d}}z . \end{aligned}$$

(36)

Making use of similar manipulations for expression II in Eq. (31), we arrive at the final form of the integrated continuity equation (see Eq. (10) in Sect. 2.1)

$$\begin{aligned} \begin{aligned} \int \limits _{h_{1}}^{h_2} \frac{\partial {\rho }}{\partial {t}}\mathrm{{d}}z&+ \underbrace{ \left\{ h_2 \left[ \frac{\partial {}}{\partial {x}}(\rho u)_2 + \frac{\partial {}}{\partial {y}}(\rho v)_2 \right] - h_1 \left[ \frac{\partial {}}{\partial {x}}(\rho u)_1 \,+\, \frac{\partial {}}{ \partial {y}}(\rho v)_1 \right] \right\} }_{I_a} \\&-\frac{\partial {}}{\partial {x}} \underbrace{ \int \limits _{h_{1}}^{h_2} \left( \rho z \frac{\partial {u}}{\partial {z}} \,+\, zu \frac{\partial {\rho }}{\partial {z}}\right) \mathrm{{d}}z }_{II_a} -\frac{\partial {}}{\partial {y}} \underbrace{ \int \limits _{h_{1}}^{h_2} \left( \rho z \frac{\partial {v}}{\partial {z}} \,+\, zv \frac{\partial {\rho }}{\partial {z}}\right) \mathrm{{d}}z }_{III_a} + \int \limits _{h_{1}}^{h_2} \frac{\partial {(\rho w)}}{\partial {z}}\mathrm{{d}}z =0 . \end{aligned} \end{aligned}$$

(37)

whereas $I_a$ contains only expressions that are evaluated at the boundaries of the fluid film, terms $II_a$ and $III_a$ need further manipulations. Inserting the fluid velocity u(z) from (6) into $II_a$, we get

$$\begin{aligned} \begin{aligned} II_a&= \underbrace{ \int \limits _{h_{1}}^{h_2} \rho z \frac{\partial {}}{\partial {z}} \left[ \frac{\partial {p}}{\partial {x}} \int \limits _{h_{1}}^{z} \frac{z}{\eta } \mathrm{{d}}z \right] \mathrm{{d}}z}_{I_b} \,+\, \underbrace{ \int \limits _{h_{1}}^{h_2} \rho z \frac{\partial {}}{\partial {z}} \left[ \frac{U_2-U_1}{F_0} \int \limits _{h_{1}}^{z} \frac{1}{\eta } \mathrm{{d}}z \right] \mathrm{{d}}z}_{II_b} \,+\, \underbrace{ \int \limits _{h_{1}}^{h_2} \rho z \frac{\partial {}}{\partial {z}} \left[ -\frac{\partial {p}}{\partial {x}} \frac{F_1}{F_0} \int \limits _{h_{1}}^{z} \frac{1}{\eta } \mathrm{{d}}z \right] \mathrm{{d}}z }_{III_b} \\&+ \underbrace{ \int \limits _{h_{1}}^{h_2} z \frac{\partial {\rho }}{\partial {z}} U_1 \mathrm{{d}}z}_{IV_b} \,+\, \underbrace{ \int \limits _{h_{1}}^{h_2} z \frac{\partial {\rho }}{\partial {z}} \left[ \frac{\partial {p}}{\partial {x}} \int \limits _{h_{1}}^{z} \frac{z}{\eta } \mathrm{{d}}z \right] \mathrm{{d}}z}_{V_b} \,+\, \underbrace{ \int \limits _{h_{1}}^{h_2} z \frac{\partial {\rho }}{\partial {z}} \left[ \frac{U_2-U_1}{F_0} \int \limits _{h_{1}}^{z} \frac{1}{\eta } \mathrm{{d}}z \right] \mathrm{{d}}z}_{VI_b} \\&+ \underbrace{ \int \limits _{h_{1}}^{h_2} z \frac{\partial {\rho }}{\partial {z}} \left[ -\frac{\partial {p}}{\partial {x}} \frac{F_1}{F_0} \int \limits _{h_{1}}^{z} \frac{1}{\eta } \mathrm{{d}}z \right] \mathrm{{d}}z }_{VII_b} . \end{aligned} \end{aligned}$$

(38)

Grouping terms and using the definitions for $F_i,G_i$ from Eq. (12) results in

$$\begin{aligned} \begin{aligned} I_b+III_b&= \frac{\partial {p}}{\partial {x}} \int \limits _{h_{1}}^{h_2} \rho z \underbrace{\left[ \frac{\partial {}}{\partial {z}} \int \limits _{h_{1}}^{z} \frac{z}{\eta } \mathrm{{d}}z \right] }_{\frac{z}{\eta }} \mathrm{{d}}z \,-\, \frac{\partial {p}}{\partial {x}} \int \limits _{h_{1}}^{h_2} \rho z \frac{F_1}{F_0} \underbrace{ \left[ \frac{\partial {}}{\partial {z}} \int \limits _{h_{1}}^{z} \frac{1}{\eta } \mathrm{{d}}z \right] }_{\frac{1}{\eta }} \mathrm{{d}}z = F_2 \frac{\partial {p}}{\partial {x}} , \\ II_b+VI_b&\,=\, \frac{U_2-U_1}{F_0}(F_3+G_2) , \\ IV_b&\,=\, G_3 U_1 , \\ V_b+VII_b&\,=\, G_1 \frac{\partial {p}}{\partial {x}} , \end{aligned} \end{aligned}$$

(39)

Inserting (39) into the integrated continuity Eq. (37) and performing analogous manipulations for expression $III_a$, the double-sided generalized Reynolds Eq. (11) is obtained.