Investigation of the tip leakage flow at turbine rotor blades with squealer cavity

Abstract

Understanding of the tip leakage flow (TLF) in turbine rotors is one key aspect in the design for improving the efficiency of turbines. This requires measurements and simulations of the TLF, especially when investigating new rotor blade designs with blade tip treatments. However, flow measurements in the tip gap of a rotating machine are highly challenging because of the small gap size of about 1 mm and the high unsteadiness of the flow requiring a high temporal resolution of about 10 μs. For this purpose, an optimized non-intrusive measurement concept based on frequency modulated Doppler global velocimetry is presented, which fulfills the requirements. Three component velocity fields of the TLF were obtained in a turbine test rig at a blade passing frequency of 930 Hz. The rotor blades were equipped with a squealer tip, and the TLF in the squealer cavity region was successfully measured. The measurement agrees well with calculated results showing gradients in the tip gap above the squealer cavity. Furthermore, the development of the tip clearance vortex was resolved at the suction side of the blades.

Appendix: coordinate transformation

According to Fig. 6, the laser incidence direction reads

$$ {\mathbf{i}} = \left( {\begin{array}{*{20}c} 0 \hfill & 0\hfill & { - 1} \hfill \\ \end{array} } \right)^{\rm T}$$

(5)

and the three observation directions are

$$ {\mathbf{o}}_{1} = \left( {\begin{array}{*{20}c} 0 \hfill & 1\hfill & 0 \hfill \\ \end{array} } \right)^{\rm T} $$

(6a)

$$ {\bf o}_{2} = \left(-\sin(\beta) \cos(\beta) 0\right)^T $$

(6b)

$$ {\bf o}_{3} = \left(+\sin(\beta) \cos(\beta) 0\right)^T $$

(6c)

with β = 35°. As explained in Sect. 2, the three measured velocity components v ₁, v ₂, v ₃ are the components of the flow velocity v = (v _x , v _y , v _z )^T = (v _t , v _r , v _a )^T along the bisecting line of the laser incidence and the observation direction:

$$ v_1 = \frac{({\bf o}_1 - {\bf i})\cdot {\bf v}}{|{\bf o}_1 - {\bf i}|} $$

(7a)

$$ v_2 = \frac{({\bf o}_2 - {\bf i})\cdot {\bf v}}{|{\bf o}_2 - {\bf i}|} $$

(7b)

$$ v_3 = \frac{({\bf o}_3 - {\bf i})\cdot {\bf v}}{|{\bf o}_3 - {\bf i}|} $$

(7c)

where $|{\bf o}_1 - {\bf i}|=|{\bf o}_2 - {\bf i}|=|{\bf o}_3 - {\bf i}|=\sqrt{2}.$ Now writing the latter equation system as

$$ (v_1, v_2, v_3)^{\rm T} = M\cdot (v_t, v_r, v_a)^{\rm T} $$

(8)

with the transform matrix

$$ M=\frac{1}{\sqrt{2}}\cdot\left( \begin{array}{lll} 0 &1 & 1\\ -\sin(\beta) & \cos(\beta) & 1\\ \sin(\beta) & \cos(\beta) & 1\\ \end{array} \right), $$

(9)

multiplying the equation system with the inverse matrix

$$ M^{-1}=\frac{1}{\sqrt{2}\cdot (1-\cos(\beta))}\cdot\left( \begin{array}{lll} 0 &-\frac{1-\cos(\beta)}{\sin(\beta)} & \frac{1-\cos(\beta)}{\sin(\beta)} \\ 2& -1 & -1\\ -2\cos(\beta) & 1 & 1\\ \end{array} \right) $$

(10)

finally yields the desired orthogonal velocity components of the flow velocity vector:

$$ (v_t, v_r, v_a)^{\rm T} = M^{-1}\cdot (v_1, v_2, v_3)^{\rm T}. $$

(11)

When calculating the tangential velocity component v _t (first row of the equation system), the components v ₂ and v ₃ are subtracted: v _t ∼(−v ₂ + v ₃). Due to this, the laser incidence direction has no effect on v _t as can be shown by inserting Eq. (7):

$$ v_t \sim(-{\bf o}_2 + {\bf o}_3)\cdot {\bf v}. $$

(12)

Hence, laser light from any direction, which illuminates the seeding particles in the measurement volume, gives the correct result. This effect occurs for the tangential and the radial velocity component and applies especially for light reflected on resting surfaces.

Furthermore, small shifts of the center frequency of the incident laser light caused by reflections on moving surfaces before entering the measurement volume also have no effect. This is true, because e.g. for $v_t$ the difference of the velocity components v ₂ and v ₃ and, thus, only the difference of the two corresponding Doppler shifts is evaluated and not their absolute value. Since the velocities of the moving surfaces are significantly below light speed, the center wavelength of the reflected light is not changed significantly and the Doppler effect remains the same [see Eq. (2)].

Altogether, this explains the missing artifacts for v _t in Fig. 12a, which do occur for v _r and v _a in Figs. 13a and 14a, respectively. Vice versa, the disappearance of the artifacts proves the reason of this disturbance: reflected light (either reflected by moving or resting surfaces) enters the measurement volume from a so far unknown direction.