# The Investigation of a New End Wall Contouring Method for Axial Compressors
## 2 New End Wall Contouring Method
### The Definition of the End Wall Contouring Units  
(1) The full-area unit  
The first type of end wall contouring unit is designed to accelerate or decelerate the cross-passage secondary flow over the end wall region. It is termed \"full-area unit\" for brevity. The definition is sketched in Figure 2. For the convenience of presentation, Figure 2a maps the \(z\)-\(\theta\) end wall surface of the blade channel to a dimensionless \(1\times 1\) standard space and sets \(\eta\), \(\bar{\zeta}\), and \(\varepsilon\) to represent the spanwise, circumferential, and streamwise directions of the compressor channel. \(\varepsilon=0\) and \(\varepsilon=1\) represent the leading edge (LE) and trailing edge (TE) lines; \(\bar{\zeta}=0\) and \(\bar{\zeta}=1\) represent the PS and SS. Then, the control equations of the full-area unit can be expressed as  
\[\mathrm{f}(\varepsilon,\bar{\zeta})=\mathrm{A}(\varepsilon)\mathrm{F}(\bar{ \zeta})=-\cos\left(\pi(\varepsilon-0.5)\right)^{3}\sin(\pi(\bar{\zeta}-0.5)) \tag{2}\]  
Note the streamwise function, i.e., \(\mathrm{A}(\varepsilon)=\cos(\pi(\varepsilon-0.5))^{3}\) acts as the amplitude of the transverse function \(\mathrm{F}(\bar{\zeta})=-\sin(\pi(\bar{\zeta}-0.5))\), and the maximum value of \(\mathrm{A}(\varepsilon)\) locates at  
Figure 2: The full-area unit. (**a**) Definition of the full-area unit; (**b**) The effect on the pressure field and the end wall secondary flow. \(\varepsilon=0.5\). To further smooth the surface, we selected five axial sections (transparent sections labeled from S1 to S5 in Figure 2) at the streamwise position of \(\varepsilon=0\), 0.3, 0.5, 0.7, and 1 in the full-area unit to construct the lofting surface. Considering that there are only five sections, the interpolation of the surface uses the cubic spline to ensure it accurately gets through all the sections, finally forming the geometry for the full-area unit. Consider that when the full-area unit is mapped to the actual end wall of the axial flow compressor, the shape of the surface causes the pressure side of the blade channel to sink and the suction side to rise, with the maximum slope at the 0.5 axial chord length (\(c_{a}\)) position. Figure 2b thus illustrates the possible variation of end wall flow according to the Bernoulli effect. In the circumferential direction, the full-area unit generates an upslope surface from the SS to the PS, just as in previous research [7,8]. So, the transverse pressure gradient (\(\
abla p_{\xi}\)) will decrease, and the cross-passage secondary flow will decelerate in the entire area. There are some side effects. On the suction side of the end wall, the variation of the streamwise pressure gradient (\(\
abla p_{\xi}\)) will stay positive from the LE to the mid-chord and then negative till the TE. Therefore, the streamwise component of end wall flow will decelerate in the front half and then accelerate in the rear half. On the pressure side of the end wall, all side effects are contrary to the suction side. It should also be noted that when applied to an actual compressor, all the above flow control, including its impact on the cross-passage secondary flow and the side effects, may turn in the opposite direction when the weight factor is less than zero. The second type of end wall contouring unit is defined to facilitate the secondary flow control in one or more particular local areas in the blade channel. It is termed the \"localized unit\" for brevity. The definition of this end wall contouring unit is shown in Figure 3a. Different from the full-area unit, the streamwise starting and ending positions \(\varepsilon_{1}\), \(\varepsilon_{2}\), and circumferential starting and ending positions \(\xi_{1}\), \(\xi_{2}\) in the localized unit are all free parameters. Varying these parameters will change the shape of the localized unit. Figure 3: The localized unit. (**a**) Definition of the localized unit; (**b**) the effect on the pressure field and the end wall secondary flow.