Yumeng Tang

Yangwei Liu
email: liuyangwei@126.com

Lipeng Lu

Huawei Lu

Ming Wang

Experimental Investigation of Flow Control Using Blade End Slots in a Highly Loaded Compressor Cascade

Keywords:

A detailed experimental investigation is conducted to suppress three-dimensional (3D) corner separation by a proposed passive control method using blade end slots in a highly loaded high-speed compressor cascade. Experiments are carried out under a wide range of incidence angles at Ma=0.59 using blades with and without blade end slots, respectively. Based on the experimental results, extensive comparisons show that the proposed method using blade end slots can efficiently suppress the 3D corner separation and broaden the effective operating range in the highly loaded high-speed compressor cascade. The total pressure loss is significantly reduced under most conditions. The reduction of total pressure loss in the measurement plane is as high as 18.4%, 20.6%, 24.3% and 39.4% at the incidence angle of -1.69˚, 0˚, 2˚ and 4˚, respectively. Furthermore, spanwise distributions of the pitch-averaged total pressure loss and deviation angle as well as the 3D flow field structures are analyzed to reveal the flow control mechanisms using blade end slots. The blade end slots can generate self-adaptive high momentum jet flow through the pressure difference from blade pressure and suction surface. These jet flows from the blade end slots effect downstream along the blade suction surface and significantly increase the flow momentum in the corner region. The main secondary vortex structures are suppressed by the high momentum jet flow; the 3D corner separation is reduced, and the two-dimesionality in the mid-span region is enhanced.

INTRODUCTION

Three-dimensional (3D) corner separation is identified as an inherent flow feature in the blade suction surface and endwall conjunction of the axial compressors [START_REF] Lei | A criterion for axial compressor hub-corner stall[END_REF]. As Denton established, the secondary flows in the blade-endwall junction causes about one-third of losses in axial compressor [START_REF] Denton | Loss mechanisms in turbomachines[END_REF]. The pressure gradient transverse to the flow direction affects both separation and loss in the compressor flow [START_REF] Taylor | Competing three dimensional mechanisms in compressor flows[END_REF]. As the development of modern jet engines, the application of highly loaded blade is desired for the purpose of reducing weight as well as lowering manufacturing and maintenance costs. However, increasing blade loading inherently aggravates the corner separation and induces higher losses. Thus, various control methods [START_REF] Gbadebo | Control of three-dimensional separation in axial compressor by Tailored boundary layer suction[END_REF][START_REF] Liu | Corner Separation Control by Boundary Layer Suction Applied to a Highly Loaded Axial Compressor Cascade[END_REF][START_REF] Li | Control of the corner separation in a compressor cascade by steady and unsteady plasma aerodynamic actuation[END_REF][START_REF] Akcayoz | Controlling corner stall separation with plasma actuators in a compressor cascade[END_REF][START_REF] Staats | Active control of the corner separation on a highly loaded compressor cascade with peridoc nonsteady boundary conditions by means of fluidic actuators[END_REF][START_REF] De Giorgi | Comparison between synthetic jets and continuous jets for active flow control: application on a NACA 0015 and a compressor stator cascade[END_REF][START_REF] Evans | Flow control in a compressor cascade at high incidence[END_REF][START_REF] Wennerstrom | Some experiments with a supersonic axial compressor stage[END_REF][START_REF] Ramzi | Numerical study of passive control with slotted blading in highly loaded compressor cascade at low Mach number[END_REF][START_REF] Hage | Control of secondary flow in a highly loaded compressor stage by means of a groove structure on the sidewalls[END_REF][START_REF] Hergt | Effects of vortex generator application on the performance of a compressor cascade[END_REF][15][START_REF] Liu | Effect of slot at blade root on compressor cascade performance under different aerodynamic parameters[END_REF] have been proposed to suppress corner separation in axial compressors, especially the highly loaded ones.

Previous studies demonstrate that the 3D corner separation in a highly loaded compressor could be suppressed through active or passive flow control methods. Some active flow control methods, such as boundary layer aspiration [START_REF] Gbadebo | Control of three-dimensional separation in axial compressor by Tailored boundary layer suction[END_REF][START_REF] Liu | Corner Separation Control by Boundary Layer Suction Applied to a Highly Loaded Axial Compressor Cascade[END_REF], plasma [START_REF] Li | Control of the corner separation in a compressor cascade by steady and unsteady plasma aerodynamic actuation[END_REF][START_REF] Akcayoz | Controlling corner stall separation with plasma actuators in a compressor cascade[END_REF] or fluidic actuators [START_REF] Staats | Active control of the corner separation on a highly loaded compressor cascade with peridoc nonsteady boundary conditions by means of fluidic actuators[END_REF], synthetic and continuous jets [START_REF] De Giorgi | Comparison between synthetic jets and continuous jets for active flow control: application on a NACA 0015 and a compressor stator cascade[END_REF], vortex generator jets [START_REF] Evans | Flow control in a compressor cascade at high incidence[END_REF], can reduce corner separation loss and improve compressor performance; however, these methods require additional devices. Compared to active flow control methods, passive flow control approach earns for its simplicity and cost effectiveness. Some passive flow control methods, such as blade center slots [START_REF] Wennerstrom | Some experiments with a supersonic axial compressor stage[END_REF] or whole span slot [START_REF] Ramzi | Numerical study of passive control with slotted blading in highly loaded compressor cascade at low Mach number[END_REF], sidewall grooves [START_REF] Hage | Control of secondary flow in a highly loaded compressor stage by means of a groove structure on the sidewalls[END_REF], and vortex generator [START_REF] Hergt | Effects of vortex generator application on the performance of a compressor cascade[END_REF] have been used and also shown some benefits. Tangential blowing over the profile suction surface is supplied by blade center slots to increase boundary layer momentum and therefore delaying or preventing separation, while the two-dimensional separation at the mid-span region is enlarged, resulting in the massive deflection at mid-span [START_REF] Wennerstrom | Some experiments with a supersonic axial compressor stage[END_REF]. The vortex generator devices produce strong vortices that enhance the mixing between the main flow and the decelerated boundary layer, reducing the corner separation and total pressure loss therefore [START_REF] Hergt | Effects of vortex generator application on the performance of a compressor cascade[END_REF]. Generally speaking, current passive flow control methods have some limitations, especially do not universally suitable for all the operating conditions.

Nerger et al experimentally studied an active flow control method and concluded that steady tangential blowing over the profile suction surface can increase boundary layer momentum and therefore delaying or preventing 3D corner separation [15], while extra blowing jet is demanded and the decrease of the total pressure loss is possible for a few operating points only. Accounting for the shortage of active flow control blowing and to utmost its advantage, Liu et al. recently proposed an adaptive passive flow control method through slot established at the end of the blade [START_REF] Liu | Effect of slot at blade root on compressor cascade performance under different aerodynamic parameters[END_REF].

Numerical investigations of the blade end slots on a lowspeed prescribed velocity distribution (PVD) cascade reveals a progressive perspective in improving compressor performance. The present paper experimentally investigates the effects of blade end slots on the 3D corner separation in a highly loaded high-speed compressor cascade under a wide range of incidence angles at Ma=0.59. Based on the experimental results, extensive comparisons are conducted to assess the effectiveness. The spanwise distributions of pitch-averaged total pressure loss and deviation angle as well as the 3D flow field structures are then analyzed to reveal the flow control mechanisms using blade end slots.

CONFIGURATIONS

The target blade investigated is a state-of-art highly loaded high turning one, which is designed for a typical subsonic Mach number that encountered in modern multistage axial compressor. Pre numerical investigations indicate that the datum blade tends to fail as a consequence of 3D corner stall on the section side under high positive incidence angles, accompanied with significant increase of underturning. While the proper designed end slotted blade provides a more uniformed outflow with less deviation angle due to the suppression of 3D corner separation, and the loss is also reduced. The high adverse pressure gradient in the cascade passage as well as the interaction between the thick boundary layer from the endwall and blade surface are the main causes for the corner separation. The geometry of the end slots section is optimized numerically before experiments under inconsistent incoming conditions. As Ashley et al. established, the endwall boundary layer state has a significant influence on the 3D corner separation [START_REF] Scillitoe | Numerical investigation of three-dimensional separation in an axial flow compressor: the influence of freestream turbulence intensity and endwall boundary layer state[END_REF], though differences on the endwall boundary layer thickness makes the designed end slots geometry is not the optimist one under the current experimental conditions, massive benefits in performance still reward at last.

Datum Blade Geometry

The datum configuration is a highly loaded high-turning compressor cascade, whose profile is cut off at the middle span of a modern compressor stator. Table 1 presents the general geometry parameters of the datum blade together with its aerodynamic design conditions. The experimental blade has an aspect ratio of 1.82 due to the height of the tunnel cross section. The turning-angle of the blade is as large as to approximately 53 degree with the Dfactor approaching 0.55. The designed Mach number is equal to 0.59, which represents a typical subsonic Mach number encountered in modern multi-stage axial compressor, so the transferability to the real turbomachines is possible.

End-Slot Blade Configurations

The end slotted blade shares the same general geometry parameters with the datum one and the experimental conditions also remains.

The 3D model of the end slotted blade is exhibited in Figure 1. In correlation to which, Figure 2 demonstrates the cut-off geometry of the datum section (in black) and the end slot section (in red). The profile of the end slots is successively derived from the datum configuration, both of them can connect seamless. Considering that the Coanda effect occurs when a free stream flow above a curved surface is entrained by a parallel high momentum wall jet tangential along the curved surface, a Coanda radius corresponding to slot throat width is integrated tangentially with the downstream profile of the suction surface. As it proved previously that the Coanda jet remains attached to the curved surface as the result of the balance between the centrifugal force around the curved surface and the sub-ambient pressure in the jet sheet [START_REF] Jones | An active flow circulation controlled flap concept for general aviation aircraft applications[END_REF]. The momentum of the jet out of the blade end slots energizes the downstream suction surface boundary layer and gains its ability to resist separation caused by the adverse pressure gradient along the curved surface. Goodhand et al. emphasized the sensitivity of geometry deviation on the loss production [START_REF] Goodhand | The impact of real geometries on three-dimensional separations in compressors[END_REF]. So all the curves are smoothly blending and the leadingedge (LE) is well-treated. The total chord length in the end slotted section remains the same with the datum one to minimize the variation in geometry. Each trailing-edge (TE) radius per slot-including section is limited to 0.1 mm to restrain aggressive wake mixing losses. Both end slots originate from the endwall and occupy 20% of the whole span height, effecting the blade-endwall conjunction only.

The end slots section is coupled with the middle datum one and is divided into three individual elements by two curve shaped slots. Those three elements are named sec1, sec2, sec3 respectively from LE to TE. The main geometric designing parameters of the end slots section are summarized in Table 2. The LE and TE positions are necessary parameters to depict slot, and the slot shapes are characterized by the axial overlap (AO) together with the throat width (t). The LE of section 1 (sec1) is contacted with the LE of the datum blade, and the TE of it is located in 25%Cx (Cx stands for axial chord length, the position is related to datum LE). The LE and TE for sec2 is seated 17.6%Cx and 50%Cx respectively, and the LE for sec3 lies in 40%Cx position. The TE of sec3 is coupled with the datum blade. AO is the axial distance between the rear airfoil LE and the indicated TE. The throat width t in the slot outlet and the slot height h are vital to the determination of adaptive mass flow rate. Small ratios of the slot throat width and radius t/R which promotes the Coanda effect is chosen to delay flow detachment and to increase total turning.

The effective entrainment effect is highly depended on the slot out position, alternatively, determined by the local pressure of the slot out on the suction side of the blade. The integral total pressure in the slot inlet region can be approximated by the incoming flow total pressure integrated from the identical height of the slot mostly, if non local separation occurs. The throat velocity in the slot outlet would be estimated through local pressure in a certain slot geometry. Defining the slot inlet total pressure as 1 t P and the pressure neighboring the slot throat as 2 P , the local tangential out flow velocity t V can be estimated as

* 2 1 1 b C ta t P VC R T P              ( 1 )
where * T is the total temperature, R is the ideal gas constant, a C and b C are constant that assigns 7 and 2/7 respectively.  is the empirical determined total pressure recovery coefficient of the slot, assuming 0.98 if no separation occurs internal the slot. The mass flow rate of the slot jet could be preliminary evaluated as

  1 22 * 1 g c t slot d f P mK t h C C T     ( 2 ) 2 * t af V CC R T   ( 3 )
where

1 2 -1 0.0404 K sm K  , d C , f
C , g C are constant assigned with 1.58, 1/6, 2.5. t and h stands for the width and height of the slot throat respectively.

The jet flow velocity and the mass flow rate of the blade end slots are adaptively based on the local flow parameters, since the flow field is determinate.

EXPERIMENTAL SETUP

The main phenomenon this study concerned is the 3D corner separation influenced by blade end slots under different incoming flow incidence conditions.

Experimental Facility

All the experiments are carried out in a high-speed linear compressor cascade tunnel. Consecutive air is supplied to the test section after cooling, which is driven by a high-pressure radial compressor. The supplied incoming flow goes through several inflowing values, a silencer, a diffusor, a settling chamber and a nozzle by step. Mach number of approximately 0.59 and Reynolds number of up to 0.72×10 6 c o u l d b e obtained.

The test section has a cross section of 100mm in height and an adjustable range more than 260mm in width. Eight arraying blades are adopted to guarantee periodic conditions in the pitchwise, and the fifth blade from the top was chosen as the testing one. The arrangement of the test rig is shown in Figure 3. The blade carriers are mounted on a ratable disk, which enables a continuous change of the incoming angle. Depending on the incoming flow angle, the vertical height of the test section would be adjusted by moving the inlet floor top and bottom. A three-hole probe with a diameter of 1.5mm is emplaced upstream to monitor the incoming flow Mach number and the real flow angle. Variation in the incidence is adjusted according to the monitoring value of the three-hole probe to guarantee the incoming flow angle actual. Adjustment for the incoming flow Mach number is achievable by the valve opening accommodations. Non aspiration devices is equipped in the test rig that the developed incoming flow boundary layer migrates directly to the blade passage along the side wall.

Experimental Measurement

Measurements are made by a calibrated five-hole probe in the plane 60% chord length behind of the TE under different incoming flow incidence angles. An extensive meshy measurements are made through the transverse of the probe along pitchwise and spanwise. More than one blade passage range, 110% pitch length, is covered. Meticulous measurement is conducted in the wake and separation regions to better capture the loss. The diameter of the probe is 2mm, occupying the proportion less than 0.05% of the full outflow area.

In the measurements, the detected five-hole probe is fixed to the movement driving by a stepping motor. A digital pressure-scanivalve system DSA3217 is adopted for pressure collection, and the detected pressure signal is automatically records by PC with special software and the probe movement is commanded by a given procedure through controlling the traverse mechanism. Those acquisitions are the valves against the atmospheric one, namely the gage pressure. For each collection, the corresponding atmosphere pressure and temperature is also recorded. The scan frequency of the scanivalve system is 5 frames per second, [START_REF] Liu | Investigation of Vortical Structures and Turbulence Characteristics in Corner Separation in a Linear Compressor Cascade Using DDES[END_REF] transducer error and the measurement error. The accuracy of the five-hole probe is within 0.1%, the pressure transducer error is about 0.05%, and the measurement error is regarded as 0.05%. Error under 0.2% exists when recovering the local total pressure and static pressure values through interpolating from the five-hole probe detected ones. Thus, the total error for the data acquisition is no more than 3% at last. The flow angle capture can be thought effective within the range of ±30 degrees.

To quantify losses, the total pressure loss coefficient is defined as Only pressure and temperature are detected in the experiments. The uncertainty for the three-hole probe or the five-hole probe highly depends on the relative probe inflow angle. Considerable precision can be assured within the inflow angle divergence of 30 degrees, thus the uncertainty is negligible under smaller angles. Since the inflow angle divergence is almost within the range of 30 degrees, the orientation error is not considered.

    1

RESULTS AND DISCUSSION

It is already described former that the end slotted blade is utilized to suppress the extent of 3D corner separation, so comparisons are made for both datum and end slotted blade under the corresponded varying incoming flow conditions. Comparisons of the typical experimental results are analyzed.

Performance over the Range of Operation

To evaluate the blade end slots effects over the operating range, performances in both design and off-design conditions are compared.

The mass-weighted averaged total pressure loss coefficient for both datum blade and the end slotted blade varying with the incoming flow angle are depicted in Figure 4.

Obvious loss reduction by the blade end slots can be seen in comparison with the datum one. The critical incidence where the loss steeply increased can be thought as the surge or choke margin of the blade, and the incidences within the margin can be regarded as effective operating range. For the datum blade, the effective operating range is narrow, mainly occupies negative incidences from -8˚ to -4˚, under which incidences the aerodynamic performance is not superior. Deterioration in the loss appears above the incidence -1.69˚ or below -8˚. The total pressure loss grows slightly when the incidence varies from -1.69˚ to 2˚, inasmuch as the transformation of 3D corner separation to mid-span separation. Relatively high benefit rewards under positive incidence conditions, where the corner stall dominates in the loss generation for the datum blade. Yielding a maximal total pressure loss of 39.4% related to the datum blade can be reduce by end slotted blade, due to the mitigation of 3D corner separation.

Spanwise Distribution of Pitch-Averaged Flow Parameters

To analyze quantitatively, comparisons are made through the spanwise distribution of pitchwise averaged total pressure loss coefficient and deviation angle.

Total Pressure Loss

The spanwise distribution of the total pressure loss coefficient for both datum blade and end slotted one are illustrated in Figure 6. Peak values of the total pressure loss seat where the corner separation flow dominates. Two relatively large total pressure loss peaks are found for both datum blade and end slotted one in accordance to the distribution of low momentum fluid in the total pressure loss contours displayed farther. More than two thirds of the passage is blocked by the low momentum flow structures in the datum blade passage, with the high loss peaks situated approximating 8% and 20% span height, while the peak location reduced to about 4% and 15% span for the end slots blade together with the proportion of high loss region significantly reduced. Loss for the end slotted blade, interval the 10% to 25% span range, is high, due to the outflow slot height which occupied 20% of the blade span, proving that this slot design scheme is not the optimist under this experimental incoming boundary layer thickness.

Loss in the mid-span region reflects the wake mixing. Boundary layers from the blade suction surface and pressure surface gather in the TE and create the wake vortex structure together. The wake vortex moves downward and dissipates to internal energy through interactions under the viscosity effect. The total pressure and velocity that the wake owes is quite small, leading to a strong mixing in the wake main-flow conjunction, together with high dissipations.

It can be clearly seen from Figure 6 that the wake loss of the end slotted blade changes little with the incidence and keeps far less than that of the datum one. As mentioned in the following chapter that the blade end slots enhances the twodimensionality in the mid-span region of the blade.

Deviation Angle

The deviation angle distribution along the spanwise demonstrates the flow deflection influenced by the 3D corner separation. Figure 7 shows the comparison of pitchwise averaged deviation angle at the measurement plane along the blade height, both datum and end slotted results are presented. Mass weighted average in the pitchwise is used in the processes. The negative valve mirrors the over turning and the positive value corresponds under turning. As it described in the classical secondary flow theory that the over turning happens close to the endwall while the deficit in turning occurs away from the endwall due to the blockage caused by corner separation. Seen from Figure 7 that the deviation angle distribution is significantly reduced and uniformed by blade end slots above 10% span height. Though enlargement in the deviation angle exists near the endwall, resulting from the end slots jets, the increased degree is not obvious. As it mentioned previously that the designed jet flow from the slot outs has the Coanda effect that keeps attached on the blade suction surface. These distribution of the deviation angle near the endwall reveals that small separation is induced additionally by the end slot jet. The blockage caused by the low momentum fluid at the corner increases the flow turning ranging from 10% to 25% span height for the end slotted blade and 10% to 35% span for the datum blade. The adaptive jet flow from the blade end slots brings the blocked low momentum fluid downstream, restraining its migration along the spanwise, thus the flux of the mid-span region is substantially improved. Conclusions can be drawn from Figure 7 that the blockage is significantly mitigated by the blade end slots, as a consequence, the main flow region with better properties enlarges and the whole capacity is improved.

Total Pressure Loss Contours

Variation in the incoming flow angle changes the flow For the datum blade under the incidence 0˚, a blockage as bulk as over two thirds in the spanwise of the passage is revealed, shown in Figure 8 (a), while the wake remains narrow. Figure 8 (b) illustrates that the blockage is reduced to lower than 25% span height through the employment of end slots, and the high loss core moves against the blade suction side. As it shown in Figure 10 (a) that the whole passage stall occurs for the datum blade under the incidence angle of 4˚ with the total pressure loss enlarges aggressively. The adoption of the blade end slots can put forward the appearance of whole passage stall.

As contoured from Figure 8 to 10, the corner separation is effectually restrained by blade end slots. The adaptive outflow from the blade end slots increase the flow momentum in the corner region, and pushes the accumulated low momentum flow downstream, restrains its upward migration along the blade suction surface, contributing to the reduction of blockage at last. Besides, it is noted from the comparison under the incidence of 2˚ and 4˚ (Figure 9 and Figure 10) that the blade end slots is conducive to enhancing the two-dimensionality in the mid-span region of the blade.

In order to gain more understanding of the insight mechanism by which the blade end slots restrain corner separation, the main secondary vortex structures featuring the corner separation are analyzed based on the measurement plane loss distributions. Through the corner separation mechanism analysis from Liu at al [START_REF] Liu | Investigation of Vortical Structures and Turbulence Characteristics in Corner Separation in a Linear Compressor Cascade Using DDES[END_REF] based on the delayed detached eddy simulation (DDES), it is thought that the passage vortex is the main secondary vortex and obviously enhances the corner separation. As Kang established in Figure 12 [21] that the passage vortex (PV), concentrated shedding vortex (SCV) and corner vortex (CV) are the main vortex structures that featuring the corner separation. The intensity of LE horseshoe vortex and the pressure gradient of the endwall cross flow contribute to the generation and evolution of passage vortex in the blade passage. With the incidence angle increasing, the vortex core of the passage vortex lifts off from the endwall. Reflecting on the development of the high loss core regions, shown from Figure 8 to Figure 10. The adaptive jet flow from the blade end slots energizes momentum to the local low momentum fluid and pushes the high loss core away from the blade suction surface and prevents it from lifting upwards. So the separation structures of the end slotted blade is pushed downstream and restrained in the blade root region, expressing as the expansion in the proportion of the main flow region. Since the position of the high loss core is so adjacent to the endwall in such incoming flow conditions that the designed end slot height is relatively higher, indicating that the reorganization of the blade-endwall corner flow is not the best. Thus, it is of great meaning to further optimize the end slots scheme.

Another high loss region above to the largest loss region results from the SCV in accordance to Figure11, is caused by the involving of the suction side flow over the accumulated low momentum corner with the attached flow from the pressure surface. The high momentum end slot jets energize the local low momentum flow and effectively restrain its accumulation at the blade-endwall corner, so that the partial deflection is impaired, reflecting in the suppression of the high loss region both in the strength and its upward migration at last. In a word, considerable decreases in the corner separation extent and relatively narrower wake rewards as a consequence.

Conclusions

The blade end slots effect on a highly loaded highturning compressor cascade under a typical subsonic Mach number conditions is investigated experimentally in the present paper. Through the detailed comparison of the typical results, conclusions can be drawn below:

• The employment of the blade end slot makes significant contribution to the reduction of corner separation. Prominent potential is shown in broadening the effective operating range positively forward, equipped the blade with better aerodynamic performance. • The blade end slot is conducive to enhance the twodimesionality in the mid-span region of the blade, resulting in the enlargement of the main flow region in the middle with the whole capacity aggressively improved. • Reduced deviation angle and uniformed deviation angle distribution along the spanwise downstream the blade is achieved by the blade end slots. • The adaptive jet flow from the blade end slots adds momentum to the local low momentum fluid and pushes the high loss core away from the blade suction surface , prevents it from lifting upwards, expressing as the expansion in the proportion of the main flow region. • The high momentum end slot jets energize the local low momentum flow, effectively restrains its accumulation at the blade-endwall corner and its spanwise migration along the suction surface, reflecting in the suppression of high loss region and its upward migration at last. Considerable benefits in the suppression of corner separation by blade end slots are validated experimentally with detailed analysis of its control mechanism. Further investigations could be conducted on the end slots scheme optimization.

Figure 1 .

 1 Figure 1. 3D Model of the End slotted blade

Figure 2 .

 2 Figure 2. Cross Section Configurations of the Datum and End Slotted Blade Geometry

Figure 3 .

 3 Figure 3. Sketch of the Test Section The experimental blades are fixed to the endwall without gap and airproofed by melt adhesive outside the endwall. The test rig of the end slotted blade is arranged the same. The end slots is symmetric on both sides of the blade with corresponded endwall filled up. Only half of the passage in spanwise was covered in the measurements due to the completely symmetric flow conditions.

  frames sampled each point and the averaged values are outputted finally. T e s t e r r o r c o m e s f r o m t h e p r o b e c a l i b r a t i o n e r r o r ,

1 tp

 1 stands for the incoming flow total pressure, 1 p is the incoming flow static pressure. The total pressure of the measured point   , xy is denoted as   , t p x y . The total pressure and the static pressure that detected by the threehole probe one at chord length ahead of the LE are used to represent incoming flow features. The total pressure loss coefficient is calculated for every measurement. Besides, the distribution of velocity and flow angle is also obtained through the measurements of the five-hole probe.Mass flow average is used when evaluating the whole losses or the pitchwise averaged losses. The density is solved under the assumption of constant total temperature. The averaged total pressure loss coefficient is obtained by

Figure 4 .Figure 5 .

 45 Figure 4. Total Pressure Loss over the Range of Operation The employment of the end slotted blade shows prominent potential in extending the effective operating range positively forward. Considerable low total pressure loss could be achieved by end slotted blade under the incidence range of -4˚ to 4˚. Figure 5 displays the averaged relative loss coefficient, whose definition is datum slot datum Yp Yp Yp Yp  

Figure 6 .

 6 Figure 6. Spanwise Distribution of Pitch-Averaged Total Pressure Loss at the Measurement Plane

Figure 7 .

 7 Figure 7. Spanwise Distribution of Pitch Averaged Deviation Angle at the Measurement Plane

Figure 8 .

 8 turning inside the blade passage, resulting in the altering in the distribution of pressure gradient along streamwise and spanwise, thus affecting the boundary layer evolution on the blade surface and endwall, making the domination of low momentum fluid varies. Considering that the location of high entropy low momentum fluid can be revealed by the total pressure loss contours, Figure8to 10 provides the total pressure loss in the measurement plane under different incoming flow incidence angles, in comparison of datum and end slotted blade. All the diagrams indicate that the highest total pressure loss occurs in the region of the corner vortices. (a) Datum 0˚ (b) End Slotted 0˚ Measurement Plane Total Pressure Loss Contours under the incidence 0˚

Figure 9 .Figure 10 .

 910 Measurement Plane Total Pressure Loss Contours under the incidence 2˚ A s t h e i n c i d e n c e i n c r e a s i n g , t h e w a k e c e n t e r o f t h e datum blade moves opposed to the blade suction side with the wake width expanded. The separation region has a tendency of growing upward from the incidence 2˚, expressing as the widening in the wake visualized by Figure9 (a). For the end slotted blade, the separation scale nearly remains, but the loss in the high loss core is reduced. (a) Datum 4˚ (b) End Slotted 4˚ Measurement Plane Total Pressure Loss Contours under the incidence 4˚

Figure 11 .

 11 Figure 11. The Main Vortex Structures in the blade-endwall conjunction[START_REF] Kang | Investigation of the Three Dimensional Flow within a Compressor Cascade with and without Tip Clearance[END_REF]

Table 1 .

 1 Geometry Parameters of the Datum Blade

	Compressor Cascade Values Units
	Chord length c	5 5	[ m m ]
	Solidity ct	1.52	[-]
	Aspect ratio hc	1.82	[-]
	Inlet mental angle	41.91	[˚]
	Stagger angle	21.7	[˚]
	Camber angle	52.94	[˚]
	Outlet mental angle	94.85	[˚]
	Design Ma number	0.59	[-]

Table 2 .

 2 Geometry Parameters of the End-Slots Section

	Section Feature Relative Position/ Values []
	sec1	LE1 TE1	0%Cx 25%Cx
		AO1	7.4%Cx
	slot1	t1 t1/R1	1 [mm] 0.043
		h1	20%H
	sec2	LE2 TE2	17.6%Cx 50%Cx
		AO2	10%Cx
	slot2	t2 t2/R2	1 [mm] 0.05
		h2	20%H
	sec3	LE3 TE3	40%Cx 100%Cx

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (No. 51676007, No. 51420105008, No. 51376001), the National Basic Research Program of China (No. 2014CB046405). Yangwei Liu is supported by the China Scholarship Council (CSC) for a one-year visit to the University of Cambridge.


Soichi Sasaki

Hisanari Hidaka

Experimental Study on Relative Flow Regime and Broadband Noise of a Propeller Fan

Keywords: Aeroacoustics, Broadband Noise, Propeller, Fan

The aim of this study is to clarify the relationship between the broadband noise from the low frequency to transition frequency domain generated from the propeller fan and the relative flow regime. In the off design operation condition, the velocity fluctuation having a lower frequency than the blade passing frequency was formed at the leading edge of the blade tip side. The low frequency broadband noise in the off design operation condition has feasibility to generate at the leading edge by adjoined blade interference of the lower frequency than the blade passing frequency. The major factor of the fan noise in the design operation condition was the broadband noise in the transition frequency. From the analysis of CFD, we clarified when the unstable wave due to Karman vortex shedding resonates with the air column resonance frequency made by the blade passage, the pressure fluctuation having a specific frequency is enhanced. These results indicated that the noise in the specific frequency band increased, even though the fan is operated in the design flow rate, when the air column resonance in the blade passage was superimposed to the span direction.

INTRODUCTION

Propeller fans are widely used in a heat exhaust fan, a ventilation fan, and an engine cooling fan, and so on. Therefore, the improvement not only to increase the aerodynamic performance but also to decrease the fan noise is one of the important technical issues. For example, the fan used for the engine cooling is driven at the off-design point in the low flow rate condition because of the resistance of the engine block [START_REF] Sortor | On-System Engine Cooling Fan Measurement as a Tool for Optimizing Cooling System Airflow Performance and Noise[END_REF] . The cooling fan used in the outdoor unit of an air conditioner is also driven in the low flow rate domain due to the resistance of the heat exchanger or the griller [START_REF] Jiang | Experimental and numerical study on aeroacoustic sound of axial flow fan in room air conditioner[END_REF] . Thus, the noise characteristics of the fan under the driving condition in the low flow rate domain, that is, fan noise in the off design operation condition have been discussed in the preceding studies.

In 2001, Jang, C.-M. et al investigated threedimensional structures of the vortical flow field in a propeller fan by experimental analysis using laser doppler velocimetry measurements [START_REF] Jang | Analysis of vortical flow field in a propeller fan by LDV measurements and LES -Part I: Three-dimensional vortical flow structures[END_REF][START_REF] Jang | Analysis of vortical flow field in a propeller fan by LDV measurements and LES -Part II: Unsteady nature of vortical flow structures due to tip vortex breakdown[END_REF] . According to the results of their investigation, the tip vortex formed near the mid chord of the blade tip suction surface convects almost in the tangential direction, thus impinging on the pressure surface of the adjacent blade. In 2004, T. Fukano et al studied that the noise due to tip clearance flow in the axial flow fans operating at the design and off-design conditions is analyzed by an experimental measurement [START_REF] Fukano | Tip clearance noise of axial flow fans operating at design and off-design condition[END_REF] . They clarified that the fan noise increased due to tip clearance flow at low flow rate condition. They also indicated that the noise was generated due to the interference on the tip leakage vortex and the pressure surface of the adjoined blade. On the other hand, since the axial fan in the design operation condition flows to the axial direction, the relative flow becomes hard to interfere with the adjacent blade. In this case, except for the discrete frequency noise generated due to mechanical irregularity of the blade row [START_REF] Fukano | The effects of tip clearance on the noise of low pressure axial and mixed flow fans[END_REF] , the broadband noise becomes the important factor of the aerodynamic noise generated from the fan. Fukano, et al have also proposed a prediction theory in which the broadband noise generated by the Karman vortex shedding based on the experiment of a rotating isolated blade [START_REF] Fukano | Periodic velocity fluctuations in the near wake of a rotating flat-plate blade and their role in the generation of broadband noise[END_REF] . In the prediction theory, the broadband noise of the predicted value in the lower frequency domain than the center frequency of the vortex shedding became small than the measured value, but the mechanism cause is not mentioned. In 2005, for the establishment of the analytical model of the broadband noise, M. Roger and S. Moreau [START_REF] Roger | Back-scattering correction and further extensions of Amiet's trailingedge noise model. Part 1: Theory[END_REF] have been suggested the analytical model further extensions of the Amiet's trailing edge model [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] . However, the analytical model is for the broadband noise in the relatively high frequency domain, that is, this analytical model is distinguished from the broadband noise in the low frequency domain.

The aim of this study is to clarify the relationship between the broadband noise in the low frequency domain generated from the propeller fan and the relative flow field. First of all, the aerodynamic characteristics and noise of the actual propeller fan are measured; the relative flow field of the blade tip at the design operation condition and the off design operation condition is measured by the phase lock method of hot-Experimental Study on Relative Flow Regime and Broadband Noise of a Propeller Fan-2 wire anemometer. In order to analyze the pressure fluctuation in the blade passage by the small calculation capacity, the relative flow field in the blade passage is analyzed by the two-dimensional blade cascade model of the annular cylinder. Based on the analysis of the relative flow field by the measurement and CFD, we discuss the mechanism of the broadband noise distributed within the low frequency to transition frequency.

NOMENCLATURE

EXPERIMENTL SETUP

Figure 1 is the schematic diagram of the test bench for the estimation of the fan performance. The diameter of the impeller is 613 mm, the hub ratio is 0.424 and number of blades is 14. The measurement method for the aerodynamic performance of the fan is shown in Fig. 1 (a). The height and width of the duct are 1 m each, while its total length is approximately 4 m. The dynamic pressure is measured 600 mm upstream of the impeller disc using a Pitot tube. From the results of the velocity distribution for the estimation of the flow rate, the flow coefficient was found to be 0.842. The flow rate is deduced from the dynamic pressure. It is controlled by an adjustable throttle at the duct exit. The static pressure of the fan is measured 400 mm upstream of the exit. The rotational speed of the driving shaft is 1200 rpm. We measured the shaft torque of the motor using a torque meter (Ono-Sokki; SS-500), and the efficiency of the fan was estimated based on the ratio of shaft power to theoretical power. The normalized characteristics of the fan are summarized as follows.

l y j h p r l r y f s s s U D L U P U D Q = - = = - = , ) 1 ( 8 2 , ) 1 ( 4 3 2 2 2 2 2 ν ν π (1)
where the φ is the flow coefficient, ψs is the static pressure coefficient, λ is the power coefficient, and η is the efficiency. The experimental setup for the fan noise measurement is shown in Fig. 1 (b). A 1/2-inch microphone (Ono-Sokki, LA-4350) is set at distance of 1.0 m from the impeller inlet on the rotational axis of the impeller. The frequency signal of the measured fan noise is provided by a Fast Fourier Transfer analyzer (Ono Sokki, CF5210). In Fig. 2, the measurement method of the flow field in the blade tip is shown. The internal flow of the fan is measured by an X type hot-wire probe (KANOMAX, 0249R-T5). The X type hot wire probe can be possible to simultaneously measure the two velocity components. The hot-wire probe is attached to the traverse machine; the arbitrary flow field is measured by controlling the stepping motor. The domain of the upstream and downstream is divided by a steel partition (SS400, 3.

mm). The clearance between the partition and the blade

Experimental Study on Relative Flow Regime and Broadband Noise of a Propeller Fan-3

tip is 7 mm. The flow regime near the blade tip is measured 160 mm in the mainstream directions and 240 mm in the radial directions. The measurement position of the wake is 5 mm behind the trailing edge of the blade. The time mean velocity of the absolute velocity and the relative velocity is expressed by the equation ( 2

). ' , ' w w w v v v + = + = (2)
where, v is the absolute velocity, w is the relative velocity, ¯ is the time mean, and ' is the fluctuation component. As shown in the velocity triangle, when the circumferential velocity of the impeller rotates at a constant velocity, the fluctuation variation of the absolute velocity and the relative velocity becomes same. The measurement method by the phase-lock method in the relative flow field of the blade tip is shown in Fig. 3. In order to measure the relative flow field in the blade tip, the flow field from -12 mm to 80 mm is measured with 4 mm interval as the base criterion of the leading edge. The three velocity components is measured by the X type hot-wire probe rotated 90 degrees. The signal synchronized with the main shaft is picked up by the photonics rotational detector (Ono-Sokki, MP-911). Then, the signal is input to the FFT analyzer as the trigger signal. The phase of the variable signal of the blade tip is locked by using a trigger signal synchronized with the rotation of the rotor axis to analyze the structure of a time-mean flow of the blade passage. At this time, when the circumferential velocity of the blade tip is multiplied to the time variable signal, the spatial relative flow can be measured from the absolute flow coordinate. The minimum grid width is set so that y + <10 (approximately 70 μm). The number of elements of this model is approximately 3.9 million. Cradle SCRYU Tetra V13 is used for the solver. From the quasi-steady state, 2048 cycles are iteratively calculated. The time interval is given so that the Courant number is about 1.0. The rotation speed of the blade row is 1200 rpm. In order to obtain the unsteady pressure on the blade surface, 26 measurement points are set on the blade surface.

' v v v + = u ' w w w + = w v ' w ' v ' u ' u

RESULTS AND DISCUSSION 2.1 Broadband noise of off design

The aerodynamic characteristics of the fan in the different rotational speed are compared in Fig. 5. The aerodynamic characteristic becomes the similarity property at a rotation speed of 1000 rpm or more. The maximum efficiency point is in the vicinity of φ = 0.3. In the following analysis, the flow rate coefficient 0.3 is employed as the design point, whereas the flow rate coefficient 0.17 is defined as the off design point. In Fig. 6, the characteristics of the fan noise are shown. On the design point, when the rotation speed becomes high, the noise level becomes large. However, the difference of the noise level to the flow rate in each rotation speed is little around the design point. On the other hand, the noise level is increased when the flow rate is decreased to the off design point. In Fig. 7, the spectral distributions of the fan noise on the design flow rate and the off design flow rate are compared. In the following analysis, for convenience, the frequency band below 500 Hz is referred to as low frequency, the band from 500 Hz to 2000 Hz as transition frequency, and the band above 2000 Hz as high frequency. The broadband noise at the off design operation condition becomes large than the design operation condition due to adjoined blade interference of the tip vortex [START_REF] Fukano | Tip clearance noise of axial flow fans operating at design and off-design condition[END_REF] . It is considered that the broadband noise in the high frequency domain is caused by trailing edge noise [START_REF] Amiet | Noise due to turbulent flow past a trailing edge[END_REF] . The discrete frequency noise (DFN) synchronized with the blade passing frequency (BPF = 280 Hz) is generated from the fan at the design operation condition. It has feasibility that the discrete frequency noise is due to the mechanical irregularity of the impeller [START_REF] Fukano | The effects of tip clearance on the noise of low pressure axial and mixed flow fans[END_REF] . The dominant factor of the fan noise in the design flow rate is the broadband noise in the low frequency to the transition frequency except for the discrete frequency noise. The broadband noise in the transition frequency is larger than the high frequency broadband noise, moreover, the property of the frequency distribution is different from the trailing edge noise in the high frequency. 8 (a)). In the case of the off design flow rate, the flow regime is expanded to the radial direction like a centrifugal fan due to the back pressure (see Fig. 8 (b)). This flow regime implies that the flow around the impeller at the off design operation condition becomes the swirl flow.

In Fig. 9, the distributions of the absolute velocity in the relative flow field at the blade tip are shown. The main flow flows from top to bottom toward the paper surface; the impeller rotates from right to left. The upper figure is the velocity distribution of the design flow rate, and the lower one is that of the off design flow rate. In the both operating condition, since the direction of the flow on the suction surface at the leading edge is deflected by the blade, the circumferential velocity on the suction surface at the leading is increased. The axial flow velocity in the vicinity of the leading edge in the off design flow rate becomes small, and its relative flow swirls to the circumferential direction. Furthermore, the wake near the trailing edge of the off design flow rate flows to the circumferential direction more than that of the design point. There is also a flow field having a small axial velocity in the vicinity of the leading edge of the design flow rate, and then the relative flow in the domain also becomes the weak swirling flow to the circumferential direction.

Broadband noise of design condition

In Fig. 11, the distribution of the deviation angle to the span direction in the vicinity of the trailing edge is shown. The measurement position is the 5 mm rear side from the trailing edge. The deviation angle is defined as the angle made by the pitch angle γ and the relative flow angle β (see Fig. 12). When the angle is positive, the flow is separated from the suction surface. Whereas, when the angle approaches to 0, the relative flow is The pressure fluctuation on the blade surface at the design operation condition calculated by CFD is shown in Fig. 14. The radial position is r / R = 0.74 ( R = 230 mm ). Fig. 14 (a) is the distribution of the overall pressure fluctuation, and Fig. 14 (b) is the spectral distribution of the sum of the pressure fluctuations of the both blade surface. The pressure fluctuations on the blade surface increase at the leading edge and the trailing edge. This result indicates that the aerodynamic noise source on the blade surface is formed not only at the trailing edge but also at the leading edge. In Fig. 14 (b), the discrete frequency synchronized with the harmonic of the blade passing frequency (BPF) and a discrete frequency of 1560 Hz are formed. The latter discrete frequency may be caused by the vortex shedding frequency of the Karman vortex. The Karman vortex is formed by the interference between the leading edge separation flow and the flow at the trailing edge. This formation of the Karman vortex becomes the factor that of an unstable wave in the blade passage. Furthermore, we can observe the broadband pressure fluctuation around 1000 Hz. This broadband pressure fluctuation is close to the property of broadband noise in the transition frequency generated from the actual propeller fan. The spectral distribution of pressure fluctuations near the leading edge is shown in Fig. 15. The measurement position is x / C = 0.23 on the pressure surface. In the spectral distribution, a humped pressure fluctuation distribution is formed in the band near 1260 Hz, which is distinguished from the blade passing frequency or the vortex shedding frequency. 3) is shown in Table 1. The band of the resonance frequency distributed to the span direction is close to the band of the transition frequency of the fan noise. These results indicate that when the air column resonance occurs in the blade passage at each span position of the propeller fan, the fan generates the broadband noise due to superimpose of the air column resonance noise.

SUMMARY

(1) The broadband noise of the propeller fan in the off design flow rate became large than that of the design operation condition in the low frequency. The velocity fluctuation having a lower frequency than the blade passing frequency was formed at the leading edge of the blade tip side. The low frequency broadband noise in the off design operation condition has feasibility to generate at the leading edge by adjoined blade interference of the lower frequency than the blade passing frequency. The low frequency broadband noise is even generated in the design operation condition.

(2) The major factor of the fan noise in the design operation condition was the broadband noise in the transition frequency except for the discrete frequency noise due to the mechanical irregularity error of the impeller. The velocity fluctuation of the relative flow at the design operation condition was widely distributed to the span direction of the impeller, and its fluctuation variation was larger than that of the off design operation condition.

(3) From the analysis of CFD, an unstable wave was generated in the blade passage due to the Karman vortex shedding. When the unstable wave resonates with the air column resonance frequency made by the blade passage, the pressure fluctuation having a specific frequency is enhanced in the blade passage. These results indicated that the noise in a specific frequency band increased, even if the fan is operated in the design flow rate, when the air column resonance in the blade passage was superimposed to the span direction.

Fig. 1

 1 Fig. 1 Schematic diagram of the test bench for the estimation of the fan performance

Fig. 2

 2 Fig. 2 measurement method of the flow field in the blade tip

  Fig. 3 Measurement method by the phase-lock method in the relative flow field of the blade tip In Fig. 4, the CFD model of a two-dimensional blade cascade of the propeller fan is shown. In this study, the two dimensional blade cascades is created in the annular cylinder with the 4 mm thickness, in which the aim of suppressing the calculation load and analyzing the pressure fluctuation of the blade cascade with high resolution. Main flow velocity estimated by the design flow rate is given to the inlet boundary and the atmospheric pressure is given to the exit side boundary.The minimum grid width is set so that y + <10 (approximately 70 μm). The number of elements of this model is approximately 3.9 million. Cradle SCRYU Tetra V13 is used for the solver. From the quasi-steady state, 2048 cycles are iteratively calculated. The time interval is given so that the Courant number is about 1.0. The rotation speed of the blade row is 1200 rpm. In order to obtain the unsteady pressure on the blade surface, 26 measurement points are set on the blade surface.

Fig. 4

 4 Fig. 4 CFD model of a two-dimensional blade cascade of the propeller fan

Fig. 5 Fig. 6

 56 Fig. 5 Aerodynamic characteristics of the fan in the different rotational speed

Fig. 7

 7 Fig. 7 Spectral distributions of the fan noise on the design flow rate and the off design flow rateThe distributions of the axial velocity component of the absolute velocity in the meridian plane are shown in Fig.8. The broken line in the figure is the position of the blade tip. The wake flow in the design operation condition flows to the axial direction (see Fig.8 (a)). In the case of the off design flow rate, the flow regime is expanded to the radial direction like a centrifugal fan due to the back pressure (see Fig.8 (b)). This flow regime implies that the flow around the impeller at the off design operation condition becomes the swirl flow.In Fig.9, the distributions of the absolute velocity in the relative flow field at the blade tip are shown. The main flow flows from top to bottom toward the paper surface; the impeller rotates from right to left. The upper figure is the velocity distribution of the design flow rate, and the lower one is that of the off design flow rate. In the both operating condition, since the direction of the flow on the suction surface at the leading edge is deflected by the blade, the circumferential velocity on the suction surface at the leading is increased. The axial flow velocity in the vicinity of the leading edge in the off design flow rate becomes small, and its relative flow swirls to the circumferential direction. Furthermore, the wake near the trailing edge of the off design flow rate

Fig. 9

 9 Fig. 8 Distributions of the axial velocity component of the absolute velocity in the meridian plane

  Fig. 10 Spectral distributions of the velocity fluctuations at the blade tip

Fig. 11

 11 Fig. 11 Distribution of the deviation angle to the span direction in the vicinity of the trailing edge

ExperimentalFig. 12

 12 Fig. 12 Definition of the deviation angleThe distributions of the relative flow to the span direction are shown in Fig.13. Fig.13 (a) is the relative flow velocity, and Fig.13 (b) is its velocity fluctuation variation. The velocity and its fluctuation of the off design operation condition are concentrated to the blade tip. This result indicates that the aerodynamic noise source in the off design operation condition is formed at the blade tip side. On the other hand, the relative flow of the design operation condition is widely distributed to the blade span, and its velocity fluctuation becomes large than the off design operation condition.

  Fig. 13 Distributions of the relative flow to the span direction

  Fig. 14 Pressure fluctuation on the blade surface at the design operation condition

Fig. 15

 15 Fig. 15 The spectral distribution of pressure fluctuations near the leading edge ( x / C = 0.23 ) Let me introduce the air column resonance frequency formed by the blade passage as in the equation (3)

Table 1

 1 Resonance frequency at each span position

	r , mm	C , mm	l , mm	Δl , mm	f , Hz
	130	72	58.3	17.5	1589
	175	92	78.5	23.6	1222
	230	101	103.2	31.0	1043
	310	122	139.1	41.7	827


Fahad B Mostafa

Revisiting the

Keywords: SVD, orthogonal matrix, orthogonal linear transformation, orthogonal projection, PCA, FDA Methodology

come

As images are very high dimensional, it is not easy to analyze them directly. Some common approaches are to reduce their diminution using principal component analysis (PCA), Fisher's discriminant analysis (FDA), and other similar methods. Both PCA and LDA are linear transformation methods. PCA yields the directions (principal components) that maximize the variance of the data, whereas LDA also aims to find the directions that maximize the separation (or discrimination) between different classes, which can be useful in pattern classification problem such as image recognition.

Our main problem is arising from 𝑌𝑌 = 𝑋𝑋𝑋𝑋 + 𝜀𝜀 where 𝑌𝑌 𝑖𝑖𝑖𝑖 𝑛𝑛 × 1 column matrix, 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑛𝑛 × 𝑝𝑝 matrix and 𝜀𝜀 is 𝑛𝑛 × 1 matrix. If we ignore the error term and compare it with system of linear equation of the standard form 𝐴𝐴𝑋𝑋 = 𝑏𝑏 then for 𝑛𝑛 > 𝑝𝑝, the system is overdetermined and we consider it as Linear least square problems (to solve it, we us Gram Schmidt process, QR factorization, Householder, etc), but when 𝑛𝑛 < 𝑝𝑝 then the system becomes underdetermined and we use singular value decomposition shortly SVD to reduce the dimension of column of matrix 𝑋𝑋 from 𝑝𝑝 to smaller dimension (say,'d' for our case). PCA is one of the central uses of SVD.

In our study, the main aim of a PCA analysis is to identify patterns in data; PCA aims to detect the correlation between variables. If a strong correlation between variables exists, the attempt to reduce the dimensionality only makes sense. In a sense PCA is all about finding the directions of maximum variance in high-dimensional data and project it onto a smaller dimensional subspace while retaining most of the information. Suppose 𝑈𝑈 be the 𝑝𝑝 × 𝑝𝑝 orthogonal matrix such that the elements of the vectors 𝑍𝑍 = 𝑈𝑈 𝑇𝑇 𝑋𝑋 are uncorrelated. The uncorrelated elements of the vector 𝑍𝑍 = 𝑈𝑈 𝑇𝑇 𝑋𝑋 are called the component of 𝑋𝑋.

So, our moto is to find this 𝑈𝑈 which can be obtained from the SVD of covariance matrix 𝐾𝐾.

Precisely, 𝑐𝑐𝑐𝑐𝑐𝑐(𝑍𝑍) = 𝑈𝑈 𝑇𝑇 𝑐𝑐𝑐𝑐𝑐𝑐(𝑋𝑋)𝑈𝑈 = 𝑈𝑈 𝑇𝑇 𝐾𝐾𝑈𝑈 = ∑. Now we can compare this SVD.

Thus 𝑍𝑍 𝑖𝑖 = 𝑈𝑈 𝑖𝑖 𝑇𝑇 𝑋𝑋 is the principal component. More elaborately (𝑍𝑍 1 , 𝑍𝑍 2 , … ) and (𝑈𝑈 1 , 𝑈𝑈 2 , … ) are the principal components and directions respectively. Here, 𝑍𝑍 1 is the largest variance and 𝑍𝑍 2 is the second largest and so on. So, we can reduce the algorithm of PCA of given data as bellows where 𝑋𝑋 denotes an independent observation vector ′𝑋𝑋′. Finding the sample covariance 𝐾𝐾 ∈ 𝑅𝑅 𝑝𝑝×𝑝𝑝 then compute the SVD of 𝐾𝐾 to obtain orthogonal matrix 𝑈𝑈 ∈ 𝑅𝑅 𝑑𝑑×𝑛𝑛 . Then we define principal directions by choosing first column of 𝑈𝑈. Thus the principal component is

𝑍𝑍 𝑑𝑑×1 = 𝑈𝑈 𝑑𝑑×𝑛𝑛 𝑇𝑇 𝑋𝑋 𝑛𝑛×1
Note that, if we have more correlated data then 𝑑𝑑 will be very smaller than 𝑛𝑛.

- A matrix captures the average separate between elements within same class is captured by the within class scatter matrix.

𝐶𝐶 1 = 𝑋𝑋 1 1 , 𝑋𝑋 2 1 , … , 𝑋𝑋 𝑁𝑁 1 1 → 𝜇𝜇 1 𝜖𝜖 𝑅𝑅 𝑛𝑛 𝐶𝐶 2 = 𝑋𝑋 1 2 , 𝑋𝑋 2 2 , … , 𝑋𝑋 𝑁𝑁 1 2 → 𝜇𝜇 2 𝜖𝜖 𝑅𝑅 𝑛𝑛 … 𝐶𝐶 𝑚𝑚 = 𝑋𝑋 1 1 , 𝑋𝑋 2 1 , … , 𝑋𝑋 𝑁𝑁 1 1 → 𝜇𝜇 𝑚𝑚 𝜖𝜖 𝑅𝑅 𝑛𝑛
𝑆𝑆 𝑤𝑤 = � ���𝑋𝑋 𝑖𝑖 𝑗𝑗 -𝜇𝜇 𝑗𝑗 ��𝑋𝑋 𝑖𝑖 𝑗𝑗 -𝜇𝜇 𝑗𝑗 � 𝑇𝑇 𝑁𝑁 𝑗𝑗 𝑖𝑖=1 � 𝑚𝑚 𝑗𝑗=1,𝑖𝑖=1 ∈ 𝑅𝑅 𝑛𝑛×𝑛𝑛
Now our goal is to find the projection, 𝑍𝑍 = 𝑈𝑈 𝑇𝑇 𝑋𝑋. The between class scatter matrix because

𝑆𝑆 𝐵𝐵 𝑍𝑍 = 𝑈𝑈 𝑇𝑇 𝑆𝑆 𝐵𝐵 𝑈𝑈
The within class scatter matrix becomes

𝑆𝑆 𝑤𝑤 𝑍𝑍 = 𝑈𝑈 𝑇𝑇 𝑆𝑆 𝑤𝑤 𝑈𝑈
Now the goal is to choose 𝑈𝑈 that maximizes the following function

𝑓𝑓(𝑈𝑈) = 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑚𝑚𝑖𝑖𝑛𝑛𝑑𝑑𝑛𝑛𝑑𝑑(𝑆𝑆 𝐵𝐵 𝑍𝑍 ) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑚𝑚𝑖𝑖𝑛𝑛𝑑𝑑𝑛𝑛𝑑𝑑(𝑆𝑆 𝑤𝑤 𝑍𝑍 ) = det (𝑈𝑈 𝑇𝑇 𝑆𝑆 𝐵𝐵 𝑈𝑈) det (𝑈𝑈 𝑇𝑇 𝑆𝑆 𝑤𝑤 𝑈𝑈) ↑ ↓ ; therefore 𝑓𝑓(𝑈𝑈) ↑
The optimal projection is 𝑈𝑈 � = 𝑑𝑑𝑑𝑑𝑎𝑎𝑚𝑚𝑑𝑑𝑎𝑎�𝑓𝑓(𝑈𝑈)�; 𝑤𝑤𝑖𝑖𝑑𝑑ℎ 𝑈𝑈 ∈ 𝑅𝑅 𝑛𝑛×𝑑𝑑 . Now, we have to find 𝑈𝑈 that the quantity 𝑈𝑈 � maximize. 𝑈𝑈 � can be solved as the generalized eigenvalue problem where we calculate 𝑆𝑆 𝐵𝐵 𝑈𝑈 � 𝑖𝑖 = 𝑑𝑑 𝑖𝑖 𝑆𝑆 𝑤𝑤 𝑈𝑈 � 𝑖𝑖 ; 𝑑𝑑 𝑖𝑖 represents the eigenvalues of the transformation matrix 𝑈𝑈. If 𝑆𝑆 𝐵𝐵 is nonsingular then we can find corresponding eigenvector by calculating 𝑈𝑈 = 𝑆𝑆 𝐵𝐵 -1 𝑆𝑆 𝑤𝑤 . In our case we will use building MATLAB code to find eigenvector.

In this study, we compute the principal component (PCA) of training images and project them down to the smaller size. That is, each training image is now represented by a small vector size 𝑘𝑘.

For the test image, we can project them down to a 𝑘𝑘 vector, using the same projection and then we find the nearest image in the training set by computing 𝑘𝑘 vectors. In case of FDA, the low dimensional projection is determined by using scatter matrices.

We will assume 𝑛𝑛 2 training images each for 𝑛𝑛 1 people in our database. The size of each image is 𝑖𝑖 1 × 𝑖𝑖 2 . These images are taken at different orientations, different facial expressions etc.

There are two parts to the procedure. One is to analyze the training images and compute a projection to their principal 𝑘𝑘 -dimensional subspace and comparing with training data. In this work we will start from the following.

Let us consider vector in a matrix size (𝑖𝑖 1 × 𝑖𝑖 2 ) × (𝑛𝑛 1 × 𝑛𝑛 2 ) is the arrangement of training images and we call it 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 . First 𝑛𝑛 2 columns are images of person 1, and next 𝑛𝑛 2 columns are images of person 2, and so on, with a total of 𝑛𝑛 1 × 𝑛𝑛 2 columns. Then we use PCA, FDA and Simple projection and we compare these three procedure.

To perform PCA from our dataset 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 , we need to calculate SVD and designate 𝑈𝑈 1 be the first 𝑘𝑘 columns of the orthogonal matrix 𝑈𝑈. The size of 𝑈𝑈 is now (𝑖𝑖 1 × 𝑖𝑖 2 ) × 𝑘𝑘. Now we will use FDA, where we still consider PCA to reduce the size of our data 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 from

(𝑖𝑖 1 × 𝑖𝑖 2 ) × (𝑛𝑛 1 × 𝑛𝑛 2 ) to 𝑑𝑑 × (𝑛𝑛 1 × 𝑛𝑛 2 ), where 𝑑𝑑 = 𝑛𝑛 1 ×𝑛𝑛 2 2
. So, we get a new matrix and let us call it 𝑌𝑌 � 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 and call the (𝑖𝑖 1 × 𝑖𝑖 2 ) × 𝑑𝑑 projection matrix 𝑈𝑈 0 . Use all the vectors from the same person as observations from the same cluster. In 𝑌𝑌 � 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 , the first 𝑛𝑛 2 columns designate the first person, the second 𝑛𝑛 2 columns designate the second person, and so on. We have two scatter matric, one is between class and another is with-in class scatter matrices, and use the generalized eigen decomposition to find 𝑘𝑘 eigen vectors that correspond to the largest eigen values. So, we get this result as a matrix form and say it submatrix 𝑉𝑉. Now we find the orthogonal columns 𝑑𝑑 × 𝑘𝑘 matrix.

Define a (𝑖𝑖 1 × 𝑖𝑖 2 ) × 𝑘𝑘 orthogonal matrix 𝑈𝑈 1 = 𝑈𝑈 0 𝑉𝑉.

We will use simple projection for comparison with PCA and FDA. In this case, we will take another projection where 𝑈𝑈 1 is simply first 𝑘𝑘 columns of (𝑖𝑖 1 × 𝑖𝑖 2 ) identity matrix. Now, we will repeat the followings for upper three cases one by one. We will use the projection 𝑌𝑌 1 = 𝑈𝑈 1 𝑇𝑇 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 with size 𝑘𝑘 × (𝑛𝑛 1 × 𝑛𝑛 2 ), which simplify each image in 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 in a reduced form of 𝑘𝑘 -dimensional vector. After using this procedure for each cases we can perform classification. We have 𝑛𝑛 2 test images per person and the test data set 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 is in the same form as the training data set 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑛𝑛 . We will take an image 𝐼𝐼 randomly from the test set which represents a random column from 𝑌𝑌 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 actually. Projection of 𝐼𝐼 can be found from 𝐼𝐼 1 = 𝑈𝑈 1 𝑇𝑇 𝐼𝐼 where 𝐼𝐼 1 is a 𝑘𝑘 × 1 vector. Using 𝑙𝑙 2 norm we can find the distance between 𝐼𝐼 1 and each column of 𝑌𝑌 1 . Now, to find the label of the column that has the smallest distance to 𝐼𝐼 1 . If this label matches the true label then our recognition is successful otherwise it is a failure. For computing the percentage of successful recognition 𝐹𝐹(𝑘𝑘), we perform the above procedure 500 times, which will also give us the average performance of upper three methods. After plotting 𝑘𝑘 against each of the three projection we can easily discuss the performance.

Result discussion:

We have summarized the result by using table and graphs. Then we will show an example of sets of images when the dimensionality parameter 𝑘𝑘 changes then the performance 𝐹𝐹(𝑘𝑘) becomes improved so that the quality of image gets better. We demonstrate it by few examples using test images and the closest images in the training set finally. analysis we report validate our claim. That is, FDA is relatively better performing than PCA.

However both has good performance under this experiment. In addition, simple projection is easy to compute but which shows quite less performance than PCA and FDA.

  Mathematically, PCA is defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by the projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate and do on. Let's say 𝑋𝑋 as, For any matrix 𝑋𝑋 ∈ 𝑅𝑅 𝑛𝑛×𝑝𝑝 , there exists an orthogonal matrices 𝑈𝑈 ∈ 𝑅𝑅 𝑛𝑛×𝑛𝑛 and 𝑉𝑉 ∈ 𝑅𝑅 𝑝𝑝×𝑝𝑝 such that 𝑈𝑈 𝑇𝑇 𝑋𝑋𝑉𝑉 = ∑ Where sigma= ∑ = diagonal (𝜎𝜎 1 , 𝜎𝜎 2 … , 𝜎𝜎 𝑛𝑛 ) ∈ 𝑅𝑅 𝑛𝑛×𝑝𝑝 𝜎𝜎 1 ≥ 𝜎𝜎 2 ≥ 𝜎𝜎 3 ≥ ⋯ ≥ 𝜎𝜎 𝑛𝑛 ≥ 0. The 𝜎𝜎 𝑖𝑖 ′𝑖𝑖 are called the singular values of 𝑋𝑋 and the columns of 𝑈𝑈 and 𝑉𝑉 are called left and right singular vectors of 𝑋𝑋 rexpectively. For 𝑛𝑛 singular values, 𝜎𝜎 1 ≥ 𝜎𝜎 2 ≥ 𝜎𝜎 3 ≥ ⋯ ≥ 𝜎𝜎 𝑑𝑑 ≥ 𝜎𝜎 𝑑𝑑+1 = 𝜎𝜎 𝑑𝑑+2 … 𝜎𝜎 𝑛𝑛 = 0. Where 𝑑𝑑 is positive singular value, others are zero. Now the rank of our original marix 𝑋𝑋 is 𝑑𝑑 (Figure-1). Range of X is the span of 𝑢𝑢 1 , 𝑢𝑢 2 , … , 𝑢𝑢 𝑑𝑑 which is equal to 𝑑𝑑. Originally we had the dimension of 𝑋𝑋 was 𝑛𝑛 × 𝑝𝑝 and now it is 𝑛𝑛 × 𝑑𝑑. Statistically, number of date is reduced from 𝑝𝑝 to 𝑑𝑑. Now, 𝑈𝑈 𝑇𝑇 𝑋𝑋𝑉𝑉 = ∑ becomes 𝑋𝑋 = 𝑈𝑈∑𝑉𝑉 𝑇𝑇 , that's meaning, in original data we had 𝑝𝑝 columns and now after SVD, we obtain 𝑑𝑑 columns.

Figure- 1 :

 1 Figure-1: Full singular value decomposition to reduced singular value decomposition

Figure- 2 :

 2 Figure-2: Principal component analysis

Figure- 3 :

 3 Figure-3: Linear Discriminant Analysis for 2-class

  Each 𝑋𝑋 𝑖𝑖 𝑗𝑗 ∈ 𝑅𝑅 𝑛𝑛 , the 𝑖𝑖 𝑡𝑡ℎ observation in the 𝑗𝑗 𝑡𝑡ℎ class. Let, 𝜇𝜇 𝑗𝑗 be the mean of observation in class 𝐶𝐶 𝑗𝑗 .A matrix captures the separation between the classes in the between class scatter matrix.𝑆𝑆 𝐵𝐵 = ��𝜇𝜇 𝑗𝑗 -𝜇𝜇��𝜇𝜇 𝑗𝑗 -

Figure- 4 :

 4 Figure-4: Paradigm of 5 different Eigen faces for test and train datasets

  becomes larger. The performance of these transforms gets better as the value of 𝑘𝑘 increases. What's the difference between the three is that while the recognition rate with PCA saturates around 11.4% to 88.8% when 𝑘𝑘 varies from 1 to 40 while the performance of simple projection vary widely from PCA and FDA. For the experiments under discussion, the average performance of correct recognition best for 𝐹𝐹 𝑃𝑃𝑃𝑃𝑃𝑃 (𝑘𝑘) = 90% when 𝑘𝑘 = 32, 𝐹𝐹 𝐹𝐹𝐹𝐹𝑃𝑃 (𝑘𝑘) = 91.6% when 𝑘𝑘 = 40, and 𝐹𝐹 𝑆𝑆𝑃𝑃 (𝑘𝑘) = 66.2% when 𝑘𝑘 = 40. From the table we can conclude that the values of 𝐹𝐹(𝑘𝑘) is increasing with the increment of 𝑘𝑘 where FDA shows relatively best performance and SP reflects weaker performance. However the performance of PCA is quite similar to FDA. To understand the result more clearly, we have added a graph of 𝑘𝑘 vs. 𝐹𝐹(𝑘𝑘) below.

Figure- 5 :

 5 Figure-5: Variation between percentage of successful recognition 𝐹𝐹(𝑘𝑘) and the number 𝑘𝑘 Here we calculate the average performance. After performing PCA, FDA and Simple projection 500 times and computing the percentage of successful recognition, we draw the graph of 𝑘𝑘 versus 𝐹𝐹(𝑘𝑘).Figure-5 shows LDA is the most proficient technique compared to other two

   shows LDA is the most proficient technique compared to other two techniques. In fact, LDA > PCA > SimPro in our experiment. The performance of FDA is strictly increasing up to almost 55% when 𝑘𝑘 runs up from 0 to 3 continuously, then there are some fluctuations of performances with some small ups and downs. FDA achieves almost 91.6% accuracy when dimensionality parameter 𝑘𝑘 becomes 40 which shows best performance in our analysis. In case of PCA, 𝐹𝐹(𝑘𝑘) represents 90% efficiency when 𝑘𝑘 is 32 and then it again declines a little and fluctuates close to 88.8% till 𝑘𝑘 becomes 40. Performance graph of FDA and PCA intersect each other as they both show similar performances. FDA and PCA both shows most of its up grading performance when dimensionality 𝑘𝑘 lays in the interval form 0 to 10 and then they stay in a constant rate almost in the rest of the interval. Hence Simple Projection (SimPro) does not represent good performance compared to other two method. However, Simple projection method also reveals a upward trend of the performance with dimensionality.To validate our numerical results in table and graphical results with real image, as we want to use our method (PCA, FDA and SP) to recognize new images and compare them with some images of known people. For this case, we have chosen different 𝑘𝑘 values such as 4, 32, 40 and take a person (say person 2) from test image and compare with our experiment. Figure-6 is indicating the perfection of our study.

Figure- 6 :

 6 Figure-6: Comparison of Eigen faces using test images with PCA, FDA and SP performed images

  ,'r--'); %Computing the FDA of Ytrain d=200/2; % define d acording to our problem given U0=U(:,1:d); % computation of the projection matrix Ytrain_new=U0'*Ytrain; % constructing the new training matrix [V,lambda]=lda(Ytrain_new); %calculation of the matrix V V=orth(V); %to make the column orthogonal for ); %index of our desired image in Ytrain data [~,ridx]=classifier(Y1,I1); % index of the recognized image if

Table 1 :

 1 Representation of average performance of correct recognition 𝐹𝐹(𝑘𝑘) versus the number 𝑘𝑘 for three cases.Observation we wish to make from Table-1 is that regarding our training datasets has to do with relative behavior of Simple projection (SP), FDA and PCA as the dimensionality parameter 𝑘𝑘

	𝑘𝑘	𝐹𝐹 𝑃𝑃𝑃𝑃𝑃𝑃 (𝑘𝑘)	𝐹𝐹 𝐹𝐹𝐹𝐹𝑃𝑃 (𝑘𝑘)	𝐹𝐹 𝑆𝑆𝑃𝑃 (𝑘𝑘)	𝑘𝑘	𝐹𝐹 𝑃𝑃𝑃𝑃𝑃𝑃 (𝑘𝑘)	𝐹𝐹 𝐹𝐹𝐹𝐹𝑃𝑃 (𝑘𝑘)	𝐹𝐹 𝑆𝑆𝑃𝑃 (𝑘𝑘)
	1	0.114	0.12	0.162	21	0.876	0.836	0.45
	2	0.414	0.228	0.19	22	0.872	0.868	0.462
	3	0.554	0.438	0.248	23	0.86	0.816	0.464
	4	0.608	0.54	0.214	24	0.86	0.862	0.486
	5	0.648	0.614	0.246	25	0.862	0.858	0.48
	6	0.676	0.622	0.334	26	0.864	0.824	0.46
	7	0.752	0.666	0.38	27	0.864	0.872	0.514
	8	0.752	0.746	0.398	28	0.882	0.868	0.52
	9	0.81	0.782	0.41	29	0.896	0.892	0.566
	10	0.846	0.778	0.426	30	0.87	0.9	0.584
	11	0.83	0.778	0.408	31	0.872	0.886	0.584
	12	0.834	0.792	0.476	32	0.9	0.888	0.578
	13	0.818	0.84	0.474	33	0.868	0.876	0.574
	14	0.868	0.838	0.504	34	0.884	0.878	0.602
	15	0.862	0.85	0.484	35	0.868	0.902	0.572
	16	0.832	0.818	0.45	36	0.872	0.912	0.602
	17	0.81	0.828	0.474	37	0.854	0.888	0.598
	18	0.842	0.856	0.48	38	0.878	0.878	0.534
	19	0.844	0.868	0.478	39	0.846	0.894	0.598
	20	0.842	0.852	0.484	40	0.888	0.916	0.662

Appendix