Patent Publication Number: US-10760424-B2

Title: Compressor rotor airfoil

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 14/469,938 filed on Aug. 27, 2014, the content of which is hereby incorporated by reference. 
    
    
     TECHNICAL FIELD 
     The application relates generally to compressor rotor airfoils and, more particularly, to compressor rotor airfoils in gas turbine engines. 
     BACKGROUND OF THE ART 
     Compressors of a gas turbine engine include a plurality of rotors mounted on a rotating shaft. Each rotor includes a hub connected to the rotating shaft and a plurality of blades radially extending from the hub. The rotors are enclosed in a casing, also known as shroud. The rotors are spaced away from the shroud by a small amount called tip clearance. Under high velocity and certain thermal conditions, the blades may lengthen, even minimally, and as a result may reduce the tip clearance and in some cases even dig into the shroud. This phenomenon is sometimes referred as plowing. The plowing may scrape the shroud and may lead to damage of the blade and the shroud, which in turn may reduce performances of the gas turbine engine. 
     SUMMARY 
     In one aspect, there is provided a compressor rotor in a gas turbine engine, the airfoil comprising a opposed pressure and suction sides joined together at chordally opposite leading and trailing edges, the pressure and suction sides extending in a span direction from a root to a tip; and a leading edge dihedral angle defined at a point on the leading edge between a tangent to the airfoil and a vertical, the leading edge dihedral angle having a span-wise distribution, the distribution having at least one inflection point. 
     In another aspect, there is provided a gas turbine engine comprising: a compressor section including a plurality of rotors, each of the plurality of rotors including a hub, the hubs being aligned axially, each of the rotors including a plurality of blades extending radially from the hub, the blades including an airfoil portion, the airfoil portion comprising: opposed pressure and suction sides joined together at chordally opposite leading and trailing edges, the pressure and suction sides extending in a span direction from a root to a tip; and a leading edge dihedral angle defined at a point on the leading edge between a tangent to the airfoil and a vertical, the leading edge dihedral angle having a span-wise distribution, the distribution having at least one inflection point. 
     In another aspect, there is provided a method of reducing a rub angle between a compressor rotor blade and a casing surrounding the blade, the method comprising: forming an airfoil having opposed pressure and suction sides joined together at chordally opposite leading and trailing edges, the pressure and suction sides extending in a span direction from a root to a tip, a leading edge dihedral angle being defined between a tangent to the airfoil and a vertical at a point on the leading edge, the leading edge dihedral angle having a span-wise distribution, the distribution of the leading edge dihedral angle having at least one inflection point. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       Reference is now made to the accompanying figures in which: 
         FIG. 1  is a schematic cross-sectional view of a gas turbine engine; 
         FIG. 2  is a schematic of a portion of a compressor of the gas turbine engine of  FIG. 1 ; 
         FIGS. 3A and 3B  are schematics of a blade for the compressor of  FIG. 2  annotated to show a sweep angle α ( FIG. 3A ) and a dihedral angle β ( FIG. 3B ); 
         FIG. 4  is a graph of a leading edge sweep angle α relative to a span Sp of a blade of the compressor of  FIG. 2 ; 
         FIG. 5  is a graph of a leading edge dihedral angle β relative to a span Sp of a blade of the compressor of  FIG. 2 ; 
         FIG. 6  is a graph of a leading edge sweep angle α relative to a leading edge dihedral angle β of a blade of the compressor of  FIG. 2 ; 
         FIG. 7  is a graph of a ratio of a leading edge sweep angle α over a leading edge dihedral angle β relative to a span Sp of the blade of the compressor of  FIG. 2 ; 
         FIG. 8  is a schematic cross-sectional view of the blade of the compressor of  FIG. 2  taken along line  8 - 8  in  FIG. 3A ; 
         FIG. 9  is a graph of a span Sp relative to axial and tangential center of gravity Xcg, Ycg components of the blade of the compressor of  FIG. 2 ; 
         FIG. 10  is the schematic cross-sectional view taken toward the hub along line  10   a - 10   a  in  FIG. 3A  (solid line) superimposed with a schematic cross-sectional view taken toward the tip (dotted line) along line  10   b - 10   b  in  FIG. 3A  of the blade of the compressor of  FIG. 2 ; and 
         FIG. 11  is a plot of a thickness distribution of different cross-sections of the blade of the compressor of  FIG. 2  versus a non-normalised chord. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates a gas turbine engine  10  of a type preferably provided for use in subsonic flight, generally comprising in serial flow communication along a centerline  11 : a fan  12  through which ambient air is propelled, a compressor section  14  for pressurizing the air, a combustor  16  in which the compressed air is mixed with fuel and ignited for generating an annular stream of hot combustion gases, and a turbine section  18  for extracting energy from the combustion gases. 
       FIG. 2  illustrates a portion of the compressor section  14  including a plurality of rotors  19  (only two of the rotors being shown). The rotor  19  is an integrally bladed rotor including a plurality of circumferentially distributed blades  20  extending radially from an annular hub  21 . The blades  20  could be integrally formed with the hub  21  or could be connected thereto. The blades  20  are supported in a circumferentially extending row around hub  21  for rotation about the centerline  11  of the engine  10  (as depicted by arrow Dr in  FIG. 2 ). As shown in  FIG. 2 , an annular compressor casing  22  (also known as shroud) surrounds the compressor blades  20 . The compressor section  14  also includes a plurality of circumferential rows or stages of stator vanes  24  disposed between the plurality of compressor blades  20  in an alternating fashion. The stator vanes  24  project radially inwardly from the compressor casing  22 . 
     Each of the blades  20  includes a root  25  joining the blade  20  to the hub  21  and an airfoil portion  26  extending from the root  25 . The airfoil portion  26  includes a tip  27  at a radially outer end thereof. The tip  27  is spaced radially from the compressor casing  22  to provide tip clearance. The hub  21  and annular casing  22  define inner and outer boundaries, respectively, for channeling a flow of air  28  through the compressor  10 . The flow of air  28  is generally aligned with the centerline  11 . The hub  21  forms with the compressor casing  22  a converging annular flow channel  29  for compressing air driven through the compressor section  14  by the blades  20 . As such, the front blades  20  (i.e. the upstream stages of compressor blades) have a longer span Sp than the rear blades  20  (i.e. the downstream stages of compressor blades, the ones just upstream of the combustor  16 ). 
     The airfoil portions  26  of the blades  20  include each a pressure side  32  and an opposed suction side  34 . The pressure side  32  and suction side  34  extend in a span direction from the root  25  to the tip  27 . The airfoil portion  26  further includes a leading edge  36  and a trailing edge  38  defined at the junction of the pressure side  32  and the suction side  34 . The airfoil portion  26  also defines the span Sp extending between the root  25  and the tip  27 , and a chord Ch extending transversally between the leading edge  36  and the trailing edge  38 . When in operation, the blade  20  rotates in the direction of rotation Dr with the suction side  34  disposed forward of the pressure side  32 . When the blades  20  are in operation connected to the hub  21 , the root  25  is commonly referred to as the hub  21 . 
     Turning to  FIGS. 3A and 3B , the airfoil portion  26  may be oriented at different positions relative to the flow of air  28  in order to optimise efficiency of the rotor  19 . The airfoil portion  26  may also be twisted or leaned. Different angles may then be used to characterise the shape of the airfoil portion  26 . 
     A sweep angle α and a dihedral angle β may thus be defined. The sweep angle α and dihedral angle β can be defined at any point P along the leading edge  36 . With reference to  FIG. 3A , at any point P along the leading edge  36  the angle between the local velocity vector Vel of the incoming flow  28  and a tangent T to the leading edge  36  may define the sweep angle α. Forward sweep may be defined when the angle (α−90°) is negative. Similarly, rearward sweep occurs when the angle (α−90°) is positive. With reference to  FIG. 3B , the dihedral angle β may be defined as the angle between a vertical V and the tangent T to leading edge  36  at any point P. The vertical V is confounded with the radial direction R (shown in  FIG. 2 ). The dihedral angle β is positive in the direction of rotation Dr. 
     Flow around the airfoil portion  26  is complex. Depending on the shape of the airfoil portion  26  and the flow conditions, transonic flow may be present in the compressor section  14  (i.e. existence of subsonic flow in some portions of the compressor section  14 , and sonic and supersonic flow in other portions of the compressor section  14 ). As a result of these flow conditions, boundary layer build up may occur at the tip  27  of the blade  20  which may influence the efficiency of the compressor section  14 . 
     Tip blade lean (in direction of rotation Dr) and forward sweep (in direction opposite to flow  28 ) may be used in the design of the blades  20  to alter the shock structure and reduce boundary layer accumulation, both of which may contribute to improvement in performance and increased stall margin. The stall initiation point may be defined as the point at which the compressor section  14  can no longer sustain an increase in pressure such that the gas turbine engine  10  stalls. 
     Having a blade that is swept forward may provide several benefits to the tip  27 . First, in terms of shock, the forward sweep may affect bow shock by sweeping the leading edge  36  while the passage shock is altered via a change in the shock location. The forward sweep thus may cause the shock to become more swallowed, which in turn, may increase the stall margin. 
     Second, increased flow toward the tip  27  may subject the tip  27  more toward negative incidence, reduce front loading and may reduce tip clearance flow. As mentioned above, tip clearance is the space defined between the tip  27  of the blade and the compressor casing  22 . The portion of the flow of air  28  which escapes from the flow channel  29  through the tip clearance may reduce the ability of the compressor section  14  to sustain pressure rise, increase downstream flow blockage and may reduce its stall margin. The downstream blades  20  may have an increased tip clearance relative to the upstream blades  20  which may increase tip clearance flow. 
     Third, forward sweep at the tip  27  may allow the tip  27  to “grab” flow sooner than other section resulting in lower axial diffusion and less boundary layer accumulation. 
     Fourth, because of the centrifugal effects produced by the rotor, there may be a migration of secondary flow along blade&#39;s  20  surface from the hub  21  to tip  27 , which may result in a thick tip boundary layer build up. While secondary flow can be affected by radial loading, any secondary flow migrating from hub  21  to tip  27  may also be reduced with forward sweep as it will likely be swept downstream before reaching tip  27 . 
     Fifth, in a multistage environment such as the one partially shown in  FIG. 2 , sweep may improve overall efficiency with improved interaction between rows of blades  20 . 
     The blade  20  having a forward sweep, flow has a positive incidence reduced compared to a blade with lesser or no forward sweep. While lesser positive incidence may improve stall margin, it may reduce flow chocking because of a reduction in effective area seen by flow. In a multistage compressor such as the one of  FIG. 2 , the throat area may be adjusted to counter this effect. Having an airfoil portion  26  that is leaned may also provide benefits to the tip  27 . Blade&#39;s lean may reduce the acute suction side dihedral angle. Blade lean may be as effective as forward sweep in reducing shock/boundary layer/tip clearance interaction. 
     A combination of sweep and blade lean may thus be adopted. In a multistage environment, certain physical spacing is required between blade rows for structural reasons. Unless the compressor length can be increased to accommodate a forward swept blade (at the expense of engine weight and cost) this imposes a limitation on how much forward sweep a rotor can employ. Thus to maximize the benefit of sweep/lean in a confined axial space, blade lean may be maximized (provided rotor remains structurally acceptable). 
     The airfoil portion  26  described herein is shaped to accommodate the structural limitations imposed by the design of the compressor section  14  while aiming at reducing at least some of the losses induced by the flow around the airfoil portions  26 . As a result, the airfoil portions  26  presented herein may have, among other design features presented below, a ratio of the sweep angle α over the dihedral angle β may be below 1. According to an embodiment, the ratio may be comprised between 0 and 1. The blade  20  shown herein may also have a dihedral tip with a reverse direction, and/or an axial component of a center of gravity of a cross-section taken chordally toward the tip of the airfoil being upstream relative to an axial component of a center of gravity of a cross-section taken chordally toward the root of the airfoil. 
     Turning now to  FIGS. 4 to 8 , several parameters defining the airfoil portion  26  will be discussed.  FIGS. 4 to 8  show only one example of parameters for the airfoil portion  26 . 
     In  FIG. 4 , the sweep angle α of the leading edge  36  of the airfoil portion  26  is plotted against the span Sp at the leading edge  36  of the airfoil portion  26 . From 0 to about 75% of the span Sp, the airfoil portion  26  is swept rearward (sweep angle α is positive). From about 75% to the tip  27 , the airfoil portion  26  is swept forward (sweep angle α is negative). The increase of sweep along the span Sp is monotonic. A transition between rearward and forward sweep may depend on the application of the rotor  19 , among many parameters rotor Mach number and pressure ratio it has to produce. In one embodiment, the forward sweep S could be between 5 and 45% of the span Sp of the airfoil portion  26 . In one embodiment, the forward sweep S could be between 10 and 35% of the span Sp of the airfoil portion  26 . 
     Forward sweep for transonic rotors may reduce secondary flow migration from the hub  21  to the tip  27 . As a result, at the tip  27 , there is less mixing loss due to interaction between the tip leakage flow, shock and secondary flow. Lower mixing losses induce lower flow blockage which could lead to improve flow capacity at high speeds. Also, forward sweep may pull flow toward the tip  27  and as a result improves rotor stall margin at both high and low (part) speeds. 
     Forward sweep for subsonic rotors may also reduce secondary flow migration from the hub  21  to the tip  27 , and mixing losses due to interaction between tip leakage flow with secondary flow. The rotor  19  may thus be less sensitive to tip clearance increase. Forward sweep may pull flow toward the tip  27  and as a result improves rotor stall margin at both high and low (part) speeds. 
     In  FIG. 5 , the dihedral angle β of the leading edge  36  of the airfoil portion  26  is plotted against the span Sp at the leading edge  36  of the airfoil portion  26 . The dihedral angle β is a positive dihedral angle, decreasing from the hub  21  (i.e. 0 on the span Sp axis) to the tip  27  (i.e. 1 on the span Sp axis). The dihedral angle β is about 72 degrees at the hub  21  and about 42 degrees at the tip  27 . The dihedral angle β may be comprised between 40 and 70 degrees. While the dihedral angle β is shown herein to decrease from hub  21  to tip  27 , it could instead increase. 
     Opposite to the sweep angle α, the dihedral angle β in this example, does not evolve monotonically along the span Sp at the leading edge  36 . In the embodiment shown in  FIG. 5 , a direction of the dihedral angle β is reversed locally twice near the tip  27 , i.e. the dihedral angle β span-wise distribution has a first inflection point P 1  and a second inflection point P 2 . From 0% span Sp to about 80% span Sp (point P 1 ), the dihedral angle β decreases. From about 80% span Sp to about 95% span Sp (point P 2 ), the dihedral angle β increases, and from 95% span Sp to 100% span Sp, the dihedral angle β decreases again. Under centrifugal force and thermal effects, the rotor  19  may expand radially. Large amounts of dihedral angle β may cause the blade  20  to rub into the casing  22  during high speed conditions. Having at least one inflection point P 1  may affect a rub angle μ at the tip  27  of the blade  20  with the casing  22 , and may be beneficial for avoiding additional deflection of the airfoil portion  26  when the blade  20  is rubbing into the casing  22 .  FIG. 8  shows a position of the tip  27  of the airfoil portion  26  relative to the casing  22 . The casing  22  is illustrated in  FIG. 8  by a line TSh tangent to the casing  22 , and the rub angle μ may be defined between a tangent Tps to the pressure side  32  and the tangent TSh to the casing  22 . While in the particular illustration of  FIG. 8 , the tangent TSh to the casing  22  seems to match a tangent to the tip  27 , it is contemplated that the tangent to the tip  27  could be at an angle with the tangent TSh to the casing  22 . Computations have determined that the rub angle μ is decreased when the blade  20  includes at least one inflection point in the dihedral angle β span-wise distribution. While the particular illustration of  FIG. 8  shows schematically a rub angle μ lesser than 90 degrees, it is contemplated that the rub angle μ could be 90 or greater than 90 degrees. 
     By having a non-monotonic decrease (or in other embodiments non-monotonic increase) of the dihedral angle β toward the casing  22 , the rub angle μ may be decreased which in turn may decrease damages or force resulting from the rubbing are decreased. In other terms, decreasing rub angle may decrease the risk of damage to the casing  22  (i.e. adrabable) during a rub by reducing the extent to which the blade elongates as a result of plowing into the casing  22  during a rub. In turn, the compressor section  14  may become more efficient. Computational Fluid Dynamics analyses supported by rig/engine test data have shown that the change to surge margin and performance may be insignificant with this type of dihedral angle β distribution. 
     A second inflection point P 2  in the dihedral angle β span-wise distribution may be used to obtain a more optimised rub angle μ than would the blade  20  have with the first inflection point P 1  only. The second inflection point P 2  may be omitted and airfoil portion  26  may have only one inflection point in the dihedral angle β span-wise distribution. The dihedral angle β span-wise distribution may also have more than two inflection points. While the inflections in dihedral angle β span-wise distribution are shown in  FIG. 5  to take place at the tip  27 , it is contemplated that the inflections could be taking place mid-span or toward the hub  21 . The inflections of the dihedral angle β span-wise distribution is shown herein applied to a blade having a ratio of sweep angle α over dihedral angle β below 1, but it is contemplated that the inflection(s) of the dihedral angle β span-wise distribution could be applied to a variety of blades not bounded to the above ratio of sweep angle α over dihedral angle β. In one embodiment, the inflection(s) of the dihedral angle β span-wise distribution may occur between 5 to 10% of the span Sp. 
     The high tip dihedral angle β may increase tensile stress at the hub  21  on the pressure side  32  of the airfoil portion  26  and compression stress on the suction side  34  at the hub  21  of the airfoil portion  26 . As discussed below with reference to  FIGS. 10 and 11 , a thickness distribution along cross-sections of the blade  20  may be determined to at least reduce these compressive stresses and tension stresses. 
       FIG. 6  shows a distribution of the leading edge dihedral angle β relative to the leading edge sweep angle α. The distribution shows two inflection points P 3 , P 4  which correspond to the two inflection points P 1 , P 2  of the dihedral angle β span-wise distribution. 
       FIG. 7  shows a ratio of the sweep angle α over the dihedral angle β as a function of the span Sp of the airfoil portion  26 . The ratio decrease monotonically from the hub  21  to the tip  27 . The ratio is below 0.5. In other embodiments, the ratio could be below 1. It is contemplated that the ratio could not be monotonic. This graph shows that the dihedral angle β is always larger than the sweep angle α for the blade  20  described herein. This ratio is in response to spatial limitations in the rotor  19 , where forward sweep is limited, as described above. 
       FIG. 9  shows the center of gravity CG of the airfoil portion  26  at different sections along the span Sp of the airfoil portion  26 . The center of gravity CG can be projected onto an axial axis (i.e. parallel to the engine axis  11 ) (Xcg) and a radial axis R (i.e. perpendicular is the engine axis  11 ) (Ycg). Xcg and Ycg are defined as the axial distance and the radial distance of the center of gravity CG of any particular span-wise section. The radial axis R is shown in  FIG. 2 . Xcg represents axial sweep, while Ycg represents tangential lean. The abscises axis of the plot represents a distance (in inches) of the center of gravity CG of a given cross-section relative to an arbitrary 0. The ordinates axis of the plot represents a position (normalised) of that given cross-section along the span Sp. Positive values on the abscises axis of the plot correspond to deviation of the center of gravity CG in the direction of flow  28 , and negative values on the abscises axis of the plot correspond to deviation of the center of gravity CG in a direction opposite to the direction of flow  28 . 
     Referring more specifically to the Xcg distribution, the Xcg of the airfoil portion  26  at the hub  21 , Xcg-hub, is disposed downstream relative to the Xcg of the airfoil portion  26  at the tip  27 , Xcq-tip. In the example shown in  FIG. 9 , the Xcg at the hub  21  (Xcg-hub) is between 0.05 and 0.1 inch on the axial axis  11 , while the Xcg at the tip  27  (Xcg-tip) is at 0.1 inch on the axial axis  11 . Starting at about 65% of the span Sp of the airfoil portion  26 , any point of the airfoil portion  26  above that is disposed upstream. The Xcg distribution shows that the Xcg at the hub  21  is downstream relative to the Xcg at the tip  27  is a consequence of the forward sweep imparted to the blade  20 . It is contemplated that blades other than the blade  20  could have such Xcg distribution. For example, radial or backward sweep rotors with hub section thicken further to the rear could have this distribution. The dashed line at 0.33 of the span is the location along the span (radial location or section) where the values of Xcg and Ycg are approximately the same. 
     Referring to  FIGS. 10 and 11 , one way to achieve the above feature of the Xcg distribution is to thicken the airfoil portion  26  toward the hub  21 . 
     Referring more specifically to  FIG. 10 , a schematic cross-section CS-hub of the airfoil portion  26  toward the hub  21  (solid line) is shown superimposed with a schematic cross-section CS-tip of the airfoil portion  26  toward the tip  27  (dotted line). A thick portion of the airfoil portion  26  may be defined by a portion along the chord Ch of a given cross-section of the airfoil portion  26  for which the thickness is at least 85% of a maximum thickness of that airfoil portion  26 , and which spans chordwise between at most between +20% and −20% from the maximum thickness. If the maximum thickness extends over an area of the chord Ch, a chordwise center of the area of maximum thickness may be used as a reference point for determining the chordwise span of the thick portion. The maximum thickness may be located at 50% of the chord Ch for a given cross-section or at a different location on that chord. 
     For the two cross-sections shown in  FIG. 10 , a thick portion ThickP_hub of the hub&#39;s cross-section CS-hub may be defined by a portion of the airfoil portion  26  along the chord Ch for which the thickness is at least 85% of the maximum thickness T_maxhub of the hub&#39;s cross-section CS-hub and which spans chordwise between +/−20% from the maximum thickness T_maxhub. Similarly, a thick portion ThickP_tip of the tip&#39;s cross-section CS-tip may be defined by a portion of the airfoil  26  along the chord Ch for which the thickness is at least 85% of the maximum thickness T_maxtip of the tip&#39;s cross-section CS-tip and spans chordwise +/−15% from the maximum thickness T_maxtip. 
     It is contemplated that the thick portion of the airfoil portion  26  could be defined by a portion of the airfoil  26  along the chord Ch of a given cross-section of the airfoil portion  26  for which the thickness is about 85% of the maximum thickness of that airfoil portion  26 , for example, 80%, 90% or even 92%. It is also contemplated that the thick portion could extend chordwise to less than +/−15% from the maximum thickness. For example, the thick portion could extend +/−10% chordwise from the maximum thickness. 
     While the cross-section CS-tip, shown herein, has a more convention airfoil shape with a thick portion being short and disposed toward the leading edge  36 , the cross-section CS-hub has the thick portion ThickP_hub extending along a longer portion of the chord Ch toward the trailing edge  38 . As a result, a center of gravity CG-hub is disposed axially downstream at Xcg-hub relative to the center of gravity CG-tip. 
     Referring now more specifically to  FIG. 11 , a thickness distribution of the cross-sections CS-hub and CS-tip are plotted along with baseline thickness distributions of the cross-sections at the hub  21  and at the tip  27 . The abscises axis of the plot represents a position along a non-normalised chord Ch and the ordinates axis a thickness of the cross-sections. 
     The plots show that the cross-section CS-hub is globally thicker than the cross-section CS-tip, with a maximum thickness T_max_hub at the hub  21  being more than twice a maximum thickness T_max_tip at the tip  27 . A distribution of the thickness at the hub  21  has been modified compared to a baseline to provide the Xcg distribution described above. In one embodiment, the thick portion ThickP_hub extends along a portion of the chord Ch comprised around between 30% and 60% of the chord Ch. In another example, the thick portion ThickP_hub extends along a portion of the chord Ch comprised around between 45% and 65% of the chord Ch. In comparison, the thick portion ThickP_tip extends along a portion of the chord Ch comprised around between 30% and 45% of the chord Ch. By designing the blade  20  with a longer thick portion ThickP_hub at the hub  21  compared to the tip  27 , the center of gravity CG-hub is disposed axially downstream relative to the center of gravity CG-tip. In another example, the thick portion ThickP_hub extends between −15% and +15% of the chord Ch percentage where the maximum thickness T_maxhub is found. The maximum thickness T_maxhub may or may not be at 50% of the chord Ch. 
     The above thickness distribution may improve performance of the gas turbine engine  10  since frontal blockage is minimized. In addition to minimize radial flow migration hub front turning is minimized. Reduction in front turning could result in small flow area. It is thus desirable to minimize frontal thickness to have maximum flow area while more thickness is added rearward to keep root stress to acceptable level. This Xcg distribution, thus, may allow more freedom to optimize the airfoil surface curvature distribution to achieve a radial pressure distribution that can result in reduced secondary flow migration. The changes in shapes of the cross-sections CS-hub to CS-tip may be done smoothly from the hub  21  to the tip  27  by decreasing smoothly (linearly or not) the length of the thick portion. 
     The above description is meant to be exemplary only, and one skilled in the art will recognize that changes may be made to the embodiments described without departing from the scope of the invention disclosed. The shapes of the airfoils described herein could be used in high speed rotors as well as in low speed rotors, and could be used in rotors that are not part of a compressor section of a gas turbine engine. The shapes of the airfoils described herein are not limited to transonic rotors. In the absence of shocks, as in subsonic designs, for rear stages of multistage compressor, both forward sweep and lean may be degrees of freedom that allow to design a blade as described above. Still other modifications which fall within the scope of the present invention will be apparent to those skilled in the art, in light of a review of this disclosure, and such modifications are intended to fall within the appended claims.