Airfoil shape for arrays of airfoils

A method for forming an airfoil section of a plurality of airfoil sections which are circumferentially spaced a distance tau .tau. apart about a rotor axis each of the airfoil sections having an inlet metal angle .beta..sub.1 *, a total camber angle .theta..sub.t *, an alpha chord angle .alpha..sub.ch, a maximum thickness t.sub.max, a leading edge, a trailing edge, a tangent line TL passing through the leading edge tangent to the path of rotation, a front chord line B.sub.f of length b.sub.f, a conical chord line B.sub.t of length b.sub.t wherein the values of .beta..sub.1 *, .theta..sub.t *, .tau., b.sub.t, the maximum thickness of the airfoil section t.sub.max, are known, comprising the steps of:

• A. establishing a cambered meanline having a concave side and a convex side and having a first arc, a second arc and a transition point TP between the first arc and the second arc, the first arc being tangent to the second arc at said transition point TP by
• Aa. determining an initial value for the alpha chord angle (.alpha..sub.chi) which is equal to the sum of the inlet metal angle (.beta..sub.1 *) and one-half of the total camber angle (.theta..sub.t *)/=, (.alpha..sub.chi =.beta..sub.1 *+.theta..sub.t *)/2,
• Ab. setting the value of the alpha chord angle chi (.alpha..sub.ch =.alpha..sub.chi),
• Ac. determining a distance l from the tangent line TL to the first covered section as measured along the conical chord line B.sub.t, the distance l being equal to the distance tau .tau. multiplied by the quantity the sine of the angle ninety degrees minus the alpha cord angle (l=.tau..sin(90-.alpha..sub.ch),
• Ad. determining the normalized distance L.sub.fcs to the first covered section by dividing the distance l by the distance b.sub.t,
• Ae. obtaining the raio of the length b.sub.f of the front chord line B.sub.f to the length b.sub.t of the conical chord line B.sub.t (b.sub.f /b.sub.t) and the ratio of the front camber angle (.theta..sub.f *) to the total camber angle .theta..sub.t *(.theta..sub.f */.theta..sub.t *) from FIG. 6 at the value L.sub.fcs of the normalized distance to the first covered section,
• Af. establishing the location of the first arc such that the arc passes through the leading edge using the values known (b.sub.t, .theta..sub.t *, .beta..sub.1 *) and the value found in step Ae. for b.sub.f and .theta..sub.f *,
• Ag. establishing the location of the second arc such that the arc passes through the trailing edge using the values known (b.sub.t, .theta..sub.t *, .beta..sub.1 *) and values found in step Ae. for b.sub.f, .theta..sub.f *,
• Ah. establishing a conical chord line B.sub.t extending between the leading edge and the trailing edge,
• Ai. determining the actual alpha chord angle .alpha..sub.cha for the cambered meanline,
• Aj. determining the difference E between the actual alpha chord angle .alpha..sub.cha and the alpha chord angle .alpha..sub.ch used to calculate the normalized location L.sub.fcs by substracting .alpha..sub.ch from .alpha..sub.cha (E=.alpha..sub.cha -.alpha..sub.ch),
• Ak. proceeding to step B if the absolute value of E is less than the predetermined value e (.vertline.E.vertline.<e) and proceeding to step Am if the absolute value of E is greater than or equal to the predetermined value e (.vertline.E.vertline..gtoreq.e),
• Al. setting the value of the alpha chord angle .alpha..sub.ch equal to the value .alpha..sub.cha (.alpha..sub.ch =.alpha..sub.cha), Am. repeating steps Ac through Aj.
• B. establishing a thickness distribution TD formed of two parts each part being disposed about the conical chord line B.sub.t, each part having a line spaced Tzn from the conical chord line B.sub.t at any point zn, the point zn being spaced a distance Lan from the leading edge on the conical chord line B.sub.t, the distance Tzn being measured along a line Zn perpendicular to the conical chord line B.sub.t, the line of the first part being TD.sub.1 and the line of the second part being TD.sub.2,
• Ba. the line of the first part TD.sub.1 being established by
• Bal. determining the distance loc mt along the conical chord line to the location TMAX of maximum thickness t.sub.max by determining the ratio loc mt/b.sub.t from FIG. 6 at the value L.sub.fcs of the normalized distance to the first covered section,
• Ba2. superimposing on the conical chord line B.sub.t a circle T.sub.max having a center on the conical chord line a distance equal to loc mt from point A and a radius R.sub.tmax equal to one-half of the maximum thickness t.sub.max of the airfoil section (R.sub.tmax =t.sub.max/2).
• Ba3. establishing on the conical chord line B.sub.t a leading edge radius circle having a radius R.sub.ler and a center on B.sub.t a distance equal to R.sub.ler from the leading edge and intersecting the leading edge at a point A, the radius R.sub.ler being equal to the quantity eighteen hundred and fifty-two ten thousandths (0.1852) multiplied by the maximum thickness t.sub.max (R.sub.ler =0.1852.t.sub.max),
• Ba4. establishing a line Q perpendicular to the conical chord line B.sub.t at a point which is a distance b.sub.f (Lan=b.sub.f) from the leading edge,
• Ba5. establishing a line F having a radius of curvature R.sub.f which is tangent to the leading edge circle at a point fl, tangent to the circle T.sub.max and which intersects the line Q at a point fq,
• Ba6. establishing a line P perpendicular to the conical chord line B.sub.t at a point which is a distance Lan equal to the quantity thirty-five thousandths multiplied by the length b.sub.t of the conical chord line (Lan=0.035b.sub.t) from the leading edge and which intersects the line F at a point fe,
• Ba7. passing the line TD.sub.1 of the first part through the points A, fe and fq such that the line of the first part is tangent to the leading edge radius circle at point A, tangent to the line F at point fe and coincident with line F between the points fe and fq,
• Bb. the line of the second part TD.sub.2 being established by
• Bb1. determining the quantity TERG from FIG. 6 at the value L.sub.fcs of the normalized distance to the first covered section and determining the radius R.sub.ter which is equal to the quantity TERG multiplied by four hundred and sixty-three thousandths (0.463) and by t.sub.max (R.sub.ter =TERG. 0.463.t.sub.max),
• Bb2. establishing on the conical chord line B.sub.t a trailing edge radius circle having a radius R.sub.ter and a center on B.sub.t spaced a distance equal to R.sub.ter from the trailing edge and intersecting the trailing edge at a point C,
• Bb3. establishing a line G having a radius of curvature Rg which is tangent to the trailing edge radius circle at a point gt and which is tangent to the line F at the point fq,
• Bb4. passing the line of the second part TD.sub.2 through the points C, gt and fq, such that the line of the second part is coincident with the trailing edge radius circle between the points C and gt and coincident with the line G between the points gt and fq;
• C. superimposing the thickness distribution on the cambered meanline by
• Ca. establishing a plurality of points zn', each point zn' being at the intersection of the line Zn and the cambered meanline,
• Cb. establishing a line Z'n perpendicular to the cambered meanline at each point zn',
• Cc. establishing a point zn" at a distance Tzn as measured along the line Z'n from the convex side of the cambered meanline at each point zn' and a point zn'" at a distance Tzn as measured along the line Z'n from the concave side of the cambered meanline at each point zn',
• Cd. establishing a concave surface passing through the leading edge and the trailing edge and through all points zn",
• Ce. establishing a convex surface passing through the leading edge and the trailing edge and through all points zn'";
• D. forming an airfoil section having a cambered meanline, a convex surface and a concave surface as established in steps A, B and C;
• wherein the thickness distribution is stretched chordwisely on the convex side and compressed on the concave side to form an airfoil section having desirable separation characteristics in a transonic aerodynamic flow field.

In a gas turbine engine of the type having an array of airfoils the array being formed of arrays of conical airfoil sections each array of conical airfoil sections having an airfoil section spaced circumferentially from a circumferentially adjacent airfoil section a distance tau (.tau.) leaving a gap therebetween, said airfoil section having a leading edge, a trailing edge, a front camber angle (.theta..sub.f *), a total camber angle .theta..sub.t *, an inlet metal angle (.beta..sub.1 *), an exit metal angle (.beta..sub.2 *), a maximum thickness t.sub.max, a length to the location of maximum thickness (loc mt), a cambered double circular arc meanline terminating at the leading edge and the trailing edge, and having a conical chord B.sub.t extending between the leading edge and the trailing edge having a length b.sub.t, an alpha chord angle (.alpha..sub.ch) between B.sub.t and a tangent line TL passing through the leading edge, a front chord having a length b.sub.f, and a gap to chord ratio (.tau./b.sub.t), the improvement which comprises:

• a conical airfoil section having
• a convex surface;
• a concave surface joined to the convex surface at the leading edge and the trailing edge;
• wherein the ratio of the front chamber angle .theta..sub.f * to the total chamber angle .theta..sub.t * is related to both the alpha chord angle .alpha..sub.ch and the gap to chord ratio .tau./b.sub.t by curve .theta..sub.f */.theta..sub.t * of FIG. 6;
• wherein the ratio of the length b.sub.f of the front chord B.sub.f to the length b.sub.t of the conical B.sub.t is related to both the alpha chord angle .alpha..sub.ch and the gap to chord ratio .tau./b.sub.t by curve b.sub.f /b.sub.t of FIG. 6;
• wherein the ratio between the length loc mt to the location of maximum thickness and the length b.sub.t of the conical chord B.sub.t is related to both the alpha chord angle .alpha..sub.ch and the gap to chord ratio .tau./b.sub.t by curve loc mt/b.sub.t of FIG. 6;
• wherein the concave surface of the airfoil section and the convex surface of the airfoil section are each spaced a distance Tzn from any point zn' perpendicular to the chambered meanline;
• and
• wherein the distance Tzn is defined by a thickness distribution TD formed of two parts generated about the conical chord line B.sub.t, each part at any point zn having a line spaced the distance Tzn from the conical chord line B.sub.t as measured along a line Zn perpendicular to the conical chord line B.sub.t passing through the point zn' and a point zn, the point zn being spaced a distance Lan from a point A on the leading edge along the conical chord line B.sub.t, the line of the first part being TD.sub.1 and the line of the second part being Td.sub.2 such that
• A. the line TD.sub.1 of the first part
• Al. intersects the leading edge at the point A,
• A2. is tangent at the point A to a circle passing through the point A the circle having a center on the conical chord line B.sub.t, and a radius R.sub.ler, the radius R.sub.ler being equal to the quantity eighteen hundred and fifth-two thoundandths (0.1852) multipled by the maximum thickness t.sub.max of the airfoil (R.sub.ler =0.1852.t.sub.max),
• A3. is tangent to a circle having a center at the location of maximum thickness TMAX on B.sub.t a distance loc mt from the point A (Lan=loc mt) and having a radius (R.sub.tmax equal to one half of the maximum thickness t.sub.max of the airfoil section (R.sub.tmax =t.sub.max /2),
• A4. is coincident with a line F at a point fe, the line F being tangent to the circle having a radius R.sub.ler at a point fl, being tangent to the circle T.sub.max and having a radius of curvature R.sub.f, the point fe being spaced from point A as measured along the conical chord line B.sub.t a distance equal to the quantity thirty-five thousandths multiplied by the distance b.sub.t (Lan=La.epsilon.=0.035.b.sub.t),
• A5. terminates at a point fq, the point fq being the point of intersection between the line of the first part TD.sub.1 and a line Q, the line Q being perpendicular to the conical chord line B.sub.t at a point which is a distance b.sub.f (Lan=b.sub.f) from the leading edge, and
• A6. has a radius of curvature R.sub.f between the point fe and the point fq;
• and
• B. the line TD.sub.2 of the second part
• B1. is tangent to the line of the first part at the point fq,
• B2. extends from the point fq having a radius of curvature Rg,
• B3. is tangent at a point gt to a circle passing through a point C on the trailing edge the circle having a center on the conical chord line B.sub.t and a radius R.sub.ter, the radius R.sub.ter being equal to the quantity TERG multiplied by four hundred and sixty-three thousandths and multiplied by the maximum thickness of the airfoil t.sub.max (R.sub.ter =TERG.0.463.t.sub.max),
• B4. is coincident with the circle having the radius R.sub.ter between the point gt and the point C.

A rotor blade having one or more airfoil sections, each airfoil section being one of a plurality of airfoil sections which are circumferentially spaced a distance tau (.tau.) apart about a rotor axis, each airfoil section having an inlet metal angle .beta..sub.1 *, a total camber angle .theta..sub.t *, an alpha chord angle .alpha..sub.ch, a maximum thickness t.sub.max, a leading edge, a trailing edge, a tangent line TL passing through the leading edge tangent to the path of rotation, a front chord line B.sub.f of length b.sub.f, and a conical chord line B.sub.t of length b.sub.t wherein the values of B.sub.1 *, .theta..sub.t *, .tau., b.sub.t, the maximum thickness of the airfoil section t.sub.max, are known, the rotor blade having one or more airfoil section geometries determined by the method steps of:

• A. establishing a cambered meanline having a concave side and a convex side and having a first arc, a second arc and a transition point TP between the first arc and the second arc, the first arc being tangent to the second arc at said transition point TP by
• Aa. determining an initial value for the alpha chord angle (.alpha..sub.chi) which is equal to the sum of the inlet metal angle (.beta..sub.1 *) and one-half of the total camber angle (.theta..sub.t */2), (.alpha..sub.chi =.beta..sub.1 *+.theta..sub.t */2),
• Ab. setting the value of the alpha chord angle .alpha..sub.chi (.alpha..sub.ch =.alpha..sub.chi),
• Ac. determining a distance l from the tangent line TL to the first covered section as measured along the conical chord line B.sub.t, the distance l being equal to the distance tau .tau. multiplied by the quantity the sine of the angle ninety degrees minus the alpha chord angle (l=.tau..sin(90-.alpha..sub.ch),
• Ad. determining the normalized distance L.sub.fcs to the first covered section by dividing the distance l by the distance b.sub.t,
• Ae. obtaining the ratio of the length b.sub.f of the front chord line B.sub.f to the length b.sub.t of the conical chord line B.sub.t (b.sub.f /b.sub.t) and the ratio of the front camber angle (.theta..sub.f.sup.*) to the total camber angle .theta..sub.t * (.theta..sub.f */.theta..sub.t *) as a function of the value L.sub.fcs of the normalized distance to the first covered section,
• Af. establishing the locations of the first arc such that the arc passes through the leading edge using the values known (b.sub.t, .theta..sub.t *, .beta..sub.1 * ) and the value found in step Ae. for b.sub.f and .theta..sub.f *,
• Ag. establishing the location of the second arc such that the arc passes through the trailing edge using the values known (b.sub.t, .theta..sub.t *, .beta..sub.1 *) and values found in step Ae. for b.sub.f, .theta..sub.f.sup.*,
• Ah. establishing a conical chord line B.sub.t extending between the leading edge and the trailing edge,
• Ai. determining the actual alpha chord angle .alpha..sub.cha for the cambered meanline,
• Aj. determining the difference E between the actual alpha chord angle .alpha..sub.cha and the alpha chord angle .alpha..sub.ch used to calculate the normalized location L.sub.fcs by subtracting .alpha..sub.ch from .alpha..sub.cha (E=.alpha..sub.cha -.alpha..sub.ch),
• Ak. proceeding to step B if the absolute value of E is less than the predetermined value e (.vertline.E.vertline.<e) and proceeding to step Am if the absolute value of E is greater than or equal to the predetermined value e (.vertline.E.vertline..gtoreq.e),
• Al. setting the value of the alpha chord angle .alpha..sub.ch equal to the value .alpha..sub.cha (.alpha..sub.ch =.alpha..sub.cha),
• Am. repeating steps Ac through Aj,
• B. establishing a thickness distribution TD having a line spaced a distance Tzn from the conical chord line B.sub.t at any point zn, the point zn being spaced a distance Lan from the leading edge on the conical chord line B.sub.t, the distance Tzn being measured along a line Zn perpendicular to the conical chord line B.sub.t ;
• C. superimposing the thickness distribution on the cambered meanline by
• Ca. establishing a plurality of points zn', each point zn' being at the intersection of the line Zn and the cambered meanline,
• Cb. establishing a line Z'n perpendicular to the cambered meanline at each point zn',
• Cc. establishing a point zn" at a distance Tzn as measured along the line Z'n from the convex side of the cambered meanline at each point zn' and a point zn"' at a distance Tzn as measured along the line Z'n from the concave side of the cambered meanline at each point zn',
• Cd. establishing a concave surface passing through the leading edge and the trailing edge and through all points zn",
• Ce. establishing a convex surface passing through the leading edge and the trailing edge and through all points zn"'.

A rotor blade having one or more airfoil sections, each airfoil section being one of a plurality of airfoil sections which are circumferentially spaced a distance tau (.tau.) apart about a rotor axis each of the airfoil sections having an inlet metal angle .beta..sub.1 *, a total camber angle .theta..sub.t *, an alpha chord angle .alpha..sub.ch, a maximum thickness t.sub.max, a leading edge, a trailing edge, a tangent line TL passing through the leading edge tangent to the path of rotation, a front chord line B.sub.f of length b.sub.f, a conical chord line B.sub.t of length b.sub.t wherein the values of .beta..sub.1 *, .theta..sub.t.sup.* , .tau., b.sub.t, the maximum thickness of the airfoil section t.sub.max, are known, the rotor blade having one or more airfoil section geometries determined by the method steps of:

• A. establishing a cambered meanline having a concave side and a convex side and having a first arc, a second arc and a transition point TP between the first arc and the second arc, the first arc being tangent to the second arc at said transition point TP by
• Aa. determining an initial value for the alpha chord angle (.alpha..sub.chi) which is equal to the sum of the inlet metal angle (.beta..sub.1 *) and one-half of the total camber angle (.theta..sub.t */2), (.alpha..sub.chi =.beta..sub.1 * +.theta..sub.t */2),
• Ab. setting the value of the alpha chord angle .alpha..sub.chi (.alpha..sub.ch =.alpha..sub.chi),
• Ac. determining a distance l from the tangent line TL to the first covered section as measured along the conical chord line B.sub.t, the distance l being equal to the distance tau .tau. multiplied by the quantity the sine of the angle ninety degrees minus the alpha chord angle (l=.tau..sin(90-.alpha..sub.ch),
• Ad. determining the normalized distance L.sub.fcs to the first covered section by dividing the distance l by the distance b.sub.t,
• Ae. obtaining the ratio of the length b.sub.f of the front chord line B.sub.f to the length b.sub.t of the conical chord line B.sub.t (b.sub.f /b.sub.t) and the ratio of the front camber angle (.theta..sub.f.sup.*) to the total camber angle .theta..sub.t *(.theta..sub.f */.theta..sub.t *) from FIG. 6 at the value L.sub.fcs of the normalized distance to the first covered section.
• Af. establishing the location of the first arc such that the arc passes through the leading edge using the values known (b.sub.t, .theta..sub.t *, .beta..sub.1 *) and the value found in step Ae. for b.sub.f and .theta..sub.f,
• Ag. establishing the location of the second arc such that the arc passes through the trailing edge using the values known (b.sub.t, .theta..sub.t *, .beta..sub.1 *) and values found in step Ae. for b.sub.f, .theta..sub.f *,
• Ah. establishing a conical chord line B.sub.t extending between the leading edge and the trailing edge,
• Ai. determining the actual alpha chord angle .alpha..sub.cha for the cambered meanline,
• Aj. determining the difference E between the actual alpha chord angle .alpha..sub.cha and the alpha chord angle .alpha..sub.ch used to calculate the normalized location L.sub.fcs by substracting .alpha..sub.ch from .alpha..sub.cha (E=.alpha..sub.cha -.alpha..sub.ch),
• Ak. proceeding to step B is the absolute value of E is less than the predetermined value e (.vertline.E.vertline.<e) and proceeding to step Am if the absolute value of E is greater than or equal to the predetermined value e (.vertline.E.vertline..gtoreq.e),
• Al. setting the value of the alpha chord angle .alpha..sub.ch equal to the value .alpha..sub.cha (.alpha..sub.ch =.alpha..sub.cha),
• Am. repeating steps Ac through Aj.
• B. establishing a thickness distribution TD formed of two parts each part being disposed about the conical chord line B.sub.t, each part having a line spaced Tzn from the conical chord line B.sub.t at any point zn, the point zn being spaced a distance Lan from the leading edge on the conical chord line B.sub.t, the distance Tzn being measured along a line Zn perpendicular to the conical chord line B.sub.t, the line of the first part being TD.sub.1 and the line of the second part being TD.sub.2,
• Ba. the line of the first part TD.sub.1 being established by
• Ba1. determining the distance loc mt along the conical chord line to the location TMAX of maximum thickness t.sub.max by determining the ratio loc mt/b.sub.t as a function of the value L.sub.fcs of the normalized distance to the first covered section,
• Ba2. superimposing on the conical chord line B.sub.t a circle T.sub.max having a center on the conical chord line a distance equal to loc mt from point A and a raduis R.sub.tmax equal to one-half of the maximum thickness t.sub.max of the airfoil section (R.sub.tmax =T.sub.max /2).
• Ba3. establishing on the conical chord line B.sub.t a leading edge radius circle having a radius R.sub.ler and a center on B.sub.t a distance equal to R.sub.ler from the leading edge and intersecting the leading edge at a point A, the radius R.sub.ler being equal to a first constant k multiplied by the maximum thickness t.sub.max (R.sub.ler =k . t.sub.max),
• Ba4. establishing a line Q perpendicular to the conical chord line B.sub.t at a point which is a distance b.sub.f (Lan=b.sub.f) from the leading edge,
• Ba5. establishing a line F having a radius of curvature R.sub.f which is tangent to the leading edge circle at a point fl, tangent to the circle T.sub.max and which intersects the line Q at a point fq,
• Ba6. establishing a line P perpendicular to the conical chord line B.sub.t at a point which is a distance Lan equal to a second constant k.sub.2 multiplied by the length b.sub.t of the conical chord line (Lan=k.sub.2 . b.sub.t) from the leading edge and which intersects the line F at a point fe,
• Ba7. passing the line TD.sub.1 of the first part through the points A, fe and fq such that the line of the first part is tangent to the leading edge radius circle at point A, tangent to the line F at point fe and coincident with line F between the points fe and fq,
• Bb. the line of the second part TD.sub.2 being established by
• Bb1. determining the quantity TERG as a function of the value L.sub.fcs of the normalized distance to the first covered section and determining the radius R.sub.ter which is equal to the quantity TERG multiplied by a third constant k.sub.3 and by t.sub.max (R.sub.ter =TERG.0.463.t.sub.max),
• Bb2. establishing on the conical chord line B.sub.t a trailing edge radius circle having a radius R.sub.ter and a center on B.sub.t spaced a distance equal to R.sub.ter from the trailing edge and intersecting the trailing edge at a point C,
• Bb3. establishing a line G having a radius of curvature Rg which is tangent to the trailing edge radius circle at a point gt and which is tangent to the line F at the point fq,
• Bb4. passing the line of the second part TD.sub.2 through the points C, gt and fq, such that the line of the second part is coincident with the trailing edge radius circle between the points C and gt and coincident with the line G between the points gt and fq;
• C. superimposing the thickness distribution on the cambered meanline by
• Ca. establishing a plurality of points zn', each point zn' being at the intersection of the line Zn and the cambered meanline,
• Cb. establishing a line Z'n perpendicular to the cambered meanline at each point zn',
• Cc. establishing a point zn" at a distance Tzn as measured along the line Z'n from the convex side of the cambered meanline at each point zn' and a point zn"'at a distance Tzn as measured along the line Z'n from the concave side of the cambered meanline at each ont zn',
• Cd. establishing a concave surface passing through the leading edge and the trailing edge and through all points zn",
• Ce. establishing a convex surface passing through the leading edge and the trailing edge and through all points zn'";
• wherein the thickness distribution is stretched chordwisely on the convex side and compressed on the concave side to form an airfoil section having desirable separation characteristics in a transonic aerodynamic flow field.